Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes

By Claude Duhr. 2012

280 citations recorded by the source corpus.

A bstractWe show how the Hopf algebra structure of multiple polylogarithms can be used to simplify complicated expressions for multi-loop amplitudes in perturbative quantum field theory and we argue that, unlike the recently popularized symbol-based approach, the coproduct incorporates information about the ζ values. We illustrate our approach by rewriting the two-loop helicity amplitudes for a Higgs boson plus three gluons in a simplified and compact form involving only classical polylogarithms.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/98f7e36e94a38285fd136dbfd5b2353a5b9f5b0a

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In Applied Mathematics

In Geometry and Topology

In Nuclear and High Energy Physics

Citations in the corpus

Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series (2018, methodology): “and the coaction on MPLs to study, and derive, relations among MPLs. We only discuss an example which highlights the main points. For a pedagogical introduction and further examples we refer to refs. [13, 76]. Consider the function f(z) = G(1,0,1,0;1 −z), (2.20) our goal is to ﬁnd a relation that al”
Polylogarithms, multiple zeta values and superstring amplitudes (2013, methodology): “While we initially used the method described in [44], it turned out that the identities produced in this way can be cast into a closed formula.”
Motivic multiple zeta values and superstring amplitudes (2012, background): “[6] the coproduct structure has been applied for a concrete physical amplitude.”
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph (2017, unknown): “05 46”
Single-valued harmonic polylogarithms and the multi-Regge limit (2012, methodology): “The result is a complicated expression involving multiple polylogarithms which can be matched to a combination of SVHPLs and zeta values by means of the symbol [20, 21, 22, 23, 24] and coproduct [59, 60, 61].”
PolyLogTools — polylogs for the masses (2019, methodology): “eed to work modulo iˇ. In applications, however, it is important to keep track of powers of ˇ. It is possible to incorporate iˇinto the construction by introducing the special rule [7] (see also ref. [28]),3 ( iˇ) = iˇ 1: (6.13) More precisely, we should work modulo iˇin all entries of the coproduct”
Soft triple-real radiation for Higgs production at N3LO (2013, methodology): “[124] an algorithm was described that allows to reconstruct these zeta-valued terms using the full Hopf algebra structure of multiple polylogarithms, augmented by some ideas by Brown [125].”
A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon (2014, background): “ot be written as products of lower-weight functions. The algebra of generalised polylogarithm functions admits a coproduct ∆ compatible with multiplication, rendering it a Hopf algebra [70] (see also [74] for a review aimed at physicists). Moreover there is a cobracket δ which squares to zero (when ”
Mathematical aspects of scattering amplitudes (2014, methodology): “hese techniques have also found there way into QCD computations, and have in particular been used to simplify the analytic expressions for the two-loop amplitudes for a Higgs boson plus three partons [30, 59, 60], light-quark contributions to top-pair production [61] as well as diboson production at”