The quantum evolutions of the diffractive transverse-momentum dependent gluon distribution

TL;DR

This work develops a first-principles treatment of the Collins-Soper-Sterman evolution for the gluon diffractive TMD within the Colour Glass Condensate framework, focusing on back-to-back diffractive di-jet production in high-energy $eA$ collisions. It presents three equivalent yet non-identical representations—$K_\perp$-space, $b_\perp$-space, and a double-logarithmic approximation—for the CSS evolution, and derives boundary conditions that encode saturation alongside BK/JIMWLK and DGLAP evolutions. Numerical solutions show a robust agreement between the $K_\perp$-space and $b_\perp$-space approaches and reveal how CSS shifts DTMD strength from the saturation plateau to higher transverse momenta, with the DGLAP boundary contributing a large-$K_\perp$ tail. A positivity issue arising from the DGLAP piece suggests the need for Y-term power corrections to maintain physical cross sections in certain kinematic regimes. The results illuminate the interplay between saturation, soft-gluon resummation, and collinear evolution, offering a principled route to diffractive jet phenomenology at upcoming facilities like the EIC.

Abstract

Using the Colour Glass Condensate description of electron-nucleus collisions at high energy, we study the diffractive production of a pair of jets with transverse momenta much larger than the nuclear saturation momentum $Q_s$. At leading order in the QCD coupling, the di-jet cross-section exhibits transverse-momentum dependent (TMD) factorisation, with a gluon diffractive TMD distribution (DTMD) which is controlled by gluon saturation and describes the transverse-momentum imbalance between the produced jets. The next-to-leading corrections generate the various quantum evolutions of the diffractive gluon distribution. We focus on the Collins-Soper-Sterman (CSS) evolution which describes the change in the gluon DTMD when increasing the ''hard scale'' (the typical transverse momentum of the di-jets). We consider two different representations for this equation, one in transverse-momentum space, the other one in transverse-coordinate space. They are not fully equivalent with each other (despite being related by a Fourier transform) because of the respective boundary conditions. These conditions encode the essential physics of gluon saturation together with the effects of two other types of quantum evolution: the BK/JIMWLK evolution over the rapidity gap (''inside the Pomeron'') and the DGLAP evolution outside the rapidity gap (''within the diffractive system''). We demonstrate that, due to gluon saturation, one can compute both the boundary conditions and the CSS solutions fully from first principles, without reference to non-perturbative physics. We numerically find a good agreement between the CSS solutions in the two aforementioned representations.

Figures (15)

• Figure 1: Left panel: Diffractive scattering with a final state composed of a hard $q\bar{q}$ dijet and a semi-hard gluon. The gluon is emitted either from the quark or from the antiquark and at a large transverse distance $R\sim 1/K_\perp$ from the small $q\bar{q}$ pair (with transverse size $r\sim 1/P_\perp$). Right panel: the TMD factorisation of the di-jet cross-section. The semi-hard gluon is now radiated by the Pomeron represented by the zig-zag line.
• Figure 2: Left: the tree-level gluon DTMD, as computed from Eqs. \ref{['Ftree']}--\ref{['gp']} together with the MV model for the dipole amplitude $\mathcal{T}_g(R)$ (with gluon saturation scale $Q_s^2=2$ GeV$^2$), is plotted as a function of the dimensionless ratio $K_\perp/Q_s$ for different values of $x$. (We omit an overall factor ${S_\perp}/{4\pi^3}$, so the plotted function $\mathcal{F}^{(0)}_g(x, K_\perp^2)$ is dimensionless.) Right: the same as in the left plot, except that the gluon DTMD is now multiplied with the measure factor $K_\perp$, to better emphasise the tail at large $K_\perp\gg Q_s$.
• Figure 3: The plots in Fig. \ref{['fig:tree']} are re-drawn in such a way to emphasise the (approximate) scaling of the tree-level gluon DTMD as a function of the variables $K_\perp$ and $x$.
• Figure 4: The FT of the tree-level gluon DTMD, as numerically obtained from Eq. \ref{['Ftree']} and after dividing out the dimensionless factor $Q_s^2 S_\perp/(4\pi^3)$. In the right plot we use a logarithmic scale on the vertical axis.
• Figure 5: The same as in Fig. \ref{['fig:F0b']}, but after dividing out the dominant $x$-behaviour of $\tilde{\mathcal{F}}_g^{(0)}(x, \mu_b^2)$ as suggested by Eqs. \ref{['Ftreesmallb']} and \ref{['xGPhigh']}, and in terms of the "scaling" ($x$-dependent) variable $\tilde{Q}_s(x)b_\perp$.