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

The Nambu--Jona-Lasinio model is utilized to investigate the pion and kaon photon leading-twist transition distribution amplitudes using proper time regularization. Separately, the properties of the vector and axial vector pion photon transition distribution amplitudes are examined, and the results meet the desired properties. Our study involves sum rule and polynomiality condition. The vector and axial vector pion photon transition form factors that are present in the $\pi^+\rightarrow \gamma e^+ \nu$ process are the first Mellin moments of the pion photon transition distribution amplitudes. The vector transition form factor originates 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. The kaon transition form factors are similar. The vector form factor's value at zero momentum transfer is determined by the axial anomaly, while this is not the case for the axial one. The vector and axial form factors, as well as the neutral pion vector form factor $F_{\pi \gamma \gamma}(t)$, are depicted. According to our findings, the pion axial transition form factor is harder than the vector transition form factor and harder than the electromagnetic form factor. We also discuss the link between $\pi - \gamma $ and $\gamma - \pi$ transitions distribution amplitudes.