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

2. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 2 Motivation 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 Harmonic oscillator of perturbative QCD: n-gluon amplitudes in pure gauge theory

3. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 3 Motivation Number of pure gluon born Feynman diagrams: nNumber of diagrams 44 525 6220 72485 834300 9559405 1010525900 [QGRAF]

4. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 4 Tree level pure gluon amplitudes Sum over non-cyclic permutations Generators of SU(N) with Tr[ T a T b ] =  ab For large N, the color structures are orthogonal: Color-ordered amplitudes are gauge independent quantities! color-ordered sub-amplitudes [?] notation:

5. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 5 Tree level pure gluon amplitudes nNumber of diagrams Number of color ordered diag. 443 52510 622036 72485133 834300501 95594051991 10105259007335  Important reduction in complexity

6. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 6 Evaluation of color ordered amplitudes Use color-ordered Feynman rules: Calculate only Feynman diagrams for fixed order of external legs (“= color-ordered”) Example: A5=A5= Reduction: 25  10 diagrams + 1,2,3,4,5

7. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 7 Nicer than Feynman diagrams: Recursion   = + External wave functions, Polarization vectors [Berends, Giele 89] colour ordered vertices off shell leg

8. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 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. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 9 Born amplitudes via recursions calculation i j

10. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 10 Born amplitudes via recursions calculation i j

11. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 11 Tree amplitudes from Berends-Giele recursion [Biedermann, Bratanov, PU] not yet fully optimized  checked with analytically known MHV amplitudes

12. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 12 Color-ordered sub-amplitudes (NLO) Leading-color amplitudes are sufficient to reconstruct the full amplitude Color structures: Leading-color structure: [?]

13. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 13 The unitarity method I Basic idea: Cut reconstruction of amplitudes: = =  [Bern, Dixon, Dunbar, Kosower 94] color-ordered on-shell amplitudes! l1l1 l2l2 [Cutkosky] Tree = × ×

14. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 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: Reformulation of the “one-loop” problem:  How to calculate the integral coefficients in the most effective way [Passarino, Veltman ’78]

15. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 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. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 16 Reduction at the integrand level: OPP  coefficients of the scalar integrals are computed from products of on-shell amplitudes

17. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 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 [Bern, Dixon, Dunbar, Kosower] No rational parts in N=4 SUSY:

18. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 18 Codes  Rocket [Giele, Zanderighi]  Blackhat / Whitehat? [Berger et al]  Helac-1Loop  Cuttools  Samurai private codes publicly available, additional input required to calculate scatteing amplitudes [Bevilaqua et al] [Ossola, Papadopoulos, Pittau] [Mastrolia et al]

19. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 19 NGluon 1.0 Publicly available code to calculate one-loop amplitudes in pure gauge theory without further input for the amplitudes Available from: http://www.physik.hu-berlin.de/pep/tools Required user input:  number of gluons  momenta  helicities External libraries: QD [Bailey et al], FF/QCDLoop [Oldenborgh, Ellis,Zanderighi ] [Badger, Biedermann,PU ’10]

20. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 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. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 21 Checks  Comparison with known IR structure  Comparison with known UV structure  Analytic formulae for specific cases  Collinear and Soft limits  test of linear combination of some triangle and box integrals  test of linear combination of bubble integrals  test of entire result  powerful test, however only applicable in soft and collinear regions of the phase space

22. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 22 Scaling test  IR and UV check always possible, however no direct test of the finite part  Comparison with analytic results of limited use 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  higher contributions in DFT not easy to interpret

23. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 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. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 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. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 25 Scaling test

26. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 26 Results: Numerical stability / accuracy ~ number of valid digits

27. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 27 Results: Numerical stability / accuracy ~ number of valid digits

28. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 28 Average accuracy

29. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 29 Bad point: bad points rule of thumb: adding one gluon doubles the fraction of bad points

30. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 30 Comparison with Giele, Kunszt, Melnikov./NGluon-demo --GKMcheck

31. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 31 Comparison with Giele & Zanderighi./NGluon-demo --GZcheck

32. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 32 Results: Runtime measurements no ‘tuned’ comparison done so far with competitors

33. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 33 Improved scaling [Giele, Zanderighi] [Badger, Biedermann, PU]

34. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 34 Comparison with proposal Ax2 as of 11/2009 What happened to the “Helac-1Loop” version announced for spring 2010? achieved for “limited field” content

35. Peter Uwer | The Harmonic Oscillator of One-loop Calculations | SFB 9./10. Dec 2010, KA | page 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