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Register a new userEfficient Fourier Transforms for Transverse Momentum Dependent Distributions
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Hadron production at low transverse momenta in semi-inclusive deep inelastic scattering can be described by transverse momentum dependent (TMD) factorization. This formalism has also been widely used to study the Drell-Yan process and back-to-back hadron pair production in $e^+e^-$ collisions. These processes are the main ones for extractions of TMD parton distribution functions and TMD fragmentation functions, which encode important information about nucleon structure and hadronization. One of the most widely used TMD factorization formalism in phenomenology formulates TMD observables in coordinate $b_perp$-space, the conjugate space of the transverse momentum. The Fourier transform from $b_perp$-space back into transverse momentum space is sufficiently complicated due to oscillatory integrands that it requires a careful and computationally intensive numerical treatment in order to avoid potentially large numerical errors. Within the TMD formalism, the azimuthal angular dependence is analytically integrated and the two-dimensional $b_perp$ integration reduces to a one-dimensional integration over the magnitude $b_perp$. In this paper we develop a fast numerical Hankel transform algorithm for such a $b_perp$-integration that improves the numerical accuracy of TMD calculations in all standard processes. Libraries for this algorithm are implemented in Python 2.7 and 3, C++, as well as FORTRAN77. All packages are made available open source.
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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.
We compute the contribution of twist-2 and twist-3 parton distribution functions to the small-$b$ expansion for transverse momentum dependent (TMD) distributions at all powers of $b$. The computation is done by the twist-decomposition method based on the spinor formalism for all eight quark TMD distributions. The newly computed terms are accompanied by the prefactor $(M^2b^2)^n$ and represent the target-mass corrections to the resummed cross-section. For the first time, a non-trivial expression for the pretzelosity distribution is derived.
We present the transverse momentum spectrum of groomed jets in di-jet events for $e^+e^-$ collisions and semi-inclusive deep inelastic scattering (SIDIS). The jets are groomed using a soft-drop grooming algorithm which helps in mitigating effects of non-global logarithms and underlying event. At the same time, by reducing the final state hadronization effects, it provides a clean access to the non-perturbative part of the evolution of transverse momentum dependent (TMD) distributions. In SIDIS experiments we look at the transverse momentum of the groomed jet measured w.r.t. the incoming hadron in the Breit frame. Because the final state hadronization effects are significantly reduced, the SIDIS case allows to probe the TMD parton distribution functions. We discuss the sources of non-perturbative effects in the low transverse momentum region including novel (but small) effects that arise due to grooming. We derive a factorization theorem within SCET and resum any large logarithm in the measured transverse momentum up to NNLL accuracy using the $zeta$-prescription as implemented in the artemide package and provide a comparison with simulations.
Transverse-momentum-dependent (TMD) gluon distributions have different operator definitions, depending on the process under consideration. We study that aspect of TMD factorization in the small-x limit, for the various unpolarized TMD gluon distributions encountered in the literature. To do this, we consider di-jet production in hadronic collisions, since this process allows to be exhaustive with respect to the possible operator definitions, and is suitable to be investigated at small x. Indeed, for forward and nearly back-to-back jets, one can apply both the TMD factorization and Color Glass Condensate (CGC) approaches to compute the di-jet cross-section, and compare the results. Doing so, we show that both descriptions coincide, and we show how to express the various TMD gluon distributions in terms of CGC correlators of Wilson lines, while keeping Nc finite. We then proceed to evaluate them by solving the JIMWLK equation numerically. We obtain that at large transverse momentum, the process dependence essentially disappears, while at small transverse momentum, non-linear saturation effects impact the various TMD gluon distributions in very different ways. We notice the presence of a geometric scaling regime for all the TMD gluon distributions studied: the dipole one, the Weizsacker-Williams one, and the six others involved in forward di-jet production.
The interplay between the small x limit of QCD amplitudes and QCD factorization at moderate x has been studied extensively in recent years. It was finally shown that semiclassical formulations of small x physics can have the form of an infinite twist framework involving Transverse Momentum Dependent (TMD) distributions in the eikonal limit. In this work, we demonstrate that small x distributions can be formulated in terms of transverse gauge links. This allows in particular for direct and efficient decompositions of observables into subamplitudes involving gauge invariant sub-operators which span parton distributions.
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