Intrinsic Transverse Motion of the Pion’s Valence Quarks
Starting with the solution to the Bethe-Salpeter equation for the pion, in a beyond rainbow-ladder truncation to QCD's Dyson-Schwinger equations, we determine the pion's $\mathcal{l}$ z = 0 and |$\mathcal{l}$ z | = 1 leading Fock-state light-front wave functions (LFWFs) [labeled by $\psi$ $\mathcal{l}$ z ($\mathcal{x}$, $\mathcal{k}$$2\atop{T}$)]. The leading-twist time-reversal even transverse momentum dependent parton distribution function (TMD) of the pion is then directly obtained using these LFWFs. A key characteristic of the LFWFs, which is driven by dynamical chiral symmetry breaking, is that at typical hadronic scales they are broad functions in the light-cone momentum fraction $\mathcal{x}$. The LFWFs have a nontrivial ($\mathcal{x}$, $\mathcal{k}$$2\atop{T}$)) dependence and in general do not factorize into separate functions of each variable. For $\mathcal{k}$$2\atop{T}$ ≲ 1 GeV 2 the $\mathcal{k}$$2\atop{T}$ dependence of the LFWFs is well described by a Gaussian; however for $\mathcal{k}$$2\atop{T}$ ≳ 10 GeV 2 these LFWFs behave as $\psi$ 0 ∝ $\mathcal{x}$(1 - $\mathcal{x}$)/$\mathcal{k}$$2\atop{T}$ and $\psi$ 1 ∝ $\mathcal{x}$(1 - $\mathcal{x}$)/$\mathcal{k}$$4\atop{T}$ and therefore exhibit the power-law behavior predicted by perturbative QCD. The pion's TMD naturally inherits many features from the LFWFs. With this being said, the TMD evolution of our result is studied using both the b* and $\zeta$ prescriptions which allows a qualitative comparison with Drell-Yan data.