Kinematical higher-twist corrections in ${\gamma }^{*}\to M_1{M}_2\gamma$: Charged meson production

Generalized distribution amplitudes (GDAs) of mesons can be probed by the reactions $e^{-}{e}^{+}\to M_1{M}_2\gamma$, which are accessible at electron-positron colliders such as BESIII and Belle II. After discussing the neutral meson production case in our previous paper, we discuss here the complementary case of the charged meson ($M^{+}{M}^{-}$) production, where one can extract the complete information on GDAs from the interference of the amplitudes of the two competing processes where the photon is emitted either in the initial or in the final state. Considering the importance of the charged meson production, we present a complete expression for the interference term of the cross section, which is experimentally accessible thanks to its charge conjugation specific property. We adopt two types of models for leading-twist $\pi \pi$ GDAs to estimate the size of the interference term in the process $e^{-}{e}^{+}\to {\pi }^{+}{\pi }^{-}\gamma$ numerically, namely, a model extracted from previous experimental results on ${\gamma }^{*}\gamma \to {\pi }^0{\pi }^0$ at Belle and the asymptotic form predicted by QCD evolution equations. We include in the calculation the kinematical power suppressed (sometimes called kinematical higher-twist) corrections up to $1/Q^2$ for the helicity amplitudes. Both models of GDAs indicate that the kinematical corrections are not negligible for the interference term of the cross section measured at BESIII, thus it is necessary to include them if we try to extract the GDAs precisely. On the other side, the kinematical corrections are very tiny for the measurements at Belle II, and the leading twist-two formula of the interference term will be good enough to describe the charge conjugation odd part of the differential cross section.

Figure 1

Description of the kinematics of the reaction $e^{-}(k_1)e^{+}(k_2)\to {\gamma }^{*}(q_1)\to M(p_1)\overline{M}(p_2)\gamma (q_2)$, in the center-of-mass frame of the meson pair; the $z$ axis is chosen along the photons’ momenta.

Figure 2

One of the leading-twist Feynman diagrams for the subprocess (a), $e^{-}{e}^{+}\to {\gamma }^{*}\to M\overline{M}\gamma$; the meson pair is produced with charge conjugation $C=+$ and the hadronization process is factorized with the help of two meson GDAs.

Figure 3

Feynman diagrams of the ISR subprocess for the reaction $e^{-}{e}^{+}\to M\overline{M}\gamma$; the meson pair is produced with charge conjugation $C=-$ and the hadronic process occurs through the meson timelike form factors.

Figure 4

The differential cross section $d\sigma /dud{W}^{2}d\cos \theta$ for the ISR process; $s$ is set as $s=12{\mathrm{GeV}}^2$ for BESIII and $s=100{\mathrm{GeV}}^2$ for Belle II. We choose (left) $u=-2$ and (right) $u=-6{\mathrm{GeV}}^2$. The solid lines denote $\cos \theta =0.2$, and the dashed lines represent $\cos \theta =0.8$.

Figure 5

The meson pair invariant mass ($W$) dependence of the cross sections $d{\sigma }_{\mathrm{G}}$ and $d{\sigma }_{\mathrm{I}}$ at $s=12{\mathrm{GeV}}^2$; the model for the $\pi \pi$ GDA [30] comes from an analysis of Belle measurements. $d{\sigma }_{\mathrm{G}}$ is depicted as solid curves, and the dashed curves represent the interference term $d{\sigma }_{\mathrm{I}}$.

Figure 6

Differential cross section of the interference term $d{\sigma }_{\mathrm{I}}$. The kinematics is set according to the BESIII experiment as $s=12$, $u=-2(-6){\mathrm{GeV}}^2$ together with $\cos \theta =0.2$ (0.8). The dashed lines are the twist-two cross sections, and the solid lines include the kinematical power suppressed contribution. The GDA model used comes from the estimate [30] using the Belle measurements.

Figure 7

Differential interference cross section $d{\sigma }_{\mathrm{I}}$; $s=100{\mathrm{GeV}}^2$ is chosen according to the Belle II experiment kinematics. The dashed curves denote the twist-two cross sections, and the solid ones include the kinematical power suppressed contribution.

Figure 8

The azimuthal angle $\phi$ dependence of the interference cross section $d{\sigma }_{\mathrm{I}}$; $s=12{\mathrm{GeV}}^2$ and the extracted GDA is used. The dashed curves denote the twist-two cross sections, and the solid ones include the kinematical power suppressed contribution.

Figure 9

Differential cross section of the interference term $d{\sigma }_{\mathrm{I}}$ at $s=12$, $u=-2(-6){\mathrm{GeV}}^2$. The asymptotic $\pi \pi$ GDA is used as the input.

Figure 10

Interference term $d{\sigma }_{\mathrm{I}}$ is dependent on the azimuthal angle $\phi$; we use the same conventions as in Fig. 8 and the asymptotic model GDA.