Abstract
I discuss the factorization-compatible definitions of transverse-momentum-dependent (TMD) pion wave functions which are fundamental theory inputs entering QCD factorization formulae for many hard exclusive processes. I will first demonstrate that the soft subtraction factor introduced to remove both rapidity and pinch singularities can be greatly reduced by making the maximal use of the freedom to construct the Wilson-line paths when defining the TMD wave functions. I will then turn to show that the newly proposed TMD definition with non-dipolar Wilson lines is equivalent to the one with dipolar gauge links and with a complicated soft function, to all orders of the perturbative expansion in the strong coupling, as far as the infrared behavior is concerned.
Highlights
I will turn to show that the newly proposed TMD definition with non-dipolar Wilson lines is equivalent to the one with dipolar gauge links and with a complicated soft function, to all orders of the perturbative expansion in the strong coupling, as far as the infrared behavior is concerned
Hard exclusive processes have significantly sharpened our understanding of the strong interaction dynamics involved in the hadronic amplitudes, due to the establishment of various QCD factorization theorems which allow for separation of the hard scattering contributions and the longdistance QCD effects
Transverse-momentum-dependent (TMD) factorization for hard exclusive processes based upon the theory of on-shell Sudakov form factor [1], with the original aim of avoiding the rapidity divergences in collinear factorization for the pion electromagnetic form factor at sub-leading power [2], turns out to be far more sophisticated conceptually due to the absence of a definite power counting scheme and to the emergence of lightcone singularities in the infrared subtraction, and this can be already partially recognized from the troubled history of TMD wave functions [3,4,5,6,7,8,9,10,11,12,13,14]
Summary
Hard exclusive processes have significantly sharpened our understanding of the strong interaction dynamics involved in the hadronic amplitudes, due to the establishment of various QCD factorization theorems which allow for separation of the hard scattering contributions and the (universal) longdistance QCD effects. Transverse-momentum-dependent (TMD) factorization for hard exclusive processes based upon the theory of on-shell Sudakov form factor [1], with the original aim of avoiding the rapidity divergences in collinear factorization for the pion electromagnetic form factor at sub-leading power [2], turns out to be far more sophisticated conceptually due to the absence of a definite power counting scheme and to the emergence of lightcone singularities in the infrared subtraction, and this can be already partially recognized from the troubled history of TMD wave functions [3,4,5,6,7,8,9,10,11,12,13,14]. It is certainly interesting to generalize the discussion presented here to understand the factorization properties of those processes taking into account the contribution from the Glauber gluons, which is crucial to construct the Wilson line structures of TMD wave functions.
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