Gluon Sivers function in dijet production at the EIC

This paper develops a revised TMD factorization formalism for dijet production with a transversely polarized target at the EIC, incorporating N$^3$LO evolution kernels to predict significant gluon Sivers asymmetries ranging from 5% to 50%.

The Gist

Imagine you are trying to understand the inside of a complex, spinning machine (a proton) by smashing it into another particle and watching the debris fly out. Specifically, you are looking at a scenario where the smash creates two jets of particles shooting out in opposite directions. This is the experiment proposed for the future Electron-Ion Collider (EIC).

This paper is a theoretical "instruction manual" for how to interpret the data from that experiment. Its main goal is to predict a specific, weird behavior called the Gluon Sivers Effect.

Here is the breakdown using simple analogies:

1. The Setting: The Spinning Top

Think of a proton not as a solid ball, but as a chaotic swarm of tiny particles (quarks and gluons) spinning inside a larger shell.

• The Gluons: These are the "glue" holding the proton together. They are invisible to the naked eye but carry most of the proton's momentum.
• The Spin: The proton is spinning like a top.
• The Mystery: The Sivers Effect asks: If the proton is spinning to the left, do the gluons inside it prefer to drift to the right?

In a normal world, you'd expect the particles to be evenly distributed. But quantum mechanics says that because of the proton's spin, the gluons might have a "handedness"—they might crowd to one side, creating an imbalance. This paper tries to predict exactly how big that imbalance will be when we smash protons at the EIC.

2. The Problem: The Foggy Lens

To see this imbalance, scientists use a mathematical tool called TMD Factorization. Think of this as a high-powered camera lens that focuses on the tiny transverse (side-to-side) movements of the particles.

However, there is a problem: The Lens is Distorted.
When particles move at near light speed, the math gets messy. The "lens" has a foggy part called Evolution. This fog represents how the particles' behavior changes depending on the energy of the collision.

• If you don't account for this fog correctly, your prediction of the Sivers effect will be wrong.
• The authors of this paper are trying to clean up the lens. They are testing two different ways to "de-fog" the math to get a clear picture.

3. The Two Schemes: Two Ways to Clean the Lens

The paper compares two methods for handling the math:

• The "CCS-Scheme" (The Old Way): This method treats the different parts of the collision (the soft glue and the hard jets) separately. It's like trying to clean a dirty window by wiping the top half, then the bottom half, then the left, then the right. It works, but it's complicated, and sometimes the edges don't match up perfectly, creating "ghosts" (mathematical errors) in the image.
• The "M-Scheme" (The New Way): The authors propose a new method where they group the messy parts together into a single "M-function" (like bundling the dirty window panes into one frame). They clean the whole frame at once.
  • The Result: The new method is simpler, avoids the "ghosts," and gives a more stable, reliable picture. It's like switching from a patchwork quilt to a single, smooth sheet of glass.

4. The Prediction: A Big Surprise

The authors ran their simulations using the new "M-Scheme" and the best available data on how these particles evolve.

• The Guess: They didn't know exactly how the gluons were distributed, so they made an educated guess: they assumed the gluons behave somewhat like the "sea" of quarks (the background particles) that we already know about.
• The Result: They predict a huge effect.
  • They expect the Sivers asymmetry (the imbalance) to be between 5% and 50%.
  • To put that in perspective: If you were betting on which way the particles would fly, and you guessed randomly, you'd be right 50% of the time. A 50% asymmetry means the particles are heavily biased to one side. It's like flipping a coin and getting heads 75% of the time.

5. Why This Matters

• The Glue is Elusive: Gluons are notoriously hard to study because they are hidden behind the more visible quarks. This paper provides a roadmap to finally "see" them clearly.
• The EIC: The Electron-Ion Collider is being built specifically to do this. This paper tells the engineers and physicists: "When you turn the machine on, look for a massive wobble in the jets. If you see it, you've proven that gluons have a strong connection to the proton's spin."
• The Uncertainty: The authors admit their prediction depends heavily on their guess about the gluons. If the gluons behave differently than the "sea quarks," the numbers might change. But even with the uncertainty, the effect is predicted to be large enough to be easily spotted.

Summary

This paper is a theoretical guidebook for the future Electron-Ion Collider. It says: "We have a new, cleaner way to do the math (the M-Scheme). Using this, we predict that when we smash protons, the gluons inside will show a massive, visible wobble (the Sivers effect) of 5% to 50%. This will be a huge discovery for understanding how the 'glue' of the universe works."

Technical Summary

1. Problem Statement

The paper addresses the challenge of extracting the gluon Sivers function ($f_{1T}^{\perp g}$), a Transverse Momentum Dependent (TMD) distribution describing the correlation between the transverse spin of a nucleon and the transverse momentum of unpolarized gluons inside it.

• Context: While quark Sivers functions are relatively well-constrained, the gluon Sivers function remains largely unmeasured and theoretically uncertain.
• Motivation: The future Electron-Ion Collider (EIC) will provide high-precision data on dijet production in Semi-Inclusive Deep Inelastic Scattering (SIDIS). Dijet production is uniquely sensitive to gluon TMDs at leading power, unlike single-jet production where gluon effects are subleading.
• Theoretical Gap: Existing formalisms for dijet factorization (specifically in Soft Collinear Effective Theory, SCET) involve complex renormalization group evolutions and rapidity divergences. Previous implementations (e.g., the "CCS-scheme") faced technical difficulties regarding the separation of soft and collinear-soft functions, leading to spurious imaginary terms and constraints on scale choices. Furthermore, there is no established non-perturbative model for the gluon Sivers function.

2. Methodology

The authors employ TMD factorization within the framework of SCET to calculate the dijet cross-section in SIDIS. Key methodological components include:

• Factorization Theorem: They derive the factorized cross-section for both unpolarized and transversely polarized targets, separating contributions from photon-gluon ($\gamma^* g$) and photon-quark ($\gamma^* q$) channels.
• Evolution Schemes: The paper introduces and compares two distinct schemes for implementing TMD evolution using the $\zeta$-prescription (implemented via the artemide code):
  1. CCS-scheme (Collinear-Soft/Soft): The traditional approach where soft and collinear-soft functions are evolved separately. The authors note this scheme suffers from angular ($c_b = \cos \phi_b$) dependence in anomalous dimensions, requiring perturbative treatment of certain terms and limiting scale choices.
  2. M-scheme (M-function): A new scheme proposed in this work. It defines an M-function as the product of the collinear-soft and soft functions ($M = C \times S$). By evolving this combined function, the angular dependence in the anomalous dimensions cancels out, simplifying the resummation and removing constraints on renormalization scales.
• Evolution Kernels: They utilize TMD evolution kernels calculated up to N3LO (Next-to-Next-to-Next-to-Leading Order). They test predictions using two different extractions of the non-perturbative Collins-Soper kernel:
  • ART23: Based on Drell-Yan data.
  • ART25: Based on Drell-Yan + SIDIS data.
• Sivers Function Modeling: Since no direct extraction of the gluon Sivers function exists, the authors adopt a controlled ansatz: they assume the gluon Sivers function is proportional to the sea-quark Sivers function extracted from previous fits (specifically the BPV model from Ref. [44]). They quantify uncertainties by varying the normalization and sign of this ansatz.

3. Key Contributions

• Proposal of the M-scheme: The primary theoretical contribution is the definition of the M-scheme. It offers a technically simpler and more robust framework for resummation compared to the CCS-scheme, avoiding spurious imaginary terms and providing better perturbative stability (smaller uncertainty bands).
• Formalism for Polarized Dijets: The paper provides a complete derivation of the factorized cross-section for transversely polarized targets, explicitly detailing the operator definitions and the weighted cross-sections required to isolate the Sivers asymmetry.
• EIC Predictions: The work generates the first detailed predictions for the gluon Sivers asymmetry at the EIC, incorporating N3LO evolution and realistic kinematic cuts ($p_T \in [5, 40]$ GeV).
• Uncertainty Quantification: The authors systematically analyze sources of uncertainty, including hard scale variations, non-perturbative parameters in the evolution kernel, and the model dependence of the gluon Sivers ansatz.

4. Results

• Scheme Comparison: The unpolarized and polarized cross-sections calculated in the M-scheme and CCS-scheme are consistent within errors. However, the M-scheme yields smaller uncertainty bands, indicating improved convergence and stability.
• Sivers Asymmetry Magnitude: The predicted Sivers asymmetry ($A_{Sivers}$) for dijet production at the EIC is large, ranging between 5% and 50% in the region of jet imbalance $r_T \sim 1-2$ GeV.
• Sensitivity to Evolution Kernels: The M-scheme shows higher sensitivity to the differences between the ART23 and ART25 evolution kernels compared to the CCS-scheme, suggesting it is a better tool for distinguishing between different non-perturbative models.
• Model Dependence:
  • The results are highly sensitive to the assumed gluon Sivers ansatz.
  • If the gluon contribution is set to zero, a residual asymmetry of a few percent remains (due to quark channels).
  • If the sign of the gluon Sivers function is reversed relative to the sea-quark assumption, the asymmetry changes significantly but remains of the same order of magnitude.
  • An observed asymmetry below the percent level would imply a non-trivial cancellation between quark and gluon contributions.

5. Significance

• Phenomenological Impact: The prediction of a large Sivers asymmetry (5–50%) suggests that the EIC will be capable of making the first definitive measurements of the gluon Sivers function, a crucial step in understanding the 3D structure of the nucleon and the origin of nucleon spin.
• Theoretical Advancement: The M-scheme provides a refined tool for TMD phenomenology. By resolving the technical issues associated with the separation of soft and collinear-soft functions, it offers a more reliable path for high-precision calculations in SCET.
• Guidance for Future Fits: The work highlights the tension between different evolution kernel extractions (ART23 vs. ART25) and emphasizes the need for global fits that include both unpolarized and polarized data to constrain the non-perturbative gluon sector.

In summary, this paper establishes a robust theoretical framework for analyzing dijet production at the EIC, proposes a superior evolution scheme (M-scheme), and predicts that the gluon Sivers effect will be a dominant and measurable feature of future polarized scattering experiments.