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Loop Integral Computation Techniques in Modern Physics

Explore the latest algorithms and methods for computing loop corrections in theoretical physics, including graph generation, reduction of integrals, and evaluation of master integrals. Learn about the Laporta algorithm, integration-by-parts identities, and the application of complex reduction relations for simplifying integrals. Discover advancements like Generalized Unitarity and Integrand Reduction for tackling complex calculations efficiently on a desktop or laptop.

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Loop Integral Computation Techniques in Modern Physics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  16. Linearly dependent propagator denominators ++=0 Use to reduce the number of Di’s. 1 loop case: ; loop momentum external momentum, only up to 4 independent ones. 4 master integrals (well known)

  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. more complex • Specify complexity of an integral • More Di’s • More positive powers of Di’s • More negative powers of Di’s • Rewrite complicated integrals by simpler ones • iteratively. O simpler

  19. Example of Step 2. Complexity: . (1) Solve in terms of Pick one identity. Apply all known reduction relations. Solve the obtained eq for the most comlex variable. Obtain a new reduction relation. Substitute to (2): Substitute to (3): Thus, are expressed by .

  20. New One-loop Computation Technologies (mainly for LHC) • Generalized unitarity (e.g. BlackHat, Njet, ...) • [Bern, Dixon, Dunbar, Kosower, 1994...; Ellis GieleKunst 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,Tramontano2010;...] • Tensor reduction (e.g. Golem, Openloops) • [Passarino, Veltman 1979; Denner, Dittmaier 2005; BinothGuillet, Heinrich, Pilon, • Reiter 2008;Cascioli, Maierhofer, Pozzorini 2011;...]

  21. Improvement 2. Many inactive IBP id’s are generated and solved in Laporta algorithm. (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 . O Manageable by a contemporary desktop/laptop PC

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