Large-Nsummation of chiral logs for generalized parton distributions

Abstract

We demonstrate that in the region of Bjorken ${x}_{\mathrm{Bj}}\ensuremath{\sim}{m}_{\ensuremath{\pi}}^{2}/(4\ensuremath{\pi}{F}_{\ensuremath{\pi}}{)}^{2}$ and/or ${x}_{\mathrm{Bj}}\ensuremath{\sim}|t|/(4\ensuremath{\pi}{F}_{\ensuremath{\pi}}{)}^{2}$ the standard $\ensuremath{\chi}\mathrm{PT}$ for the pion generalized parton distributions (GPDs) fails and one must perform all order resummation of $\ensuremath{\chi}\mathrm{PT}$. We perform such resummation in the large-$N$ limit of the $O(N+1)$ extension of the chiral theory. Explicit resummation allows us to reveal novel phenomena---the form of the leading chiral correction to pion parton distribution functions (PDFs) and GPDs depends on the small $x$ asymptotic of the pion PDFs. In particular, if the pion PDF in the chiral limit has the Regge-like small $x$ behavior $q(x)\ensuremath{\sim}1/{x}^{\ensuremath{\omega}}$, the leading large impact parameter (${b}_{\ensuremath{\perp}}\ensuremath{\rightarrow}\ensuremath{\infty}$) asymptotics of the quark distribution in the transverse plane has the form (${m}_{\ensuremath{\pi}}=0$) $q(x,{b}_{\ensuremath{\perp}})\ensuremath{\sim}1/{x}^{\ensuremath{\omega}}{ln}^{\ensuremath{\omega}}({b}_{\ensuremath{\perp}}^{2})/{b}_{\ensuremath{\perp}}^{2(1+\ensuremath{\omega})}$. This result is model independent and it is controlled completely by the all order resummed $\ensuremath{\chi}\mathrm{PT}$ developed in this paper. This asymptotic interweaves with small-$x$ behavior of usual PDFs, hence it depends on the scale, at which the corresponding PDF is defined. This is a new and interesting result in which the chiral expansion meets the QCD evolution.