Matthew D. Sievert

Calculating TMDs of a large nucleus: Quasi-classical approximation and quantum evolution

Abstract We set up a formalism for calculating transverse-momentum-dependent parton distribution functions (TMDs) of a large nucleus using the tools of saturation physics. By generalizing the quasi-classical Glauber–Gribov–Mueller/McLerran–Venugopalan approximation to allow for the possibility of spin–orbit coupling, we show how any TMD can be calculated in the saturation framework. This can also be applied to the TMDs of a proton by modeling it as a large “nucleus.” To illustrate our technique, we calculate the quark TMDs of an unpolarized nucleus at large-x: the unpolarized quark distribution and the quark Boer–Mulders distribution. We observe that spin–orbit coupling leads to mixing between different TMDs of the nucleus and of the nucleons. We then consider the evolution of TMDs: at large-x, in the double-logarithmic approximation, we obtain the Sudakov form factor. At small-x the evolution of unpolarized-target quark TMDs is governed by BK/JIMWLK evolution, while the small-x evolution of polarized-target quark TMDs appears to be dominated by the QCD Reggeon.

Helicity evolution at small x

A bstractWe construct small-x evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the g1 structure function. These evolution equations resum powers of αs ln2(1/x) in the polarization-dependent evolution along with the powers of αs ln(1/x) in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in the large-Nc and large-Nc & Nf limits. As a cross-check, in the ladder approximation, our equations map onto the same ladder limit of the infrared evolution equations for the g1 structure function derived previously by Bartels, Ermolaev and Ryskin [1].

Sivers Function in the Quasi-Classical Approximation

In this brief article we summarize the derivation of the Sivers function of a heavy nucleus, which we obtain by generalizing the quasi-classical McLerran-Venugopalan model to incorporate the role of spin and orbital angular momentum. In doing so we obtain a new channel which is capable of generating the Sivers function of the nucleus from the orbital motion of its nucleons. An essential role is played in this channel by the multiple rescatterings on spectator nucleons which screen the distribution function during the initial- or final-state interactions. The combination of orbital angular momentum together with multiple rescattering yields a new interpretation of the sign flip relation between the Sivers function measured in semi-inclusive deep inelastic scattering and the Drell-Yan process.

Small-

We construct a numerical solution of the small-x evolution equations derived in our recent work [J. High Energy Phys. 01 (2016) 072.JHEPFG1029-847910.1007/JHEP01(2016)072] for the (anti)quark transverse momentum dependent helicity TMDs and parton distribution functions (PDFs) as well as the g_{1} structure function. We focus on the case of large N_{c}, where one finds a closed set of equations. Employing the extracted intercept, we are able to predict directly from theory the behavior of the quark helicity PDFs at small x, which should have important phenomenological consequences. We also give an estimate of how much of the protons spin carried by the quarks may be at small x and what impact this has on the spin puzzle.

x

We present the first steps in an effort to incorporate the physics of transverse spin asymmetries into the saturation formalism of high energy QCD. We consider a simple model in which a transversely polarized quark scatters on a proton or nuclear target. Using the light-cone perturbation theory the hadron production cross section can be written as a convolution of the light-cone wave function squared and the interaction with the target. To generate the single transverse spin asymmetry (STSA) either the wave function squared or the interaction with the target has to be T-odd. In this work we use the lowest-order q → q G wave function squared, which is T-even, generating the STSA from the T-odd interaction with the target mediated by an odderon exchange. We study the properties of the obtained STSA, some of which are in qualitative agreement with experiment: STSA increases with increasing projectile xF and is a non-monotonic function of the transverse momentum kT. Our mechanism predicts that the quark STSA ...

asymptotics of the quark helicity distribution

We determine the small-

SINGLE SPIN ASYMMETRY IN HIGH ENERGY QCD

x

Gluon Helicity Distribution at Small x

asymptotics of the gluon helicity distribution in a proton at leading order in perturbative QCD at large

Helicity Evolution at Small

N_c

x

. To achieve this, we begin by evaluating the dipole gluon helicity TMD at small