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1. Advanced Computing and Analysis Techniques in Physics ResearchFebruary 22-27, 2010, Jaipur, India Methodology of Computations in Theoretical Physics --- Summary --- Peter Uwer

2. Statistics 5 + 4 + 6 = 15 presentations 450 min = 7.5 h In total 367 transparencies, 1.2 min / slide Average number of transparencies: 24.4 / talk Extreme values: min: 10, max: 56

3. Where the speakers came from Europe: 8, Japan: 2, Russia: 2, US: 2

4. Main topics • Automation of higher order corrections • Techniques for loop integrals • Computational aspects • Real corrections and subtractions • Computer Algebra • Various topics 6 2 2 2 3

5. Automation of higher order corrections What is the basic problem ? arbitrary unphysical scale [Daniel Le Maitre]

6. Automation of higher order corrections Example: Top-quark production + 1 Jet Born approximation LHC Next-to-leading order (NLO) O(10) diagrams 350 diagrams  Born approximation is not reliable, need to go beyond leading-order

7. Automation of higher order corrections Generic one-loop calculation ∫ 2 Leading-order, Born approximation n-legs ∫ * ∫ 2 + + Next-to-leading order (NLO) 2Re x virtual corrections (n+1)-legs, real corrections IR divergent IR divergent

8. Automation of higher order corrections Bottleneck in one-loop calculation: • Calculation of the virtual corrections • Many diagrams • Each with complicated analytical structure • Numerical stability and speed • Combination of virtual corrections with real ones • Cancellation of IR singularities, conceptually solved, but cumbersome if done by hand Need for new methods and automation Algorithms crucial for automation

9. Automation of higher order corrections A typical one-loop diagram  complicated function of many variables How do we calculate this efficiently?

10. Techniques for loop integrals Different approaches: Refinement of mixed approaches Improved integration methods New algorithms

11. Techniques for loop integrals [Tord Riemann] Basic idea: Make use of the fact that all the scalar one-loop integrals are known analytically  Derive reduction avoiding leading Gram determinants in the denominator Explicit reduction formulae are implemented in Computer code Exceptional configuration with vanishing Gram determinants are handled by special reduction (extrapolation)

12. Techniques for loop integrals [Tord Riemann] x allows to test numerical stability

13. Techniques for loop integrals [Giovanni Ossola] General structure of one-loop amplitude:

14. Automation of higher order corrections [Giovanni Ossola] Reduction at the integrand level Structures in red vanish after integration and their form is known  finite number of terms  Determine coefficients by solving linear system of equations OPP method (Ossola, Pittau, Papadopoulos)

15. Automation of higher order corrections [Giovanni Ossola] OPP method very powerfull Available as Fortran program CutTools Can be combined with automated amplitude generation (combination with HELAC already done) Many new results recently (pp->Wjjj,pp->ttbb, pp->ttjj) One-loop amplitudes solved ?

16. Techniques for loop integrals [Elise de Doncker] Basic idea: Recursive (deterministic) integration over Feynman parameter, combined with extrapolation use DQAGE from QUADPACK recursively

17. Techniques for loop integrals [Elise de Doncker] Six-point scalar integrals are reduced algebraically to 3- and 4-point scalar integrals 3- and 4-point integrals are then evaluated numerically Technique also applicable to tensor integrals Recursive integration might be interesting also for other fields

18. Techniques for loop-integrals [Mikhail Tentyukov] Sector decomposition to isolate singularities to find the decomposition of a complicated integrand highly non-trivial

19. Techniques for loop-integrals [Mikhail Tentyukov] FIESTA Feynman Integral Evaluation by a Sector decomposiTion Approach Computer algebra part in Mathematica combined with numerical integration routine Important: Publicly available Different Algorithms for sector decomposition Applicable to multi-loop integrals Important new 4-loop results Circumvent memory problem in Mathematica Own interpreter to process formulae of TeraByte length

20. Techniques for loop-integrals [Toshiaki Kaneko] Alternative algorithm for sector decomposition Sector decomposition based on computational geometry  implementation underway

21. Techniques for loop-integrals Methods rely on Increased computational power Increased main memory Parallelization is used frequently

22. Automation of higher order corrections [Theodoros Diakonidis] Apply reduction scheme presented by Tord Riemann to ggttgg @ 1-loop O(1000) Feynman diagrams with complex structure  Automation needed

23. Automation of higher order corrections DIANADiagram construction Output (form) Color2fortran.frm SUn.prc hex_m.frm : bub_m.frm 50 different Structures color.F FORTRAN OPT cFi_rtSum3(1…4) : cS_rtSum23(1…4) MAPLE INPUT cRank0.m(1…4) : cRank5.m(1…4) OPTIMIZATION ggttgg.m hex_mf.frm : bub_mf.frm gm(line,n1,…,n9) Spinor structures Passrt_hex.F : Passrt_bub.F Hex(Sum_6(4)) : Bubble(Sum_2(4)) MADGRAPHmomenta.dat Mainfortran program (QgrafFormMapleFortran) [Theodoros Diakonidis] Steering with shell scripts, process specific

24. Automation of higher order corrections Process independent automation based on Feynman diagrams and tensor reduction [Thomas Hahn]

25. Automation of higher order corrections [Thomas Hahn] Many new features in the Feynarts-System Tweaking model files Diagram selection Linear combination of fields (mass vs gauge states) Computational aspects Efficient Fortran code generation, abbreviations to remove common subexpressions Parallelization of parameter scans  very powerful tool but:

26. Real corrections and subtractions Generic one-loop calculation ∫ 2 Leading-order, Born approximation n-legs ∫ * ∫ 2 + + Next-to-leading order (NLO) 2Re x (n+1)-legs, real corrections IR divergent IR divergent

27. Real corrections and subtractions Problem: Phase space integration cannot be done in d dim. • Add and subtract a counterterm which is easy enough to be integrated analytically: Can be done numerically Construction of subtraction for real corrections more involved, Fortunately a general solution exists: Dipole subtraction formalism

28. Real corrections and subtractions Generic form of individual dipol: Leading-order amplitudes Vector in color space universal ! ! Color charge operators, induce color correlation Color charge operators, induce color correlation Color charge operators, induce color correlation Spin dependent part, induces spin correlation Spin dependent part, induces spin correlation Exampleggttgg: 36 (singular) dipoles  Automation required

29. Real corrections and subtractions [Rikkert Frederix] Automation of NLO subtraction terms Two different methods: Catani-Seymour subtraction Frixione-Kunszt-Signer subtraction  useful to interface with MC@NLO  Fully automated based on Madgraph: MadFKS

30. Real corrections and subtractions [Paolo Bolzoni] Extension of subtraction method to NNLO Much more involved due to double unresolved configuration Analytic integration of subtraction terms highly non trivial  Solution using: Mellin-Barnes representation Special summation algorithms for nested sums (XSummer in Form)

31. Computer Algebra -- Form Standard Tool in Theoretical Particle Physics if large expressions are encountered: Form by Jos Vermaseren et al. Important features: Expression size only limited by disk space (TB) Only local operations i.e. no factorization  Many ongoing developments Talks by Irina Pushkina and Mikhail Tentyukov

32. Computer Algebra -- Form [Irina Pushkina, Mikhail Tentyukov] New features: • Architecture independent file storage (32bit vs 64bit) • Checkpoints to save intermediate states • Steps towards open source (summer 2010?) • Two approaches to parallelisation: • Parform based on MPI for cluster • Tform based on threads for multi-core machines • Improved load balancing • Link to Grace system

33. Computer Algebra -- Form [Mikhail Tentyukov] Speed up in Form:

34. Techniques for Event generator tuning Tuning framework Professor [James Monk] Multi-dimensional interpolation Change Generator Parameters on the fly through interpolation in parameter space

35. Remarks Main programming language in Theoretical Particle Physics Fortran Powerful mixed approach: Analytic part is combined with numerical part, a chain of different tools is connected using scripts Numerical instabilities Switch to quadrupel and higher accuracy  Very active field, many new and important developments recently

36. Remarks Important progress concerning the automation of one-loop amplitudes  OPP Method

37. Final Remarks Apologies that not everybody could be mentioned Many thanks to all the speakers Many thanks to the audience of track 3 for their contribution in many lively discussions All talks are uploaded, if you want to see the details check Indico

38. Thank You

40. Automation of higher order corrections Traditional approach to tensor reduction [Passarino, Veltman 78] Major problem: (spurious) numerical instabilities for exceptional momentum configurations vanishing Gram determinants (integral bases degenerates “0/0”) in the example above: