Pion-photon and kaon-photon transition distribution amplitudes in the Nambu--Jona-Lasinio model

Pion and kaon photon leading-twist transition distribution amplitudes are investigated within the framework of the Nambu--Jona-Lasinio model using proper time regularization. The properties of the vector and axial vector pion photon transition distribution amplitudes are examined separately, the results satisfy the desirable properties. The sum rules and polynomiality condition are studied, the first Mellin moments of the pion and kaon photon transition distribution amplitudes correspond to the vector and axial vector pion and kaon photon form factors $F_V(t)$ and $F_A(t)$, which appear in the $\pi^+\rightarrow \gamma e^+ \nu$ process. The vector transition form factor comes from the internal structure of hadrons, the axial current can be coupled to a pion, this pion is virtual, and its contribution will be present independently of the external hadrons, kaon transition form factors are similar. The value at zero momentum transfer of the vector form factor is fixed by the axial anomaly, while this is not the case for the axial one. The diagrams of the two form factors are plotted, in addition, the neutral pion vector form factor $F_{\pi \gamma \gamma}(t)$. We find that the axial vector transition form factor is harder than the vector transition form factor and harder than the electromagnetic form factor. The results are listed in our table, we also discuss the relationship of $\pi - \gamma $ and $\gamma - \pi$ transitions distribution amplitudes.