[Paper Review] Light-by-light scattering in SANC

D.Y. Bardin, L. V. Kalinovskaya+2 more|ArXiv.org|Nov 14, 2006

Quantum Chromodynamics and Particle Interactions4 references3 citations

TL;DR

This paper presents the implementation of one-loop light-by-light scattering in quantum electrodynamics (QED) with massive fermion loops within the SANC (Sewage for Amplitude and NLO Calculations) framework. It derives covariant and helicity amplitudes, computes Passarino–Veltman scalar functions $D_0$, $C_0$, and $B_0$ for massive fermions, and provides analytical results for the differential and total cross sections, validated against known literature in the massless limit.

ABSTRACT

In this paper we describe the implementation of the QED process $γγ oγγ$ through a fermion loop into the framework of SANC system. The computations of this process takes into account non-zero mass of loop-fermion. We briefly describe additional precomputation modules used for calculation of massive fermion-box diagrams. We present the covariant and helicity amplitudes for this process and also some particular cases of $D_0$ and $C_0$ Passarino-Veltman functions. Whenever possible, we compare the results with those existing in the literature.

Motivation & Objective

• To implement the one-loop QED process $\gamma\gamma \to \gamma\gamma$ with massive fermion loops in the SANC framework for high-precision collider physics.
• To compute covariant and helicity amplitudes for light-by-light scattering, including non-zero fermion mass effects.
• To provide analytical expressions for Passarino–Veltman scalar functions $D_0$, $C_0$, and $B_0$ in the massive case.
• To derive and tabulate integrals over the scattering angle necessary for computing the total cross section.
• To validate results against known literature in the massless limit and ensure consistency with existing QED calculations.

Proposed method

• Utilizes the SANC system to compute one-loop amplitudes from the Standard Model Lagrangian, focusing on $\gamma\gamma \to \gamma\gamma$ via fermion box diagrams.
• Derives the covariant amplitude tensor structure using Dirac matrices and external momenta, parameterized by form factors.
• Applies the Passarino–Veltman reduction scheme to express one-loop integrals in terms of scalar functions $D_0$, $C_0$, and $B_0$.
• Computes the helicity amplitudes using the spinor helicity formalism, valid for both massive and massless fermion loops.
• Performs analytical integration over the scattering angle $\theta$ using a table of integrals involving logarithmic and polylogarithmic functions.
• Uses the Mandelstam variables $s$, $t$, $u$ and the energy $\omega$ in the center-of-mass frame to express the differential cross section.

Experimental results

Research questions

• RQ1 How can the one-loop QED process $\gamma\gamma \to \gamma\gamma$ with massive fermion loops be systematically implemented in the SANC framework?
• RQ2 What are the analytical expressions for the covariant and helicity amplitudes of light-by-light scattering when the loop fermion has non-zero mass?
• RQ3 How do the Passarino–Veltman scalar functions $D_0$, $C_0$, and $B_0$ behave in the massive fermion case, particularly in the high-energy and low-energy limits?
• RQ4 What is the total cross section for $\gamma\gamma \to \gamma\gamma$ after integrating over the scattering angle, and how does it compare to known results in the massless limit?
• RQ5 What are the key integrals over $\cos\theta$ required to compute the total cross section, and how are they evaluated analytically?

Key findings

• The covariant amplitude for $\gamma\gamma \to \gamma\gamma$ is derived in terms of form factors, with explicit expressions for the massive fermion loop case.
• The helicity amplitudes are computed in both massive and massless limits, with the latter matching known results from the literature.
• Analytical expressions for $D_0$, $C_0$, and $B_0$ Passarino–Veltman functions are provided for the massive fermion loop, including limiting cases.
• The differential cross section is expressed as $d\sigma = \frac{1}{128\pi\omega^2}|\mathcal{A}|^2 d\cos\theta$, with $\omega$ the photon energy in the CMS.
• The total cross section is obtained by integrating over $\cos\theta$, using a table of 11 specific integrals involving logarithmic and polylogarithmic terms.
• In the massless limit, the result for $D_0$ matches the known expression involving $\ln^2(-M^2/t)$, $\pi^2/2$, and $\text{Li}_2$ functions, confirming consistency with previous calculations.