The Harmonic Oscillator of One-loop Calculations

1. SFB meeting, 09.12.2010 – 10.12.2010, Karlsruhe The Harmonic Oscillator of One-loop Calculations B5 Peter Uwer Work done in collaboration with Simon Badger and Benedikt Biedermann arXiv 1011.2900, http://www.physik.hu-berlin.de/pep/tools

2. Motivation Harmonic oscillator of perturbative QCD: n-gluon amplitudes in pure gauge theory Why should we study the Harmonic Oscillator ? Simple system which shares many properties with more complicated systems Allows to focus on the interesting physics avoiding the complexity of more complicated systems very well understood  ideal laboratory to apply and test new methods  no complicated field content, only gauge fields in particular no fermions  general structure of one-loop corrections  well known IR structure, UV structure, color decomposition… Despite the simplifying aspects, n-gluon amplitudes are still not trivial

3. Motivation Number of pure gluon born Feynman diagrams: [QGRAF]

4. Tree level pure gluon amplitudes color-ordered sub-amplitudes [?] Generators of SU(N) with Tr[TaTb] = dab Sum over non-cyclic permutations notation: For large N, the color structures are orthogonal: Color-ordered amplitudes are gauge independent quantities!

5. Tree level pure gluon amplitudes  Important reduction in complexity

6. Evaluation of color ordered amplitudes Use color-ordered Feynman rules: Calculate only Feynman diagrams for fixed order of external legs (“= color-ordered”) 1,2,3,4,5 1,2,3,4,5 Example: A5= + Reduction: 25  10 diagrams

7. Nicer than Feynman diagrams: Recursion [Berends, Giele 89] colour ordered vertices off shell leg S S + = External wave functions, Polarization vectors

8. Born amplitudes via recursions Remark: Berends-Giele works with off-shell currents BCF, CSW “on-shell” recursions use on-shell amplitudes  on-shell recursions useful in analytic approaches, in numerical approaches less useful since caching is less efficient Berends-Giele: caching is trivial:

9. Born amplitudes via recursions calculation i j

10. Born amplitudes via recursions calculation i j

11. Tree amplitudes from Berends-Giele recursion not yet fully optimized [Biedermann, Bratanov, PU]  checked with analytically known MHV amplitudes

12. Color-ordered sub-amplitudes (NLO) [?] Color structures: Leading-color structure: Leading-color amplitudes are sufficient to reconstruct the full amplitude

13. The unitarity method I [Bern, Dixon, Dunbar, Kosower 94] Basic idea: = × = Tree Tree l1l1 = Tree Tree l2l2 [Cutkosky] color-ordered on-shell amplitudes! Cut reconstruction of amplitudes: × Tree Tree 

14. The unitarity method II [Badger, Bern, Britto, Dixon, Ellis, Forde, Kosower, Kunszt, Melnikov, Mastrolia, Ossala, Pittau, Papadopoulos,…] After 30 years of Passarino-Veltman reduction: [Passarino, Veltman ’78] Reformulation of the “one-loop” problem:  How to calculate the integral coefficients in the most effective way

15. Reduction at the integrand level: OPP Study decomposition of the integrand [Ossola,Papadopoulos,Pittau ‘08]  put internal legs on-shell  products of on-shell amplitudes

16. Reduction at the integrand level: OPP  coefficients of the scalar integrals are computed from products of on-shell amplitudes

17. Rational parts Doing the cuts in 4 dimension does not produce the rational parts Different methods to obtain rational parts: Recursion working in two different integer dimensions specific Feynman rules SUSY + massive complex scalar No rational parts in N=4 SUSY: [Bern, Dixon, Dunbar, Kosower]

18. Codes Rocket[Giele, Zanderighi] Blackhat / Whitehat? [Berger et al] Helac-1Loop Cuttools Samurai private codes [Bevilaqua et al] publicly available, additional input required to calculate scatteing amplitudes [Ossola, Papadopoulos, Pittau] [Mastrolia et al]

19. NGluon 1.0 [Badger, Biedermann,PU ’10] Publicly available code to calculate one-loop amplitudes in pure gauge theory without further input for the amplitudes http://www.physik.hu-berlin.de/pep/tools Available from: Required user input: number of gluons momenta helicities External libraries: QD [Bailey et al], FF/QCDLoop [Oldenborgh, Ellis,Zanderighi]

20. Some technical remarks Written in C++, however only very limited use of object oriented Operator overloading is used to allow extended floating point arithmetic i.e. double-double (real*16), quad-double (real*32) using qd Extended precision via preprocessor macros instead of templates Scalar one-loop integrals from FF [Oldenborgh] and QCDLoop [Ellis,Zanderighi] Entire code encapsulated in class NGluon NGluon itself thread save, however QCDLoop, FF most likely not

21. Checks Comparison with known IR structure  test of linear combination of some triangle and box integrals Comparison with known UV structure  test of linear combination of bubble integrals Analytic formulae for specific cases  test of entire result Collinear and Soft limits  powerful test, however only applicable in soft and collinear regions of the phase space

22. Scaling test IR and UV check always possible, however no direct test of the finite part Comparison with analytic results of limited use higher contributions in DFT not easy to interpret Independent method to assess the numerical uncertainty: Scaling test Basic idea: in massive theories masses needs to be rescaled as well, renormalization scale needs also to be rescaled

23. Scaling test Scaling can be checked numerically i.e. we calculate the same phase space point twice ? How can we learn something from this test For the mantissa of all rescaled floating point numbers will become different  different arithmetics at the hardware level  different rounding errors results will differ in digits which are numerically out of control

24. Scaling test Remark: test is not cheap: doubles runtime, however it gives reliable estimate of the numerical uncertainty, for cases where no analytic results are available In practical applications test should be used if: high reliability is requested (“luxury level”) previous (cheaper tests) indicate problems  may help saving runtime

25. Scaling test

26. Results: Numerical stability / accuracy ~ number of valid digits

27. Results: Numerical stability / accuracy ~ number of valid digits

28. Average accuracy

29. Bad point: bad points rule of thumb: adding one gluon doubles the fraction of bad points

30. Comparison with Giele, Kunszt, Melnikov ./NGluon-demo --GKMcheck

31. Comparison with Giele & Zanderighi ./NGluon-demo --GZcheck

32. Results: Runtime measurements no ‘tuned’ comparison done so far with competitors

33. Improved scaling [Giele, Zanderighi] [Badger, Biedermann, PU]

34. Comparison with proposal Ax2 as of 11/2009 achieved for “limited field” content What happened to the “Helac-1Loop” version announced for spring 2010?

35. Summary NGluon allows the numerical evaluation of one-loop pure gluon amplitudes without additional input Publicly available www.physik.hu-berlin.de/pep/tools Improved scaling behavior Fast and stable (12-14 gluons)  can compete with other private codes Can be used as framework for further developments Outlook: add massless quarks (internal/external) add massive quarks