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

Henn, Johannes M.; Smirnov, Alexander V.; Smirnov, Vladimir A.

doi:10.1007/JHEP03(2014)088

Title: Evaluating single-scale and/or non-planar diagrams by differential equations

Journal Article · Thu Mar 20 00:00:00 EDT 2014 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/JHEP03(2014)088· OSTI ID:1598516

Henn, Johannes M. ^[1]; Smirnov, Alexander V. ^[2]; Smirnov, Vladimir A. ^[3]

Inst. for Advanced Study, Princeton, NJ (United States)
Moscow State Univ. (Russia)
Moscow State Univ. (Russia). Skobeltsyn Inst. of Nuclear Physics; Institut für Theoretische Teilchenphysik, KIT, Karlsruhe (Germany)

We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of the differential equations to single-scale Feynman integrals. Moreover, the information about singular limits significantly simplifies finding boundary constants for the differential equations. To illustrate these points we consider two families of three-loop integrals. The first are form-factor integrals with two external legs on the light cone. We introduce one more scale by taking one more leg off-shell, $$p_2^2\neq 0$$. We analytically solve the differential equations for the master integrals in a Laurent expansion in dimensional regularization with $$\epsilon=(4-D)/2$$. Then we show how to obtain analytic results for the corresponding one-scale integrals in an algebraic way. An essential ingredient of our method is to match solutions of the differential equations in the limit of small $$p_2^2$$ to our results at $$p_2^2\neq 0$$ and to identify various terms in these solutions according to expansion by regions. The second family consists of four-point non-planar integrals with all four legs on the light cone. We evaluate, by differential equations, all the master integrals for the so-called $$K_4$$ graph consisting of four external vertices which are connected with each other by six lines. We show how the boundary constants can be fixed with the help of the knowledge of the singular limits. We present results in terms of harmonic polylogarithms for the corresponding seven master integrals with six propagators in a Laurent expansion in $$\epsilon$$ up to weight six.

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Research Organization:: Institute for Advanced Study, Princeton, NJ (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: SC0009988

OSTI ID:: 1598516

Journal Information:: Journal of High Energy Physics (Online), Vol. 2014, Issue 3; ISSN 1029-8479

Publisher:: Springer BerlinCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 67 works

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97 MATHEMATICS AND COMPUTING
71 CLASSICAL AND QUANTUM MECHANICS
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differential equations: solution
family: 2
regularization: dimensional
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Laurent expansion
Feynman graph
light cone
asymptotic expansion
form factor
propagator
off-shell
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singularity

Title: Evaluating single-scale and/or non-planar diagrams by differential equations

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