Double-Higgs boson production in the high-energy limit: planar master integrals

TL;DR

This paper computes the planar two-loop master integrals for $gg\to HH$ in the high-energy limit by expanding in the small top-quark mass $m_t$, and solves the associated differential equations to obtain analytic results in terms of harmonic polylogarithms up to weight 6, extended to $m_t^{16}$. Two complementary strategies are used to determine boundary conditions: a $t$-dependent method with small-$m_t^2$ expansion and a boundary-value approach from the $m_t^2\to 0$ limit, with boundary data derived via the method of regions and Mellin-Barnes representations. The reduction to a minimal planar master set is achieved using FIRE/LiteRed with cross-checks, yielding 161 two-loop planar masters (plus 10 one-loop masters) and exact $m_t$ dependence, and the resulting master integrals are validated against numerical benchmarks and provided in ancillary progdata for easy reuse in high-energy approximations and Padé-based reconstructions. The results enable independent cross-checks of exact calculations and offer compact, analytic inputs for phenomenological studies of double Higgs production in the high-energy regime.

Abstract

We consider the virtual corrections to the process $gg\to HH$ at NLO in the high energy limit and compute the corresponding planar master integrals in an expansion for small top quark mass. We provide details on the evaluation of the boundary conditions and present analytic results expressed in terms of harmonic polylogarithms.

Figures (4)

• Figure 1: ${ {\rm d}\sigma }/{ {\rm d} {{\theta}} }$ as a function of $\sqrt{s}$ for fixed ${{\theta}} =\pi/2$.
• Figure 2: Real and imaginary part of the $\epsilon^0$ term of the two master integrals $G_{6}(1,1,1,1,1,1,1,0,0)$ and $G_{20}(1,1,1,1,1,2,1,0,0)$. For convenience we multiply by powers of $m_t$ and $s$ as indicated above the plot.
• Figure 3: One-loop master integrals. Solid and dashed lines represent massive and massless scalar propagators, respectively. The external (thin) lines are massless. The four master integrals which are not shown are obtained by crossing.
• Figure 4: Two-loop planar master integrals. Solid and dashed lines represent massive and massless scalar propagators, respectively. The external (thin) lines are massless. The planar master integrals form (\ref{['eq::MI2l']}) which are not shown are obtained by crossing.