Transverse distributions of the pion cloud of the proton

The pion cloud of a proton is generated by its fluctuations into pion/baryon pairs. For fast-moving protons, this cloud is contracted into a disk transverse to the proton's longitudinal momentum. We construct a two-body light cone wave function $\psi (y, k_\perp)$ to represent a fluctuation in which a pion has longitudinal momentum fraction $y$ and transverse momentum $k_\perp$. We then use a 2D Bessel transform to calculate the transverse spatial probability distribution $\rho_\pi(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.