Modeling the TMD shape function in $J/ψ$ electroproduction

TL;DR

This work develops a TMD-factorized description of $J/\psi$ electroproduction at low $P_T$, computing the NLO hard function $H_{[n]}$ and modeling the TMD shape function (TMDShF) that encodes soft-gluon effects in NRQCD. The TMDShF is defined at the operator level and evolved in $b_T$-space with perturbative and nonperturbative components, whose interplay with the unpolarized gluon TMDPDF is explored through the convolution that governs the cross-section. Phenomenological predictions for the EIC demonstrate how the TMDShF, governed by NP parameters $A_S$ and $B_S$, can suppress or shape the low-$P_T$ distribution, with LDME choices introducing additional uncertainties; at higher $Q^2$, perturbative and NP contributions become comparable. The study highlights the potential of future EIC measurements to constrain gluon TMDs and soft-gluon dynamics, and discusses the need for a factorization framework valid in the $Q^2 \gg M^2$ regime and extensions to photoproduction.

Abstract

The next-to-leading order hard function for quarkonium electroproduction is calculated within the framework of transverse-momentum-dependent (TMD) factorization in the low-transverse-momentum regime. The structure of the TMD shape function in quarkonium leptoproduction is analyzed through its operator-level definition. Particular attention is given to the convolution of the unpolarized TMD gluon distribution with the TMD shape function, thereby illustrating the latter's phenomenological role. Building on this framework, we provide predictions for the unpolarized differential cross-section of $J/ψ$ electroproduction at the future Electron-Ion Collider in the region of small transverse momentum.

Figures (9)

• Figure 1: Evolution of the $\, ^1S_0^{[8]}$ LDME as a function of $b_T$. The middle band shows the perturbative region ($b_{T\text{min}} \leq b_T \leq b_{T\text{max}}$), and the horizontal black line corresponds to the value of the extracted $\, ^1S_0^{[8]}$ LDME. The dashed lines represent the evolved LDME considering the pair of values $\left( \left\langle \, ^1P_1^{[1]} \, \right\rangle , \left\langle \, ^1P_1^{[8]} \, \right\rangle \right)$.
• Figure 2: $q_T$-spectrum of the theoretical uncertainty of the $\, ^1S_0^{[8]}$ TMDShF for several levels of accuracy and for $\zeta_{B,b} \in \{2,1,0.5\}$ (left) and $\zeta_{B,b} = 1$ (right). Bands are obtained by varying the renormalization scale and the rapidity scale by a factor of 2. The black line represents the default value.
• Figure 3: $q_T$-spectrum of the $\, ^1S_0^{[8]}$ TMDShF considering $S_{\text{NP}} = \mathcal{S}_{\text{NP}}$ (left) and $S_{\text{NP}} = \mathcal{D}_{\text{NP}}$ (right) for several values of the NP parameters.
• Figure 4: $b_T$-spectrum of $e^{-S_A^g}$ and of the third lane of Eq. (\ref{['eq:conv']}) for several values of $\mu_H$ and for $b_{T\text{max}} = [0.5,1,2]$ GeV$^{-1}$. Bands are obtained by varying $\zeta_{B,b}$ by a factor of 2.
• Figure 5: $q_T$-spectrum of $q_T \cdot \mathcal{C}[f_1^g S_{\,^1S_0^{[8]}}]$ for several values of the hard scale and for $x_A = [10^{-3}, 10^{-2}, 10^{-1}]$. Bands are obtain by varying the NP parameters of the TMD shape function. We plot for $b_{T\text{max}} = 1$ GeV$^{-1}$.