⚛️ nuclear experiments

Relativistic corrections and high-energy resummation for exclusive heavy quarkonium photoproduction

This paper computes the $O(v^2)$ relativistic correction to the high-energy resummed coefficient function for exclusive heavy quarkonium photoproduction, finding that while the correction itself is negligible at the physical mass scale, it significantly improves prediction robustness by partially canceling the factorization scale dependence of the leading-order relativistic correction.

Original authors: Maxim Nefedov

Published 2026-07-17

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Maxim Nefedov

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 universe as a giant, bustling construction site where the tiniest building blocks, called quarks, are constantly being glued together to form heavier structures like protons and neutrons. To understand how these blocks stick together, scientists use a set of rules called Quantum Chromodynamics (QCD), which is like the instruction manual for the "glue" (gluons) that holds everything in place. But here's the tricky part: sometimes these instructions get messy when things move incredibly fast or when we look at them from very different angles. One of the biggest headaches for physicists is trying to predict exactly what happens when a high-energy beam of light (a photon) smashes into a heavy particle, like a charmonium or bottomonium (which are essentially heavy-duty versions of the proton made of a heavy quark and its anti-quark twin).

For a long time, scientists had a good way to calculate these collisions, but it started to break down at super-high energies. It was like trying to use a ruler to measure the distance across the ocean; the tool just wasn't designed for that scale. The calculations became unstable, swinging wildly depending on how you set your measuring tape (a concept called the "factorization scale"). To fix this, physicists developed a special "high-energy" version of the rules that accounts for the fact that at these speeds, the particles are moving so fast that time and space look different to them. This paper dives into a specific, tiny correction to those rules—a "relativistic correction"—to see if it helps smooth out the wobbly predictions and make our cosmic construction manual more reliable.

The story this paper tells is about polishing a very specific part of that cosmic instruction manual. The author, Maxim Nefedov, is tackling a problem where two different ways of fixing the "wobbly" predictions were already known, but they hadn't been combined yet. One fix involved accounting for the fact that the heavy quarks inside the particle aren't perfectly still; they jiggle around a bit (relativistic motion). The other fix involved a "high-energy resummation," which is like stacking up a huge pile of tiny, repeated corrections that become important when the collision energy is massive.

The paper calculates what happens when you combine these two fixes: the "jiggle" of the quarks and the "high-energy stacking." The author does this by performing a complex mathematical dance, using a framework called NRQCD (Non-Relativistic QCD) to match the messy, real-world physics with the clean, theoretical equations. The goal was to see if adding the "jiggle" correction to the "high-energy stack" would make the predictions for how often these heavy particles are created by light beams much better.

Here is the surprising twist the paper finds: when the author crunches the numbers using the actual physical mass of the heavy particles (rather than just theoretical estimates), the new correction turns out to be surprisingly small. It's like adding a pinch of salt to a giant pot of soup; while it's there, it doesn't change the flavor much on its own. Specifically, at the scale of the particle's mass, this new correction is negligible compared to other known effects.

However, the paper argues that this tiny correction is still worth keeping in the recipe, but not for the reason you might think. Even though the correction itself is small, it acts like a counterweight. It partially cancels out a different, larger instability in the calculations related to how the "measuring tape" (the factorization scale) is set. By including this small correction, the overall prediction becomes more robust and less sensitive to arbitrary choices in the math. It's like adding a small, precise weight to a scale to stop it from wobbling, even if the weight itself isn't heavy.

The paper also rules out the idea that this correction is a massive, game-changing discovery on its own. It explicitly states that the correction is numerically small and doesn't drastically alter the final numbers for how often these particles are produced. Instead, its value lies in making the existing theory more stable and reliable. The author concludes that while this specific correction is a "non-trivial dynamical effect" (meaning it comes from a complex interplay of forces), it is best viewed as a fine-tuning tool that improves the precision of the model rather than a new discovery that changes the whole picture. The work suggests that future studies should include this correction to get the most accurate possible predictions, but it doesn't claim to have solved the entire mystery of high-energy particle collisions.

Technical Summary: Relativistic Corrections and High-Energy Resummation for Exclusive Heavy Quarkonium Photoproduction

Problem Statement
The exclusive photoproduction of heavy vector quarkonia ( $J/\psi$ , $\Upsilon$ ) off hadrons serves as a primary channel for probing gluon distributions. While the process is well-described within collinear factorisation (CF) at Next-to-Leading Order (NLO) in $\alpha_s$ , high-energy collisions ( $\sqrt{s} \gg M_V$ ) reveal a strong dependence on the factorisation scale $\mu_F$ . This instability arises because the region $\xi \ll |x| \ll 1$ (where $\xi \sim M_V^2/2s$ is the skewness parameter) dominates the convolution integral, leading to large logarithmic terms $\alpha_s^n \ln^{n-1}(x/\xi)$ .

To address this, High-Energy Factorisation (HEF) with Double-Logarithmic Approximation (DLA) resummation has been employed to cure the instability and improve predictive power. However, a complete theoretical description requires the inclusion of relativistic corrections of order $O(v^2)$ , where $v$ is the relative velocity of the heavy quarks. While the $O(v^2)$ correction to the Leading Order (LO) CF coefficient function was recently computed, the corresponding $O(v^2)$ correction to the high-energy resummed coefficient function had not been determined. This paper addresses that gap, computing the $O(v^2 \alpha_s^n \ln^{n-1}(x/\xi))$ corrections to the DLA-HEF resummed amplitude.

Methodology
The computation utilizes the Non-Relativistic QCD (NRQCD) factorisation formalism. The methodology proceeds in two main stages:

Impact Factor Calculation: The authors compute the $O(v^2)$ correction to the process-dependent impact factor (projectile side) for the partonic subprocesses $\gamma + g \to Q\bar{Q} + g$ and $\gamma + q \to Q\bar{Q} + q$ in the Regge limit. This involves evaluating one-loop box diagrams where the heavy quark-antiquark pair is produced in a $^3S^{[1]}_1$ state. The calculation employs the standard NRQCD projector and expands the amplitude in powers of the relative momentum $k$ (where $v^2 \sim k^2/m_Q^2$ ).
Resummation and Matching: The calculated impact factor is convoluted with the universal HEF resummation function in Mellin space. The authors derive the resummed coefficient function in the DLA limit, retaining terms up to $O(v^2)$ .
Scheme Conversion: A critical aspect of the methodology is the treatment of the heavy quark mass. The authors compute results in two schemes:
- Mixed-Mass (MM) Scheme: Expresses results in terms of the pole mass $m_Q$ and the physical meson mass $M_V$ .
- Physical-Mass (PM) Scheme: Uses the Gremm–Kapustin relation to express all dependence on $m_Q$ in terms of the physical mass $M_V$ and the velocity parameter $\langle v^2 \rangle$ . This eliminates scheme ambiguity related to mass renormalisation and aligns the result with physical observables.

Key Contributions and Results

Derivation of the $O(v^2)$ Impact Factor: The paper derives the explicit form of the $O(v^2)$ correction to the HEF impact factor, denoted as $h^{(v^2, S)}(q_T^2)$ . The authors find that this function is concentrated primarily in the low transverse momentum region ( $q_T^2 \lesssim 0.1 M_V^2$ ).
Resummed Coefficient Function: The $O(v^2)$ correction to the DLA-HEF resummed coefficient function is expressed in Mellin space and subsequently transformed back to $\rho$ -space ( $\rho = \xi/|x|$ ). The result is shown to be related to derivatives of the LO resummed function with respect to the factorisation scale $\mu_F$ .
Numerical Magnitude in the PM Scheme: When expressed in the Physical-Mass scheme using the relation $m_Q = \frac{M_V}{2}(1 - \langle v^2 \rangle/2)$ , the $O(v^2)$ correction to the resummed coefficient function is found to be numerically negligible at the scale $\mu_F = M_V$ . Specifically, the correction factor is approximately $(1 - \langle v^2 \rangle/6)$ , making it five times smaller than the corresponding correction to the LO coefficient function (which is governed solely by the coefficient $b_0^{(PM)} = -5/6$ ).
Cancellation of Scale Dependence: Despite its small numerical magnitude in the PM scheme, the paper demonstrates that the $O(v^2)$ correction to the HEF-resummed amplitude partially cancels the $\mu_F$ dependence of the $O(v^2)$ correction to the LO amplitude. This cancellation improves the robustness of the cross-section predictions against scale variations.
Matching Procedure: The authors propose an extension of the subtractive matching formula to include these relativistic corrections, ensuring a consistent description across moderate and high-energy regimes while avoiding double counting of asymptotic terms.

Significance and Claims
The paper claims that the computation of the $O(v^2)$ correction to the high-energy resummed coefficient function is necessary for a consistent theoretical framework, even if the correction is numerically small in the Physical-Mass scheme. The primary significance lies in the improvement of the robustness of predictions: by including this term, the strong dependence of the cross-section on the factorisation scale $\mu_F$ is reduced.

The authors emphasize that the smallness of the correction in the PM scheme is a non-trivial dynamical consequence of the specific $q_T$ dependence of the impact factor, rather than a generic suppression. They conclude that while the $O(v^2)$ correction to the LO coefficient function (computed in Ref. [16]) is phenomenologically significant, the corresponding correction to the resummed amplitude is essential for stabilizing the scale dependence of the total prediction, thereby justifying its inclusion in future phenomenological analyses of exclusive heavy quarkonium photoproduction at the LHC and future Electron-Ion Collider (EIC).

More like this