Multiple polylogarithms with algebraic arguments and the two-loop EW-QCD Drell-Yan master integrals

TL;DR

The paper tackles the challenge of evaluating two-loop EW-QCD Drell-Yan master integrals whose symbol letters include unrationalizable roots, by proving linear reducibility for key topologies and constructing an $\\epsilon$-decoupled basis. It develops a symbol-matching framework to express root-valued integrals in terms of conventional multiple polylogarithms, and introduces region-specific analytic continuation and basis optimization to avoid explicit $+i0$ prescriptions while ensuring numerical stability. The authors succeed in obtaining a complete weight-four MPL representation for the most intricate six-line master integral $\\boldsymbol{m}_{32}$ and demonstrate fast, precise numerical evaluation, highlighting the method's applicability to broader problems with algebraic letters. Collectively, the work extends MPL techniques to challenging Feynman integrals with algebraic symbol letters and provides practical tools for precise phenomenology of Drell-Yan processes at colliders. The approach holds promise for enabling efficient, accurate predictions in the presence of root-valued letters across a range of high-energy processes.

Abstract

We consider Feynman integrals with algebraic leading singularities and total differentials in $ε\,\mathrm{d}\ln$ form. We show for the first time that it is possible to evaluate integrals with singularities involving unrationalizable roots in terms of conventional multiple polylogarithms, by either parametric integration or matching the symbol. As our main application, we evaluate the two-loop master integrals relevant to the $αα_s$ corrections to Drell-Yan lepton pair production at hadron colliders. We optimize our functional basis to allow for fast and stable numerical evaluations in the physical region of phase space.

Figures (6)

• Figure 1: Planar top-level topology for the two-loop QED corrections to Bhabha electron-positron scattering (left) and the most complicated top-level topology for the mixed EW-QCD two-loop corrections to Drell-Yan lepton production (right). Thick lines denote massive propagators and massive legs, thin lines are used in the massless case. Both of the integral topologies depicted above admit $\epsilon\,\mathrm{d} \ln$ differential equations.
• Figure 2: Feynman integrals relevant to the two-loop QED corrections to Bhabha electron-positron scattering (left) and the two-loop mixed EW-QCD corrections to Drell-Yan lepton production (right), which have problematic root-valued leading singularities.
• Figure 3: The $(w,z)$ representation of the physical phase space has two main components which merge at the point of non-analyticity $w = z = -1$, corresponding to the two-mass threshold at $s = 4\,m^2$. The line segments on the negative real axes of $w$ space (left) and $z$ space (right) are half-open intervals which correspond to the above-threshold region; the points $w = 0$ and $z = 0$ are approached in the $s \rightarrow \infty$ limit. The semi-circular domains in the upper $w$ and $z$ half-planes correspond to the below-threshold region; the point $w = 1$ corresponds to the phase space boundary point $s = 0$ and $z = 1$ corresponds to the phase space boundary point $s = -t = 2\,m^2$. Note that the upper endpoint of $z$ depends parametrically on $w$ both above and below the two-mass threshold.
• Figure 4: The path in $w$ space prescribed by Eq. \ref{['eq:analycontpath']} for $w_{(0)} = 1/6$ and $\delta = 1/17$ (left) induces a nearly semi-circular path in $s$ space (right) which respects the $+i\,0$ prescription for $s$. Note that, for fixed $t$ and $m^2$, this straight-line path in $w$ determines a nearly straight-line path in $z$ as well.
• Figure 5: The path in $w$ space prescribed by Eq. \ref{['eq:analycontpath']} for $w_{(0)} = 1/6$ and $\delta = 1/17$ (left) induces an essentially hyperbolic path in $w^2$ space (right) which crosses the negative real axis at $-\delta^2$.