Modeling the TMD shape function in $J/\psi$ electroproduction

This paper calculates the next-to-leading order hard function for $J/\psi$ electroproduction within TMD factorization, analyzes the operator structure and phenomenological role of the TMD shape function, and provides predictions for the unpolarized differential cross-section at the future Electron-Ion Collider in the low-transverse-momentum regime.

The Gist

The Big Picture: Catching a Ghost in the Machine

Imagine you are trying to understand how a specific type of heavy, exotic ball (called a J/ψ meson) is formed when you smash a high-speed electron into a proton. This is a bit like firing a bullet (the electron) at a cloud of dust and gas (the proton) and watching a specific, heavy snowflake (the J/ψ) form out of the collision.

Physicists have known for a long time how to calculate the hard part of this crash—the moment the bullet hits the gas. But they have struggled to understand the "soft" part: how the debris settles down to form that heavy snowflake. This paper is about building a better map for that "settling down" phase.

The Problem: The "Blurry" Photo

For decades, physicists used a standard recipe (called NRQCD) to predict how these snowflakes form. This recipe works great when the snowflake flies off at high speed. But when the snowflake is created almost standing still (low speed), the recipe starts to fail. It's like trying to take a photo of a hummingbird's wings with a slow shutter speed; the picture comes out blurry, and you can't see the details.

The "blur" in this physics recipe comes from soft gluons. Think of gluons as the invisible "glue" that holds quarks together. When the heavy quarks are forming a bond, they emit these soft, low-energy gluons. The old recipe didn't account for how these "glue whispers" affect the final shape of the snowflake.

The Solution: The "Shape Function"

The authors of this paper introduce a new tool called the TMD Shape Function.

• The Analogy: Imagine you are throwing a ball into a wind tunnel.
  • The Old Way: You calculated the throw perfectly but ignored the wind. You predicted the ball would go in a straight line.
  • The New Way: You realize the wind (the soft gluons) pushes the ball sideways and changes its spin. The Shape Function is a mathematical description of exactly how that wind distorts the ball's path.

In this paper, the authors:

1. Calculated the "Wind" precisely: They did complex math (Next-to-Leading Order calculations) to figure out exactly how the electron-proton collision creates the initial conditions.
2. Modeled the "Distortion": They created a model for the Shape Function, which describes how the heavy quarks "dress up" in a cloud of soft gluons before becoming a stable J/ψ particle.
3. Tested it: They used this new model to predict what will happen at the Electron-Ion Collider (EIC), a massive new machine being built to smash electrons and protons together.

Why Does This Matter?

The authors found that ignoring this "Shape Function" leads to wrong predictions.

• Without the Shape Function: The theory predicts too many J/ψ particles at low speeds. It's like saying the wind doesn't exist, so you expect the ball to go further than it actually does.
• With the Shape Function: The prediction changes. The "wind" (soft gluons) actually suppresses the number of particles formed at very low speeds.

This is crucial because the J/ψ particle is a unique probe for gluons. While we know a lot about how quarks behave inside a proton, we know very little about how gluons move and spread out. By studying the J/ψ, we are essentially using it as a flashlight to see the "glue" inside the proton.

The Future: The Electron-Ion Collider (EIC)

The paper concludes with predictions for the future EIC.

• The Goal: The EIC will take incredibly precise photos of these collisions.
• The Prediction: The authors say, "If you look at the data with our new 'Shape Function' map, you will see a specific pattern that the old maps missed."
• The Payoff: If the EIC data matches their new prediction, it proves that our understanding of how heavy particles form is correct. It will also help us finally measure the "glue" (gluon distributions) inside the proton with high precision.

Summary in a Nutshell

• The Issue: Old physics formulas fail to predict how heavy particles form when they are moving slowly.
• The Fix: The authors added a new ingredient called the TMD Shape Function, which accounts for the "soft glue" (gluons) that surrounds the forming particle.
• The Result: This new model changes the predictions for future experiments, specifically at the EIC. It suggests that the "soft glue" acts like a brake, slowing down the formation of particles at low speeds.
• The Takeaway: This work bridges the gap between the messy, complex world of quantum glue and the clean, mathematical world of particle physics, giving us a better map to explore the universe's building blocks.

Technical Summary

1. Problem Statement

The production of heavy quarkonia (specifically $J/\psi$) serves as a critical probe for understanding the interplay between perturbative QCD (pQCD) and non-perturbative (NP) dynamics. While the production of a heavy quark-antiquark pair is calculable via pQCD, the formation of the bound state involves NP physics.

• The Gap: Standard Non-Relativistic QCD (NRQCD) factorization successfully describes high-transverse-momentum ($p_T$) production but faces challenges at low $p_T$. In the low-$p_T$ regime, soft gluon radiation becomes significant, requiring a Transverse-Momentum-Dependent (TMD) factorization framework.
• Specific Issues:
  1. Previous studies on TMD factorization in quarkonium production yielded conflicting definitions for the TMD Shape Function (TMDShF), often due to ambiguous choices of the hard scale ($\mu_H$).
  2. The Next-to-Leading Order (NLO) hard function for $J/\psi$ electroproduction within the TMD framework had not been explicitly calculated, limiting the precision of theoretical predictions.
  3. There is a need to quantify the phenomenological impact of the TMDShF on the differential cross-section, particularly for future measurements at the Electron-Ion Collider (EIC).

2. Methodology

The authors employ a rigorous TMD factorization framework combined with NRQCD effective field theory (specifically $v$NRQCD).

• Theoretical Framework:

  • Factorization: The cross-section is factorized into a hard function ($H$), a gluon TMD Parton Distribution Function (TMDPDF, $G_{g/N}$), and the TMD Shape Function ($S^{[n]\to J/\psi}$).
  • Hard Scale Definition: The authors derive a specific hard scale $\mu_H = (M^2 + Q^2)/M$ (where $M$ is the $J/\psi$ mass and $Q^2$ is the photon virtuality) to resolve discrepancies in previous literature regarding the $Q^2$ dependence of the TMDShF.
  • Evolution: The TMDShF and TMDPDF are evolved using Renormalization Group (RG) equations and Collins-Soper evolution, resumming large logarithms to NNLL' (Next-to-Next-to-Leading Logarithmic prime) accuracy.

• Calculations:

  • NLO Hard Function: The authors performed a one-loop virtual QCD correction calculation for the partonic process $\gamma^* + g \to c\bar{c}$ in the color-octet channel ($^1S_0^{[8]}, ^3P_J^{[8]}$). This is the first explicit NLO calculation of the virtual contribution for electroproduction in this context.
  • TMDShF Modeling: The TMDShF is modeled using an operator-level definition. It is separated into perturbative (calculable) and non-perturbative (modeled) components using the $b_*$-prescription. The NP part is parameterized with Gaussian functions involving parameters $A_S$ and $B_S$.
  • LDME Evolution: The Long-Distance Matrix Elements (LDMEs) are evolved from the extraction scale to the hard scale, accounting for off-diagonal mixing between S-wave and P-wave states.

• Numerical Implementation:

  • Predictions are generated for EIC kinematics ($\sqrt{s} = 45$ GeV and $140$ GeV).
  • Various sets of LDMEs (SV, BBXW, CMSWZ, BCKL) are tested to assess theoretical uncertainty.
  • The convolution of the gluon TMDPDF and the TMDShF is computed in impact-parameter space ($b_T$) and transformed to transverse momentum space ($q_T$).

3. Key Contributions

1. Explicit NLO Hard Function: The paper provides the first calculation of the NLO virtual corrections for $J/\psi$ electroproduction within the TMD framework, establishing a consistent hard scale $\mu_H$ that eliminates unphysical $Q^2$ dependencies found in previous matching analyses.
2. Resolution of TMDShF Discrepancies: By adhering to the operator-level definition and the derived hard scale, the authors reconcile previous conflicting results regarding the TMDShF structure.
3. Systematic Evolution Analysis: The study details the evolution of the TMDShF, including the impact of the cusp anomalous dimension and the rapidity anomalous dimension, demonstrating how the shape function evolves from the non-perturbative scale to the hard scale.
4. Phenomenological Modeling: The authors introduce a consistent non-perturbative model for the convolution of the gluon TMDPDF and the TMDShF, identifying key parameters ($A, B_S$) that govern the width and magnitude of the distribution.

4. Results

• Cross-Section Predictions: The authors present differential cross-sections ($d\sigma/dP_{\psi\perp}$) for EIC kinematics. The distributions exhibit a peaked structure at low $P_{\psi\perp}$, consistent with TMD resummation expectations.
• Impact of TMDShF:
  • The inclusion of the TMDShF significantly modifies the cross-section compared to standard NRQCD predictions.
  • The parameter $B_S$ (controlling the NP width of the shape function) is the dominant source of uncertainty. A larger $B_S$ leads to a suppression of the cross-section at very low $P_{\psi\perp}$.
  • The ratio $R(P_{\psi\perp})$ (cross-section with TMDShF / cross-section without TMDShF) shows that for non-zero $B_S$, the TMDShF can either suppress or enhance the cross-section depending on the momentum region, with a peak enhancement observed around $P_{\psi\perp} \sim 5-6$ GeV for intermediate $B_S$ values.
• Uncertainty Analysis:
  • Scale Uncertainty: Varying renormalization and rapidity scales by a factor of 2 yields uncertainty bands that narrow as the resummation accuracy increases (from NLL to NNLL').
  • LDME Uncertainty: The variation between different LDME sets (SV, BBXW, etc.) produces a wider spectrum of predictions than the NP parameter uncertainty, highlighting the need for better LDME extraction, particularly in the low-$p_T$ region.
  • Breakdown of Factorization: At large $Q$ (e.g., $Q=9$ GeV) and large $x_B$, the model predicts unphysical negative cross-sections near $P_{\psi\perp} \approx \mu_H/2$, signaling the breakdown of the TMD expansion and the potential need for a specific TMD framework valid for $Q \gg M$.

5. Significance

• EIC Physics: This work provides the first robust theoretical baseline for $J/\psi$ electroproduction at the future Electron-Ion Collider. It enables the extraction of gluon TMDs and the study of soft-gluon dynamics in a regime where standard collinear factorization fails.
• Theoretical Consistency: By resolving the hard scale ambiguity and providing the NLO hard function, the paper establishes a consistent framework for future high-precision quarkonium studies.
• Non-Perturbative Dynamics: The analysis underscores the critical role of the TMD Shape Function in governing the low-$p_T$ behavior of quarkonium production. It suggests that precise EIC data could constrain the non-perturbative parameters ($A, B_S$) and improve the extraction of LDMEs, thereby refining our understanding of heavy-quark hadronization.
• Future Directions: The paper identifies the need for a dedicated TMD factorization framework for the $Q \gg M$ regime and calls for further experimental constraints on the non-perturbative parameters of the gluon TMD and the shape function.