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News from FormCalc and LoopTools Thomas Hahn, Michael Raucha MPP-2006-7 aMax-Planck-Institut fu¨r Physik Föhringer Ring 6, D–80805 Munich, Germany 6 0 0 TheFormCalcpackageautomatesthecomputationofFeynArtsamplitudesuptooneloopincludingthe 2 generation of a Fortran code for the numerical evaluation of the squared matrix element. Major new or n enhancedfeaturesinVersion5are: iterativebuild-upofessentially arbitrary phase-spacesincludingcuts, a convolutionwithdensityfunctions,anduniformtreatmentofkinematicalvariables. TheLoopToolslibrary J supplies the one-loop integrals necessary for evaluating the squared matrix element. Its most significant 0 extensions in Version 2.2 are thefive-point family of integrals, and complex and alternate versions. 3 1 1. Introduction Enhanced cache functionality, v • 8 FormCalc [1] is a Mathematica package for Betterinternalaccuracy,andacomplete 4 • the calculation of Feynman amplitudes. Am- quadruple precision build available on 2 1 plitudes generated by FeynArts [2] are sim- Intel x86/ifort platforms. 0 plified analytically and converted to a self- 6 contained Fortran code for the computation 2. FormCalc 0 of the squared matrix element. The present / 2.1. Phase-space h article describes the following features added p or enhanced in Version 5: FormCalc 5 no longer uses hand-tailored - phase-space parameterizations like its prede- p Iterative build-up of essentially arbi- cessor versions, but builds up the n-particle e • h trary phase-spaces with various cuts, phase-spaceiteratively by nested integrations : overthe invariantmassM andsolidangleΩ v Convolution with density functions for i i i • of each outgoing particle i. This procedure is X external particles, encoded in the subroutine Split: r a Uniform treatment of kinematic vari- • ables, Ω1 m1 √s Better modularity and code reusabil- m2 • ity, i.e. less cross-talk between program Ω2 modules. M 1 TheLoopToolslibrarysuppliestheimplemen- call Split mn 1 − tations ofthe scalarandtensorone-loopinte- M2 Ωn 1 grals necessary for running the Fortran code ...− generated by FormCalc. The extensions in Ω n Version 2.2 are: m n Additionofthefive-pointfamilyoffunc- • tions, Counting the degrees of freedom, there are (n 1) M-integrations and n Ω-integrations. A collective two-point function Bget − • Thecorrespondingphase-spaceparameteriza- similar to Cget and Dget, tion is Complex versions of most integrals, • 1 √s−m1 k1 Run-time selection of alternate versions 2√sZ dM1dΩ1 2 • for checking, m2+···+mn M1 m2 k − 2 A command-line interface, ×Z dM2dΩ2 2 • m3+ +mn ··· 2Thomas Hahn, Michael Rauch MPP-2006-7 Var(v) = the actual value of v. ×··· • Mn−2−mn−1 kn 1 Show(v)=the valueprintedinthe out- ×Zmn dMn−1dΩn−1 2− • put – to print e.g. t instead of cosθ. k dΩ n (1) Lower(v), Upper(v), Step(v) = the ×Z n 2 • lower limit, upper limit, and step width where dΩ = dcosθ dϕ . The particle’s mo- ofv. Ifthestepiszero,thecross-section i i i mentum k and cosθ are given in the respec- is integrated over v. i i tive decay’s rest frame. The ϕ1-integration CutMin(v),CutMax(v)= the lowerand is trivial because of axial symmetry. From • upper cuts on v. the practical point of view this looks as follows(this codeis takenalmostverbatimfrom There are two special variables: FIXED for FormCalc’s 2to3.F): fixed values, i.e. no integration, and TRIVIAL for trivial integrations. p = 0 ex = 0 2.3. Cuts ey = 0 Split allows to place cuts on each M- and ez = 1 cosθ-integration. The ϕ-integration is not minv = sqrtS modified in the present setup. msum = MASS3 + MASS4 + MASS5 Cuts restricting M i call Split(5, dble(MASS5), Cut on Key & p, ex,ey,ez, minv, msum, fac, 0, M CUT_MREM i & Var(XMREM5), Var(XCOSTH5), Var(TRIVIAL)) E CUT_MREM_E i k CUT_MREM_K i call Split(4, dble(MASS4), E CUT_MREM_ET & p, ex,ey,ez, minv, msum, fac, 0, T,i k CUT_MREM_KT & Var(FIXED), Var(XCOSTH4), Var(XPHI4)) T,i y CUT_MREM_RAP i η CUT_MREM_PRAP call VecSet(3, dble(MASS3), p, ex,ey,ez) i One starts with the initial reference direction Cuts restricting cosθ i in(ex,ey,ez)andnoboost,p = 0. Theavail- Cut on Key ableenergyisgiveninminvandthesumofex- cosθ CUT_COSTH i ternal masses in msum. The Split subroutine cosθˆ CUT_COSTHCMS i is then called (n 1) times for an n-particle E CUT_COSTH_E − i finalstate. Thereferencedirection,theboost, k CUT_COSTH_K i minv, and msum are automatically adjusted along the way for the respective remaining In practice, the application of cuts works as subsystem and ultimately determine the re- follows: maining n-th vector unambigously, which is key = 0 then simply set by VecSet. About the integration variables more will CutMin(XMREM5) = E5MIN be said in the next section. For the moment, key = key + Cut(CUT_MREM_E, CUT_MIN) note that the X in XMREM5 refers to the ratio, i.e. XMREM5runs from0 to 1. The actual inte- CutMin(XCOSTH5) = -(1 - COSTH5CUT) CutMax(XCOSTH5) = +(1 - COSTH5CUT) gration borders are determined internally by key = key + Split. & Cut(CUT_COSTH, CUT_MIN + CUT_MAX) 2.2. Variables FormCalc 5 introduces a new homogeneous call Split(5, dble(MASS5), & p, ex,ey,ez, minv, msum, fac, key, system for all (potential) integration vari- & Var(XMREM5), Var(XCOSTH5), Var(TRIVIAL)) ables. Each variable is referred to by a pre- ... processorconstant,e.g.SQRTSorXCOSTH.The followingpartscanbe accessedviapreproces- The value of the cut is deposited in CutMin sor macros: or CutMax and the cut registered by adding News from FormCalc and LoopTools 3 an identifier for the cut to the integer key, Kinematics definitions: • e.g. Cut(CUT_MREM_E, CUT_MIN) specifies a 1to2.F, cut on the energy (CUT_MREM_E) from be- 2to2.F, low (CUT_MIN) which is used to restrict the 2to3.F, invariant-mass integration (CUT_MREM_E). Note that these cuts really restrict the in- Convolution with PDFs: • tegration limits. They do not introduce veto lumi_parton.F, functions (1 in wanted, 0 in unwanted areas) lumi_hadron.F, intotheintegrand,whichcanseverelyhamper lumi_photon.F, convergence. Model initialization: Inaddition,foreachexternalparticleisev- • model_sm.F, eral kinematical quantities are available: model_mssm.F, momspec(SPEC_M,i) model_thdm.F. • – mass m, There is almost no cross-talk between differ- momspec(SPEC_E,i) ent modules which are in that sense ‘univer- • – energy E, sal.’ Also, the main program has been rad- ically slimmed down in FormCalc 5. It now momspec(SPEC_K,i) only scans the command line and invokes • – momentum k, call ProcessIni(...) momspec(SPEC_ET,i) call ParameterScan(...) • – transverse energy E , T All further action is decoupled from the main momspec(SPEC_KT,i) program and can easily be called from any • – transverse momentum k , application. It is thus straightforward to use T FormCalc-generatedcode in own programs. momspec(SPEC_RAP,i) • – rapidity y, 3. LoopTools momspec(SPEC_PRAP,i) LoopToolsisalibraryfortheone-loopinte- • – pseudo-rapidity η, grals. ItisbasedonFF[5]andhasaFortran, C/C++,andMathematicainterface. Itisref- momspec(SPEC_DELTAK,i) erenced by the FormCalc-generated code but • – the difference E k. − canof course be used also without FormCalc. 2.4. Convolution 3.1. Five-point functions Withthenewvariablesystem,theconvolu- The most significant addition is the five- tion with arbitrary parton distribution func- point family of functions – the scalar inte- tions can easily be achieved. Three modules gralE0 and the tensor coefficients up to rank are already included in FormCalc 5: four. The default versions use the Denner– Dittmaier decomposition [6] in which one in- lumi_parton.F = initial-state partons, • verse power of the Gram determinant is can- no convolution. celled and thus has considerably better nu- lumi_hadron.F= initial-state hadrons, merical stability than the Passarino–Veltman • convolution with hadronic PDFs from [7] decomposition, which is also available for the LHAPDF library [3]. comparison. 3.2. Combined two-point function lumi_photon.F= initial-state photons, • The two-point functions have been united convolutionwithCompAZspectrum[4]. into the Bgetfunction which workssimilar to 2.5. Modularity and Code-Reusability itsCget,Dget,andEgetcounterparts,inpar- The choice of parameters is directed as in ticularitcachesitsresults. Compatibilityrou- previous versions by the two files process.h tinesfortheoldB0,B1,etc.areofcourseavail- and run.F, which include one each of able. The reasonis mainly cache efficiency in 4Thomas Hahn, Michael Rauch MPP-2006-7 view of the five-point decomposition: 3.6. Environment variables Version key, debug key, and debug range calls Eget 5Dget −→ canbesetalsofromthe‘outside,’i.e.through call (5 4)Cget environment variables, thus making it unnec- −→ · essary to recompile the code. Unfortunately, call (5 4 3)Bget (2) the KeyX and DebugX labels cannot be used −→ · · 3.3. Complex versions ontheshellcommand-line,but-1offersacon- Versions of the LoopTools functions for venient way to set all bits of an integer, thus complex parameters have been added as far havingthesameeffectasKeyAllorDebugAll. as they are contained in FF, i.e. currently For example: only some special cases for the complex D0 setenv LTVERSION -1 are available. setenv LTDEBUG -1 real versions complex versions setenv LTRANGE 4711-4715 A0, A00 A0C, A00C 3.7. Command-line Interface B0, B1, ... B0C, B1C, ... The new command-line interface is useful B0i, Bget B0iC, BgetC in particular for testing and debugging. It C0, C0i, Cget C0C, C0iC, CgetC liststheN-pointscalarandtensorcoefficients D0, D0i, Dget D0C, D0iC, DgetC corresponding to the number of arguments, E0, E0i, Eget E0C, E0iC, EgetC i.e. 3 arguments = B, 6 arguments = C, 3.4. Alternate versions etc. Currently only the real versions are ac- For some functions alternate versionsexist, cessible through the command-line interface, mostofwhicharebasedonanimplementation mainly because it is not straightforward syn- by Denner. The user can choose at run-time tactically to specify complex parameters on whether the default version ‘a’ (mostly FF) the command-line. or the alternate version ‘b’ (mostly Denner) is used and whether checking is performed. 3.8. LoopTools caches This is determined by the version key: The internal caching mechanism is mean- while used by the following functions: Bget, 0*key compute version ‘a’, Cget, Dget, Eget, BgetC, CgetC, DgetC, and 1*key compute version ‘b’, EgetC. Obviously the former system with 2*key compute both, compare, return ‘a’, its getcachelast and setcachelast call for 3*key compute both, compare, return ‘b’. each individual cache was no longer practica- Usage is as in ble for all those caches. call setversionkey(2*KeyC0 + 3*KeyD0) The new cache-management functions op- erate on all caches simultaneously: The following keys for alternate versions are currently available: KeyA0, KeyBget, KeyC0, call clearcache KeyD0,KeyEget,KeyEgetC.KeyAllcomprises • – clears all caches, all of these. call markcache 3.5. Debugging • – marks the current position, Debugging output can be turned on simi- larly with e.g. call restorecache • – reverts to the last marked position. call setdebugkey(DebugC + DebugD) Forcompatibility,LoopToolsstillincludestwo Identifiers range from DebugB to DebugE and routines getcachelast and setcachelast. are summarized by DebugAll. The integrals Theyworkonlyapproximatelyasbefore,how- are listed in the output with a unique serial ever, and are therefore deprecated. number. If the list of integrals becomes too Furthermore, the cache mechanism itself long,onecanselectonlyarangeofserialnum- has been improved and now uses a binary bers for viewing, as in search method. Cache lookups are thus now call setdebugrange(4711, 4715) faster, but as this has not often been a bot- This makes it easy to monitor ‘suspicious’ in- tleneck, the impact on program performance tegrals. will typically be minor in most applications. News from FormCalc and LoopTools 5 3.9. Accuracy alternate versions, by program call or The accuracy of the tensor reduction has through an environment variable. been improved through the use of an LU de- In the same manner, debugging output compositionfortheGrammatrixandquadru- • canbe turnedonandoffindividuallyat ple precision at strategic points internally. run-time. Thanks to the Intel ifort compiler, the latter is now widely available on the x86 platform. The cache system has been extended A complete quadruple-precision build can • and improved. be chosen with an alternate makefile. In particular the quadruple-precision version of the A new command-line interface is useful • frontend can be useful to check cases where a for testing and debugging. severe loss of precision is suspected. Theaccuracyofthetensorreductionhas • been enhanced. 4. Summary A complete quadruple-precision build FormCalc 5 is the current release of the • canbechosenwithanalternatemakefile FormCalcpackagewiththefollowingnewfea- (e.g. make -f makefile.quad-ifort). tures and extensions: New kinematics routines can build up Both packages are available as open source • almost arbitrary phase-spaces with a and stand under the GNU Lesser General wide range of cuts possible. PublicLicense(LGPL).Theycanbeobtained from the Web sites FormCalc’s driver code consists of in- http://www.feynarts.de/formcalc, • tuitively programmed, concise modules http://www.feynarts.de/looptools. withalmostnocross-talkbetweenthem. Calling FormCalc-generated code from Acknowledgments other applications has been much sim- We thank S. Dittmaier for significant help plified. with the implementation and comparison of Convolution with arbitrary distribution the five-point functions and C. Schappacher • functions is possible. The following for intensive beta-testing. common cases are available out-of-the- box (if FormCalc came in one): REFERENCES – partons (no convolution), 1. T. Hahn, M. Pérez-Victoria,Comp. – hadrons (uses LHAPDF), Phys. Commun. 118 (1999) 153 [hep-ph/9807565]. – photons (uses CompAZ). 2. T. Hahn, Comp. Phys. Commun. 140 All possible integration variables are (2001) 418 [hep-ph/0012260]. • treateduniformlyanddetailedkinemat- 3. M.R. Whalley, D. Bourilkov, icalinformationisavailableforallexter- R.C. Group, hep-ph/0508110, nal legs. http://hepforge.cedar.ac.uk/lhapdf. 4. A.F. Zarnecki, Acta Phys. Polon. B34 LoopTools 2.2 is the latest version of the (2003) 2741 [hep-ex/0207021]. LoopTools library with the following novel- 5. G.J. van Oldenborgh, ties: J.A.M. Vermaseren, Z. Phys. C 46 (1990) 425. The scalar five-point function including • 6. A. Denner, S. Dittmaier, Nucl. Phys. B its tensor coefficients up to rank four is 658 (2003) 175 [hep-ph/0212295]. provided. 7. G. Passarino,M.J. Veltman, Nucl. Phys. Complex versionsare available as far as B 160 (1979) 151. • implemented in FF. Checking can be enabled at run-time, • individually for all integrals which have