# Dispersion relations for $\gamma^*\gamma^*\to\pi\pi$: helicity amplitudes, subtractions, and anomalous thresholds

## TL;DR

This paper develops a detailed dispersion relation framework for the doubly-virtual gamma-gamma to pion-pion process, addressing analytic structures, subtraction constants, and anomalous thresholds to improve hadronic light-by-light scattering calculations.

## Contribution

It introduces a comprehensive method for solving partial-wave dispersion relations with subtractions, accounting for anomalous thresholds, and applies this to predict the doubly-virtual response of the $f_2(1270)$ resonance.

## Key findings

- Derived kernel functions for Roy-Steiner equations.
- Formulated solutions using Omnès functions with subtractions.
- Predicted the doubly-virtual response of the $f_2(1270)$ resonance.

## Abstract

We present a comprehensive analysis of the dispersion relations for the doubly-virtual process $\gamma^*\gamma^*\to\pi\pi$. Starting from the Bardeen-Tung-Tarrach amplitudes, we first derive the kernel functions that define the system of Roy-Steiner equations for the partial-wave helicity amplitudes. We then formulate the solution of these partial-wave dispersion relations in terms of Omn\`es functions, with special attention paid to the role of subtraction constants as critical for the application to hadronic light-by-light scattering. In particular, we explain for the first time why for some amplitudes the standard Muskhelishvili-Omn\`es solution applies, while for others a modified approach based on their left-hand cut is required unless subtractions are introduced. In the doubly-virtual case, the analytic structure of the vector-resonance partial waves then gives rise to anomalous thresholds, even for space-like virtualities. We develop a strategy to account for these effects in the numerical solution, illustrated in terms of the $D$-waves in $\gamma^*\gamma^*\to\pi\pi$, which allows us to predict the doubly-virtual responses of the $f_2(1270)$ resonance. In general, our results form the basis for the incorporation of two-meson intermediate states into hadronic light-by-light scattering beyond the $S$-wave contribution.