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

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

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