We demonstrate that in the region of Bjorken $x_{\mathrm{Bj}}\sim m_{\pi }^2/(4\pi F_{\pi }{)}^2$ and/or $x_{\mathrm{Bj}}\sim |t|/(4\pi F_{\pi }{)}^2$ the standard $\chi \mathrm{PT}$ for the pion generalized parton distributions (GPDs) fails and one must perform all order resummation of $\chi \mathrm{PT}$. 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 $q(x)\sim 1/x^{\omega }$, the leading large impact parameter ($b_{\perp }\to \infty$) asymptotics of the quark distribution in the transverse plane has the form ($m_{\pi }=0$) $q(x,b_{\perp })\sim 1/x^{\omega }{ln}^{\omega }(b_{\perp }^2)/b_{\perp }^{2(1+\omega )}$. This result is model independent and it is controlled completely by the all order resummed $\chi \mathrm{PT}$ 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.

• Received 12 November 2008

DOI:https://doi.org/10.1103/PhysRevD.79.014028