Light-by-light scattering at three loops in massless QCD and QED: amplitudes and cross sections

Abstract

We present the calculation of three-loop massless QCD and QED helicity amplitudes for light-by-light scattering. We make use of Lorentz tensor decomposition in the 't Hooft-Veltman dimensional regularisation scheme to reduce the complexity of the computation. Our analytic amplitude results are remarkably compact and can be efficiently evaluated numerically. We employ them to compute the corresponding NNLO differential cross-section predictions in the invariant mass and rapidity distributions of the di-photon system, for which we find agreement with the experimental ATLAS data from ultra-peripheral heavy-ion collisions.

Figures (6)

• Figure 1: The only diagram type contributing to the one-loop amplitude.
• Figure 2: Representative diagrams for the two-loop amplitude. All other diagrams can be obtained via permutations of the external legs. All diagrams in the two-loop amplitude correspond to the same colour and flavour factors and are proportional to $\alpha^2 \alpha_s N_c C_F \tilde{n}_q^{(4)}$.
• Figure 3: Representative diagrams for the gauge invariant parts of the three-loop amplitude. Other members in each set are obtained by permutations of the external momenta and of the attachments of the internal gluons to the fermion lines. Diagrams contributing to $\mathcal{N}^{(3,2)}$ are purely non-abelian and therefore give no contribution to the QED corrections to the amplitude.
• Figure 4: QCD finite remainders with coupling constants, $8\alpha^2\, \left({\alpha_s}/{\pi}\right)^{L-1}\, f^{(L)}_{\text{ren},{\vec{\lambda}}}$, as function of $x=-t/s$ for specific values of $\alpha=1/137$, $\alpha_s=0.118$ and with $n_q=5$ quark flavours.
• Figure 5: Differential distribution in (a) $d\sigma/dm_{\gamma \gamma}$ and (b) corresponding $K$-factor at LO, NLO and NNLO with bin sizes corresponding to experimental ATLAS data ATLAS:2020hii.