HadCalc manual
Contents
1. MSL parameter mass parameter of the left handed sleptons MSE parameter mass parameter of the right handed selectron like sleptons At parameter trilinear coupling of the sup like squarks Ab parameter trilinear coupling of the sdown like squarks A tau parameter trilinear coupling of the selectron like sleptons M_1 parameter U 1 y gaugino mass M_2 parameter SU 2 gaugino mass MG1 parameter SU 3 c gaugino mass SQRTS parameter square root of the hadronic center of mass energy SQRTSHAT parameter square root of the partonic center of mass energy THETA parameter angle between the two outgoing particles in degrees THETACUT parameter cut on the angle between the two outgoing particles in degrees K50 8 parameter energy of the third outgoing particle K50CUT parameter cut on the energy of the third out going particle PTRANS parameter transverse momentum PTRANS3CUT parameter cut on the transverse momentum of particle 3 PTRANS4CUT parameter cut on the transverse momentum of 2only for 2 2 processes particle 4 21 3only for differential cross sections 2 3 Allowed tokens in input files token type description PTRANSSCUT parameter cut on the transverse momentum of particle 5 RAPID 3 parameter rapidity RAPID3CUT parameter cut on the rapidity of particle 3 RAPID4CUT parameter cut on the rapidity of particle 4 RAPID5CUT parameter cut on the rapidity of2. 2 2 2 PDF parameters The set used for the calculation of the parton distribution functions is chosen by this submenu The layout and choices presented depends on whether LHAPDF or PDFlib is used For PDFlib three numbers must be entered The first de notes the type of parton distribution functions and is 1 for proton PDFs The second number specifies the respective group which has performed the fit to the experimental data and the third number chooses a specific PDF set When using LHAPDF astring must be entered that directly specifies the filename of the PDF set in the LHAPDF subdirectory 16 Chapter 2 Manual of the HadCalc Program 2 2 3 Integration parameters This submenu chooses the integration routine and sets its parameters Currently there are six integration routines available GAUSSKR One dimensional Gauss Kronrod algorithm GAUSSAD One dimensional adaptive Gauss algorithm DCUHRE Multi dimensional adaptive Gauss algorithm VEGAS Monte Carlo integration algorithm SUAVE Subregion adaptive Monte Carlo integration algorithm DIVONNE Monte Carlo integration via stratified sampling The last four algorithms are part of the CUBA library 15 In the following only a short description of the possible parameter settings is given The technical details of these algorithms and the precise impact of the variables are described in the CUBA manual and shall not be repeated here The GAUSSKR and GAUSSAD algorithms can only handle one dimension
3. April 2002 http durpdg dur ac uk lhapdf manual htm 31 32 13 14 15 16 Bibliography S Heinemeyer W Hollik and G Weiglein Comput Phys Commun 124 76 89 2000 S Eidelman et al Phys Lett B592 1 2004 T Hahn Comput Phys Commun 168 78 95 2005 T Hahn FormCale 4 User s Guide March 2005 http www feynarts de formcalc FC4Guide ps gz
4. MSUSY common SUSY mass scale MSQ Mmg mass parameter of the left handed squarks MSU Mi mass parameter of the right handed sup like squarks MSD M mass parameter of the right handed sdown like squarks MSL mr mass parameter of the left handed sleptons MSE Me mass parameter of the right handed selectron like sleptons At A trilinear coupling of the sup like squarks 24 Chapter 2 Manual of the HadCalc Program Name Parameter description Ab Ap trilinear coupling of the sdown like squarks A tau A trilinear coupling of the selectron like sleptons M1 M U 1 y gaugino mass M_2 M SU 2 gaugino mass MG1 IM gluino mass SQRTS VS square root of the hadronic center of mass energy SQRTSHAT v3 square root of the partonic center of mass energy THETA 0 angle between the two outgoing particles in de grees THETACUT Oeut cut on the angle between the two outgoing particles in degrees K50 7 k8 energy of the third outgoing particle K50CUT 7 elos cut on the energy of the third outgoing particle PTRANS Diranis transverse momentum PTRANS3CUT Panse Cut on the transverse momentum of particle 3 PTRANSACUT Panse ut on the transverse momentum of particle 4 PTRANSSCUT Prranscuy Cut on the transverse momentum of particle 5 RAPID in rapidity RAPID3CUT 73 cut on the rapidity of particle 3 RAPIDACUT n u cut on the rapidity of particle 4 RAPID5CUT 72 cut on the rapidity of particle 5
5. angles of the two particles in the transverse plane Its main use are exclusive hadronic cross sections where final state jets shall be observed explicitly It mimics the behavior of jet clustering algorithms There two jets which are separated by a jet separation below a certain limit are seen in the reconstruction as a single jet which has kinematic properties that are averaged over the two final state partons 1 4 Cuts 7 For the first two cut parameters rapidity and transverse momentum it is possible to translate these cuts into a limit on the integration parameters of the phase space The most general case is assumed here that cuts on both the rapidity Ycut and the transverse momentum Preu Of a particle shall be applied Using eq 1 15 the transverse momentum cut can be translated into a cut on c and one obtains 2 2 min _ fy _ PT cut 1 PT cut max 1 20 c z7 lt C lt E 6 p p Opr Opr Likewise the cut on the rapidity can also be turned into a cut on cg via eq 1 10 yielding 2 1 2 cg gt fit tanh u lt In ial ce 5 2 T Y P m 1 a max C lt 1 a tanh Ucut 2 ln C F 1 21 To shorten the notation the abbreviation 1 22 is used in the following Again the momenta and mass used in the equations all refer to the particle whose phase space should be constrained Applying both cuts requires that the conditions on cg are all fulfilled simul taneously This also restricts the integr
6. k T 0 kl Mi m tm A 9 Bibliography 1 2 10 11 12 D J Gross and F Wilczek Phys Rev Lett 30 1343 1346 1973 H D Politzer Phys Rev Lett 30 1346 1349 1973 R Brock et al Rev Mod Phys 67 157 248 1995 R Brock et al Handbook of perturbative QCD Version 1 1 September 1994 FERMILAB PUB 94 316 1994 J D Bjorken and E A Paschos Phys Rev 185 1975 1982 1969 R P Feynman Phys Rev Lett 23 1415 1417 1969 E D Bloom et al Phys Rev Lett 23 930 934 1969 M Breidenbach et al Phys Rev Lett 23 935 939 1969 J I Friedman and H W Kendall Ann Rev Nucl Part Sci 22 203 254 1972 J C Collins D E Soper and G Sterman Adv Ser Direct High Energy Phys 5 1 91 1988 G Altarelli and G Parisi Nucl Phys B126 298 1977 C Meier yZ Produktion in P P Kollisionen mit elektroschwachen 1 Schleifen Korrekturen Diplom thesis Universit t Karlsruhe July 2001 J Kublbeck M Bohm and A Denner Comput Phys Commun 60 165 180 1990 T Hahn Comput Phys Commun 140 418 431 2001 T Hahn and C Schappacher Comput Phys Commun 143 54 68 2002 T Hahn and M Perez Victoria Comput Phys Commun 118 153 165 1999 T Hahn New developments in FormCalc 4 1 hep ph 0506201 2005 H Plothow Besch PDFLIB User s Manual Version 8 04 CERN ETT TT 2000 04 17 W Giele and M Whalley LHAPDF version 4 Users Guide
7. shows all token names that may appear in an input file together with its associated type The tokens are not case sensitive Thereby parameter means that the variable can either be followed by four comma separated values that denote the lower and upper bound whether the increase is linear or loga rithmic and the number of intermediate intervals or a single number that is the fixed value of this parameter boolean means that a specific behavior is switched on There is a corresponding separate token that switches the same behavior off again double and integer tokens take a single double precision or integer value respectively as input string assigns the remainder of the line to the parameter Finally preselected takes special values as an argument The possible choices for each of these ones were discussed during the description of the menus given in section 2 2 Any settings referring to particle 5 are relevant only for 2 3 processes and will be silently ignored otherwise Chapter 2 Manual of the HadCalc Program token type description MAO parameter mass of the CP odd Higgs boson TB parameter ratio of the Higgs vacuum expecta tion values MUE parameter js parameter in the Higgs sector MSusy parameter common SUSY mass scale MSQ parameter mass parameter of the left handed squarks MSU parameter mass parameter of the right handed sup like squarks MSD parameter mass parameter of the right handed sdown like squarks
8. system is fixed by the relation 2 1 2 c i gt tanh s 3 ln 1 10 Pp where the second term in the argument of tanh originates from the boost from the hadronic center of mass system which is the laboratory frame to the partonic one in which the partonic subprocess is calculated This leads to oc _ m 1 14 1 11 Oy 7 cosh y In 2 ur For processes with three or more particles in the final state the formula is very similar Additional phase space integrals appear for the further particles but otherwise eq 1 9 stays unchanged In the following equation the differential cross section for a 2 3 process is given do A Oc N 1 12 dy ws al ons f di Jas en Ta k2dh Oy The parametrization of the three particle phase space is described in appendix A 2 1 3 3 Transverse Momentum The last implemented differential hadronic cross section is the one with respect to the transverse momentum pr ypz p of one of the final state particles For 2 2 processes it is defined as do dL do dc 1 1 dpr ow ae deg Opr ee with 065 1 1 14 Opr Boz l pa P which follows from 5 Pr q 1 5 1 15 Here two possible solutions arise because of the sign ambiguity when taking the square root In principle both solutions have to be taken into account and added 6 Chapter 1 Hadronic Cross Sections up unless they are excluded by other constraints as shown below The lower
9. two outgoing particles for those with three final particles it denotes the energy k of the fifth particle which is the third final state particle The menu items 8 12 14 and 15 which refer to the fifth particle are ignored for 2 2 processes and cannot be changed The settings of the parameters are possible in the same way as already de scribed in the previous item Scale parameters This menu sets the renormalization and factorization scale of the process in the same way as described above A negative number for the renormalization scale has a special meaning Then the sum of the masses of the final state particles is taken multiplied with the absolute value of the setting and this number is taken as the renormalization scale Additionally it can be chosen that both renormalization and factorization scale are always set to the same value Show ModelDigest FormCalc Finally this choice invokes FormCalc s ModelDigest function which takes the pa rameters as an input and calculates the physical masses of the particles Thereby it applies lower bounds on the masses established by experiment and refuses the calculation if these bounds are violated The calculated cross section will also be zero in that case The FormCalc manual contains a more detailed explana tion of this function There it is also described how the check for the violation of experimental bounds can be switched off by flipping a switch in FormCalc s process h
10. DELTAR34CUT AR cut on the distance between particles 3 and 4 DELTAR35CUT AR cut on the distance between particles 3 and 5 DELTAR45CUT ARY cut on the distance between particles 4 and 5 RENSCALE HR renormalization scale FACTSCALE ur factorization scale MhO Mpo mass of the lighter CP even Higgs boson MHO MHo mass of the heavier CP even Higgs boson MHpm Ma mass of the charged Higgs boson MCha 1 Ma mass of the lighter chargino MCha 2 Mya mass of the heavier chargino MNeu 1 MAS mass of the lightest neutralino MNeu 2 Myo mass of the second lightest neutralino MNeu 3 Mas mass of the second heaviest neutralino MNeu 4 Myo mass of the heaviest neutralino MG1 Mg mass of the gluino MSn 1 Mg mass of the electron sneutrino only relevant for the computation of hadronic cross sections 5only relevant for the computation of partonic cross sections and differential hadronic cross section with respect to invariant mass Sonly for 2 2 processes Tonly for 2 3 processes 2 4 Allowed variable names for outputstring 25 Name Parameter description MSn 2 Mo mass of the muon sneutrino MSn 3 Ma mass of the tau sneutrino MS1 1 Ma mass of the lighter selectron MS1 2 Mi mass of the lighter smuon MS1 3 MA mass of the lighter stau MSL 1 M s mass of the heavier selectron MSL 2 a mass of the heavier smuon MSL 3 nz mass of the heavier stau MSu 1 mass o
11. HadCalc manual Michael Rauch January 2006 Contents 1 Hadronic Cross Sections 1 1 Parton Model 1 2 Integrated Hadronic Cross Sections 2222 1 3 Differential Hadronic Cross 1 3 1 Invariant Mass 1 3 2 Rapidity Sections 004 133 Transverse Momentum 0 004 14 Cuts 15 HadCale 2 Manual of the HadCalc Program 2 1 Prerequisites and Compilation 2 1 1 Prerequisites 2 1 2 Configuration and Compilation 2 2 Running the program 2 2 1 Physics parameters 2 2 2 PDF parameters 2 2 3 Integration parameters 2 2 4 Amplitude switches 225 Input Output options wos aa regel eu 2 2 6 Amplitude calculation 2 ss cn ta 2 3 Allowed tokens in input files 0 2 4 Allowed variable names for outputstring 22 2222 00 A Phase space parametrization A 1 Two particle phase space A 2 Three particle phase space Bibliography Oon oaoam m e E 27 27 28 30 Chapter 1 Hadronic Cross Sections The cross sections which are obtained by applying the Feynman rules contain amongst other particles quarks and gluons The leading interaction between these particles is the strong interaction which is described by quantum chromo dynamics QCD This theory possesses two characteristic properties asymptotic freedom 1 and confinement Asymptotic freed
12. al integrals If multi dimensional integrals are attempted to be computed VEGAS is used as a fallback In contrast the DCUHRE and DIVONNE algorithms can not handle one dimensional integrals There the GAUSSKR algorithm is used instead In both cases a warning is printed on the screen All integration routines share these two variables e relative error the desired relative error e absolute error the desired absolute error Additionally the following variables are available for one or more of the routines Which ones these are is denoted in square brackets after the entry e maximum of points the maximum number of function evaluations used GAUSSAD DCUHRE VEGAS SUAVE DIVONNE e of points for starting the initial number of points per iteration VEGAS e increase in of points the number of points the previous number is incre mented for the next iteration VEGAS e of points for subdivision the number of points used to sample a subdi vision SUAVE e flatness for splits the type of norm used to compute the fluctuation of a sample SUAVE e of passes the number of passes after which the partitioning phase ter minates DIVONNE 2 2 Running the program 17 e key 1 determination of sampling in the partitioning phase DIVONNE e key 2 determination of sampling in the final integration phase DIVONNE e key 3 sets the strategy for the refinement phase DIVONNE e maximum y for subregion the maximum x val
13. al on which appears in the parton luminosity given in eq 1 2 In total the z interval divides into five different regions which will be labeled by roman numbers First the two cases where both cuts cannot be fulfilled simultaneously are considered because the lower limit of one cut lies above the upper limit of the other one i l r Region I 9 lt o uy re e ap SA 1 23 min max Yout 1 u zn Region V cj gt on gt x gt yTe EIv 1 24 These two regions are excluded and the cross section vanishes there For specifying the other regions first two special cases are considered where the lower limits on cg and the upper limits respectively coincide For these cases 8 Chapter 1 Hadronic Cross Sections the according value of x is determined 1 m n 5 l r max Yeut i Gr aire te Tp Tmax 1 26 Gg a gt eye Oy Opr Using these two definitions the other intermediate regions can be specified The ranges for cg which are deduced from these following regions specify the allowed area where the cuts are fulfilled and therefore the cross section does not vanish The next two regions handle the cases where the limits on cg from rapidity and transverse momentum overlap and one limit is given by the rapidity cut and the other one by the transverse momentum cut Region IT Co El Sot se SS Gr a mola ine by der by dor 1 27 Region IV Ze hoe lt lt gt Mira Tmar lt l
14. allowed interval for cz sl ee Ca 1 36 with 2 1 2 comin 114 tanh yot ln 1 37 Dy a2 2 7 P 2 1 2 x 14 Z3 tanh l yar im 1 38 0y Fi 2 T Again this leads to a corresponding change in the limits of the x integration which are given by gt M lt a lt min yree ET 1 1 39 1 7 1 r NR max T y Te Yen with 1 40 This again corresponds to eqs 1 32 and 1 22 where instead of the cut on the transverse momentum Preu its fixed value pr which is an argument to the differential cross section is taken 10 Chapter 1 Hadronic Cross Sections 1 5 HadCalc For the numerical evaluation of the cross sections presented in the following chap ters a program called HadCalc was developed to facilitate this task It is based on the established program packages FeynArts 8 and FormCalc 9 10 which are used to generate the partonic cross sections The main task of HadCalc then consists of the convolution with the PDFs that are taken from the PDFlib 11 or LHAPDF 12 library packages that include PDF fits from various groups With this program it is possible to calculate both totally integrated and dif ferential hadronic cross sections of processes with up to three particles in the final state The latter ones can be differential with respect to the partonic center of mass energy or the rapidity or the transverse momentum of one of the outgoing particles Several cuts can be appli
15. and if this fails probes the existence of PDFlib After having run configure a call to make compiles the program When it successfully finishes a binary called HadCalc has been created in the current path 14 Chapter 2 Manual of the HadCalc Program 2 2 Running the program The program is simply started by running HadCalc It will then present a menu which allows one to tune various settings and start the calculation of cross sections The following subsections describe the possible settings in detail An item is chosen by typing the number shown in brackets before the item and pressing Enter In every menu 0 exits the submenu or for the top level menu quits the program Invalid input is ignored and an error message is written on the screen 2 2 1 Physics parameters This submenu sets the parameters of the MSSM and related things and is divided in three further submenus MSSM parameters Here all values which correspond to parameters of the MSSM are set First let us look at menu item 16 This decides whether the program should use a common mass Msysy in the sfermion sector or if individual values for the left handed squarks and sleptons and the right handed sups sdowns and selectrons are allowed Depending on this flag either the MS variables cannot be set because they are fixed at Msysy or Msusy itself cannot be set because it is irrevelant and not used in the computation When choosing a common SUSY mas
16. anti quark d 2 up anti quark u 3 strange anti quark 5 4 charm anti quark 5 bottom anti quark b 6 top anti quark t Table 2 1 PDG flavor codes 2 1 2 Configuration and Compilation First the partonic subprocess must be generated and prepared by following the instructions in the FormCalc4 manual Especially the definitions in process h have to be updated correctly as HadCalc relies on those It is not necessary to fill in correct MSSM parameters or tune integration parameters however Then the distribution file HadCalc 0 5 tar gz should be unpacked As next step change into its subdirectory and run configure from there The following configure options are mandatory with partonprocess DIR This is the location of the FormCalc generated partonic output with processtype mn By this option the processtype is fixed specified by the number of incoming par ticles m and the number of outgoing par ticles n Note that m and n form a sin gle number i e for a 2 2 process one would write with processtype 22 Currently 2 gt 1 2 2 and 2 3 is implemented and can be entered here with partonl i The type of the first parton is specified by i given as the PDG flavor code 14 see table 2 1 with parton2 i Similarly this is the PDG flavor code for the second parton 2 1 Prerequisites and Compilation 13 Additionally the following options are recognized by configure and enable optional features ena
17. ble antiproton1 enable antiproton2 with condor DIR with feynhiggs DIR with looptools DIR with lhapdf DIR with pdflib DIR Only one of the last two options can be given on the command line If neither with lhapdf nor with pdflib was given configure first tries to find LHAPDF Hadron 1 is an anti proton instead of a proton Hadron 2 is an anti proton instead of a proton Link the final code with the Condor workload management system libraries If the binary is not in the standard search path of your shell its location can be specified with the optional DIR argu ment Link the final code with the Feyn Higgs library This is mandatory if the FormCalc option to compute the MSSM Higgs sector via FeynHiggs is chosen The optional DIR specifies the location of the FeynHiggs library libFH a if it is not in the standard search path of the compiler If LoopTools is not in the standard search path of the compiler its location can be specified here Use LHAPDF for the parton distribution functions If the LHAPDF library is not in the standard search path its location can be given by the optional DIR argu ment The PDF data is assumed to be found at the same place Use PDFlib for the parton distribution functions If the PDFlib library is not in the standard search path and the CERN lib environment variables CERN and CERN_LEVEL are not set the DIR argument designates where it can be found
18. bprocess have momentum xP with x 0 1 As normally the mass of the hadrons is small compared to their kinetic energy one can assume P 0 The interaction of an electron and a hadron or of two hadrons among each 2 Chapter 1 Hadronic Cross Sections other can be split into two parts Because of Lorentz contraction and time dilation the interaction time of the two incoming particles in the laboratory frame is very short Therefore effectively a static hadron is seen For the hard scattering process interactions between partons of the same hadron need not be considered Also the process of hadronization after the interaction happens on time scales which are much larger than the interaction itself From this the theorem of factorization 5 follows immediately It states that all diagrammatic contributions to the structure functions can be separated into a product of two functions C and f which depend on two mass scales ur and up Hr is the renormalization scale up is the so called factorization scale and sepa rates the long distance from the short distance effects Slightly simplifying one can say that every parton propagator which is off shell by up or more contributes to C while those which are below this value contribute to f 1 2 Integrated Hadronic Cross Sections The hard scattering process C therefore can be calculated in perturbation theory by Feynman rules using partons as incoming particles It is independent of long distance e
19. define the beam axis and carry a center of mass energy of ys For each final state particle an integral over its three momentum k occurs in the calculation of integrated cross sections The energy k of the particle is fixed by the on shell condition k 4 k m where m denotes the mass of the particle Four of these integrals are eliminated by global energy momentum conservation In the following sections the parametrizations of the two and three particle phase space are shown A 1 Two particle phase space With two particles in the final state labeled by the subscripts 3 and 4 in the following the phase space integral can be written in terms of two angles They are the azimuth angle and the polar angle 0 with respect to the beam axis Because of rotational invariance around the beam axis the integration over amp is trivial and amounts to a factor of 27 So the integral over the two particle phase space is given by Jar fa des LE A 1 s m3 mi 2m3s 2m3s 2mzmj 4s denotes the squared absolute value of the three momentum of the final state particles m3 and m4 are their respective masses and ys specifies the center of mass energy of the incoming particles where ka Ku A 2 27 28 Appendix A Phase space parametrization Figure A 1 Graphical representation of the variables used in the parametrization of the 2 3 phase space The figure is taken from ref 16 A 2 Three
20. ection to a frame with a velocity P the rapidity transforms as y y artanh P Thus the shape of the rapidity distribution T stays unchanged More generally the sum of two rapidities when the momenta point into the same direction is given by the rapidity of the sum of the momenta added via the formula for the relativistic pitp2 1 p1p2 a slightly different measure the pseudo rapidity 7 is used It is derived from the standard rapidity by taking the limit of a vanishing mass of the particle and is defined as addition of velocities y p y pe y In experimental analyses often 1 1 co n lt In ae 1 7 In the HadCalc program both normal rapidity and pseudo rapidity are imple mented As conversion between both variables can be performed by the simple transformation h m2 artan 1 tanh 1 8 in the following only the shorter expressions for the standard rapidity are given The ones for pseudo rapidity can then be deduced from them Using the above mentioned definition of the rapidity the differential hadronic cross section with respect to the rapidity for 2 2 processes then reads do f dL d 9 a ee di 1 dy dr dc Oy 1 TO 1 3 Differential Hadronic Cross Sections 5 The momenta and masses given in the formulae always refer to the particle for which the rapidity distribution is calculated The angle cg between the particle and the beam axis in the partonic center of mass
21. ed to the phase space HadCalc operates either in batch mode where the parameters are read from a file and the cross sections are written back to disk allowing for easy post processing with e g a tool that generates plots It can also be used in interactive mode where in and output are done via keyboard and screen and which allows the user for example to tune the parameters most easily Chapter 2 Manual of the HadCalc Program For the calculation of hadronic cross sections a computer code called HadCalc was written see chapter 1 5 In this appendix the manual of the program is presented 2 1 Prerequisites and Compilation 2 1 1 Prerequisites The following programs are required for compiling and running HadCalc and must be installed e a Fortran compiler compliant with the Fortran77 standard a C compiler conforming to ANSI C GNU make FormCalc 4 9 one of the two following packages that include sets of parton distribution functions from various groups PDFLIB CERN Computer Program Library entry W5051 11 or LHAPDF 12 Additionally support for the following two programs is integrated into HadCalc e FeynHiggs 2 1beta or newer 13 e Condor workload management system for compute intensive jobs 11 12 Chapter 2 Manual of the HadCalc Program PDG flavor code Particle 0 gluon g 1 down quark d 2 up quark u 3 strange quark s 4 charm quark c 5 bottom quark b 6 top quark t 1 down
22. f the lighter sup MSu 2 Ma mass of the lighter scharm MSu 3 mM mass of the lighter stop MSU 1 M z mass of the heavier sup MSU 2 CS mass of the heavier scharm MSU 3 my mass of the heavier stop MSd 1 M mass of the lighter sdown MSd 2 AN mass of the lighter sstrange MSd 3 Ma mass of the lighter sbottom MSD 1 M mass of the heavier sdown MSD 2 ile mass of the heavier sstrange MSD 3 7M mass of the heavier sbottom TREE o tree level cross section LOOP iat one loop cross section TREEERR 0 00 integration error of the tree level cross section LOOPERR 01 integration error of the one loop cross section TREEPROB x o 00 probability of the integration error of the tree level cross section LOOPPROB x 0 0 probability of the integration error of the one loop cross section NREGIONS number of regions used for integration NEVAL number of function evaluations used for integration FAIL a non zero value indicates that the desired accuracy could not be reached Sonly relevant for some integration routines Appendix A Phase space parametrization In this appendix the parametrization of the phase space for 2 2 and 2 gt 3 processes as it was used for the calculations of this thesis is presented It is the same parametrization which is also used in FormCalc 9 10 16 The parametrization is performed in the center of mass system of the two incoming particles which
23. ffects and especially from the type of the colliding hadron The parton distribution function PDF f x wr contains the long distance effects It is independent of the underlying scattering process but depends on ur and the type of hadron h It is normalized such that it can also be interpreted as a probability density namely the probability of finding the parton 7 in the hadron h with a momentum xP Its behavior as a function of the parameters is determined by the Altarelli Parisi integro differential equations 6 Its numerical value however cannot be calculated a priori from the theory At a single reference point it must be determined by experiments Therewith one obtains the expression 2 e Opp fintx gt dr Imn fin TS as urR 1 1 may 2 for an integrated hadronic cross section with the parton luminosity me dr gt 14 6mn Im x ur fa p E nr Info Ur fm p E nr 42 Here VS denotes the hadronic center of mass energy i e the one of the two colliding protons and Gmn fin the partonic cross sections of the subprocesses where the two incoming partons m and n produce some final state labeled fin The sum includes all possible parton combinations m and n where the order of 1 3 Differential Hadronic Cross Sections 3 appearance is not taken into account The integration variable 7 relates the par tonic and hadronic center of mass energies with each other More specifically YT can be
24. iables that are either invariant under Lorentz transformations or at least have very simple trans formation properties In this thesis three differential hadronic cross sections are presented which are also implemented in the HadCalc program that is described below in section 1 5 They are cross sections differential with respect to the in variant mass of the final state particles the rapidity of one final state particle and thirdly the transverse momentum 1 3 1 Invariant Mass The first differential hadronic cross section is the one with respect to the invariant mass of the final state particles The invariant mass of a process is equivalent to the partonic center of mass energy v VTS of the process or in other words 4 Chapter 1 Hadronic Cross Sections the sum of the final state momenta of the outgoing particles The differential cross section takes the form ds Vs dE A O Dra 1 5 m n TS where fin again labeles a general final state 1 3 2 Rapidity The rapidity y of a particle is defined as y artanh E no 1 6 p 2 PP where p p cg denotes the fraction of the particle s three momentum p that goes in the direction of the beam axis labeled z The mass of the particle will later be referred to as m Using the rapidity instead of directly taking the angle 0 between the particle and the beam axis possesses some advantages because the rapidity of a particle has a few useful properties Under a boost in the z dir
25. interpreted as the part of the hadronic center of mass energy which takes part in the partonic subprocess as the partonic center of mass energy is given by v VrS The lower limit of the integral 7 is determined by the kinematic configuration YroS is the minimal energy which is necessary to produce the final state fin and therefore denotes the production threshold The formula given above is valid for processes with two or more particles in the final state For hadronic cross sections it is also possible to calculate integrated cross sections for 2 1 processes One first obtains for the partonic cross section of the process mn f T d mnf 1M gi mn gt f a po p p 1 3 ETNEA N Again m and n specify the incoming partons f denotes the outgoing particle my its mass and p the energy of the respective particle i Pm indicates the three momentum of particle m in the partonic center of mass system and M y is the matrix element When convoluting with the parton distribution functions the single remain ing function in the partonic cross sections solves the 7 integral in eq 1 1 analytically Thus one obtains for the integrated hadronic cross section dL Opp f y dr m n T M mn gt f aa 1 4 T u 2M S Pm 1 3 Differential Hadronic Cross Sections Additionally one can define hadronic cross sections that are differential in one or more parameters For these parameters it is useful to take var
26. limit of the 7 integral 7 must be adjusted such that sg is always inside its co domain 0 1 2 ym Pht ond 23 To 3 l 1 16 f and f denoting the two final state particles For 2 3 processes the extension to include the third final state particle is straightforward The lower limit for 7 in these processes is 2 y mi T Pr mn T mp 15 1 1 TO S 7 when the cross section is differential in the particle fi Therefore the expression for the differential cross section reads Oc k5 3 f i a 1 1 T f ars ak ow w ren Opr en 1 4 Cuts In order to improve the ratio of the signal process cross section to that of the background processes it can be useful to place appropriate cuts on the final state particles Also experimental techniques used in the reconstruction of events like jet clustering algorithms can mandate the use of cuts in theoretical predictions so that the behavior of these techniques is emulated there In the HadCale program cuts on three different properties of the final state particles are implemented 7 The first two are cuts on the rapidity and the transverse momentum of a particle The definition of these two variables was already presented in the previous section The third one is a mutual property of two particles the jet separation AR which is defined as Ay denotes the rapidity difference between the two particles and j and Ad the difference in the azimuthal
27. mber of points for starting VINCREASE integer increase in number of points SNNEW integer number of points for subdivision 2 4 Allowed variable names for outputstring 23 token type description SFLATNESS integer flatness number for splits MAXDPASS integer number of passes in partitioning phase DKEY1 integer Divonne key 1 DKEY2 integer Divonne key 2 DKEY3 integer Divonne key 3 DBORDER double border of the integration region MAXDCHISQ double maximum x for subregion MINDDEV double minimum deviation for split VERBOSITY integer verbosity of integration output PDFTYPE double type of the PDF PDFlib PDFGROUP double group of the PDF PDFlib PDFSET double set of the PDF PDFlib PDFPATH string path where the PDF files are LHAPDF PDFNAME string name of the PDF LHAPDF ScreenQutput boolean print output on the screen OUTPUTFILE string print output into file OUTPUTSTRING string parameters to print in output see section 2 4 2 4 Allowed variable names for outputstring The following list shows all variable names that may appear in outputstring The individual entries are separated from each other by spaces Variables with the dimension of a mass are output in GeV Note that these names are case sensitive Name Parameter description MAO IMA mass of the CP odd Higgs boson TB tan 8 ratio of the Higgs vacuum expectation values MUE pl u parameter in the Higgs sector MSusy
28. om describes the behavior of the theory at small distances In this region the interaction is weak and the coupling constant gets smaller with decreasing distance or equivalently with rising energy At large distances confinement appears because the interaction becomes strong and binds the particles tightly together If the space between them becomes even larger it is energetically favorable to form new quark anti quark pairs One consequence of this behavior is that quarks and gluons cannot be observed as free particles but only as constituents of hadrons i e mesons which are quark anti quark pairs and baryons which are states of three quarks or three anti quarks An example for these hadrons are protons which are the colliding particles at the LHC To make theoretical predictions it is necessary to relate the interactions at the parton level to the interactions at the hadron level 2 The basis for doing this is the parton model 3 which will be described in the next section 1 1 Parton Model The parton model describes the inner structure of hadrons in hard collisions It starts from the assumption that every observable hadron consists of constituents the so called partons which can be identified as quarks and gluons Experimental evidence for this assumption comes from the observation of scaling 4 in deep inelastic electron proton scattering If the hadron carries some momentum P the partons which take part in the partonic su
29. particle 5 DELTAR34CUT parameter cut on the distance between parti cles 3 and 4 DELTAR35CUT parameter cut on the distance between parti cles 3 and 5 DELTAR45CUT parameter cut on the distance between parti cles 4 and 5 RENSCALE parameter renormalization scale FACTSCALE parameter factorization scale CommonSUSYMassScale boolean choose a common SUSY mass scale NoCommonSUSYMassScale boolean do not choose a common SUSY mass scale CommonRenFactScale boolean choose a common remormalization and factorization scale NoCommonRenFactScale boolean do not choose a common remormal ization and factorization scale AMPLITUDE preselected choose which amplitude s to calcu late Ptrans3 gt cut boolean require the transverse momentum of particle 3 to be larger than the cut Ptrans3 lt cut boolean require the transverse momentum of particle 3 to be smaller than the cut Ptrans3nocut boolean disable cut on the transverse mo mentum of particle 3 Rapid3 gt cut boolean require the rapidity of particle 3 to be larger than the cut Rapid3 lt cut boolean require the rapidity of particle 3 to be smaller than the cut Rapid3nocut boolean disable cut on the rapidity of parti cle 3 Ptrans4 gt cut boolean require the transverse momentum of particle 4 to be larger than the cut Chapter 2 Manual of the HadCalc Program token type description Ptrans4 lt cut boolean require the transverse momentum of particle 4
30. particle phase space For the three particle phase space where the outgoing particles are labeled by the indices 3 4 and 5 five independent integration variables remain after global energy momentum conservation has been applied They are the energies ks and k9 the azimuth angle and the polar angle 9 of the fifth particle with respect to the beam axis and the angle 7 which rotates particle 3 out of the plane defined by particle 5 and the beam axis A graphical representation of the angles is given in Fig A 1 The four momenta of the outgoing particles have the following explicit form k3 RD ks e3 k4 a W i ks Be ks ks ksl amp 5 A 3 with CoC Se SoC Se 63 SSE Es 0 A 4 Cole SoCHSe Co The angle 6 which is also plotted in the figure is defined over Vs k K9 mi kal ks 2 ka ks Cg A 2 Three particle phase space 29 m again denotes the mass of the respective particle and ys is the center of mass energy of the initial state particles Due to axial symmetry the trivial integration over can be carried out immediately and yields a factor of 27 Then the parametrization of the three particle phase space takes the following form Keynes A Koyma i 2pi 1 dl a ans dk f do f dj A 6 where the integration limits are given by kg Y a y A 7 and 1 gt gua lo r mimo E lisy r ma r m2 A8 using o ys
31. rules above a warning message is printed and their content is ignored Furthermore some integration routines offer the possibility to write interme diate results or progress report to the screen This is turned on with Verbose integration output For hadronic cross sections this also enables writing PDFlib statistics on the screen at the end of the calculation Finally one can choose whether the calculation results will be written to the screen or into a file In the latter case the variable outputstring describes which elements should be written to the output file The form of this variable together with the valid tokens is described in section 2 4 The output file format starts with a quoted header with a file identification and the content of outputstring Then each on a line by itself for every scanned parameter point the values defined in outputstring are written separated from each other by a space 2 2 6 Amplitude calculation This submenu finally allows one to choose the cross section one wants to compute and does the calculation During the following integration the process may be interrupted with Ctrl C after which it aborts the current calculation and jumps back into the main menu Due to restrictions imposed by Condor this feature is not available if HadCalc was configured with the option with condor Here pressing Ctrl C aborts the whole HadCalc program 2 3 Allowed tokens in input files The following list
32. s are ignored Lines starting with the character after optional white space are com ment lines and ignored The first token which is separated by white space from the rest of the line is extracted This token has to be a token from the list of valid tokens in section 2 3 If the token type is boolean its associated parameter is set If the token type is integer an attempt to read an integer value is made and if it succeeds this is assigned to the associated parameter If the token type is double an attempt to read a double precision floating point number is made and if it succeeds this is assigned to the associated parameter If the token type is string the second token is assigned to the associated parameter If the token type is parameter the following actions happen An attempt to read four comma separated double precision floating point numbers is made If this attempt succeeds the four numbers are assigned to lower bound upper bound log and number of intermediate intervals of the param eter log means linear increase if this variable is zero and exponential one otherwise If this does not succeed an attempt to read a single double precision floating point number is made If this succeeds this number is the constant value of the parameter If this also does not succeed the line is flagged as not parsable 2 3 Allowed tokens in input files 19 e For lines not parsable by the
33. s scale the settings in the MS variables are retained and restored when deselecting this option All other parameter settings can be in two states They can either have a fixed setting then this value is used for all calculations Or their value can be running In this case a lower and upper bound and the number of intermediate intervals must be chosen Then the computation of the cross section is done intermediate intervals 1 times with the value of the parameter increasing from lower bound to upper bound The distance between two values is equal for the setting linear and exponentially increasing for the setting logarithmic i e the values are closer at the lower bound and they have equal distance again when plotting them on a logarithmic scale A behavior vice versa with values closer at the upper bound can be easily achieved by exchanging lower and upper bound If more than one parameter is chosen to be running the iteration loops are nested with the first parameter varying fastest Despite its name the lower bound can be mathematically larger than the upper bound then the value of the parameter is decreasing 2 2 Running the program 15 Kinematic parameters In this menu all parameters are set which are related to kinematic variables of the process The underlying parameter of items 3 and 4 depends on the type of process For processes with two particles in the final state this is the angle 0 between the
34. t Ty Opr Oy Opr Oy 1 28 Finally the definition of the last region is the case whether one cut gives a range on cg that completely lies inside the other one Depending on which cut this is the limits on x are different Region III a Che mn lt e lt RZ SS the lt T lt Emax Opr Oy 0 Oy Opr 1 29 Region III b ASA LASA e Cmax LE lt Emin Oy Opr 0 Opp Oy 1 30 In addition to those regions the original constraint for x for a hadronic cross section without cuts applies rl 1 31 Combining the result of all regions one can see that no holes in the integration over x or c appear and the final borders of the integration routine can be simplified to max 7 11 lt x lt min zy 1 1 32 and max cg Co E mine Co 1 33 1 4 Cuts 9 For a cross section which is differential with respect to the rapidity of a final state particle the cut on the transverse momentum yields a restriction on cg in the same way as in eq 1 33 Ge OSG 1 34 The constraint on x must then be adjusted such that cj is always inside this allowed interval yielding 1 1 max r Te Y o lt x lt min yre T 2 A 1 1 35 r p which corresponds to eq 1 32 where the rapidity cut yeu is replaced by its value y given as an argument to the cross section Similarly for cross sections that are differential in the transverse momentum of a final state particle a cut on the rapidity puts a further constraint on the
35. to be smaller than the cut Ptrans4nocut boolean disable cut on the transverse mo mentum of particle 4 Rapid4 gt cut boolean require the rapidity of particle 4 to be larger than the cut Rapid4 lt cut boolean require the rapidity of particle 4 to be smaller than the cut Rapid4nocut boolean disable cut on the rapidity of parti cle 4 DeltaR34 gt cut boolean require the jet separation between particles 3 and 4 to be larger than the cut DeltaR34 lt cut boolean require the jet separation between particles 3 and 4 to be smaller than the cut DeltaR34nocut boolean disable the cut on the jet separation between particles 3 and 4 DeltaR35 gt cut boolean require the jet separation between particles 3 and 5 to be larger than the cut DeltaR35 lt cut boolean require the jet separation between particles 3 and 5 to be smaller than the cut DeltaR35nocut boolean disable the cut on the jet separation between particles 3 and 5 DeltaR45 gt cut boolean require the jet separation between particles 4 and 5 to be larger than the cut DeltaR45 lt cut boolean require the jet separation between particles 4 and 5 to be smaller than the cut DeltaR45nocut boolean disable the cut on the jet separation between particles 4 and 5 INTMETHOD preselected choose the integration routine EPSABS double absolute integration error EPSREL double relative integration error MAXPTS integer maximum number of points VSTARTPTS integer nu
36. ue a single subregion is allowed to have in the final integration phase DIVONNE e minimum deviation for split a bound which determines whether it is worth while to further examine a region that failed the x test DIVONNE 2 2 4 Amplitude switches This submenu sets the type of diagrams used for the computation and how the cuts should be applied The value of the cuts is set in the parameter section and was already described there The first entry decides whether the tree level and the one loop result shall be computed in one go or only one of them Possible choices are Tree only Tree Loop and Loop only Which way is better depends on the concrete circumstances and features of the problem Computing both at the same time might save computation time but the integration routine has to optimize its choices for both at the same time which might lead to sub optimal performance On the other hand it is not too likely that there are problematic regions in the tree level cross section which are no longer present in the one loop computation so normally this procedure gives satisfactory results If only one cross section is computed the value of the other one is set to zero The remaining entries decide if and how the cuts on rapidity transverse mo mentum and jet separation should be applied It is either possible to have the particle or a pair of particles in case of the jet separation fulfill a cut violate it or ignore the c
37. ut altogether Since HadCalc relies on FormCalc for the partonic process and implementation details for the cuts for particle three in the 2 2 case and particle five in the 2 3 case it cannot be chosen that the rapidity and transverse momentum cut is violated but they always have to be fulfilled They can however be switched off by setting the relevant entry in the parameters section to zero 2 2 5 Input Output options This submenu allows one to read in a set of parameters from a file and specify where and how to write the calculation output To read in a set of parameters a parameter specification must have been written into a file and this filename then has to be entered here All possible variables which can be set in such a file are given in section 2 3 There are three 18 Chapter 2 Manual of the HadCalc Program basic types of variables Those which specify a parameter can either take four comma separated values that are the lower and upper bound the behavior with respect to increments i e linear or logarithmic and the number of intermediate intervals or a single number denoting its fixed value The ones of type boolean turn on a certain switch and take no arguments All remaining ones take a single argument and the variable is assigned to this parameter In the following also a formal definition of the parsing rules is given The file is read line by line White space at the beginning of a line is ignored Empty line
Download Pdf Manuals
Related Search