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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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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
Reduction of loop integrals to master integrals Y. Sumino (Tohoku Univ.)
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
Integration-by-parts (IBP) Identities Chetyrkin, Tkachov In dim. reg. Ex. at 1-loop:
(3-loop) 21-dim. space Reduction by Laporta algorithm O
(3-loop) 21-dim. space Reduction by Laporta algorithm O
(3-loop) 21-dim. space Reduction by Laporta algorithm O
(3-loop) 21-dim. space Reduction by Laporta algorithm O
(3-loop) 21-dim. space Reduction by Laporta algorithm O
(3-loop) 21-dim. space Reduction by Laporta algorithm O
(3-loop) 21-dim. space Reduction by Laporta algorithm O
(3-loop) 21-dim. space Reduction by Laporta algorithm O
(3-loop) 21-dim. space Reduction by Laporta algorithm O
(3-loop) 21-dim. space Reduction by Laporta algorithm O Master integrals
Out of only 12 of them are linearly independent. An improvement Evolution in 12-dim. subspace O
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)
In the case of QCD potential 1 loop: 1 master integral 2 loop: 5 master integrals 3 loop: 40 master integrals
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
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 .
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;...]
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