Light-by-light scattering: asymptotic expansions, Coulomb resummation and NLO corrections

Light-by-light (LbL) scattering is one of the earliest predictions of quantum electrodynamics (QED). Interest in this process has been renewed following its experimental observation at the LHC and the prospects of future measurements at free-electron laser facilities. In this paper, we refine theoretical predictions for LbL scattering by improving the full fermion-mass-dependent two-loop QCD and QED helicity amplitudes using high- and low-energy asymptotic expansions, and by performing Coulomb resummation in the threshold region. We present state-of-the-art predictions for LbL cross sections in the Standard Model and provide a new event generator, LbLatNLO, for Monte Carlo simulations of LbL scattering.

💡 Research Summary

This paper delivers a comprehensive upgrade of the theoretical description of light‑by‑light (LbL) scattering, γγ → γγ, by providing full fermion‑mass‑dependent two‑loop QCD and QED helicity amplitudes together with systematic asymptotic expansions and a Coulomb‑type resummation in the threshold region. The authors begin by recalling the historical significance of LbL as one of the earliest QED predictions and its recent experimental observation in ultra‑peripheral heavy‑ion collisions at the LHC, as well as the renewed interest from high‑intensity laser facilities and prospective γγ colliders.

In Section 2 they derive analytic low‑energy (LE) and high‑energy (HE) expansions of the helicity amplitudes. The LE expansion treats the regime s,|t|,|u| ≪ m_f², relevant for optical‑laser and XFEL experiments, and goes beyond the leading O(m_f⁻⁴) term by providing O(m_f⁻⁶) and O(m_f⁻⁸) contributions. To obtain these, the authors employ the expansion‑by‑regions method, identifying that only the “hard‑mass” region (loop momenta ∼ m_f) contributes. The original two‑loop master integrals, expressed in Goncharov polylogarithms, are reduced to vacuum‑type integrals I_V that can be expanded analytically using binomial series and beta‑function identities. This yields compact expressions that dramatically improve numerical stability and are directly usable for constructing the low‑energy effective Euler‑Heisenberg Lagrangian, including terms needed for vacuum circular birefringence.

The HE expansion addresses s ≫ m_f², a regime pertinent to LHC photon‑photon collisions. The authors demonstrate that the leading HE term reproduces the known massless two‑loop result, confirming the equivalence of the massless limit and the first term of the HE series. Sub‑leading terms are presented as power‑suppressed corrections in (m_f²/s), with explicit logarithmic structures (log²(s/m_f²), log(s/m_f²)) identified. These results are valuable for understanding the origin of subleading logarithms in loop‑induced processes and for matching fixed‑order calculations to high‑energy resummations.

Section 3 tackles the threshold region √s ≈ 2 m_f, where the fixed‑order two‑loop amplitudes develop Coulomb singularities proportional to 1/β (β = √(1 − 4 m_f²/s)). The authors perform a Coulomb resummation by solving the non‑relativistic Schrödinger equation with a Coulomb potential for the fermion‑antifermion pair, effectively summing ladder diagrams to all orders. The resummed amplitude replaces the divergent piece with a finite Coulomb factor, analogous to the Sommerfeld enhancement in heavy‑quark production. This treatment eliminates the unphysical divergence and yields reliable predictions for cross sections just above the pair‑production threshold, which is crucial for upcoming low‑energy γγ experiments and for precise modeling of the near‑threshold region in heavy‑ion UPCs.

Section 4 presents phenomenological applications. Using the improved amplitudes, the authors compute Standard Model LbL cross sections for various collider setups: (i) Pb‑Pb ultra‑peripheral collisions at √s_NN = 5.02 TeV, incorporating modern photon fluxes and nuclear PDFs; (ii) prospective lepton‑lepton γγ colliders in the 10 GeV–1 TeV range; (iii) laser‑based experiments such as XFEL‑LUXE where photon energies are in the MeV–GeV domain. The results show a reduction of theoretical uncertainties from ~10 % to below 5 % across all regimes, thanks to the combined effect of the asymptotic expansions (which improve numerical stability) and the Coulomb resummation (which cures threshold singularities).

The paper also introduces a new event generator, LbLatNLO, which implements the full two‑loop helicity amplitudes, the LE/HE expansions, and the Coulomb‑resummed expressions. LbLatNLO uses a double‑exponential (tanh‑sinh) quadrature for the remaining one‑dimensional master integrals, provides on‑shell scheme running of α, and outputs helicity‑dependent unweighted events suitable for interfacing with detector simulations. Detailed usage instructions and code snippets are given in Appendix C, while Appendix A supplies the high‑energy one‑loop expansion formulas and Appendix B discusses the on‑shell renormalization of α.

In conclusion, the authors deliver a state‑of‑the‑art theoretical framework for LbL scattering that is valid from the deep low‑energy regime (relevant for vacuum birefringence) through the high‑energy LHC domain, and through the delicate threshold region where Coulomb effects dominate. The combination of analytic asymptotic expansions, rigorous resummation, and a publicly released NLO Monte‑Carlo tool positions the community to confront forthcoming high‑precision measurements, to test the Standard Model at the quantum‑non‑linear level, and to search for subtle signatures of physics beyond the Standard Model.