The twist-3 gluon contribution to Sivers asymmetry in $J/ψ$ production in semi-inclusive deep inelastic scattering

This paper presents the first calculation of the twist-3 gluon contribution to the Sivers asymmetry in $J/\psi$ production via semi-inclusive deep inelastic scattering, demonstrating that this observable is ideal for constraining the $C$-even twist-3 gluon distribution and providing numerical predictions for future electron-ion collider experiments.

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 tiny messengers called gluons that zip around, holding everything together. Scientists have long known that these gluons have a "spin" (like a spinning top) and a "path" (like a car driving on a road). But there's a mystery: how do these gluons move in three dimensions, and how does their motion affect the way they spin?

This paper is like a new, high-resolution map that helps us see a specific, hidden part of this city's traffic. Here is the breakdown of what the authors, Chen, Xing, and Yoshida, have discovered:

1. The Mystery of the "Wobble" (The Sivers Asymmetry)

In the world of particle physics, when scientists smash particles together, they sometimes see a strange "wobble." If they shoot a beam of electrons at a proton that is spinning in a specific direction, the resulting debris doesn't scatter evenly. It leans to one side. This is called the Single Transverse-Spin Asymmetry (SSA).

Think of it like throwing a ball at a spinning merry-go-round. If the merry-go-round is spinning, the ball might bounce off to the left more often than the right. This "wobble" tells us about the hidden orbital motion of the particles inside.

2. The "Twist-3" Glue

For a long time, scientists used two different rulebooks to explain this wobble:

• Rulebook A (TMD): Looks at the gluons as if they are driving on a 3D highway with side-to-side movement.
• Rulebook B (Twist-3): Looks at the gluons as if they are part of a complex, multi-lane traffic jam where they interact in groups.

This paper focuses on Rulebook B, specifically a "twist-3" calculation. Imagine "twist-3" as a way of looking at the traffic where you don't just see one car, but you see how three cars interact with each other in a specific, twisted pattern. The authors wanted to see if this "twisted" view could explain the wobble when creating a J/ψ particle (a heavy, short-lived particle made of a charm quark and an anti-charm quark).

3. The "Magic Cancellation"

The authors did the math (a very complex calculation involving thousands of terms) to see how the gluons behave when creating a J/ψ particle. They found something surprising and very helpful:

• The "Bad" Noise Disappears: In previous studies, there were two types of "twisted" gluon interactions: one that was "C-even" (symmetric) and one that was "C-odd" (antisymmetric). Usually, both types mix together, making it hard to tell which one is causing the wobble.
• The Filter: The authors discovered that when creating a J/ψ particle, the "C-odd" noise completely cancels out. It's like having a radio station with static, but suddenly, the static disappears, leaving only the clear music.
• The Result: This means the wobble (SSA) in J/ψ production is a pure signal of the "C-even" twist-3 gluon distribution. It is a clean, unfiltered view of how these gluons are moving.

4. The "Ghost" of the Heavy Particle

Usually, when a heavy particle like a J/ψ is formed, it involves a messy process called "hadronization" (where quarks glue themselves together to form a new particle). This process usually adds a lot of "fog" to the data, making it hard to see the underlying physics.

However, the authors found that for the J/ψ wobble, this "fog" also cancels out.

• Analogy: Imagine trying to measure the wind speed by watching a kite. Usually, the shape of the kite and the string affect how it flies, confusing the measurement. But in this specific case, the authors found that the kite's shape and the string's tension cancel each other out perfectly. What you are left measuring is purely the wind (the gluon distribution), not the kite.

5. The Future: The Electron-Ion Collider (EIC)

The paper doesn't just do the math; it also ran simulations for a future machine called the Electron-Ion Collider (EIC). This machine will be like a super-microscope for the proton city.

The authors simulated what the data would look like under different assumptions about how the gluons move. They found that:

• Different types of "twisted" gluon interactions leave different "fingerprints" on the data.
• By measuring the J/ψ wobble at the EIC, scientists can finally pin down exactly which type of gluon motion is dominant.
• This is crucial for understanding how gluons move at very small scales (low "x" values), a region that is currently a "dark zone" in our understanding of the proton.

Summary

In simple terms, this paper is a breakthrough because it found a clean window into the proton's interior. By studying the creation of J/ψ particles, the authors showed that the confusing background noise and the messy formation process disappear. This leaves scientists with a crystal-clear view of a specific type of gluon motion (the C-even twist-3 distribution) that was previously impossible to isolate. It's like finally finding a way to hear a single instrument in a symphony without the rest of the orchestra drowning it out.

Technical Summary

Technical Summary: The Twist-3 Gluon Contribution to Sivers Asymmetry in J/ψ Production

Problem Statement
The internal structure of the nucleon, particularly the role of gluons and their orbital angular momentum, remains a central subject in Quantum Chromodynamics (QCD). While Transverse-Momentum-Dependent (TMD) factorization has successfully described large single transverse-spin asymmetries (SSA) in various processes, the twist-3 collinear factorization framework offers an alternative description of incoherent multi-parton scattering. Although the equivalence between these frameworks is established in certain kinematic regions, a unified picture of SSA origins requires consistent calculations across different processes.

Heavy-flavored hadron production is a critical probe for gluon structure, as heavy quarks primarily originate from gluon fusion. While TMD-based studies of $J/\psi$ SSA exist, the twist-3 gluon contribution to $J/\psi$ production in Semi-Inclusive Deep Inelastic Scattering (SIDIS) had not been calculated prior to this work. Furthermore, previous twist-3 calculations for heavy mesons (e.g., $D$-mesons) did not address the specific cancellations and survival mechanisms of twist-3 distributions in the color-singlet channel of $J/\psi$ production.

Methodology
The authors perform a leading-order (LO) calculation of the twist-3 gluon contribution to the SSA in the process $e(\ell) + p^\uparrow(p, S_\perp) \to e(\ell') + J/\psi(P_{J/\psi}) + X$ within the collinear factorization framework.

1. Theoretical Framework: The calculation utilizes Non-Relativistic QCD (NRQCD) to describe the hadronization of the charm-anticharm pair into a $J/\psi$, focusing specifically on the color-singlet channel ($^3S_1^{[1]}$).
2. Twist-3 Formalism: The authors define and incorporate both kinematical twist-3 functions (related to the first $k_T^2/M_N^2$ moment of the gluon Sivers function) and dynamical twist-3 functions. The dynamical functions are categorized into C-even ($N(x_1, x_2)$) and C-odd ($O(x_1, x_2)$) types, defined via matrix elements of three gluon field strength tensors.
3. Calculation of Cross Sections:
   • The unpolarized cross section is derived using standard LO diagrams.
   • The twist-3 polarized cross section is derived by evaluating the interference between the hard scattering amplitudes and the twist-3 gluon distribution functions. This involves calculating diagrams where an extra gluon is attached to the hard part, including soft-gluon pole contributions.
   • The authors explicitly contract the hadronic tensor with the relevant Lorentz structures to isolate the spin-dependent terms.
4. Numerical Simulation: To assess the phenomenological viability, the authors perform numerical simulations for kinematics accessible at the future Electron-Ion Collider (EIC). They utilize two simple phenomenological models for the C-even function $N(x_1, x_2)$ to estimate the magnitude of the normalized structure functions.

Key Contributions and Results

1. First Calculation: This paper presents the first calculation of the twist-3 gluon contribution to the $J/\psi$ SSA in SIDIS.
2. Cancellation of C-Odd Contributions: A significant finding is that the C-odd twist-3 gluon contribution ($O(x_1, x_2)$) is exactly canceled in the spin-dependent cross section. This is attributed to the charge neutrality of the charm-anticharm pair.
3. Survival of C-Even Contributions:
   • The derivative terms of the C-even function, specifically $\frac{d}{dx}N(x, x)$ and $\frac{d}{dx}N(x, 0)$, are found to be exactly canceled in the color-singlet channel. This result aligns with previous statements regarding soft-gluon pole cancellations in SIDIS color-singlet processes.
   • Crucially, the authors demonstrate that non-derivative terms $N(x, x)$ and $N(x, 0)$ survive in the cross section. Additionally, contributions from "hard-pole" type terms ($N(x, Ax)$, $N(x, (1-A)x)$, etc.) are identified and calculated.
4. Hadronization Cancellation: In the ratio defining the SSA, the long-distance matrix element (LDME) associated with the color-singlet hadronization of $J/\psi$ completely cancels out. Consequently, the spin-dependent structure functions directly reflect the behavior of the C-even twist-3 gluon distribution without contamination from non-perturbative hadronization effects.
5. Numerical Observations:
   • The simulations show that the five distinct components of the C-even function ($N(x,x)$, $N(x,0)$, etc.) exhibit different qualitative behaviors across the normalized structure functions.
   • Comparing two models for the $x$-dependence of $N(x_1, x_2)$, the authors observe that while the qualitative behavior remains similar, the magnitude of the asymmetry is enhanced in the model with stronger small-$x$ singularity. This suggests that $J/\psi$ SSA measurements could provide information on the small-$x$ behavior of twist-3 gluon distributions.

Significance and Claims
The paper claims that the $J/\psi$ SSA in SIDIS serves as an ideal observable for pinning down the C-even twist-3 gluon distribution function ($N(x_1, x_2)$). The primary advantages highlighted are:

• Direct Connection: The observable is directly linked to the C-even twist-3 distribution, which has a known relationship to the gluon TMD Sivers function.
• Purity: The cancellation of the C-odd contribution and the LDME (hadronization effect) in the color-singlet channel isolates the twist-3 gluon dynamics, offering a "clean" probe compared to other processes.
• EIC Potential: The results provide a theoretical basis for future investigations at the EIC. By measuring the $J/\psi$ SSA over a wide range of Bjorken-$x$, the EIC can constrain the poorly known twist-3 gluon distribution functions and shed light on the orbital motion of gluons inside the proton.

The authors conclude that their derived LO cross-section formula enables these future investigations and that the distinct qualitative behaviors of the various C-even function components could help identify the dominant contributions to the observed asymmetry.