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

Y. Sumino

(Tohoku Univ.)

Reduction of loop integrals to

master integrals

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

Integration-by-parts (IBP) Identities

In dim. reg.

Ex. at 1-loop:

Chetyrkin, Tkachov

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

(3-loop)21-dim. space

O

Reduction by Laporta algorithm

O

(3-loop)21-dim. space

Reduction by Laporta algorithm

O

(3-loop)21-dim. space

Master integrals

Reduction by Laporta algorithm

O

Evolution in12-dim. subspace

Out of only 12 ofthem are linearly independent.

An improvement

+ + =0×

1𝐷1𝐷2𝐷3𝐷4

×1𝐷1

Linearly dependent propagator denominators

1 loop case:

;

loop momentum external momentum, only up to 4 independent ones.

4 master integrals(well known)

Use to reduce the number of Di’s.

In the case of QCD potential

1 loop: 1 master integral

2 loop: 5 master integrals

3 loop: 40 master integrals

𝑞

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 Di’s

b. More positive powers of Di’s

c. More negative powers of Di’s

2. Rewrite complicated integrals by simpler ones

iteratively.

O

simpler

more complex

Example of Step 2.

{(1) Solve in terms of

𝑧=−𝑥−2 𝑦⋯ (3 )

Substitute to (2):

𝑥− 𝑦+3 (−𝑥−2 𝑦 )=−2 𝑥+5 𝑦=0

∴ 𝑦=25𝑥

Substitute to (3):

𝑧=−𝑥−2×25𝑥=−

95𝑥

Thus, are expressed by .

Complexity: .

Pick one identity.

Apply all known reduction relations.

Solve the obtained eq for the most comlex variable.

Obtain a new reduction relation.

𝑥− 𝑦+3 𝑧=0

𝑥− 𝑦+3 (−𝑥−2 𝑦)=0

−2 𝑥+5 𝑦=0

𝑦=25𝑥

• 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)

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 generatedand solved in Laporta algorithm.

Manageable by a contemporary desktop/laptop PC