The twist-3 gluon contribution to $A_N$ in $J/\psi$ production in $pp$ collisions

This paper presents the first rigorous collinear factorization calculation of the twist-3 gluon contribution to the single transverse-spin asymmetry in $J/\psi$ production, demonstrating that the $C$-even gluon distribution drives a sizable asymmetry at RHIC and LHC energies and offering a unique probe for the three-dimensional motion of gluons within the proton.

The Gist

Imagine the proton, the tiny particle at the heart of every atom, not as a solid marble, but as a bustling, chaotic city. Inside this city, there are two main types of residents: quarks (the famous ones) and gluons (the glue that holds everything together). For a long time, scientists knew the quarks were there, but the gluons were like a mysterious, invisible crowd whose movements were hard to track.

This paper is a new map drawn by physicists Longjie Chen and Shinsuke Yoshida to help us understand how these gluons move, specifically how they "spin" or orbit inside the proton.

Here is the story of their discovery, broken down into simple concepts:

1. The Mystery of the "Wobble"

In the 1970s, scientists noticed something strange. When they smashed protons together, the resulting particles didn't just fly out randomly; they had a slight "wobble" or preference to fly to one side. This is called Single Transverse-Spin Asymmetry (SSA).

Think of it like spinning a top. If you spin a top perfectly, it goes straight. But if the top is slightly lopsided, it wobbles and veers off to the side. In particle physics, this "wobble" was a huge mystery because the old rules of physics couldn't explain it. It suggested that the particles inside the proton (the gluons) weren't just sitting still; they were orbiting and moving in complex ways.

2. The Two Ways to Look at the City

To solve this mystery, scientists have developed two different "lenses" or theories to look at the proton:

• The TMD Lens: This looks at the proton as if you are taking a high-speed photo, capturing the exact sideways motion of the particles.
• The Twist-3 Lens: This looks at the proton as a complex dance where particles interact in groups of three or more, rather than just one-on-one.

For a long time, we had a good map for how quarks moved using these lenses. But for gluons (the glue), and specifically for creating a heavy particle called J/ψ (which is like a heavy, exotic car made of two charm quarks), we were missing the map. We knew the data existed from experiments done over a decade ago at the RHIC (Relativistic Heavy Ion Collider), but we didn't have the math to explain why the gluons were causing that wobble.

3. The New Map: Finding the "C-Even" Glue

Chen and Yoshida finally did the heavy lifting. They calculated the "Twist-3" contribution for gluons in J/ψ production.

Here is the big discovery they made, using a simple analogy:
Imagine the gluons inside the proton have two different "personalities" or "types" of movement, which the scientists call C-even and C-odd.

• The C-odd type: This is like a ghost. The authors found that when you do the math for J/ψ production, this type of movement completely cancels itself out. It's there, but it doesn't leave a trace in this specific experiment.
• The C-even type: This is the star of the show. The paper shows that only this type of gluon movement contributes to the wobble (SSA) in J/ψ production.

This is a huge deal because it means J/ψ production is a perfect "magnifying glass" to study the C-even gluons. It's a direct line to understanding how gluons orbit inside the proton.

4. The Simulation: What the Data Says

The authors didn't just stop at the math; they ran simulations to see what this would look like in real life at two major particle accelerators: RHIC (in the US) and LHC (in Europe).

They used a simple model to guess how strong these gluon movements might be. Their results showed something interesting:

• Different from the usual: In lighter particles (like pions) or D-mesons, the "wobble" gets stronger as you look at particles flying at certain angles.
• The J/ψ Surprise: For the heavy J/ψ particle, the "wobble" didn't follow that same pattern. The part of the math that usually drives the wobble in other particles was very small here.

This suggests that the mechanism causing the wobble in J/ψ is different from the one causing wobbles in lighter particles. It's like driving a heavy truck versus a sports car; even on the same road, they handle turns differently.

5. Why This Matters

The paper concludes that because the "ghost" (C-odd) cancels out and only the "star" (C-even) remains, measuring the wobble of J/ψ particles is a key tool for scientists.

• It confirms the dance: The fact that RHIC already saw a non-zero wobble means gluons are definitely orbiting inside the proton.
• It guides the future: This new calculation gives scientists a solid foundation to interpret future experiments. It helps them understand the "gluon Sivers effect" (a fancy term for how gluons are distributed in a spinning proton) much better.

In a nutshell: This paper provides the first complete mathematical recipe to explain why heavy J/ψ particles wobble when protons collide. It reveals that this wobble is caused by a specific type of gluon movement (C-even) and proves that heavy particles behave differently than light ones, offering a new, clearer window into the hidden, swirling motion of the glue that holds our universe together.

Technical Summary

Technical Summary: The Twist-3 Gluon Contribution to $A_N$ in $J/\psi$ Production in $pp$ Collisions

Problem Statement
Despite experimental data on the single transverse-spin asymmetry ($A_N$) in $J/\psi$ production in proton-proton ($pp$) collisions being reported by the RHIC experiment over a decade ago, a rigorous theoretical calculation based on collinear factorization for the dominant gluonic contribution has remained unavailable. While the transverse-momentum-dependent (TMD) framework has successfully described large $A_N$ in various processes, and twist-3 calculations exist for light hadrons and $D$-mesons, the specific twist-3 gluon contribution to $J/\psi$ production in $pp$ collisions had not been derived. Furthermore, the role of the C-even versus C-odd twist-3 gluon distributions in this specific heavy-flavor channel was unclear.

Methodology
The authors employ the collinear factorization framework combined with the Non-Relativistic QCD (NRQCD) formalism to describe the hadronization of the charm quark pair into a $J/\psi$. The calculation focuses on the color-singlet contribution ($^3S_1^{[1]}$) as the leading order mechanism.

The core of the methodology involves:

1. Formalism: Utilizing the modern non-pole technique to derive the polarized cross-section formula for $pp$ collisions, accounting for both Initial State Interactions (ISI) and Final State Interactions (FSI).
2. Distribution Functions: Defining and utilizing twist-3 gluon distribution functions, specifically distinguishing between kinematical functions (related to $G_T^{(1)}$ and $\Delta H_T^{(1)}$) and dynamical functions. The dynamical functions are categorized into C-even ($N(x_1, x_2)$) and C-odd ($O(x_1, x_2)$) types.
3. Diagrammatic Calculation: Calculating the hard scattering cross-sections by evaluating 12 leading-order (LO) diagrams for the unpolarized case and 86 diagrams involving an additional gluon line for the twist-3 polarized contribution. The authors explicitly handle the complex pole structures arising from ISI and FSI.
4. Symmetry Analysis: Investigating the cancellation properties of the C-odd and C-even contributions within the polarized cross-section.
5. Numerical Simulation: Performing numerical simulations for $A_N$ at RHIC ($\sqrt{S} = 200$ GeV) and LHCspin ($\sqrt{S} = 115$ GeV) energies. Due to the lack of experimental constraints on the C-even function $N(x_1, x_2)$, the authors adopt a simple model where all relevant twist-3 functions are proportional to the unpolarized gluon distribution $G(x)$, specifically $N(x, x) = N(x, 0) = \dots = 0.002xG(x)$.

Key Contributions

• First Derivation: This paper presents the first rigorous calculation of the twist-3 gluon contribution to the single transverse-spin asymmetry in $J/\psi$ production within the collinear factorization framework.
• Selection Rule: The authors demonstrate that the contribution from the C-odd type twist-3 gluon distribution function, $O(x_1, x_2)$, is completely canceled in the polarized cross-section for this process. Consequently, the $J/\psi$ SSA is driven solely by the C-even type function, $N(x_1, x_2)$.
• Relation to TMD: The paper establishes a direct relationship between the relevant C-even twist-3 function and the gluon Transverse-Momentum-Dependent (TMD) distribution function (specifically the gluon Sivers function), positioning $J/\psi$ production as a unique probe for three-dimensional gluon motion.
• Hard Scattering Coefficients: The authors provide explicit analytical expressions for the hard cross-sections ($\hat{\sigma}_U, \hat{\sigma}_D, \hat{\sigma}_{ND1,2}, \hat{\sigma}_{H1,2,3}$) and the complex color factors associated with the 86 twist-3 diagrams, which are detailed in the Appendix.

Results

• Cancellation of C-odd Terms: Similar to Semi-Inclusive Deep Inelastic Scattering (SIDIS), the C-odd function $O(x_1, x_2)$ does not contribute to the $J/\psi$ SSA.
• Behavior of Derivative Terms: Unlike light hadron and $D$-meson production, where the derivative terms of the twist-3 distribution functions (e.g., $\frac{d}{dx}N(x,x)$) dominate the asymmetry and lead to an increase in $A_N$ with increasing Feynman-$x$ ($x_F$), the authors find that these derivative terms are small over the entire $x_F$ range in $J/\psi$ production.
• Numerical Predictions: The simulations indicate that a sizable $A_N$ can still be generated, but through a mechanism distinct from that in light hadron or $D$-meson production. The results show a non-zero asymmetry in the positive $x_F$ region at RHIC energies, consistent with existing experimental data points, and predict behavior at LHCspin energies where the kinematic coverage is wider.
• Kinematic Independence: The derived result is shown to be independent of the arbitrary light-like vector $n$ used in the definition of the distribution functions.

Significance
The paper claims that the $J/\psi$ SSA serves as a key observable to isolate and understand the C-even type twist-3 gluon distribution function, which is directly related to the gluon Sivers function. The absence of the C-odd contribution simplifies the interpretation of the asymmetry in terms of gluon orbital angular momentum. The authors suggest that their results provide a necessary theoretical basis for future analyses of RHIC and LHCspin data, potentially leading to a phenomenological understanding of the poorly known gluon Sivers effect. The work highlights that the mechanism generating $A_N$ in heavy quarkonium production differs fundamentally from that in light hadron production, offering a new perspective on the orbital motion of gluons inside the proton.