The Massless Higher-Loop Two-Point Function

By F. Brown. 2008

270 citations recorded by the source corpus.

We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to multiple zeta values. The criterion depends only on the topology of G, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with 6th roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph K3,4 at one edge.

Subfields: Applied Mathematics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e68295b4fa35f7c5054e8da775ae07db2766f842

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In Applied Mathematics

In Nuclear and High Energy Physics

In Statistical and Nonlinear Physics

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].”
From polygons and symbols to polylogarithmic functions (2011, background): “[81, 95], differing from our definition only by an overall normalization (F. Brown, in a text in preparation on the representation theory of polylogarithms, uses essentially the same operators, referring to ref.”
Feynman integrals and hyperlogarithms (2015, methodology): “For example, it is known to suffice to prove linear reducibility of many vacuum graphs up to six loops [53] and also for plenty of massless on-shell four-point functions up to three loops [127].”
Polylogarithms, multiple zeta values and superstring amplitudes (2013, background): “1) was discussed in [33] and - in a physics context - in [34].”
Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism (2017, background): “r every x2Uwe can write f(x) = Xn k=0 f k(x) logk(x x 0); (A.1) where nis a positive integer and the functions f k are holomorphic on U. The regularised value of fat x 0 is dened by (see, e.g., ref. [53]) Reg x 0 f(x) f 0(x): (A.2) The map Reg x 0 has the following properties: 1. If fhas at most log”
Four Loop Massless Propagators: an Algebraic Evaluation of All Master Integrals (2010, methodology): “• Recently, 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 ([86, 87]).”
Minimally subtracted six loop renormalization of $O(n)$-symmetric $\phi^4$ theory and critical exponents (2017, methodology): “These were obtained in a Feynman diagram computation, that became feasible through the automatization of new techniques [30, 32, 130] to calculate Feynman integrals, very briefly summarized in section III.”
The Multiple Zeta Value data mine (2009, background): “Euler sumsζ~a at given weight and depthw,d are calledindependent if there exists no relation between them, cf. Sect.”
On the periods of some Feynman integrals (2009, background): “llaries 95 and 91 we either reduce to the trivial graph or K3,3. Corollary 147. All primitive-divergent graphs in φ4 theory up to 6 loops, bar K3,4, are linearly reducible (this is the main result of [9]). 12.3. The case K3,4. Consider the bipartite graph K3,4 with the given edge labelling. It has v”
Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals (2014, methodology): “We close this gap and provide a complete implementation of the method [14] of symbolic integration using hyperlogarithms in the computer algebra system Maple [22].”
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].”
Soft triple-real radiation for Higgs production at N3LO (2013, background): “[96] to determine whether an integral can be performed recursively in terms of multiple polylogarithms.”