The two-loop helicity amplitudes for $gg \to V_1 V_2 \to 4~\mathrm{leptons}$

TL;DR

This work computes the two-loop massless QCD corrections to gluon-fusion production of electroweak vector-boson pairs with off-shell final states and leptonic decays, expressing the result through a tensor decomposition and a set of master integrals. The authors construct the gg -> V1 V2 partonic current, reduce its 138 tensor structures to 20 scalar form factors, and recast the 16 helicity configurations of gg -> V1 V2 -> 4 leptons in terms of nine independent form factors, applying UV renormalization and IR subtraction in a q_T-based scheme. They provide analytic expressions for the finite remainders, validate their results against an independent calculation, and deliver a public C++ implementation (VVamp) for fast numerical evaluation using master integrals from Gehrmann et al. This work lays the groundwork for NLO corrections to gluon-induced diboson processes and helps refine theoretical uncertainties relevant to Higgs width constraints at the LHC. Future extensions to massive loops (e.g., top quarks) are anticipated to broaden the phenomenological impact.

Abstract

We compute the two-loop massless QCD corrections to the helicity amplitudes for the production of two electroweak gauge bosons in the gluon fusion channel, $gg \to V_1 V_2$, keeping the virtuality of the vector bosons $V_1$ and $V_2$ arbitrary and taking their decays into leptons into account. The amplitudes are expressed in terms of master integrals, whose representation has been optimised for fast and reliable numerical evaluation. We provide analytical results and a public C++ code for their numerical evaluation on HepForge at http://vvamp.hepforge.org .

Figures (3)

• Figure 1: Example Feynman diagrams for the process $gg \to V_1 V_2$ at the two-loop level, where the vector bosons couple to the same fermion loop, $[A]$, to different fermion loops, $[B]$, or to an intermediate vector boson, $[F_V]$. The sum of all type $[B]$ contribution and the sum of all type $[F_V]$ contributions vanish, respectively.
• Figure 2: Real parts of the two loop form factors $E_j^{(2),LL\,[A]}$ for the process $gg\to V_1V_2$. The plots illustrate their dependence on the velocity, $\beta_3$, and the cosine of the scattering angle, $\cos\theta_3$, of the vector boson $V_1$, where $p_4^2 = 2 p_3^2$ is chosen for the vector boson virtualities.
• Figure 3: Real parts of the two loop form factors $E_j^{(2),LR\,[A]}$ for the process $gg\to V_1V_2$. The plots illustrate their dependence on the velocity, $\beta_3$, and the cosine of the scattering angle, $\cos\theta_3$, of the vector boson $V_1$, where $p_4^2 = 2 p_3^2$ is chosen for the vector boson virtualities.