Advanced Techniques in Loop Corrections and Integral Reduction
Explore cutting-edge algorithms and methodologies for computing loop corrections, dimensional regularization, graph generation, integral reduction, and more in high-energy physics calculations. Learn about integration-by-parts identities, reduction of loop integrals to master integrals, and advanced techniques for evaluating master integrals in various dimensions.
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Todays 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 ?1+ ?2+ ?3+ ?4=0 1 ?1?2?3?4 1 1 1 1 + + + = 0 ?2?3?4 ?1?3?4 ?1?2?4 ?1?2?3 1 1 1 1 1 ?1 + + + = 0 2?3?4 2?2?4 2?2?3 ?1?2?3?4 ?1 ?1 ?1 Use to reduce the number of Di s. 1 loop case: ??= ?2+ 2 ?.??+ ?? ; ? = 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 ?(0,0,0) ?(1,0,0) 0 0 ? = 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 1. Specify complexity of an integral a. More Di s b. More positive powers of Di s c. More negative powers of Di s 2. Rewrite complicated integrals by simpler ones iteratively. simpler O
Example of Step 2. { ? + 2? + ? = 0 ? ? + 3? = 0 1 (2) Complexity: ? < ? < ? . (1) Solve ? in terms of ?,? Pick one identity. ? = ? 2? 3 ? ? + 3? = 0 Substitute to (2): Apply all known reduction relations. ? ? + 3 ? 2? = 2? + 5? = 0 ? ? + 3( ? 2?) = 0 ? =2 Solve the obtained eq for the most comlex variable. 5? Substitute to (3): 2? + 5? = 0 Obtain a new reduction relation. ? = ? 2 2 5? = 9 5? ? =2 5? Thus, ?,? are expressed by ?.
New One-loop Computation Technologies (mainly for LHC) 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;...]
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
