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2020

Massive Conformal Symmetry and Integrability for Feynman Integrals

Loebbert, Florian, Miczajka, Julian, Müller, Dennis et al.

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In the context of planar holography, integrability plays an important role for solving certain massless quantum field theories such as N=4 SYM theory.

In this letter we show that integrability also features in the building blocks of massive quantum field theories.
At one-loop order we prove that all massive n-gon Feynman integrals in generic spacetime dimensions are invariant under a massive Yangian symmetry.
At two loops similar statements can be proven for graphs built from two n-gons.

Massive Conformal Symmetry and Integrability for Feynman Integrals · Around

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Note that the p j 2 p_{j}^{2} are unconstrained and the m j m_{j} are generic; we have x j μ = x 1 μ − ∑ k < j p k μ x^{\mu}_{j}=x^{\mu}_{1}-\sum\displaylimits_{k<j}p^{\mu}_{k} and x n + 1 μ = x 1 μ x_{n+1}^{\mu}=x_{1}^{\mu}

Cited in the paper.

In these cases one acts on massive legs with the generators of the massive symmetry and on massless legs with the usual dual-conformal generators obtained by dropping the mass terms in ( 2

Cited in the paper.

Switching on the internal mass m 0 m_{0} , only the combination P ^ \mathaccentV h a t 05 E μ + P ^ extra \mathaccentV h a t 05 E μ \mathrm{\mathaccent 866{P}}^{\mathaccentV{hat}05E\mu}+\mathrm{\mathaccent 866{P}}^{\mathaccentV{hat}05E\mu}_{\text{extra}} is a symmetry

Cited in the paper.

Note that this equation arises from the differential operator P ^ extra \mathaccentV h a t 05 E μ \mathrm{\mathaccent 866{P}}^{\mathaccentV{hat}05E\mu}_{\text{extra}} , while the rest of P ^ \mathrm{\mathaccent 866{P}} trivially annihilates I 2 I_{2} . This is a generic feature for two-point invariants

Cited in the paper.

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Niklas Beisert, Aleksander Garus, and Matteo Rosso, “Yangian Symmetry and Integrability of Planar N=4 Supersymmetric Yang-Mills Theory,” Phys. Rev. Lett. 118

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Dmitry Chicherin, Vladimir Kazakov, Florian Loebbert, Dennis Müller, and De-liang Zhong, “Yangian Symmetry for Bi-Scalar Loop Amplitudes,” JHEP 05

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Andrei I. Davydychev, “Four-point function in general kinematics through geometrical splitting and reduction,” Proceedings, 18th International Workshop on Advanced Computing and Analysis Techniques in Physics Research (ACAT 2017): Seattle, WA, USA, August 21-25, 2017 , J. Phys. Conf. Ser. 1085

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