1. Numerical approach to multi- loop integrals K. Kato (Kogakuin University) with E. de Doncker, N.Hamaguchi, T.Ishikawa, T.Koike, Y. Kurihara, Y.Shimizu, F. Yuasa The XXth International Workshop High Energy Physics and Quantum Field Theory September 25, 2011 Sochi, Russia

2. motivation Theoretical prediction for High Energy Physics can be done by the perturbative calculation in Quantum Field Theory. (QFTHEP) Sometimes important information comes from multi-body final states. Experimentalists presents (after hard work) high-statistic data. This requires higher order calculation.  Large scale calculation is inevitable.

3. How to handle large scale computation? It is beyond man-power. Automated systems to perform perturbative calculation in QFT have been developed. Many systems are successfully working in tree and 1-loop level. GRACE, CompHEP, CalcHEP, FeynArt/Calc, FDC,…

4. Next generation of systems should manage 2-loop and higher orders. One of the essential components is the general multi-loop calculation library. Formulae for 2-loop integrals are given for many cases: However, it seems to be difficult to write ‘general solution’.

5. multi-loop integrals(scalar) Introduce Feynman parameters to combine denominators Integrate by loop momenta Polynomials of x’s: depends on masses and momenta TARGET

6. Integration of singular function 01 0101 Analytic DCMContour deformation Numerical would be hard for multi-x case example

7. Direct Computation Method(DCM) Target Simple example Analytic If D has zero in the integration domain, we keep finite. If D has no zero in the integration domain, take and perform numerical integration. Denominator is positive: Numerical evaluation is possible.

8. DCM( cont.), extrapolation Wynn’s algorithm ( Math. magic) Input Even k terms give good estimation DCM= regularized integration + series extrapolation

9. Examples: 2-loop box (Yuasa) Following loop diagrams are successfully calculated by DCM. Mostly scalar integrals, but inclusion of numerator will be straightforward since DCM is based on numerical integration. 1-loop : 3, 4, 5, 6 – point functions 2-loop: 2, 3, 4 – point functions Following slides are the results for 2-loop box.

10. Numerical results of Two-loop planar box with masses m=50 GeV, M = 90 GeV, t = -100 2 GeV 2 ACAT2011 5-9 September 2011F.Yuasa/KEK x 10 -12

11. Numerical results of Two-loop non-planar box with masses x 10 -12 ACAT2011 5-9 September 2011F.Yuasa/KEK Re. fsCPU time 6.0 16 hours 7.02 days 10.0 1 week Intel(R) Xeon(R) CPU X5460 @ 3.16GHz m=50 GeV, M = 90 GeV, t = -100 2 GeV 2

12. extrapolation control (Koike) Example Prepare integral values for m=0 and j=0,..,140. The first term is of m=0,..,120. The 21 terms starting from j=m are the target of extrapolation. The choice of epsilons

13. 13 Best region Analytical value Values after extrapolation Error in extra- polation JPS 17Sept. 2011 T.Koike

14. Real part, M-dependence JPS 17Sept. 2011 T.Koike

15. separation of singularity (de Donker) This integral might have IR divergence and/of UV divergence as pole(s) of. We need double extrapolation for both when V has zero in the integral region. Separation of IR poles is successful even for double-pole cases.

16. ACAT2011 5-9 September E. de Doncker analytic 1-loop vertex with IR Each term is obtained after extrapolation extrapolation (linear)

17. summary Direct computation method(DCM) is a unique numerical method to calculate loop integrals for general masses and momenta. Some items remain before it will become an important component in an automated system for higher order radiative corrections. - Study the validity of the method for wider class of mass configuration - Numerical handling of UV/IR divergence - Improve parameter selection technique for iterated computation - Accelerate computation using modern IT technology