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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

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

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

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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.

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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)

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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

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