A first computation of three-loop master integrals for the production of two off-shell vector bosons with different masses

TL;DR

This work advances three-loop QCD computations by delivering analytic master-integral expressions for four integral families describing two off-shell vector bosons with different masses, expressed up to $\varepsilon^6$ in terms of real-valued multiple polylogarithms. The authors construct canonical differential equations with pure bases, determine a complete alphabet, and fix boundary conditions via numerical AMFlow data and PSLQ, enabling explicit MPL representations and efficient semi-numerical evaluation with DiffExp. They validate the analytic results against finite-field IBP and AMFlow benchmarks, and provide a comprehensive performance comparison between analytic and semi-numeric approaches, including integration paths that yield real-valued MPLs in the physical region. The results are a key step toward N3LO QCD predictions for $pp \to V_1V_2$, with plans to optimize MPL reduction, implement fast numerical codes, and extend the methodology to remaining planar and non-planar three-loop four-point families, potentially revealing deeper algebraic structures such as cluster algebras. Overall, the study expands the frontier of exact three-loop calculations for multi-leg processes with unequal external masses and offers practical routes for high-precision collider phenomenology.

Abstract

We present analytic results on physical kinematics for four integral families that are relevant to the production of two off-shell vector bosons with different masses. Our study consists of a ladder-box, a tennis-court, and two reducible ladder-box-like families. The results for the master integrals of these families are expressed up to order six in the dimensional regulator in terms of real-valued multiple polylogarithms. Furthermore, a semi-numeric solution is provided, employing generalized power series expansions using the package DiffExp.

Figures (2)

• Figure 1: The two irreducible integral families. The upper one is $\mathrm{PL}_1$ and the lower one is $\mathrm{PT}_4$. The black lines represent massless particles, while the two curled color lines correspond to the two vector bosons. The particle with mass $m_4$ ($m_3$) is represented by a blue (red) color.
• Figure 2: The two reducible integral families. The left one is $\mathrm{RL}_1$ and the right one is $\mathrm{RL}_2$. The black lines represent massless particles, while the two curled color lines correspond to the two vector bosons. The particle bearing mass $m_3$ ($m_4$) is drawn by a red (blue) color.