Transverse-momentum-dependent wave functions of the pion from lattice QCD

We present a lattice QCD calculation of the transverse-momentum-dependent wave function (TMDWF) of the pion using large-momentum effective theory. Numerical simulations are based on one ensemble with $2+1+1$ flavors of highly improved staggered quarks with lattice spacing $a=0.121\mathrm{fm}$ from the MILC collaboration, and one with $2+1$ flavor clover fermions and tree-level Symanzik gauge action generated by the CLS collaboration with $a=0.098\mathrm{fm}$. As a key ingredient, the soft function is first obtained by incorporating the one-loop perturbative contributions and a proper normalization. Based on this and the equal-time quasi-TMDWF simulated on the lattice, we extract the TMDWF. The results for both lattice ensembles are compatible and a comparison with a phenomenological parametrization is made. Our studies provide a first attempt of ab initio calculation of TMDWFs which will eventually lead to crucial theory inputs for making predictions for exclusive processes under QCD factorization.