Pion light-cone wave function and pion distribution amplitude in the Nambu–Jona-Lasinio model

Abstract

We compute the pion light-cone wave function and the pion quark distribution amplitude in the Nambu--Jona-Lasinio model. We use the Pauli-Villars regularization method and as a result the distribution amplitude satisfies proper normalization and crossing properties. In the chiral limit we obtain the simple results, namely ${\ensuremath{\varphi}}_{\ensuremath{\pi}}(x)=1$ for the pion distribution amplitude, and $\ensuremath{\int}{d}^{2}{k}_{\ensuremath{\perp}}{\ensuremath{\Psi}}_{\ensuremath{\pi}}(x,{k}_{\ensuremath{\perp}}{)k}_{\ensuremath{\perp}}^{2}=〈{k}_{\ensuremath{\perp}}^{2}〉=\ensuremath{-}M〈\ifmmode \bar{u}\else \={u}\fi{}u〉{/f}_{\ensuremath{\pi}}^{2}$ for the second moment of the pion light-cone wave function, where M is the constituent quark mass and ${f}_{\ensuremath{\pi}}$ is the pion decay constant. After the QCD Gegenbauer evolution of the pion distribution amplitude good end-point behavior is recovered, and a satisfactory agreement with the analysis of the experimental data from CLEO is achieved. This allows us to determine the momentum scale corresponding to our model calculation, which is close to the value ${Q}_{0}=313\mathrm{MeV}$ obtained earlier from the analogous analysis of the pion parton distribution function. The value of $〈{k}_{\ensuremath{\perp}}^{2}〉$ is, after the QCD evolution, around $(400\mathrm{MeV}{)}^{2}.$ In addition, the model predicts a linear integral relation between the pion distribution amplitude and the parton distribution function of the pion, which holds at the leading-order QCD evolution.