Transverse Distributions of the Pion Cloud in a Chiral Light Cone Perturbation Theory Model

Because of the Heisenberg uncertainty principle, protons are allowed to briefly fluctuate into a pion and a nucleon or a pion and a delta. Our goal is to calculate the splitting of the proton into these separate particles while the proton is moving at relativistic speeds, where its spatial extent becomes nearly two-dimensional, a disk of pion cloud. We use a pion 2D momentum distribution function $f_\pi _N(y,t) $, derived from chiral light cone perturbation theory, in which $y$ is the fraction of proton momentum carried by the pion and the momentum transfer $t $ depends on $y$ and $k_\perp$, the transverse momentum of the pion. To find transverse momentum distributions we calculate $f_\pi _N $ as a function of $y$ and $k_\perp$ for a range of physically reasonable values of the form factors and coupling constants on which it depends. We then use a 2D Bessel transform of $f_\pi _N$ to calculate the transverse spatial probability distribution $\rho_\pi _N(y,b) $ with $b$ the transverse position coordinate. We compare our results to the expected spatial extent of the cloud, $\sim 1/m_\pi$, and to other theoretical transverse spatial distributions.