Master Integrals For Massless Two-Loop Vertex Diagrams With Three Offshell Legs

TL;DR

<3-5 sentence high-level summary>The paper tackles the analytic computation of master integrals for massless two-loop vertex diagrams with three off-shell legs, a key ingredient for NNLO QCD predictions in processes like H → V*V* and off-shell triple-gluon/quark-gluon vertices. The authors employ the differential-equation method, solving systems in external invariants with carefully chosen boundary conditions and expanding each master integral in ε. A central advance is the introduction of an extended set of two-dimensional harmonic polylogarithms, including quadratic-denominator structures, to faithfully represent the vertex kinematics and phase-space boundaries. They provide explicit ε-expansions for all relevant two-loop topologies up to six propagators and demonstrate how these results enable massless two-loop 2→2 amplitudes with two off-shell legs, while outlining remaining topologies for future work.

Abstract

We compute the master integrals for massless two-loop vertex graphs with three off-shell legs. These master integrals are relevant for the QCD corrections to H to V*V* (where V = W, Z) and for two-loop studies of the triple gluon (and quark-gluon) vertex. We employ the differential equation technique to provide series expansions in epsilon for the various master integrals. The results are analytic and contain a new class of two-dimensional harmonic polylogarithms.

Figures (10)

• Figure 1: The phase space for the vertex graph with three off-shell legs. The shaded region corresponds to the case where $p_3^2 > p_1^2,~p_2^2$. The solid line marks the boundary where $\lambda = 0$.
• Figure 2: The one-loop Master Integral, $\mathrm{BB}(q^2)$
• Figure 3: The one-loop Master Integral, $F_0(m_1^2,m_2^2,m_3^2)$.
• Figure 4: The two-loop Master Integral, $\mathrm{SS}(q^2)$.
• Figure 5: The two-loop Master Integral, $\mathrm{GL}(q^2)$.