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CalcHEP Calculator for High Energy Physics A - Indico

1. 0 200 100 lt events_1 txt gt plot_1 txt will read the events from the file events_1 txt generate the M b B distri bution with minimum value of 0 maximum value of 200 and 100 bins and write the plot data to the file plot_1 txt The resulting plot can be viewed by use of the program plot_view which is stored in the bin directory This will display the plot using the Plot Viewer see Section 3 5 13 Simpson Integration In the case of 2 2 processes the integration is one dimensional and stan dard one dimensional integration techniques are used This method is very fast and very accurate It can be accessed via the 1D integration menu item which brings the user to Menu 8 see Fig 9 We note however that certain user settings are ignored when using the 1D integration method The structure functions are not included the center of mass rapidity is set to zero the regularizations are ignored and all the cuts are ignored In fact the only cut allowed in the Simpson integration is on the cosine of the angle between the third particle the first outgoing particle and the first incoming particle We call this cos13 The user can modify the minimum and maximum values of this kinematical variable via the Cos13 min and the Cos13 max menu items This cut is often necessary to remove T channel singularities from massless propagators such as a T channel photon in etet etet The default cut is 0 999 lt cos13 lt 0 999 Upon su
2. An example of a Constraints table with this comment is Name Expression Local where the refers to other entries in the Constraints table All the public constraints are compiled and calculated together and sep arately from the squared amplitude code Thus passing of parameters via global variables between functions involved in the calculation of the depen dent parameters in user defined code is possible The public parameters ap pear in the menus of the interactive sessions and can be used in the definition of the QCD scale and in the limits of the cuts and histograms The local constraints on the other hand are only calculated when needed by the squared amplitude The code for these parameters is attached to the squared amplitude code 8 3 Particles The particles are defined in the Particles table which consists of 11 columns Each particle anti particle pair is described by one row of the table The columns are 1 Full name The full name of the particle can be entered here It is not used directly by CalcHEP It is used to clarify what the short particle names mean 2 amp 3 A and Ac These columns are where the particle name and antiparti cle name belong More precisely these columns contain the quantum field and its C conjugate The field operator acting on the vacuum is understood to create the corresponding anti particle Self conjugate fields such as photons and Majorana neutrinos should co
3. L 6t 100000 10 1000000 o CalcHEP will run vegas with the specified settings in the background Other scenarios can easily be imagined We note that the blind mode is used for the automatic width calculation When CalcHEP need the numerical value of an automatically calculated particle width it begins by running the symbolic session in blind mode It then compiles the code as a shared library which is then attached to the currently running program The micrOMEGAs 42 package is based on this idea too In this case all the processes of Dark Matter annihilation are generated on the fly using the blind mode of the symbolic session 7 2 Shell Scripts It is possible to write universal shell scripts based on the blind mode which accept user input create the keystroke sequence string and start calchep in blind mode This relieves the user from the burden of determining the keystroke sequence string but allows the user to run CalcHEP in batch mode 72 We provide several such shell scripts for common tasks in the CALCHEP bin directory which is symbolically linked to the WORK bin directory In this section we describe them If any of these scripts require parameters but are called without any parameters the script first prints a message to screen informing the user of the required parameters and then quits We begin by describing s_blind which does a symbolic calculation It is called from the users WORK directory es_blind
4. When the initial state particles are massless the user may request polar ized beams The way this is accomplished is by adding the symbol to the end of a massless particle name For example entering the process ef E gt A A will cause CalcHEP to generate the code for the annihilation of polarized et and e beams to produce two photons In the current version we only 36 consider initial states which consist of mixture of left and right polarizations We do not currently support linear polarizations After the process has been entered CalcHEP allows the user to enter any particles he she would like excluded from the internal lines of the diagrams It does this by displaying the text Exclude diagrams with followed by a text input box The syntax for this entry is as follows Exclude diagrams with Pi gt ni P2 gt n2 where P1 P2 are particle names and n1 n2 specify the maximum number of internal lines that can contain these particles In other words diagrams where the number of internal lines containing P1 is greater than n1 or where the number of internal lines containing P2 is greater than n2 and so on will not be constructed For example Exclude diagrams with W gt 1 will cause CalcHEP to only construct diagrams with zero or one W internal line The input P gt 0 can be shortened to P and is understood by CalcHEP to mean that P can not appear in any internal lines If the user leaves this text entry blank the
5. val_NN where name_1 name_2 name_N should be the set of names of independent model parameters while val_11 val_1iN are val ues for the respective parameters to be used for the first point and val_Ni val_NN are values for these parameters for the last grid point of the calculation Note that this script does not do summa tion over the subprocesses i e it will do the grid calculation only for chosen subprocess in the menu The results of the calculation are printed in the terminal in the format 76 name_1 name_2 name_N val_11 val_i2 val_iN res_i val_Ni val_N2 val_NN res_N or can be redirected into some file e g results txt with par_scan lt data txt gt results txt command epar_scan_sum lt data txt calculates the cross sections according to the grid for names and parameters given in data txt file similarly to the par_scan one but in addition it performs a summation over all available subprocesses egen_events Nevents This script can by launched after successul end of run_vegas script with active second Vegas loop Parameter Nevents events defines number of events to generate If any of these scripts ends with an error a message is printed to stderr and the return value of the script can be seen by issuing echo on the shell A description of the possible error codes can be found in the CalcHEP manual 7 3 Batch interface Although the shell scripts of the previous subsection greatly improve
6. Belyaev C Leroy R Mehdiyev A Pukhov Leptoquark single and pair production at LHC with CalcHEP CompHEP in the complete model JHEP 09 2005 005 arXiv hep ph 0502067 A Belyaev et al Technicolor Walks at the LHC Phys Rev D79 2009 035006 arXiv 0809 0793 doi 10 1103 PhysRevD 79 035006 H J He et al LHC Signatures of New Gauge Bosons in Minimal Higgsless Model Phys Rev D78 2008 031701 arXiv 0708 2588 doi 10 1103 PhysRevD 78 031701 N Christensen P de Aquino C Degrande C Duhr B Fuks M Her quet F Maltoni S Schumann A Comprehensive approach to new physics simulations Eur Phys J C71 2011 1541 arXiv 0906 2474 doi 10 1140 epjc s10052 011 1541 5 A Datta K Kong K T Matchev Minimal Universal Extra Di mensions in CalcHEP CompHEP New J Phys 12 2010 075017 arXiv 1002 4624 doi 10 1088 1367 2630 12 7 075017 169 30 31 32 33 34 35 36 37 B A Dobrescu D Hooper K Kong R Mahbubani Spinless photon dark matter from two universal extra dimensions JCAP 0710 2007 012 arXiv 0706 3409 doi 10 1088 1475 7516 2007 10 012 B A Dobrescu K Kong R Mahbubani Leptons and photons at the LHC Cascades through spinless adjoints JHEP 07 2007 006 arXiv hep ph 0703231 doi 10 1088 1126 6708 2007 07 006 G Burdman B A Dobrescu E Ponton Six dimensional gauge the ory on the chiral square JHEP 02 2006 033 arXiv hep ph 0506334 doi 10 10
7. The Axodraw syntax is straight forward and the user can modify the CalcHEP output to fine tune the picture Other changes the user can make are to the line width the scale of the picture and the size of the characters The ETFX and Azodraw instructions are located at the beginning of the output For example small letter size control SetWidth 0 7 line width control SetScale 1 0 line scale control unitlength 1 0 pt text position control Note that the SetScale instruction influences the position of the lines whereas the unitlength variable is responsible for the position of text Con sequently if the user would like to change the scale of the picture he she has to modify both of these settings in a consistent manner For instance to in crease the size of the picture by a factor of two use the following SetScale 2 0 picture size control unitlength 2 0 pt picture size control In the case of Feynman diagram output CalcHEP substitutes the BT EX names of the particles as they are defined in the Particle table see Section 8 3 B Self check of the CalcHEP package We have included a suite of tools for testing the CalcHEP package Positive results from these tests signal that the CalcHEP package is working properly 130 The tests described here involve the symbolic calculation and have been realized with the help of the Reduce 45 symbolic manipulation system These test routines are stored in the CAL
8. and SEG 5 a D1 dc p2 dG23 p3 Again the factor 27 d p1 p2 p3 i farazaz is substituted by CalcHEP Thus this interaction may be implemented in the Vertex table as g 2m 5 p1 po ps py i fsa Al A2 A3 A4 Factor Lorentz part G C Ge G GG pl m3 where G C and G c are the Faddeev Popov anti ghost c and ghost c respectively 4 gluon interaction In addition to the 3 gluon interaction of QCD the Lagrangian also contains the following 4 gluon interaction 2 Sig Fg g Baar 8 CH GY 2 y Cy 2 GY aaa Fourier transformation and functional differentiation lead us to an ex pression which contains three different SU 3 color structures SaGa 5 GM p1 OGN pa GUS p3 SGU pa R oO ei tae oe F aafaa gg gews guana 31 The complicated color structure of this vertex cannot be directly written down in the CalcHEP Vertex table To implement this vertex we must use the following trick We introduce the auxiliary tensor field t x and the following Lagrangian for its interaction with the gluon field M 4 FS ta 0 G2 2 GY e sf wa e de g 2r 6 p1 po ps pa bce X 144 It can be seen that functional integration over the auxiliary field t 2 reproduces the 4 gluon interaction in the partition function et 4c pon II dti 2 LOL H V For each colored vector particle CalcHEP automatically adds a
9. int pCode which allows the user to write his her own kinematical functions to be used in the cuts and distributions When the user enters U lt name gt for a cut or a histogram CalcHEP calls usrfun lt name gt For example if the user enters Uatb2 in a cut table CalcHEP calls usrfun alb2 It is then the responsibility of the user to write the code that calculates the kinematical variable and returns the value In the absence of user code CalcHEP contains a dummy version of this function which prints an error message to stderr and terminates the numerical session After the user writes these functions the user must add the full path to his her code in the Libraries table of the model Details can be found in Section 4 1 If applicable the environment variables CALCHEP and WORK can be used as part of the path which are defined by CalcHEP Other environment variables can also be used We will now describe the other parameters of these functions nIn and nOut are the number of incoming and outgoing particles respectively pvect is a one dimensional array that contains the momenta of the external parti cles The jth component of the ith particle s momentum is given by p pvect 4 it j The energy is the Oth component E pvect 4 i and the momentum along the axis of collision is given by pz pvect 4 i 3 pName i and pCode i give the name and PDG code of the ith particle respectively Two more functions which the u
10. p2 M where A is the conjugate of A and M is a mass of the gauge boson again we consider Feynman gauge Auxiliary tensor field Whereas the Faddeev Popov ghosts and Gold stone bosons are standard elements of modern quantum field theory this auxiliary tensor field was invented by the original CalcHEP authors in order to construct complicated color vertices such as the four gluon vertex These auxiliary fields are automatically generated whenever a particle is defined with a nontrivial SU 3 color representation by adding t and T to the particle name Two auxiliary tensor fields are generated automatically and are typically used for a constraint and a Lagrange multiplier These auxiliary fields are commutative and satisfy the same conjugation rule as the parent particle while it is Lorentz transformed like a tensor field The propagator is point like j 4 1 j lt O T A t p1 A t p2 0 gt 27 ti 5 p1 pa g g m 13 Further information about the use of the Faddeev Popov ghosts Gold stone bosons and auxiliary tensor fields can be found in Appendix E 8 8 The SLHAplus package 8 9 LanHEP automatic generation of models 8 10 FeynRules 112 9 CalcHEP as a generator of matrix elements for other packages Here we present tools which allow to compile Squared Matrix Elements for different processes and calculate them for needed values of input parame ters and momenta Actually we presents tools which allo
11. the Plot Viewer displays the plot and waits for a signal from the keyboard or the mouse If the mouse is clicked inside the plot the x coordinate of the mouse is displayed at the bottom of the window Additionally the value of the function or histogram at that x coordinate value is shown In the case of a two dimensional histogram density plot both the x coordinate and the y coordinate along with the histogram height are shown If a key is pressed a menu appears see Fig 5 allowing the user to control some aspects of the plot as well as to export the plot data The available options are e Y max which allows to set the maximum height of the plot e Y min which allows to set the minimum height of the plot e Y scale which allows whether the vertical axis should be linear or logarithmic However note that the logarithmic scale is only available if the lower limit is positive and the ratio of upper and lower limits is greater than ten 25 Figure 4 An example of a histogram plot in CalcHEP min 5 84 Y scale Log Save plot in file LaTex file Mathematica plot Figure 5 An example of a continuous plot in CalcHEP for a 2 2 process 26 e Save plot in file which allows to save the plot data to the file plot_ txt where is an integer This file also includes example plot instructions for gnuplot and PAW Additionally the file plot_ gnu is created which has gnuplot instructions and can be used by issuing
12. the diagrams can be scrolled up and down by pressing the PgUp and PgDn keys The Home and End keys move to the beginning and end of the diagrams respectively To move directly to the diagram with index n the user may press the key type n and press enter Exiting the Diagram Viewer can be done by pressing the Esc key Alternatively these commands can be accomplished by clicking on the command labels located at the bottom left of the Diagram Viewer with the mouse The Diagram Viewer may also have some optional functions which de 24 pend on the context These are e Delete D which allows the user to turn off all the diagrams e On off 0 which allows the user to toggle the currently highlighted diagram between on and off e Restore R which allows the user to turn on all the diagrams e Latex L which allows the user to write all the diagrams that are not turned off to file in BTgX axodraw 44 syntax e Ghosts G which displays the current diagram along with all other diagrams which are related to the current one by replacing gauge bosons with their associated ghosts and Goldstone bosons This command is only available for squared diagrams These commands can be invoked by pressing the key marked in parentheses or by clicking on the label along the top left of the Diagram Viewer window 7 Plot Viewer The Plot Viewer is designed to display histograms see Fig 4 and continuous curves see Fig 5 After being launched
13. the results are combined and stored with names unique to that param eter point for easy retrieval Both the symbolic calculations and the numerical calculations are par allelized Each subprocess and each parameter point are run as separate jobs and run on all available cpu cores The number of cores available is set by the user as is the type of cluster software used Multicore machines PBS cluters and LSF clusters are currently supported The progress of the calculation is stored in a series of html files which can be viewed in a web browser These html pages contain informa tion about the progress of the calculation as well as the results of the calculations which are already finished The final event files are linked as are the session dat and prt files which give the full details of each individual calculation Pure text versions of the progress pages are also created for situations where a web browser is not convenient Once the user creates the batch file and runs the batch interface no user input is required until it finishes It can be run in the background and checked periodically After the user has created their batch file they would typically run the batch interface from their CalcHEP work directory as calchep_batch batch_file 78 where batch_file is the name of their batch file which can be named any thing the user likes The batch interface will start by printing a message to the shell which will contain the loc
14. website https mcdb cern ch Some special features of LHE file generated by CalcHEP are e A history of each decay is presented for each event The information about the parent particles and their mean life time is included This information can be used for proper hadronisation and detector simula tion e When connecting decays event_mixer uses a Breit Wigner virtual mass distribution where we assume that the matrix elements of the subprocesses do not depend strongly on the off shell momentum Our procedure does not break momentum conservation e According to the LHE file format accord the header marked by lt header gt and lt header gt section can be used for auxiliary information event_mixer places the following in the header a lt hepml gt section see below a lt slha gt section with information about quantum numbers masses widths and decays of non SM particles and a lt calchep batch gt sec tion for the run_details txt file e Information about the process such as a list of the subprocesses kine matical cuts model name number of generated events cross sections and model parameters is stored in the lt hepm1 gt section For instance lt files gt lt file gt lt eventsNumber gt 1000 lt eventsNumber gt lt crossSection unit pb gt 0 254087 lt crossSection gt lt file gt lt files gt 67 This information is recognised when the LHE files are being uploaded in the MCDB data base an
15. wert where W7C and YE CT C 40 TY In the basis used by CalcHEP the charge conjugation operator is given by C 7 The Lagrangian can then be written in the form L rAvyyP 1t 7 UeF ANI VP 9 r e NUT YP 1 7 VF AVIVA HUF uU Direct implementation of the definition 8 4 gives us the Vertex table Al A2 A3 A4 Factor Lorentz part u je F lambda 1 G5 E U F lambda 1 G5 By means of equation 10 we can rewrite this table in the equivalent form A1 A2 A3 A4 Factor Lorentz part e u F lambda 1 G5 U E F lambda 1 G5 F Color string basis CalcHEP performs averaging summation over color states of incoming outgoing quarks and gluons But in order to describe the hadronization of outgoing particles one needs to specify the color states in more detail CalcHEP passes the problem of hadronization to other programs such as PYTHIA 54 PYTHIA performs the hadronization in the framework of the color string model According to this model pairs of outgoing partons with opposite color are jointed into a colorless object called strings Partons are attached at the ends of the string and usually move in different directions When the distance between the partons becomes large the color string breaks creating two strings each with smaller energy than the parent string This 1
16. 27 8 q 74 gt 1 ag ae Ela S1 g dI q S2 2T 2T 35 The above formula expresses a multi particle volume in terms of two particle one the volumes dI q1 S1 and dI q2 S2 with a reduced number of particles and the virtual squared masses s1 S2 of clusters S1 Sb Recursive application of this formula allows one to express the multi particle phase space in terms of two particle phase space In its turn the two particle phase space is explicitly described by spherical angle Q of motion of the first decaying particle in the rest frame of initial state 58 dT q 1 2 kQ Qn A T where k is the absolute value of three dimensional momentum of outgoing particles in the rest frame Thus applying recursively 35 and 36 to 33 we obtain an explicit expression for the phase space volume in terms of the squared masses s of virtual clusters and the two dimensional spherical angles Q where j is an ordinal number of decay 36 k o n 1 kj a0 n 1 dala T JP L I ea a ds 37 Here k is a momentum of outgoing clusters produced by decay of the j cluster in its center of mass The expression 37 means some sequential 1 gt 2 decay scheme which starts from incoming state and finishes with outgoing particles of the pro cess For example the integration domain for s parameters depends on this 158 scheme Below we present two such schemes for a process with four outgoing particles P
17. Beamstrahlung ISR Initial State Radiation is a process of photon radiation by the incom ing electron due to its interaction with other collision particle The resulting spectrum of electron has been calculated by Kuraev and Fadin 64 In Com pHEP we realize the similar expression by Jadach Skrzypek and Ward 65 F x exp 6 3 4 Euler 1 z 1 2 B 1 3a In w 2 1 x 2 2V 1 8 151 where a 1 137 0359895 is the fine structure constant B a 2In SCALE m 1 a m 0 00051099906 is the electron mass Euler 0 5772156649 is the Euler constant r is the gamma function SCALE is the energy scale of reaction In the Kuraev and Fadin article the parameter SCALE equals to the total energy of the process because they considered the process of direct ete annihilation In order to apply this structure function to another processes we provide the user with a possibility to define this parameter Beamstrahlung is a process of energy loss by the incoming electron due to its interaction with the electron positron bunch moving in the opposite direction The key parameter of beamstrahlung is 5a NE Yy z 6m3o 6 dy where N is number particles in the bunch On Oye Ge are sizes of bunch E is a center of mass momentum The effective energy spectrum of electrons we use approximated formulas of P Chen 66 Namely for T lt 1 we use formulas 17 18 and 22 23 otherwise Pi
18. CalcHEP notation The names are separated by commas or spaces and can be specified in any order int slhaDecayPrint char pname FILE FD Calculates the width and branching ratios of particle pname and writes down the result in SLHA format The return value is the PDG particles code In case of problem for instance wrong particle names this function returns zero This function just present another output format for pWidth printTxtList 9 5 Compilation of new processes Generic procedure for matrix element compilation reads numout getMEcode int twidth flag which forces Breit Wigner form of t channel propagators int UG flag which forces Unitary gauge char Process process name char excludeVirtual list of particles which forbidden in propagators char excludeOut list of particles which forbidden as outgoing ones char libName name of shared library without terminating so In case of success result of the compilation is stored in the library aux so generated libName so If the library libName already exists it is not recompiled and the corre spondence between the contents of the library and the Process parameter 116 is not checked libName is also inserted into the names of routines in the libName so library Thus libName can not contain symbols that cannot be used in identifiers for example the symbols x The getMEcode return address of computer memory where compiled code and auxilary p
19. Carlo simulation menu also allows to generate kinematic distributions and generate events It is described in further detail in Subsection 5 11 For 2 gt 2 processes with fixed energies of incoming particles the phase space integral is one dimensional and can be integrated using traditional Riemann approach In this case one can chose the 1D integration which is the last item in the menu This option is described in Subsection 5 13 For the 1 gt 2 case we have zero dimension phase space and this option allows a fast summation over channels and calculation of branchings 5 2 Bookkeeping Each time any parameters are changed which affect the numerical calculation CalcHEP increases the session number by one and clears the statistics The current session number is displayed at the top of the interactive session screen These parameters not only include the dependent model parameters but also include the choice of subprocess incoming momenta parton distribution functions QCD coupling and cuts During a Vegas session the full set of parameters for the current session is stored in the file session dat located in the results subdirectory This file changes to match the current session If the user quits the interactive session and restarts it later CalcHEP will read the parameters from the session dat file The user can then continue from where he she left off The full set of parameters for each session is also stored in the file prt_N where N is
20. Houches event files Comput Phys Commun 176 2007 300 304 arXiv hep ph 0609017 doi 10 1016 j cpc 2006 11 010 G Belanger F Boudjema A Pukhov A Semenov micrOMEGAs2 0 A program to calculate the relic density of dark matter in a generic model Comput Phys Commun 176 2007 367 382 arXiv hep ph 0607059 doi 10 1016 j cpc 2006 11 008 G Belanger et al Indirect search for dark matter with mi crOMEGAs2 4 Comput Phys Commun 182 2011 842 856 arXiv 1004 1092 doi 10 1016 j cpc 2010 11 033 J A M Vermaseren Axodraw Comput Phys Commun 83 1994 45 58 doi 10 1016 0010 4655 94 90034 5 A C Hearn REDUCE 2 USERS MANUALSTAN CS 70 181 G P Lepage A New Algorithm for Adaptive Multidimensional Integra tion J Comput Phys 27 1978 192 doi 10 1016 0021 9991 78 90004 9 W Press S Teukolsky W Vetterling B Flannery Numerical recipes in C the art of scientific computing Cambridge University Press 1999 J Bjorken S Drell Relativistic quantum mechanics International series in pure and applied physics McGraw Hill 1964 U Baur J A M Vermaseren D Zeppenfeld Electroweak vector boson production in high energy e p collisions Nucl Phys B375 1992 3 44 doi 10 1016 0550 3213 92 90332 6 Y Kurihara D Perret Gallix Y Shimizu e e e anti electron neutrino u anti d from LEP to linear collider energies Phys Lett B349 1995 367 374 arXiv hep ph 9412215 doi 10 1016 0370 2
21. Q2 k2 52 P2 Qi ky S3 D3 03 k3 P4 In the case of CompHEP project such decay scheme is defined by the user via the Kinematics menu see Section 5 9 I 1 2 Polar vectors To complete phase space parameterization we must fix a polar coordinate system choosing the polar and the azimuthal angles for each of decays PA d cos O jd 38 We have an ambiguity in the choice of polar coordinate Let us remind that our goal is not only parameterization of phase space but also regulariza tion of the squared matrix element in the phase space manifold The main idea of such regularization is a cancellation of integrand sharp peaks by the phase space measure Originally the phase space measure 37 has no cancel lation factors but we can create them by means of a Jacobian of transformed variables To get an appropriate Jacobian we need to have the initial phase space variables related to poles of the squared matrix element In their turn the poles of squared matrix element are caused by virtual particle propagators and generally have one of the forms 4 5 or 6 Sec tion 5 10 depending on a squared sum of momenta Variables s in 37 are also equal to squared sums of momenta So the parameterization 37 allows us to smooth some peaks of the matrix element It appears to be that the polar coordinates can be chosen in such a way that all cos have simple linear relations to the squared sums of momenta 69 70 The polar angle
22. W C Faddeev Popov anti ghosts are anti commuting scalar fields The nonzero propagators for these fields are lt O T A c p1 A C p2 0 gt lt O T A C p1 A c p2 0 gt Ac pi p2 M where A is the conjugate of A and M is the mass of the parent particle we are assuming Feynman gauge The reason CalcHEP introduces the Faddeev Popov ghosts at tree level is that it sums over the unphysical polarizations of the gauge bosons in the external states as well as the physical polarizations see Appendix C in order to reduce precision loss due to large cancellations The Faddeev Popov ghosts and the Goldstone bosons for a broken gauge theory are required to cancel the unphysical polarizations See 48 for further details Goldstone boson are related to broken symmetries In the case of bro ken gauge symmetries they become the longitudinal degrees of freedom of the gauge boson CalcHEPautomatically generates these fields for massive vector bosons by appending a f to the end of the gauge boson name For example the W and W gauge bosons have the Goldstone bosons W f and W f associated with them These Goldstone bosons are commuting scalar The well known spin statistics relation is not valid for unphysical fields 111 fields that satisfy the same conjugation rules as the gauge boson they belong with For example W f W f The nonzero propagators for these fields are T A f p1 A f p2 Ac pi
23. aJian ATG ees Te de a but now each integration has only a single peak It is easy to extend this method for an arbitrary number of peaks The branching method was used in 71 to separate peaks which came from various diagrams In that paper there was also proposed to use the expression 42 where f x is replaced by a fi x with a subsequent search for optimal coefficients a CompHEP passes on this weight optimization to Vegas combining two integrals in one Vegas hypercube As was mentioned above CompHEP automatically searches for a polar vector for each angle integration in order to reach a linear relation between cosO and one of the squared sum of momenta which is responsible for the peak It could happen that various peaks need different polar vectors for the same decay In this case CompHEP uses the branching method again but now for the whole two dimension sphere integration In other words we use the branching equation 42 where z is the two dimensional sphere angle 69 70 161 I 2 Adaptive Monte Carlo integration package Vegas This section contains a short description of the adaptive Monte Carlo pro gram VEGAS See for details 46 47 The Monte Carlo method reduces a task of integral evaluation to the task of mean value calculation Let g x is a density function satisfying then JEE de f 9 a g x de lt f g gt Jim Y F ei 9 s N where points x are sampled with the probability density g x dz T
24. about the dependent parameters The user can follow the formula with a and then a comment describing the parameter Public and local dependent parameters Some models can contain thousands of dependent parameters For a particular process only a small subset of these is used For this reason CalcHEP attempts to reduce the file size by only including the dependent parameters that are used in the numerical code that it generates The way it does this is that it divides the dependent parameters into two groups which we will call the public param eters and the local parameters The public parameters are those parameters that are required to calculate all the particle masses and widths all pa rameters that depend on external functions except the standard C math functions and all dependent parameters above any of these In other words all the parameters from the top of the Constraints table down to the last parameter required for the calculation are public and are included in the nu merical code All dependent parameters below this are defined as local and are not included in the numerical code If the user would like to force CalcHEP to include a larger subset of the dependent parameters in the numerical code he she can place the comment Local in the Constraints table in the first column CalcHEP will always include the parameters up to at least this point All the parameters 101 above the Local line will always be considered public
25. ad ea RS eke Se aS ws 132 C 2 Incoming and outgoing ghosts 4 24 s256 74 b86 133 C 3 Massless vector particle case 1 ee 136 C 4 Summation of ghost diagrams in CalceHEP 136 C 5 Gauge symmetry and cancellations 138 D Feynman rules in CalcHEP 139 D 1 Lorentz part of diagram 2444 64 8 2 6 Oe Sew 6 139 D2 Color TactOr s ee eat eee eS em ES eR Bee Ge a 140 D3 Common factors 222 4 0h bebe Se She SSS 8 ee SS SS 142 E Examples of model realization 142 E 1 Implementation of QCD Lagrangian 142 E 2 Neutrino as a Majorana fermion 145 E 3 DLeptoguarkSi 2 4 2 464 484 2 od ee alk oe bee oe he a 148 F Color string basis 149 G Distribution functions and beam spectra G 1 Backscattered photon spectrum G 2 Weizsaecker Williams approximation G 3 ISR and Beamstrahlung 2 nd wee eee ee ee eR H PDT Particle Distribution Tables in CalcHEP H 1 CTEQ and MRST parton distributions 2 6 6 4 4 a6 H 2 Format of parton distribution tables 2 I Monte Carlo phase space integration I 1 Parameterization of multi particle phase space 1 1 1 Parameterization via decay scheme Li Polar vectors s aa eoa 88 43 exe de a oh og Iie Smoothing s sop Sh ee ee ee ee r I 2 Adaptive Monte Carlo integration package Vegas 1 2 1 Importance sampling 2 2 26 52 seus L22 Stratified sampling 22442442666 44845
26. also supported Many models of particle interactions have been implemented in CalcHEP Among them are the minimal supersymmetric extension of the standard model MSSM 20 21 the next to minimal supersymmetric extension of the standard model NMSSM 22 the cp violating minimal supersymmetric extension of the standard model CPVMSSM 23 Little Higgs Models 24 a Lepto quark model 25 a Technicolor model 26 a Higgsless Model 27 28 and models with extra dimensions 29 30 31 32 33 34 Some of these models are available on the CalcHEP website for the users convenience A new model of particle interactions is implemented into CalcHEP by writing a set of pure text model files which contain all the details of the model including the properties of its particles parameters and vertices Although it is possible to do this by hand especially for simple models new models are typically implemented using a dedicated implementation package such as LanHEP 35 or FeynRules 36 which automatize the process of calculating the Feynman rules and writing the CalcHEP model files This can be especially important for models with a large number of new particles and complicated Lagrangians The CalcHEP package consists of three parts which perform the symbolic numerical and batch calculations The first two parts are written in the C programming language The symbolic part produces codes for squared matrix elements which are then used in the num
27. and presents it as a sum of pole terms e sum_int red which combines the expressions fromt the squared dia grams and integrates it over the phase space and presents it as a total cross section sum_22 red and sum_int red only work with 2 2 processes These files are stored in the CALCHEP utile directory Suppose the symbolic output symb1 red for yu scattering A m gt A m is prepared 7 We copy the sum_cd red sum_22 red and sum_int red files into our WORK directory and launch the Reduce program from within the results directory We display the outcome of using Reduce with each of these example programs If the electron is massless it leads to a divergence in the total cross section For the purposes of this illustration we therefore use a muon instead of an electron 121 Example 1 If we use the sum_cd red program we will get in sum_cd red to load the summation package in symb1 red to read the contributions of the diagrams sum to write the answer 32 ee 4 Q p1 p2 4 Axp1 p2 3 pl1 p3 3 pl1 p2 2 pl p3 2 Q pl p2 2 p1 p3 xmm 2 p1 p2 p1 p3 3 2 p1 p2 p1 p3 2 mm 2 pl p3 2 mm 4 propden p1t p2 mm 0 2 propden p2 p3 mm 0 2 Example 2 If we use the sum_22 red program we will get in sum_22 red to load the summation package in symb1 red to read the contributions of the diagrams sum to write the answer 2 eex 4 4 sp mm 2 4mm 4 8xsp mm up
28. are empty then the minimum or maximum limits respectively are not applied These limits may contain numerical values model parameters standard algebraic expressions including and and functions defined in the C math library such as sqrt sin and cos If the process contains identical outgoing particles the cut is applied to each particle combination For example for the process pp AA the cut I Parameter gt Min bound lt gt Max bound lt ECA 20 100 is equivalent to Parameter gt Min bound lt gt Max bound lt E A 100 E_ A 20 59 On the other hand the same cut but with an exclamation mark in the first column Parameter gt Min bound lt gt Max bound lt ECA 20 100 demands the absence of photons with energy in the 20 100 GeV interval For processes with several identical particles a cut marked by does not mean the mathematical negation of the condition without the exclamation mark It means the negation of each individual cut for each particle combination If a cut contains particles which are not included in the current subpro cess they are ignored until the user starts to work with a subprocess which does include them 5 9 Kinematics The Kinematics menu subitem of the Phase space mapping menu item al lows to display and change the phase space parameterization used in the Monte Carlo integration The way this is
29. begins with the parameter name val0 and then increases it by size step until N steps are completed As in 75 the previous case distributions are stored in the files distr_k and can be viewed using the disp_dist program esubproc_cycle L Nmax This script calculates the cross section and gener ates events for each subprocess When it is finished it adds the cross sections together and prints the total cross section to the screen If there are distributions specified then they are added together and the resulting distribution is stored in the file distr_j1_j2 where ji is the first session number and j2 is the final session number and where j2 j1 1 is equal to the number of subprocesses It also generates unweighted events for each subprocess The number it generates is equal to the smaller of the cross section times the lu minosity which is specified by L and Nmax It writes these events to the files events_k txt where k is the session number when it was generated and again j1 lt k lt j2 These events can be combined using the program event_mixer which is stored in the WORK bin di rectory The cuts regularization and histograms must apply to all subprocesses and the outgoing particles must be identical epar_scan lt data txt calculates the cross sections according to the grid for names and parameters given in data txt file The format of data txt is supposed to be name_1 name_2 name_N val_11 val_12 val_iN val_N1 val_N2
30. button 18 highlighted and the user can move between the items by pressing the Up and Down keys or by clicking on the desired menu item with his her mouse Once the desired menu item is highlighted the user can activate it by pressing the Enter key or by clicking on it again with the mouse IN state Hodel parameters Constraints QCD coupling Breit Wigner cuts Phase space mapping Figure 1 An example of the CalcHEP graphical user interface with a menu displayed If the menu is too long to fit in the graphical user interface PgUp and or PgDn will appear at the respective top and or bottom of the menu The user can scroll to the other available menu items by pressing the PgUp PgDn keys or by clicking on the PgUp PgDn symbols on the menu As the user s calculation progresses and he she moves through the menus he she can always return to a previous menu by pressing the Esc key or by clicking on the lt symbol at the top left corner of the menu border When a menu is present CalcHEP is also sensitive to the Function keys F1 F2 F10 A list of currently active Function keys is displayed at the bottom of the graphical user interface and depends on the currently active On Macintosh operating systems these keys can be activated within CalcHEP by holding the Fn key down while pressing them 19 menu The typical Function keys that are available are F1 Help displays a help message about the highlighted menu ite
31. colliding protons which however are composed of quarks antiquarks and gluons In this section we describe routines that combine the results from different subprocesses 6 1 Distribution Summation The bin directory contains the program sum_distr which combines distribu tions from different CalcHEP sessions typically from different subprocesses The way it works is that it sums the distributions that have exactly the same kinematical variable specification In other words M b B and M B b would not be combined although they are the same distribution M u d would not be combined with M u s although they are both the invariant mass of two jet particles M Jet Jet would be combined from different subprocesses however because they have exactly the same specification The user must make sure each distribution specification is the same for each subprocess and that there is no ambiguity The distributions from each CalcHEP session are stored in the files distr_N where N is the CalcHEP session number To sum the distributions from the subprocesses the user would typically run CALCHEP bin sum_distr lt distr_Ni gt lt distr_N2 gt gt lt distr_out gt where lt distr_N1 gt lt distr_N2 gt and are the distribution files to be com bined and lt distr_out gt is the file where the results should be written The program show_distr can be used on the output file in the same way as for the distribution files written during the inte
32. continue his her work by pressing any key or by clicking on any part of the graphical user interface with the mouse The second kind of message is in the form of a dialog box This message ends with Y N and the user is required to make a choice by pressing the Y or the N key The user can also click on the Y or the N in the dialog with the mouse 4 String Editor At times e g the input of a new process the user is required to enter a textual string At these moments CalcHEP provides a 20 text box where the user can enter and modify his her text If text has already been entered in this text box in the past CalcHEP will often remember this string and present it as the default value in the text box for convenience If the user s first input is a letter number or other textual symbol the old string will be removed and a new string will be started If on the other hand the first input is non textual such as a mouse click a tab or an arrow key the old string will be kept and the user will be able to modify it To edit this text string the user can use the left right arrow keys or the mouse to move the cursor to the desired position The Delete and Backspace keys both remove a character to the left and move the cursor back one position to the left When the user is finished with his her text string he she can press enter to accept it If the user wishes to cancel he she can press the Esc key 5 Table Editor CalcHEP uses a table
33. distributions for a range of masses for the W This can be done with the Run parameter Run begin Run step size and Run n steps keyphrases Here is an example Run parameter MWP Run begin 400 Run step size 50 Run n steps 17 83 This will generate the events and or distributions for the model with the mass of the W set to 400GeV 450GeV 500GeV 1200GeV As many runs as desired can be specified including zero For each run all four keyphrases have to be specified Furthermore if there is more than one run all four keyphrases have to be specified together Here is an example with two runs Run parameter MWP Run begin 400 Run step size 50 Run n steps 17 Run parameter MF Run begin 2000 Run step size 200 Run n steps 11 This example will run over both parameters MWP and MF QCD The parameters of the QCD menu of the numerical session can be specified as in the following example parton dist alpha ON alpha MZ 0 118 alpha nf 5 alpha order NLO mb mb 4 Mtop pole 174 alpha Q M45 The default values are the ones in the interactive session Not all the key words have to be included in the batch file It is sufficient to include the ones that need to be changed For example if only the QCD scale needs to be changed it can be specified as alpha Q Mt 2 The QCD scale can be specified in terms of the invariant mass of certain final state particles as in Mij which means th
34. fermion is purely left and right handed respectively This can only be applied to massless fermions The effect of this is that when CalcHEP averages over the spin of the incoming fermion it takes into account that there is only one polarization for this particle This is used for example for the SM neutrinos g declares that the vector particle is treated as a gauge bo son In this case t Hooft Feynman gauge is used for the vector boson propagator and the ghost fields A c and A C where A is the name of the vector boson as well as the Goldstone boson A f can contribute to the Lagrangian A massless vertor particle must be treated as a gauge boson In the absence of g in this column the unitary gauge is always used for massive vector bosons and the ghosts and Goldstones associated with it are not used in Feynman diagrams The Formulaes for the particle s propagators are presented in Section 8 6 The Aux column can also be used to specify a particle s electric charge This charge is required by many external packages CalcHEP already knows the charge of the SM particles and assigns it according to the particle s PDG code It can determine the charges of many BSM parti cles by analyzing the Feynman rules and assuming they conserve elec tric charge However for some particles this will not be sufficient to determine their charge For this reason CalcHEP allows to specify the charge of any BSM particle Specifically three
35. full set of diagrams will be constructed At any time during the process entry the user may press the Esc key to return to the previous input and the F1 key to get process input help After the process entry is complete CalcHEP generates all the Feynman diagrams satisfying the user s constraints If no diagrams are allowed CalcHEP dis plays a warning message and returns the user to the beginning of the process entry to try again If one or more diagrams are constructed CalcHEP ad vances to Menu 5 which we describe next 4 4 Squared Diagrams and Symbolic Calculation Menu 5 This menu appears on the screen after the construction of the Feynman diagrams along with information about the number of diagrams and subprocesses generated The first menu item is View diagrams and allows the user to view a graphical representation of the generated Feynman diagrams via the Diagram Viewer In addition to viewing the diagrams as described in Section 3 the user can remove some of the diagrams before they are squared and can generate BTFX output for the diagrams which are not removed If more than one subprocess is generated CalcHEP will first present the user with a list of the subprocesses when he she chooses View diagrams 37 Each subprocess will be listed along with the number of diagrams for that subprocess The user can move among the subprocesses by using the PgUp and PgDn keys or by using the mouse If the F7 key is pressed while on this
36. in anti proton x grid After this keyword the file has to contain a sequence of numbers which specify the X grid The number of points has exceed 2 Numbers must increase from point to point and belong to interval 0 1 q grid Specifies the beginning of Q grid The data must be positive and in crease This item is optional Generally we can consider parton distri butions which do not depend on Q alpha decignates beginning of data for alpha Q corresponding to Q grid The item is optional and can not precede the q grid item 1i parton 2 parton These keywords mark the beginnings of data for distribution functions They can not precede the x grid and q grid items because the power of data for these items must be the production of the x grid and q grid powers Date corresponding to i position of x grid and jt position of q grid is disposed on the i j 1 n place where n is the power of x grid The reader of pdt file finishs its work when it has read the needed parton item Thus all other items described below should preceed the parton one These items are auxilaries and if it needs can be specified separately before each n parton item x min q min specifies the minimum boundaries of x and q intervals where result of 156 interpolation should be correct The correspoding number must follow to the keyword These keywords are optional If one of them is absent the first point of the correspondi
37. lt lt symb1 m In 3 sum 4 4 3 4 2 Out 3 32 EE 2 SC pi p2 4 SC pi p2 SC p1 p3 Mm SC pi p3 2 2 gt SC p1 p2 2 Mm SC p1 p3 SC pi p3 2 2 gt SC p1 p2 SC p1 p3 2 Mm 3 SC p1 p3 2 2 gt propDen p1 p2 Mm 0 propDen p2 p3 Mm 0 Example 2 If we use the sum_22 m program we will get 123 In 1 lt lt sum_22 m In 2 lt lt symb1 m In 3 sum 4 2 2 4 4 4 2EE Mm s 2 EE t 8 EE Mm Out 3 2 2 2 2 2 Mm s Mm s Mm s t 4 4 4 2 4 2 2 3 EE Mm 6EE Mm s EE s gt Sa se Gem ite me mc es em ma ach mel me mem sk Tec Se ele em sm ec Na a fr Sm a a tel a 2 2 Mm s Mm s t Example 3 If we use the sum_int m program we will get In 2 lt lt sum_int m In 3 lt lt symb1 m In 4 sum 4 8 6 4 2 2 3 4 Out 4 FE Mm 2 Mm s 16 Mm s 14Mm s s 2 2 4 2 2 Mm 2 2 3 gt 2s 3 Mm 6 Mm s s Logl 16 pis Mm s s 124 10 4 Form 10 5 Reduce program The original output of CalcHEP was for the Reduce program CalcHEP would generate the squared diagram expressions and have Reduce expand con tract indices trace gamma matrix chains and simplify the expressions Later on when the CalcHEP build in symbolic calculator was created the Re duce output became unnecessary However we still keep this output for testing purposes The user does not see the in
38. mm mm 4 4 sp mm mm 2 sp mm t 4 up mm 2 mm 4 5 up mm mm 2 up mm s 1 where s pl p2 t pl p3 and u pl p4 are the Mandelstam variables and the functions sp tp up are defined as sp 1 s 2 tp a 1 t 2 up x 1 u 2 Example 3 If we use the sum_int red program we will get in sum_int red to load the summation package in symb1 red to read the contributions of the diagrams sum to write the answer for total cross section eex 4x 2 s 4 log s mm 2 s 4 12 5 3 log s mm 2 tmm 2 14 8 34 mm 2 6 s 2 log s mm 2 mm 4 16 s 24mm x4 2xs xmm 6 mm 8 16 s xx2kpi s 3 3kS 2 mm 2 3eSemm 4 mm 6 122 Sometimes the result for the total cross section includes a complicated expression of kinematic variables under the square root which appears as a result of the evaluation of the integrand limits be_ tmax tmin s where s and t are Mandelstam variables Substitution for be_ is not done Instead variable beSquared be_ 2 is defined 10 3 Mathematica examples There following packages for Mathematica sum_cd m sum_22 mand sum_int m perform the same task as the Reduce packages did see the previous subsec tion The following output was performed in Mathematica 3 0 SC pi pj is the scalar product of momentum pi and pj Example 1 If we use the sum_cd m program we will get In 1 lt lt sum_cd m In 2
39. nModel Process nOutput This command runs s_calchep in blind mode and generates the squared amplitude code for the pro cess Process which must be enclosed in quotation marks as in e E gt m M The model is specified by nModel which must be an integer and corresponds with the position of the model in the model list The first model is specified by 1 the second by 2 and so on nOutput determines the format of the ouput and is also required to be an integer The supported outputs are C code Reduce code Mathematica code and Form code which are specified by 1 4 5 and 6 respectively There are some further things the user should keep in mind when using this shell script Because the keystroke sequence required for this shell script depends on the previous session this script first removes the previous results and starts a fresh session Feynman gauge is always used by this shell script There is no possibility to use composite definitions remove particles or choose diagrams with this script This script is used by MicrOMEGAs 21 to generate numerical code at run time We now describe scripts which are designed to work with the numeri cal session These are called from the directory where the numerical code n_calchep is stored Typically this is the WORK results directory but this directory can be renamed or moved by the user For each of these scripts all session parameters are kept fixed except for the ones explicitly described as be
40. nm Nm Composite W W W As many composite particles as necessary can be specified These definitions can be used in cuts and distributions as well as in the processes and decays The default is not to have any composite definitions PDF The PDF of a proton or antiproton can be specified with the pdf1 and pdf2 kewords which correspond to the pdfs of the first and second incoming particles respectively Choices for these keywords are cteq6l anti proton cteq6l1 proton mrst2002lo anti proton mrst2002lo proton cteq6m anti proton cteq6m proton cteq5m anti proton cteq5m proton mrst2002nlo anti proton mrst2002nlo proton None An example for the LHC is pdfi pdf2 cteq6l proton cteq6l1 proton 81 The default is None These keywords can also be used for electron positron colliders For this process the available pdfs are e ISR e ISR amp Beamstrahlung e Equiv Photon e Laser photons e None The following proton electron collider pdf is also available e Proton Photon All of these pdfs must be typed exactly or copied into the batch file If ISR amp Beam is chosen then the following beam parameters may be specified Bunch x y sizes nm 550 Bunch length mm 0 45 Number of particles 2 1E 10 The default values are the default values in CalcHEP and correspond roughly with the ILC If Equiv Photon is chosen for the pdf then the following parameters may be specified Photon p
41. numbers which accompany particle symbols are codes of Monte Carlo particle numbering scheme CalcHEP version 2 3 Type 2 gt 2 Initial_state P1_3 1 000000E 02 P2_3 1 000000E 02 StrFunl ISR 1 00S 5 Beamstr OFF 11 StrFun2 ISR 1 00S 5 Beamstr OFF 11 PROCESS 11 e 11 E gt 5 b 5 B MASSES 0 00000E 00 0 00000E 00 4 62000E 00 4 62000E 00 Cross_section Width 1 700425E 01 Number_of_events 1000 164 After that the table of events is written The first column presents weights of events Normal value for weight is 1 The reason of appearance weight 1 is explained in the section 1 3 After that the columns which specify the momenta of particles are presented The first line of the table contains titles for the columns Say P3_2 means the second component of momentum of the third particle The zero energy components are not presented because they can be calculated using the information about the particle masses For incoming particles only the third momentum component is presented because other ones are zero After description of momentum the event record contains information about color flows Each color flow is presented by a couple of particle numbers enclosed into brackets It corresponds to propagation of color 3 from the first particle to the second one See section F for algorithm of color flows generation Information about color flows is need to PYTHIA 54 to generate the correct fragmentation of colo
42. numerical sessions are supported in the batch session The instructions for the batch computation are written in a text file The batch interface reads the instructions and performs them in the background noninteractively The batch session enables the user to e perform the symbolic and numerical calculations which are available in the interactive mode e view the results of the calculation s progress via a series of html pages e perform the calculations in parallel on a multicore machine or on a computer cluster e combine events from multiple production and decay events e write the final events to an LHE file 41 for further processing e perform scans over multiple physical parameters 1 2 Acknowledgments The CalcHEP package contains codes written by the CompHEP group We would like to thank to V Ilyin D Kovalenko A Kryukov V Edneral A Semenov for permission to use their code in CalcHEP During the last decade much of the CalcHEP development was motivated by the development of the micrOMEGAs package 42 43 which was done by A Pukhov in collaboration with G Belanger F Boudjema and A Semenov Many new models for CalcHEP were implemented as part of this collabora tion The work of A Semenov developing the LanHEP program has been cru cial for CalcHEP We are very thankful to G Belanger F Boudjema A Semenov for this collaboration This work was supported by PICS 397 of CNRS Calcul en physique en des partiqules We would a
43. resonace are entered in the second and third columns They can include numerical values model parameters and algebraic 57 expressions Typically they should simply be the model parameters which specify the mass and width of the resonant particle The last column is for the power of the denominator of the propagator that appears in the squared matrix element which can be either 1 or 2 Of course in a squared matrix element resonant propagators appear to the second power However there are times that the gauge cancellations allow the exponent to be effectively decreased to 1 This typically only happens when the resonance is stable its width is 0 If the resonant particle is unstable and its width is nonzero CalcHEP uses 2 and ignores the user s input 5 11 Monte Carlo simulation The integration of the multiparticle phase space is done by the Vegas Monte Carlo routine 46 47 see Section I 2 for further details The Monte Carlo simulation menu allows the user to control the integration It also allows the user to set up histograms to be filled during the integration explained below and generate unweighted events see the Subsection 5 12 Vegas runs Nees sessions After each session if the grid is not frozen Vegas improves the grid so that the integral of the next session converges more efficiently The number of sessions run by Vegas is controlled by the menu item nSess The greater this number the more likely Vegas is to fi
44. started by a keyword A keyword is started from the symbol Numeri cal data following a keyword must be separated by white space characters The appearance of new keywords designates the end of the previous item of information The keywords are e distribution After this word the title of distribution function is disposed The title has been surrounded by the quotation marks symbols It can con tain space symbols inside After that the incoming particle number and numbers for partons are disposed All particles and partons are numer ated according to the Monte Carlo numeration scheme 58 Incoming particle and its parton are separated by the gt string Partons can be combined into groups by the the brackets For example distribution cteq6m proton 2212 gt 5 5 4 4 3 3 1 2 21 2 1 All partons in one group have the same distibution Position of a parton or of a group in the list correspond to number of table presented below 131 2 3 4 5 are numbers of d u c s b quaks 21 is the gluon number 155 Say b and b quark distributions will be disposed after the 1 parton keywords One file can contain several usually two distribution items It al lows to use one file both for the particle and for the anti particle For example the cteq6m pdt file contains also distribution cteq6m a proton 2212 gt 5 5 4 4 3 3 1 2 21 2 1 So 7 parton item describes u quark in proton and wu
45. structure to store information about the model s parameters particles and vertices as well as for the cuts and distributions of a numerical calculation For each of these cases one row stores the information for one parameter one particle one vertex one cut and one distribution respectively CalcHEP has a table editor which allows the user to view and at times to modify the contents of these tables An example of a table can be seen in Fig 2 The top line of a table window displays the title of the table Below this the table is surrounded by a frame box The columns of the table are separated by vertical lines The first table row contains the column names and below this the table data is displayed One cell the intersection of a row and a column is highlighted at atime This is the current cell and contains a cursor if the table is open for editing The current line number is shown in the top right corner of the window To change the position of the highlighted cell and or the cursor the user can use the arrow keys the Tab key and the click of the mouse Pressing any printing symbol will enter that symbol in the cuurent cell at the cursor position if in edit mode The PgUp and PgDn keys allow the user to scroll the table up and down The F1 and F2 keys provide help information about the current table and the table editor respectively Exiting the table is achieved by pressing the Esc key There are some further auxiliary commands
46. the users ability to run their desired processes in batch mode there are still some limitations when doing large complex calculations involving scans over parameter space many subprocesses and parallelization To overcome these challenges we have written a Perl script which we call the batch interface The main features of this Perl interface are e The input is a pure text file we call the batch file It consists of a series of keywords together with values for those keywords with each keyword on a separate line Most of the options available in the interactive session are supported by keywords in the batch file and thus most calculations can be done using the batch interface 77 A library of subprocess numerical codes is utilized Each time the batch interface is run it first checks whether the subprocess numerical code exists If it does it reuses it and skips the often long process of code generation Any requested numerical codes not in the library are then generated and added to the library If the model changed the numerical codes are regenerated as appropriate The numerical phase space integration is done and events are generated for each subprocess and the results are combined Production and decay events are connected and the final event output is an LHE file with all the events fully decayed which can be used directly by Pythia or other software Multiple parameters can be scanned over For each parameter point
47. the CalcHEP package is to generate C code that numerically calculates the squared matrix element The next menu item Make amp Launch n_calchep performs the symbolic calculation as described in the previous paragraph It then writes the C code for those squared dia grams compiles it and executes the resulting interactive numerical code for the generated squared diagrams It also advances to the next menu 40 The next menu item is Make n_calchep which causes CalcHEP to per form the symbolic calculation of the squared matrix elements write the C code and compile it However in distinction to Make amp Launch n_calchep it does not execute the resulting code Also in distinction it closes the graphical user interface and performs these steps in the background When it is finished it prints the message n_calchep is created to stdout and quits The executable can be found in the results subdirectory of the user s work directory A lock file is stored in the user s work directory to prevent CalcHEP from having multiple instances running at the same time and interfering with each other The Reduce program menu item creates a version of the squared diagrams that is formatted for the Reduce program 45 Each squared diagram is put in a separate file pm_n red where m is the subprocess number and n is the squared diagram number These files are not used further by CalcHEP but can be useful when the user would like a symbolic expression for the s
48. the command gnuplot lt plot_ gnu The file plot_ kumac is also created and contains PAW instructions which can be used by issuing the command paw b plot_ kumac Alternatively the user can load the data interactively in either gnuplot or PAW e Math file which allows to save the plot data to the file plot_ math in Mathematica syntax It can be read into a Mathematica session using Get lt file gt where lt file gt is replaced with the file name and path in parantheses This file also includes example plotting commands which the user can copy and modify to suit his her purpose e LaTeX file which allows to save the plot in the form of a BTFX axodraw 44 file The axodraw style file is required for compilation and can be found in the CALCHEP utile directory In the case of a two dimensional histogram the Plot Viewer represents the differential cross section of each bin by a solid black rectangle where the height and width of the black rectangle is given by S 0 9 x bin Size x F Fmax 1 where S is the height or width of the rectangle binSize is the size of the vertical or horizontal bins F is the differential cross section for that bin Fmaz is the maximum differential cross section in the histogram and P is a parameter chosen by the user to achieve the best resolution It defaults to the value 1 If a two dimensional histogram is being viewed the first three menu options are replaced with e S F P which allo
49. the options available in this file are the text font whether to use color or black and white and whether to use a sound to signal certain events e calchep is the shell script which is normally used to start the symbolic session It is invoked as calchep e calchep_batch is the shell script which is normally used to start a batch session It is invoked as calchep_batch lt batch file gt where 15 lt batch file gt is the name of the file which contains the batch instruc tions An example batch file is stored in CALCHEP utile batch_file The user can copy it and modify as necessary for their calculation When run the batch program creates the following directories Processes is where the binaries for the individual processes are generated and stored and where the numerical calculations are performed Events is where the event files in LHE format and distributions are stored html is where a system of html files are stored that inform the user of the progress and results of the calculation This directory also contains a rich set of help files which explain the details of how to use the batch system These files can be opened in the users web browser The file html index html contains links to all the other html files 2 7 Potential problems in compilation X11 The most frequent compilation problem is due to the absence of the X11 include files on the user s computer Usually these files are stored in the di
50. the session number This file also contains the results of the Monte Carlo integration It is useful if the user would like to determine what parametershe she used in an earlier calculation and what the results were When CalcHEP generates events they are stored in the file events_N where N is the session number Moreover distributions are stored in distr_N where N is the session number Other results are written by CalcHEP to files with the session number N as part of the file name and will be described in later sections 46 5 3 Parton Distribution Functions The first items in the In state menu are S F 1 and S F 2 which control the structure functions of the first and second incoming particles respectively Each of these menu items opens a new menu which allows to choose whether the user wants the structure functions OFF or whether he she wants the PDT structure functions or the LHAPDF structure functions The LHAPDF 40 sets require separate installation described below After making this choice CalcHEP presents the user with a list of the available structure functions and then finally allows to choose any free parameters of the structure functions Only structure functions allowed for the incoming particles are listed Examples of the list of parton distribution functions and of setting the properties of a LHAPDF structure function are presented in Fig 5 3 The PDT structure functions are stored in the directory CALCHEP pdTables CalcH
51. times the charge should be entered in the Aux field For example a particle with electric charge of 1 would be enetered as 3 a particle with electric charge of 2 3 would be entered as 2 and so on This charge must be written 104 before other symbols in this column if there are any We reiterate that this charge is not used to define the Feynman rules of the photon in calculations done by CalcHEP The interactions of the photon are entered in the Vertices table along with all other Feynman rules see Subsection 8 4 The electric charge defined in the Aux column of the Particles table is only used to communicate with other programs that require it 10 amp 11 LaTeX A and LaTeX A This is where the BTFX symbol for the particle and antiparticle are entered These symbols are used when CalcHEP produces TeX output for the Feynman diagrams that it constructs 8 4 Interaction Vertices The Vertices table contains the Feynman rules for the model The first four columns A1 A2 A3 and A4 specify the particles and antiparticles involved in the interaction These must be the particle and antiparticle names defined in the Particles table The last of these A4 may be empty which specifies a three point vertex The first three columns must be nonempty The propagators are not specified in this table They are hard coded Section 8 6 contains further details The last two columns Factor and LorentzPart define the vertex If S is the a
52. 0 p2 4000 HHHHHHHHHHHHHHHHHHHHHHHHHHHEHHEHEH Parameter Info HHHHHHHHHHHHHHHHHHHHHHHHHHHEHHEHEY Parameter Mtp 172 5 92 FERPA TE TETERE ETE ETE HE AE PETE TET EET EE HEHE BE ATE EE Run Info SERETETETETERE TERE HEHE AE TE TE TET EE AE BEE ATE Run parameter Mh Run begin 120 Run step size 5 Run n steps 3 FEHEHETETE TE TERE PETETEEE SE HE HEHE BE PE TEE EEE HE EEE QCD Running Info TERETE TE TETETETE TERE HEHE BE PETE TEETER HEHEHE AE BE PETE TEE EEE HEHE alpha Q M45 HHTHHHHHHHHHHEHHEHHEE HEHEHE H REAR RES Cut Info HEHHHHHHHHHHHHHHHHHHEHHHHEHERHEHE HEHEHE Cut parameter M b B Cut invert False Cut min 100 Cut max Cut parameter J jet jet Cut invert False Cut min 0 5 Cut max Cut parameter T jet Cut invert False Cut min 20 Cut max Cut parameter N jet Cut invert False Cut min 2 5 Cut max 2 5 HHEHHHHHHHHHEHHEHHEE HAAR HEE HERE R 93 Kinematics Info HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEHEHEHH Kinematics 12 gt 3 45 Kinematics 45 gt 4 5 FEHR HETETE PETE TE PETETE TEE HE HEHE AE PE TEE EEE HEE Regularization Info TEER HETE RE TE TETEPEAETEHE SE HEHE BE TE TEE EEE EEE Regularization momentum 45 Regularization mass Mh Regularization width wh Regularization power 2 HHEHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEH HEH Distribution Info HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH HEH Dist parameter M W b Dist min 100 Dist max 200 Dist n bins 100 Dist title
53. 4 and E e Vertices with a non trivial color structure for example the four gluon vertex of the SM are implemented by means of an unphysical tensor auxiliary field The vertices involving this auxiliary field are treated in the same way as the Fadeev Popov ghosts and Goldstone bosons The Feynman diagrams involving these auxiliary fields are not constructed at this point They are restored after the diagrams are squared See Sections 8 7 8 4 E for further explanation The second menu item is Squaring and causes CalcHEP to create squared diagrams CalcHEP uses these squared diagrams for subsequent calculations of squared matrix elements See Section C for the details The Write down processes menu item creates the file list_prc txt in the results subdirectory This file contains a list of the constructed subprocesses Menu 6 The View squared diagrams menu item is similar to the View diagrams of the previous menu however it displays the squared diagrams Each squared diagram is a graphical representation of AB where A and B are Feynman diagrams constructed in the previous step 39 We summarize some features of the squared diagrams in CalcHEP e CalcHEP does not construct both AB and BA Instead it only generates AB and calculates its contribution to the squared matrix element as 2Re AB This results in smaller more efficient code e CalcHEP constructs only one representative of a set of squared dia grams which can be tr
54. 4 ees I 3 Generation of events ooo a 0202s I4 Format of event files oe 4 4 2 ue eh eee Oe ee J Table of exit codes 150 150 150 151 153 153 155 157 157 157 159 160 162 162 163 163 164 166 1 Preface 1 1 Introduction CalcHEP 1 is a package for the automatic calculation of elementary par ticle collisions and decays in the lowest order of perturbation theory the tree approximation The main idea of CalcHEP is to provide an interac tive environment where the user can pass from the Lagrangian to the final distributions effectively with a high level of automation Other packages created to solve similar problems are GRACE 2 3 4 HELAS 5 CompHEP 6 7 FeynArts FormCalc 8 9 10 MADGRAPA 11 12 HELAC PHEGAS 13 14 15 O MEGA 16 WIHIZARD 17 and SHERPA 18 19 The interactive session of CalcHEP is graphical and menu driven and guides the user through the calculation by breaking the calculation up into a series of steps At each step CalcHEP presents the user with the available options allowing him her to control the details of their calculation in an intuitive way Moreover at each menu contextual help is available which explains the details of the current choices The batch session of CalcHEP is controlled by a set of scripts which perform common tasks noninteractively After initializing the desired calcu lation the user runs one of the scripts in the background In some cases parallelization is
55. 42 bee deb ee 105 8 5 External functions and libraries 108 8 0 Propagatois s agas a malia Uae bee Ole ae Aw eRe 8 109 8 7 Ghost and Goldstone fields propagators 110 8 8 The SLHAplus package 2442448444 28 444 44 444 amp 4 112 8 9 LanHEP automatic generation of models 112 8 10 FeynRules a aaa be Oe oS a Se 2 Es 112 CalcHEP as a generator of matrix elements for other pack ages 113 9 1 Choosing of model 0404 4 64 46 4 6 ee oe oa 113 9 2 Setting of parameters and calculation of constraints 114 9 3 Testing of particle contents 115 9 4 Decay widths and branching fractions 115 9 5 Compilation of new processes oo ooo a 116 9 6 Calculation of matrix elements 117 10 CalcHEP output for Reduce Mathematica and Form 119 10 1 General structure 64 64 665 24 26 heh dS ee ES 119 10 2 Reduce examples c o 4 des sost soa Oe ee Oe Be T21 10 3 Mathematica examples 2624 86 bo ew ee es 123 104 FOTM ias 6 eie a E e OO AD e om N e e a 125 10 5 Redu e progran 2444244488244 beri toin ead ao 125 Appendix 130 A PTRX output 130 B Self check of the CalcHEP package 130 B 1 Check of the built in symbolic calculator 2 131 B 2 Comparison of results produced in two different gauges 131 C Ghost fields and the squared diagram technique for the t Hooft Feynman gauge 132 ol Whe problenm s s e maaie ww
56. 49 process continues until the energy of each string is low enough to form a stable composite particle and it is then treated as a meson If we consider a QCD amplitude diagram without external gluons we can use the rules from 27a 28a to transform the amplitude s color diagram to one where separated quark lines connect gq pairs This corresponds to the color string picture described above and used by PYTHIA It should be noted that these color states are orthogonal only in the Ne gt oo limit The orthogonality is required to treat the squared basis coefficients as mutually independent probabilities of producing the corresponding color strings Thus the color string model should be considered in the framework of the 1 N approximation In the same approximation one can consider the gluon color state as a qq state 28a Therefore the gluon is a particle where one color string is finished and another on begins During event sampling CalcHEP generates a phase space point according to the exact N 3 matrix element At the same time it also calculates the leading coefficients of the 1 N expansion for the matrix element over the color flow basis The generated phase space point is accompanied with a color flow with a probability proportional to the squared basis coefficient G Distribution functions and beam spectra G 1 Backscattered photon spectrum This function describes the spectrum of photons scattered backward from the inte
57. 693 95 00298 Y 171 51 52 53 54 55 56 57 58 59 60 61 E Boos M Dubinin L Dudko Higgs boson production under the resonance threshold at LEP II Int J Mod Phys A11 1996 5015 5026 arXiv hep ph 9602220 doi 10 1142 S0217751X96002315 E Byckling K Kajantie Particle kinematics Wiley 1973 V A Ilyin D N Kovalenko A E Pukhov Recursive algorithm for the generation of relativistic kinematics for collisions and decays with regu larizations of sharp peaks Int J Mod Phys C7 1996 761 arXiv hep ph 9612479 doi 10 1142 S0129183196000648 T Sjostrand High energy physics event generation with PYTHIA 5 7 and JETSET 7 4 Comput Phys Commun 82 1994 74 90 doi 10 1016 0010 4655 94 90132 5 J Alwall et al A Les Houches Interface for BSM Generator sarXiv 0712 3311 doi 10 2172 921331 S Belov L Dudko D Kekelidze A Sherstnev HepML an XML based format for describing simulated data in high energy physics Comput Phys Commun 181 2010 1758 1768 arXiv 1001 2576 doi 10 1016 j cpe 2010 06 026 G Belanger N D Christensen A Pukhov A Semenov 2010 arXiv 1008 0181 Pardicle data group URL http pdg 1b1l gov S Wolfram Mathematica A System for Doing Mathematics by Com puter 1988 L Baulieu Perturbative Gauge Theories Phys Rept 129 1985 1 doi 10 1016 0370 1573 85 90091 2 T P Cheng L F Li GAUGE THEORY OF ELEMENTARY PART
58. 88 1126 6708 2006 02 033 G Belanger M Kakizaki A Pukhov Dark matter in UED the role of the second KK levelarXiv 1012 2577 G Belanger A Pukhov G Servant Dirac Neutrino Dark Matter JCAP 0801 2008 009 arXiv 0706 0526 doi 10 1088 1475 7516 2008 01 009 A Semenov LanHEP a package for the automatic generation of Feyn man rules in field theory Version 3 0 Comput Phys Commun 180 2009 431 454 arXiv 0805 0555 doi 10 1016 j cpc 2008 10 012 N D Christensen C Duhr FeynRules Feynman rules made easy Comput Phys Commun 180 2009 1614 1641 arXiv 0806 4194 doi 10 1016 j cpe 2009 02 018 P Z Skands et al SUSY Les Houches Accord Interfacing SUSY Spectrum Calculators Decay Packages and Event Genera tors JHEP 07 2004 036 arXiv hep ph 0311123 doi 10 1088 1126 6708 2004 07 036 B C Allanach et al SUSY Les Houches Accord 2 Comp Phys Com mun 180 2009 8 25 arXiv 0801 0045 doi 10 1016 j cpc 2008 08 004 V M Budnev I F Ginzburg G V Meledin V G Serbo The Two photon particle production mechanism Physical problems Applica tions Equivalent photon approximation Phys Rept 15 1975 181 281 doi 10 1016 0370 1573 75 90009 5 170 40 41 42 43 44 45 46 47 48 49 50 M R Whalley D Bourilkov R C Group The Les Houches Accord PDFs LHAPDF and LhagluearXiv hep ph 0508110 J Alwall et al A standard format for Les
59. ATION OF FEYNMAN GRAPHS AND AMPLI TUDES Comput Phys Commun 60 1990 165 180 doi 10 1016 0010 4655 90 90001 H T Hahn Generating Feynman diagrams and amplitudes with FeynArts 3 Comput Phys Commun 140 2001 418 431 arXiv hep ph 0012260 doi 10 1016 S0010 4655 01 00290 9 167 10 13 14 15 16 17 18 19 ie 20 T Hahn Automatic loop calculations with FeynArts FormCalc and LoopTools Nucl Phys Proc Suppl 89 2000 231 236 arXiv hep ph 0005029 doi 10 1016 S0920 5632 00 00848 3 F Maltoni T Stelzer MadEvent Automatic event generation with MadGraph JHEP 02 2003 027 arXiv hep ph 0208156 J Alwall M Herquet F Maltoni O Mattelaer T Stelzer Mad Graph 5 Going Beyond JHEP 06 2011 128 arXiv 1106 0522 doi 10 1007 JHEP06 2011 128 A Kanaki C G Papadopoulos HELAC A package to compute elec troweak helicity amplitudes Comput Phys Commun 132 2000 306 315 arXiv hep ph 0002082 doi 10 1016 S0010 4655 00 00151 X C G Papadopoulos PHEGAS A phase space generator for automatic cross section computation Comput Phys Commun 137 2001 247 254 arXiv hep ph 0007335 doi 10 1016 S0010 4655 01 00163 1 A Cafarella C G Papadopoulos M Worek Helac Phegas a generator for all parton level processes Comput Phys Commun 180 2009 1941 1955 arXiv 0710 2427 doi 10 1016 j cpc 2009 04 023 M Moretti T Ohl J Reuter O Mega An optimizing
60. By default CalcHEP uses the double numerical type to store the initial and intermediate parameter values and double precision functions to work with them The user can optionally choose to compile CalcHEP for high precision calculations The long double type is part of the C99 standard and realized on all modern compilers however one has to note that usually the long double type is implemented with 80 bit precision In this case calculations will be as fast as with the standard double type but the increase in preci sion is not significant To enable the long double type the compiler option D_LONG_ needs to be added to the FlagsForSh file and CalcHEP needs to be recompiled If CalcHEP has already been compiled gmake clean needs to be run first but FlagsForSh should be kept If CalcHEP has not been compiled yet gmake flags should be run first in order to create FlagsForSh The Intel C compiler has a _Quad type quadruple precision for 128 bit real numbers To use this type in CalcHEP calculations the compiler option D_QUAD_ has to be added to FlagsForSh One should note that currently only the Intel compilers support _Quad type and that the Intel compilers re quire further options Here is an example of the CFLAGS line of FlagsForSh for the Intel compiler CFLAGS D_QUAD_ fPIC fsigned char Qoption cpp extended_float_type To implement other numeric types the user should edit the file include nType h 14 2 6 User insta
61. CHEP utile directory To use these tests the user should first copy the test files into his her working directory The commands to run the tests should be executed from the same directory as the location of the test files in the users working directory B 1 Check of the built in symbolic calculator The first check is a comparison of the results of the CalcHEP symbolic calculator see Section 10 against the results of the Reduce calculator see Section 10 5 We take agreement between these results to mean that the CalcHEP symbolic calculator is working correctly We note that the results of the Reduce code created by CalcHEP for the test can be viewed step by step as the calculation is performed by Reduce The result can then be compared with the result of the CalcHEP symbolic calculator This check is realized by means of the program check red which must be started from within a Reduce session by the command in check red When doing this it is assumed that the Reduce code for the diagrams and the corresponding symbolic expressions generated by CalcHEP are stored in the results directory in advance The results of this check are saved in the message file It consists of a list where each line contains a diagram number accompanied by the label OK or Error depending on the result of the comparison B 2 Comparison of results produced in two different gauges A comparison of results produced in unitary gauge and t Hooft Feynman gau
62. CalcHEP Calculator for High Energy Physics A package for the evaluation of Feynman diagrams integration over multi particle phase space and event generation A Pukhov A Belyaev N Christensen User s manual for version 3 30 XXXX 2011 Based on Pukhov et al hep ph 9908288 calchep googlegroups com Contents 1 Preface 1 1 Introduction 0 0000200 00000000 a a 1 2 Acknowledgments 4 s d a 5 amp wd kew Ob eS a we Installation procedure 2 1 CalcHEP Web Site 2 x 24k ope coke 2B oe eB EES 2 2 Galchep License 2 2 ge 8 ee ee ee ee 2 8 Howto get the code 46 6202 eee ee wR ee 24 Compilation procedure 6 224665 445 555445 2 5 Compilation for High Precision calculations 2 6 User installation and start of the CalcHEP session 2 7 Potential problems in compilation Elements of the user interface Menu system for symbolic calculation 4 1 Model Choice and Manipulation 4 2 Numerical Evaluations 2 22000 4 3 Process Tipu eas acd sag Db be ah ew Od OS 2S 4 4 Squared Diagrams and Symbolic Calculation 4 5 Output of results and launching of the numerical calculation 4 6 Switehes ci eek A Se oo Boe ES Oe eee Se Oke eS Numerical session 5 1 Sketch of the menu system 2 42245 240 4 e oe ee we 5 2 Bookkeeping 0 2 444 624 b 2 6a hee ee eee EE ES 5 3 Parton Distribution Munctions 64 4444 a 404 ba QCD copnne s
63. CalcHEP first asks whether the user would like to save his her modifications If the user answers N CalcHEP returns to Menu 2 and the modifications are lost If however the user answers Y CalcHEP first performs the following series of tests on the model which can also be initiated by choosing the CHECK MODEL menu item e Do all particle and parameter names satisfy the CalcHEP naming requirements Are all numbers entered correctly Have all parameters been declared before they are used in any expres sions Are all particles used in a vertex defined Are the algebraic expressions correct Are all the Lorentz indices correctly contracted e For any vertex is its conjugate vertex included If all these tests are passed after pressing the Esc key and choosing to save the model the model is saved to disk and CalcHEP returns to Menu 2 If on the other hand any of the tests are failed CalcHEP stops the tests after the first detected error and displays a message describing this error along with the table and position in that table where the error can be found In this case the user remains in Menu 3 and is allowed to fix the 32 mistake in his her input The error message can be viewed again by pressing E in any of the model tables After the user fixes the mistake he she can press the Esc key and try to save the model again 4 2 Numerical Evaluations After choosing Numerical Evaluations on Menu 2 the user is taken to M
64. EP comes with a set of PDT tables These include structure functions for initial state radiation ISR for incoming electrons Weizsaecher Williams structure functions for photons and structure functions for backscattering laser photons which are described further in Section G Additionally a set of parton distribution functions for the proton and anti proton are included as shown on the left of Fig 5 3 New PDT tables can be added as described in Appendix H PDT cteq6m proton PDT cteq61 anti proton PDT cteq61 proton PDT CTEQ5H anti proton cteq61 LHgrid PDT CTEQ5M proton PDT mrst2002nlo anti proton PDT mrst2002nlo proton PDT mrst200210 anti proton PDT mrst20021o proton Set 0 0 39 Figure 10 PDT structure functions left and LHA settings right To use the LHAPDF structure functions the LHAPDF library must be installed and linked to the model The way this is handled in CalcHEP is that it comes preinstalled with a dummy version of the LHAPDF routines which inform it that the LHAPDF structure functions are not used When the user installs and links the true LHAPDF structure functions they are AT used in place of the dummy version that comes with CalcHEP The de sired structure functions along with the LHAPDF libraries according to the LHAPDF instructions see 40 To use the LHAPDF structure function in the CalcHEP first of all one has to add L lt path_to_lhapdf gt 1LHAPDF line to Li
65. ES from LHE files An example of using this program is CALCHEP bin nt_maker lt events lhe gt where lt events lhe gt is the file containing the LHE events 68 7 Batch Mode Initially CalcHEP was designed for interactive calculations with a graphical user interface However there are times when a batch system is ideal For example when a calculation takes a very long time or the user is interested in doing scans over parameter space usters of computers where parallelization is possible In these situations it is preferrable to set up the program with a set of batch instructions and run it in the background CalcHEP provides the user with the ability to record their keystrokes and then use it as an instruction set in a batch mode The recording step is done by using the blind option When this option is used CalcHEP opens in interactive mode and records all the keystrokes the user performs When the user exits CalcHEP prints to the screen a string which represents the sequence of keystrokes The user can then call the program with the blind option along with the keystroke string to initiate the same calculation in batch mode All the results of the calculations are stored in the same places as in interactive mode allowing the user to use the results in the same way as in the interactive mode This batch option has proved to be very powerful but also very difficult for users to take full advantage of Some challenges are e Ift
66. I CLE PHYSICSOxford Uk Clarendon 1984 536 P Oxford Science Publications 172 62 63 64 65 69 A P Kryukov A Y Rodionov COLOR PROGRAM FOR CAL CULATION OF GROUP WEIGHTS OF FEYNMAN DIAGRAMS IN NONABELIAN GAUGE THEORIES Comput Phys Commun 48 1988 327 334 doi 10 1016 0010 4655 88 90052 5 I F Ginzburg G L Kotkin V G Serbo V I Telnov Colliding gamma e and gamma gamma Beams Based on the Single Pass Accelerators of Vlepp Type Nucl Instr Meth 205 1983 47 68 doi 10 1016 0167 5087 83 90173 4 E A Kuraev V S Fadin On Radiative Corrections to e e Single Photon Annihilation at High Energy Sov J Nucl Phys 41 1985 466 472 Yad Fiz 41 733 742 1985 M Skrzypek S Jadach Exact and approximate solutions for the elec tron nonsinglet structure function in QED Z Phys C49 1991 577 584 doi 10 1007 BF01483573 P Chen Differential luminosity under multi photon beamstrahlung Phys Rev D46 1992 1186 1191 doi 10 1103 PhysRevD 46 1186 J Pumplin et al New generation of parton distributions with un certainties from global qcd analysis JHEP 07 2002 012 arXiv hep ph 0201195 A D Martin R G Roberts W J Stirling R S Thorne Un certainties of predictions from parton distributions 1 Experimen tal errors Eur Phys J C28 2003 455 473 arXiv hep ph 0211080 doi 10 1140 epjc s2003 01196 2 V A Ilyin D N Kovalenko A E Pukhov Recu
67. O can be unambiguously fixed by the polar vector Pole whose space components in the rest frame of decay correspond to the 159 O 0 direction Let qj and qjz be the momenta of the first and the second clusters produced by the jt decay Then Pole qj1 Pole g Pole a1 2cos Pole g Pole qj2 Pole qjo Pole Ge 2cos Pole Go Thus in order to get cos related to a squared sum of some particle momenta we may construct the polar vector as a sum of particle momenta 69 70 For the non contradictory construction we need to set the decays in some order with a natural requirement that the sub decays of clusters produced by the jt decay have the ordinal numbers larger than j In giving such ordering we can construct a polar vector for each decay based on the incoming momenta and on those of particles produced by decays possessing smaller ordinal numbers The following statements can be proved In the framework of any ordered scheme of decays and for any sum P of particle momenta one can find the decay number j such that either P s or P might be represented as Pole qj where q is the momentum of one of the clusters in the j decay and Pole is a polar vector constructed according to the above rule In other words any of poles 4 5 6 can be expressed either in terms of s parameters or in terms some of cosO for an appropriate choice of the pol
68. W W W W e ne ne e e ne ne e gt 0 gt gt gt gt 0 gt gt 0 gt W W W W lt lt lt 0 lt gt lt lt gt A C W C W C A C A c W c W c A c c d Figure 15 Ghost diagrams 137 expanded diagram As a result we might expect that after being summed together the total expression is roughly the same size as each individual expression CalcHEP takes advantage of this by summing up the expanded diagram expressions and writes just one numerical code for this set C 5 Gauge symmetry and cancellations Cancellation of diagram contributions is an essential point both for sym bolic and numerical calculations because a relatively small variation of one diagram contribution may lead to a significant error Such variations can be caused either by finite precision calculations of floating point operations or by modifications of Feynman rules for instance by including particle widths or by removal of some diagram subset from the calculation We would like to stress the importance of these challenges to the user and encourage him her to think carefully about his her calculation There are two well known examples of gauge cancellations The first is the ultraviolet cancellation of quickly growing terms originating from the propagators of massive vector particles This problem can be resolve
69. account This is the reason mb mb and Mtop pole are included in this menu e order the loop order of the running Choices are LO NLO and NNLO e mb mb the pole mass of the bottom quark 48 e Mtop pole the pole mass of the top quark e Q GeV the scale of the calculation More details can be found below The QCD scale typically depends on the momenta of the particles In CalcHEP it can be defined as an algebraic expression which includes floating point numbers model parameters and the following primitive phase space functions e Sij gives p p the invariant mass squared of particles i and j which must satisfy i j lt Nin Nout e Mij gives p p the invariant mass of particles i and j which must satisfy either Nin lt i j lt Nin Now O i j lt Nin e Ti gives p p the transverse momentum of particle i e Mi gives M the mass of particle i e Wij gives the transverse mass of particles i and j which must satisfy i j gt nin The definition of the transverse mass can be found in Section 5 6 For example a popular choice of scale based on the Mandelstam variables is 2stu s t u and can be realized by the following function in CalcHEP Q GeV sqrt 2 S12 S13 S14 S1272 S13 2 S81472 The min and max functions can also be used with an arbitrary number of arguments as can standard mathematical functions Whatever function is defined for the scale o
70. age The Lorentz indices of the fields in the vertex are labeled by a m for the first index and a M for the second index followed by the particle number for that vertex For example a vector field in the third column would have Lorentz index m3 while a tensor field in the second column would have Lorentz indices m2 and M2 The momenta use the symbol p followed by the same number For example a scalar field in column 1 would have momentum pi A dot is placed between two momenta a momentum and its Lorentz index and between two Lorentz indices for the metric tensor Here are some examples pi p2 means pi ph pi M2 means p mi m2 means Jm mz Dirac y matrices are written with a G and the momentum or Lorentz index in parentheses while the ys matrix has a 5 without parentheses For example we have G m1 means y G p2 means po G5 means 5 The ys matrix is defined by Ys t WNI 106 The anti commutation relation for the gamma matrices in CalcHEP notation is G v1 G v2 G v2 G v1 2 vi v2 where vi and v2 are either momenta or Lorentz indices In the case of anti commuting fields the functional derivative in Equa tion 8 4 is assumed to act from the right The number of fermion fields in a vertex must be either two or zero If the user would like to implement a four fermion interaction he she must use an unphysical auxiliary field with a point like propagator see Subsections 8 3 and 8 6 for further det
71. ails CalcHEP interprets the anti particle spinor field as a C conjugated parti cle field rather than the Dirac conjugated field These definitions are related to each other by apa dye dy which is the reason for the appearance of the C matrix in Eq 8 4 The particle and anti particle fields can appear in the vertices in any order Vertices can also contain two particle fields or two antiparticle fields In other words vertices that violate fermion number are allowed Any fermion vertex can be written in two forms which depend on the order of the fermion fields After permutation of the fermion fields the LorentzPart is transformed according to 9 G v1 G v2 G5 G un gt G un G5 G v2 G w1 10 where the order of the gamma matrices is reversed and each gamma matrix with a Lorentz index gets a sign change while the ys matrix does not get a sign change We note that the definition in Eq 8 4 the LorentzPart has the ap propriate symmetry property when identical particles appear in the vertex This symmetry is not checked by CalcHEP and its absence will lead to the wrong results Equation 10 can be used to check this symmetry in the case of two identical Majorana fields in one vertex It should also be noted that in the case of n identical particles the functional derivative 8 4 gets a corresponding factor of n which should be included in the vertex 107 The totally antisymmet
72. alues for the particles of the SM that have been dis covered For the Higgs particle and particles beyond the SM event_mixer will assume that all the decay channels are present in the event files and estimate the total width from them The run_details txt file contains information about the events and is placed by event_mixer in the header of the resulting event file For example this file could contain the parameter values the center of mass energy the parton distribution functions and so on event_mixer randomly mixes production events and their decays accord ing to their cross sections and branching ratios It does this until the re quested number of events is generated or until it runs out of production events in any of the files Before it mixes the events it writes to screen the final cross section and the maximum number of events that can be generated For example 2 368E 01 total cross section pb 10098 maximum number of events To get this number before mixing the events simply request 0 events The results of event_mixer are stored in the file event_mixer lhe This file is written in the LHE file format with an XML header 41 and additional 66 sections written in HepML 56 format This format allows to automatically upload the LHE file to the CERN Monte Carlo Database MCDB using the command upload2mcdb_hepml pl header hepml event_mixer lhe where the upload2mcdb _hepm1 p1 script can be downloaded from the MCDB
73. ameter belongs It can contain up to 11 characters The first character must be a letter The others may be either letters or digits The underscore symbol is also permitted and CalcHEP is sensitive to the case of the characters There is a set of reserved names which cannot be used for parameter names e i is reserved for the imaginary unit e Sqrt2 is reserved for V2 e pl p2 p3 are reserved for particle momenta emi M1 are reserved for Lorentz indices G5 is used for the y Dirac matrix 99 There is another subtelty that should be considered when naming pa rameters Although CalcHEP is sensitive to the case of the parameters Reduce is not Therefore if the user would like to use the CalcHEP re sults in Reduce he she should distinguish all names by more than case Additionally although CalcHEP allows underscores as part of param eter names the underscore is treated differently by Mathematica So if the user would like to use the CalcHEP results in Mathematica he she shouldnot use underscores in the parameter names Further more CalcHEP allows parantheses in parameter names but Reduce and Mathematica do not The user should name their parameters ac cordingly 2 Value This is where the numerical value for the parameter is stored Dimensionful parameters should be in powers of GeV 3 Comment This is where the user can enter a description of the pa rameter It is ignored by CalcHEP and is purely for informat
74. ams with A All Constraints Exclude X particles G Delete model Masses Widths Br View squared diagrams C code for num calc C compiler Symbolic calculation View diagrams Edit linker REDUCE code Make amp Launch n_calchep Square diagrams MATHEMATICA code FORM code Write down processes Enter new process E S C progran Figure 7 Menu flow for the interactive symbolic session 30 Menu 2 The first item on this menu allows the user to enter a physical process and will be explained further in Subsection 4 3 The second item on this menu is Force Unit Gauge and allows the user to use Unitary gauge in his her calculation even if the model is imple mented in t Hooft Feynman gauge Generally we recommend to use t Hooft Feynman guage whenever possible as the ultra violet cancellations between diagrams are much better in this case Further information can be found in Section C The third item on this menu is Edit Model and allows the user to view and modify the current model and will be described further when we discuss Menu 3 below We also note that after the user reaches Menu 5 where a process is entered he she can still view the model by pressing the F3 key but cannot modify it The fourth item on this menu is Numerical Evaluation and allows the user to view the value of the dependent variables as well as the masses widths and branching ratios of the particles Further details can be found in Subsection 4 2 Th
75. an be obtained by cc gt interface gt sqme nsun GG pvect amp err where cc is a pointed for compiled process nsub number of subprocess GG is strong coupling agea GG 47 pvect array of momenta err 117 interger variable which contains error code after execution Sumation over outgoing polarization and avaraging over incomings ones is performed 118 10 CalcHEP output for Reduce Mathemat ica and Form 10 1 General structure In addition to writing C code which has already been described CalcHEP can output the results of the built in symbolic calculation in a format suitable for the Reduce package 45 the Mathematica package 59 and the Form package We have attempted to present the results in a form which can easily be used for different purposes All the squared diagram contributions for one subprocess are stored in one file The subprocess number is appended to the file name For example the symb1 red and symb1 m files contain the symbolic expressions for the squared diagrams in the first subprocess in the syntax of Reduce and Mathematica respectively The output files have the following structure Initial declarations initSum First diagram code addToSum Second diagram code addToSum finishSum Initial declarations includes the declaration of variables for the momenta and conservation law relations for them the declaration of the independent parameters involved in the cal
76. anching fractions but the 63 user can switch between the two via the Show Branchings Show Partial widths menu item The total width for the particle the sum of the partial widths is displayed at the top of the screen The dependence of the branching ratios and the width on the model parameters can be viewed by using the Parameter dependence menu item This will open a menu allowing to choose either the total width or one of the decay branching ratios It will then allow to choose the model parameter and display the dependence using the Plot Viewer The width of the Higgs particle depends sensitively on the effective quark masses which can accumulate higher order QCD corrections to their Yukawa couplings In the built in version of the Standard Model we assume that these masses depend on the model parameter Q The correct width is obtained if Q is set equal to the mass of the particle that is decaying e g the Higgs This is the default value of Q for a decay But the user can adjust this or view the dependence of the width on this parameter as desired For other models to enjoy this feature the masses must be implemented in a similar fashion 64 6 Collecting Subprocesses The CalcHEP interactive sessions are designed to run one subprocess at a time However many typical collider processes contain many subprocesses that differ only by the initial state and or final state particles For example at the LHC the initial states are two
77. ansformed into one another by permutations of identical outgoing particles The needed symmetrization for these par ticles is performed during the symbolic or numerical calculations of the squared matrix element Again this results in smaller more efficient code e Each squared diagram represents a set of squared diagrams where some physical particles are replaced by their associated ghosts Goldstone bosons and or auxiliary fields in all possible ways according to the Feynman rules defined for the model This set of squared diagrams can be viewed at this stage by pressing the G key while the desired squared diagram is highlighted Just as for the diagrams of the previous menu the squared diagrams can be deleted by the user while in the Diagram Viewer Furthermore if the squared diagrams have already been calculated each diagram will contain one of CALC ZERO Out of memory or Del They mean respectively that the squared diagram is calculated is identically zero the calculation ran out of memory or the squared diagram was deleted by the user The second menu item is Symbolic calculation and instructs CalcHEP to begin the symbolic calculation of the squared matrix element using the gen erated squared diagrams This is done by the built in symbolic calculator During this calculation CalcHEP displays the current status of the calcu lation which includes which diagram is currently being worked on and how many are left The main goal of
78. ar to a chess sacrifice The main idea is to include unphysical polarizations in the incoming and outgoing states and cancel them by use of the Fadeev Popov ghosts and Goldstone bosons We consider the Faddeev Popov ghosts and Goldstone boson states as new polarizations similar to the temporal polarization see Eq 16 We note that the Faddeev Popov ghost states have a negative norm as does the temporal polarization whereas the Goldstone boson state has a positive norm 48 As a result the unphysical fields can give both a positive and a negative contribution to the polarization sum 134 The main point is that the sum of the contributions of all unphysical states and unphysical polarizations to the squared matrix element vanishes 60 As a result L AA D o AAT 22 i Sphys te Sail where 7 is a multi index for the polarization states A is the amplitude of the process Sphys is the set of physical polarization states S is the full set of physical polarizations unphysical polarizations and unphysical states and o i 1 depending on the signature of the Hilbert space norm for the polarization state 2 A drawback of enlarging the set of polarization states is that we have a much greater number of squared matrix element terms for evaluation and subsequent summation However to see the advantage of this trick let us sum the contributions from the temporal polarization Eq 16 and the longitudinal polarization Eq 21 We not
79. ar vector 69 70 In CompHEP the ordering is arranged automatically so that all sub decays of the first cluster have smaller numbers than those of the second cluster Polar vectors are also constructed automatically according to the list of peaks prepared by the user 1 1 3 Smoothing The general idea of the integrand smoothing is trivial Let us need to evaluate b J F x dx 39 and let F x have a peak like f x where f x is a simple symbolically integrable function in contrast to F x g Fear 40 Now we may represent the integral 39 as 9 F g y I eee E FT i where g y is the inverse function for g x The integrand is a smooth function now We face very often squared matrix elements which have several poles in one of variables For example the y gt b b Z b b and H b b vir tual subprocesses may contribute just to the same amplitude Although in this case we can evaluate the integral function g x symbolically the in verse function g y can be computed only as a numerical solution of the corresponding equation To bypass the calculation of inverse function Com pHEP uses the multi channel Monte Carlo branching method to smooth a sum of peaks The idea of the branching method is the following Let F x have two peaks one is similar to f x and another to fo x fi a and f x are singular but elementary functions Then instead of one integration 39 we could perform two ones J EO
80. article e IQlmax 150 Choices for the Photon particle keyphrase are mu e e mu The default is e The default for the keyword Q max is the same as in the CalcHEP interactive session If Proton Photon is chosen then the following may be specified Incoming particle mass 0 937 Incoming particle charge 1 Q 2 max 2 1 Pt cut of outgoing proton 0 11 The defaults are the same as in the CalcHEP interactive session 82 Momenta The momentum of the incoming states can be specified with the keywords p and p2 and are in GeV as in pi 7000 p2 7000 These are the default values for the momenta Parameters The default parameters of the model are taken from the varsN mdl file in the models directory Other parameter values can be used if specified using the Parameter keyword Here is an example Parameter EE 0 31 Parameter SW 0 481 Parameter MZ 91 1884 Parameter wW 2 08895 This gives a convenient way of changing the default values of the param eters Simply open CalcHEP in symbolic mode choose to edit the model and change the values of the indepenedent parameters These new values will then become the default values used by this batch program There is no need to redo the process library Scans In some models it is useful to scan over a parameter such as the mass of one of the new particles For example if there is a new W gauge boson it may be desireable to generate events and or
81. article the order does not matter in the case of a Majorana fermion vertex and the Lorentz piece is inserted directly The same is true for a C conjugate operator Once the fermion chain is finished the spur switch is restored and Re duce performs the y matrix trace CalcHEP multiplies this trace by 4 The 4 is required because Reduce leaves this out of the gamma matrix trace The 1 is due to the Feynman rules The next step is the multiplication of the other vertices and the contrac tion of the Lorentz indices The Lorentz indices that need to be contracted are declared by the Index instruction before the multiplication An example of this fragment of code is Index m2 Vrt_3 Vrt_3 Vrt_5 RemInd m2 This code instructs Reduce that vertex 3 is multiplied by vertex 5 and that Lorentz index m2 is contracted The result of this multiplication is stored in the variable Vrt_3 In order to make the Reduce code produced by CalcHEP manage memory better we declare the symbols m as vectors indices step by step as we perform the calculation If two vertices are connected by the propagator of a massive vector par ticle treated in unitary gauge see Eq 11 the code is organized in the following way CalcHEP first writes the code for the product with the in dices contracted the g piece of the propagator It then writes the same product but with the p p M piece of the propagator It then adds the pieces together Below we present an
82. at the QCD scale is taken to be the 84 invariant mass of particles i and j Or it can be specified as a formula in terms of the parameters of the model as in Mt 2 which means half of the top quark mass When specifying the scale in terms of the invariant mass of final state particles the numbers are taken from the way the processes are entered with the Process keyword So if the process is specified as p p gt j 1 n M45 means the invariant mass of the lepton and neutrino 1 n The batch script will take care of renumbering if the subprocesses have the final state particles in a different order It is also sometimes useful to use a different scale for different processes For example suppose the two processes p p gt j 1 n and p p gt j j 1 n are specified in the batch file the scales could be specified as in this example alpha Q 1 M45 alpha Q 2 M56 The number between the specifies which process to apply this scale and corresponds to the order in which the user specified the processes If more than one process is specified but the same non default scale is desired for all of them this can be specified as in alpha Q Mt 2 This specification will apply the same scale Mt 2 to all processes Cuts Cuts are specified with the keywords Cut parameter Cut invert Cut min and Cut max and use standard CalcHEP notation except for Cut invert which can be either True or False These cuts are only applied to the production proce
83. ation of the html progress reports which the user can simply copy and paste into their browser url window The first time the user runs the batch interface they can also run the following from the work directory calchep_batch help which will complain that no batch file was present create a series of html help files and quit The location of the html help files will be printed to screen This html help file can be opened in a web browser and contains all the details that are presented here In the following subsection we describe each keyword available for the batch file and how to use it An example batch file is stored in CALCHEP utile batch_file 7 3 1 Structure and keywords of the bacth file Comments Any line beginning with a is ignored by run_batch The has to be at the very beginning of the line Some examples are This is ignored Model Standard Model This is ignored Model Standard Model CKM 1 This is not ignored Model The first section of the batch file should contain the specification of the model This is done by model name and should match exactly the name in the CalcHEP model list So if you want to run the Standard Model CKM 1 you would specify this with the batch file line Model Standard Model CKM 1 There is no default for this line It must be included The gauge of the calculation should also be specified in this section Choices are Feynman and unitary gauge CalcHEP is much better
84. au tomatically to accomodate the new string Del D removes the current row of the table and stores it in the buffer Pressing the Enter key creates a new row with buffer contents Size S allows the user to change the width of the current column Colunm sizes are also increased automatically if the user enters a string which is too wide for the current column ErrMess E redisplays the most recent error message associated with editing a table One can use the Enter key to creates a new row 6 Diagram Viewer The Diagram Viewer is designed to display multiple Feynman diagrams at a time The viewer splits the window into rectangular cells and puts one diagram in each cell Each cell is enclosed by a frame while the current diagram is highlighted by having a thicker frame than the 23 Delete On foff Restore Latex F1 Help F2 Man PgUp PgDn Home End Esc Figure 3 An example of the CalcHEP Diagram Viewer others The index of the current diagram is displayed at the top right of the window along with the total number of diagrams An example can be seen in Fig 3 The number of diagrams which can be displayed simultaneously depends on the size of the window The user can increase this number by increasing the size of the window using his her window manager typically by dragging an edge or corner of the window The currently highlighted diagram can be changed by using the Ar row keys and or the mouse Furthermore
85. braries model item It is enough to compile executable results n_calchep but may be not enough to launch it If your lt path_to_lhapdf gt is disposed in system area then problems are not expected An executable has to know a location of shared libraries used at time of launching of CalcHEP Stan dard paths like usr lib or usr local 1lib are checked by default An arbitrary library location can be passed to executable via environment export LD_LIBRARY_PATH LD_LIBRARY_PATH lt path_to_lhapdf gt You can define this enviroment before you start to work with CalcHEP in clude it to you startup file batchrc or add it to calchep and calchep_batch scripts In case of of adding it to calchep it is better to use the other variable LD_RUN_PATH which provides the generated n_calchep with the property to be launched independently without setting environment variables For calchep_batch run we recommend to define LD_LIBRARY_PATH variable 5 4 QCD coupling The value of the strong coupling constant depends on the scale of the calcu lation CalcHEP runs the strong coupling constant The parameters of this running can be set in the QCD coupling menu and include e alpha MZ the strong coupling value at Mz e nf the maximum number of quark flavors in the running At small energies CalcHEP takes into account threshold effects caused by charm and bottom quarks and changes nf If nf 6 then the top quark mass threshold is also taken into
86. calculation Moreover the precision of this approach is not too bad especially in cases of QCD processes where loop corrections are at the same order See Section for more details about this option 141 D 3 Common factors The final squared diagram is multiplied by a set of factors which are e SymmFact n d where n 1 if the left part of the squared diagram is the same as the right part Otherwise n 2 The denominator d equals the number of symmetric permutations of identical outgoing particles e AverFact 1 N were N is the number of polarization and color states of the incoming particles e FermFact 1 4 were Ny is a total number of incoming fermions and anti fermions E Examples of model realization E 1 Implementation of QCD Lagrangian 3 gluon vertex The QCD Lagrangian contains the following 3 gluon vertex Sza J 8 G2 8 G f3 G9Cld x g J DGA grs fa agGe2Gedte where Gi is the gluon field and g is the strong coupling constant Applying the Fourier transformation 8 we get Ssa 2m g i 5 pitpotps i forazas P g GR pi G03 p2 G83 p3 d pid pod ps This vertex contains three identical fields so the calculation of the func tional derivatives gives us six terms Da Ga p Ga2 p2 0G ps glp ga _ ph giis phe ger _ gg a peg t p gt gr 27 8 p1 p2 p a Toes In notations of section 10 5 142 Comparing this expression with it s vertex
87. ccessful integration the cross section is displayed on the screen The precision of the calculation can be set via the Set precision menu item The default value is 1074 The angular dependence of the differential cross section can be viewed via the Angular dependence menu item Plots of 62 CalcHEP num Incoming particle Show Branchings QCD Scale Q M1 Hodel parameters Constraints Parameter dependence Figure 11 An example of the Easy 1 gt 2 menu for the two particle decays of the SM Z boson the dependence of the cross section on the model parameters or the center of mass energy can be viewed by the Parameter dependence menu item This menu also provides sigma v plots for v x a v at v gt 0 Here v is relative velocity of particles This characteristics is useful for astroparticle applications in particular for the estimations of the elastic scatteting of Dark Matter cadidates 5 14 Two Particle Decays 1 2 decays do not require numerical integration at all It can be done symbolically In this case CalcHEP displays the Easy 1 gt 2 menu item which allows to view the widths and branching ratio An example of this screen can be seen in Figure 11 If multiple decay channels are generated during the symbolic session for example if the user specifies the process as Z gt 2 x each nonzero decay channel will be listed on this screen along with either its partial width or branching fraction The default listing is with br
88. cond the events are generated Details of the algorithms used and the format of the event files can be found in Appendices F I 3 and 1 4 Generator preparation The efficiency of the event generator depends on the number of phase space cubes and the estimation of the maximum differential cross section or partial width in each cube Since the differential cross section or partial width can very greatly from phase space point to point especially around a resonance a larger number of phase space cubes allows for a more efficient generator The menu item Event cubes allows to modify this parameter On the other hand for each phase space cube it is very important to have a good estimate of the maximum for that cube The maximum is searched for during Vegas integration sessions when the grid is frozen The more phase space points Vegas generates the more likely the estimate of the maximum is accurate On the other hand the greater the number of generator phase space cubes the longer Vegas will have to run to find the maximum for each cube When a user would like to generate events he she will typically begin by improving the Vegas grid as described in Subsection 5 11 Once the grids are optimized they should be frozen and Vegas should be run again After each Vegas session the current estimate of the event generation efficiency will be displayed along with the cross section or particle width Vegas should be 60 run until the event genera
89. ction for a particular vertex the vertex can be obtained by functionally differentiating with respect to the fields in the vertex as in 6S 5 Al fmy D1 5A2mo P2 8A3ma p3 FAS pnaj P4 7 27 04 pi p2 ps pa Cea ColorStructure Factor LorentzPart where p and m refer to the 4moment and Lorentz indices if any The square brackets and denote parts of the expression which only appear in some vertices but not others The Fourier transform is defined as Ate f a PAR 8 The other pieces of Equation 8 4 will be discussed below 105 The Factor column is where the Factor from Equation 8 4 belongs This must be a rational monomial constructed from the model parameters integer number and the imaginary unit i It is best to factor as much as possible from the LorentzPart since the LorentzPart of the Feynman diagrams is usually the most time consuming and memory intensive part of the calculation The LorentzPart column is where the LorentzPart from Equation 8 4 belongs It must be a Lorentz tensor or a Dirac y matrix expression The coefficients of the terms in this expression can be polynomials of the model parameters and scalar products of the momenta The division operator is forbidden from this column It must be transferred to the Factor column or into a model parameter The notation for Lorentz indices momenta contractions and the metric tensor are similar to those in the Reduce pack
90. cture 18 presents comparision of implemented approach with Monte Carlo simulation by Guinea Pig program for CLIC3000 design Be cause singuarity 1 x 3 is expected for beamstrahlung we have defived spectra on this factor for proper comparision in x 1 region Convolution of beamstrahlung and ISR spectra is done automatically ISR affect smooths spectra and improve agreement of P Chen formulas with Monte Carlo sim ulation The Beamstrahlung spectrum cannot be integrated by the current CompHEP version because it contains a 6 function 152 Beamstrahlung for CLIC 3000 0 35 0 3 0 25 F x 1 x 2 3 0 2 0 15 0 1 0 05 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 08 0 9 1 X Figure 18 Comparison of P Chen approximed formula implemented in the CalcHEP solid line with Guinea Pig Monte Carlo for CLIC3000 dashed line H PDT Particle Distribution Tables in CalcHEP H 1 CTEQ and MRST parton distributions Both CTEQ and MRST groups store information about parton distributions in two dimensional tables and interpolate these tables CalcHEP has its own file format for parton tables but uses interpolation procedures of CTEQ and MRST Thus CalcHEP produces exactly the same results as original CTEQ MRST functions The information about interpolation procedure is stored in CalcHEP tables and is detected automatically Besides of parton distributions CalcHEP tables contain data for amp s q 153 which correspond
91. culation and their numerical values the decla ration of the dependent parameters and their substitution rules and finally the declaration of the process The momenta are named p1 p2 p3 where p1 is the momentum of the first particle in the process p2 is the momentum of the second particle and so on The signs of momenta are defined in such a way that the sum of incoming momenta is equal to the sum of the outgoing momenta The list of substitutions of numerical values for the independent parameters is stored in the variable parameters The list of substitutions for 119 e M M e gt lt lt gt P1 P4 P4 P1 E Aim m A E lt Q 1 Q gt gt Q 2 lt P2 P5 P3 P3 P6 P2 Figure 14 An example of a pseudo graphic representation of a squared dia gram found in the symbolic expression code the dependent parameters is stored in the variable substitutions The lists of incoming and outgoing particles is stored in the variables inParticles and outParticles respectively After the initial declarations the function initSum is called Then the expression for each squared diagram is presented After each squared diagram code the function addToSum is called After all the squared diagrams are finished the funciton finishSum is called These three procedures initSum addToSum and finishSum must be written by the user and loaded before the the process co
92. d by the use of t Hooft Feynman gauge in the calculation of the squared matrix element as described in Appendix C The second example is the cancellation of double pole terms of the t channel photon propagator of the form t For example there is a wide class of processes where the incoming electron continues in the forward direc tion emitting a virtual photon as in Fig 17 These diagrams have a 1 t pole where t is the squared momentum of the virtual photon For the described kinematics the photon appears very close to its mass shell t 0 hence this configuration gives a respectively large contribution to the cross section However after summing the diagrams in the squared matrix element we expect the 1 t pole to be reduced to a 1 t pole 39 in the zero electron mass limit This cancellation is caused by electro magnetic U 1 gauge invariance If diagrams similar to that in Fig 17 contribute to your process we strongly recommend to set the Gauge invariance switch to ON see Section 5 5 to prevent the breaking of the gauge symmetry by width terms Another way to solve this problem is by using the Weizsaecker Williams approximation see Appendix G 2 138 D Feynman rules in CalcHEP D 1 Lorentz part of diagram Fermion loop calculation Our algorithm for the evaluation of fermion loops takes into account the possible appearance of vertices with fermion number violation CalcHEP chooses an arbitrary direction for the multi
93. d is a small correction far from the pole this propagator is well approx imated by replacing p with T I m which gives the Breit Wigner propagator denominator 1 p m ilm This is the propagator denominator used in CalcHEP However because we are using a width which comes from higher order corrections in a tree level calculation it has the potential to violate gauge invariance in the calculation and ruin the large cancellations that sometimes occur between diagrams There are three different regimes to consider In the first regime the particle is exactly on shell In this regime the calculation is exactly gauge invariant and there is no problem In the second regime the particle is off shell but not very far from on shell In this regime the process is still dominated by the resonant diagrams and the effect of gauge invariance breaking is still small In the third regime the width is not needed to regularize the integral So gauge invariance can be satisfied by not including the width 50 The Breit Wigner menu allows to adjust the properties of the propaga tor denominators used in the phase space integrals The first menu item is BreitWigner range which allows to set the regions where the width is used by adjusting the value of R CalcHEP then uses the width in the propaga tors for p m lt RmI No width is used for p m gt VR Imr And in the intermediate region the propagator is replaced
94. d is used for automatically creating the article in MCDB The routine lhe2tab histograms the events in an LHE file It is called in a similar fashion to events2tab described in Subsection 5 12 For example CALCHEP bin lhe2tab lt var gt lt min gt lt max gt lt N gt lt lt lhe gt gt lt plot txt gt where lt var gt is the kinematical variable lt min gt and lt max gt are the minimum and maximum values for the distribution lt N gt is the number of bins lt lhe gt is the LHE event file and lt plot txt gt is the output file where the histogram data should be written The only difference with respect to events2tab is that the PDG number of the particles should be used rather than the names of the particles For example M 5 5 should be used in place of M b B for the invariant mass of the b quark and the anti b quark The output file also contains a line which records the largest deviation from energy momentum conservation An example is lost_momenta_max Etot 7 9E 11 1 3E 12 1 3E 12 8 0E 11 Typical value should be approximately 1071 because the original event files recorded 11 digits for the particle momenta This allows the user to test whether energy and momentum were conserved in the mixing The CalcHEP batch interface see Section 7 event_mixer is automati cally called at the end of the batch session to construct the resulting event file 6 3 N tuples CalcHEP contains a program that creates PAW NTUPL
95. de is loaded In this way the user can decide how the code from the different diagrams is combined Now we shall explain the structure of each squared diagram contribution Each squared diagram begins with a pseudo graphic image of the diagram such as in the ete pty example found in Fig 14 This is followed by the assignments e totFactor is a rational function depending on the model parameters e numerator is a polynomial of the model parameters and scalar products of the momenta e denominator is a product of the propagator denominators propDen P Mass Width where P Mass and Width are respectively the momentum mass and width of propagating particle In the case that Width 0 propDen 120 should be defined as Mass P The user can treat the Width argu ment as he she likes The total contribution of a squared diagram is then numerator totFactor ___ _ denominator We note that as was mentioned in Section 4 4 the result obtained by the summation of the diagrams must be symmetrized in the case of identical outgoing particles This can be done for example by the finishSum procedure 10 2 Reduce examples We have prepared some example programs to work with the Reduce output They are e sum_cd red which combines the expressions from the squared diagrams and presents it in a form with a single common denominator e sum_22 red which combines the expressions from the squared diagrams
96. directly to it e Scans over parameter space are not directly supported by this keystroke sequence batch mode In the present version of CalcHEP we have attempted to streamline this process and provide the user more powerful and convenient ways to use CalcHEP in batch mode In this section after reviewing how to use the batch mode for the symbolic and numerical sessions we give details of a va riety of shell scripts which allow the user to control CalcHEP in blind mode without the need to write keystroke sequence strings The final subsection details a new Perl interface to CalcHEP which automates the procedure of scanning over parameters and parallelizing the calculations Before continuing with the details of the improvements we note that the presence and interelated nature of the interactive and the batch regime facilitates gives the user the ability to set everything up in the interactive regime where the user can see the results and check that everything is working properly and then run the long calculation in batch mode This combines the advantages of the interactive and batch modes 7 1 Blind mode As mentioned CalcHEP has an option blind which allows the user to run in batch mode It can be used with both the symbolic and numerical code as in s_calchep blind STRING n_calchep blind STRING where STRING is a string which represents the sequence of keystrokes required which would be used in an interactive session to achie
97. done in CalcHEP is that it con tinually splits the remaining particles into two sets until each particle set contains one particle In this framework the multi particle phase space is parameterized by the invariant masses of each particle set and by the two dimensional spherical angles of the 1 2 splitting 52 53 Since the choice of the kinematics influences the phase space mapping it also affects the con vergence of the Monte Carlo integration In other words a mapping choice that is related to the physical problem is more likely to converge efficiently than one that is not See Appendix I 1 for further details Upon entering the kinematics menu item the current phase space map ping scheme is displayed along with a dialogue asking the user whether he she would like to change it If the user answers Y CalcHEP will re quest the splittings one at a time For each splitting CalcHEP presents the user with the particle set that requires splitting or 12 if the first splitting which means all the final state particles The user then splits these parti cles into two groups and enters them separated by a comma For example suppose we are considering a 2 4 process The default splitting is 12 gt 3 456 456 gt 4 56 56 gt 5 6 56 However the user might change this to 12 gt 34 56 34 gt 3 4 56 gt 5 6 where the 34 is what the user enters 5 10 Regularization Generally speaking the Monte Carlo integratio
98. e each is added to the distribution the distribution is a sum of each possibility The user can continue to run Vegas until he she is satisfied with the uncertainties in the distribution The distributions can be viewed by en tering the Display Distributions menu A list of the distributions will be presented and the user can choose the one he she would like to view After allowing the user to choose the number of bins for the histogram CalcHEP will display it using the Plot Viewer described in Section 3 An example of a distribution is presented in Fig 4 The distribution data is stored in the file distr_N where N corresponds with the CalcHEP session number not the Vegas session number The 59 user can view the distributions from previous CalcHEP sessions by use of the show_distr program located in the bin subdirectory of the user s work directory For example CALCHEP bin show_distr distr_1 would display the distributions from the CalcHEP session number 1 5 12 Event Generation CalcHEP can generate unweighted events These events are useful in sim ulations of particle physics collisions and can be passed to other programs for further analysis For example it is often desirable to pass events through PYTHIA 54 which hadronizes colored final states and adds radiation to the event Event generation is done in the Monte Carlo simulation menu It consists of two steps In the first step the generator is prepared and in the se
99. e each of them separately grow as k m but together that k m growth cancels 100 0 a 000 0 0 0 _ 3 3 _ C e 6 000 0 23 000 1 which can be achieved in a different basis 0 0 3 3 _ 1010 38 Cee eo 6 64 24 where e 1 0 0 0 3 e 0 0 0 1 In this new polarization basis none of the polarization vectors grow with energy As a result by inclusion of the temporal polarization the sum over polarization squares see Eq 17 becomes ee ene Ei ee ys 25 which again does not lead to the bad high energy growth and loss of pre cision In the case of an unpolarized calculation we simply replace the sum 135 over polarizations by g as in Eq 25 In a gauge invariant theory the unphysical polarizations are canceled if we add Faddeev Popov ghosts and Goldstone bosons to the external states and we get the same result as if we only included the physical polarizations However now we have better behavior at high energy and much less loss from finite numerical precision C 3 Massless vector particle case Now that we have discussed the massive vector boson case we turn to the massless gauge boson It also has a gauge symmetry which must be satisfied in the calculation of the squared matrix element The Feynman gauge leads in this case to the propagator of Eq 15 with m 0 As in the case of the massive gauge bosons the massless gauge bosons also have Faddeev Popov ghosts 48 However in di
100. e final menu item is Delete model which removes the current model and returns CalcHEP to Menu 1 where a new model can be chosen Before removing the model a warning dialog appears and allows the user to cancel the model deletion Menu 3 The model details are stored in five text files located in the models subdirectory of the users work directory These model files can be edited with any text editor However the user must not include any Tab symbols and must keep the column structure of the file unchanged We recommend that the user take advantage of the built in model editor which can detect mistakes in the user s input This menu items appearing for this menu are e Parameters Edit the independent parameters of the model e Constraints Edit the dependent parameters of the model e Particles Edit the particle properties of the model e Vertices Edit the Feynman interaction vertices of the model 31 e Libraries Edit the prototypes for external functions and the external libraries which should be linked to the numerical code e RENAME Edit the name of the model e CHECK MODEL Checks whether the model passes a set of tests described below The detailed format and requirement for these files is described in Section 8 An explanation of the Table Editor which is used to modify the model files can be found in Section 3 When the user is satisfied with his her modifications he she can press the Esc key When this occurs
101. e full font string of the desired font Other fonts available on the users system can be obtained by the command xlsfonts however only non proportional fonts should be used for CalcHEP We recommend fonts from the Courier family The default value should give the user some guidance 1 On line Help During interactive symbolic and numerical sessions the user can press the F1 key to be shown context sensitive help This help is in the form of a textual message explaining the currently available options to the user If the text is longer than can be seen at one time in the graphical user interface there will be a PgDn symbol in the bottom right corner of the help screen By pressing the PgDn key or by clicking on the PgDn symbol the user can advance the text to the next page The user can close the help window by pressing the Esc key or by clicking the asterisk in the top left corner of the help screen 2 Menu The details of a calculation in CalcHEP are controlled by a series of menus which allow the user to set the properties of the calculations These menus appear on the right side of the graphical user interface as a vertical list of the available options see Fig 1 The current menu item is Use the Ctr1 sequence if the Esc key is missing on your keyboard 3On some keyboards Prev and Next replace PgUp and PdDn CalcHEP is only sensitive to the release of the left mouse button It is not sensitive to the press of the mouse
102. e is an example 87 Dist parameter M e E Dist min 0 Dist max 200 Dist n bins 100 Dist title p p gt l 1l Dist x title M 1 1 GeV The value for the keyphrase Dist n bins has to be one of 300 150 100 75 60 50 30 25 20 15 12 10 6 5 4 3 or 2 These are the values allowed by the CalcHEP histogram routines The values given for the titles have to be pure text No special characters are currently allowed Gnuplot must be installed for plots to be produced on the fly and included in the html progress reports More than one distribution can be specified however each distribution must be unambiguous and apply in exactly one way for each subprocess Also distributions will work even if no events are requested For this to work the distributions have to be unambiguous and apply to all subprocesses the same way For example if a process is p p gt 1 1 1 and the distribution M 1 1 is given then this routine will not know which two leptons to apply the distribution to and the results are unpredictable If the process is p p gt 1 1 where l e E m M and the distribution M e E is desired this distribution will only apply to some of the subprocesses and give unpredictable results Make sure your distribution is unambiguous and applies in exactly one way to each subprocess If this is done it should work Nevertheless check each distribution carefully to make sure it is being done correctly Events The number of ev
103. e of integration into a large number of sub volumes and calculate integrals separately in each sub volume This method produces a smaller uncertainty comparing with the direct Monte Carlo method because here the uncertainty is caused only by a function variance in the sub volumes while the integrand variation from one sub volume to another does not contribute to the uncertainty The stratified sampling method is used to estimate the integral for any VEGAS iteration The larger number Ncall is chosen the smaller size of sub volume becomes available and consequently the more successfully the stratified sampling works 1 3 Generation of events CalcHEP generates events according to the Von Neumann algorithm See 58 p 202 Let the probability density f x is smaller that an easily gen erated density F x Then one can generate x according to distribution F x and accept this event with probability f x F x This procedure is repeated in cycle until the needed number of events is generated To built F x CalcHEP divides the space volume on large number of sub cubes and in each sub cube sets F x a constant which equals to max f z CalcHEP has two strategies of detecting the corresponding maxima First one is a random search The program generates random points in each sub cube and tests f a in these points The second one is a search by the simplex method 47 Here the program analyzes function in vertices of some simplex and tries to shif
104. e trick it should be mentioned that the factor 2 kills contributions of non pole diagrams in the p m point 51 51 5 6 Kinematical Functions CalcHEP allows a very large class of kinematic functions that can be used for cuts and distributions Firstly it defines many popular kinematical functions which we describe in this section Secondly it allows the user to code any kinematical function in a C code file see Section 5 7 In this way any cut and or distribution can be achieved The built in kinematical functions are called with the syntax Name _ P1 P2 P3 where Name is one capital letter the and _ are optional and will be de scribed below and P1 P2 etc are anti particles The available functions are A A P1 P2 gives the angle between P1 and the combined momentum pp2 pp3 If only one particle is specified as in A P1 the angle between P1 and the first incoming particle is returned The angle is given in degrees C C P1 P2 gives the cosine of the angle defined above for AC P1 P2 J J P1 P2 gives the jet cone angle between P1 and P2 The jet cone angle is defined as Ay Ay where Ay is the difference in pseudo rapidity and Ay is the difference in azimuth angle between P1 and P2 E E P1 P2 gives the energy of the combined momentum pp Pp2 M M P1 P2 gives the invariant mass of the combined momen tum pp pp2 P P P1 P2 P3 fir
105. ealizations of the Standard Model in CalcHEP s format one with a full and another with a diagonal CKM matrix work which is a used to initialize a directory for the users calculations Compilation procedure In order to compile the CalcHEP source code you need a C compiler the X11 graphics library and the X11 include files The compilation is launched by running CalcHEP scripts automatically set the environment variable CALCHEP which contains the path to the CalcHEP root directory CALCHEP 12 gmake from the calchep_3 30 directory CalcHEP Makefiles are written for gmake If the gmake command is absent on the users computer then make should also work If the compiler is detected and the sources are compiled successfully you will see the message CalcHEP is compiled successfuly and can be started Otherwise the corresponding error message is printed on screen See Sec tion 2 7 for a discussion of possible problems The size of the installed package is approximately 5Mb To clean all the files created during compi lation and any files created in calchep_3 30 work the user can issue the command gmake clean This command asks whether the user would like to delete the FlagsForSh file which contains user modifiable compiler names and compiler flags The user may choose to keep this file if he she wishes to use it in subsequent compilations Compilation tuning The CalcHEP compilation procedure consists of two step
106. ed see saw mechanism which can potentially explain why the neutrino mass is so small Using this mechanism the Yukawa constant responsible for the value of my can be of the same order as for the other fermions If M is very large and the mixing angle a is very small the light neutrino mass m will be tiny and the heavy neutrino mass my will be on the order of M Roughly we will have mym2 my To get the Feynman rules in terms of the Majorana fermions we simply substitute the new fermion basis into the interaction Lagrangian CalcHEP only supports 4 component fermion notation In this notation a Majorana fermion is given by Full name A A 2 spin mass width color aux neutrino MN MN 1 0 0 1 In terms of the Dirac 4 component spinor field Y a neutrino appears in the Standard Model Lagrangian in the following way see Eq e D 5 Ei E V 4 Zt U 4 sin O cos O e Wta yp U WETA aTe 82 m yH D WETA 7 Ge 32 where W is the electron field To rewrite this Lagrangian in terms of a Majorana neutrino let us perform the substitution 1 1 A Fha J9 e where Y 1 Y U Vo Ye go e sin Ow g e cos Ow 147 where y and w are the Majorana fermions Omitting the Lagrangian for wv there are no interactions for and applying the following identities for Majorana fermions Beye x OM di y
107. ently a special treatment of these vertices by CalcHEP is not needed This also applies to other vertices The 139 combinatorial factors in the case of identical particles should already be present in the Vertices table We also note that a loop of Faddeev Popov ghosts gives an extra factor 1 For convenience CalcHEP uses a vector propagator with the wrong sign to achieve this The total sign of a diagram is corrected at the end of the evaluation Namely the diagram is multiplied by GhostFact 1 9 where n is the number of Faddeev Popov loops and n is the number of vector particles D 2 Color factor We here explain the method 62 of color factor evaluation that is used by CalcHEP in the case of the QCD SU N group We shall use the desig nations gluon g quark q and anti quark q for colored particles which belong to the adjoint fundamental and conjugate fundamental representa tions respectively Following section 8 4 the color part of the ggg vertex is given by the group structure constant 7 fabe whereas the color part of the gqq vertex is given by the fundamental representation matrix 7 of the groups By means of the basic relations TaTb ToTa t Tabete and A E 1 rT 5 ab the ggg vertex can be expressed in terms of the gqq vertex i fave 2 tr a 2 Te 27 This vertex relation can also be achieved by a diagram substitution where the ggg vertex is replaced by tw
108. ents is specified with the keyphrase Number of events This specifies the number of events to produce after all subprocesses are combined and decayed If a run over a parameter is specified this keyphrase determines the number of events to produce for each value of the run param eter The number of events requested can be zero In this case the cross sections are determined and the distributions generated but no events are produced Here is an example Number of events 1000 88 The name of the file can be specified using the Filename keyword If specified all the files will begin with this name Here is an example Filename pp ll If nt_maker has been installed in the bin directory PAW ntuples can be made on the fly by setting NTuple to True as in NTuple True The default is False The keyword Cleanup determines whether the intermediate files of the calculation are removed This can be useful if many large intermediate files are created and space is an issue On the other hand it can be useful to keep the files when debugging is necessary If this keyword is set to True the intermediate files are removed If set to False then they are not removed Here is an example Cleanup False Parallelization The parallelization mode is set using the keyphrase Parallelization method and can be either local pbs or lsf In local mode the jobs run on the local computer in pbs mode the jobs are run on a pbs cluster and in 1lsf m
109. enu 4 where numerical evaluations of the dependent parameters masses and widths can be done The default independent parameters are those defined in the model files However the user can change the values of the independent parameters which are used for these numerical evaluations by choosing Parameters on this menu This will cause CalcHEP to display a menu which lists all the independent parameters and allow the user to change them one by one Furthermore CalcHEP displays READ_FROM_FILE at the top of this menu If the user chooses this CalcHEP opens the File Search Engine and allows the user to choose a file Afterwards it reads the file and updates all the independent parameters used for the numerical evaluations This parameter file must be written in the form of two columns separated by whitespace The first columns must contain the parameter name and the second column must contain the numerical value Each parameter must be on a separate line Here is an example for the SM EE 3 1223E 01 alfSMZ 1 172E 01 SW 4 81E 01 Ml 1 777 Mtp 175 MZ 91 1884 Mh 120 These new independent parameter values are only used for the numerical evaluations done in this menu The default values used for other numerical calculations via Enter Process on Menu 2 are those defined in the model files Once the user is satisfied with the values of the independent parameters he she can initiate the numerical evaluation of the dependent parameters by choosing Al
110. ependent masses widths and branching ratios and write them in a file in the results subdirectory of the work directory with the filename decaySLHAN txt where N is an integer The format of this file fol lows the SLHA 37 convention and should be suitable for other programs that follow this convention When calculating the decay widths and branching ratios CalcHEP first calculates the contribution from 1 2 decays If the resulting width is zero it then calculates the contribution from 1 3 decays If still zero it calculates the contribution from 1 4 decays However if the model is defined in terms of an SLHA file and that file contains the widths and branching ratios CalcHEP does not calculate them but uses the values specified in the SLHA file 34 Calc HEP version 2 0 Model Std Model Feyn gauge List of particles antiparticles photon ne Ne e neutrino tau lepton k W H H boson Enter process jupi Meau of composit p consists of PRAL Rae composit ap consists of RIA Rae Exclude diagrams with G G gluon lel Es e E electron mn Nm m neutrino u quark s quark Higgs Figure 8 Example of the input of a process in the interactive symbolic session 4 3 Process Input After choosing Enter process on Menu 3 the user is presented with the Process Input screen see Fig 8 where he she can enter the physical pro cess he she would like to calcu
111. erical calculations The last part is written in Perl It calls the routines of the first two parts and collects the results to obtain final cross sections and events for multichannel processes typical of modern collider physics The symbolic session of CalcHEP enables the user to interactively e load a new model of particle interactions e modify a model of particle interactions e check a model for sytax errors e read parameter information from an SLHA file 37 38 e calculate dependent parameters e calculate decay widths and branching ratios e choose between Feynman and unitary gauge e choose a collision or decay process by specifying the incoming and out going particles e specify particles exclusions for the diagrams e generate Feynman diagrams e display Feynman diagrams e generate BIFX output for the diagrams e remove particular diagrams from the calculation e generate and display the squared Feynman diagrams e remove particular squared diagrams from the calculation e calculate analytic expressions for the squared diagrams using the built in symbolic calculator e export the resulting squared diagram expressions to Reduce Mathe matica or Form format for further symbolic manipulations in those packages e generate optimized C codes for the squared matrix elements e compile the generated code e launch the resulting numerical session e generate numerical libraries of squared matrix elements for other
112. esses Lib PID Time hr wD gt W bB 4 S c gt W b B c S gt W b B C s gt W b B W gt E ne W gt M nm gt e Ne W gt m Nm Widths SO SUSENS SSR SORES NOS b CalcHEP Events Library Date LHE Sat Mar 10 12 52 39 2012 pp Wbb Inbb Mh120 lhe Sat Mar 10 12 53 09 2012 pp Wbb Inbb Mh125 Ihe Sat Mar 10 12 53 432012 pp Wbb Inbb Mh130 lhe plain Ntuple a Figure 12 Monitoring of the CalcHEP batch session using through the web browser CalcHEP Numerical Details Done Runs Mh120 Mh125 Mh130 sigma fb 8 9460e 02 8 8420e 02 8 7380e 02 Running 0 13 0 13 0 13 Finished 13 13 13 13 13 13 96 Time hr 0 02 0 02 0 02 N events 1000 1000 1000 Numerical Sessions Home Symbolic Results Standard Model CKM 1 Numerical Results Events Library Done Process Library Help gt z ice Processes o fb PID Time hr N events Details Thank you for using wD gt W b B 1202 5 16747 0 00 383 383 prt_1 session dat CalcHEP U d gt W b B 644 99 16750 0 00 220 220 prt 1 session dat See ong nocd d U gt W b B 646 43 17063 0 00 220 220 pt1 session dat CASN D u gt W b B 1203 5 17065 0 00 383 383 prt 1 session dat s C gt W b B 81 777 17379 0 00 40 40 pt 1 session dat S c gt W b B 81 762 17382 0 00 40 40 pt 1 session dat cS gt W b B 82 08 17695 0 00 40 40 prt 1 session dat C s gt W b B 82 073 17698 0 00 40 40 pt 1 session dat Total 4025 1 1366 1366 Decays T GeV PID Time hr N events D
113. etails W gt E ne 0 22339 18011 0 00 5101 5100 pt 1 session dat W gt M nm 0 22339 18013 0 00 5101 5100 pt 1 session dat W gt e Ne 0 22339 18555 0 00 5101 5100 pt 1 session dat W gt m Nm 0 22339 18564 0 00 5101 5100 pt 1 session dat Widths PID Time hr Details Widths 19099 0 00 session dat Total 894 6 0 02 Distributions P p gt H b B asa 300 258 3 200 a 2 150 i gt 100 58 a 180 128 140 168 188 208 ihe b gt Gev Figure 13 Monitoring of the CalcHEP batch session using through the web browser 97 7 3 4 Results Storage After the events and or distributions are generated they are stored in the Events directory The prefix of the files is the name specified in the batch file plus either single if no scans were specified or a string specifying the run parameter values if one or more scans are specified We will assume this is filename in the following If events are requested they will be stored in the files filename lhe filename nt where filename lhe is the event file in Les Houches format and filename nt is in PAW ntuple format The ntuple file is only created if the keyword NTuple is set to True and nt_maker is present in the bin directory If distri butions are requested they will be stored in the files filename distr filename_1 png filename_2 png where filename distr is the raw distribution data and can be read by show_distr in the bin directory The distributions generated on the f
114. example of such a piece of code Index m1 Vrt_0 Vrt_1 Vrt_2 RemInd mi Vrt_L Vrt_1 Vrt_R Vrt_2 Vrt_L Vrt_L where mi gt P2 P3 MZ Vrt_R Vrt_R where m1 gt P2 P3 MZ Vrt_0 Vrt_O Vrt_L Vrt_R The end of the code for any diagram contains the instruction 128 numerator_ numerator_ DiagramFactor GhostFact Vrt_1 which adds it s contribution to the total It also multiplies by the ghost factor if any where GhostFact 1 where l is the number of loops of Faddeev Popov ghosts and v is the number of vector field lines in the diagram The 1 factor appears because the evaluations described above correspond to substituting gv k k M for the propagator of the vector field whereas the correct expression has the opposite sign The last step of the program is the assignment of the variable denominator_ It is expressed as a product of the terms propDen P Mass Width as ex plained in Subsection 10 Finally the total result for the evaluated squared diagram is given by numerator_ totFactor denominator_ 129 Appendix A BIFX output CalcHEP uses the Azodraw package by J A M Vermaseren 44 to create diagrams and plots in BT X format To use this package the Azxodraw style should be included in the documentstyle statement For example documentstyle axodraw article With kind permission of the author we include a copy of the axodraw sty file in the CALCHEP directory
115. f the ascii image is usually small compared to the rest of the code The default is ON e Widths in t channels OFF ON This switch determines whether CalcHEP includes the particle width in the propagator on t channel lines where there is no chance that the particle will go on shell The default is OFF which means that these t channel widths are not included The user may change this to ON in which case the user must also turn this feature 42 on in the Breit Wigner menu of the numerical session described in the next section 43 5 Numerical session 5 1 Sketch of the menu system In Section 4 we described the process of generating and compiling C code for a collision or decay process for any particle physics interaction model In this section we describe how to use the resulting executable to calculate the collision cross section or the decay width in interactive mode In Fig 9 we present a schematic view of the menu system of the interactive numerical session An example of the first menu Menu 1 is shown in Fig 1 We will now describe each of these menu items The first menu item is Subprocess which allows the user to choose which subprocess to work on if more than one was generated during the symbolic session The current subprocess is displayed at the top of the numerical session screen see Fig 1 If a combination of the subprocesses is desired we direct the user to the tools described in Sections 6 and 7 The next menu
116. f the calculation CalcHEP sets the minimum value as 1 GeV The last menu item in the QCD coupling menu is Alpha Q plot which allows to see a plot of the strong coupling over a range of QCD scales When parton distribution functions are used for the initial state particles it is preferable to use the strong coupling constant defined in the parton distribution function instead of the internal CalcHEP value This can be controlled by using the parton dist alpha item of the QCD coupling menu It can take the value ON or OFF The default is ON the strong coupling 49 constant defined in the parton distribution function is used However if the parton distribution functions are not being used or if they do not define the strong coupling constant CalcHEP displays ON in place of ON and uses its own internal value of the strong coupling constant 5 5 Breit Wigner propagator The propagator denominator of a particle at tree level is given by 1 pe m where m is the particle mass and has a pole at p m If this pole is inside the phase space volume being integrated over it causes the integral to diverge At higher order the propagator denominator is modified to become 48 1 pP m il p m where T m is the width of the particle the inverse of the particle s mean lifetime This removes the pole and renders the integral convergent Since the I p terms only dominates this propagator denominator near the pole an
117. fter the vector declarations comes the mass shell declarations for the incoming and outgoing particles This is followed by substitution rules that use momentum conservation to replace the internal momenta with the external momenta The next thing in the file is a multiplicative factor for the diagram which includes a symmetry factor a factor for averaging over the initial state helic ities a factor for the number of incoming fermions a QCD factor and some model parameters that were factored out of the diagram expression For example in the case of ete pt p we have ke aaa Factors SymmFact 1 1 Diagram symmetry factor AverFact 1 4 h Normalization factor of polarization average FermFact 1 1 number of in fermion particles ColorFact 1 1 QCD color weight of diagram totFactor_ EE 4 totFactor_ totFactor_ SymmFact AverFact FermFact ColorFact For the most part these declarations are self explanatory except perhaps for SymmFact Generally SymmFact N D where N is equal to 1 when the two diagrams in the squared diagram are identical Otherwise this factor equals 2 D is a factorial connected with the presence of identical outgoing particles and partially reduced by a number of various possibilities to assign the momenta of the outgoing particles to the corresponding diagram lines Each file includes both the main squared diagram and any that are related to it by replacement of a gauge b
118. ge is a very important nontrivial test Not only does this test check the CalcHEP symbolic calculator but also the model implementation In fact we suggest that a test of gauge invariance always be performed for any new model implementation To perform this test we evaluate the symbolic sum of squared diagrams in the two supported gauges unitary and t Hooft Feynman for the SM and 131 compare the results If the difference between these two calculations is non zero there is a mistake in the symbolic calculator or the model The symbolic summation of the squared diagrams is performed by Reduce This summation is the most difficult step of comparison because the sum of diagrams can be an extremely large complicated expression which is not easily simplified To perform this test a process should be calculated in unitary gauge and the results should be stored in results_ symb1 red Then the same process should be calculated in t Hooft Feynman gauge and the results should be stored in results symb1 red Finally Reduceshould be started from the work directory results_ and results should be subdirectories of the work directory and the program cmp red should be read in as in in cmp red This program will read in the expressions from results_ symb1 red and sum the squared diagrams It will then read the expressions from results symb1 red and sum the squared diagrams It will then take the difference of the two expressions If after s
119. hat some jobs are run on the pbs or Isf queue computer even on the pbs or Isf cluster Here is an example Nice level 19 Level 19 is the default Vegas The number of vegas calls can be controlled with the keywords nSess_1 nCalls_1 nSess_2 and nCalls_2 The values are the same as in CalcHEP Here is an example nSess_i 5 nCalls_1 100000 nSess_2 5 nCalls_2 100000 The defaults are the same as in CalcHEP Generator The following parameters of the event generation can be modified sub cubes 1000 random search 100 Simplex search 50 MAX N 2 find new MAX 100 The defaults are the CalcHEP defaults 91 7 3 2 Example of the bacth file This example generates 1000 events for each Mh of the process p p gt W b B for the model Standard Model CKM 1 HHTHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEHE Model Info HHEHHHHHHHHHHHHHHHHHHHHHHHHHHH RHEE Model Standard Model CKM 1 Model changed False Gauge Feynman HHHHHHHHHHHHHHHHHHHHHHHHHHHEHHEHEH Process Info HHHHHHHHHHHHHHHHHHHHHHHHHHHEHHEHEH Process p p gt W b B Decay W gt le n Composite p u U d D s 5 c C b B G Composite W W W Composite le e E m M Composite n ne Ne nm Nm Composite jet u U d D s 5 c C b B G HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEHEH PDF Info HHHHHHHHHHHHHHHHHHHHHHHHHHHHEHEHEY pdfi cteq6l proton pdf2 cteq6l proton HHHHHHHHHHHHHHHHHHHHHHHHEHHEHEHEE HE Momentum Info HEHHHHHHHHHHHHHHHHHHHHHHHHHEHHHEEH pi 400
120. he and press enter to choose the directory The initial path can also be started with the environment variables CALCHEP WORK and lt user_name gt 28 vj a CalCHEP SY MD File Search IMPORT MODEL results bin Events html Processes calchep ini calchep calchep_batch batch_file allbin lock Figure 6 An example of the CalcHEP File Search Engine 4 Menu system for symbolic calculation The flow of menus in the symbolic calculation session is presented schemati cally in Fig 7 4 1 Model Choice and Manipulation Menu 1 This menu presents a list of available models and allows the user to choose among them for his her calculations Addittionally at the bottom of this list is the entry IMPORT MODEL which allows the user to import a new model of particle interactions into CalcHEP When IMPORT MODEL is chosen the File Search Engine will open allowing the user to specify the directory where the new model is stored It will then allow the user to choose among the models in that directory and choose a new name for the model if desired afterwhich the model will be imported and appear on the list of models in this menu 29 SELECT MODEL Parameters Constraints IMPORT OF MODELS FILE SEARCHILIST Particles Libraries Enter Process RENAME CHECK MODEL Force Unit gauge OFF stella ere Enter processes p p gt W 2 x Parameters Numerical Evaluations composite p consists of u U d D Exclude diagr
121. he uncertainty oy of lt f g gt estimation by N sample points is propor tional to square root of function s variance divided over N on lt gt lt f 9 gt 2 N VEGAS uses two techniques which allow to decrease the uncertainty of Monte Carlo calculation namely the importance sampling and the stratified sam pling 1 2 1 Importance sampling The idea of importance sampling technique is based on diminution of variance by a proper choice of the density function g x The general solution of this problem could be in choosing g a F 0 JIE ae However this solution is useless because it returns us to the problem of evaluation of f x integral and requires a generation of sampling points for complicated density function To bypass these problems VEGAS seeks this function in the factored form Qs 2 En g1 21 Gala dan Onley The optimal functions g x could be easily evaluated in terms of f x 46 47 VEGAS is an adaptive program For the first iteration it puts g 1 The 162 information about f x which VEGAS gets during the iteration is used to refine the density function Generally VEGAS performs several iterations improving the density function after each of them The following parameters manage VEGAS work 1 tmz is a number of iterations 2 Ncallis a number of integrand calls for one iteration 1 2 2 Stratified sampling The idea of stratified sampling method is to divide a volum
122. he user wishes to perform a complicated calculation that takes a very long time it is not possible to quickly create the key sequence string using the blind option In this situation the user must become adept at writing or modifying the key sequence string to achieve the desired results e The symbolic CalcHEP session does not always begin at the same point The entry point depends on the previous session Correspondingly the key sequence required to achieve the desired result depends on the previous session and can be very difficult for the user to control e CalcHEP sometimes presents the user with a dialog that requires some form of input such as Press any key or Yes No In batch mode all such dialogs are skipped and Yes No questions are automatically answered with Yes e If the user has a mistake in their keystroke sequence string or if there is a problem in the setup of the session the batch mode can not interact 69 with the user to fix the problem CalcHEP simply quits with an error code meant to inform the user about the problem e Some interactive menus depend on the physical problem being ana lyzed For example the position of the t quark mass could be in differ ent locations in the parameter menu and different keystroke sequences would be required to change its value In the present version we have solved this by implementing a Find menu option which allows the user to type the name of the parameter and be taken
123. i ee eee be ota be hes Re ERS Ske 5 5 Breit Wigner propagator 225 4 as 0h epee ea as 5 6 Kinematical Functions 0 084 5 7 User Defined Functions lt 2 2 4 6 24s a a a ewe eR Ewe DO CUS rawe ree ek a ae Oe Oe Oe Se a 59 Kimematics 2 iw aeae amp eed jaw geek Gov oe OG Gs ee a 5 10 Reg larizati i e s s s aad ace Mala ee 8 ee Ee Bee RE 5 11 Monte Carlo simulation 0 0 0 0 0 00484 10 11 it 11 11 12 14 15 16 18 29 29 33 35 37 41 42 5 12 Event Generation 0 0 0 0 0 0 0000 es 60 5 13 Simpson Integration lt i osy eroe csetera dpon a Sew 62 5 14 Two Particle Decays aoaaa a 63 Collecting Subprocesses 65 6 1 Distribution Summation ooo a a 65 6 2 Event Mixing and LHEF 2 2 69 644 aaa a 65 Oe NAUpl S 2 2304 24486 d a a na aE aA aA 68 Batch Mode 69 Gl Blind Wide a sirs as samos rama saaa ia es 70 T2 Shell Cpe sar ee oe os gS ee Be a a a a EO ow ES 72 7 3 Batch interface 2 244 44 46 62 2546 Ae ws ee oe od TT 7 3 1 Structure and keywords of the bacth file 79 7 3 2 Example of the bacth file a aoaaa aa 92 7 3 3 Monitoring of the calchep batch session 2 95 oe Results Storage gt s ccc u Od ua dpa deni 98 Particle Interaction Model Implementation 99 8 1 Independent Parameters oaoa ooa a a 99 8 2 Dependent Parameters o oo a a a a 100 Bio Particles s samaca a a a a S OES RSS SE OS HE a ZG 102 8 4 Interaction Vertices ooa 0544 62
124. ibution must provide the current licence for users This License does not permit any commercial profit making or pro prietary use or re licensing or re distributions Persons interested in a for profit use should contact the Authors The Authors of CalcHEP do not guarantee that the program is free of errors or meets its specification and cannot be held responsible for loss or consequential damage as a result of using it 2 3 How to get the code If you agree with the license above you may download the CalcHEP code from the CalcHEP web site The current filename is 11 calchep_3 30 tar gz which corresponds to the current CalcHEP version 3 30 The next step is to unpack this file by doing gtar xzf calchep_3 30 tar gz As a result a directory named calchep_3 30 will be created Below we shall refer to this directory as CALCHEP This directory contains the following subdirectories 2 4 c_sources which is used for the source codes of the CalcHEP package lib which is used for the libraries generated during the CalcHEP compilation bin which contains the CalcHEP executable scripts and binary files include which contains some header files pdTables which contains tables of partons distribution functions help which contains text files used in the interactive session for the contextual help utile which contains auxiliary routines which are described in the utile README file models which contains two r
125. ile Here is an example Kinematics 12 gt 34 56 Kinematics 34 gt 3 4 Kinematics 56 gt 5 6 If multiple processes are specified using a single colon as in the previous example will apply the kinematics to all processes If different kinematics are desired for each process then the n notation can be used as in 86 Kinematics 1 12 gt 34 56 Kinematics 1 34 gt 3 4 Kinematics 1 56 gt 5 6 Kinematics 2 12 gt 3 456 Kinematics 2 456 gt 45 6 Kinematics 2 45 gt 4 5 where n corresponds with the process number as entered in the batch file Regularization When a narrow resonance is present in the signal it is a good idea to specify the Regularization This is done with the same notation as in CalcHEP Here is an example Regularization momentum 34 Regularization mass MW Regularization width wW Regularization power 2 Regularization for as many resonances can be specified as desired Further more different resonances can be specified for each process using the n notation as in Regularization momentum 1 34 Regularization mass 1 MW Regularization width 1 wW Regularization power 1 2 Regularization momentum 2 45 Regularization mass 2 MZ Regularization width 2 wZ Regularization power 2 2 Distributions Distributions are only applied to the production process The decays are ignored Standard CalcHEP notation is used for the distribution parameter Her
126. implification the difference is zero it will print OK in the message file Otherwise it will write Error C Ghost fields and the squared diagram tech nique for the t Hooft Feynman gauge C 1 The problem Whenever we implement a new model with higher spin we encounter a prob lem with asymptotic behavior of it s propagator at high energy This can be solved in the case of a spin 1 gauge boson by use of the Goldstone bosons and Faddeev Popov ghosts associated with it as we describe in this appendix The unitary gauge propagator for a massive vector boson is given by i Qu Kyky m Q27 m k where the gw kykv m projects out the unphysical polarizations and leaves only the physical polarizations This projection is necessary because we have used a 4 component vector field to describe a particle with 3 degrees 132 of freedom Unfortunately the k k m term leads to a quadratic growth of the amplitude at high energies which if left uncancelled leads to a violation of unitarity and renormalizability This problem is solved for vector fields in the framework of gauge field theories where the gauge symmetry leads to a cancellation between diagrams of the bad high energy growth 48 However numerical calculations have finite precision If the cancellation is very large although the cancelation is perfect in theory the finite precision arithmetic can lead to partial can cellation and consequently incorrect re
127. ine the second loop calculations The nCubes parameter defines number of sub cubes which are used in the second loop for a proper fitting of integrand for efficient event generation Note that parameters of second loop can not be defined in graphic interface mode The parameters are stored in session dat file erun_vegas This script launches subsequently two loops of Vegas Monte Carlo integration of the phase space using parameters defined by set_vegas At first loop we allow Vegas to adopt the integration grid After that all obtained results for cross section and histogram are cleaned and we launch second loop with fixed grid which gener ates final statistics and prepare integrand fitting for event generation If it1 0 or N1 0 then the only the second loop is running In the same manner only the first loop is running if it2 0 or N2 0 If only one loop is active intermediate clearing of results is not applied run_vegas is used by pcm_cycle name_cycle subproc_cycle and 74 par_scan scripts presented below eset_momenta p1 p2 This script updates the momenta of the incoming particles to p1 and p2 and then quits eset_param name1 value1 name2 value2 This script changes the nu merical values of one or more of the independent model parameters name1 name2 etc to valuel value2 etc respectively and then quits eset_param File In this case this script changes the numerical values of the independent model parameters as
128. ing changed Typically the user would start n_calchep in interactive mode and set all the session parameters as desired The user would then quit the interactive session and run one of these scripts 73 erun_vegas iti N1 it2 N2 This script runs the Vegas Monte Carlo inte gration of the phase space It runs Vegas it1 times with N1 calls each and then it2 times with N2 times each The results are cleared between these two runs If the user calls this script with all of itt N1 it2 and N2 nonzero Vegas will run it1 times clear the results and then run it2 times more the idea being that the first it1 calls allow Vegas to adapt the grid but the second it2 calls achieve the actual integration If it1 and N1 are nonzero but it2 or N2 are zero it will run Vegas it1 times and quit This would typically be used if the user is satisfied with the adaptation of the Vegas grid and has already cleared the results but wants to add more statistics to their integration If it1 or N1 are zero but it2 and N2 are nonzero it will first clear the results and then run Vegas it2 time This would typically be used if the user has been adapting the grid but is now satisfied and wants to perform the integration after clearing the statistics eset_vegas iti N1 it2 N2 nCubes This script sets parameters of two loops Vegas calculation which drive script run_vegas Here it1 is number of Vegas runs and number of integrand calls for the first loop it2 and nCall2 def
129. ional purposes 8 2 Dependent Parameters The table Constraints contains all the dependent parameters of the model It consists of two columns 1 Name This is where the name of the parameter belongs The re strictions on the names are the same as for the independent parameter names 2 Expression This is where the formula belongs which defines the value of this dependent parameter The formula can contain the fol lowing e integer and float point numbers e independent parameter names contained in the Parameters table dependent parameter names defined above the current row e parentheses and arithmetic operators the symbols i and Sqrt2 100 e standard functions from the C mathematics library such as sqrt x and sin x The full list of these functions is contained in the CALCHEP include extern h file e functions from the SLHAplus package e the function if x y z which returns y if x gt 0 and z otherwise e any user defined functions The code containing these functions should be included in the Libraries table Their prototypes can also be included in the Libraries table If their prototypes are not included CalcHEP assumes they return double type A list of the resulting auto prototyped functions appears in the results autopot h file after compilation of the numerical code Additionally anything after the symbol is considered a comment and ig nored This can be used to enter a comment
130. ions it is possible to write keystroke sequence strings that perform any desired calculation that can be achieved in an interactive session However it can be very difficult in practice to create a keystroke sequence string from scratch For this reason another option blind was created and is used as in s_calchep blind n_calchep blind 71 This will open an interactive session where each keystroke the user makes is stored internally When the user quits the entire keystroke sequence string is printed to the screen The user can then copy or modify this string and use it with the blind option As an example of this process suppose the user would like to run vegas with 6 iterations of 100 000 calls clear the statistics and then run 10 iter ations of 1 000 000 calls each This could take a long time depending on the complexity of the process A simple way of achieving this is to start n_calchep with the blind option When the interactive session starts the user would use their keyboard as usual to move through the menus and change the values of nSess_1 nCalls_1 nSess_2 and nCalls_2 but to smaller values which will finish in a reasonable time for example 1 20000 1 and 20000 respectively After quitting CalcHEP prints the following to the screen CCUCCLC t1 20000f 1 20000 o The user can then simply change the numerical values to those they desire and run with the blind option as in n_calchep blind
131. istributions This is done via the Clear statistics menu item The grid is usually well adjusted when the Monte Carlo uncertainty stabilizes at or below approximately 1 Once the grid is improved the user can optionally freeze the grid via the Freeze grid menu item so that Vegas does not further adjust it Another benefit of freezing the grid is that the event generator will be prepared during the Vegas sessions In some cases it is desirable to start with a fresh grid This can be done by the Clear grid menu item CalcHEP has facilities to generate kinematical distributions during the Vegas sessions The user can specify which distributions he she would like by choosing Set Distributions A table with 6 columns will open The first column is where the user specifies the kinematical variable to be histogrammed The kinematical variables available are described in Subsec tion 5 6 The second and third columns are the minimum and maximum values of the histogram If a 1 dimensional distribution is desired the last three columns should be left blank If a 2 dimensional distribution is desired another kinematical variable a minimum and a maximum can be entered in the last three columns Multiple distributions can be entered one per line The minima and maxima can contain numbers model parameters algebraic expressions and standard math functions from the C math library If there are multiple ways the final state particles fit the kinematical variabl
132. item is In state which allows to enter the momenta of the incoming particles their polarizations and their parton distribution func tions We note that in order to set the polarizations the incoming particles must be massless and the symbol had to be used in the symbolic session see Section 4 3 If this was done the polarization can be set anywhere between the maximum and minimum helicity value for the particle For example for a fermion the helicity must be set between 1 2 and 1 2 For a vector boson the helicity must be set between 1 and 1 and so on The parton distribution functions will be described further in Subsection 5 3 The next menu item is Model parameters which allows to modify the numerical value of the independent parameters which are used in the numer ical calculations The following menu item Constraints is the same as in the symbolic session and has been described in Subsection 4 2 However the QCD strong coupling GG is not included under either of these menus since it depends on the scale of the interactions Its value is controlled by the next menu item QCD coupling which is described in Subsection 5 4 The Cuts menu item allows to set cuts on the Monte Carlo phase space integration and event generation Details of the cuts specification can be found in Subsection 5 8 The Breit Wigner menu item allows to modify the behavior of the propagators for the unstable particles Further details can be found in Subsectio
133. ively Further manipulations of the symbolic expressions can be performed in those programs as desired by the user The user could for ex ample sum over the squared diagrams perform substitutions evaluate the expression numerically or calculate the total cross section Using these ex pressions in these external programs is reasonable when the number and size of the diagrams are small Further details about this output can be found in Section 10 4 6 Switches There are some switches which influence the results of the symbolic calculator and C output They are controled by a menu which can be obtained by pressing the F5 key The items on this menu are e Number of QCD colors 3 inf This switch has two possible val ues 3 or inf If this switch is set to 3 CalcHEP performs the usual SU 3 quantum chromodynamic calculations including all the terms If this switch is set to inf on the other hand CalcHEP only calculates the leading term in the large Ne expansion This removes many inter ference diagrams which only contribute at higher order and reduces the size of the code Of course the numerical value of N is still taken as 3 in the final results The default is 3 e Diagrams in C output ON OFF This switch determines whether CalcHEP writes an ascii image of the diagram in the C code of the squared dia grams This can be useful when analyzing the C code The size of the code can be reduced by turning this off However the size o
134. l outgoing particles For example CalcHEP creates only one diagram for the SM et e y 7 process whereas a textbook would present two diagrams The reason for this is that CalcHEP has not yet as signed the momenta in the diagrams so the representative diagram is sufficient 38 e At this stage CalcHEP does not generate diagrams with the Fadeev Popov ghosts or the Goldstone bosons associated with gauge symme try breaking These fields are restored when the diagrams are squared This can be done because the vertices with the Fadeev Popov ghosts and Goldstone bosons are related to the vertices with the gauge bosons After squaring each squared diagram with a gauge boson in it gives rise to a set of squared diagrams with the gauge bosons replaced with the Fadeev Popov ghosts and Goldstone bosons as determined by the Feyn man rules In some cases there is no gauge boson vertex corresponding to a vertex with Fadeev Popov ghosts or Goldstone bosons such as the G5 vertex where G is the Goldstone boson eaten by the Z boson In these cases CalcHEP produces a diagram as if the corresponding gauge boson vertex existed such as a Z vertex After squaring this squared diagram simply represents the ones with the Fadeev Popov ghosts and Goldstone bosons as determined by the Feynman rules The actual Zt diagram is dropped while the ones with the Fadeev Popov ghosts and or Goldstone bosons are kept Further details can be found in Sections 8 7 8
135. l Constraints By default CalcHEP will only calculate the independent parameters up to and including any dependent masses If the 33 user would like further independent parameters calculated he she can add the keyword Local to the dependent parameter definitions CalcHEP will then calculate all independent parameters up to 4Local An example of how to include the Local keyword is Constraints Name Expression Local where the line with the represents a list of dependent parameter defini tions The calculated dependent parameters will appear in a menu which the user can scroll through The user can also choose one of the dependent parameters in this menu in order to view its dependence on the independent parameters This is accomplished by displaying another menu that allows the user to choose the independent parameter and then choose the beginning value the ending value and the number of points to evaluate The results will be plotted on screen in the Plot Viewer The final menu item of Menu 4 is Masses Widths Branch This item will bring up a new menu which lists all the particles in the model Choosing any particle in this menu will cause CalcHEP to calculate its mass if de pendent as well as its width and branching ratios if any and display them onscreen along with other particle information Additionally the user may choose ALL PARTICLES at the top of this menu which will cause CalcHEP to calculate all the d
136. late At the top of this screen CalcHEP dis plays a list of the model particles Each entry contains the particle name followed by the antiparticle name in parentheses and ends with the full de scriptive name for the particle If the list of particle is too long to fit on the screen the user may press the PgUp and PgDn buttons to view the other particles Below the particle list CalcHEP displays the prompt Enter process and presents the user with a text entry box where he she can enter his her de sired process The syntax for this entry is Pi P2 gt P3 P4 P5 where the incoming particles and outgoing particles are separated by gt and P1 P5 are anti particle names The total number of anti particles should not exceed 6 For example the input u U gt G G specifies the annihilation of a u quark and an anti u quark into two gluons In place of anti particle names the user can enter N x after the gt where N is an integer CalcHEP replaces this with all possible combinations of N particles and antiparticles from the X particles list The default is for this list to contain all the particles and antiparticles from the model For example theinputu U gt G G 2 x specifies the annihilation into two gluons plus any other two particles from the model The user can however limit which particles are included in the X particles list If N x is used CalcHEP presents the user with the text Exclude X particles follo
137. lating processes with incoming 133 or outgoing massive vector particles we meet a similar problem in the exter nal states Each diagram is multiplied by the vector boson polarization vec tors These polarization vectors e e e constitute an orthonormal basis in the sub space orthogonal to the vector boson momentum k By considering the relation eney ee ene kyky m gw 17 we can see that at least one of the polarization vectors grows as k m for any choice of polarization basis Let vector k have the components k ym tp 0 0 p 18 Then the polarization vectors can be chosen as e 0 100 19 e 1001 0 20 e p m 0 0 1 p2 m 21 The first two vectors correspond to spatially transverse polarizations while the third corresponds to the longitudinal polarization We see that the lon gitudinal polarization grows as k m and becomes large for large momentum It may imply a rapid increase in the cross section in processes with longitu dinal polarizations at high energies for effective theories or the appearance of large cancellations between various diagrams Indeed for a gauge theory the second case is realized and hence we again have a problem with finite precision calculations As in the previous case we see that the problem is related to the projection operator in Eq 17 and wonder whether we might use gauge invariance to solve this problem too One solution to this problem is simil
138. llation and start of the CalcHEP session After compilation of the CalcHEP package the user should install a work directory where they perform their calculations This is created with the mkUsrDir script which takes as its only argument the directory name where the user would like to do his her calculations mkUsrDir lt dir Name gt The directory name can include path information as appropriate Here is an example that creates a work directory named work in the user s home directory mkUsrDir work The user can create several work directories for different calculations if they like The mkUsrDir script will create the directory lt dir Name gt and copy or link the following directories and files to it e bin is a symbolic link to the CALCHEP bin directory and contains all the scripts and binaries required for calculations with CalcHEP e models is where the particle interaction model files belong It is initial ized with the default models contained in CALCHEP models but the user can add further models to this directory e tmp is where CalcHEP stores temporary files during symbolic calcula tions e results is where the output of the symbolic session is written In particular this is where the symbolic session creates the numerical code n_calchep for the user to perform their numerical calculations e calchep ini is a text file which allows the user to specify his her pref erences for the graphical user interface Among
139. lso like to thank A Datta and K C Kong for numerous sug gestions discussions and testing of CalcHEP as well as to Patrik Svantesson recent help debugging help For the last two years A Pukhov s work on CalcHEP has been supported by the Royal Society grant JP090598 N Christensen has been supported by the US National Science Founda tion under grants PH Y 0354226 and PHY 0705682 and PITT PACC 10 2 2 1 Installation procedure CalcHEP Web Site The CalcHEP code and manual can be found at the following Web site 2 2 http theory sinp msu ru pukhov calchep html Calchep License Non profit Use License Agreement This Agreement is to be held between the Authors of the CalcHEP program and any Party which acquires the program On acquiring the program the Party agrees to be bound by the terms of this Agreement 1 This License entitles the Licensee one person and the Licensee s re search group to obtain a copy of the source code of CalcHEP and to use the acquired program for academic research and education or other non profit purposes within the research group or it entitles the Li censee a company organization or computing center to install the program and allow access to the executable code to the members of the Licensee for academic research and education or other non profit use No user or site will re distribute the source code or executable code to a third party in the modified form Any re distr
140. ly by the batch script are stored in the files ending in png 98 8 Particle Interaction Model Implementation A model of particle interaction in CalcHEP is stored in five tables named Parameters Constrains Particles Vertices and Libraries These ta bles are stored in the respective files varsN mdl funcN mdl prtclsN mdl lgrngN mdl extlibN md1 which are located in the models sub directory of the users work directory The N in these file names refers to the model number Each model has a unique N For all of these tables a at the be ginning of any row means that that row is a comment and CalcHEP ignores it We describe each of these tables in this section The SLHAplus 57 library contains external functions that allow CalcHEP to read parameters from a SLHA file We include it in the CalcHEP package A brief description can be found in Subsection 8 8 Although it is possible to implement a new model of particle interactions directly using the table definitions described here for complicated models with a large number of particles parameters and Feynman rules it is a good idea to use an external program to generate the model files We briefly describe two programs that do this job at the end of this section LanHEP 35 and FeynRules 36 8 1 Independent Parameters The table Parameters contains all the independent parameters of the model It consists of the following three columns 1 Name This is where the name of the par
141. m F2 Manual displays information for using the graphical user interface F3 Models displays the current model of particle interactions F4 Diagrams displays the Feynman diagrams for the current process F6 Results displays and allows the user to delete the output files F9 Ref displays the CalcHEP website the required citation for using the CalcHEP package and other acknowledgements F10 Quit quits the CalcHEP session The function key functionality can also be activated by clicking the mouse on the function key name at the bottom of the graphical user interface They can also be activated by pressing the numeric key corresponding to the Function key For example 3 will be interpreted as F3 and so on The only caveat is that 0 initiates the F10 key Another helpful feature of the menu is the menu search Some menus can be quite long For example some model parameter lists are much longer than can be shown in one screen In situations such as this the user can press the f F or F key and a text box will open The user can then type the desired menu item and press enter The menu will immediately skip ahead to and highlight the menu item matching the users string 3 Messages At various times CalcHEP displays a message on the screen There are two kinds of messages The first is informational and does not require a response from the user The informational message ends with Press and key The user can
142. matrix element generatorarXiv hep ph 0102195 W Kilian T Ohl J Reuter WHIZARD Simulating Multi Particle Processes at LHC and ILC 2007 arXiv 0708 4233 T Gleisberg et al SHERPA l alpha a proof of concept ver sion JHEP 02 2004 056 arXiv hep ph 0311263 doi 10 1088 1126 6708 2004 02 056 T Gleisberg et al Event generation with SHERPA 1 1 JHEP 02 2009 007 arXiv 0811 4622 doi 10 1088 1126 6708 2009 02 007 A S Belyaev A V Gladyshev A V Semenov Minimal supersym metric standard model within CompHEP software packagearXiv hep ph 9712303 168 21 22 23 24 25 26 27 28 29 G Belanger F Boudjema A Pukhov A Semenov micromegas A program for calculating the relic density in the mssm Comput Phys Commun 149 2002 103 120 arXiv hep ph 0112278 G Belanger F Boudjema C Hugonie A Pukhov A Semenov Relic density of dark matter in the NMSSM JCAP 0509 2005 001 arXiv hep ph 0505142 doi 10 1088 1475 7516 2005 09 001 G Belanger F Boudjema S Kraml A Pukhov A Semenov Relic density of neutralino dark matter in the MSSM with CP violation Phys Rev D73 2006 115007 arXiv hep ph 0604150 doi 10 1103 PhysRevD 73 115007 A Belyaev C R Chen K Tobe C P Yuan Phenomenology of littlest Higgs model with T parity including effects of T odd fermions Phys Rev D74 2006 115020 arXiv hep ph 0609179 doi 10 1103 PhysRevD 74 115020 A
143. menu all the diagrams for the highlighted subprocess are removed On the other hand if the F8 key is pressed all the diagrams are restored for the highlighted subprocess If the Enter key is pressed while on this menu or the mouse clicks on the highlighted process the Diagram Viewer will open with the diagrams for the highlighted process We will now list some details of the visual representation of Feynman diagrams used in CalcHEP e Incoming particles are drawn on the left side of the diagrams while the outgoing particles are shown on the right e CalcHEP uses dotted lines for scalar particles spin 0 dashed lines for other bosonic particles spins 1 and 2 and solid lines for fermionic particles spins 1 2 and 3 2 e Charged particles are represented by lines with arrows The arrow in dicates the direction of the particle not the anti particle propagation e Incoming and outgoing particles are labeled by their names at the end of their lines Virtual particles are labeled by their names at the middle of their lines If a particle is not self conjugate the particle s name is used for the labeling not the anti particle s name e In the case of scattering processes the first scattering particle enters at the top of the diagram while the second scattering particle enters at the bottom e CalcHEP produces only one representative diagram for a set of dia grams which can be transformed into one another by replacing identica
144. mprove the generator further and try again We recommend the user test the generator with a small number of events before generating the full desired set of events Once the user is satisfied with the events he she can answer y to accept them The events will be written in plain text to the file events_N txt where N is the CalcHEP session number A full description of the event format can be found in Appendix I 4 These events can be converted to the Les Houches Accord format by use of the event_mixer program found in the bin directory event_mixer is further described in Section 6 2 Event analysis In addition to many external programs that can analyze events CalcHEP contains a program which can histogram the events and generate a distribution It s name is events2tab and it is stored in the bin directory For example it can be run as 61 CALCHEP bin events2tab lt var gt lt min gt lt max gt lt N gt lt lt evnts gt gt lt plt gt where lt var gt is the kinematical variable to be histogrammed and must be in quotation marks lt min gt and lt max gt are the minimum and maximum val ues of the distribution and lt N gt is the number of bins for the histogram The events are given to this program via the redirection operator lt and lt evnts gt is the event file The output is the distribution data and would typically be redirected by the gt operator to the file lt plt gt For example CALCHEP bin events2tab M b B
145. n 5 5 The Phase space mapping menu item opens 44 S F 1 OFF S F 2 OFF Momentum P1 GeV 1000 Momentum P2 GeV 1000 First particle unpolarized Second particle unpolarized OFF ISR amp Beamstralung Laser photon beam Equiv Photon Appr PDT LHA parton dist alpha OFF alpha MZ 0 1172 nf 5 Order NLO mb mb 4 20 Mtop pole Qfact 91 197 Qren Qfact Alpha Q plot Menu 1 Subprocess IN state Model parameters Constrains QCD coupling Breit Wigner Cuts Phase space mapping Monte Carlo simulation 1D integration nSess 5 nCalls 10000 Set Distributions Start integration Display distributions Clear statistics Freeze grid OFF Clear grid Event Cubes 10000 Generate Events Breit Wigner range 2 7 T channel widths OFF ON GI in t channel OFF ON GI in s channel OFF ON Kinematics Poles Set precision Angular dependence Parameter dependence Number of events 10000 Launch generator Regenerate events ON Figure 9 Schematic diagram of the menu system for the numerical session 45 up into a menu with two items They are Kinematics and Regularization and are described in Subsections 5 9 and 5 10 respectively They allow to modify the mapping of phase space to improve Monte Carlo integration The menu item Monte Carlo simulation allows to run the Vegas 46 47 Monte Carlo integration of the multiparticle phase space to determine the collision cross section or the decay width The Monte
146. n external code These lines should include the full function prototype including the semicolon at the end on one line in the syntax of the C programming language These functions can be used in the definitions of the dependent parameters in the Constraints table see Subsection 8 2 External code and libraries can be linked to the numerical code by us ing this table as well The user should enter a list of the external code 108 libraries and any flags necessary for his her model in this table Some typ ical examples of external code are user defined kinematical variables which can be used in cuts and histograms see Subsection 5 7 and the LHAPDF libraries see Subsection 5 3 All lines which do not start with or extern are concatenated and passed to the linker which creates the executable for numerical calculations These lines can make use of environment variables CalcHEP defines two in it s startup scripts calchep and calchep_batch that the user can make use of They are CALCHEP which is the path to the CalcHEP root directory and WORK which is the path to the user s working directory The user can also make use of his her own environment variables These environment variables can be used with or without parentheses either CALCHEP or CALCHEP is acceptable CalcHEP will translate between the two depending on whether they are used in a Makefile or in a shell environment 8 6 Propagators CalcHEPdefines the propagators fo
147. n hours the job can run for If this time is exceeded the jobs are killed by the Isf cluster Memory specifies the maximum amount of memory in G that the jobs can use If this memory is exceeded by a job the Isf cluster will kill the job email specifies which email to send any messages t The default for all of these is whatever is the default on the Isf cluster Sleep time specifies the amount of time in seconds the batch script waits before checking which jobs are done and updating the html progress reports If a very short test run is being done then this should be low say a few seconds However if the job is very large and will take several hours or days this should be set very high say minutes or tens of minutes or hours This will reduce the amount of cpu time the batch program uses Here is an example setting the sleep time to 1 minute sleep time 60 The default is 3 seconds When jobs are run on the local computer the keyword Nice level spec ifies what nice level the jobs should be run at If other users are using the same computer this allows the job to be put into the background and run at 90 lower priority so as not to disturb the other users This should be between 0 and 19 where 19 is the lowest priority and the nicest Typically it should be run at level 19 unless the user is sure it will not disturb anyone The nice level should be set both for a local computer and for a pbs or lsf batch run The reason is t
148. n in the Vertex table then CalcHEP enters the Lorentz part directly in the expression If on the other hand the fermion comes before the antifermion in the Vertex table then CalcHEP first transforms the vertex by means of the rule given in Eq 10 and writes a comment stating that it has done this For exam ple consider the vertex of a u d and a W Suppose it is encoded in the Vertex tableas D u W and its hermitian conjugate is encoded as lU lda W If CalcHEP is moving against the fermion flow direction on a u quark line and because the u comes after the D in the Vertex table the Lorentz piece is inserted without change If however the vertex was encoded as u ID IW then the Lorentz piece would first be reversed using Eq 10 before being inserted into the expression If on the other hand CalcHEP is moving with the fermion flow direction perhaps after a Majorana or fermion number violating vertex and the in coming antifermion comes after the fermion in the vertex CalcHEP again enters the Lorentz part directly If however the antifermion comes before the fermion CalcHEP first transforms the vertex by means of Eq 10 and writes a comment about the transformation Using the same example ver tex above if CalcHEP is moving with fermion flow on a u quark line and the U comes before the d as in U ld W the Lorentz piece is first 127 reversed before insertion Because a Majorana fermion is its own antip
149. n of squared matrix elements does not converge well because of the presence of the singular propagators Even after including the widths the convergence may not be optimal It can be improved by doing a phase space transformation which smooths the sharp peaks of the squared matrix element see Appendix I 1 for further details The resonances can come in the forms a a Game 5 HET 6 where m I and p are the mass width and momentum of the virtual particle respectively In order for CalcHEP to regularize these propagators it needs to know the position of the resonances These can be entered via the Regularization subitem of the Phase space mapping menu item which opens a table edi tor allowing to enter the position of these resonances This table has four columns allowing to specify the momentum mass width and power of the resonance denominator The momentum of the resonant particle can be expressed as a sum of the external momenta The user can tuype the numbers of the particles which should be added together to get the resonant momentum Initial state momenta are added to each other and final state momenta are added to each other but initial and final state momenta are subtracted from each other For example the entry of 12 means the resonance occurs at p p2 m the entry of 34 means the resonance occurs at p3 p4 m and the entry of 134 means the resonance occurs at p p3 pa i The mass and width of the
150. nd a satisfactory grid Typical values for nSess are between 5 and 10 The default is 5 During each session Vegas calculates the integrand Neas times This is controlled by the menu item nCalls Greater values of Neas give better estimates for the integral and allow for better improvement of the grid The optimal value depends on the process being analyzed Processes with more final state particles typically need larger values of Nea whereas processes with fewer final state particles converge with smaller Neas The default value of NCalls is 10000 Vegas begins calculating the integral and improving the grid when the menu item Start integration is chosen During the integration the status of each session is displayed along with the integration results of previous sessions These results include the integral estimate the uncertainty estimate and the estimated efficiency of the event generator if the grid is frozen When Vegas finishes it makes a final estimate of the total integral uncertainty 58 and x It is usually a good idea to achieve uncertainties of approximately 1 or better for individual sessions After Vegas finishes the user can adjust the Vegas parameters and or run Vegas again until satisfactory results are obtained Unless the statistics are cleared the new Vegas results are combined with the previous results After the grid is improved it is a good idea to clear the statistics before calculating the final integral and d
151. ng grid is user to define the limit These limits do not influent on the work of the program They are used to collect statistics of points out of limits These statistics also count the number of points where Q exeeds the last point of q grid e mass This keyword allows to introduce the mass of the composite particle The corresponding numerical value must follow the keyword Default value for mass parameter is 1 GeV e Interpolation It defines interpolation procedure For current version the following interpolations are available CTEQ4 CTEQ6 MRST2001 CTEQ6 interpolation procedure depends on Agcp parameter which must be written after CTEQ6 e q threshold Defines thresholds for c and b quarks Used by MRST I Monte Carlo phase space integration I 1 Parameterization of multi particle phase space 1 1 1 Parameterization via decay scheme The element of phase space volume for a n particle state is equal to 58 iO ln EET Oe ap 33 i l The same expression is valid for both the decay of unstable particle with momentum q and the interaction of two particles with momenta q and q2 such that q q2 q For further discussion we need a designation for a MCTEQ5 interpolation procedure is identical to CTEQ4 157 phase space volume of some subset S of the full n particle set According to 33 ara 8 e ta Er ee Hi 34 Let S and S be two disjoint particle subsets then dI q Sy U S2 J as dsa
152. ntain the same name in both columns Any printing character can be used in the particle name except white space parentheses and the percent symbol The length of the particle name can not exceed 8 symbols For long particle names we should note that the graphical representation of the diagrams might contain overlapping symbols 102 PDG This is where the PDG code 58 belongs This number is used mainly for interfacing with other packages For example these codes are included in event files 41 in order to communicate the particle flavor to other programs The parton distribution functions are also applied according to this number The conventional PDG codes should be used for SM particles For other particles the user should ensure that the code is not reserved for another particles such as a meson or baryon Otherwise conflicts could arise when passing events to other programs such as Pythia 2 Spin This is where the spin of the particle is specified It should be entered as the integer equal to twice the spin In other words 0 should be entered for a scalar field 1 for a spin 1 2 fermion 2 for a vector boson 3 for a spin 3 2 fermion and 4 for a spin 2 boson Spin 3 2 and 2 particles should be massive Mass This is where the mass of the particle is entered If massless O can be entered Otherwise it must be a parameter name which is defined in either the Parameters table or the Constraints table Width This i
153. o diagrams with a quark loop 2 2 27a Be Aa 2 were Aq are the Gell Mann matrices 140 The QCD group also has the following Fiertz identity j l Glas 5010 any oot 28 which at the diagram level can be presented with a substitution which removes gluon lines connecting quarks 3 ON gt 4 28a The final result is that we have only closed separated lines of quark color flows which are easily evaluated O tr N 29 However there is a complication in the separate determination of the color and the Lorentz parts of the ggg vertex This vertex contains a symmetry corresponding to the permutation of identical legs Altogether it contains a product of an anti symmetric Lorentz part and an anti symmetric color part Thus in the case of separate calculations of the Lorentz and the color parts one has to take care that the orientation of the legs in the color and the Lorentz diagrams are identical In the Ne oo limit the tree level squared matrix element has the asymptotic form Neera C KOUN 30 where ng Ng Nq are the numbers of quarks ant quarks and gluons in the diagram CalcHEP provides the user with the possibility to calculate matrix elements in the N oo limit where only the leading term of Eq 30 is used After the extraction of the leading term we substitute N 3 A significatn subset of the interference diagrams are zero in this limit Thus this approach simplifies the
154. ode the jobs are run on an Isf cluster If run from a pbs or Isf cluster the terminal should be on the computer with the pbs or Isf queue Here is an example of setting the batch to run in pbs mode Parallelization mode pbs Local mode is the default If run in pbs mode there are several options that may be necessary for the pbs cluster All of them can be left blank in which case they will not be given to the pbs cluster Here is an example of the options available Que brody Walltime 1 5 Memory 1 email name address 89 The que keyword specifies which pbs queue to submit the jobs to Walltime specifies the maximum time in hours the job can run for If this time is exceeded the jobs are killed by the pbs cluster Memory specifies the maximum amount of memory in G that the jobs can use If this memory is exceeded by a job the pbs cluster will kill the job email specifies which email to send a message to if the job terminates prematurely The default for all of these is whatever is the default on the pbs cluster If run in lsf mode there are several options that may be necessary for the Isf cluster All of them can be left blank in which case they will not be given to the lsf cluster Here is an example of the options available Que brody Walltime 1 5 Memory 1 email name address Project project_name The que keyword specifies which Isf queue to submit the jobs to Walltime specifies the maximum time i
155. or even during one session but any time he changes the model codes of matrix elements ob tained before will be cleaned 113 9 2 Setting of parameters and calculation of constraints Three functions can be used to set the value of independent parameters int assignVal char name double val void assignValW char name doube val assigns value val to parameter name The function assignVal returns a non zero value if it cannot recognize a parameter name while assignValW writes an error message int readVar char fileName reads parameters from a file The file should contain two columns with the following format name value readVar returns zero when the file has been read successfully a negative value when the file cannot be opened for reading and a positive value corresponding to the line where a wrong file record was found After parameter assignment is completed in has to call int calcMainFunc void routine with calculates public constraints Zero return value of this routine means that all constraints where sucsessfully calculated It non zero err value is returned than there is a problem in calculation of varNames err parameter The following routines are used to display the value of independent and constrained public parameters int findVal char name double val finds the value of variable name and assigns it to parameter val It returns a non zero value if it cannot recognize a parameter name double findValW char name just re
156. oson with a Faddeev Popov ghost or a Goldstone boson according to the Feynman rules for the model In unitary gauge these are removed We note that all these diagrams have the same denominator The evaluation of the expression for the diagram is started with an initialization of a variable for the sum of the numerators for these diagrams 126 numerator_ 0 This is followed by each diagram in this set The code for each diagram is preceded by a pseudo graphical representation of the squared diagram for example see Fig 14 In these pseudo graphical representations the name of each particle and it s corresponding momentum are written down near it s line It s Lorentz index is written in the line The code for each diagram is started with the fermion loop evaluation CalcHEP moves along each fermion line multiplying the vertex and propa gator terms The nospur instruction is declared before each loop evaluation to prevent Reduce from prematurely tracing the gamma matrix chains If the result of this multiplication contains Lorentz indices which can be contracted the program declares the corresponding vectors as indices by means of the index instruction CalcHEP begins moving along the fermion lines in the direction opposite to the direction of the fermion flow arrows When it encounters a vertex it looks it up in the Vertex table If it is moving against the fermion flow di rection and the incoming fermion comes after the antifermio
157. p p gt W b B Dist x title M W b GeV HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHEHE HEE Event Info HHHHHHHHHHHHHHHHHHHHHHHHEHHHHEHEE HE Number of events 1000 Filename pp Wbb lnbb HHHHHHHHHHHHHHHHHHHHHHHHEHHEHHHEE HE Parallelization Info HHHHHHHHHHHHHHHHHHHHHHHHHHHEHHEHEH Parallelization method local Max number of cpus 2 sleep time 3 HHHHHHHHHHHHHHHHHHHHHHHHEHHEHEHEE HE 94 Vegas Info FERETETETETELE TERE EHH TE PETE TET ETE HE BEE PETE nSess_i 5 nCalls_1 100000 nSess_2 5 nCalls_2 100000 7 3 3 Monitoring of the calchep batch session After the start of calchep bacth session with calchep_batch batch_file command the following information appears on the screen Processing batch Progress information can be found in the html directory Simply open the following link in your browser file WORK html1 index html You can also view textual progress reports in WORK html index txt and the other txt files in the html directory where WORK denote the path to calchep working directory Using browser user can monitor the progress of all stages of the calchep batch session and check CalcHEP batch detals Fig 12 a details of symbolic session Fig 12 b the progress in numerical session Fig 12 c as well as the progress on event generation Fig 12 d Further details on numerical session can be checked by clicking on par ticular value of the running parameter Mp in our example which opens the windo
158. pack ages The numerical session of CalcHEP enables the user to interactively e convolute the squared matrix element with structure functions and beam spectra The CTEQ and MRST parton distribution functions the ISR and Beamstrahlung spectra of electrons the laser photon spec trum and the Weizsaecker Williams photon structure functions are available 39 for muons electrons and protons Linking with LHAPDF 40 is also supported e modify physical parameters such as total energy coupling constants masses etc involved in the process e set the polarization of incoming massless particles e set the QCD scale for the evaluation of the QCD coupling constant and optionally its running and for the parton distribution functions e automatically calculate particle widths including both 1 gt 2 and 1 gt 3 decay processes e apply various kinematical cuts e define the kinematic scheme phase space parameterization for effec tive Monte Carlo integration e introduce a phase space mapping in order to smooth the sharp peaks of a squared matrix element or of structure functions e perform a Monte Carlo phase space integration by use of Vegas e generate unweighted events e display distributions for various kinematic variables e create graphical and YTRX output for histograms e save the histogram data to file for further analysis and or plotting by gnuplot PAW or Mathematica Most of the features of the symbolic and
159. plication of vertices on a fermion line The vertices and propagators are multiplied according to this direction in order to evaluate the gamma matrix trace For those vertices which have a fermion line coming in the second fermion is taken in the form it is presented in the Vertices table see Sub section 8 4 Otherwise we first rewrite them according to Eq 10 There are two kinds of fermion propagators lt pl y p2 gt and lt w pl w p2 gt In the case of Dirac or Majorana fermions both of them are equal to lt U pl b p2 gt lt Y p1 Y p2 gt p M A p1 p2 M were A is the scalar part of the propagator The propagators thus provide a factor of p M to a fermion line The sign of p depends on the line direction There is an exception to this rule in the case of a pure left right handed massless fermion which has the propagator b 6 C 0 9 p P2 lt Wyr pl bijr p2 gt and lt Yili gt PEED 9 p m 0 Furthermore the result of the trace evaluation is multiplied by 1 We should also remark that in the case that a fermion vertex contains two identical fermions the Wick contraction can be done in two ways We remind the user see Section 8 4 that the expressions presented in the Vertices table table of the model corresponds to the functional derivative of the Lagrangian Therefore the symmetry property and the factor of 2 should already be present there Consequ
160. quared matrix element Moreover they can be used to check the results of the CalcHEP symbolic calculator CalcHEP includes some tools for checking the symbolic calculator using Reduce More details can be found in Appendix B 4 5 Output of results and launching of the numerical calculation Menu 7 This menu occurs after the symbolic calculations have been performed The first menu item is C code which causes CalcHEP to write C code for the squared diagrams to the results subdirectory of the work directory After the C code has been written the user can execute the second menu item C compiler which cause CalcHEP to compile the C code and create the executable n_calchep If the compilation is successful it will launch the resulting interactive numerical session which should appear on the user s screen If there are problems with the linking the user can modify the libraries linked by using the Edit Linker menu item Any changes made using Edit Linker will be added to the model definition for later use Details of the numerical interactive session are covered in Section 5 This menu has three more items allowing the user to export the squared matrix element expressions to formats appropriate for other programs Each Al writes the expressions to files in the results subdirectory of the work di rectory The REDUCE code MATHEMATICA code and FORM code menu items write the symbolic expressions to Reduce Mathematica and Form for mats respect
161. r particles of spin less than or equal to two These propagators are hard coded and not modifiable in by the user unless specified below Spin 0 The spin 0 propagator is given by 10 pi p2 27 pj M lt 0 T A p1 A p2 0 gt Ac pi p2 M Spin 1 2 The spin 1 2 propagator is given by lt O T A p1 A p2 0 gt pr T M A p1 p2 M where p pyu If the fermion is defined to be purely left or right handed see Subsection 8 3 the propagator is defined as fil ys A c M 9 p P2 Spin 1 In unitary gauge the propagator is given by lt O T A p1 A p2 0 gt g PFE A p1 p2 M 11 109 while in t Hooft Feynman gauge it is given by g A p1 Pa M We remind the user that a massless vector particle must be defined as a gauge boson see Subsection 8 3 Spin 3 2 The spin 3 2 propagator is given by M1 pmo p lt 0 T A p A p 0 gt 3 p Mr EF 12 M m m2 ETNA Mr 2 Alp p M M Spin 2 The spin 2 propagator is given by at M pma m muz j nl mimoz P P H H2 pp lt OT AM p A 2 p 0 gt g 2a si 2 3 gH p 2 yt p gp Pd 4 GMM yb phd 4 gi 1 p p M 3 g gare gages mm gtt cl p M Auxiliary propagators When massive particles are marked as auxiliary fields see Subsection 8 3 by putting a in the Aux column the momentum dependence of the propagator i
162. raction of laser light with the high energy electron beam f0 for gt Zmar IOS Ni a a eee OE ee where zo 4 82 Umax Lo 1 zo N is a normalization factor The above spectrum corresponds to the special initial condition when unpolarized photons are created See 63 for more details G 2 Weizsaecker Williams approximation Weizsaecker Williams approximation is used to describe processes of electro production in the case of small angle of charged particle scattering In this 150 case the virtual photon emitted by the scattering particle appears near to the mass shell see Fig 17 It gives a possibility to reduce the process of electro production to the photo production one with an appropriate photon spectrum f x a 2m log 1 2 x76 1 1 2 x 2 1 z 6x z where a is the fine structure constant q is a charge of incoming particle m is its mass 6 m Qmaz Qmax Sets out the region of photon virtuality P gt Q2 ar which contributes to the process It is assumed that region of large virtuality can be taken into account by direct calculation of electro production As a rule this contribution is small enough Figure 17 Example of process with the 1 t pole cancellation Parameters q m and Qmar are defined by the user See 39 for the further explanations In the case of CompHEP the Weizsaecker Williams photon spectrum is available for charged leptons only G 3 ISR and
163. ractive session 6 2 Event Mixing and LHEF As described in Subsection 5 12 the interactive numerical session can write events for each subprocess and for each decay However it is often desirable to combine these events and connect production events with decay events so that the final events are fully decayed The program event_mixer does this 65 The parameters of this program are the number of final events to produce and a list of the directories where the event files can be found For example CALCHEP bin event_mixer lt N gt lt dir1i gt lt dir2 gt where lt N gt is the total number of events to produce lt dir1 gt lt dir2 gt and are the directories where the event files produced by the interactive numerical session are stored event_mixer searches these directories for event files to mix Two other files are used by event_mixer The first is the file decaySLHA txt which contains a list of the particles masses quantum numbers widths and decay channels in SLHA format 37 55 The user can generate this file during the interactive session by use of the Constraints then the Masses Widths Branching and then the All Particles menu items See Subsection 4 2 for further de tails The decaySLHA txt file is used to determine the total widths of the particles that are decaying in order to determine the branching ratios and the final cross section If this particle is missing event_mixer will use the current experimental v
164. re expression to a single line When the line is too long for the interactive session of CalcHEP it will print the following error message to the shell Error in model file models xx Length of record exceeds the maximum defined by the parameter STRSIZ 4096 which is defined in c_sources chep_crt include syst h The user can solve this problem by using the compiler option DSTRSIZ lt needed_number gt where lt needed_number gt is the line length required by the model This option should be set in the FlagsForSh file and CalcHEP should be recompiled gmake clean gmake However this time FlagsForSh should not be removed 17 3 Elements of the user interface In this section we would like to discuss the general elements of the CalcHEP graphical user interface Among these elements are the on line help the menu messages the string editor the table editor the diagram viewer and the plot viewer The user can control them using the Arrow keys the Enter key the Esc key the Backspace key the PgUp PgDn keys and mouse clicks Additionally the user can control whether colors and sounds are used and can choose the font used in the graphical user interface The user can set all of these preferences by editing the calchep ini file located in his her work directory This file contains three lines one for each of these settings The color and sound are set by simply toggling between on and off The font is set by specifying th
165. rectory usr include X11 but CalcHEP checks other locations as well If the X11 header files are not found CalcHEP still compiles however it only runs in non interactive mode If the user attempts to launch calchep in this case it will immediately close and print the following error message to the shell Error You have launched the interactive session for a version of CalcHEP that has been compiled without the X11 library Presumably the X11 development package is not installed on your computer Nevertheless all the non interactive functionality including the batch ses sion should still work If the user would like to use CalcHEP in interactive mode he she should install the following additional package 1ibX11 devel for Fedora Scientific Darwin MAC 1ibX11 dev for Ubuntu Debian xorg x11 devel for SUSE 16 After installing the X11 development libraries the user should recompile CalcHEP by issuing the commands gmake clean gmake When gmake clean asks whether to remove FlagsForSh the user should answer yes since CalcHEP needs to regenerate it with the X11 library infor mation Limits on size of texual strings Some CalcHEP models need very long lines in the model files for their implementation Typically this happens when the model is generated automatically by LanHEP or FeynRules Al though some of the dependent parameter definitions could in principle be split over several lines these packages write the enti
166. red quarks and gluons 165 J Table of exit codes For both s_ and n calchep 0 normal termination 2 16 the process was killed by the corresponding signal 59 error in edittab c 65 error in writing on the disk 80 can not open X11 display 81 can not find fixed X11 font 100 LOCK file was not removed 101 end of command sequence before end of program For s_calchep only 20 exit code for restart caused by user break or problem with memory 22 exit code for restart to realize Make n calchep 55 runtime error in colorf c 60 error in lagrangian detected in the time of symbolic evaluation 62 error in model detected in the time of symbolic evaluation 70 not enough memory 90 debug exit for read_func c 99 A needed directory say results is absent and can t be created 102 LOCK file in results forbids to continue symbolical session 110 error in input of process 111 process of the type specified is absent For n calchep only 50 error in evaluation of QCD scale 51 can not recognize position of singularity 52 error in kinematics 53 runtime error caused by regularization a pole in the phase space 54 usrfun was not defined but is used 1217 wrong name of variable 122 dependences can not be evaluated NaN is produced 123 energy is too small 124 can not evaluate cuts limlts 125 wrong format of table of regularizations The symbol marks the exit codes which can be prod
167. representation in CalcHEP 8 4 where the ColorFactor is i faiazaz We get Factor LorentzPart g pi p8 g p pt p ph gt CalcHEP uses the notation GG for the strong coupling constant g So for the 3 gluon vertex in CalcHEP format we have A1 A2 A3 A4 Factor Lorentz part G IG IG GG m1 m2 pl p2 m3 m2 m3 p2 p3 m1 m3 m1 p3 p1 m2 Quark gluon interaction The interaction of a gluon with a quark is described by the following action Soa 9 Gola ale d z Applying the Fourier transformation and substituting g q4 we get Sona 9 2n f 5 b1 pa ps G3 ps F P1 faa p2 A pid pad ps and S QG Ogi p1 O p2 dGE ps The factor 2m 5 p1 po p3 ta j is substituted by CalcHEP auto matically The factor C7 appears in 8 4 according to 9 and is also substituted by CalcHEP Thus the quark gluon interaction is implemented in the CalcHEP Vertex table as g 2r 6 p1 po p3 a y Al A2 A3 A4 Factor Lorentz part Q iq G GG G m3 where q and Q are the designations for a quark and an antiquark Interaction of ghosts with gluon This interaction is described by the term Gs J Ga 2 O F9 G8 x e a ate 143 Fourier transformation and subsequent evaluation of the functional deriva tives gives Seog g 2n 5 p1 p2 Ps ea P1 i pei ps 368 p3 Pa d pid pad ps
168. ric Levi Civita tensor can be used in vertices It is given by eps v1 v2 v3 v4 where v1 v2 v3 and v4 are either momenta or Lorentz indices The ColorStructure from Eq 8 4 is not included in the Vertices table CalcHEP substitutes it in automatically according to the following rules If all the particles in the vertex are color singlets CalcHEP inserts 1 If the vertex contains one fundamental and one antifundamental 3 x 3 the identity matrix is inserted If the vertex contains two color octet fields 8 x 8 the identity matrix is inserted If the vertex contains three color octet fields 8 x 8 x 8 it inserts i f al a2 a3 where f 2 3 is the structure constant of SU 3 and the color adjoint indices al a2 and a3 are taken in the same order they appear in the Vertices table If the vertex contains a fundamental an antifundamental and a color adjoint field 3 x 3 x 8 CalcHEP inserts 1 ae SAG j a where A i j a are the Gell Mann matrices Other color structures are not implemented in CalcHEP however it is possible to construct them by means of an unphysical auxiliary field see Subsections 8 3 8 6 and 13 for further details 8 5 External functions and libraries The Libraries table is used to link external code and declare external func tions Lines beginning with a are comments and are ignored by CalcHEP Lines beginning with the keyword extern are considered to be prototypes of external functions defined i
169. rograms are stored If compilation was not successful then return value is NULL It can be in case when input process is absent in the model User can find definition of struct numout in file dynamic_cs h We will not explain this structure but give some examples of it usage Codes of matrix elements are not related directly with parameters of the model describe above So before we start to work with SQME codes we have to export numerical values of model variables to code of matrix elements Let cc is a variable of numout type obtained by getMEcode procedure Then export of parameters can be done by the command for i 1 i lt cc gt interface gt nvar it if cc gt link i cc gt interface gt va i cc gt link i Number of compiled subprocesses and number of incoming and outgoing particles for these subprocesses can be detected by int procInfol nuout cc int nproc int nin int nout Particle contents for given subprocess 1 lt nsub lt ntot can be obtained by int procInfo2 numout cc nsub char pName REAL Masses Here pName and Masses are arrays of nin nout elemennts which present particle names and particle masses 9 6 Calculation of matrix elements To calculate matrix element one has to fill particle momenta Momenta are presented by one dimention array of type REAL which contains in turn 4 momenta of incoming and outgoing partices 4 momenta are started from energy zeroth component Value of squared matrix element c
170. rsive algorithm for the generation of relativistic kinematics for collisions and decays with regu larizations of sharp peaks Int J Mod Phys C7 1996 761 arXiv hep ph 9612479 doi 10 1142 S0129183196000648 D N Kovalenko A E Pukhov Multiparticle phase space integration with arbitrary set of singularities in CompHEP Nucl Instrum Meth A389 1997 299 300 doi 10 1016 S0168 9002 97 00102 2 173 71 F A Berends R Pittau R Kleiss Excalibur A Monte Carlo pro gram to evaluate all four fermion processes at LEP 200 and beyond Comput Phys Commun 85 1995 437 452 arXiv hep ph 9409326 doi 10 1016 0010 4655 94 00138 R 174
171. rticle mass Basic variables which defines particles are int nModelParticles ModelPrtclsStr ModelPrtcls Structure ModelPrtclsStr is defined in CALCHEP include VandP h 9 4 Decay widths and branching fractions The calculation of particle widths decay channels and branching fractions can be done by the function double pWidth char pName txtList branchings returns directly the particle width If the 1 gt 2 decay channels are kinemat ically accessible then only these channels are included in the width If not pWidth compiles all open 1 gt 3 channels and use these for computing the 115 width If 1 gt 3 channels are closed too then 1 gt 4 channels are considered An improved routine with a better matching between the 1 gt 2 1 gt 3 and 1 gt 4 calculations is kept for the future The returned parameter branchings gives an address where information about the decay channels is stored The txtList type is presented in CALCHEP c_sources dynamic_me include dynamic_cs h If SLHA file with decay information was used before pWidth call then pWidth will not calculate widths using CalcHEP matrix elements but will display information stored in the file void printTxtList txtList branchings FILE f writes to open file branching fractions obtained by pWidth doouble findBr txtList branchings char pattern finds the branching fraction for a specific decay channel specified in pattern a string containing the particle names in the
172. s During the first step CalcHEP looks for compilers and com piler flags The results are written in the FlagsForSh file in the form of bash assignment instructions For instance CC gcc CFLAGS g signed_char Wall If the resulting parameters satisfy the CalcHEP requirements then the file FlagsForMake which contains the same assignments but written in make format is created This file is included in all Makefile files used during CalcHEP compilations In this way the user can tune the file FlagsForSh to fit his her computing environment and recompile CalcHEP The FlagsForSh file contains comments which explain the available parameters The option to save the FlagsForSh file during the gmake clean procedure is implemented to save the users modifications if desired for the recompilation The command gmake flags 13 generates the FlagsForMake file and stops Both and Fortran compilers are defined in FlagsForSh The For tran compiler is not used for CalcHEP compilation but it can sometimes be required for compilation of programs used by CalcHEP interaction mod els For example the CalcHEP implementation of the MSSM needs either SuSpect Isajet or SoftSUSY to calculate the particle spectrum All of these programs require a Fortran compiler So although CalcHEP does not re quire a Fortran compiler some problems are expected in some models in the absence of a Fortran compiler 2 5 Compilation for High Precision calculations
173. s is increased in comparing with CTEQ case be cause MRST tables don t contain the corresponding information Note that name is the identifier of distribution that you will see in CalcHEP menu If the last four parameters are not specified then a will not be included in the table See MRST documentation to find the proper parameters For simple checks of pdt files one can use the checkpdt program The source of this program is stored in CALCHEP utile Compilation instruction is 12nf 5 always the order is included in the file name 154 cc o checkpdt checkpdt c pdf c 1m The usage checkpdt file pdt parton x q where parton is a parton symbol G for gluon u d s c b for quarks U D S C B for anti quarks This program writes on the screen the corresponding parton density and a q See code checkpdt c to create more extended test H 2 Format of parton distribution tables The structure of pdt file in CalcHEP is closed to the CTEQ one but is more flexible and complete Generally distribution function depends on two arguments They are the Feynman parameter X and the energy scale Q X is unitless and runs in 0 1 interval Q gt 1GeV and traditionally is presented in GeV units Thus we describe grids for X and Q variables and tables of parton distributions corresponding to the grids CalcHEP pdf file contains several items of information Each item is
174. s removed A p p2 M is replaced with p p2 27r ti M and all terms proportional to the particle momentum p in the numerator are dropped Auxiliary particles cannot appear as incoming or outgoing states They are only used to implement point like interactions 8 7 Ghost and Goldstone fields propagators In addition to the fields enumerated in the Particles table the Lagrangian can depend on a few other fields In particular gauge theories have Faddeev Popov ghosts 48 and if broken Goldstone bosons Furthermore complex color structures require a special tensor auxiliary field All of these fields are automatically generated by CalcHEPwhere appropriate by adding a final c C f t or T as described below 110 Faddeev Popov ghosts and anti ghosts are generated for any gauge vector particle which is marked by a g in the Auz column of the Particles table see Subsection 8 3 The names of the Faddeev Popov ghosts and anti ghosts are constructed by adding a c and C respectively to the particle name For example if the gluon is named G the gluonic ghost is named G c and the gluonic anti ghost is named G C The ghosts and anti ghosts corresponding with the W and W gauge bosons are W c W C W c and W C Hermitian conjugation transforms a Faddeev Popov ghost into a ghost with the same sign whereas it changes the sign of the anti ghost For example Ge Ge G C G C W c0 t W c W C
175. s where the width of the particle is entered If the particle is stable 0 can be entered Otherwise it must be a parame ter name In this case however this parameter can be defined in the Parameters table the Constraints table or it can be preceeded with the symbol If it is preceeded with the symbol CalcHEP will automatically calculate it when needed In this case the parameter should not appear in either the Parameters table or the Constraints table When automatically calculating the width CalcHEP first at tempts the 1 2 decays If none are found it attempts the 1 gt 3 decays If none are still found it attempts the 1 4 decays If none are found at this point it takes the width as zero Color This is where the color SU 3 representation is specified Supported representations are the singlet specified by a 1 the fun damental triplet specified by a 3 and the octet specified by a 8 If the particle is specified as a triplet the antiparticle is treated as an anti triplet the 3 representation 103 9 Aux This field is used to modify the propagator of the field For most fields this column will be left blank The other possibilities are specifies that the propagator should be point like all mo mentum dependence is dropped This can be used to construct 4 fermion propagators for example These fields can not appear as external states of processes l andr are used to specify that a
176. scribe how this is handled for cuts and distributions Here we describe the use of the and _ For any kinematic function causes the highest value to be returned while _ causes the lowest value to be returned For our three photon example E A returns the highest of the three energies and E_ A returns the lowest of the three energies There are two additional types of kinematical variables that do not follow the patterns described so far The first is M12 which returns the invariant mass of the two initial state particles 4 p p2 The second is user defined kinematical variables User defined functions always begin with U The rest of the name can be anything the user likes For example the user could define 53 the function xyz in which case the kinematical variable would be written as Uxyz Further details can be found in Section 5 7 5 7 User Defined Functions There are two C code functions that the user can write to modify the results of the numerical session The first prototype is double usrFF int nIn int nOut double pvect char pName int pCode which multiplies the squared matrix element for each phase space point This could for example be used to implement a K factor to approximate the effect of loops If the user does not define this function CalcHEP uses a dummy version of this function which always returns 1 The second function prototype is double usrfun char name int nIn int nOut double pvect char pName
177. ser may find helpful in writing his her code are 54 int findval char name double value int qnumbers char pname int spin2 int charge3 int cdim The first gives the value of the model parameter specified by its name Both independent and dependent parameter values can be obtained in this way If the parameter given by name is found in the model then findval returns O and fills value with the numerical value of the parameter The qnumbers function gives the particle s quantum numbers The particle is specified by name and the pointers spin2 charge3 and cdim are filled with twice the par ticle s spin three times the particle s charge and dimension of the particle s color representation This function returns the particle s PDG code If the parameter name cannot be found it returns 0 The CalcHEP utile directory contains examples of the usrFF and usrfun functions which can be modified to suit the needs of the user 5 8 Cuts Cuts can be entered by choosing the Cuts menu item This opens a table where the cuts can be defined There are four columns in this table The first column can contain a a or can be empty A means that the cut should be ignored A means the inverse cut for each particle combination The second column accepts a kinematical function see Section 5 6 The third and fourth columns take the minimum and maximum values respectively for the kinematical variable If the third or fourth columns
178. specified in the file File File must have each model parameter on a separate line with the name coming first followed by the new numerical value separated by white space epcm_cycle pcm0 step This script scans the cross section over the center of mass energy For each point in the scan it updates the momenta of the initial state particles and then runs the Vegas Monte Carlo integration When it is finished it writes the resulting cross sections to the file pcm_tab_j1_j2 where j1 is the session number when the script began and j2 is the session number when it finished where N j2 j1 1 It begins its calculations with the momenta of the initial state particles equal to pcmO and pem0 and increases in steps of size step for a total of N steps If there are distributions specified then they are stored in the files distr_k where k corresponds with the session number when it was generated In general j1 lt k lt j2 These distributions can be viewed using the program disp_dist contained in the WORK bin directory ename_cycle name val0 step N This script scans the cross section over a model parameter s value For each point in the scan it updates the parameter name and then calculates the cross section When it is fin ished it writes the resulting cross sections to the file name_tab_j1_j2 where name is the name of the parameter j1 is the session number when the script began and j2 is the session number when it finished Again N j2 j1 1 The scan
179. ss G3 Gn G3 ee CO e 0 Ste Gt lG s fe Gy Gy Ga Ga Oy feat G Oy Figure 16 Splitting of four gluon vertex right handed neutrinos the Yukawa interaction can give mass to the neutrino and the left and right handed components of the neutrino can be written as one Dirac 4 component spinor like the other SM fermions However in distiction to the other SM fermions the right handed neutrino has zero U 1 hypercharge and is in fact a singlet under the SM gauge group As a result a mass term involving only the right handed neutrinos is also allowed 1 The diagonalization of this combination of a Dirac mass term and a Majorana mass term splits the neutrino into two Majorana eigenstates with different masses To see this explicitly we express the 4 component Dirac neutrino field in terms of two Majorana fields y and Wy 1 1 W 51 a SL 7 ee In terms of the chiral fields the mass terms are _ 1 1 _ E _ my ULV zMU _ 15 UF 5 My did F My VrY T 2M vbr were my is the mass generated by the Yukawa interaction Choosing the basis which diagonalize the mass matrix p cos a y sin a y WU sin a cos a y 10Note that Y ysr 0 146 gives us the mixing angle a and the masses tan 2a 2my M M a AN Nae ae ee M 2 m my tana gt Thm Negative masses are allowed and can be removed by the transformation p 51 This is the so call
180. sses They are not applied to the products of the decays Here is an example Cut parameter T le Cut invert False Cut min 20 Cut max For each cut all four keyphrases have to be present As many cuts as desired can be included Including Cut min or Cut max but leaving the value blank 85 will leave the value blank in the CalcHEP table If the cut should only be applied to a certain process then the colon can be changed to n where n is the process number So for example we could do Cut parameter T 1 Cut invert True Cut min Cut max 20 Cut parameter T j Cut invert False Cut min 20 Cut max Cut parameter 2 J j j Cut invert False Cut min 2 0 4 Cut max 2 This set of cuts will apply a pT cut to leptons and jets in all processes but a jet cone angle cut only to process 2 The numbering of the processes corresponds to the order in which the processes are entered in the batch file Composite particle names can be used as long as they are defined by the keyword Composite in the process section Note that both of the transverse mass cuts apply a pr gt 20GeV cut in this example Kinematics As the number of final state particles increases it can be very helpful to specify the kinematics which helps CalcHEP in the numerical integration stage This is done in exactly the same notation as in CalcHEP The num bering corresponds to the order the particles are entered in the process in the batch f
181. st boosts into the cms frame of P1 P2 P3 and then takes the cosine of the angle between P1 in the cms frame and the boost direction T TC P1L P2 gives the transverse momentum of the combined momentum pp pp2 52 Y Y P1 P2 gives the rapidity of the combined momentum pri t ppa tpos N N P1 P2 gives the pseudo rapidity of the combined mo mentum ppi pea W W P1 P2 gives the transverse mass of the particle set S P1 P2 given by E yin oe wt ieS iES where m and p are the mass and transverse momentum respectively of the ith particle For example M e E returns the invariant mass of an electron and a positron in the final state The keyword Jet can be used instead of a anti particle name and is an alias for a gluon or any of the first 5 quarks If the kinematical lists the same particle more than once such as in the case of J Jet Jet or M e e CalcHEPconstructs the kinematical variable for distinct final state particles So J Jet Jet means the jet cone angle of two distinct Jet particles with different momenta and M e e means the invariant mass of two electrons with different momenta For some processes and kinematical variables there are multiple ways the final state particles can be assigned For example consider the process p p A A A and the kinematical variable E A which could be applied to any of the photons Sections 5 8 and 5 11 de
182. stinction to the massive case there are no Goldstone bosons associated with the massless gauge boson and the longi tudinal polarization becomes unphysical like the temporal polarization We can still extend the summation over physical polarizations to an enlarged set of polarization as in Eq 22 see the proof in 61 C 4 Summation of ghost diagrams in CalcHEP CalcHEP uses the squared diagram technique with a summation over the polarizations According to our counting one squared diagram corresponds to gt AA 26 i Sau in a squared matrix element where A and A are the amplitudes of two Feynman diagrams in the set of all Feynman diagrams and 7 counts the polarizations and unphysical states as described in the previous subsections Strictly speaking when gauge bosons are replaced by unphysical states we have a much larger set of diagrams For example consider one squared diagram for the process e y ne W The W is associated with two ghost and one Goldstone boson Altogether there are four diagrams see Fig 15 associated with this one parent diagram see Fig 15a They all have the same topology and in fact they all have the same denominators The numerators of these expanded diagrams are polynomials in scalar products of the momenta and the powers of these polynomials are the same for each 136 e ne ne e e ne ne e gt gt gt gt gt 0 gt gt gt
183. suited to calculation in Feynman gauge but there may be times that unitary gauge is useful This can be specified using the keyword Gauge as in 79 Gauge unitary The default is Feynman Process Processes are specified using the Process keyword and standard CalcHEP notation as in Process p p gt j l l Multiple processes can also be specified as in Process p p gt E ne Process p p gt M nm As many processes as desired can be specified When more than one process is specified the processes are numbered by the order in which they are specified in the batch file So in this example p p gt E ne is process 1 and p p gt M nm is process 2 This numbering can be useful when specifying QCD scale cuts kinematics regularization and distributions allowing these to be specified separately for each process There is no default for this keyword It must be specified Decays are specified using the Decay keyword and are also in standard CalcHEP notation as in Decay W gt l nu Again multiple decays can be specified as in Decay W gt l nu Decay Z gt 1 1 The default is to not have any decays Cuts kinematics regularization and distributions do not apply to decays It is sometimes convenient to specify groups of particles as in the particles that compose the proton or all the leptons This can be done with the keyword Composite as in 80 Composite p u d U D G Composite l e E m M Composite nu ne Ne
184. sults An accompanying problem is that a symbolic calculation can lead to complicated expressions with the appearance of mutually cancelling terms On the other hand there is a freedom in the formulation of the Feynman rules for gauge theories resulting from an ambiguity in the gauge fixing terms 48 These terms modify the quadratic part of the Lagrangian and conse quently may improve the vector particle propagator Indeed in the case of t Hooft Feynman gauge the propagator for vector particles takes the form 1 Juv Grim k p 15 where the term with the bad high energy growth k k m is absent The price for this solution is the appearance of three additional unphysical particles in the model They consist of two Faddeev Popov ghosts and one Goldstone boson All three of them have scalar type propagators however the ghosts have the opposite sign with the same mass m as the gauge boson In contrast to Eq 14 the propagator given by Eq 15 is not orthogonal to the temporal polarization state de k m 16 which is another way to see the appearance of additional unphysical state The main principle of gauge invariance guarantees 48 that an expression for the amplitude should be the same for any gauge if only physical incoming and outgoing states are considered C 2 Incoming and outgoing ghosts Generally the t Hooft Feynman gauge solves the problem of large cancella tions for the internal states But when calcu
185. t wy 0 which can be obtained by means of Eq 10 we get i h Holy 8L _ Talta L g pi 8dr yp Taaa Zup Vr i E gg Weber ih WERA 7H The first term is the free Lagrangian for a massless Majorana fermion while the following terms define the interaction Using the definition 8 4 we can write them in the CalcHEP Vertex Al A2 A3 A4 Factor Lorentz part MN MN Z EE 2 SW CW a E MN W EE 2 Sqrt2 SW G m3 1 G5 MN le W EE 2 Sqrt2 SW G m ee We note that there are two identical neutrino fields in the Lagrangian term which describes the interaction of neutrinos with a Z boson It leads to an additional factor of 2 and to the symmetry property of the corresponding vertex One of the typical mistakes users make in the realization of this vertex is the erroneous introduction of a G m3 1 G5 term which breaks the symmetry property Correct evaluation of the functional derivative 8 4 with the help of the identity 10 does not produce such a term E 3 Leptoquarks In this section we present an example of a Lagrangian which contains C conjugated fermions These terms appear in interactions which violate fermion number conservation Let Y and Y be the fermion fields of the electron 148 and the u quark respectively and let these fields interact with a complex scalar leptoquark field F with Lagrangian L AVE 1 75 VF AU 1 7
186. t one vertex of this simplex to increase the function This 15We assume f x and F x are not normalized 163 method leads to fast converges to local maximum but one has to take into account that the distribution function can have several local maxima on the cub cube boundary Thus preliminary random search needs to define a good start point for the search by the simplex method The number of calls for random search and the number of steps for simplex search are defined by the user In general the detected maxima are lower than the true ones To satisfy the inequality f x lt F x the function F x based on the detected maxima may be multiplied by some factor say 2 Of course it decreases the efficiency of the generator just on the same factor Nevertheless in some sub cubes were the variance of the function is large this factor may be not enough If CalcHEP finds a point x where f x gt F x it accompany point with an integer weight w This weight is the integral part of f x F ax plus one with the probability equal to the fraction part of f x F x From view point of calculation of various distributions one event with integer weight w should be treated as w independent events with identical parameters But for the evaluation of statistical uncertainties a more careful treatment is needed I 4 Format of event file In general all needed comments are attached to the file See below an exam ple of header of such file The
187. tensor field with the same color to the internal list of quantum fields The propa gator for this field 13 corresponds to the Lagrangian t p ta Consequently in order to realize the 4 gluon interaction we must introduce the following vertex for the interaction of the gluon with this tensor field a Sica A Hytal a GR GUa ate Siaa Gis PL G33 2 iP 27 6 p1 P2 p3 i Jorazas g H2H3 H1 H3 H1H3 pH2H3 x 7 A g gga In the CalcHEP Vertex table this vertex looks like A1 A2 A3 A4 Factor Lorentz part G G Gt GG Sqrt2 m2 m3 m1 M3 m1 m3 m2 M3 where G t is the CalcHEP notation for the auxiliary tensor field t associ ated with the vector field G Capital M denotes the second Lorentz index of the tensor field From the viewpoint of Feynman diagrams this realization of the 4 gluon interaction means that instead of one 4 gluon vertex we substitute three sub diagrams as shown in Fig 16 The contribution of each of these diagrams corresponds to one of the terms of expression 31 E 2 Neutrino as a Majorana fermion In the Standard Model only the left handed component of the neutrino field takes part in the gauge interactions In principle the right handed component can be omitted However such a model can not describe neutrino oscillations which have been detected at experiments In the framework of the SM with 145 Gis gee Gu Cre
188. ternal details of the CalcHEP symbolic calcu lator and so it appears as a black box On the other hand the Reduce code that CalcHEP produces is written in a form that can be understood by humans and followed step by step in Reduce The results of the Reduce cal culation can then be compared with the results of the CalcHEP symbolic calculator as a test of its correctness see Appendix B for details about the self tests In this subsection we describe the structure of the Reduce output The Feynman rules used in this output can be found in Appendix D CalcHEP generates a separate file for each squared diagram The files are named pNNN MMM red where NNN is the subprocess number and MMM is the diagram number Each file begins with a declaration of the momenta and Lorentz indices For example VARIABLES vector A pl p2 p3 p4 p5 p6 p7 p8 p9 p10 pil p12 ZERO_ vector mi m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12 m13 m14 m15 m16 has Mass shell declarations MASS P1 O MSHELL P1 MASS P2 O MSHELL P2 MASS P3 Mm MSHELL P3 MASS P4 Mm MSHELL P4 4 Momentum substitutions p1 p2 p3 Let p5 p1 p2 Oo ct ge D ll 125 Let p6 p1 p2 The vector A is used by the Reduce package to construct the y matrix as in y G 1n A The vectors whose names begin with p are used for the momenta The vectors whose names begin with m are used for the Lorentz indices A
189. tion efficiency converges If the final efficiency is too low the number of phase space cubes should be increased via the Event cubes menu item and Vegas should be run again This process can be continued until a satisfactory efficiency is achieved The user must balance obtaining a high efficiency against the time it takes to estimate the maxima for a large number of phase space cubes Event generation When the generator is prepared and the efficiency is acceptable the user can enter the Generate Events menu under Monte Carlo Simulation The first item on this menu Number of events allows to specify the number of events to generate Event generation is started by the menu item Launch generator During event generation if an event is ever produced with a differential cross section or partial width which is greater than the estimated maximum value in that cube then two things happen The first is that the event is split into multiple unweighted events The sec ond is that the maximum for that phase space cube is increased in order to prevent this from occuring in the future When CalcHEP finishes generating the events it displays an informa tional message which states the number of events generated the actual ef ficiency the number of multiple events generated and the number of events with negative weight generated The user is asked whether he she would like to accept these events If there are many multiple events it is a good idea to i
190. to the given parton set and this function is available to CalcHEP user See section 5 4 The files containing parton distributions must have the pdt extension n_calchep searches such files in the directories CALCHEP pdTables and Usually the last two directories are the user s working directory and its sub directory results CALCHEP pdTables contains the following parton sets CTEQ6L CTEQ6M 67 andlo2002 mrst2002nlo mrst2002nnlo 68 We pass to the user the routines which transform CTEQ and MRST data files to the CalcHEP format By means of them the user can add other distri bution to the list The primary c files are stored in the CALCHEP pdTables directory In case of CTEQ the compilation instruction is cc o cteq2pdt cteq2pdt c alpha c 1m The usage is cteq2pdt lt cteq_file tbl gt calchep_file pdt for example cteq2pdt lt cteq6m tbl gt cteq6m pdt The name of pdt file doesn t play a role The cteq2pdt routine can be ap plied to any CTEQ4 CTEQ5 CTEQ6 file It automatically detects version and a formula stored in the CTEQ file In the case of MRST file the corresponding compilation instruction is cc o mrst2pdt msrt2pdt c alpha c 1m The usage is mrst2pdt name lt mrst_file dat gt calchep_file pdt or mrst2pdt name nf order a MZ lt mrst_file dat gt calchep_file pdt For example mrst2pdt mrst2002nlo 5 nlo 0 1197 lt mrst2002nlo dat gt mrstnlo pdt The number of parameter
191. turns the value of variable name and writes an error message if it cannot recognize a parameter name The variables accessible by these commands are all free parameters and the con strained parameters of the model in file model func1 md1 treated as public For treating of independent and constrained physical parameters the basic variables are int nModelVars int nModelFunc char varNames contains nModelVarstnModelFuncti elements The zero one is not used 114 REAL varValues contains nModelVars nModelFuncti elements type REAL is defined in Type REAL is defined in CALCHEP include nType h By default REAL means double 9 3 Testing of particle contents char pdg2name int nPDG returns the name of the particle whose PDG code is nPDG If this particle does not exist in the model the return value is NULL int pNum char name returns PDG code of particle defined by name If the input parameters does not corresponds to any particle then return value is zero int gqNumbers char pName int spin2 int charge3 int cdim returns the quantum numbers for the particle pName Here spin2 is double spin of the particle charge3 is three times the electric charge cdim is the dimension of the representation of SU 3 it can be 1 3 3 or 8 The value returned is the PDG code If pName does not correspond to any particle of the model then qNumbers returns zero double pMass char pName returns numerical value of the pa
192. uced only in the blind mode The symbol marks exits included in the time of debugging They are not expected and presented only for completeness 166 References 1 A Pukhov CalcHEP 2 3 MSSM structure functions event generation batchs and generation of matrix elements for other packages 2004 arXiv hep ph 0412191 H Tanaka T Kaneko Y Shimizu Numerical calculation of Feyn man amplitudes for electroweak theories and an application to e e W W gamma Comput Phys Commun 64 1991 149 166 doi 10 1016 0010 4655 91 90058 S F Yuasa et al Automatic computation of cross sections in HEP Sta tus of GRACE system Prog Theor Phys Suppl 138 2000 18 23 arXiv hep ph 0007053 G Belanger et al Automatic calculations in high energy physics and Grace at one loop Phys Rept 430 2006 117 209 arXiv hep ph 0308080 doi 10 1016 j physrep 2006 02 001 H Murayama I Watanabe K Hagiwara HELAS HELicity amplitude subroutines for Feynman diagram evaluationsKEK 91 11 A Pukhov et al CompHEP A package for evaluation of Feynman diagrams and integration over multi particle phase space User s manual for version 33 1999 arXiv hep ph 9908288 E Boos et al CompHEP 4 4 Automatic computations from La grangians to events Nucl Instrum Meth A534 2004 250 259 arXiv hep ph 0403113 doi 10 1016 j nima 2004 07 096 J Kublbeck M Bohm A Denner FEYN ARTS COMPUTER AL GEBRAIC GENER
193. ve the desired result 70 Up Down Enter Escape Special keys where NN is the hexadecimal value Function keys or numeric input SrA a 2 T Ke Table 1 Characters used in blind mode for Up Down Enter and Escape Other special keys are specified by their hexadecimal number NN The interpretation of numeric characters depends on when they are initiated as discussed in the text All other characters which print to screen are used directly In this string each printing character e g alphabetical characters is rep resented by itself while up down enter and escape are represented by and respectively The response of the numerical characters depends on where in the interactive session they appear If alphanumeric input is required of the user such as the input of a process cut distribution etc a numeric characters is treated as the number it represents However if al phanumeric input is not required then the numeric characters signal function keys For example in this situation 0 corresponds with the F10 key which signals the end of the session and instructs CalcHEP to quit For this reason keystroke sequence strings are usually terminated with a 0 All other special keystrokes are represented by NN where NN is the hexadecimal value for the character For example the Tab key is represented by 08 For reference we also include these characters in Table 1 With these definit
194. w to construct such packages as micrOMEGAs where matrix elements generated by CalcHEP are used for calculation of different observables related to Dark Matter We as sume that user writes a corresponding main program which can be compiled by CALCHEP bin make_main mainProgram c See example of such main program in CALCHEP utile main_22 c Af ter compilation executable program mainProgram has to appear Because CalcHEP uses run time generated code for width calculation a tool which works with CalcHEP matrix elements has to have an opportunity to generate new matrix element If so it is quite naturally to provide user an option to generate any matrix element in run time In the same time the user can link preliminary generated matrix elements as well 9 1 Choosing of model The first command of the main routine should be int setModel char modelFilesDisposition int modelNumber For example if ones prefer to work in CalcHEP working directory the com mand can be setModel models 1 The setModel command generates aux subdirectory which is organised as CalcHEP working directiry with subrirectories models results tmp and directory so_generated to store compiled code of matrix elements set Model has to generate VandP so file in directory so_generated which con tains compiled constrains of the model list of variables and list of particles In case of problems not rezo value is returned The user can change a model from session to session
195. w with detailed information shown in Fig 13 This page also present the requested distributions Moreover the results shown in html browser as also recorded in the ascii files located in the WORK htm1 direc tory For example the results for numerical session are recorded in the file WORK html numerical txt as well as in the files in the WORK html1 runs directory containing further details For example after sucessful run of the batch_file given in the example above the user should get the WORK htm1 numerical txt with the following info 95 Home Symbolic Results Numerical Results Events Library Process Library Help Thank you for using CalcHEP Please cite arXiv 0000 0000 Home Symbolic Results Numerical Results Events Library Process Library Help Thank you for using CalcHEP Please cite arXiv 0000 0000 CalcHEP Batch Details Standard Model CKM 1 Done Finished Time hr Symbolic 12 12 0 00 o 3 3 0 05 Events 3 3 0 03 a Numerical Sessions Standard Model CKM 1 Done Runs fb Running Finished Time hr N events Mh 120 894 6 0 13 Mh 125 884 2 0 13 Mh 130 873 8 0 13 13 13 0 02 1000 13 13 0 02 1000 13 13 0 02 1000 0 05 c Home Symbolic Results Numerical Results Events Library Process Library Help Thank you for using CalcHEP Please cite arXiv 0000 0000 Home Symbolic Results Numerical Results Events Library Process Library Symbolic Sessions Standard Model CKM 1 Proc
196. wed by a text entry box where the user can list any desired particle and antiparticle limitations The syntax for this entry is Exclude X particles P1i gt ni P2 gt n2 where P1 P2 are particle names and n1 n2 are quantity limits This instructs CalcHEP to remove diagrams with more than n1 particles of type P1 n2 particles of type P2 and so on in the part of the final state specified by N x The specification P gt 0 can be shortened to P and forbids the appearance of the particle P among the X particles The user may also enter an alias for multiple particles such as a p for the partons in a proton j for the particles that produce jets or 11 for leptons The user can use any short name he she likes as long as it is different than the names of the anti particles defined in the model When an alias is used in the process CalcHEP requests its definition For example if the user enters the process p ap gt W b B and there is no p or ap defined in the model CalcHEP will display the prompt composit p consists of followed by a text entry where the user can specify which particles and or antiparticles he she would like included in the alias definition of p In this example the same is done for the ap entry This specifies the collision of any particles in the definition of p against any particles in the definition of ap and producing the particles W b B Aliases can be used both for the incoming and the outgoing particles
197. which can help the user when 21 2 CalCHEP SY MD Clr Del Size Read ErrHes Al A2 3 124 l gt Lorentz part cE c IGG m2 pl p2 m3 m2 m3 IG z GG Sqrt2 H3 m2 m3 m1 m3 m2 H3 IH EE m2 p1 p2 m3 m2 m3 IH EE CH SH m1 m2 p1 p2 m3 m2 m3 H EE CH SH 2 2 m1 m2 m3 m4 m1 m3 m2 H EE SH 2 2 m1 m2 m3 m4 m1 m3 m2 EE 2 CH SH 2 m1 m2 m3 m4 m1 m3 m2 EE 2 2 m1 m2 m3 m4 m1 m3 m2 EE HH SH EE SH CH 2 HH 3 2 EE Mh 2 MH SH 3 4 EE Mh MH SH 2 1 2 EEZ SH CH 2 1 2 EE SH 2 EE Mm 2 MW SH EE H1l 2 HMNW SH EE Mc 2 MHW SH l l l Vertices EE Hb 2 HHW SH EE Ht 2 HMH SH EE Figure 2 An example of a CalcHEP table working with tables These commands are achieved by holding down the Control key while pressing another key or by clicking on the command label displayed on the table border with the mouse These commands are e Xgoto X allows the user to enter a position and then moves the cursor to that position in the current cell e Ygoto Y allows the user to enter a line number and then moves the cursor to that line Entering causes CalcHEP to move to the end of the table e Find F allows the user to enter a string or comma separated list of strings to be searched for in the table CalcHEP will search the table for the search string s and move the cursor to the position of the ne
198. with a constant that interpolates between the two regions We find that the default value of R 2 7 leads to a difference of 0 2 in the integral of the squared propa gator between the modified propagator described in this paragraph and the propagator with a constant width for the entire momentum range The second menu item is T channel widths and allows to turn the width on in the t channel propagators The default is to not include these since they are not required to regularize the integration and these propagators never go on shell Note that by default the symbolic session does not include the widths in the t channel propagators In order to turn this on the user must also turn it on during the symbolic session see Section 4 6 The last two menu items GI in t channel and GI in s channel con trol whether CalcHEP applies another method of restoring gauge invariance described in 49 50 The diagrams which do not contain the resonant prop agator are multiplied by the factors p m P m2 n 2 And in diagrams which contain only the single power of propagator this propagator is replaced by p m OF me mp a expression This modification corresponds to the symbolic summation of all diagram contributions at a common denominator expression with subsequent substi tution of the width term into the factored denominator The trick allows to keep all gauge motivated cancellations As a defect of th
199. ws to choose the scaling of the size of the rectangle relative to the differential cross section as described above Large values for P tend to increase the resolution for large histogram bins but reduce the resolution for small histogram bins Small values for P do the opposite When finished viewing the plot the user can exit the Plot Viewer by pressing the Esc key 27 In addition to using the Plot Viewer in the interactive numerical session the user can also view plots that have been saved to file by using the exe cutable plot_view located in the CALCHEP bin directory This command takes one option which is the file where the plot data has been exported as in CALCHEP bin plot_view plot_ txt This is a stand alone version of the Plot Viewer 9 File Search Engine When CalcHEP requests a file or directory from the user such as when importing a new model it opens a text box where the user can enter the filename The user can enter the full file name including the path or he she can enter part of the path and end his her text with When this is done CalcHEP opens a menu where the contents of the directory are listed see Fig 6 The user can use the menu functionality to choose the directory or file he she wants after which the file or directory name in the text box will be updated accordingly This can be continued until the desired file or directory is found If the final target is a directory the user must finally remove t
200. xt occurance of the string s starting from the current position of the cursor If the user enters a comma separated list of strings CalcHEP will search for a row containing an instance of each of the strings where the order of the strings is not important Spaces are matched as well as printing characters Pressing F again after a search will result in 22 the cursor moving again to the next occurance of the string A new search string can be entered by changing the position of the cursor first and then pressing F The original purpose of table searching was to facilitate finding particular vertices in the vertex table however it is available in all tables Write W allows the user to enter a filename and then writes the contents of the current highlighted cell to that file If the content of the cell is very long then new line symbols are inserted automatically The above commands are available in both modes of the Table Editor and are displayed along the bottom border of the table The following commands are available only if the table is open for editing and appear allong the top border of the table Clr C clears the contents at the cursor position and to the right of the cursor position in the current cell Read R allows the user to enter a filename and then reads the con tents of that file and enters it into the current highlighted cell Spaces and new line symbols are ignored The size of the cell is increased

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