Two-Loop QCD Corrections for Three-Photon Production at Hadron Colliders

TL;DR

The study delivers a complete two-loop QCD computation for triphoton production at hadron colliders, including subleading-color non-planar contributions. It advances the methodology by combining numerical unitarity with embedding-space syzygy for surface terms and a spinor-helicity based analytic reconstruction, producing analytic results and a public library for immediate use. The results show that subleading-color effects can significantly reduce two-loop corrections, influencing NNLO predictions, while maintaining numerical stability and efficiency. This work enables precise phenomenology of triphoton processes at the LHC and informs broader NNLO calculations with non-planar topologies.

Abstract

We complete the computation of the two-loop helicity amplitudes for the production of three photons at hadron colliders, including all contributions beyond the leading-color approximation. We reconstruct the analytic form of the amplitudes from numerical finite-field samples obtained with the numerical unitarity method. This method requires as input surface terms for all relevant five-point non-planar integral topologies, which we obtain by solving the associated syzygy problem in embedding space. The numerical samples are used to constrain compact spinor-helicity ansätze, which are optimized by taking advantage of the known one-loop analytic structure. We make our analytic results available in a public C++ library, which is suitable for immediate phenomenological applications. We estimate that the inclusion of the subleading-color contributions will decrease the size of the two-loop corrections by about 30% to 50%, and the NNLO cross sections by a few percent, compared to the results in the leading-color approximation.

Figures (5)

• Figure 1: Representative diagrams for the different contributions to eq. \ref{['eq:coulourDec-gen']}.
• Figure 2: Non-planar diagram corresponding to the propagators in \ref{['eq:propExample']}.
• Figure 3: Distributions of correct digits $R$ (see eq. \ref{['eq:correct-digits']}) that characterize the numerical performance of the C++ implementation of our analytic results in FivePointAmplitudesFivePointAmplitudes. The phase-space sample of $100k$ phase-space points is defined in ref. Kallweit:2020gcp and generated by MATRIX v2Grazzini:2017mhc. The renormalization scale is set to $\mu = m_{\gamma\gamma\gamma}$.
• Figure 4: Numerical stability of the rational functions contributing to the finite remainder $R^{(2,0)}_{-++}$ in two different representations: one in terms of Mandelstam invariants and the other in terms of spinor-helicity variables. At each phase-space point, we plot the lowest number of correct digits $R$ over the pentagon-function coefficients $r_i$ of \ref{['eq:finite-remainder-as-weighted-sum']}. See the caption of \ref{['fig:numerical-stability']} for details of the phase-space definition.
• Figure 5: Distribution of separate subleading-color corrections $\delta \mathcal{H}_x$ defined in \ref{['eq:subleading-color-corrections']} to the the hard function $\mathcal{H}$ alongside the total correction $\delta \mathcal{H}$. The phase-space sample of $100k$ phase-space points is defined in ref. Kallweit:2020gcp and generated by MATRIX v2Grazzini:2017mhc. The renormalization scale is set to $\mu = m_{\gamma\gamma\gamma}$.