[Paper Review] Sunrise integral with two internal masses and pseudo-threshold kinematics in terms of elliptic polylogarithms

Johanna Campert, Francesco Moriello+1 more|arXiv (Cornell University)|Nov 3, 2020

Black Holes and Theoretical Physics101 references4 citations

TL;DR

This paper presents a complete analytical solution for a two-loop sunrise Feynman integral with two different internal masses at pseudo-threshold kinematics, expressed in terms of elliptic multiple polylogarithms (eMPLs) to all orders in the dimensional regulator $\varepsilon$. The result generalizes previous finite-order expansions by providing an exact all-orders Laurent series in $\varepsilon$, enabling precise computations in non-relativistic QCD and QED phenomenology.

ABSTRACT

We consider a two-loop sunrise integral with two different internal masses at pseudo-threshold kinematics and we solve it in terms of elliptic polylogarithms to all orders of the dimensional regulator.

Motivation & Objective

• To derive an all-orders analytic expression for a two-loop sunrise integral with two different internal masses at pseudo-threshold kinematics.
• To extend the applicability of elliptic multiple polylogarithms (eMPLs) to physical processes involving multiple mass scales.
• To provide a closed-form solution in terms of eMPLs that avoids truncation in the $\varepsilon$-expansion.
• To support high-precision calculations in non-relativistic QCD and QED, such as heavy quarkonium and $t\bar{t}$ production.

Proposed method

• The sunrise integral is expressed in terms of a generalized hypergeometric function ${}_3F_2$, which is then recast into a double integral representation.
• The double integral is systematically reduced using iterated integration techniques on the torus, leveraging the algebraic structure of elliptic curves.
• The solution is constructed using elliptic multiple polylogarithms (eMPLs), defined via iterated integrals on the complex torus with appropriate branch cuts and monodromy properties.
• The dimensional regularization parameter $\varepsilon$ is treated exactly, allowing the full Laurent series in $\varepsilon$ to be captured in terms of eMPLs.
• The method relies on a careful regularization scheme that preserves the analytic structure of the integral across all orders in $\varepsilon$.
• The final result is verified through consistency checks with known hypergeometric representations and leading-order $\varepsilon$-expansions.

Experimental results

Research questions

• RQ1 Can a two-loop sunrise integral with two distinct internal masses and pseudo-threshold kinematics be solved exactly in all orders of the dimensional regulator $\varepsilon$?
• RQ2 How can elliptic multiple polylogarithms (eMPLs) be systematically applied to represent Feynman integrals beyond the multiple polylogarithm (MPL) framework?
• RQ3 What is the structure of the Laurent expansion of the sunrise integral in $\varepsilon$ when the integral is finite at $\varepsilon=0$?
• RQ4 How do the geometric and algebraic properties of the underlying elliptic curve manifest in the final eMPL representation?
• RQ5 Can the solution be expressed in a form compatible with standard physical representations, such as Feynman parameter integrals?

Key findings

• The finite sunrise integral $J_{1,2,2}(m^2, M^2)$ is solved exactly in terms of elliptic multiple polylogarithms (eMPLs) to all orders in the dimensional regulator $\varepsilon$.
• The solution is expressed as a combination of eMPLs of weight four, with arguments and indices derived from the kinematic invariants and mass ratios.
• The result is valid for arbitrary $\varepsilon$, providing a complete Laurent series expansion without truncation.
• The integral is shown to be expressible in terms of iterated integrals on the torus, confirming its elliptic nature.
• The structure of the solution reveals a non-trivial dependence on the ratio $m^2/M^2$, encoded in the arguments of the eMPLs.
• The derivation establishes a consistent framework for handling two-loop sunrise integrals with multiple mass scales in dimensional regularization.