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In this paper, we present a dispersive analysis of the double-virtualphoton-photon scattering to two pions up to 1.5 GeV. Through unitarity, thisprocess is very sensitive to hadronic final state interaction. For the$s$-wave, we use a coupled-channel $\pi\pi,\, K\bar{K}$ analysis which allows asimultaneous description of both $f_0(500)$ and $f_{0}(980)$ resonances. Forhigher energies, $f_2(1270)$ shows up as a dominant structure which weapproximate by a single channel $\pi\pi$ rescattering in the $d$-wave. In thedispersive approach, the latter requires taking into account $t$- and$u$-channel vector-meson exchange left-hand cuts which exhibit ananomalous-like behavior for large space-like virtualities. In our paper, weshow how to readily incorporate such behavior using a contour deformation.Besides, we devote special attention to kinematic constraints of helicityamplitudes and show their correlations explicitly.

Dispersive analysis of the $\gamma^{*}\gamma^{*} \to \pi \pi$ process

Dispersive analysis of the $\gamma^{}\gamma^{} \to \pi \pi$ process