Relativistic corrections to exclusive photoproduction of Quarkonia near-threshold

This paper utilizes Non-relativistic QCD within the Generalized Parton Distribution framework to calculate large relativistic corrections for exclusive near-threshold vector quarkonium photoproduction, revealing a breakdown of the GPD moment expansion for $J/\psi$ and the presence of endpoint divergences away from the threshold regime.

The Gist

Imagine the nucleus of an atom not as a solid marble, but as a bustling city filled with tiny, fast-moving particles called "partons" (quarks and gluons). Physicists want to take a high-resolution photo of this city to understand how it's built. One way to do this is to fire a beam of light (photons) at a proton (a type of nucleus) and watch what happens when it bounces off a heavy particle called a "quarkonium" (like a J/ψ or an Υ).

This paper is about taking a very specific, difficult photo: one where the collision happens just barely hard enough to create the heavy particle. This is called the "near-threshold" region.

Here is the story of what the researchers found, explained simply:

1. The "Slow-Motion" Mistake

For a long time, physicists have used a set of rules called NRQCD (Non-Relativistic QCD) to calculate these collisions. Think of this like using a map that assumes everyone in the city is walking slowly. It works great for heavy, slow-moving cars.

However, in this specific "near-threshold" crash, the heavy particles are moving much faster than the map assumes. They are zooming around at speeds where Einstein's theory of relativity starts to matter. The authors realized that if you ignore these "relativistic corrections" (the fact that the particles are actually moving fast), your map is wrong.

2. The "Blurry" Photo

The researchers tried to use a common technique called the GPD Moment Expansion. Imagine you are trying to describe a complex painting. Instead of looking at the whole picture, you only look at the average color of the top half, then the bottom half, and try to guess the rest. This usually works okay.

But in this specific crash, the authors found that this "average color" method breaks down completely.

• The Problem: When they added the "fast-moving" (relativistic) corrections to their calculation, the math started to scream. The "higher moments" (the finer details of the painting) became huge, swamping the simple average.
• The Result: If they only used the simple "average" method, their prediction for how often these collisions happen was off by a factor of 5. It was like predicting a car crash would happen once a year, but it actually happens five times a year.

3. The "Double-Edge" Sword

When they fixed the math to include the full speed of the particles and the full complexity of the painting (using the full GPD functions instead of just averages), the numbers changed dramatically.

• For the J/ψ (a lighter heavy particle), the relativistic corrections were massive. They caused a huge cancellation effect, drastically reducing the predicted number of collisions.
• For the Υ (a much heavier particle), the particles move slower relative to their size. Here, the "fast-moving" corrections were small, and the old, simple maps worked much better.

4. The "Edge" Problem

The paper also discovered a mathematical "glitch" at the very edges of the calculation.

• Imagine trying to count the number of people in a room, but the door is so narrow that people are getting stuck right at the threshold. The math gets "divergent" (infinite) at these edges.
• The authors found that these "endpoint divergences" appear when calculating these corrections. They didn't solve this glitch in this paper; they just pointed it out and said, "Hey, this is a problem we need to fix in the future."

5. The Takeaway

The main message is: If you want to understand the structure of the proton by smashing it near the energy limit, you cannot use the old, slow-motion rules.

• For the J/ψ, the "relativistic" (fast) effects are so big that they are just as important as other major corrections. Ignoring them gives a completely wrong picture.
• For the Υ, the old rules still hold up reasonably well.
• The "simple average" method (Moment Expansion) fails for the J/ψ near the threshold, so scientists must use the full, complex description of the proton's interior to get accurate results.

In short, this paper is a warning label for future experiments (like those at the Electron-Ion Collider): "Don't trust the simplified maps when you are driving near the speed limit; the terrain is much more complex than you think."

Technical Summary

Technical Summary: Relativistic Corrections to Exclusive Photoproduction of Quarkonia Near-Threshold

Problem Statement
The paper addresses the theoretical description of exclusive photoproduction of vector quarkonia ($J/\psi$ and $\Upsilon$) in the near-threshold region ($W^2 \sim M_V^2 \gg -t$). While Generalized Parton Distributions (GPDs) provide a framework to encode the spatial distribution of partons in nucleons, and leading power (LP) calculations have successfully factorized the amplitude into gluon GPDs and hard matching coefficients, the validity of these approximations near threshold requires scrutiny. Specifically, the authors investigate the magnitude of relativistic corrections (order $v^2$ in the quark-antiquark relative velocity) for heavy vector mesons (HVM). Previous studies have focused on high-energy/small-$x$ limits or electroproduction; this work targets the near-threshold photoproduction regime relevant to current experiments like GlueX and future facilities like the Electron-Ion Collider (EIC).

Methodology
The calculation employs a hybrid framework combining Non-Relativistic QCD (NRQCD) for the heavy quarkonium formation and collinear factorization for the hard scattering process.

1. Kinematics: The analysis is performed in the Ji frame, utilizing light-cone coordinates. The near-threshold limit is defined by the hierarchy $M_V \gg M_N$ (nucleon mass) and $1-\xi \sim M_N/M_V$, where $\xi$ is the skewness parameter.
2. Factorization: The amplitude is expanded in powers of the heavy quark relative velocity $v$. The leading power (LP) term corresponds to the color-singlet, spin-triplet operator $\langle O_1(^3S_1) \rangle$. The relativistic correction (Next-to-Leading Power, NLP) is derived from the higher-order NRQCD matrix element $\langle P_1(^3S_1) \rangle$, which involves covariant derivatives acting on the quark fields.
3. Hard Matching: The authors calculate the hard matching coefficients $C^{(0)}$ (LP) and $C^{(2)}$ (NLP) by projecting the partonic amplitude $\gamma g \to Q\bar{Q}g$ onto the appropriate NRQCD states. This involves evaluating six Feynman diagrams.
4. GPD Convolution: The amplitude is expressed as a convolution of the hard coefficients with the gluon GPD correlator $F^g(x, \xi, t)$. The authors analyze two approaches:
   • Moment Expansion: Expanding the hard coefficient in powers of $x/\xi$ and retaining only the zeroth moment, which relates the amplitude to gluon gravitational form factors (GFFs).
   • Full GPD: Using model-dependent GPD functions (specifically a modified GUMP model, denoted GUMP') to perform the full convolution without truncating the moment series.

Key Contributions and Results

• Large Relativistic Corrections: The study finds that relativistic corrections are substantial for $J/\psi$ production. The ratio of the NLP to LP amplitude is governed by $\langle v^2 \rangle_{J/\psi} \approx 0.22$. When included, these corrections lead to a significant cancellation with the LP term, reducing the predicted cross-section by a factor of approximately 5 in the moment expansion approach.
• Breakdown of Moment Expansion: A critical finding is the failure of the GPD moment expansion near threshold when relativistic corrections are included. While the zeroth moment dominates at LP, the NLP correction is highly sensitive to higher moments. The authors estimate that higher-order moments contribute $\sim 78\%$ of the NLP contribution, rendering the zeroth-moment approximation insufficient and leading to a breakdown of the expansion.
• Endpoint Divergences: The calculation reveals that the NLP hard matching coefficient $C^{(2)}(x, \xi)$ contains double poles at $x = \pm \xi$, unlike the single poles in the LP term. This leads to endpoint divergences in the convolution integral, particularly in the transition region between the DGLAP and ERBL domains. The authors note that these divergences are expected to be resolved by a proper organization of subleading power factorization (involving soft contributions), which is left for future work.
• Cross-Section Predictions:
  • $J/\psi$: Using the full GPD model (GUMP') and a cutoff to regulate the endpoint divergence, the NLP corrections still reduce the cross-section significantly (by a factor of $\sim 2$) compared to the LP result, though less drastically than the moment expansion suggested. The results are highly sensitive to the charm quark mass ($m_c$) and the choice of the cutoff parameter $\lambda$.
  • $\Upsilon$: Predictions for $\Upsilon$ production show that relativistic corrections are much smaller ($\langle v^2 \rangle_{\Upsilon} \approx 0.013$) due to the larger bottom quark mass, making the LP approximation more robust for this system.

Significance and Claims
The paper claims that relativistic corrections are as important as Next-to-Leading Order (NLO) perturbative QCD contributions for $J/\psi$ photoproduction near threshold. Consequently, neglecting these corrections leads to a poor description of experimental data (specifically from the GlueX collaboration).

The authors emphasize that the standard GPD moment expansion, often used to simplify threshold calculations, is inadequate for NLP corrections in this regime. They argue that reliable extraction of gluon GPDs and gravitational form factors from near-threshold data requires:

1. The inclusion of full GPD dependence rather than truncated moment expansions.
2. Higher-order perturbative corrections ($\alpha_s$) at both LP and NLP to reduce sensitivity to the heavy quark mass.
3. A complete factorization theorem that resolves the endpoint divergences found in the NLP calculation.

The work provides a necessary step toward a full power correction analysis for heavy quarkonium production, highlighting the complexity of the near-threshold regime and setting the stage for future precision studies at the EIC.