Transverse momentum distribution in hadrons: What can we learn from QCD?

Abstract

We discuss some strict QCD constraints on the light-cone \ensuremath{\pi} meson wave function \ensuremath{\psi}(k${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\perp}}}^{2}$,x). The analysis is based on such general methods as dispersion relations, duality, and partial conservation of axial-vector current. We calculate the asymptotic behavior of the wave function (WF) at the end-point region (x\ensuremath{\rightarrow}1 and k${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\perp}}}^{2}$\ensuremath{\rightarrow}\ensuremath{\infty}) by analyzing the corresponding large nth moments in transverse 〈k${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\perp}}}^{2\mathit{n}}$〉\ensuremath{\sim}n! and longitudinal 〈(2x-1${)}^{\mathit{n}}$〉\ensuremath{\sim}1/${\mathit{n}}^{2}$ directions. This information fixes the asymptotic behavior of the WF at large k${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\perp}}}^{2}$ (which turns out to be the Gaussian commonly used in phenomenological analyses). We discuss one particular application of the obtained results. We calculate the nonleading ``soft'' contribution to the pion form factor at intermediate momentum transfer. We argue that, due to the specific properties of \ensuremath{\psi}(k${\ensuremath{\rightarrow}}_{\mathrm{\ensuremath{\perp}}}^{2}$,x), the corresponding contribution can temporarily simulate the leading twist behavior in the extent region of ${\mathit{Q}}^{2}$: 3 ${\mathrm{GeV}}^{2}$\ensuremath{\le}${\mathit{Q}}^{2}$\ensuremath{\le}40 ${\mathrm{GeV}}^{2}$, where ${\mathit{Q}}^{2}$F(${\mathit{Q}}^{2}$)\ensuremath{\sim}const. Such a mechanism, if it is correct, would be an explanation of the phenomenological success of the dimensional counting rules at available, very modest energies for many different processes. We discuss some inclusive amplitudes (such as Drell-yan and deep inelastic) where the intrinsic \ensuremath{\pi} meson structure might be essential. The relation to the valence quark model is also discussed.