Linearly polarized small-$x$ gluons in forward heavy-quark pair production

We use the color glass condensate (CGC) framework to study the production of forward heavy quark-antiquark pairs in unpolarized proton-nucleus or proton-proton collisions in the small-$x$ regime. In the limit of nearly back-to-back jets, the CGC result simplifies into the transverse-momentum dependent (TMD) factorization approach. For massless quarks, the TMD factorization formula involves three unpolarized gluon TMDs: the Weizsäcker-Williams gluon distribution, the adjoint-dipole gluon distribution, and an additional one. When quark masses are kept nonzero, three new gluon TMDs appear—each partnered to one of the aforementioned distributions—which describe the distribution of linearly polarized gluons in the unpolarized small-$x$ target. We show how these six gluon TMDs emerge from the CGC formulation, and we determine their expressions in terms of Wilson line correlators. We calculate them analytically in the McLerran-Venugopalan model, and further evolve them toward smaller values of $x$ using a numerical implementation of Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) evolution.

Figure 1

Amplitude for quark-antiquark production in the CGC formalism. The pair can be radiated before (left) or after (right) the interaction with the target. The two terms come with a relative minus sign.

Figure 2

One of the leading order diagrams for inclusive heavy-quark pair production in $p+A$ collisions.

Figure 3

Momentum dependence of gluon TMDs in the initial MV configurations. The linear vertical scale is rescaled by the factor $2{\pi }^3{g}^2{L}^{-2}$. The logarithmic momentum scale is in inverse lattice spacing units. $\mathcal{T}(y,k_t)$ is a shorthand for the labels of the gluon TMDs displayed in the figure.

Figure 4

Momentum dependence of gluon TMDs near the lower bound (top) and upper bound (bottom) of the geometric scaling window. The logarithmic momentum scale is in inverse lattice spacing units. $\mathcal{T}(y,k_t)$ is a shorthand for the labels of the gluon TMDs displayed in the figure.

Figure 5

“Staple” gauge link used in the gauge-invariant definition of the Weizsäcker-Williams gluon TMD.