Dispersive analysis of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>γ</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:msup><mml:mi>γ</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:mi>π</mml:mi><mml:mi>π</mml:mi></mml:math> process
Igor Danilkin, Oleksandra Deineka, Marc Vanderhaeghen
Abstract
We present a dispersive analysis of the double-virtual photon-photon scattering to two pions up to 1.5 GeV. Through unitarity, this process is very sensitive to hadronic final-state interaction. For the $s$-wave, we use a coupled-channel $\ensuremath{\pi}\ensuremath{\pi}$, $K\overline{K}$ analysis which allows for a simultaneous description of both ${f}_{0}(500)$ and ${f}_{0}(980)$ resonances. For higher energies, ${f}_{2}(1270)$ shows up as a dominant structure which we approximate by a single-channel $\ensuremath{\pi}\ensuremath{\pi}$ rescattering in the $d$-wave. In the dispersive approach, the latter requires taking into account $t$- and $u$-channel vector-meson exchange left-hand cuts which exhibit an anomalouslike behavior for large spacelike virtualities. We show how to readily incorporate such behavior using a contour deformation. Besides, we devote special attention to kinematic constraints of helicity amplitudes and show their correlations explicitly.