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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mn>2</a:mn><a:mo>+</a:mo><a:mn>1</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:math> flavors of highly improved staggered quarks with lattice spacing <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>a</c:mi><c:mo>=</c:mo><c:mn>0.121</c:mn><c:mtext> </c:mtext><c:mtext> </c:mtext><c:mi>fm</c:mi></c:math> from the MILC collaboration, and one with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mn>2</e:mn><e:mo>+</e:mo><e:mn>1</e:mn></e:mrow></e:math> flavor clover fermions and tree-level Symanzik gauge action generated by the CLS collaboration with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>a</g:mi><g:mo>=</g:mo><g:mn>0.098</g:mn><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>fm</g:mi></g:math>. 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 calculation of TMDWFs which will eventually lead to crucial theory inputs for making predictions for exclusive processes under QCD factorization.