1. Today’s algorithm for computation of loop corrections Dim. reg. Graph generation QGRAF, GRACE, FeynArts Reduction of integrals IBP id., Tensor red. Evaluation of Master integrals Diff. eq., Mellin-Barnes, sector decomp. Lots of mathematics

2. Y. Sumino (Tohoku Univ.) Reduction of loop integrals to master integrals

3. Loop integrals in standard form Express each diagram in terms of standard integrals 1 loop 2 loop 3 loop Each can be represented by a lattice site in N-dim. space NB: is negative, when representing a numerator. e.g. A diagram for QCD potential

4. Integration-by-parts (IBP) Identities In dim. reg. Ex. at 1-loop: Chetyrkin, Tkachov

5. O (3-loop) 21-dim. space Reduction by Laporta algorithm

6. O (3-loop) 21-dim. space Reduction by Laporta algorithm

7. O (3-loop) 21-dim. space Reduction by Laporta algorithm

8. O (3-loop) 21-dim. space Reduction by Laporta algorithm

9. O (3-loop) 21-dim. space Reduction by Laporta algorithm

10. O (3-loop) 21-dim. space Reduction by Laporta algorithm

11. O (3-loop) 21-dim. space Reduction by Laporta algorithm

12. (3-loop) 21-dim. space O Reduction by Laporta algorithm

13. O (3-loop) 21-dim. space Reduction by Laporta algorithm

14. O (3-loop) 21-dim. space Master integrals Reduction by Laporta algorithm

15. O Evolution in 12-dim. subspace Out of only 12 of them are linearly independent. An improvement

16. Linearly dependent propagator denominators 1 loop case: 4 master integrals (well known) Use to reduce the number of D i ’s.

17. In the case of QCD potential 1 loop: 1 master integral 2 loop: 5 master integrals 3 loop: 40 master integrals

18. More about implementation of Laporta alg. cf. JHEP07(2004)046 IBP ids = A huge system of linear eqs. Laporta alg. = Reduction of complicated loop integrals to a small set of simpler integrals via Gauss elimination method. 1.Specify complexity of an integral a.More D i ’s b.More positive powers of D i ’s c.More negative powers of D i ’s 2.Rewrite complicated integrals by simpler ones iteratively. O simpler more complex

19. Example of Step 2. Substitute to (2): Substitute to (3): Pick one identity. Apply all known reduction relations. Solve the obtained eq for the most comlex variable. Obtain a new reduction relation.

20. Generalized unitarity (e.g. BlackHat, Njet,...) [Bern, Dixon, Dunbar, Kosower, 1994...; Ellis Giele Kunst 2007 + Melnikov 2008; Badger...] Integrand reduction (OPP method) (e.g. MadLoop (aMC@NLO),GoSam) [Ossola, Papadopoulos, Pittau 2006; del Aguila, Pittau 2004; Mastrolia, Ossola, Reiter,Tramontano 2010;...] Tensor reduction (e.g. Golem, Openloops) [Passarino, Veltman 1979; Denner, Dittmaier 2005; Binoth Guillet, Heinrich, Pilon, Reiter 2008;Cascioli, Maierhofer, Pozzorini 2011;...] New One-loop Computation Technologies (mainly for LHC)

21. Improvement 2. O (1) Assign a numerical value to temporarily and complete reduction. (2) Identify the necessary IBP identities and reorder them; Then reprocess the reduction with general. Many inactive IBP id’s are generated and solved in Laporta algorithm. Manageable by a contemporary desktop/laptop PC