Three-loop master integrals for ladder-box diagrams with one massive leg

TL;DR

The paper solves the analytic evaluation of the complete set of three-loop ladder-box master integrals with one massive leg by casting the differential equations for the integrals into a canonical form using Magnus exponential-based basis changes. All 85 three-loop MIs (plus subtopologies) are obtained as an $\epsilon$-expanded series up to weight six, expressed through uniform-weight multiple polylogarithms, with boundary data fixed by regularity at pseudothresholds. The results underpin next-to-next-to-next-to-leading order virtual corrections for processes like $V^{*}\to jjj$ and $pp\to Hj$, and are validated against numerical methods, with explicit canonical matrices and the scalar-box integral provided. The approach clarifies the analytic structure of high-loop integrals and delivers practical tools (ancillary files) for implementing these results in phenomenological calculations.

Abstract

The three-loop master integrals for ladder-box diagrams with one massive leg are computed from an eighty-five by eighty-five system of differential equations, solved by means of Magnus exponential. The results of the considered box-type integrals, as well as of the tower of vertex- and bubble-type master integrals associated to subtopologies, are given as a Taylor series expansion in the dimensional regulator parameter epsilon = (4-d)/2. The coefficients of the series are expressed in terms of uniform weight combinations of multiple polylogarithms and transcendental constants up to weight six. The considered integrals enter the next-to-next-to-next-to-leading order virtual corrections to scattering processes like the three-jet production mediated by vector boson decay, V* -> jjj, as well as the Higgs plus one-jet production in gluon fusion, pp -> Hj.

Figures (5)

• Figure 1: The two-loop and three-loop ladder box diagram, with one off-shell leg: the solid lines stand for massless particles; the dashed line represents a massive particle. Momentum conservation is $\sum_{i=1}^4p_i = 0$, with $p_i^2 = 0$$(i=1,\ 2,\ 3)$ and $p_4^2 = m^2$.
• Figure 2: Two-loop Master Integrals $\{ {\cal T}_i \}_{i=1,\ldots,18}$. The solid lines stand for massless particles; the dashed line represents a massive particle; dots indicate squared propagators; numerators may appear as indicated ($p_{ij}\equiv p_i+p_j$).
• Figure 3: Three-loop Master Integrals $\{ {\cal T}_i \}_{i=1,\ldots,30}$. The solid lines stand for massless particles; the dashed line represents a massive particle; dots indicate squared propagators; numerators may appear as indicated ($p_{ij}\equiv p_i+p_j$). See also Figs. \ref{['Fig:3loopMIs2']} and \ref{['Fig:3loopMIs3']}.
• Figure 4: Three-loop Master Integrals $\{ {\cal T}_i \}_{i=31,\ldots,59}$. The solid lines stand for massless particles; the dashed line represents a massive particle; dots indicate squared propagators; numerators may appear as indicated ($p_{ij}\equiv p_i+p_j$). See also Figs. \ref{['Fig:3loopMIs1']} and \ref{['Fig:3loopMIs3']}.
• Figure 5: Three-loop Master Integrals $\{ {\cal T}_i \}_{i=60,\ldots,85}$. The solid lines stand for massless particles; the dashed line represents a massive particle; dots indicate squared propagators; numerators may appear as indicated ($p_{ij}\equiv p_i+p_j$). See also Figs. \ref{['Fig:3loopMIs1']} and \ref{['Fig:3loopMIs2']}.