Abstract
We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure math- ematics and string theory. We then focus on the equal-mass and non-equal-mass sunrise integrals, and we develop a formalism that enables us to compute these Feynman integrals in terms of our iterated integrals on elliptic curves. The key idea is to use integration-by-parts identities to identify a set of integral kernels, whose precise form is determined by the branch points of the integral in question. These kernels allow us to express all iterated integrals on an elliptic curve in terms of them. The flexibility of our approach leads us to expect that it will be applicable to a large variety of integrals in high-energy physics.
Highlights
With the discovery of the Higgs boson at the Large Hadron Collider (LHC) at CERN and the absence of signals of physics beyond the Standard Model (SM), we have entered a new era of precision physics
We have presented an algorithm for computing Feynman integrals which involve square roots of quartic polynomials in terms of iterated integrals on the corresponding elliptic curves
Since our kernels have only simple poles, they define a class of functions on the elliptic curve which deserve to be called elliptic multiple polylogarithms
Summary
With the discovery of the Higgs boson at the Large Hadron Collider (LHC) at CERN and the absence of signals of physics beyond the Standard Model (SM), we have entered a new era of precision physics. Bloch and Vanhove have shown that the sunrise integral with three equal masses in two dimensions can naturally be written in terms of a generalization of the dilogarithm to an elliptic curve [23] The latter is a special case of a more general class of functions, called elliptic multiple polylogarithms (eMPLs) [34,35,36], and they have recently appeared in the context of superstring amplitudes at one loop [37,38,39].
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