Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

Burkhard Eden; Vladimir A. Smirnov

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Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

Burkhard Eden, Vladimir A. Smirnov

High Energy Physics - Theory · Submitted 21 Jul 2016

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The authors' main claim Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms.

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arXiv ID1607.06427 (version 1)

Pith ID6MRMOBEA, Pith's permanent record of this paper and its review

CitationBurkhard Eden, Vladimir A. Smirnov. "Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations." arXiv:1607.06427 (2016). https://pith.science/paper/6MRMOBEA

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We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals.	contribution 0.89	0.96	0.31	asserted 1.00
s1	To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals.	method 0.99	0.75	0.04	asserted 0.96
s2	We find a solution to these equations up to weight eight in terms of multiple polylogarithms.	result 0.97	0.97	0.18	asserted 1.00
s3	Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms.	result 0.99	0.96	0.53	asserted 1.00
s4	As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.	result 0.97	0.96	0.16	asserted 1.00

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Original abstract

We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.

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References

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Energy-energy correlations at next-to-next-to-leading order
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Ten dimensional symmetry of N=4 SYM correlators
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A conjectured ten-dimensional symmetry organizes all R-charged four-point correlators in planar N=4 SYM into one generating function, and its light-like limit equals a Coulomb-branch scattering amplitude.
Gluing Ladders into Fishnets
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Every m-by-n fishnet diagram in planar phi^4 theory is conjectured, with strong evidence, to equal the determinant of an m-by-m matrix of ladder integrals with rational coefficients fixed by Steinmann relations.
Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA
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CANONICA automates the transformation of multi-scale Feynman integral differential equations into epsilon form, and proves that such canonical forms are unique up to a constant rotation.
Transforming differential equations of multi-loop Feynman integrals into canonical form
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A recursive algorithm, built on a multivariate partial-fraction decomposition, finds the rational change of basis that puts multi-scale multi-loop Feynman integral differential equations into canonical epsilon form.

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11 arXiv papers cite this in 14 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 7 sentences without a confident label

Uses its method or tool 2 papers

In [ref] the differential equations method for master integrals of uniform transcendentality weight [ref] was applied in this context, for reasons of simplicity and comparability to begin with on a linearly reducible test case.

Half-BPS half-BPS twist two at four loops in N=4 SYM · Introduction · 0.82
[ref] , we obtain

Gluing Ladders into Fishnets · Ladders, Steinmann, and all that · 1.00

Cites it as background 5 papers

Several methods to find a canonical form have already been discussed [ref] .

Transforming differential equations of multi-loop Feynman integrals into canonical form · Introduction · 0.94
They belong to a broader family of conformal integrals which has attracted much attention over the years [ref] .

Gluing Ladders into Fishnets · Introduction and Main Result · 0.99
Modern techniques have been used to evaluate off-shell correlator integrals up to three loops [ref] , and partially at four loops [ref] .

Perturbative Correlation Functions and Scattering Amplitudes in Planar $\mathcal{N}=4$ Supersymmetric Yang-Mills · Conclusions · 0.93

What it cites · 21 sentences without a confident label

Builds on 1 paper

We believe that the strategy outlined in [ref] can be adjusted to the case of conformal integrals as well.

Iterative structure of finite loop integrals · Introduction · 0.92

Uses the method or tool of 19 papers

To solve these linear differential equations we follow the strategy suggested by Henn [ref] (see also [ref] ) and first applied in [ref] and then in many other papers.

Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation · Introduction · 1.00
To solve these linear differential equations we follow the strategy suggested by Henn [ref] (see also [ref] ) and first applied in [ref] and then in many other papers.

Lectures on differential equations for Feynman integrals · Introduction · 1.00
To solve these linear differential equations we follow the strategy suggested by Henn [ref] (see also [ref] ) and first applied in [ref] and then in many other papers.

Evaluating single-scale and/or non-planar diagrams by differential equations · Introduction · 1.00

Compares with 1 paper

Our result ([ref]) is in agreement with a calculation by different means [ref] about which we knew in advance and with a calculation[Thanks to Erik Panzer for the comparison.] based on HyperInt [ref] .

Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals · Solving canonical differential equations · 0.99

Cites as background 3 papers

These two developments were brought together by the construction of an integrable system [ref] for the so-called remainder function in amplitudes.

Thermodynamic Bubble Ansatz · Introduction · 0.85
These two developments were brought together by the construction of an integrable system [ref] for the so-called remainder function in amplitudes.

Space-time S-matrix and Flux-tube S-matrix at Finite Coupling · Introduction · 0.85
The recent paper [ref] considers the leading terms in an asymptotic expansion for the entire set of integrals.

Extracting OPE coefficient of Konishi at four loops · Introduction · 0.88

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