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FormCalc 8 User's Guide

1. Normalized gt True Ej Ej 1 InvSimplify controls whether CalcFeynAmp should try to simplify combinations of invari ants as much as possible Antisymmetrize determines whether Dirac chains are antisymmetrized Note that this does not affect Weyl chains i e has an effect only if FermionChains gt Chiral or VA is chosen in CalcFeynAmp Take care that antisymmetrized and non antisymmetrized Dirac chains have an indistinguishable notation Needless to say one should not change the antisymmetriza tion flag between CalcFeynAmp and HelicityME 4 3 Clearing Combining Selecting CalcFeynAmp needs no declarations of the kinematics of the underlying process it uses the information FeynArts hands down This is convenient but it also requires certain care on the side of the user because of the abbreviations FormCalc automatically introduces in the result see Sect 4 5 Owing to the presence of momenta and polarization vectors abbrevi ations introduced for different processes will in general have different values even if they have the same analytical form To ensure that processes with different kinematics cannot be mixed accidentally CalcFeynAmp refuses to calculate amplitudes belonging to a process whose kinematics differ from those of the last calculation unless ClearProcess is invoked in between ClearProcess removes internal definitions before calculating a new process The Combine function combines amplitud
2. computation is very similar to that of an RC MassRC f MassRC f f2 FieldRC f FieldRC f f2 TadpoleRC f WidthRC f the mass RC My the mass RC Ms f the field RC 5Z the field RC 5Zf f the tadpole RC T the width Ps The explicit formulas for computing the RCs are given in the following For compact ness of notation f refers to a fermion and B to a boson field and 0 m is short for 5 4 Renormalization Constants 71 OE K2 Ok 9 _ 2 5My Re Li my Ly SM Re Eza m5 1 5Mp B 5 Re Es m3 Epa m XVE OF me m m Z Re JF ir pa 5Zp Red pp m3 Leg OL Mp Mp Mf 2 E mp mp m 2 Lap y 7 Re hhl far Mas mp OZB B z Re Ep 8 m5 Mi Mp Lef M5 Mp M4 ms ms Tz Eg m2 7 Es ss Tf Im Esp mp 1 1 P PB ao Esg mi 7 where LP ma B m a E m FBE m BLP Mm a LR m 5 4 2 Calculation of renormalization constants For the actual calculation of the RCs FormCalc provides the two functions CalcRenConst and WriteRenConst Both functions search the expressions they receive as arguments for symbols which are RCs CalcRenConst returns the calculated RCs as a list of rules whereas WriteRenConst writes them to a program file CalcRenConst expr calculate the RCs appearing in expr and return the results as a list of rules WriteRenConst expr dir the same but write the results to a progr
3. option default value SortDen True whether to sort the denominators of the loop integrals PaVeReduce False whether to reduce tensor to scalar integrals SimplifyQ2 True whether to simplify q in the numerator OPP 100 the N in N point function above which OPP loop integrals are emitted OPPQSlash False whether to introduce fi also on q Gamma5Test False whether to substitute Ys gt ys5 1 Gamma5Test D 4 Gamma5ToEps False whether to substitute Ys gt F Euvoo YY YY in fermion traces NoExpand Ep sums containing any of the symbols in the list are not expanded NoBracket symbols not to be included in the bracketing in FORM MomRules E extra rules for transforming momenta PreFunction Identity a function applied to the amplitudes before any simplification PostFunction Identity a function applied to the amplitudes after all simplifications FileTag amp the middle part of the temporary FORM file s name RetainFile False whether to retain the temporary FORM input file EditCode False whether to display the FORM code in an editor before sending it to FORM CalcLevel is used to select the desired level in the calculation In general a diagram can have both Classes and Particles insertions The default value Automatic selects the deepest level available i e Particles level if available otherwise Classes level Dimension specifies the space time dimension in which to perform the calculation It can take the values D
4. expected if one uses HelicityME on results from process A after having computed am plitudes from process B with CalcFeynAmp Since this requires at least one invocation of ClearProcess though it is unlikely to happen accidentally 4 9 Colour Matrix Elements 31 option default value Dimension 4 the dimension to compute in MomElim taken from how to apply momentum conservation CalcFeynAmp DotExpand taken from whether to expand terms collected for CalcFeynAmp momentum elimination RetainFile False whether to retain the temporary FORM command file EditCode False whether to display the FORM code in an editor before sending it to FORM Dimension is used as in CalcFeynAmp Only the value 0 has the effect of actually computing in D dimensions however since for D and 4 the limit D 4 has already been taken in CalcFeynAmp The dimensional dependence of the result is expressed through Dminus4 and Dminus4Eps where the latter represents the Dminus4 arising from the contraction of Levi Civita tensors For testing and comparison the default equivalence Dninus4Eps Dminus4 can be unset The options MomElim DotExpand RetainFile and EditCode are used in the same way as for CalcFeynAmp see page 13 4 9 Colour Matrix Elements Diagrams involving quarks or gluons usually contain objects from the SU N algebra These are simplified by CalcFeynAmp using the Cvitanovic algorithm Cv76 in an extended version of the impl
5. gt ui j gt Hj takes legs i j off shell by substituting the rela tion p u for the true on shell relation p m This is unlike the CalcFeynAmp option OnShell gt False which takes all legs off shell by not using p m at all Finally the following two functions serve to add factors to particular diagrams MultiplyDiagrams f amps multiply the diagrams in amps with the factor returned by the function f TagDiagrams amp multiply each diagram in amp with the identifier Diagram n where n is the diagram s number MultiplyDiagrams f amp multiplies the diagrams in amp with factors depending on their contents The factor is determined by the function f which is applied to each diagram either as f amplitude for fully inserted diagrams or f generic amplitude insertion For example to add a QCD enhancement factor to all diagrams containing a quark mass the following function could be used as MultiplyDiagrams QCDf actor Lamps QCDfactor args__ QCDenh FreeQ args Mf 3 4 __ _QCDfactor 1 18 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS TagDiagrams is a special case of MultiplyDiagrams and multiplies each diagram with an identifier of the form Diagram n where n is the diagram s running number This provides is very simple mechanism to identify the origin of terms in the final amplitude 4 4 Ingredients of Feynman amplitudes The results of CalcFeynAmp contain the following symbol
6. model_sm F h initialization of the Standard Model parameters model_mssm F h initialization of the MSSM parameters model_thdm F h initialization of the two Higgs doublet model parameters lumi_parton F luminosity computation for a partonic process lumi_hadron F luminosity computation for a hadronic process lumi_photon F luminosity computation for a photonic process MtoN F the kinematics fora M N process MtoN h declarations for MtoN F configure a script to find out library locations and compiler flags makefile in the makefile minus the flags set by configure mktm a script to set up Mathematica interfacing code internal use data a script for extracting the actual data out of the log files pnuglot a script for plotting data files using gnuplot sfx a script for making a self extracting archive turnoff a script to turn on or off the evaluation of code modules submit a script to submit parallel jobs do a script to automate computations tools tools for the numerical evaluation ReadData tm a program for reading data files into Mathematica reorder c a utility to reorganize parameters in data files ScanGraphics m special Mathematica graphics functions for parameter scans bfunc m the explicit expressions for the one and two point functions btensor m the explicit tensor reduction for the one and two point functions in Mathematica 5 1 Generating code 41 5 1 Generating code In most cases the numerical evaluation of an amplitude directly i
7. 1 j is Kronecker s delta 5j IndexDiff 7 j is 1 dj Maplf f expr is equivalent to Map except that it does not modify the conditional parts if expr is an IndexIf BlockSplit var gt expr split expr into subexpressions with leaf count less than BlockSize FileSplit exprlist mod writemod writeall split exprlist into batches with leaf count less than FileSize call writemod to write a each module to file and finally writeall to generate a master subroutine which invokes the modules ToArray s take the symbol s apart into letters and digits e g X123 X 123 Renumber expr v U2 renumber all v nJ v2 n in expr MaxDims args find the maximum indices of all functions in args BlockSize the size of each block for BlockSplit FileSize the size of each file for FileSplit BlockSplit var gt expr tries to split the calculation of expr into subexpressions each of which has a leaf count less than BlockSize FileSplit exprlist mod writemod writeall splits exprlist into batches with leaf count less than FileSize If there is only one batch writemod batch mod is invoked to write it to file Otherwise writemod batch modN is invoked on each batch where modN is mod suffixed by a running number and in the end writeall mod res is called where res is the list of writemod return values The optional writeall function can be used e g to write out a master subroutine
8. 1 a set number beginning with which parameters and data are stored ParaHead Para the head under which parameters are stored DataHead Data the head for the data storage LogFile oa the log file to save screen output in Polarizations determines the polarizations of the external particles specified as in the stand alone version i e a string of characters for each external leg U unpolarized L longitudinal polarization T transversely polarized left handed polarization right handed polarization Serial gives the range of serial numbers for which to perform the calculation specified as serialfrom serialto serialstep The concept of serial numbers used to parallelize parameter scans is described in Sect 5 2 9 This option applies only to parameters scanned by do loops in the parameter statements Parameters read from Mathematica are unaffected by this option SetNumber specifies the set number beginning with which parameters and data are stored see next Section ParaHead gives the head under which parameters are stored i e parameters are retrievable from parahead setnumber see next Section DataHead gives the head under which data are stored i e data are retrievable from datahead setnumber see next Section LogFile specifies the log file to save screen output in An empty string indicates no output redirection i e the output will appear on screen 5 3 3 Return values Storage
9. Boo p2 mi m3 i a A O 1 Dow 001 a o O 1 Dooooi e O 1 By st O 1 m2 m2 m2 p p p Coool p p3 p m mB m3 MER PERPE A o Alternatively UV finiteness may be checked numerically by varying the parameters that regularize the infinity in the loop integrals u and A on which the final result must not depend This is of course limited to the achievable numerical precision 38 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS 4 14 Useful Functions DenCollect expr collects terms in expr whose denominators are identical up to a numerical constant DenCollect expr wrap additionally applies wrap to the collected numerators TagCollect expr tag f apply f to the part of expr tagged by tag TermCollect expr combines terms with common factors i e does ab ac d a b c d Pool expr same as TermCollect but different method Poollexpr wrap additionally applies wrap to the b c part DotSimplify f1 f2 expr simplify expr with f if free of any NoBracket items else with fo OnSize fas expr returns f expr if LeafCount expr lt n fo lexpr if LeafCount expr lt nz etc and faet expr if the expression is still larger ApplyUnitaritylexpr U d s simplify expr by exploiting the unitarity of the d dimensional matrix U The optional argument s specifies the simplification function to use internally default FullSimplify ExpandSums expr turns all pieces of expr multiplied
10. CW2 S 2 SW2 Abb1 Alfa2 CW2 S T U SW2 8 AbbSum29 Alfa2 CW2 S T U SW2 The first line stands for the tensor coefficient function C12 M34 My s Miy M3 Me which is multiplied with a linear combination of abbreviations like Abb1 or AbbSum28 with certain coefficients The coefficients of the abbreviations contain kinematical variables in this case the Mandelstam variables S T and U and parameters of the model here e g Alfa2 or MW2 This particular excerpt of code happens to be from a process without external fermions otherwise spinor chains abbreviated as Fn would appear too The abbreviations like Abb1 or AbbSum29 can drastically reduce the size of an amplitude particularly so because they are nested in three levels Consider AbbSum29 from the example above which is an abbreviation of about average length AbbSum29 Abb2 Abb22 Abb23 Abb3 Abb22 Pairi Pair3 Pair6 Pair3 Pair e 3 k 1 Without abbreviations the result would for each AbbSum29 contain Pair e 1 1 e 2 1 Pair e 3 x 11 Pairle 4 x 1 Pair e 1 e 2 Pair e 3 k 2 Pairle 4 x 1 Pair e 1 e 2 Pair e 3 x 1 1 Pairle 4 x 2 Pairfe i e 2 Pair e 3 1 x 21 Pair e 4 1 x 2 1 24 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS The size reduction effect can be quantified by comparing the LeafCount of the expressions in Mathematica The leaf count is a measure for the size of an expression more precisely it
11. CalcFeynAmp performs the following simplifications e indices are contracted as far as possible e fermion traces are evaluated e open fermion chains are simplified using the Dirac equation e colour structures are simplified using the SU N algebra e the tensor reduction is performed e local terms are added e the results are partially factored abbreviations are introduced The output of CreateFeynAmp can be fed directly into CalcFeynAmp Technically this means that the arguments of CalcFeynAmp must be FeynAmpList objects CalcFeynAmp is invoked as amps CreateFeynAmpl result CalcFeynAmp amps The results are returned in the form Amplin gt out r1 1 In D dimensions the divergent integrals are expanded in e 4 D 2 up to order e and the 1 poles are subtracted The E poles give rise to local terms when multiplied with D s from outside the integral e g from a g In 4 dimensions local terms are added depending on the contractions of indices of the tensor integrals according to the prescription of constrained differential renormalization dA98 4 1 CalcFeynAmp 9 The lists in and out in the head of Amp specify the the external particles to which the result belongs The presence of a particle s mass in the header does not imply that the amplitudes were calculated for on shell particles The actual result is split into parts r1 r2 such that index sums marked by SumOver alway
12. abbreviations 27 RegisterSubexpr 27 regularization method 10 reorder c 40 RetainFile 10 31 35 run F 40 s 19 s n 18 30 SA2 22 Samurai 12 SB2 22 SBA2 22 ScanGraphics m 40 selecting diagrams 17 SetupCodeDir 41 sfx 40 short hands 23 Sij 19 SimplifyQ2 10 size of the amplitude 24 SMSimplify 21 SortDen 10 spin 29 Spinor 20 21 INDEX spinor chain 11 28 squared matrix element 33 SquaredME 33 SU N objects 19 32 SubExpr 27 subexpressions registering 27 submit 40 SumLegs 35 SUNN 19 SUNT 19 32 SUSYTrigExpand 22 SUSYTrigReduce 22 SUSYTrigSimplify 22 SW2 21 T 19 TagCollect 38 TagDiagrams 17 TB2 22 tensor coefficients 12 tensor reduction 8 test y5 12 T 19 ToComponents 36 tools 40 transversality 15 Transverse 15 turnoff 40 U 19 UChaC 22 ultraviolet divergences 37 unpolarized case 29 unsquared 36 useful functions 38 US C 22 UV divergences 10 UVDivergentPart 37 UVSeries 37 INDEX VA 11 VChaC 22 VecSet 36 vector 36 Weyl 11 WeylChain 21 WriteRenConst 41 WriteSquaredME 41 xsection F h 40 ZNeuC 22 93
13. by process basis For example one can calculate the main process and the RCs in one program such that the options chosen for the former e g for InsertFields automatically apply to the latter The definitions of the RCs can be made anywhere before one of the functions that calculates them is invoked A particularly convenient location is in the model file to which the RCs belong This is the case for the model files that come with the current versions of FeynArts where the definitions of the RCs are implemented according to the on shell scheme of De93 The definition of an RC may make use of the following functions They may be used in the model file or anywhere else even if FormCalc is not loaded SelfEnergy li gt f m DSelfEnergy i gt f m TreeCoupling i gt f LVectorCoeff expr RVectorCoeff expr LScalarCoeff expr RScalarCoeff expr ReTilde expr ImTilde expr the self energy Ej k m the derivative dLj k dk Es the tree level contribution to i gt f without external spinors the coefficient of Kw in expr the coefficient of kw in expr the coefficient of w in expr the coefficient of w in expr takes the real part of loop integrals in expr takes the imaginary part of loop integrals in expr Most RCs fall into one of the following categories for which special functions exist to save typing and reduce errors The width is of course not an RC but is also listed here as its
14. compilers on all platforms and the possibility to link with existing code This first step of writing out a subroutine from the calculated amplitudes can be im plemented in a reasonably general way and is handled by FormCalc itself see Sect 5 1 2 What FormCalc generates is just the subroutine for computing the squared matrix el ement Of course this subroutine has to be provided with the proper kinematics the model parameters have to be initialized etc In other words a driver program is re quired to invoke the generated subroutine Unfortunately it is not really possible to 40 5 TOOLS FOR THE NUMERICAL EVALUATION write one single fully automated driver program that fits each and every purpose or rather such a program would be difficult to use beyond its designed scope The driver programs that come with FormCalc together are a powerful tool for comput ing cross sections but moreover they have expressly been written to give an example of how the generated code can be used Extending or modifying this code for a related purpose should be fairly straightforward The following files in the FormCalc distribution are used for the numerical evaluation drivers driver programs for running the generated code main F the main program command line processing xsection F the code for computing the cross section xsection h declarations for xsection F process h the process specification run F the parameters for a particular run
15. for them In calculations with non scalar external particles where such objects are ubiquitous code produced from the FormCalc output say in Fortran can be significantly shorter and faster than without the abbreviations FormCalc can work in D and 4 dimensions In D dimensions it uses standard dimensional regularization to treat ultraviolet divergences in 4 dimensions it uses the method of con strained differential renormalization which at the one loop level is equivalent to dimen sional reduction Details on these methods can be found in Ha98 A one loop calculation generally includes three steps e Create the topologies Diagram e Insert fields FeynArts generation e Apply the Feynman rules e Paint the diagrams yl e Contract indices Algebraic e Calculate traces simplification e Reduce tensor integrals e Introduce abbreviations FormCalc e Convert Mathematica output Numerical to Fortran C code evaluation e Supply a driver program e Implementation of the integrals LoopTools The automation of the calculation is fairly complete in FormCalc i e FormCalc can eventually produce a complete program to calculate the squared matrix element for a given process The only thing the user has to supply is a driver program which calls the generated subroutines The FormCalc distribution includes a directory of tools and sample programs which can be modified and or extended for a particular application To de
16. function using rawexpr for determining the summation indices AbbrevDolexpr i introduce abbreviations for subexpressions of expr where 1 is either a level or a function as above AbbPrefix the prefix for abbreviations introduced by Abbreviate Abbreviate introduces abbreviations for subexpressions starting from a given depth of the expression Depending on this starting level the expression will be more or less thoroughly abbreviated A starting level of 1 represents the extreme case where the result of Abbreviate is just a single symbol Currently only sums are considered for abbreviations The alternate invocation with a function as second argument introduces abbreviations for 26 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS all subexpressions for which this function yields True like Select This is useful for ex ample to get a picture of the structure of an expression with respect to a certain object as in Abbreviatela b c d e x FreeQ x amp MinLeafCount gt 0 which gives Sub2 Sub1 x thus indicating that the original expression is linear in x The functionality of Abbreviate is actually separated into a pair of functions AbbrevSet and AbbrevDo i e Abbreviate internally runs AbbrevSet to define AbbrevDo and then executes AbbrevDo The expression given to AbbrevSet is not itself abbreviated but used for determin ing the summation indices it could be e g a raw amplitude This is particularly imp
17. of Data The return value of the generated function is an integer which records how many parame ter and data sets were transferred Assigning parameter and data sets as the data become 5 3 The Mathematica Interface 67 available has several advantages e the return value of run is an integer rather than a large bulky list e the parameters corresponding to a particular data set are easy to identify e g Para 4711 contains the parameters corresponding to Data 4711 e most importantly if the calculation is prematurely aborted the parameters and data transferred so far are still accessible Both the starting set number and the heads of the parameter and data assignments can be chosen with the options SetNumber ParaHead and DataHead respectively The parameters which are actually returned are chosen by the user in the PRINTn statements in run F in much the same way as parameters are selected for printout in the stand alone code To specify that TB and MAO be returned one needs the definitions define PRINT1 call MmaPutReal TB TB define PRINT2 call MmaPutReal MAO MAO Notwithstanding parameters can still be printed out in which case they end up in the log file or on screen if no log file is chosen To transfer e g TB to Mathematica and print it out one would use define PRINT1 call MmaPutReal TB TB define PRINT2 SHOW TB TB An analogous subroutine exists of course for integer and complex parameter
18. of the calculation that are already finished so all the user has to do to resume an aborted calculation is start the program again with the same parameters 5 2 11 Shell scripts turnoff switches off and on the evaluation of certain parts of the amplitude which is a handy thing for testing For example turnoff box switches off all parts of the ampli tude with box in their name Invoking turnoff without any argument restores all modules sfx packs all source files but not object executable or log files in the directory it is invoked in into a mail safe self extracting archive For example if sfx is invoked in the directory myprocess it produces myprocess sfx This file can e g be mailed to a collaborator who needs to say myprocess sfx x to unpack the contents pnuglot produces a high quality plot in Encapsulated PostScript format from a data file in just one line In fact pnuglot does not even make the plot itself it writes out a shell script to do that thus pnuglot mydata creates mydata gp1 which then runs gnuplot ATEX and dvips to create mydata eps The advantage of this indirect method is that the default gnuplot commands in mydata gp1 can subsequently be edited to suit the user s taste Adding a label or choosing a different line width is for example a pretty trivial matter Needless to say all labels are in ATEX and Type 1 fonts are selected to make the EPS file nicely scalable pnuglot by d
19. symmetry From the prac tical point of view this looks as follows this code is taken almost verbatim from FormCalc s 2t03 F p 0 ex ey ez ROO minv sqrts msum MASS3 MASS4 MASS5 call Split 5 dble MASS5 amp p ex ey ez minv msum fac O amp Var XMREM5 Var XCOSTH5 Var TRIVIAL call Split 4 dble MASS4 amp p ex ey ez minv msum fac O 54 5 TOOLS FOR THE NUMERICAL EVALUATION amp Var FIXED Var XCOSTH4 Var XPHI4 call VecSet 3 dble MASS3 p ex ey ez One starts with the initial reference direction in ex ey ez and no boost p 0 The avail able energy is given in minv and the sum of external masses in msum The Split subroutine is then called n 1 times for an n particle final state The reference direction the boost minv and msum are automatically adjusted along the way for the respective remaining subsystem and ultimately determine the remaining n th vector unambigously which is then simply set by VecSet About the integration variables more will be said in the next section For the moment note that the X in XMREM5 refers to the ratio i e XMREM5 runs from 0 to 1 The actual integration borders are determined internally by Split 5 2 3 Variables The kinematic input variables are organized in a homogeneous system Each variable is referred to by a preprocessor constant e g SQRTS or XCOSTH variables starting with X are generally scaled i e run from 0
20. theory 44 5 TOOLS FOR THE NUMERICAL EVALUATION Folder specifies the name of the subdirectory relative to the code directory into which the generated code is written When using this option the default makefile will usually have to be adapted accordingly FilePrefix specifies a string which will be prepended to the names of all generated files This complements the Folder option as another way of making files unique SymbolPrefix is a way of avoiding symbol collisions when more than one process generated by WriteSquaredME is linked together The given string must be acceptable to Fortran or C as it is prepended to all global symbols like the names of subroutines and common blocks FileHeader specifies a string to be used as file header This string may contain f d and t which are substituted at file creation by file name description and time stamp respectively SubroutineIncludes gives declarations to be inserted in the declaration section of each sub routine usually in the form of include statements In the most general form a list of three strings is given the first of which included before the second after local variable declara tions and the third at the end of the routine It is admissible to provide a list with fewer strings or just a string without a list in which case the unspecified ones remain empty FileIncludes gives declarations to be inserted at the beginning of each generated file It accepts the same
21. to 1 The following parts can be accessed via preprocessor macros e Var i the actual value of i e Show the value printed in the output to print e g t instead of cos 0 e Lower i Upper i Step i the lower limit upper limit and step width of i If the step is zero the cross section is integrated over i If the step is 999 the variable is considered spurious i e used for output only not integrated over or stepped through e CutMin CutMax 1 the lower and upper cuts on i There are two special variables FIXED for fixed values i e no integration and TRIVIAL for trivial integrations 5 2 4 Cuts The subroutine Split allows to place cuts on each M and cos integration The q integration is not modified in the present setup 5 2 Running the Generated Code Cuts restricting M Cuts restricting cos 0 Cut on Key Cuton Key Mi CUT_MREM cos 6 CUT_COSTH E CUT_MREM_E cos CUT_COSTHCMS ki CUT_MREM_K Ei CUT_COSTH_E Er CUT_MREM_ET ki CUT_COSTH_K kri CUT_MREM_KT Yi CUT_MREM_RAP Mi CUT_MREM_PRAP In practice the application of cuts works as follows key 0 CutMin XMREM5 ESMIN key key Cut CUT_MREM_E CUT_MIN CutMin XCOSTH5 CutMax XCOSTH5 1 COSTH5CUT 1 COSTH5CUT key key Cut CUT_COSTH CUT_MIN CUT_MAX call Split 5 dble MASS5 amp p ex ey ez minv msum fac key amp Var XMRE
22. unpolarized particle for a transversely polarized particle for right circular polarization for left circular polarization and 0 for longitudinal polarization For a scalar particle the polarization parameter must be present on the command line al though its value is ignored The energies are specified in GeV If an energy is below thresh old run issues a warning and sets the energy to the threshold 0 01 GeV A serial number range may optionally be added as a third argument restricting the param eter scans This is normally used only internally by the submit script see below Important If a particular polarization was already fixed during the algebraic simplification e g if _Hel 0 was set in FormCalc to calculate with unpolarized external fermions the code thus produced can never compute a cross section for a different polarization no mat ter how the executable is invoked Running such code with an incompatible polarization will in general only result in a wrong averaging factor 5 2 8 Parallelization and Vectorization The assembly of the squared matrix element in code generated by FormCalc can be sketched as in the following figure where the helicity loop sits at the center of the calculation 5 2 Running the Generated Code 59 Loop s over ys amp model parameters Loop over helicities A1 An o ECM An MUAT Aa The helicity loop is not only strategically the mo
23. whe 6 5 2 7 Compiling and running the code o o o oo a 5 2 8 Parallelization and Vectorization 5 2 9 Scans over parameter space 5 2 10 Log files Data files and Resume 5 2 11 Shell scripts The Mathematica Interface Setting up the Interface The Interface Function in Mathematica Return values Storage of Data Using the Generated Mathematica Function Renormalization Constants Definition of renormalization constants 5 4 2 Calculation of renormalization constants Infrared Divergences and the Soft photon Factor Post processing of the Results 6 1 Reading the data files into Mathematica 6 2 Special graphics functions for Parameter Scans Low level functions for code output File handling Type conversion Writing Expressions Variable lists and Abbreviations Compatibility Functions CONTENTS 1 General Considerations With the increasing accuracy of experimental data one loop calculations have in many cases come to be regarded as the lowest approximation acceptable to publish the results in a re spected journal FormCalc goes a big step towards automating these calculations FormCalc is a Mathematica package which calculates and simplifies tree level and one loop Feynman diagrams It accepts diagrams generated with FeynArts 3 Ha00 and returns the results in a way well suited for further numerical or analytical evaluation Internally FormCalc performs most of the hard work e g working out fermionic traces in FOR
24. which take precedence over the default ones in DriversDir To illustrate the concept consider generating code for a particular scattering process The first time no customized drivers exist and hence FormCalc copies the default drivers into the code directory The user then modifies say process h To preserve the changes he copies the file to a local directory e g drivers and specifies this directory with the Drivers option Now even if the code directory is deleted generating the code anew will recover all modifications because the customized version of process h in drivers supersedes the default process h in DriversDir The function WriteSquaredME assumes that there is a tree level amplitude M tree and a one loop amplitude M1 loop and writes out a subroutine called SquaredME which computes the two expressions M reel and 2 Re M eeM 1 loop When there is no tree level contribution 0 42 5 TOOLS FOR THE NUMERICAL EVALUATION and IM1 to0p are returned and when there is no one loop contribution IM tree and 0 are returned WriteSquaredME tree loop mat abbr dir generate code for a subroutine SquaredME in the directory dir this subroutine computes the squared matrix element composed of the amplitudes tree for the tree level part and loop for the one loop part The tree and loop arguments are results of CalcFeynAmp and are trailed by all other objects necessary for the computation o
25. with SumOver into an actual sum DenCollect works essentially like Mathematica s Collect except that it takes the denomina tors appearing in the expression to collect for Note that this refers to the real denominators i e parts of the expression with negative powers not to the symbol Den p m which stands for propagator denominators the latter can easily be collected with Collect TagCollect collects an expression with respect to powers of a tag and applies a function to the term linear in the tag This function is typically used to apply a function to the tagged part of an expression such as is possible with the MultiplyDiagrams function TermCollect and Pool combine terms with common factors Unlike Factor they look at the terms pairwise and can thus do ab ac d gt a b c d fast Unlike Simplify they do not modify b and c DotSimplify simplifies an expression with two different functions depending on whether it contains any of the objects listed in the NoBracket option i e during the execution of CalcFeynAmp or PolarizationSum 39 The OnSize function is similar to the Switch statement but for the size of the expression This can be useful when simplifying expressions because Simplify even FullSimplify is fast and efficient on short expressions but tends to be slow on large ones For example OnSize 100 FullSimplify 500 Simplify DenCollect would apply FullSimplify if the expressi
26. worry about actually calling SquaredME For certain applications e g Monte Carlo generators it might nevertheless be attractive to invoke SquaredME directly Its declaration is subroutine SquaredME result helicities flags double precision result 2 integer 8 helicities integer flags The results are returned in result helicities encodes the helicities to include and flags is an integer specifying flags for the computation The results of SquaredME are result 1 Mee and result 2 2ReM M 1 loop X cept when there is no tree level contribution in which case result 1 0 and result 2 My ioop is returned 5 1 Generating code 49 The helicities are encoded bitwise in the integer argument helicities For each external particle five bits represent from most to least significant bit right circular spin 2 or 3 2 right circular spin 1 or 1 2 longitudinal left circular spin 1 or 1 2 and left circular spin 2 or 3 2 polarization O Cy gt DN Y O Mons O helicities Dsm 1 Dsn 2 bsn 3 bsn 4 bsn 5 eee Da bs bz by bo saa R Particle 1 Particle n For instance a left and right circularly polarized fermion scattering into an unpolarized two vector boson final state would be given by helicities 00010 01000 0111001110 7419010 Some compilers allow binary notation directly as B 00010 01000 01110 01110 this is non portable however Currently two flags are p
27. 1 To control vectorization by hand one needs to specify the preproces sor variable SIMD in distrib h rather than including the file simd h generated by configure SIMD specifies the length of the vector sensible values are 2 for AVX 1 for SSE3 and 0 other wise In Fortran the vector length can in principle been chosen arbitrarily though a speedup is expected only for the mentioned values commensurate with the hardware On top of this the calculation automatically detects and omits helicity combinations con tributing negligibly This feature is controlled by the two environment variables FCHSELN and FCHSELEPS in the following way for the first FCHSELN phase space points the compu tation runs over all helicity combinations h and the absolute value of the squared matrix element is added up in an array t h For subsequent phase space points then only helicity combinations with a t value larger than FCHSELEPS max t h are actually computed Un less one intentionally begins sampling e g on the borders of phase space the default values 60 5 TOOLS FOR THE NUMERICAL EVALUATION FCHSELN 10 and FCHSELEPS 107 should be sufficient In case of doubt about the valid ity of the result the value of FCHSELN should be increased or set to zero which turns the detection off If algebraic relations between helicities are known from analytic considerations these can be specified in Mathematica before calling WriteSquaredME and will r
28. 10 2986891221231292 0 015013326141014818 0 0 DataRow 1500 0 99 10 5071238252157443 0 012260927614082411 0 0 gt DataRow v r e a row of data with kinematic variables v cross section or decay rate results r and integration error e The DataRow v r e function has three arguments e the unintegrated kinematic variables v ys cos 0 above e the cross section or decay rate results r tree level result one loop correction above and e the respective integration errors e 0 0 above as this example originates from the computation of a differential cross section where no integration is performed 5 3 4 Using the Generated Mathematica Function To the Mathematica novice it may not be obvious how to use the function described above to analyse data produce plots etc As an example let us produce a contour plot of the cross section in the M o tan f plane It is assumed that the function run has the two parameters MAO and TB in its argument list 5 4 Renormalization Constants 69 Install run xs sqrtS_ MAO_ TB_ run sqrtS sqrtS MAO TB Data 1 1 2 gt ContourPlot xs 500 MAO TB MAO 100 500 TB 5 50 The function xs runs the code and selects the data to plot The first argument of run sqrtS sqrtS instructs the code to compute the total cross section for just one point in energy We then select the first and only DataRow in the ou
29. 173 Abb174 Abb187 are replaced by AbbSum637 gt Abb109 Abb187 AbbSum504 gt AbbSum637 Abb173 Abb174 AbbSum566 gt AbbSum637 Abb173 Abb174 4 6 More Abbreviations 25 Then in a second step common subexpressions are eliminated thereby simplifying the last lines further to AbbSum637 gt Abb109 Abb187 Opti gt Abb173 Abb174 AbbSum504 gt AbbSum637 Opti AbbSum566 gt AbbSum637 Opti Optimizing the abbreviations may take some time but can also speed up numerical com putations considerably The prefix for the new abbreviations introduced by OptimizeAbbr i e the Opt in the OptN in the example above can be chosen through the global variable 0ptPrefix SubstSimpleAbbr expr abbr removes simple abbreviations the list of abbreviations abbr and substitutes them back into expr Abbreviations of the form number and number symbol are considered simple Such expressions may show up in the abbreviation list e g after further simplification 4 6 More Abbreviations The abbreviations above are introduced automatically by FormCalc and for specific quanti ties only There is also the possibility to abbreviate arbitrary expressions with the Abbreviate command Abbreviate expr lev introduce abbreviations for subexpressions of expr starting at level lev Abbreviatelexpr f introduce abbreviations for subexpressions of expr for which f returns True AbbrevSet rawexpr set up the AbbrevDo
30. FormCalc 8 User s Guide February 12 2015 Thomas Hahn Abstract FormCalc is a Mathematica package which calculates and simplifies tree level and one loop Feynman diagrams It ac cepts diagrams generated with FeynArts 3 and returns the results in a way well suited for further numerical or analytical evalua tion The dreadful legal stuff FormCalc is free software but is not in the public domain Instead it is covered by the GNU library general public license In plain English this means 1 We don t promise that this software works But if you find any bugs please let us know 2 You can use this software for whatever you want You don t have to pay us 3 You may not pretend that you wrote this software If you use it in a program you must acknowledge somewhere in your publication that you ve used our code If you re a lawyer you will rejoice at the exact wording of the license at http gnu org licenses 1gp1 html FormCalc is available from http feynarts de formcalc FeynArts is available from http feynarts de FORM is available from http nikhef n1 form LoopTools is available from http feynarts de looptools If you make this software available to others please provide them with this manual too There exists a low traffic mailing list where updates will be announced Contact hahn feynarts de to be added to this list If you find any bugs or want to make suggestions or just write fan mail address i
31. M by Jos Vermaseren Ve00 A substantial part of the Mathematica code indeed acts as a driver that threads the FeynArts amplitudes through FORM in an appropriate way The concept is rather straightforward the symbolic expressions of the diagrams are prepared in an input file for FORM then FORM is run and finally the results are read back into Math ematica The interfacing is completely shielded from the user and is handled internally by the FormCalc functions The following diagram shows schematically how FormCalc interacts with FORM Mathematica PRO friendl Y O user friendly CON slow on large expressions user interface FeynArts FormCalc amplitudes results input file MathLink internal FormCalc functions FORM PRO very fast on polynomial expressions CON not so user friendly FormCalc combines the speed of FORM with the powerful instruction set of Mathematica and the latter greatly facilitates further processing of the results Owing to FORM s speed FormCalc can process for example the 1000 odd one loop diagrams of W W scattering in the Standard Model in a few minutes on ordinary hardware 6 1 GENERAL CONSIDERATIONS One important aspect of FormCalc is that it automatically gathers spinor chains scalar prod ucts of vectors and antisymmetric tensors contracted with vectors and introduces abbre viations
32. M5 Var XCOSTH5 Var TRIVIAL The value of the cut is deposited in CutMin or CutMax and the cut registered by adding an identifier for the cut to the integer key e g Cut CUT_MREM_E CUT_MIN specifies a cut on the energy CUT_MREM_E from below CUT_MIN which is used to restrict the invariant mass integration CUT_MREM_E Note that these cuts really restrict the integration limits They do not introduce veto func tions 1 in wanted 0 in unwanted areas into the integrand which can severely hamper convergence In addition for each external particle i several kinematical quantities are available e momspec SPEC_M 1 mass m e momspec SPEC_E 1 energy E 56 5 TOOLS FOR THE NUMERICAL EVALUATION e momspec SPEC_K 1 momentum k e momspec SPEC_ET 1 transverse energy Er e momspec SPEC_KT 1 transverse momentum kr e momspec SPEC_RAP 1 rapidity y e momspec SPEC_PRAP 1 pseudo rapidity n e momspec SPEC_DELTAK i the difference E k 5 2 5 Convolution With the new variable system the convolution with arbitrary parton distribution functions can easily be achieved Three modules are already included in FormCalc 5 e lumi_parton F initial state partons no convolution e lumi_hadron F initial state hadrons convolution with hadronic PDFs from the LHAPDPF library WhBGO5 e lumi_photon F initial state photons convolution with CompAZ spectrum Za03 5 2 6 Integration pa
33. SA2 CB2 SB2 TB2 CBA2 SBA2 USfC UChaC VChaC ZNeuC MG12 MS 2 MCha2 MNeu2 MhO2 MHH2 MAO2 MG02 MHp2 MGp2 MSSMSimplify expr SUSYTrigExpand expr SUSYTrigReduce expr SUSYTrigSimplify expr 2 2 cos a sin cos B sin and tan B cos B a sin 6 a the complex conjugates of various mixing matrix elements the squares of various masses Simplify with SUSY trigonometric identities e g SBA2 1 CBA2 express various trigonometric symbols SB2 S2B etc through ca sa cb sb substitute back ca sa cb sb perform trigonometric simplifications by applying SUSYTrigExpand Simplify and SUSYTrigReduce in sequence Often one wants to neglect certain variables typically masses Directly defining say ME 0 may lead to problems however for instance if ME appears in a negative power or in loop integrals where neglecting it may cause singularities A better way is to assign values to the function Neglect e g Neglect ME 0 which allows FormCalc to replace ME by zero whenever this is safe Watch out for the built in definitions mentioned above since e g ME 2 is automatically replaced by ME2 one has to assign Neglect ME Neglect ME2 0 in order to have also the even powers of the electron mass neglected Neglect s 0 replace s by 0 except when it appears in negative powers or in loop integrals To a certain extent it is also possible to use
34. VALUATION sets all model parameters to their initial values This subroutine is invoked before the user gets to set the input parameters subroutine ModelConstIni fail integer fail initializes the constant model parameters i e the ones that do not depend on vys the centre of mass energy If the routine determines that the input parameters yield unphysical or otherwise unacceptable model parameters it must return a non zero value in fail subroutine ModelVarIni fail sqrtS integer fail double precision sqrts is called after ModelConstIni to initialize the model parameters that depend on ys typically a and derived quantities Finally the subroutine ModelDigest is invoked to print a digest of the model parameters Note one detail some model constants are declared in model_x h with parameter state ments i e as numerical constants WriteSquaredME can optimize the generated code based on the information which symbols are numerical constants to the compiler and which are variables The trick is to move the numerical constants to the front of each product so that the compiler can combine these constants into one number thus reducing the number of multiplications at run time For example if WriteSquaredME knows Alfa2 and SW2 to be numerical constants it will write 3 Alfa2 8 DO SW2 2 Abb1 cint44 MH2 instead of Mathematica s de fault ordering 3 Abb1 A1fa2 cint44 MH2 8 D0 SW2 2 While this may seem a m
35. Y SIMPLIFYING DIAGRAMS DiracChain the antisymmetrized product of Dirac matrices The actual Dirac matrix with which an argument is contracted is not written out e g k 1 stands for Yu k 1 inside a DiracChain inside a DiracChain taa 5 y7 6 the chirality projector Pr w 1 ys 2 7 the chirality projector P w 1 y5 2 i antisymmetrized chain i 1 5 6 7 as above Lor n a Lorentz index connecting two DiracChains Spinor p m 1 particle spinor u p for which p m u p 0 Spinor p m 1 antiparticle spinor v p for which y m v p 0 It should be especially noted that DiracChains unlike WeylChains are antisymmetrized starting from FormCalc Version 6 Antisymmetrized chains are typically more convenient for analytical purposes for example there is the correspondence DiracChain 1 u v Oyy Note that the antisymmetrization does not extend to the chirality projector or ys e g w 31 Mia VuVoVv Y vYuYp YvYpYu VpVuVv YoYvYu DiracChain 6 u v p Weyl chains are quite similar in notation to the Dirac chains The constituents of a Wey1Chain always alternate between upper o and lower indices 0 3 0 The arguments 6 and 7 hence fix the index positions for the entire chain WeylChain the product of sigma matrices The actual sigma matrix with which an argument is contracted is not written out e g k 1 stands for o k 1 inside a WeylChain inside a We
36. ally write out a master subroutine which calls the individual parts To further automate the code generation such that the resulting code needs few or no changes by hand many ancillary functions are available 7 1 File handling Type conversion MkDir dirs make sure the directory dirs exists creating subdirectories as necessary OpenCode file open file for writing code TimeStamp return a string with the current date and time ToCode expr return a string with the Fortran or C form of expr ToSymbol args concatenate args into one symbol ToList expr return a list of summands of expr 7 2 Writing Expressions 79 MkDir dir dir3 makes sure the directory dir dir exists creating the indi vidual subdirectories dirz dir as necessary It works roughly as mkdir p in Unix OpenCode file opens file for writing code TimeStamp returns a string with the current date and time ToCode expr returns the Fortran or C form of expr as a string ToSymbol args concatenates its arguments into a new symbol e g ToSymbol a 1 x y gives alxy ToList expr returns a list of summands of expr i e turns a sum of items into a list of items 7 2 Writing Expressions PrepareExpr exprs prepare exprs for write out to code WriteExpr file exprs write exprs to file CodeExpr v t exprs the expressions exprs with variables v and temporary variables tin a form ready for write out by Wr
37. am in the code directory dir WriteRenConst writes out the subroutine CalcRenConst as well as the declarations of the RCs in renconst h It shares a number of options regarding the technicalities of code gen eration with WriteSquaredME In the following table if the default value coincides with the option name this means that the value is taken from the WriteSquaredME options 72 5 TOOLS FOR THE NUMERICAL EVALUATION option default value Folder renconst the subdirectory of the code directory into which the generated code is written FilePrefix FilePrefix a string prepended to the filenames of the generated code SymbolPrefix SymbolPrefix a string prepended to global symbols to prevent collision of names when more than one process is linked FileHeader FileHeader the file header FileIncludes FileIncludes per file include statements SubroutineIncludes Subroutine per subroutine include statements Includes At this point it is necessary to discuss how selections of diagrams can be made for the self energies The functions SelfEnergy and DSelfEnergy which are used in the definitions of the RCs are invoked when the RCs are calculated They generally use what options of InsertFields CreateFeynAmp and CalcFeynAmp are in place at that time Additionally they provide a hook for the user to customize the exact use of InsertFields That is SelfEnergy and DSelfEnergy do not call InsertFields directly but a functi
38. an It also makes it easier to pinpoint errors since the error message appears right next to the corrupted data va hia The real data are marked by a in column 1 and there exists a simple shell script data to extract the real data from the log file For example data run UUUU 00200 creates a data file run UUUU 00200 data containing only the real data arranged in three columns where the first column is the scattering angle in radians the second column is the tree level cross section and the third column is the one loop correction For an integrated 5 2 Running the Generated Code 63 cross section there are five columns ys the tree level cross section the one loop correction and the integration errors for the tree level and one loop cross sections Cross sections are computed in picobarn The log file management also provides an easy way to resume an aborted calculation This works as follows when running through the loops of a parameter scan the log file for a particular serial number e may not exist then it is created with execute permissions e may exist but have execute permissions then it is overwritten e may exist and have read write permissions then this serial number is skipped The execute permissions which serve here merely as a flag to indicate an ongoing calcula tion are reduced to ordinary read write permissions when the log file is closed In other words the program skips over the parts
39. and 4 This is a question of how UV divergent expressions are treated The essential points of both methods are outlined in the following For a more thorough 4 1 CalcFeynAmp 11 discussion see Ha98 e Dimension gt D corresponds to dimensional regularization tH72 Dimensionally regularizing an expression involves actually two things analytic continuation of the momenta and other four vectors in the number of dimensions D and an extension to D dimensions of the Lorentz covariants Yu Suv etc The second part is achieved by treating the covariants as formal objects obeying certain algebraic relations Prob lems only appear for identities that depend on the 4 dimensional nature of the objects involved In particular the extension of y5 to D dimensions is problematic FormCalc employs a naive scheme Ch79 and works with an anticommuting ys in all dimen sions e Dimension gt 4 selects constrained differential renormalization CDR dA98 This technique cures UV divergences by substituting badly behaved expressions by deriva tives of well behaved ones in coordinate space The regularized coordinate space ex pressions are then Fourier transformed back to momentum space CDR works com pletely in 4 dimensions At one loop level it has been shown Ha98 to be equivalent to regularization by dimensional reduction Si79 which is a modified version of di mensional regularization while the integration momenta are still D dimensiona
40. ange exactly cancel those in the emission amplitude of soft i e low energetic photons off the external legs Soft photon ra diation is always present experimentally because photons are massless they can have arbi trarily low energies and therefore escape undetected So while the unphysical A dependence cancels the final result depends instead on a detector dependent quantity EX the maxi mum energy a soft photon may have without being detected In a one loop calculation one needs to take into account the bremsstrahlung off the Born diagrams to be consistent in the order of Dealing with soft photons is relatively easy since the soft photon cross section is always proportional to the Born cross section a A PELE dQ SB dQ Born where SB stands for soft bremsstrahlung The soft photon factor dsp is implemented in main F using the formulas of De93 It is controlled by the following two preprocessor statements in process h c define BREMSSTRAHLUNG define ESOFTMAX 1DOx sqrtS The first flag BREMSSTRAHLUNG is commented out by default To switch on soft photon cor rections take out the c before the define The second line determines the energy cutoff max Eoft i 6 Post processing of the Results 6 1 Reading the data files into Mathematica The format in which data are written out by main F is directly practical only for making plots of the cross section over the energy or some phase space variable In many circum sta
41. assed e Bit 0 is the reset flag if 1 the abbreviations must be re calculated completely The lat ter is the case when the centre of mass energy or the model parameters have changed e Bit 1 is the loop flag if 1 the loop corrections are computed This flag allows e g to integrate phase space separately for tree level and one loop part The following picture shows the default constellation of driver programs and code modules generated by WriteSquaredME m cross sections decay rates asymmetries Sa parameters for this run driver programs process definition run F xsection F MtoN F process h user level code included in FormCalc R ER generated code black box quare master subroutine abbr0_s F abbreviations calculated only when necessary abbr0_angle F form factors 50 5 TOOLS FOR THE NUMERICAL EVALUATION 5 2 Running the Generated Code The code produced by WriteSquaredME needs in addition a driver program which supplies it with the necessary parameters kinematics etc The default driver program that comes with FormCalc is organized in a very modular way and spreads out over several files Technically the main file is run F i e this is the file the compiler sees as main program However run F is just a clever way of defining parameters at the right places and the real work is done in other files which are inclu
42. ata set to be preceded by two lines of the form MT 160 00 MH 100 00 Such a scan can be a real CPU hog but on the other hand the calculation can be performed completely independently for each parameter set and is thus an ideal candidate for par allelization The real question is thus not how to parallelize the calculation but how to automate the parallelization The obstacle to automatic parallelization is that the loops are user defined and in general nested A serial number is introduced to unroll the loops serial 0 LOOP1 LOOP2 serial serial 1 if serial not in allowed range goto 1 calculate cross section 1 continue As mentioned before the serial number range can be specified on the command line so that it is quite straightforward to distribute patches of serial numbers on different machines Most easily this is done in an interleaved manner since one then does not need to know to which upper limit the serial number runs i e if there are N machines available send serial numbers 1 N 1 2N 1 on machine 1 send serial numbers 2 N 2 2N 2 on machine 2 etc This procedure is automated in FormCalc The user once creates a submitrc file in his home directory and lists there all machines that may be used one on each line The only require ment is that the machines are binary compatible because the same executable will be started on each In the case of multi processor machines the number of processors is
43. ave more than one instance of squaredme or renconst e g when computing an hadronic cross section to which several partonic processes contribute Both WriteSquaredME and WriteRenConst have the Folder option with which a unique folder name can be chosen and the Symbo1Prefix option with which the symbols visible to the linker can be prefixed with a unique identifier This identifier is inserted only at compile time and can easily be changed in the sub makefile at any time The util library is a collection of ancillary routines which currently includes e System utilities log file management e Kinematic functions Pair Eps e Diagonalization routines DIAG library Ha06 e Univariate integrators Gauss Patterson e Multivariate integrators CUBA library Ha04 Older versions of FormCalc used to include these directly into the main program The library version has the advantage that the linker selects only the routines that are actually needed and furthermore it is straightforward to add new code The util a library is compiled once when FormCalc is installed and then copied to the main code directory thus avoiding unnecessary compiles Specifying numerical parameters Numerical parameters are specified in the model initialization file e g model_sm F with declarations in model_sm h Each such initialization file must implement the following three subroutines subroutine ModelDefaults 46 5 TOOLS FOR THE NUMERICAL E
44. bexpr subexpr register a list of subexpressions RegisterAbbr registers a list of abbreviations e g the output of Abbr in a previous session such that future invocations of CalcFeynAmp will make use of them Note that abbreviations introduced for different processes are in general not compatible RegisterSubexpr registers a list of subexpressions e g the output of Subexpr in a previ ous session such that future invocations of Abbreviate will make use of them For long running calculations the Keep function is helpful to store intermediate expressions such that the calculation can be resumed after a crash As a side effect the intermediate 28 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS results can be inspected easily even while a batch job is in progress Keep expr name path loads path name m if it exists otherwise evaluates expr and stores the result together with the output of Abbr and Subexpr in that file path is optional and defaults to KeepDir Keep expr name same as Keep expr name KeepDir Keep lhs rhs same as lhs Keep rhs lhs KeepDir the default directory for storing intermediate expressions Keep has two basic arguments a file path and name and an expression If the file exists it is loaded If not the expression is evaluated and the results stored in the file thus creating a checkpoint If the calculation crashes it suffices to restart the very same program which w
45. cients of 71 and e work out to zero In LoopTools for example this is effected by setting LTLAMBDA 1 and 2 respectively The local terms would quite obviously spoil this cancellation and must be set to zero during 4 4 Ingredients of Feynman amplitudes 19 the check To make this possible CalcFeynAmp multiplies the local terms it generates with the symbol Finite such that Finite 1 normally but Finite 0 when checking finiteness S Mandelstam variable s x 1 k 2 Mandelstam variable t k 1 k 3 U Mandelstam variable u k 2 k 3 Sij invariant of the invariant mass type Sij ki k j Tij invariant of the momentum transfer type Tij x i x j1 SU N structures are always simplified so that only chains of generators SUNT remain SUNT a b i j the product T T of SU N generators where a b are gluon indices 1 8 andi and j are colour indices 1 3 note that SUNT i j represents the identity 6 in colour space SUNT a b 0 0 the trace In ttm The N in SU N is specified with the variable SUNN variable default value SUNN 3 the N in SU N Fermionic structures like spinor chains are returned as a DiracChain or WeylChain object Lorentz indices connecting two DiracChains are denoted by Lor n all other Lorentz in dices are implicit e g in the contraction with a vector 20 4 ALGEBRAICALL
46. computing in D dimensions however since for D and 4 the limit D 4 has already been taken in CalcFeynAmp The dimensional dependence of the result is expressed through Dminus4 and Dminus4Eps where the latter represents the Dminus4 arising from the contraction of Levi Civita tensors For testing and comparison the default equivalence Dminus4Eps Dminus4 can be unset GaugeTerms drops terms containing the gauge dependent auxiliary vector n More precisely the n terms are actually introduced at first to let potential cancellations of n s in the numer ator against the denominator occur but set to zero later The options MomElim DotExpand NoBracket RetainFile and EditCode are used in the same way as for CalcFeynAmp see page 13 Note PolarizationSum uses internal definitions set up by CalcFeynAmp It is therefore not advisable to mix the evaluation of different processes For instance wrong results can be expected if one uses PolarizationSum on results from process A after having computed amplitudes from process B with CalcFeynAmp Since this requires at least one invocation of ClearProcess though it is unlikely to happen accidentally 36 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS 4 12 Analytic Unsquared Amplitudes The smallest object appearing in the output of CalcFeynAmp is a four vector i e FormCalc does not normally go into components Those are usually inserted only in the numerical part This has advantag
47. counts the number of subexpressions or leaves on the expression tree AbbSum29 has a leaf count of 1 since it is just a plain symbol In comparison its fully expanded contents have a leaf count of 77 Abbr the list of abbreviations introduced so far Abbr patt the list of all abbreviations including or excluding if preceded by Not the pattern patt OptimizeAbbr a optimize the abbreviations a 0ptPrefix the prefix for additional abbreviations introduced by OptimizeAbbr SubstSimpleAbbr expr abbr remove simple abbreviations from abbr and substitute them back into expr The definitions of the abbreviations can be retrieved by Abbr which returns a list of rules such that result Abbr gives the full unabbreviated expression Needless to say if one wants to use the results of CalcFeynAmp outside of the present FormCalc session the abbreviations have to be saved too e g with Abbr gt gt abbr Abbr patt retrieves only the subset of abbreviations matching at least one of the pat terns patt To exclude rather than include a pattern precede it by Not For example Abbr a b returns all abbreviations with a but not b OptimizeAbbr a optimizes the list of abbreviations a The optimization is done in two steps First redundant parts are removed e g the abbreviations AbbSum637 gt Abb109 Abb187 AbbSum504 gt Abb109 Abb173 Abb174 Abb187 AbbSum566 gt Abb109 Abb
48. ctors are numbered sequentially like in the following figure ky m 1 k3 m3 3 k4 Ma 4 k2 M2 2 In addition to the external particles the following items are defined in process h e a colour factor COLOURFACTOR to account for coloured particles in the initial state e acombinatorial factor IDENTICALFACTOR to account for identical particles in the final state e a wave function renormalization WF_RENORMALIZATION if a non onshell renormaliza tion scheme is used e whether soft bremsstrahlung shall be added BREMSSTRAHLUNG and the maximum en ergy a soft photon may have ESOFTMAX see Sect 5 5 for details 5 2 2 Building up phase space FormCalc builds up the n particle phase space iteratively by nested integrations over the invariant mass M and solid angle Q of each outgoing particle i This procedure is encoded in the subroutine Split 5 2 Running the Generated Code 53 vs a m ma SE call Split Mn 1 Nea Lem an Counting the degrees of freedom there are n 1 M integrations and n Q integrations The corresponding phase space parameterization is 1 fvs m k EA aM do 2 s M Mn 2 M m ko x 1 dM gt dQ a m3 Mn 2 Mises Kn 1 2 Mn 2 Mn 1 x I dM 1 dO 1 Mn k x dQ o where dQ d cos 6 dp The particle s momentum k and cos 0 are given in the respective decay s rest frame The p integration is trivial because of axial
49. ded by run F There is almost no cross talk between different modules which are in that sense universal Also the main program has been radically slimmed down in FormCalc 5 It now only scans the command line and invokes call ProcessIni call ParameterScan All further action is decoupled from the main program and can easily be called from any application It is thus straightforward to use FormCalc generated code in own programs The distribution of the code over the various include files and their interdependencies is shown in the following figure where arrows indicate code inclusion via include model_x F model_x h i I f E user h main F ucil la xsection F lt xsection h looptools h A renconst h process h A MtoN F WI MtoN h lumi_y F lt lumi_y h Organizing the code in this seemingly entangled but highly modular way makes it possible for one program to perform many different tasks simply by setting preprocessor flags The different modules have the following duties e run F defines a run e g the ranges over which to scan model parameters e process h defines all process dependent parameters 5 2 Running the Generated Code 51 main F does the command line parsing xsection F contains the kinematics independent code xsection h contains
50. different colour factor But this is precisely the idea of helicity matrix elements applied to the SU N case Because of this close analogy the combinations of SUNi and SUNj are called colour matrix elements in FormCalc and are written accordingly as Mat SUNi SUN The function which computes them is ColourME It is invoked just like HelicityME ColourME M M3 calculate the colour matrix elements for all combinations of SUNn appearing in My and M3 where as the x indicates M3 is the part to be complex conjugated All used as an argument of ColourME instead of selecting the SUNs which appear in an expression simply take all SUNs currently in abbreviations Mat SUNi SUNj the colour matrix element resulting from the SU N objects SUNi and SUN The core function behind ColourME can also be used directly to simplify colour structures ColourSimplifylexpr simplify colour objects in expr ColourSimplify expr conj simplifies the colour objects in expr conj Furthermore FormCalc implements a special case of FeynArts s DiagramGrouping function in ColourGrouping which groups Feynman diagrams according to their colour structures The correct grouping can only be done with fully simplified colour structures which is why this function is part of FormCalc not FeynArts ColourGrouping ins group the inserted topologies output of InsertFields according to their colour structures 4 10 P
51. e to start numbering the sets with n use head hpara instead of Para and use head haata instead of Data Data n the nth data set Para n the nth parameter set In addition ReadData has a handy way of applying operations to each line of data it does not in fact put each line of numbers in a list but in a function called DataRow By default DataRow is equal to List Now consider that instead of the absolute values of the cross section which is what main F calculates by default you want e g the relative deviation of the one loop and the tree level cross section All it needs to achieve this is to redefine the DataRow function as follows Clear DataRow DataRow x_ tree_ loop_ x loop tree This works because main F puts the tree level result in the second column and the one loop correction in the third column function default value DataRow List the function which is applied to each row of data in a file read by ReadData 6 2 Special graphics functions for Parameter Scans 77 6 2 Special graphics functions for Parameter Scans A common way of displaying information from scans over the parameter space is either as a three dimensional plot or as a density plot Usually the height of the figure represented by levels of grey in the case of a density plot then determines exclusion limits or similar information The standard Mathematica plotting functions Plot3D and DensityPlot require funct
52. ecting the Fs which appear in an expression simply take all Fs currently in the abbreviations Mat Fi Fj the helicity matrix element resulting from the spinor chains in Fi and Fj Hel n the helicity A of the nth external particle s n the helicity reference vector of the nth external particle Note that HelicityME does not calculate the full expression M1 M3 but only the combina tions of Fs that appear in this product These are then called Mat Fi Fj and depend on the helicities and helicity reference vectors of the external particles Hel n and s n See Sect 4 10 on how to put together the complete expression M M3 If possible specific values for the Hel n should be fixed in advance since that can dramat ically speed up the calculation and also lead to much more compact results For example as mentioned before unpolarized matrix elements can be obtained by putting the helicities An 0 and multiplying by 2 for each external fermion Therefore for calculating only the unpolarized amplitude one could use _Hel 0 mat HelicityMEL which is generally much faster than HelicityME Hel _ gt 0 Don t forget that the matrix elements obtained in this way have yet to be multiplied by 2 for each external fermion Note HelicityME uses internal definitions set up by CalcFeynAmp It is therefore not ad visable to mix the evaluation of different processes For instance wrong results can be
53. ed without reference to a concrete representation for the spinors The point is that in terms like M or 2 Re M M3 only products FiFj of spinor chains appear and these can be calculated using the density matrix for spinors panama 204 ci 5 1 Ays m for massive fermions where A and s is the helicity reference vector corresponding to the momentum p s is the unit vector in the direction of the spin axis in the particle s rest frame boosted into the CMS It is identical to the longitudinal polarization vector of a vector boson and fulfills s p 0ands 1 In the unpolarized case the A dependent part adds up to zero so the projectors become Y up p m le p m A Technically one can use the same formula as in the polarized case by putting A 0 and multiplying the result by 2 for each external fermion FormCalc supplies the function HelicityME to calculate these so called helicity matrix ele ments FiF tn the limit E gt m the vector s becomes increasingly parallel to p i e s p m hence pp m p mp m p m pp m p my m p m 1 Ays p m 1 AYs5 ptm 1 Ay5 p 30 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS HelicityME M M3 calculate the helicity matrix elements for all combinations of Fn appearing in My and M3 where as the indicates MM is the part to be complex conjugated All used as an argument of HelicityME instead of sel
54. efault uses commands only available in gnuplot version 3 7 or higher This version can be obtained from http www gnuplot info 64 5 TOOLS FOR THE NUMERICAL EVALUATION pnuglot opts file file make a plot of the data files file file options o outfile how to name the output file the plotting script will be called outfile gp1 and the actual plot outfile eps The default is outfile file 2 use only columns 1 2 usually the tree level cross section for plotting 3 use only columns 1 2 3 usually the one loop corrected cross section for plotting 5 3 The Mathematica Interface The Mathematica interface turns the stand alone code into a Mathematica function for eval uating the cross section or decay rate as a function of user selected model parameters The benefits of such a function are obvious as the whole instrumentarium of Mathematica com mands can be applied to them For example it is quite straightforward using Mathematica s FindMinimum to determine the minimum or maximum of the cross section over a piece of parameter space Interfacing is done using the MathLink protocol The changes necessary to produce a Math Link executable are quite superficial and affect only the file run F where the user has to choose which model parameters are interfaced from Mathematica To make the obvious even clearer the cross section is not evaluated in Mathematica but in Fortran or C and only the nu
55. ement as a Mathematica expression The function which does this is SquaredME SquaredME M M3 calculates M M3 taking care to put the Mat 7 j in the right places SquaredME is called in much the same way as HelicityME Because of the number of terms that are generated this function is most useful only for rather small amplitudes SquaredME does not insert the actual values for the Mat 7 j This can easily be done later by applying the output of HelicityME and ColourME which are lists of rules substituting the 34 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS Mat i j by their values That is to say SquaredME and HelicityME or ColourME perform complementary tasks SquaredME builds up the squared amplitude in terms of the Mat 7 j whereas HelicityME and ColourME calculate the Mat 7 j 4 11 Polarization Sums In the presence of external gauge bosons the output of SquaredME will still contain polar ization vectors in general implicitly i e through the abbreviations For unpolarized gauge bosons the latter can be eliminated by means of the identities 2 kk y E k Aev k A guv ai for massive particles A 1 2 2k k k k ph Ej K Ajey k A 8uv in ee oe e for massless particles In the massless case the polarization sum is gauge dependent and n is an external four vector which fulfills n 0 and n k 4 0 For a gauge invariant quantity the n dependence should ultimately cancel FormCa
56. ementation in Ve96 The idea is to transform all SU N objects to products of generators Th which are generically denoted by SUNT in FormCalc In the output only two types of objects can appear e Chains products of generators with external colour indices these are denoted by SUNT a b i j T T j where i and j are the external colour indices and the a b are the indices of external gluons This notation includes also the identity in colour space as the special case with no external gluons 5 SUNT Z j e Traces over products of generators these are denoted by SUNT a b 0 0 Terese Diagrams generated with the SM mod model file contain no SU N objects since in the electroweak sector colour can be taken care of by a trivial factor 3 for each quark loop 32 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS The situation is much the same as with fermionic structures just as an amplitude contains open spinor chains if external fermions are involved it also contains SUNTs if external quarks or gluons are involved For the SUNT objects in the output FormCalc introduces abbreviations of the type SUNn These abbreviations can easily be evaluated further if one computes the squared amplitude be cause then the external lines close and the Cvitanovic algorithm yields a simple number for each combination of SUNi and SUN One can think of the squared amplitude being decom posed into parts each of which is multiplied by a
57. ep ex 0207021 Acknowledgements FormCalc would not be able to calculate in four dimensions without the hard work of Manuel P rez Victoria and his deep understanding of constrained differential renormalization The fermionic matrix elements would not be half as well implemented without the relentless testing of Christian Schappacher Index t Hooft Veltman scheme 13 AbbPrefix 25 DriversDir 41 Editor 13 EditorModal 13 FormAbbrDepth 14 FormCmd 7 0ptPrefix 24 1to2 F h 40 2to2 F h 40 2to3 F h 40 Abbr 24 AbbrevDo 25 Abbreviate 25 abbreviations 6 23 30 registering 27 AbbrevSet 25 Alfa 21 Alfa2 21 Alfas 21 Alfas2 21 A11 30 32 analytic amplitudes 36 antisymmetrization 16 Antisymmetrize 15 ApplyUnitarity 38 Automatic 10 bfunc m 40 Breitenlohner Maison scheme 13 btensor m 40 C code 41 CA2 22 CalcFeynAmp 8 14 CalcLevel 10 cancellation of terms 11 CB2 22 CBA2 22 Chiral 11 chirality projector 20 Classes 10 ClearProcess 16 code generation 41 colour indices 19 32 ColourGrouping 32 ColourME 32 ColourSimplify 32 Combine 16 compiling 7 components 36 configure 40 constrained differential renormalization 6 10 CutTools 12 Cvitanovic algorithm 32 CW2 21 data 40 debugging options 13 DeclareProcess 14 demo programs 6 Den 18 DenCollect 38 denominator 18 density matrix 29 Deny 26 diagram genera
58. es It works before and after CalcFeynAmp i e on either FeynAmpList or Amp objects When trying to combine amplitudes from different processes Combine issues a warning only but does not refuse to work as CalcFeynAmp Combine amp amp2 combines amp amp The following two functions are helpful to select diagrams 4 3 Clearing Combining Selecting 17 FermionicQ d True if diagram d contains fermions DiagramType d returns the number of propagators in diagram d not belonging to the loop FermionicQ is used for selecting diagrams that contain fermions i e ferm CalcFeynAmp Selectlamps FermionicQ DiagramType returns the number of propagators not containing the integration momentum To see how this classifies diagrams imagine a 2 gt 2 process without self energy insertions on external legs i e in an on shell renormalization scheme There DiagramType gives 2 for a self energy diagram 1 for a vertex correction diagram and 0 for a box diagram so that for instance vert CalcFeynAmp Selectlamps DiagramTypel 1 calculates all vertex corrections DiagramType is of course only a very crude way of clas sifying diagrams and not nearly as powerful as the options available in FeynArts like ExcludeTopologies Individual legs can be taken off shell with the function OffShe11 OffShell amp i gt H enforce the relation p u on the amplitudes amp OffShelllamp i
59. es for example the analytical expression does not reflect a particular phase space parameterization One can also obtain an analytic expression in terms of kinematic invariants but no four vectors by squaring the amplitude and computing the polarization sums as outlined above This has the advantage of being independent of the representation of the spinors and vectors but of course the size of the expression is roughly doubled by squaring As a third alternative one can obtain an analytical expression for the unsquared amplitude This requires to go into components however To this end one has to load the extra package VecSet lt lt FormCalc tools VecSet and for each external vector invoke the function VecSet which has the same syntax as its Fortran namesake e g VecSet 1 m1 p1 0 O 1 VecSet n m p ey ey ezt set the momentum polarization vectors and spinors for particle n with mass m and three momentum p ex ey ez The amplitude is then evaluated with the function ToComponents e g ToComponents amp This delivers an expression in terms of the phase space parameters used in VecSet ToComponents lamp p1p2 Pu ToComponents amp p1 p2 Punt evaluate amp by substituting four vectors by their component wise representation using external polarizations p1 Pn ToComponents amp pol actually plugs the components of the four vec
60. estrict the sum over helicities accordingly For example Hel 2 Hel 1 WriteSquaredME 5 2 9 Scans over parameter space To perform a scan the actual calculation needs to be enclosed in a number of do loops in which the values of the scanned parameters are changed i e a structure of the form do 1 paral do 1 para2 etc calculate the cross section 1 continue run F provides several preprocessor variables LOOP1 LOOP15 which may be defined to contain the do statements that initiate the loops Each loop must terminate on statement 1 just as above For example the lines define LOOP1 do 1 MT 160 180 5 define LOOP2 do 1 MH 100 300 100 result in a scan of the form do 1 MT 160 180 5 do 1 MH 100 300 100 calculate the cross section 1 continue The loops are nested inside each other with LOOP1 being the outermost loop It is not manda tory to declare a loop with each LOOPn also a fixed value may be given as in the following definition define LOOP1 MH 115 When scanning over parameter space it is further necessary to keep track of the parameter values for each data set This is done by defining PRINT commands in the preprocessor 5 2 Running the Generated Code 61 variables PRINT1 PRINT5 in run F The PRINT command takes pairs of arguments first a string identifying the variable and then the variable itself Defining for example define PRINT1 PRINT MT MT MH MH causes each d
61. ether polarization vectors should be orthogonal to the corresponding momentum i e k u 0 Normalized True whether the polarization vectors should be assumed normalized i e e E n 1 InvSimplify True whether to simplify combinations of invariants as much as possible Antisymmetrize True whether Dirac chains are antisymmetrized OnShe11 determines whether the external particles are on their mass shell i e it sets k m where k is the ith external momentum and m the corresponding mass The special value ExceptDirac works like True except that the Dirac equation is not applied to on shell momenta i e K inside fermion chains are not substituted by m Invariants gt True instructs CalcFeynAmp to introduce kinematical invariants In the case of a 2 gt 2 process these are the familiar Mandelstam variables Transverse gt True enforces orthogonality of the polarization vectors of external vector bosons and their corresponding momentum i e it sets e k 0 where k and e are the ith external momentum and polarization vector respectively Normalized gt True allows CalcFeynAmp to exploit the normalization of the polarization vectors i e set Ep er a 1 The last four options can concisely be summarized as 16 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS Option Action OnShell gt True ki k m Mandelstam gt True ki kj 5 s t i m m Transverse gt True k 0
62. f a function is given it is queried for each variable assignment and its out put True False or a string individually determines generation of the debug statement Debugging messages are usually generated for the expressions specified by the user only To extend them to intermediate variables e g the ones introduced for optimization use DebugLines A11 or A11 string or A11 func The actual debugging statements are constructed from the items in DebugCmd which is a list of four strings The first two strings provide commands issued before and after the debug ging statement the second two strings provide the opening and closing of the debugging statement The debugging statement has two arguments a string the variable name and a number the variable value The default setting is DebugCmd ifdef DEBUG n endif n DEB Hence debugging is activated at compile time by defining the preprocessor variable DEBUG and the preprocessor macro DEB might be defined in Fortran as 82 7 LOW LEVEL FUNCTIONS FOR CODE OUTPUT define DEB a x print a x FinalTouch gives a function which is applied to each final subexpression just before write out to file ResetNumbering resets the internal numbering of variable names created by PrepareExpr i e variables of the form dupn from Optimize and tmpn from Expensive start atn 1 It may be necessary to disable this reset e g if more than one generated expression shall be p
63. f the squared matrix element like helicity matrix elements colour matrix elements and the abbreviations An empty list is used to indicate that the tree or loop argument is missing Since WriteSquaredME has a rather complicated invocation it is perhaps best to give an example born CalcFeynAmp bornamps self CalcFeynAmp selfamps vert CalcFeynAmp vertamps box CalcFeynAmp boxamps col ColourME A11 born dir SetupCodeDir fortran WriteSquaredME born self vert box col Abbr dir The tree and loop arguments of WriteSquaredME have the somewhat unorthodox feature that the naming of the code modules depends on the variable names chosen for the components of the amplitude like born self etc in the example above That is WriteSquaredME tries to name the code modules like the variables e g the amplitude contained in the variable self will be written to files of the form self F This is only possible if the variable names are plain symbols in all other cases the modules are named Tree1 F Tree2 F or Loop1 F Loop2 F etc The following options can be used with WriteSquaredME 5 1 Generating code 43 option default value ExtraRules Ep extra rules to be applied to the expressions before the loop integrals are abbreviated TreeSquare True whether the square of the tree level amplitude is added to the result LoopSquare False whether the square of the one loop amplitude is added to the re
64. gh for Fortran RuleAdd var expr is equivalent to var gt var expr DebugLine s translates to a debugging statement for variable s print out of variable s in the code SymbolPref ix is a string prepended to all externally visible symbols in the generated code to avoid symbol collisions 7 2 Writing Expressions 81 PrepareExpr option default value Optimize False whether to introduce temporary variables for common subexpressions Expensive objects which should be hoisted from inner loops as far as possible MinLeafCount 10 the mininum leaf count above which a subexpression becomes eligible for abbreviationing DebugLines False whether to generate debugging statements for every variable FinalTouch Identity a function which is applied to the final subexpressions ResetNumbering True whether to number dupn and tmpn variables from n 1 Optimize determines whether variables should be introduced for subexpressions which are used more than once Expensive lists patterns of objects which are expensive in terms of CPU time and should therefore be hoisted from inner do loops if possible MinLeafCount specifies the minimum leaf count a common subexpression must have in or der that a variable is introduced for it DebugLines specifies whether debugging statements are generated for each expression If instead of True or False a string is given debugging messages are generated and prefixed by this string I
65. given after the host name Empty lines and lines beginning with are treated as comments Furthermore a line of the form nice n determines the nice value at which the remote jobs are started For example submitrc for FormCalc for the institute cluster 62 5 TOOLS FOR THE NUMERICAL EVALUATION start jobs with nice 10 nice 10 pcxeonl 8 pcxeon2 8 pcjoe pcath6 pcath7 The executable compiled from FormCalc code typically called run is then simply prefixed with submit For instance instead of run uuuu 500 1000 the user invokes submit run uuuu 500 1000 The submit script uses ruptime to determine the load of the machines and ssh to log in Handling of the serial number is invisible to the user 5 2 10 Log files Data files and Resume Due to the parallelization mechanism a single output file is not sufficient Instead FormCalc creates a directory for each invocation of run e g run UUUU 00200 for a differential cross section or run UUUU 00200 00500 00010 for an integrated cross section where 00010 is the step width in the loop over the energy opens one log file for each serial number in this directory and redirects console output to this file Each log file contains both the real data and the chatter progress warning and error mes sages This has the advantage that no unit numbers must be passed between subroutines every bit of output is simply written to the console unit in Fortr
66. he executable Its default name is run which is quite natural because the Fortran compiler sees run F as the main program as mentioned before The advantage is that for a different run one can make a copy of run F say run1 F with different parameters This new run is compiled with make run1 and results in the executable runi Thus one can have several executables for different runs in one directory The way the makefile compiles the code is also convenient if one wants to use the generated subroutine SquaredME alone i e without the surrounding driver programs The necessary object files are all placed in the library squaredme a so that only a single file needs to be linked It is possible to build just this library with 58 5 TOOLS FOR THE NUMERICAL EVALUATION make squaredme a The executables run run1 etc are able to calculate differential and integrated cross sections for particles of arbitrary polarization depending on the command line run p1 pn sgrtS serialfrom serialto serialstep compute the differential cross section at a centre of mass energy sqrtS with the external particles polarized according to pj Pn run p1 pn sqrtSfrom sgrtSto sqrtSstep serialfrom serialto serialstep 11 compute the integrated cross section in the energy range sgrtSfrom sqrtSto in steps of sqrtSstep pz Pn being the polarizations of the external particles The p can take the following values for an
67. ill then load all parts of the calculation that have been completed and resume at the point it left off The syntax of Keep is constructed so as to make adding it to existing programs is as painless as possible For example a statement like amps CalcFeynAmpL simply becomes Keeplamps CalcFeynAmp Needless to say this logic fails to work if symbols are being re assigned i e appear more than once on the left hand side as in Keeplamps CalcFeynAmp virt Keeplamps Joinlamps CalcFeynAmp counter Due to the first Keep statement the second will always find the file keep amps m and never execute the computation of the counter terms And there are other ways to confuse the system mixing intermediate results from different calculations changing flags out of sync with the intermediate results etc In case of doubt i e if results seem suspicious remove all intermediate files and re do the calculation from scratch 4 8 Fermionic Matrix Elements When FermionChains gt Chiral or VA is chosen an amplitude involving external fermions will contain DiracChains abbreviated as Fi e g 4 8 Fermionic Matrix Elements 29 F1 gt DiracChain Spinor k 2 ME 1 6 Lor 1 Spinor k 1 ME 1 DiracChain Spinor k 3 MT 1 6 Lor 1 Spinor k 4 MT 1 In physical observables such as the cross section where only the square of the amplitude or interference terms can enter these spinor chains can be evaluat
68. in the internal multiplication brackets in FORM This bracketing is done for performance but prevents the symbols from partaking in further evaluation MomRules specifies a set of rules for transforming momenta The notation is that of the final amplitude i e k1 kn for the momenta el en for the polarization vectors PreFunction and PostFunction specify functions to be applied to the amplitude before and after all simplifcations have been made These options are typically used to apply a function to all amplitudes in a calculation even in indirect calls to CalcFeynAmp such as through CalcRenConst RetainFile and EditCode are options used for debugging purposes The temporary file to which the input for FORM is written is not removed when RetainFile gt True is set The name of this file is typically something like fc amp 1 frm The middle part amp can be chosen with the FileTag option to disambiguate files from different CalcFeynAmp calls EditCode is more comfortable in that it places the temporary file in an editor window before sending it to FORM The command line for the editor is specified in the variable Editor EditCode gt Modal invokes the EditorModal command which is supposed to be modal non detached i e continues only after the editor is closed thus continuing with possibly modified FORM code In truly obnoxious cases the function ReadFormDebug bits can be used to enable debug ging output on stde
69. inor rearrangement the increase in execution speed can be 30 To tell WriteSquaredME which symbols are numerical constants to the compiler simply as sign the symbol a numerical value like in this statement is near the end of FormCalc m Scan N Random amp Alfa Alfa2 MW MW2 Details of the code generation The following excerpt is from a Fortran code generated by WriteSquaredME subroutine self 5 1 Generating code 47 implicit none include vars h Cloop 1 Cloop 1 amp 3 Alfa2 8 D0 SW2 2 Abbi cint44 MH2 amp 9 Alfa2 8 D0 SW2 2 Abb1 cint5 MH2 2 amp Alfa2 4 D0 SW2 2 Abbi cint41 MH2 6 MW2 amp Al fa2 MW2 SW2 2 Abb1 cint43 S amp Al fa2 MW2 2 DO CW2 SW2 2 Abbi xcint46 S amp A1fa2 8 DOx CW2 SW2x x 2 amp Abb1 cint45 6 MW2 CW2 MH2 Note the use of the C preprocessor in statements like include Because of this the code files have the extension F which is recognized by almost all Fortran compilers WriteSquaredME pulls the calculation of several kinds of objects out of the actual computa tion of the amplitude That is these objects are calculated first and stored in variables and then the amplitude is computed Because the replacement of more complex objects by scalar variables shortens the expressions these objects are referred to as abbreviations The following objects are treated as abbreviations 1 variables replacing s
70. input as SubroutineInclude except that there is no subroutine relative to which the placement can be made Instead the first list element goes into the central include file vars h the other two into the individual code files after the include vars h statement SetLanguage Fortran set the output language to Fortran default SetLanguage C set the output language to C Libraries and Makefiles The code is organized in FormCalc into a main code directory which contains the main pro gram and all its prerequisite files and subsidiary folders subdirectories to the main code directory The default setup looks like this main code directory created by SetupCodeDir L squaredme generated by WriteSquaredME L renconst generated by WriteRenConst util comes with FormCalc 5 1 Generating code 45 Each folder is equipped with its own makefile which makes a library of the same name e g the makefile in util makes the library util a These sub makefiles are orchestrated by the master makefile Libraries required for the main program are listed in the LIBS variable and built automatically by invoking the sub makefiles LIBS squaredme a renconst a util a The master makefile is no longer overwritten in the code generation process and is treated like the other driver programs i e a customized copy can be saved in the local drivers direc tory Occasionally it is useful to h
71. ions to be plotted not arrays of numbers Although this can be circumvented by using interpolating functions there are two considerable disadvantages of this method first the grid used by the plotting function in general does not represent the calculated points second there is no control over missing values For this reason two graphics functions for 3D and density plots have been included in Form Calc in the ScanGraphics package They are especially designed for parameter scan plots in that they use a grid which precisely matches the computed values and that they can deal with missing values ScanPlot3D vj v2 n opts ScanDensityPlot v v2 n opts ScanContourPlot v v2 n opts make a 3D density or contour plot of a quantity which has been computed for different values of v7 and vz with v und v2 displayed on the x and y axes The data to be plotted are assumed to be in the data sets Data 1 Data n The argument opts may contain additional 3D 2D graphics options Both functions cooperate closely with the ReadData function described in the last section The argument n which specifies the number of data sets to be taken is usually just the output of ReadData ScanPlot3D and ScanDensityPlot determine the grid on a democratic vote basis they make a list of all the spacings Av Av2 between any two points and take the values of Av and Av that occur most often This of course assumes that the g
72. iteExpr RuleAdd a b sameasa gt a b DebugLine s a debugging statement for variable s SymbolPrefix a string prepended to all externally visible symbols to avoid symbol conflicts PrepareExpr var gt expr var gt expr2 prepares a list of variable assignments for write out to a code file Expressions with a leaf count larger than BlockSize are split into several pieces as in var part1 var var part2 thereby possibly introducing temporary variables for subexpressions SumOver DoLoop and IndexIf objects are properly taken into account as do loops and if statements The output is a CodeExpr vars tmpvars exprlist object where vars are the original and tmpvars the tem porary variables introduced by PrepareExpr WriteExpr file exprlist writes a list of variable assignments to file The exprlist can either be a CodeExpr object ie the output of PrepareExpr or a list of expressions of the form var gt expr var gt expr2 which is first converted to a CodeExpr object using PrepareExpr WriteExpr returns a list of the subexpressions that were actually written 80 7 LOW LEVEL FUNCTIONS FOR CODE OUTPUT CodeExpr vars tmpvars exprlist is the output of PrepareExpr and contains a list of expres sions ready to be written to a file where vars are the original variables and tmpvars are tem porary variables introduced in order to shrink individual expressions to a size small enou
73. l name returns a string with the declaration of the subroutine name VarDecl v1 02 t returns a string with the declaration of v1 v2 as variables of type t Any other strings are output verbatim e g ifdef COND n endif n VarDecl Common c v1 v2 t declares v7 v2 to be members of common block c VarDecl NameMap c v7 v2 t works much like Common but puts only arrays into the common block one for each data type together with preprocessor statements mapping the V1 02 onto array elements VarDecl arguments wrapped in NotEmpty are output only if at least one of its variable lists is non empty DoDecl i m returns a string with the declaration of a loop over i from 1 to m DoDecl i a b returns the same for a loop from a to b DoDecl i invokes Dim i to de termine the upper bound on i CallDecl sub sub returns a string with the invocations of the subroutines sub sub taking into account possible loops indicated by DoLoop Dim i returns the highest value the index i takes on as determined from the amplitude currently being processed 7 5 Compatibility Functions 87 7 5 Compatibility Functions ToOldBRules rules to convert to the conventions for two point functions of LoopTools 2 2 and before ToNewBRules rules to convert to the conventions for two point functions of LoopTools 2 3 or later ToO1dBRules and ToNewBRules are two sets of transf
74. l as in dimensional regularization all other tensors and spinors are kept 4 dimensional Al though the results are the same it should be stressed that the conceptual approach in CDR is quite different from dimensional reduction e Dimension gt 0 keeps the whole amplitude D dimensional No rational terms are added and the D dependency is expressed through Dminus4 MomElim controls in which way momentum conservation is used to eliminate momenta False performs no elimination an integer between 1 and the number of legs eliminates the specified momentum in favour of the others and Automatic tries all substitutions and chooses the one resulting in the fewest terms DotExpand determines whether the terms collected for momentum elimination are expanded again This prevents kinematical invariants from appearing in the abbreviations but typi cally leads to poorer simplification of the amplitude NoCostly switches off simplifications in the FORM code which are typically fast but can cause endless computations on certain amplitudes FermionChains determines how fermion chains are returned Wey1 the default selects Weyl chains Chiral and VA select Dirac chains in the chiral P Pr and vector axial vector 1 y5 decomposition respectively Note that to fully evaluate amplitudes containing Dirac chains helicity matrix elements must be computed with HelicityME For more details on the conceptual treatment of external fermions in FormCa
75. lc see Ha02 Ha04a 12 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS FermionOrder specifies an ordering for the external spinors in Dirac chains i e makes sense only in combination with FermionChains gt Chiral or VA Possible values are None Fierz Automatic Colour or an explicit ordering e g 2 1 4 3 corresponding to fermion chains of the form 2 TP 1 4 T 3 None applies no reordering Fierz applies the Fierz identities Ni05 twice thus simplifying the chains but keeping the original order Colour applies the ordering of the external colour indices after simplification to the spinors Automatic chooses a lexicographical ordering small numbers before large numbers The Evanescent option toggles whether fermionic operators are tracked across Fierz trans formations by emitting terms of the form Evanescent original operator Fierzed operator InsertionPolicy specifies how the level insertions are applied and can take the values Begin Default or an integer Begin applies the insertions at the beginning of the FORM code this ensures that all loop integrals are fully symmetrized Default applies them af ter simplifying the generic amplitudes this is fastest An integer does the same except that insertions with a LeafCount larger than that integer are inserted only after the ampli tude comes back from FORM this is a workaround for the rare cases where the FORM code aborts due to very long insertions SortDen determi
76. lc provides the function PolarizationSum to apply the above identities PolarizationSum expr sums expr over the polarizations of external gauge bosons It is assumed that expr is the squared amplitude into which the helicity matrix elements have already been inserted Alternatively expr may also be given as an amplitude directly in which case PolarizationSum will first invoke SquaredME and HelicityME with _Hel 0 to obtain the squared amplitude PolarizationSum cannot simplify Weyl chains such as CalcFeynAmp introduces with FermionChains gt Weyl the default 4 11 Polarization Sums 35 option default value SumLegs All which external legs to include in the polarization sum Dimension 4 the dimension to compute in GaugeTerms True whether to drop the n dependent terms MomElim taken from how to apply momentum conservation CalcFeynAmp DotExpand taken from whether to expand terms collected for CalcFeynAmp momentum elimination NoBracket taken from symbols not to be included in the CalcFeynAmp bracketing in FORM RetainFile False whether to retain the temporary FORM command file EditCode False whether to display the FORM code in an editor before sending it to FORM SumLegs allows to restrict the polarization sum to fewer than all external vector bosons For example SumLegs gt 3 4 sums only vector bosons on legs 3 and 4 Dimension is used as in CalcFeynAmp Only the value 0 has the effect of actually
77. ld mean that results are returned which are no longer correct In such a case the cached values can be removed with ClearSE Similar to the functions FormSub FormDot etc there are simplification wrappers specifically for RCs RCSub subexpr a function applied to subexpressions extracted by FORM RCInt intcoeff a function applied to the coefficients of loop integrals in the FORM output 5 5 Infrared Divergences and the Soft photon Factor Infrared divergences appear in processes with charged external particles They originate from the exchange of virtual photons More precisely they come from diagrams containing structures of the form Such diagrams are IR divergent because the photon is massless if the photon had a mass A the divergent terms would be proportional to log A A note on the numerical implementation In LoopTools it is not necessary to introduce a pho ton mass by hand if a requested integral is IR divergent LoopTools automatically regularizes the divergence with a photon mass 1 but treats A as an infinitesimal quantity which means that terms of order A or higher are discarded In practice A is a user definable numerical constant Since the final result should not depend on it after successful removal of the IR divergences it can be given an arbitrary numerical value despite its infinitesimal character 74 6 POST PROCESSING OF THE RESULTS The divergences in the diagrams with virtual photon exch
78. merical results are transferred back to Mathematica One thing one cannot do thus is to increase the numerical precision of the calculation using Mathemat ica commands like SetPrecision 5 3 1 Setting up the Interface The model parameters are specified in the file run F Typical definitions for stand alone code look like here from an MSSM calculation with TB tan 6 and MAO M yo define LOOP1 do 1 TB 5 50 5 define LOOP2 MAO 500 These lines declare TB to be scanned from 5 to 50 in steps of 5 and set MAO to 500 GeV To be able to specify TB in Mathematica instead the only change is 5 3 The Mathematica Interface 65 define LOOP1 call MmaGetReal TB Such invocations of MmaGetReal and its companion subroutines serve two purposes At compile time they determine with which arguments the Mathematica function is generated for details see below and at run time they actually transfer the function s arguments to the specified variables MmaGetInteger i read the integer real complex parameter i r c MmaGetReal r from Mathematica MmaGetComplex c MmaGetIntegerList 1 n read the integer real complex parameter list r c MmaGetRealList r n of length n from Mathematica MmaGetComplexList c n Note that without a separate MmaGetReal call MAO would still be fixed by the Fortran state ment i e not be accessible from Mathematica Once the makefile detects the presence of these subroutines it automatically ge
79. monstrate how a full process is calculated several non trivial one loop calculations in the electroweak Standard Model are included in the FormCalc package It is nevertheless important to realize that the code is generated only at the very end of the calculation if at all i e the calculation proceeds analytically as far as possible At all intermediate stages the results are Mathematica expressions which are considerably easier to modify than Fortran or C code 2 Installation To run FormCalc you need Mathematica 5 or above a Fortran compiler and gcc the GNU C compiler FormCalc comes in a compressed tar archive FormCalc n m tar gz To install it create a directory for FormCalc and unpack the archive there e g gunzip c FormCalc n m tar gz tar xvf cd FormCalc n m compile The last line compiles the C programs that come with FormCalc The compile script puts the binaries in a system dependent directory e g Linux A single FormCalc installation can thus be NFS mounted on different systems once compile has been run on each FormCalc by default invokes the executable form to run FORM If FORM is installed under a different name on your system edit FormCalc m and change the following line FormCmd form 3 Generating the Diagrams FormCalc calculates diagrams generated by FeynArts Version 3 or above Do not use the Feyn Arts function ToFA1Conventions to convert the output of CreateFeynAmp to the old conve
80. n tions of FeynArts 1 FormCalc can be loaded together with FeynArts into one Mathematica session The FeynArts results thus need not be saved in a file before calculating them with FormCalc FormCalc can deal with any mixture of fully inserted diagrams and generic diagrams with insertions The former are produced by FeynArts if only one level is requested via InsertionLevel AmplitudeLevel or PickLevel see the FeynArts manual While both types of input eventually produce the same results it is important to understand that it is the generic amplitude which is the most costly part to simplify so using the latter type of diagrams where many insertions are derived from one generic amplitude can significantly speed up the calculation Usually it is also helpful to organize the diagrams into classes like self energies vertex cor rections and box corrections it will also make the amplitudes easier to handle since it reduces their size 8 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS 4 Algebraically Simplifying Diagrams 4 1 CalcFeynAmp FeynArts always produces purely symbolic amplitudes and refrains from simplifying them in any way so as not to be restricted to a certain class of theories Needless to say the resulting expressions cannot be used directly e g in a Fortran program but must first be simplified algebraically The function for this is CalcFeynAmp CalcFeynAmp a 42 calculate the sum of amplitudes a1 a2
81. n Mathematica is too slow so the FormCalc results need to be converted to a Fortran or C program The simplest way to do this in Mathematica is something like FortranForm result gt gt file f Unfortunately this leaves a lot to be done manually FormCalc contains two much more sophisticated functions for generating code WriteSquaredME and WriteRenConst The philosophy is that the user should not have to modify the generated code This eliminates the source of many errors but of course goes well beyond simple translation to code the code has to be encapsulated i e no loose ends the user has to bother with and all necessary subsidiary files such as include files and a makefile have to be produced Also the code has to be chopped into pieces small enough that the compiler will accept them Before using WriteSquaredME or WriteRenConst a directory must be created for the code and the driver programs copied into this directory This is done with SetupCodeDir SetupCodeDir dir create a directory dir and install the driver programs necessary to compile the generated code in this directory Note that SetupCodeDir never removes or overwrites the directory or the driver programs so a code directory once created stays put The Drivers option can be used to load cus tomized driver programs from a local directory Option default value Drivers drivers a directory containing customized versions of the driver programs
82. nces one needs to make more sophisticated figures in which the data are either arranged differently or are run through some filter or calculation Consider for example performing a scan over some region in parameter space and then wanting to plot the band of allowed values of the cross section i e the minimum and maximum with respect to the parameter variations for each point in phase space Given the variety of operations one could conceivably perform on the data a Fortran or C program would sooner or later prove too inflexible Instead there is a utility program with which it is possible to read the data into Mathematica where it is much easier to further process the data sets and eventually produce plots of whatever function of whatever data set one is interested in 6 1 Reading the data files into Mathematica 75 The data files are read into Mathematica with the MathLink program ReadData which is used in the following way In 1 Install ReadData In 2 ReadData run UUVU 00300 01000 data Out 2 24 The data are stored in Mathematica as separate data and parameter sets These sets are ac cessed through an integer argument e g Data 1 and Para 1 ReadData returns the num ber of sets read 24 in this example The data files have to be roughly in the format used by main F i e a number of lines begin ning by in which the parameters are given followed by columns of data The following example shows this fo
83. nerates inter facing code and compiles a MathLink executable For a file run F the corresponding Math Link executable is also called run as in the stand alone case This file is not started from the command line but used in Mathematica as Install run 5 3 2 The Interface Function in Mathematica After loading the MathLink executable with Install a Mathematica function of the same name is available For definiteness we will call this function run in the following since run F is the default parameter file This function has the arguments run sqrtS argi arg options run sqrtSfrom sqrtSto sqrtSstep arg argo options The first form computes a differential cross section at y s sqrtS The second form computes a total cross section for energies ys varying from sqrtSfrom to sqrtSto in steps of sqrtSstep This is in one to one correspondence with the command line invocation of the stand alone executable The arg1 argo are the model parameters declared automatically by the presence of their MmaGet Real Complex calls see above They appear in the argument list in the same order as the corresponding MmaGet Real Complex calls 66 5 TOOLS FOR THE NUMERICAL EVALUATION Other parameters are specified through the options Default Value default value Polarizations UUUU the polarizations of the external particles Serial ly the range of serial numbers to compute SetNumber
84. nes whether the denominators of loop integrals shall be sorted This is usually done to reduce the number of loop integrals appearing in an amplitude Sorting may be turned off for testing and in few cases may even lead to shorter amplitudes PaVeReduce governs the tensor reduction False retains the one loop tensor coefficient func tions Raw reduces them to scalar integrals but keeps the Gram determinants in the denomi nator in terms of dot products True simplifies the Gram determinants using invariants SimplifyQ2 controls simplification of terms involving the integration momentum q squared If set to True powers of q in the numerator are cancelled by a denominator except for OPP integrals where conversely lower N integrals are put on a common denominator with higher N integrals to reduce OPP calls as in N2 DoD1 N3 DoDi1D2 gt N2D2 N3 DoD1D2 OPP specifies an integer N starting from which an N point function is treated with OPP methods For example OPP gt 4 means that A B C functions are reduced with Passarino Veltman and D and up with OPP A negative N indicates that the rational terms for the OPP integrals shall be added analytically whereas otherwise their computation is left to the OPP package CutTools or Samurai The integration momentum q starts life as a D dimensional object In OPP any q surviving SimplifyQ2 is substituted by q fi after which q is considered 4 dimensional The di mensionful
85. ome function call Pairn dot products Epsn contracted epsilon tensors cintn iintn loop integrals 2 products or sums of variables of type 1 Abb AbbSumn 3 matrix elements Mat Type i j All of these abbreviations except the cintn and iintn are in fact the ones already introduced by FormCalc These abbreviations share the general feature of being costly in CPU time in particular the loop integrals which means that for optimal performance they should be calculated as few times as possible WriteSquaredME splits them into 2 x 3 categories e abbr0_cat F tree level and e abbr1_cat F one loop e abbri_s F abbreviations which depend only on ys e abbri_angle F abbreviations which depend also on other phase space variables but not on the helicities nad e abbri_hel F helicity dependent abbreviations 48 5 TOOLS FOR THE NUMERICAL EVALUATION Needless to say each of these abbri_cat modules is invoked only when necessary All modules generated by WriteSquaredME are invoked in the proper fashion by the master subroutine SquaredME as shown in the following figure subroutine SquaredME result helicities flags TRUE call abbr_s FALSE 1 call abbr_angle call abbr_hel 1 calculate form factors c call born self vert loop over helicities As long as one sticks to the default driver programs one does not have to
86. on s leaf count is less than 100 Simplify if it is between 100 and 500 and DenCollect above ApplyUnitarity uses unitarity of a given matrix to simplify an expression The two uni tarity relations uut lor Via u Ue 56 and UU lor ae Uj Uj dj are used to substitute sums of UU elements with more than d 2 terms For example ApplyUnitarity expr CKM 3 might replace CKM12CKM CKM22CKM3z by CKM32CKM3z ExpandSums turns expressions where index summations are denoted by a SumOver 7 r mul tiplied with the part of the expression to be summed over index i such as delivered by CalcFeynAmp into ordinary Mathematica sums ExpandSuns expr h uses head h instead of Plus 5 Tools for the Numerical Evaluation The numerical evaluation has to take care of two important issues and therefore resolves into two conceptual steps 1 Almost invariably the numerical evaluation in Mathematica itself is too slow Even though the speed of numerical computations has been improved greatly in Mathemat ica 4 the sheer amount of number crunching e g in a scan over parameter space places the numerical evaluation safely in the domain of a compiled language This makes it necessary to translate the Mathematica expressions resulting from CalcFeynAmp HelicityME etc into a high level language program For FormCalc Fortran has been chosen because of its proven track record for fast and reliable number crunching the availability of good
87. on called InsertFieldsHook whose default definition is InsertFieldsHook tops_ proc_ InsertFields tops proc which simply inserts fields into the topologies tops for the process proc The function InsertFieldsHook can be redefined either generally or for a specific process e g InsertFieldsHook tops_ f1_F gt 2_F InsertFields tops f1 gt 2 ExcludeParticles gt V 1 would exclude photons in fermion self energies One could also imagine more complicated definitions e g involving DiagramSelect Furthermore it is possible to have the diagrams painted by setting the variable PaintSE PaintSE can be True False or a string which indicates that the output should be saved in a PostScript file instead of being displayed on screen If a string is given itis used as a prefix for the name of the PostScript file hence it is possible to specify e g a path Finally it is possible to specify options for individual RCs as in Options dMWsqi Restrictions gt NoSUSYParticles 5 5 Infrared Divergences and the Soft photon Factor 73 These options are taken as InsertFields options which apply only to the calculation of this particular renormalization constant For performance reasons SelfEnergy and DSelfEnergy remember the values they have computed In unlikely cases it may happen that the user changes settings that affect the gen eration of diagrams but are not recognized by SelfEnergy and DSelfEnergy This wou
88. ormation rules that convert between the old BO B1 and new B0i b0 b1 conventions for two point functions in LoopTools 88 REFERENCES References Ch79 M Chanowitz M Furman and I Hinchliffe Nucl Phys B159 1979 225 Cv76 P Cvitanovic Phys Rev D14 1976 1536 dA98 F del Aguila A Culatti R Mu oz Tapia and M P rez Victoria Nucl Phys B537 1999 561 hep ph 9806451 De93 A Denner Fortschr Phys 41 1993 307 Ha98 T Hahn and M P rez Victoria Comp Phys Commun 118 1999 153 hep ph 9807565 Ha00 T Hahn Comp Phys Commun 140 2001 418 hep ph 0012260 Ha02 T Hahn Nucl Phys Proc Suppl 116 2003 363 hep ph 0210220 Ha04 T Hahn Comp Phys Commun 168 2005 78 hep ph 0404043 Ha04a T Hahn Nucl Phys Proc Suppl 135 2004 333 hep ph 0406288 Ha06 T Hahn physics 0607103 Ni05 C C Nishi Am J Phys 73 2005 1160 hep ph 0412245 Si79 W Siegel Phys Lett B84 1979 193 tH72 G t Hooft and M Veltman Nucl Phys B44 1972 189 Ve00 J A M Vermaseren math ph 0010025 See also http www nikhef n1 form Ve96 J A M Vermaseren The use of computer algebra in QCD in H Latal W Schweiger Proceedings Schladming 1996 Springer Verlag ISBN 3 540 62478 3 WhBG05 M R Whalley D Bourilkov R C Group hep ph 0508110 http hepforge cedar ac uk lhapdf Za03 A F Zarnecki Acta Phys Polon B34 2003 2741 h
89. ortant in cases where partial expressions will be given to AbbrevDo and where the summation in dices may not be correctly inferred because AbbrevDo does not see the full expression The definition of AbbrevDo is not automatically removed so be careful not to execute AbbrevDo with an expression that is not a subexpression of the one given to AbbrevSet The prefix of the abbreviations are given by the global variable AbbPrefix option default value MinLeafCount 10 the mininum leaf count above which a subexpression becomes eligible for abbreviationing Deny k q1 symbols which must not occur in abbreviations Fuse True whether to fuse adjacent items for which the selection function is True into one abbreviation Preprocess Identity a function applied to subexpressions before abbreviationing MinLeafCount determines the minimum leaf count a common subexpression must have in order that a variable is introduced for it Deny specifies an exclusion list of symbols which must not occur in abbreviations Fuse specifies whether adjacent items for which the selection function returns True should be fused into one abbreviation It has no effect when Abbreviate is invoked with a depth For example Abbreviate a Pair 1 2 Pair 3 4 MatchQ _Pair amp Fuse gt False MinLeafCount gt 0 introduces two abbreviations one for Pair 1 2 and one for Pair 3 4 whereas with 4 7 Resuming Previous Sessions 27 Fuse g
90. patterns in the argument of Neglect Simple patterns like _ and __ work always Since FORM s pattern matching is far inferior to Mathe matica s though it is not at all difficult to come up with patterns which are not accepted by FORM Determining the mass dimension is an easy way of checking consistency of an expression The function MassDim substitutes all symbols in the list MassDimO by a random number all symbols in MassDim1 by Mass times a random number and all symbols in MassDim2 by Mass times a random number Symbols not in MassDim 0 1 2 are not replaced The random numbers are supposed to guard against accidental cancellations An expression consistent in the mass dimension should end up with just one term of the form number Mass 4 5 Handling Abbreviations 23 MassDim expr replace the MassDimn symbols in expr by random number Mass MassDimO a list of symbols of mass dimension 0 MassDim1 a list of symbols of mass dimension 1 MassDim2 a list of symbols of mass dimension 2 Mass asymbol representing the mass dimension 4 5 Handling Abbreviations CalcFeynAmp returns expressions where spinor chains dot products of vectors and Levi Civita tensors contracted with vectors have been collected and abbreviated A term in such an expression may look like COi cc12 MW2 MW2 S MW2 MZ2 MW2 4 AbbSum16 Alfa2 CW2 MW2 S SW2 32 AbbSum28 Alfa2 CW2 S72 SW2 4 AbbSum30 Alfa2 CW2 S 2 SW2 8 AbbSum7 Alfa2
91. r ToDoLoops list ifunc splits list into patches which must be summed over the same set of indices ifunc is an optional argument ifunc expr must return the indices occurring in expr DoLoop Lind expr is asymbol introduced by ToDoLoops indicating that expr is to be summed over the set of indices ind ToIndexIf expr turn the IndexDelta and IndexDiff in expr into IndexIf IndexIf cond a b same as If cond a b except that the expressions a and b are not held unevaluated IndexDeltali j Kronecker s IndexDiff i j 1 dij Maplf f expr maps f over the expr except for the conditional parts of an IndexIf ToIndexIf expr converts all IndexDelta and IndexDiff objects in expr to IndexIf which will be written out as if statements in the generated code ToIndexIf expr patt operates only on indices matching patt IndexIf cond a b is the same as If cond a b except that expressions a and b are not held unevaluated IndexIf cond a is equivalent to IndexIf cond a 0 i e the else part defaults to 0 Several conditions can be combined as IndexIf cond a1 cond2 a2 which is equivalent to IndexIf cond a1 IndexIf cond2 a2 Despite its name the statement is not restricted to index nor to integer comparisons It is furthermore written out as a regular if statement i e if cond then a 7 3 Variable lists and Abbreviations 85 else b endif IndexDelta
92. r n 20 INDEX Lorentz index 20 lumi_hadron F 40 lumi_parton F 40 lumi_photon F 40 MAO2 22 main F h 40 makefile in 40 Mandelstam variables 15 19 23 mass shell 15 MassDim 22 MassDim0O 22 MassDim1 22 MassDim2 22 Mat 30 32 33 MB2 21 MC2 21 MCha2 22 MD2 21 ME2 21 MGO2 22 MG12 22 MGp2 22 MhO2 22 MH2 21 MHH2 22 MHp2 22 MinLeafCount 26 mktm 40 ML2 21 MLE2 21 MM2 21 MNeu2 22 model parameters 21 model dependent symbols 23 model_mssm F 40 model_sm F 40 model_thdm F 40 MomElin 10 31 35 91 momenta 18 momentum conservation 11 MomRules 10 MQD2 21 MQU2 21 MS2 21 MS 2 22 MSSMSimplify 22 MT2 21 MU2 21 MultiplyDiagrams 17 MW2 21 MZ2 21 Neglect 22 neglecting masses 22 NoBracket 10 35 NoCostly 10 NoExpand 10 non expansion of terms 13 normalization 15 Normalized 15 numbering of momenta 18 numerator function 12 numerical evaluation 6 39 OffShel1 17 on shell particles 15 OnShel1 15 OnSize 38 open fermion chains 8 OPP 12 OPP 10 OPPQSlash 10 OptimizeAbbr 24 ordering 11 Pair 18 Particles 10 PaVeReduce 10 92 pnuglot 40 polarization sum 34 polarization vector 15 18 PolarizationSum 14 34 Pool 38 PostFunction 10 PreFunction 10 Preprocess 26 process h 40 propagator denominator 18 rational terms 12 ReadData tm 40 ReadFormDebug 13 RegisterAbbr 27 registering
93. rameters Depending on the integrand the actual integration can be fairly tricky to carry out numeri cally 2t03 F employs the CUBA library Ha04 which offers four integration routines The CUBA parameters are chosen in run F as preprocessor variables define METHOD DIVONNE define RELACCURACY 1D 3 define ABSACCURACY 1D 7 define VERBOSE 1 define MINEVAL 0 define MAXEVAL 50000 for Vegas define NSTART 1000 define NINCREASE 500 for Suave define NNEW 1000 5 2 Running the Generated Code 57 define FLATNESS 50 for Divonne define KEY1 47 define KEY2 1 define KEY3 1 define MAXPASS 5 define BORDER 1D 6 define MAXCHISQ 10 define MINDEVIATION 25D0 for Cuhre define KEY O The integration algorithm is selected with METHOD which can take the values VEGAS SUAVE DIVONNE and CUHRE The other preprocessor variables determine parameters of the integra tors and may should be tuned for a particular integrand for details see Ha04 5 2 7 Compiling and running the code The code produced by WriteSquaredME is compiled with the commands configure make The configure script searches for the compilers and necessary libraries and writes out a makefile by adding the appropriate locations and flags to makefile in The usual envi ronment variables like FC Fortran compiler and FFLAGS Fortran flags can be used to force particular choices Based on the makefile the make command then builds t
94. rid is equidistantly spaced Points on this grid which do not correspond to a data point are missing points If missing points are detected both graphics functions issue a warning and deposit the coordinates of the missing points in the variable MissingPoints for checking In fact if the plotting function uses a grid which differs very slightly from the grid of data points the plot may even display some funny bumps which are artifacts of the interpolation 78 7 LOW LEVEL FUNCTIONS FOR CODE OUTPUT 7 Low level functions for code output FormCalc s code generation functions used internally e g by WriteSquaredME can also be used directly to write out an arbitrary Mathematica expression as optimized code The basic syntax is very simple 1 handle OpenCode file F opens file F as a Fortran file for writing 2 WriteExpr handle var gt expr writes out Fortran code to calculate expr and store the result in var 3 Close handle closes the file again The code generation is fairly sophisticated and goes well beyond merely applying Mathe matica s FortranForm The generated code is optimized e g common subexpressions are pulled out and computed in temporary variables Expressions too large for Fortran are split into parts as in var part1 var var part2 If the expression is too large even to be sensibly evaluated in one file the FileSplit function can distribute it on several files and option
95. rmat TB 1 50 MUE 1000 00 MSusy 1000 00 MAO 700 00 M_2 100 00 160 790000000000 0 264226254743272 0 502096473139054 190 482475649966 10 7918866098049 9 58435238790029 211 567524166608 11 8170727823835 10 3862900941921 4 TB 1 50 MUE 1000 00 MSusy 1000 00 MAO 700 00 M_2 400 00 160 790000000000 0 264226254743272 0 502510502734037 190 482475649966 10 7918866098049 9 60119470492217 211 567524166608 11 8170727823835 10 4047052203442 The output of ReadData for this file is 2 hence there are then two data and two parameter sets defined Para 1 4 TB gt 1 5 MUE gt 1000 MSusy gt 1000 MAO gt 700 M 2 gt 100 76 6 POST PROCESSING OF THE RESULTS Data 1 160 79 0 264226 0 502096 190 482 10 7919 9 58435 211 568 11 8171 10 3863 Para 2 TB gt 1 5 MUE gt 1000 MSusy gt 1000 MAO gt 700 M 2 gt 400 Data 2 160 79 0 264226 0 502511 190 482 10 7919 9 60119 211 568 11 8171 10 4047 Note that the numbers have not suffered in accuracy it is a peculiarity of Mathematica to display real numbers with no more than six digits by default Note also that the underscore in M_2 which is not allowed in symbol names in Mathematica has been replaced by a ReadDatalf n Mpara Adata read the data file f into parameter and data sets in Mathematica with the optional parameters n para and hgata one can choos
96. rr for subsequent CalcFeynAmp calls according to the bits bit pattern ReadFormDebug bits file writes the output to file instead For currently defined bit patterns and their meaning please see the header of ReadForm tm 14 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS FormPre amp a function to set up the following Form simplification functions amp is the raw amplitude FormSub subexpr a function applied to subexpressions extracted by FORM FormDot dotprods a function applied to combinations of dot products by FORM FormMat matcoeff a function applied to the coefficients of matrix elements Mat in the FORM output FormNum numfunc a function applied to numerator functions in the FORM output OPP only FormQC gcoeff a function applied to loop momentum independent parts of the OPP numerator in the FORM output FormQF qfunc a function applied to loop momentum dependent parts of the OPP numerator in the FORM output FormAbbrDepth minimum depth an expression has to have to be abbreviated CalcFeynAmp wraps the above functions around various parts of the FORM output for sim plification upon return to Mathematica These are typically relatively short expressions which can be simplified efficiently in Mathematica The default settings try to balance exe cution time against simplification efficiency Occasionally though Mathematica will spend excessive time on simplification and in this case one or several of the above
97. s Momenta and polarization vectors are consecutively numbered i e the incoming momenta are numbered k 1 k nin and the outgoing momenta k nin 1 k Min Nout k n nth external momentum e n ec n nth polarization vector and its conjugate eT n eTc n nth polarization tensor and its conjugate Pair p q scalar product of the four vectors p and q Eps p q r s1 i uvpop q r s where e is the totally antisymmetric Levi Civita tensor in 4 dimensions with sign convention 0128 1 Den k m the denominator 1 k m Deltali j the Kronecker delta 6 IGram d the denominator arising from the reduction of a tensor integral equivalent to 1 d Finite a symbol multiplied with the local terms resulting from D divergent integral Note the extra factor i in the definition of Eps which is included to reduce the number of explicit i s in the final result About the use of Finite Whenever a divergent loop integral is multiplied by D coming e g as gi from a self contracted coupling local terms arise because the e in D 4 2e cancels the 1 e divergence of the integral schematically D loop integral 4 loop integral local term Local refers to the fact that these terms contain no loop integral anymore In dimensional regularization a popular way of checking IR finiteness of an amplitude is to substitute the loop integrals by their divergent parts and test whether the coeffi
98. s too MmaPutInteger s i transfer the integer real complex parameter i r c MmaPutReal s r to Mathematica under the name s MmaPutComplex s c MmaPutIntegerList s i n transfer the integer real complex parameter list MmaPutRealList s r n 1 r cof length n to Mathematica under the name s MmaPutComplexList s c n The parameters are stored in the form of rules in Mathematica i e as name gt value The first argument specifies the left hand side of this rule It need not be a symbol in the strict sense but can be an arbitrary Mathematica expression But note that in particular the underscore has a special meaning in Mathematica and may not be used in symbol names The second argument is then the right hand side of the rule and can be an arbitrary Fortran expression containing model parameters kinematic variables etc 68 5 TOOLS FOR THE NUMERICAL EVALUATION The following example demonstrates the form of the parameter and data assignments Shown are results of a differential cross section for a 2 gt 2 reaction at one point in MSSM parameter space Within the data the varied parameter is cos 6 the scattering angle Para 1 Data 1 TB gt 1 5 MUE gt 1000 MSusy gt 1000 MAO gt 700 M2 gt 100 DataRow 1500 0 99 10 10592302458950732 0 016577997941111422 0 0 DataRow 1500 0 33 10 16495552191438356 0 014989931149150608 0 0 DataRow 1500 0 33
99. s apply to the whole of each part It is possible to extend this splitting also to powers of coupling constants such that part r has a definite coupling order To this end one needs to wrap the coupling constants of interest in Power0f for example CalcFeynAmplamp g gt g Power0f g The function Power0f is there only to keep track of the coupling order and can be replaced by 1 at the end of the calculation The full result the sum of the parts can trivially be recovered by applying Plus to the outcome of CalcFeynAmp i e Plus CalcFeynAmp amps CalcFeynAmp has the following options option default value CalcLevel Automatic which level of the amplitude to select Automatic selects Particles level if available otherwise Classes level Dimension D the space time dimension in which the calculation is performed D 4 or 0 MomElim Automatic how to apply momentum conservation DotExpand False whether to expand terms collected for momentum elimination NoCostly False whether to turn off potentially time consuming simplifications in FORM FermionChains Weyl how to treat external fermion chains Weyl Chiral or VA FermionOrder Automatic the preferred ordering of external spinors in Dirac chains Evanescent False whether to keep track of fermionic operators across Fierz transformations InsertionPolicy Default how the level insertions are processed 10 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS
100. scale fi enables the OPP libraries to reconstruct the R3 rational terms OPPQSlash extends this treatment to the q on external fermion chains i e also substitutes y gt q iysf where odd powers of fi are eventually set to zero 4 1 CalcFeynAmp 13 Gamma5Test True substitutes each ys by ys 1 Gamma5Test D 4 and it can be tested whether the final result depends on the variable Gamma5Test which it shouldn t Gamma5ToEps gt True substitutes all ys in fermion traces by F Euvpo Y VY This effec tively implements the t Hooft Veltman Breitenlohner Maison y5 scheme External fermion chains are intentionally exempt since at least the Weyl formalism needs chiral chains Take care that due to the larger number of Lorentz indices the computation time may increase significantly NoExpand prohibits the expansion of sums containing certain symbols In certain cases ex pressions can become unnecessarily bloated if all terms are fully expanded as FORM always does For example if gauge eigenstates are rotated into mass eigenstates the couplings typ ically contain linear combinations of the form U c1 U 2c2 It is not difficult to see that the number of terms generated by the full expansion of such couplings can be considerable in particular if several of them appear in a diagram NoExpand turns off the automatic expan sion in this example one would select NoExpand gt U NoBracket prevents the given symbols to be included
101. should be rede fined e g set to Identity 4 2 DeclareProcess For the calculation of an amplitude many definitions have to be set up internally This happens in the DeclareProcess function DeclareProcessl a 47 set up internal definitions for the calculation of the amplitudes a1 a2 Usually it is not necessary to invoke this function explicitly as CalcFeynAmp does so All DeclareProcess options can be specified with CalcFeynAmp and are passed on Functions that need the internal definitions set up by DeclareProcess are CalcFeynAmp HelicityME and PolarizationSum Invoking DeclareProcess directly could be useful e g if one needs to change the options between calls to CalcFeynAmp and HelicityME or if one wants to call HelicityME in a new session without a previous CalcFeynAmp 4 2 DeclareProcess 15 The output of DeclareProcess is a FormAmp object FormAmp proc lamps a preprocessed form of the FeynArts amplitudes amps for process proc DeclareProcess has the following options They are also accepted by CalcFeynAmp in di rect invocations and passed on to DeclareProcess but cannot be set permanently using SetOptions CalcFeynAmp as as they are not CalcFeynAmp options option default value OnShell True whether the external momenta are on shell i e k m Invariants True whether to introduce kinematical invariants like the Mandelstam variables Transverse True wh
102. sion WriteExpr writes out declarations of that type for the given expressions Otherwise no declarations are produced TmpType is the counterpart of Type for the temporary variables TmpType gt Type uses the settings of the Type option IndexType likewise determines the generation of declarations for do loop indices RealArgs gives a list of functions whose numerical arguments must be of a guaranteed type usually real double precision For example if the function foo expects a single real argu ment it must be invoked as foo 0D0 in Fortran not foo 0 RealArgs foo defines the actual conversion for foo The default is to map NArgs over all arguments NArgs args returns args with all integers turned into reals Note that NArgs is not quite the same as N while it changes 1 to 1 it leaves e g m 1 intact so that array indices remain integers Newline specifies a string to be printed after each statement 7 3 Variable lists and Abbreviations OnePassOrder list order list such that the definition of each item comes before its first use MoveDepsRight 71 move variables among the lists r1 rn such that a definition does not depend on r further to the right MoveDepsLeft r1 fn move variables among the lists 77 7 such that a definition does not depend on r further to the left OnePassOrder r orders a list of interdependent rules such that the definition of each item var g
103. st desirable but also the most obvious candi date for concurrent execution FormCalc can distribute the helicity loop in two ways paral lel execution via fork wait on the cores of the local CPU i e SMP and vectorization using the CPU s SIMD instructions that is the squared matrix element is computed for several helicity combinations at once or both The include file distrib h chooses the distribution properties of the compiled code Distribution of scans over parameter space is described in Sect 5 2 9 Parallel execution with fork wait is switched on by setting the preprocessor variable PARALLEL in distrib h to 1 The drawback of this method is that it competes for compute cores in particular with Cuba In the usual settings of few e g 4 or 8 cores per system the cores should be assigned to Cuba since it computes entire phase space points in parallel not just the helicity loop The number of cores is determined from the environment variable FCCORES or if that is not set from the number of cores reported by the system minus the current load average Vectorization is turned on automatically by the configure script if it detects that the CPU and compiler are capable of it If this is in conflict with plans to send the executable on a cluster of computers with heterogeneous SIMD capabilities run configure with the generic op tion Then again SIMD instructions are typically available on recent computers SSE3 since 2004 AVX since 201
104. sult Folder squaredme the subdirectory of the code directory into which the generated code is written FilePrefix ey a string prepended to the filenames of the generated code SymbolPrefix ia a string prepended to global symbols to prevent collision of names when more than one process is linked FileHeader if On the file header pin din generated by FormCalc 8 4 t n endif n n SubroutineIncludes FileIncludes per subroutine include statements FileIncludes per file include statements include decl h include inline h n include contains h n ExtraRules specifies additional transformation rules that are applied to the modules of the amplitude before the loop integrals are abbreviated TreeSquare determines whether the square of the tree level amplitude is added to the result This is the case unless one needs to compute only the interference term usually for testing LoopSquare determines whether the square of the one loop amplitude is added to the one loop result i e whether the SquaredME subroutine shall compute 2 Re Mfi M Loop IM1 100p rather than only 2Re Mir M1 100p Without the LM 100p term which com pletes the square Miree M4100 the one loop result may become negative if Mee is very small The contribution of the same order from the two loop result 2 Re M treeM 2100p is much smaller than M1 100p in this case so that there is no inconsistency with the order in perturbation
105. t True only one abbreviation for the product is introduced Preprocess specifies a function to be applied to all subexpressions before introducing ab breviations for them The abbreviations introduced with Abbreviate are returned by Subexpr This works sim ilar to Abbr and also the OptimizeAbbr function can be applied in the same way Subexpr the list of abbreviations introduced for subexpressions so far Subexpr args executes Abbreviate args locally i e without registering the subexpressions permanently and returns list of subexpressions abbreviated expression Introducing abbreviations for subexpression has three advantages e The overall structure of the abbreviated expression becomes clearer e Duplicate subexpressions are computed only once e When writing out code for the abbreviations with WriteSquaredME the abbreviations are sorted into categories depending on their dependence on kinematical variables and are thus computed only as often as necessary The combined effect of the latter two points can easily lead to a speed up by a factor 3 4 7 Resuming Previous Sessions Loading a list of abbreviations from a previous FormCalc session does not by itself mean that CalcFeynAmp or Abbreviate will use them in subsequent calculations To this end they must be registered with the system There are two functions for this RegisterAbbr abbr register a list of abbreviations RegisterSu
106. t value comes before its use in the right hand sides of other rules When OnePassOrder detects a recursion among the definitions of a list it deposits the offending rules in an inter nal format in 0nePassDebug as debugging hints MoveDepsRight 71 rn shuffles variable definitions var gt value among the lists of rules r such that the definitions in each list do not depend on definitions in r further to the left For example MoveDepsRight a gt b b gt 5 produces b gt 5 a gt b i e it moves a gt b to the right list because it depends on b MoveDepsLeft 1 rn shuffles variable definitions var gt value among the lists of rules 84 7 LOW LEVEL FUNCTIONS FOR CODE OUTPUT r such that the definitions in each list do not depend on definitions in r further to the right For example MoveDepsLeft a gt b b gt 5 produces b gt 5 a gt b 1 Le it moves b gt 5 to the left list because that depends on b SplitSums expr split expr into a list of expressions such that index summations apply to the whole of each part ToDoLoops list categorize list into patches that must be summed over the same set of indices DoLoopl expr ind a do loop of expr over the indices ind SplitSums expr splits expr into a list of expressions such that index sums marked by SumOver always apply to the whole of each part SplitSunms expr wrap applies wrap to the coefficients of the SumOve
107. t to Thomas Hahn Max Planck Institut fiir Physik Werner Heisenberg Institut Fo hringer Ring 6 D 80805 Munich Germany e mail hahn feynarts de CONTENTS Contents 1 General Considerations 2 Installation 3 Generating the Diagrams 4 Algebraically Simplifying Diagrams Al lt CalereynAmpre tire he ah A OS Y E RIA ed 4 2 DecltareProces8 oe atta aces a a a0 eae eS Re ES ng ae a 4 3 Clearing Combining Selecting aa tes Bah wh EA de a 4 4 Ingredients of Feynman amplitudes o ooo 0000004 4 5 Handling Abbreviations eos Se A a 4 6 More Abbreviations aoaaa aaa eee ee ee eee 4 7 Resuming Previous Sessions 2 ead e RA 4 8 Fermionic Matrix Elements 00 ee eee eee 4 9 Colour Matrix Elements 00 eee ee ee eee 4 10 Putting together the Squared Amplitude o o ooo o o 4 11 Polarization Sums 2 200000 eee ee ee eee 4 12 Analytic Unsquared Amplitudes o ooo oo o 4 13 Checking Ultraviolet Finiteness ccs gah ee eae a 4 14 Useful Functions sipeer et ai e a eee ane ee ee 5 Tools for the Numerical Evaluation Bak Generating COS ssh A IS A ae ia 5 2 Running the Generated Code na Ben 52t Process definition ion oa rl ho Bcc ey oR te tok WR Be ee 5 2 2 Building up phase space a hy eR Gia MR 523 Variables szp daen a A A E E e LSS 52A NCUS e E a a ey A e Y E A A 5 2 5 ACORDAR A A een 5 2 6 Integration parameters prin Baw AA A
108. the declarations for xsection F MtoN F contains the kinematics dependent code for a M gt N process MtoN h contains the declarations for MtoN F model_x F currently one of model_sm F model_mssm F model_thdm F initializes the model parameters model_x h contains the declarations for model_x F lumi_y F currently one of lumi_parton F lumi_hadron F lumi_photon F calcu lates the parton luminosity lumi_y h contains the declarations for lumi_y F util h contains the declarations for the functions in the util library looptools h is the LoopTools include file renconst h declares the renormalization constants see Sect 5 4 In this setup the choice of parameters is directed by the two files process h and run F which include one each of Kinematics definitions 1to2 F 2to2 F 2to3 F Convolution with PDFs lumi_parton F lumi_hadron F lumi_photon F Model initialization model_sm F model_mssm F model_thdm F 5 2 1 Process definition All process specific definitions are given in process h The external particles are declared in the lines 52 5 TOOLS FOR THE NUMERICAL EVALUATION define TYPEI t define MASSI mi define CHARGE c where each t is one of the symbols SCALAR FERMION VECTOR PHOTON or GLUON PHOTON is the same as VECTOR except that longitudinal polarization states are not allowed and GLUON is just an alias for PHOTON As in FormCalc the momenta masses and polarization ve
109. tion 7 DiagramType 17 Dimension 10 31 35 dimension 22 dimensional reduction 6 10 regularization 6 10 89 90 Dirac equation 8 Dirac matrix 20 DiracChain 20 Divergence 37 Dminus4 11 31 35 37 Dminus4Eps 31 35 dot product 18 DotExpand 10 31 35 DottedSpinor 21 Drivers 41 drivers 40 e n 18 EditCode 10 31 35 Eps 18 Evanescent 10 ExceptDirac 15 ExpandSums 38 expansion 13 external fermions 28 external spinors 11 fermion chain 11 fermion traces 8 FermionChains 10 fermionic matrix elements 28 FermionicQ 17 FermionOrder 10 FeynArts 7 Fierz transformation 11 FileTag 10 Finite 18 FormAmp 15 FormDot 14 FormMat 14 FormPre 14 FormQC 14 FormQF 14 FormSub 14 Fortran code 41 INDEX four vector 36 Fuse 26 Gamma5Test 10 Gamma5ToEps 10 GaugeTerms 35 generators of SU N 19 generic amplitude 7 gluon indices 32 Gram determinant 18 Hel n 30 helicity 29 helicity matrix elements 29 helicity reference vector 29 HelicityME 11 14 30 IGram 18 index contractions 8 InsertionPolicy 10 insertions 12 installation 7 internal definitions 16 Invariants 15 invariants 15 simplification 16 inverse Gram 18 InvSimplify 15 k n 18 Keep 28 kinematical simplification 12 kinematics 16 18 leaf count 24 levels in diagrams 7 10 Levi Civita tensor 18 local terms 8 loop integral symmetrized 12 Lo
110. tors and spinors into the amplitude amp The external polarizations pol can be given either as a string with el ements O for right handed left handed and longitudinal polarization or as a list of integers 1 1 0 4 13 Checking Ultraviolet Finiteness 37 4 13 Checking Ultraviolet Finiteness One way of checking ultraviolet finiteness is to replace the one loop integrals by their di vergent parts and see if the coefficient of the divergence adds up to zero The function UVDivergentPart takes the 1 D 4 term of each one loop integral This means that non divergent integrals are set to zero UVDivergentPart expr replace all loop integrals in expr by their UV divergent part UVSeries expr similar to UVDivergentPart but perform a series expansion in Dminus4 Divergence asymbol representing the dimensionally regularized divergence A 2 4 D Dminus4 asymbol representing D 4 To assert UV finiteness of an expression one can check that the following expression is true uvcheck UVDivergentPart expr Simplify FreeQ uvcheck Divergence Note that models may have hidden parameter relations e g CW2 MW2 MZ2 in the Stan dard Model which need to be inserted in order to find analytical cancellation of divergences The UV divergent integrals are Ap m mA O 1 Co 5 O 1 Ago m ne O 1 Cooi 2 0 1 By A O 1 Cooii a O 1 By 2 00 Cooij at O 1
111. tput and choose its second argu ment the cross section results Data 1 1 2 This example can be extended a little to produce a one dimensional plot where e g for each value of tan f the minimum and maximum of the cross section with respect to M40 is recorded lt lt Graphics FilledPlot xsmin sqrtS_ TB_ FindMinimum xs sqrtS MAO TB MAO 1003 1 xsmax sqrtS_ TB_ FindMinimum xs sqrtS MAO TB MAO 1007 11 FilledPlot xsmin 500 TB xsmax 500 TB TB 5 507 5 4 Renormalization Constants FormCalc provides a number of functions to facilitate the computation of renormalization constants RCs FormCalc makes a conceptual distinction between the definition and the calculation of RCs 5 4 1 Definition of renormalization constants FormCalc regards those symbols as RCs for which a definition of the form RenConst rc exists The purpose of the definition is to provide the functional relationship between the RC and the self energy from which it is calculated for example RenConst dMHsqi ReTilde SelfEnergy S 1 gt S 1 MH 70 5 TOOLS FOR THE NUMERICAL EVALUATION Note that this definition is quite generic in particular it contains no details about the selec tion of diagrams It is intentional that the specifics of the diagram generation and calculation are chosen only when the RCs are actually calculated because this allows the user to make se lections on a process
112. ut in the same program unit Note these options may also be given to WriteExpr which simply hands them down to PrepareExpr They cannot be set using SetOptions WriteExpr however WriteExpr option default value HornerStyle True whether to order expressions according in Horner form FinalCollect False whether to collect factors in the final expression FinalFunction Identity a function to apply to the final expression before write out Type False the type of the expressions False to omit declarations TmpType ComplexType the type of temporary variables IndexType False the type of indices RealArgs A0 A00 B0 B1 functions whose arguments must be BOO B11 B001 of a guaranteed type default real B111 DBO DB1 DBOO DB1i1 BOi C0i D0i E0i FOi Bget Cget Dget Eget Log Sqrt Newline un a string to be printed after each expression HornerStyle specifies whether expressions are ordered in Horner form before writing them out as code FinalCollect chooses a final collecting of common factors after going over to Horner form 7 3 Variable lists and Abbreviations 83 just before write out to code FinalFunction specifies a function to be applied to the final expressions just before write out to code This function can be used to apply language specific translations Type determines whether declarations shall be generated for the variables and of which type If a string is given e g Type gt double preci
113. utting together the Squared Amplitude 33 4 10 Putting together the Squared Amplitude Now that CalcFeynAmp has calculated the amplitudes and HelicityME and ColourME have produced the helicity and colour matrix elements the remaining step is to piece together the squared matrix element M or more generally products like M1M3 For example M1 could be the sum of the one loop diagrams and M3 the sum of the tree diagrams This is non trivial only if there are helicity or colour matrix elements Mat 7 j which have to be put in the right places Specifically if M and M are written in the form My ay Fi SUN1 a21 F2 SUN1 Y a Fi SUN and ij M2 by F1 SUN1 bz F2 SUN1 bj Fi SUN ij MM becomes M1M3 a b Mat F1 F1 Mat SUN1 SUN1 a71bj Mat F2 F1 Mat SUN1 SUN1 oS ajxb Mat Fi Fj Mat SUNK SUNC ijkl The coefficients aj and bj are known as form factors For efficiency they are usually com j y ney y puted separately in the numerical evaluation so that the final expression for the squared matrix element is easily summed up e g in Fortran as do 1 i 1 ofFsin M do 1 j 1 ofFsin M2 do 1 k 1 Hof SUNs in My do 1 1 1 Hof SUNs in M2 result result a i k DCONJG b j 1 MatF i j MatSUN kx 1 1 continue While this is arguably the most economic way to evaluate a squared amplitude numerically it is also possible to directly obtain the squared matrix el
114. which invokes the individual modules If mod is given as a dot product name delim the delimiter is used to separate the N suffix For example foo _ will evaluate to foo for a single file and to foo_1 foo_2 etc for multiple files ToArray s turns the symbol s into an array reference by taking it apart into letters and digits e g Var1234 becomes Var 1234 ToArray expr s1 2 turns all occurrences of the symbols s NNN s NNN etc in expr into s NNN s NNN etc Renumber expr var var2 renumbers all var n var gt n in expr 86 7 LOW LEVEL FUNCTIONS FOR CODE OUTPUT MaxDims Largs returns a list of all distinct functions in args with the highest indices that appear e g MaxDims foo 1 2 foo 2 1 returns foo 2 217 7 4 Declarations SubroutineDecl name the declaration for the subroutine name VarDecl vars type the declaration of vars as variables of type type Nos aa a VarDec1 Common com a the declaration of common block com with individual variables a a as above VarDecl NameMap com a the declaration of common block com with an array plus a preprocessor map of the a onto array elements a as above VarDecl NotEmpty a output only if at least one variable list in a is non empty DoDecl i range the declaration of a do loop of i over range CallDec1 subs the invocations of subroutines subs Dim i the highest value the index i takes on SubroutineDec
115. y1Chain 6 signifies that the following sigma matrix has upper spinor indices 7 signifies that the following sigma matrix has lower spinor indices 1 the spinor metric e na a Spinor p m s an undotted 2 spinor DottedSpinor p m s a dotted 2 spinor 4 4 Ingredients of Feynman amplitudes FormCalc also introduces some model dependent symbols It can be argued that this is not a good practice but the advantage of doing so is just too great to ignore as it can speed up calculations by as much as 15 This is because e g MW2 is an ordinary symbol while MW 2 is a non atomic expression the internal representation is Power MW 2 To change or add to these definitions edit FormCalc m and find the section beginning with the line global definitions for specific models Currently the following model dependent symbols are introduced by FormCalc for the Standard Model Alfa Alfa2 Alfas Alfas2 CW2 SW2 MW2 MZ2 MH2 MLE2 ME2 MM2 ML2 MQU2 MU2 MC2 MT2 MQD2 MD2 MS2 MB2 SMSimplify expr ed the fine structure constant a 7 and its square 2 the strong fine structure constant a and its square the spelling with an f was chosen so as not to collide with the CERNIib function ALPHAS2 cos w and sin 0w the squares of various masses Simplify with SW2 gt 1 CW2 and CW2 MW2 MZ2 22 4 ALGEBRAICALLY SIMPLIFYING DIAGRAMS for the MSSM CA2

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