⚛️ phenomenology

The effect of TMD evolution on the Sivers asymmetry in back-to-back $J/\psi+\gamma$ and $J/\psi+\text{jet}$ production at the Electron-Ion-Collider

This paper predicts a sizable and robust Sivers asymmetry in back-to-back $J/\psi$ -photon and $J/\psi$ -jet production at the Electron-Ion Collider using a TMD factorization framework with CSS evolution, demonstrating that the asymmetry serves as a reliable probe of the gluon Sivers function largely independent of the specific long-distance matrix element sets used.

Original authors: Arghya Jana, Asmita Mukherjee, Sangem Rajesh

Published 2026-07-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Arghya Jana, Asmita Mukherjee, Sangem Rajesh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the proton, a tiny particle inside an atom, not as a solid marble, but as a bustling, three-dimensional city. For a long time, scientists only had a flat, two-dimensional map of this city, showing how much "traffic" (particles called partons) was moving in one direction. But they knew there was more to the story: the traffic also had a side-to-side wobble, or "intrinsic transverse momentum."

This paper is like a team of cartographers trying to draw a new, 3D map of this proton city, specifically looking for a very tricky kind of traffic pattern called the Sivers effect.

The Big Picture: The Electron-Ion Collider (EIC)

The authors are doing their calculations for a future super-microscope called the Electron-Ion Collider (EIC). Think of the EIC as a giant, high-speed racetrack where they smash electrons into protons. The goal is to see how the pieces inside the proton fly apart after the crash.

The "Back-to-Back" Dance

The scientists are looking at a specific scenario: a "back-to-back" dance. They smash an electron into a proton, and out pops a heavy particle called a J/ψ (a charm-anticharm quark pair) and something else—either a photon (a particle of light) or a jet (a spray of particles).

In this dance, the J/ψ and the photon/jet fly off in almost exactly opposite directions. This specific setup is crucial because it acts like a magnifying glass, allowing the scientists to use a special mathematical tool called TMD factorization. This tool helps them separate the "hard" crash from the "soft" internal structure of the proton, letting them see the hidden 3D map.

The Mystery: The Gluon Sivers Function

Inside the proton city, there are two main types of traffic: quarks and gluons. Gluons are the "glue" that holds the quarks together.

The Problem: We have a pretty good map of how quarks wobble (the quark Sivers function). But the map for gluons is mostly blank. We don't know how they wobble or if they have a preferred direction when the proton is spinning.
The Goal: This paper tries to predict what that missing gluon map looks like by calculating a specific "asymmetry" (a lopsidedness) in how the J/ψ and photon/jet fly out. If the gluons are wobbly in a specific way, the J/ψ will fly slightly more to the left than the right, depending on how the proton was spinning.

The "Evolution" of the Map

Here is where the paper gets technical but uses a great analogy. The authors talk about TMD evolution.

Imagine you are looking at a blurry photo of a spinning top. If you zoom in (increase the energy of the collision), the blur changes. The "wobble" of the particles looks different at different energy levels.

Old way: Previous studies often ignored how this "blur" changes as you zoom in.
This paper's way: The authors use a sophisticated method (called the CSS formalism) to account for how the map "evolves" or changes as the energy scale changes. They are essentially saying, "We need to adjust our map for the zoom level to get an accurate picture."

They found that even after adjusting for this "zoom" effect, the wobble (the asymmetry) is still strong enough to be seen.

The Results: What Did They Find?

The team ran simulations for two different scenarios:

J/ψ + Photon: This is a cleaner, simpler process.
J/ψ + Jet: This is messier, involving more types of particles.

Key Findings:

The Signal is Strong: Even with the "evolution" adjustments, the predicted wobble (asymmetry) is large enough to be measured. They estimate it could be around 15% for one version of their model and 3% for another. That's a big signal in the world of particle physics!
It Doesn't Matter Which "Rulebook" You Use: In the J/ψ + Photon case, the result is the same regardless of which specific mathematical "rulebook" (called LDMEs) you use to describe how the J/ψ forms. This makes the result very reliable.
Gluons Rule: In the J/ψ + Jet case, even though there are many different ways the particles can form, the gluon channel is so dominant that the result is still mostly telling us about the gluons, not the quarks.
The "Back-to-Back" Advantage: Because the particles fly in opposite directions, the quark contributions are tiny compared to the gluon contributions. This makes J/ψ production a perfect "gluon detector."

The Conclusion

In simple terms, this paper says: "If we build the EIC and smash electrons into protons to create these back-to-back J/ψ particles, we will be able to finally see how gluons wobble inside a spinning proton."

They have provided a reliable prediction that says, "Yes, the signal is there, it's strong, and it doesn't depend on uncertain details." This gives experimentalists at the future EIC a clear target to aim for to finally fill in the missing pieces of the proton's 3D map.

Technical Summary: The Effect of TMD Evolution on the Sivers Asymmetry in Back-to-Back $J/\psi + \gamma$ and $J/\psi + \text{jet}$ Production at the Electron-Ion Collider

Problem Statement
Transverse momentum dependent (TMD) parton distribution functions (TMDs) are essential for mapping the three-dimensional spin-orbit structure of hadrons. While quark TMDs, particularly the quark Sivers function, have been extensively studied, the gluon Sivers function (GSF) remains poorly constrained. The GSF describes the correlation between the intrinsic transverse momentum of unpolarized gluons and the transverse spin of the proton. A critical challenge in extracting the GSF is the lack of experimental data and the theoretical necessity of incorporating TMD evolution to connect different energy scales reliably. Furthermore, the process dependence of gluon TMDs, arising from complex gauge link structures, requires specific hard processes to isolate them. This paper addresses the need to estimate the Sivers asymmetry in specific electron-proton ($ep$) scattering channels at the kinematics of the upcoming Electron-Ion Collider (EIC) while rigorously accounting for TMD evolution effects.

Methodology
The authors employ a TMD factorization framework within the non-relativistic Quantum Chromodynamics (NRQCD) formalism to study the semi-inclusive deep inelastic scattering (SIDIS) process:
$e(l) + p^\uparrow(P) \to e(l') + J/\psi(P_\psi) + \gamma(p_\gamma)/\text{jet}(p_j) + X$
The study focuses on the "back-to-back" kinematic region where the transverse momentum of the $J/\psi$ and the recoiling photon or jet are nearly equal and opposite ( $|q_T| \ll |K_\perp|$ ), a condition required for the validity of TMD factorization.

Key methodological components include:

Production Mechanism: The $J/\psi$ $J / ψ$ production is modeled using NRQCD, factorizing the cross-section into a perturbative hard part and non-perturbative long-distance matrix elements (LDMEs). Two LDME sets (BK11 and SV13) are utilized to test sensitivity.
- For $J/\psi + \gamma$ , the leading-order (LO) process is dominated by the gluon channel ( $\gamma^* + g \to J/\psi + \gamma$ ) via the color-octet (CO) mechanism.
- For $J/\psi + \text{jet}$ , both gluon ( $\gamma^* + g \to J/\psi + g$ ) and quark/antiquark channels contribute, though the gluon channel is found to dominate.
TMD Evolution: The evolution of TMDs from the initial scale $\mu_b$ $μ_{b}$ to the hard scale $\mu = \sqrt{M_\psi^2 + Q^2}$ $μ = M_{ψ}^{2} + Q^{2}$ is calculated using the Collins-Soper-Sterman (CSS) formalism. This involves:
- Transforming TMDs to impact parameter ( $b_T$ ) space via Fourier transform.
- Applying the $b_*$ -prescription to separate perturbative (small $b_T$ ) and non-perturbative (large $b_T$ ) regions, ensuring the calculation remains within the perturbative regime.
- Incorporating the perturbative Sudakov factor (at Next-to-Leading Logarithmic accuracy) and a non-perturbative Sudakov factor extracted from existing SIDIS data (JLab, HERMES, COMPASS).
GSF Parametrization: Since the GSF has not yet been extracted with TMD evolution, the authors utilize two parametrizations for the normalization function $N_g(x)$ $N_{g} (x)$ :
- TMD-a: $N_g(x) = [N_u(x) + N_d(x)]/2$ .
- TMD-b: $N_g(x) = N_d(x)$ .
  These are based on fits to quark Sivers data, adapted for the gluon sector.

Key Contributions and Results
The paper presents numerical estimates for the unpolarized differential cross-sections and the Sivers asymmetry ( $A_{\sin(\phi_S - \phi_T)}$ ) at EIC center-of-mass energies of $\sqrt{s} = 45$ GeV and $140$ GeV.

Cross-Section Behavior:
- The unpolarized cross-section for $J/\psi + \text{jet}$ is significantly larger than that for $J/\psi + \gamma$ .
- The $J/\psi + \text{jet}$ cross-section shows dependence on the chosen LDME set (SV13 yields roughly twice the cross-section of BK11 due to the negative CO $^3P_0$ contribution in BK11).
- The $J/\psi + \gamma$ cross-section is dominated by the CO $^3S_1$ state, making it roughly 1.73 times larger for the BK11 set compared to SV13.
- In both processes, the gluon channel dominates over the quark channel, validating these processes as probes for gluon TMDs.
Sivers Asymmetry:
- Magnitude: The estimated Sivers asymmetry is sizable. For $J/\psi + \gamma$ , the TMD-b parametrization predicts an asymmetry of approximately 15%, while TMD-a predicts about 3%. For $J/\psi + \text{jet}$ , the predictions are roughly 13% (TMD-b) and 3% (TMD-a).
- LDME Independence: A significant finding is that the Sivers asymmetry for $J/\psi + \gamma$ is independent of the LDME set due to the exact cancellation of the amplitude modulation factor in the numerator and denominator. For $J/\psi + \text{jet}$ , the asymmetry remains largely unaffected by the LDME choice because the gluon channel dominates the asymmetry calculation.
- Kinematic Dependence:
  - The asymmetry increases linearly with $q_T$ at small values before peaking and decreasing.
  - The dependence on the inelasticity $y$ and the energy fraction $z$ is governed by the $x$ -dependence of the Sivers function ( $N_g(x) \sim x^\alpha(1-x)^\beta$ ). The asymmetry exhibits a peak at intermediate $y$ and $z$ values, with the peak position shifting based on the center-of-mass energy.
- Sign: The calculated asymmetry is positive. The authors note this contrasts with some previous studies (which reported negative asymmetries) and attribute the difference to the specific definition of the Sivers function used ( $f_{1T}^{\perp g}$ vs. $\Delta N f_{g/p^\uparrow}$ ).

Significance and Claims
The paper claims that the Sivers asymmetry in back-to-back $J/\psi + \gamma$ and $J/\psi + \text{jet}$ production serves as a "robust probe" of the gluon Sivers function at the future EIC. The primary significance lies in the demonstration that:

Even after incorporating TMD evolution (which typically suppresses asymmetries), the Sivers asymmetry remains sizable and observable.
The asymmetry is largely independent of the uncertainties associated with the non-perturbative long-distance matrix elements (LDMEs) in NRQCD, particularly for the $J/\psi + \gamma$ channel.
These processes are dominated by the gluon channel, making them ideal for isolating the less-known gluon Sivers function from the better-constrained quark contributions.

The authors conclude that these theoretical estimates provide a solid foundation for future experimental programs at the EIC to constrain the gluon Sivers function and test the process dependence of gluon TMDs.