Abstract
We investigate the predictive power of transverse-momentum-dependent (TMD) distributions as a function of the light-cone momentum fraction $x$ and the hard scale $Q$ defined by the process. We apply the saddle point approximation to the unpolarized quark and gluon transverse momentum distributions and evaluate the position of the saddle point as a function of the kinematics. We determine quantitatively that the predictive power for an unpolarized transverse momentum distribution is maximal in the large-$Q$ and small-$x$ region. For cross sections the predictive power of the TMD factorization formalism is generally enhanced by considering the convolution of two distributions, and we explicitly consider the case of $Z$ and $H^0$ boson production. In the kinematic regions where the predictive power is not maximal, the distributions are sensitive to the non-perturbative hadron structure. Thus, these regions are critical for investigating hadron tomography in a three-dimensional momentum space.
Highlights
The theoretical study and experimental exploration of the internal structure of nucleons are of fundamental importance to science [1,2,3]
For cross sections the predictive power of the transverse momentum dependent distribution functions (TMDs) factorization formalism is generally enhanced by considering the convolution of two distributions, and we explicitly consider the case of Z and H0 boson production
In this article we extend those arguments to the context of the modern TMD factorization formalism [9], linking the predictive power of the TMDs to their double scale evolution, i.e., the ultraviolet and rapidity renormalization scales to be defined below
Summary
The theoretical study and experimental exploration of the internal structure of nucleons are of fundamental importance to science [1,2,3]. The study of the kinematic dependence originates from the work of Parisi and Petronzio [37] and Collins, Soper, and Sterman [5], which focused on the value of the hard scale of the process compared to the infrared scale of QCD (ΛQCD) It has been shown at the level of the cross section [17,29,38,39] that the light-cone momentum fraction x, which is effectively a measure of available phase space for the parton shower, could play an important role in determining the relevance of the nonperturbative corrections.
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