Kinematical higher-twist corrections in <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:mi>γ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>M</mml:mi><mml:mover accent="true"><mml:mi>M</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover></mml:math>
Cédric Lorcé, B. Pire, Qin-Tao Song
Abstract
We estimate kinematical higher-twist (up to twist 4) corrections to the ${\ensuremath{\gamma}}^{*}({q}_{1})\ensuremath{\gamma}({q}_{2})\ensuremath{\rightarrow}M({p}_{1})\overline{M}({p}_{2})$ amplitudes at large ${Q}^{2}=\ensuremath{-}{q}_{1}^{2}$ and small $s=({q}_{1}+{q}_{2}{)}^{2}$, where $M$ is a scalar or pseudoscalar meson. This process is known to factorize at leading twist into a perturbatively calculable coefficient function and generalized distribution amplitudes (GDAs). The kinematical higher-twist contributions of order $s/{Q}^{2}$ and ${m}^{2}/{Q}^{2}$ turn out to be important in the cross section, considering the kinematics accessible at Belle and Belle II. We present numerical estimates for the cross section for ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ with the $\ensuremath{\pi}\ensuremath{\pi}$ GDA extracted from Belle measurements and with the asymptotic $\ensuremath{\pi}\ensuremath{\pi}$ GDA as inputs to study the magnitude of the kinematical corrections. To see how the target mass corrections of order ${m}^{2}/{Q}^{2}$ affect the cross section, we also perform the calculation for ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\eta}$ by using a model $\ensuremath{\eta}\ensuremath{\eta}$ GDA. In the range $s>1\text{ }\text{ }{\mathrm{GeV}}^{2}$, the kinematical higher-twist corrections account for $\ensuremath{\sim}15%$ of the total cross section, an effect which is not negligible. Since $\ensuremath{\pi}\ensuremath{\pi}$ GDAs are the best way to access the pion energy-momentum tensor (EMT), our study demonstrates that an accurate evaluation of EMT form factors requires the inclusion of kinematical higher-twist contributions.