Intrinsic Transverse Motion of the Pion's Valence Quarks

Abstract

Starting with the solution to the Bethe-Salpeter equation for the pion, in a beyond rainbow-ladder truncation to QCD's Dyson-Schwinger equations (DSEs), we determine the pion's $l_z=0$ and $|l_z|=1$ leading Fock-state light-front wave functions (LFWFs) [labeled by $\psi_{l_z}(x,k_T^2)$]. The leading-twist time-reversal even transverse momentum dependent parton distribution function (TMD) of the pion is then directly obtained from these LFWFs. A key characteristic of the LFWFs, which is driven by dynamical chiral symmetry breaking, is that at typical hadronic scales they are broad functions in the light-cone momentum fraction $x$. The LFWFs have a non-trivial $(x,k_T^2)$ dependence and in general do not factorize into separate functions of each variable. The $l_z=0$ LFWF is concave with a maximum at $x=1/2$, whereas orbital angular momentum effects causes the $|l_z|=1$ LFWF to have a slight {\it double-humped} structure for quark transverse momentum in the range $0.5 \lesssim k_T^2 \lesssim 5\,$GeV$^2$. For $k_T^2 \lesssim 1\,$GeV$^2$ the $k_T^2$ dependence of the LFWFs is well described by a Gaussian, however for $k_T^2 \gtrsim 10\,$GeV$^2$ these LFWFs behave as $\psi_0 \propto x(1-x)/k_T^2$ and $\psi_1 \propto x(1-x)/k_T^4$, and therefore exhibit the power-law behavior predicted by perturbative QCD. The pion's TMD inherits many features from the LFWFs, where for $k_T^2 \lesssim 1\,$GeV$^2$ the $k_T^2$ dependence is well described by a Gaussian, and for large $k_T^2$ the TMD behaves as $f_\pi^q \propto x^2(1-x)^2/k_T^4$. At the model scale we find the average transverse momentum, defined by a Bessel-weighted moment with $b_T=0.3\,$fm, to equal $\langle k_T^2\rangle = 0.19\,$GeV$^2$. The TMD evolution of our result is studied using both the $b^*$ and $\zeta$ prescriptions which allows a qualitative comparison with existing Drell-Yan data.