On the periods of some Feynman integrals

By F. Brown. 2009

215 citations recorded by the source corpus.

We study the related questions: (i) when Feynman amplitudes in massless $\phi^4$ theory evaluate to multiple zeta values, and (ii) when their underlying motives are mixed Tate. More generally, by considering configurations of singular hypersurfaces which fiber linearly over each other, we deduce sufficient geometric and combinatorial criteria on Feynman graphs for both (i) and (ii) to hold. These criteria hold for some infinite classes of graphs which essentially contain all cases previously known to physicists. Calabi-Yau varieties appear at the point where these criteria fail.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/aadf07772464363a1ef2fea2f75d5cae48d2feab

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Citations in the corpus

A quasi-finite basis for multi-loop Feynman integrals (2014, methodology): “(3)The integration via hyperlogarithms implemented in HyperInt was first proposed in [21] and the method by which the program decides whether a given Feynman integral is in fact linearly reducible is based on the compatibility graph method introduced in [22].”
Minimally subtracted six loop renormalization of $O(n)$-symmetric $\phi^4$ theory and critical exponents (2017, methodology): “Our approach rests on a significantly improved understanding of the parametric representation of Feynman integrals [30, 31], symbolic integration algorithms based on hyperlogarithms [130] and the Hopf algebra [57] of renormalization underlying the BPHZ scheme [32].”
Graphical functions and single-valued multiple polylogarithms (2013, methodology): “Brown on integration in parametric space [10].”
PolyLogTools — polylogs for the masses (2019, unknown): “Out[4]:= a*(-(G[0,1,z]/a) - G[0,1,z]/(-a+z) + G[a,1,z]/a) + G[a,0,1,z] Out[5]:= G[0,1,z] z/(a-z)^2”
Mathematical aspects of scattering amplitudes (2014, background): “There are criteria that allow one to determine a priori if there is an order of the integration variables such that this procedure succeeds [69, 70].”
Four Loop Massless Propagators: an Algebraic Evaluation of All Master Integrals (2010, methodology): “a deﬁnite class of massless p-integrals was proven to be expressible in terms of the multiple zeta values for all orders of expansions in D − 4, and a direct method of their evaluation was suggested ([87, 88]). In our case (four-loop p-integrals) it predicts that the result is expressible through ζn”
Feynman integrals and hyperlogarithms (2015, methodology): “Francis Brown originally developed this for the moduli spaces M0,n of the Riemann sphere Ĉ = C ∪ {∞} with n marked points [51].”
Collider physics at the precision frontier (2020, methodology): “The direct evaluation of Feynman integrals in parameter space can become very cumbersome with increasing numbers of parameters. However, this approach has seen major progress due to the work of Brown [716] and Panzer [717], who introduced the concept of linear reducibility. It can be applied to Feyn”