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Running coupling effects in the anti-collinear resummation in high energy evolution

This paper investigates how the running of the QCD coupling affects anti-collinear resummation in JIMWLK evolution, deriving a resummed BFKL kernel that reveals a significant reduction in the Pomeron intercept due to combined resummation and coupling effects, despite a surprising cancellation that leaves the characteristic function at $\gamma=1$ unchanged.

Original authors: Alex Kovner, Michael Lublinsky, Maxim Nefedov, Vladimir Skokov

Published 2026-07-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alex Kovner, Michael Lublinsky, Maxim Nefedov, Vladimir Skokov

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, chaotic dance floor where subatomic particles called gluons are the dancers. When these dancers collide at incredibly high speeds, they don't just bump into each other; they scream out a cascade of new dancers, creating a massive, expanding cloud of energy. Physicists call this "high-energy evolution," and they use a set of mathematical rules (the JIMWLK Hamiltonian) to predict how this cloud grows.

For a long time, scientists tried to predict this growth using a "fixed" rulebook, assuming the strength of the interaction between dancers never changed. But in reality, the "strength" of the glue holding them together (the QCD coupling) changes depending on how close the dancers are. This is called the running coupling.

This paper is like a team of physicists (Kovner, Lublinsky, Nefedov, and Skokov) stepping in to rewrite the rulebook, adding the "running" rule to the most complex part of the dance: the anti-collinear resummation.

The Problem: The "Too-Fast" Dance

In the world of high-energy physics, there's a specific scenario called the "anti-collinear" regime. Imagine one dancer (the projectile) is zooming in from very far away, while the target dancer is standing still but surrounded by a huge, fuzzy cloud of other dancers. The math gets messy here. If you don't fix the rules, the prediction for how fast the cloud grows becomes unstable and produces nonsense results, like infinite energy.

Previous work fixed the math for a "fixed" strength, but the authors realized that ignoring the changing strength of the glue was like trying to drive a car with a speedometer that doesn't work. They set out to fix the math by including the running coupling.

The Big Discovery: A Surprising Magic Trick

The team did two main things:

They figured out the right "speed limit" (scale) for the glue. They argued that the strength of the interaction shouldn't just be an average; it depends on the specific geometry of the dance. If the dancers are far apart, the glue is weaker; if they are close, it's stronger. They determined that the correct scale to use is the smallest distance involved in the interaction (or the largest momentum).
They calculated the new growth rate.

Here is the most surprising part of their finding:
When they added the running glue to the math, they expected the whole system to slow down significantly. And it did! The "Pomeron intercept" (a fancy number that tells you how fast the cross-section grows) dropped significantly, from a high theoretical value down to a much more realistic 0.3. This is great news because a value of 0.3 matches what we see in real-world experiments.

However, there is a twist.
The authors found that at one very specific point in their math (where a variable called $\gamma$ equals 1), the running glue effect completely vanished.

Why? It wasn't a mistake. It was a "perfect cancellation." The effect of the glue running in the main equation canceled out exactly with the effect of the glue running in the resummation equation.
The Result: At this specific point, the value of the growth function is exactly the same as if the glue strength had never changed at all. The authors call this a "surprising" and "intricate" cancellation. They don't know why nature chose to do this exact cancelation, but the math shows it happens.

What They Ruled Out

The paper explicitly argues against a few things:

The "Naive" Scale Choice: They reject the idea that you can just pick an average distance for the glue strength. They show that for certain observables (like angular correlations), using the wrong scale leads to unphysical artifacts, like fake "poles" (mathematical explosions) in the results.
The "Fixed" Reality: They demonstrate that ignoring the running coupling leads to an overestimation of the growth rate. The "fixed" version is too fast and doesn't match the "anti-collinear" reality.

How Sure Are They?

The authors are very confident in their mathematical derivation. They have proved the cancellation at $\gamma = 1$ within their specific framework. They have derived the new equations for the running coupling.

However, when it comes to the final number for the growth rate (the Pomeron intercept of 0.3), they are more cautious. They describe their result as "semi-quantitative."

Why? Their calculation involves a "saddle-point approximation," which is a clever shortcut to solve a very hard integral.
The Caveat: They admit their method ignores some "non-perturbative" effects (the messy, non-mathematical parts of the dance that happen at very low energies). They had to artificially cut off a part of their calculation to avoid getting imaginary numbers, which means the 0.3 is a strong estimate, not a final, unchangeable law of the universe.

The Takeaway

Think of this paper as the team that finally fixed the engine of a race car that was running too hot.

They found that the engine (the glue) changes its power based on the track conditions (the distance between particles).
They discovered that at one specific corner of the track, the engine's power change magically cancels itself out, leaving the speed exactly as it was before.
But for the rest of the race, the engine is much slower than they thought, bringing the car's top speed down to a level that actually makes sense for the real world.

They haven't solved the entire mystery of the universe's dance floor, but they've fixed a major glitch in the rulebook and shown us that the "running" nature of the glue is essential for getting the right answer.

Technical Summary: Running coupling effects in the anti-collinear resummation in high energy evolution

Problem Statement
High-energy QCD evolution, described by the JIMWLK equation and its linear BFKL limit, suffers from instabilities when the projectile and target possess vastly different transverse momentum scales ( $Q_P \ll Q_T$ ). In this "anti-collinear" regime, large transverse logarithms of the form $\alpha_s \ln(Q_T/Q_P)$ appear, rendering the evolution unstable and leading to unphysical cross-sections. While previous work (Ref. [16]) established a framework for resumming these anti-collinear logarithms within the JIMWLK Hamiltonian, it treated the strong coupling constant $\alpha_s$ as fixed. This paper addresses the critical extension of that framework to include the running of the QCD coupling, $\alpha_s(\mu^2)$ , which is essential for a realistic description of high-energy evolution.

Methodology
The authors extend the anti-collinear resummation formalism by incorporating running coupling effects in two distinct locations where the coupling enters the evolution:

The JIMWLK Kernel: The prefactor of the resummed Hamiltonian and the kernel itself.
The DGLAP Resummation Equation: The evolution equation for the dressed Wilson lines (resolution scale $Q$ ) which generates the resummation functions.

A central methodological challenge was determining the appropriate scale $\mu_\star$ for the running coupling in the JIMWLK kernel. The authors revisit the scale choice proposed in Ref. [1] ( $\mu^2 \sim g(X^{-2})g(Y^{-2})/4\pi$ ) and argue that while adequate for leading terms, it fails for subleading configurations where $X^2 \gg Y^2$ (or vice versa). By analyzing the interplay between the real and virtual diagrams and utilizing the Collins-Soper-Sterman (CSS) evolution equation for Transverse Momentum Distributions (TMDs), they derive a corrected scale choice:
$\mu_\star^2(x, y, z) = \min(X^{-2}, Y^{-2})$
where $X$ and $Y$ are the transverse distances from the emission point to the sources. This choice is physically motivated by the fact that in asymmetric kinematics, the coupling is controlled by the largest transverse distance scale.

The authors then:

Linearize the resummed JIMWLK Hamiltonian to derive the momentum-space BFKL kernel.
Solve the DGLAP-like evolution equations for the resummation functions $R^{(2,1)}_Q$ with a running coupling.
Compute the generalized characteristic function $\chi(n, \gamma)$ in momentum space, defining it via the action of the kernel on test functions, acknowledging that these are no longer strict eigenfunctions when $\alpha_s$ runs.
Construct an approximate closed-form expression for the BFKL Green's function using a Borel transform technique and a saddle-point approximation to extract the Pomeron intercept.

Key Contributions and Results

Scale Setting and CSS Interpretation: The paper establishes that the correct scale for the running coupling in the anti-collinear JIMWLK kernel is the minimum of the inverse squared distances to the sources. This result is interpreted through a modified CSS evolution where the double-logarithmic term is subtracted (as it is already included in the LO JIMWLK evolution), leaving a single-logarithmic term that leads to an enhancement of the field amplitude rather than suppression. This resolves the apparent contradiction between standard TMD evolution and the required scale setting.
Generalized Characteristic Function: The authors derive the generalized characteristic function $\chi(n, \gamma)$ including both anti-collinear resummation and running coupling effects.
- Suppression: Generally, the running of the coupling further suppresses the characteristic function compared to the fixed-coupling case, slowing down the evolution.
- Exact Cancellation at $\gamma=1$ : A surprising and significant result is that at $\gamma = 1$ , the effects of the running coupling in the BFKL kernel and the running coupling in the DGLAP resummation equation exactly cancel. Consequently, the value of the characteristic function at $\gamma = 1$ remains identical to the fixed-coupling result derived in Ref. [16]:
  $\chi(\gamma=1) = \frac{\pi}{\alpha_s N_c} \times \begin{cases} 4/3 & n_F \neq 0 \\ 12/11 & n_F = 0 \end{cases}$
  This cancellation is non-trivial and occurs only at this specific point; for $\gamma < 1$ , the cancellation is only partial.
Pomeron Intercept: Using a saddle-point approximation on the derived Green's function, the authors calculate the Pomeron intercept $\omega_P$ in the anti-collinear regime.
- They find that the combined effects of anti-collinear resummation and the running coupling significantly reduce the intercept.
- Numerical results indicate that the intercept can drop to values around $\omega_P \sim 0.3$ , which is phenomenologically consistent with the "hard Pomeron" observed in experiments.
- The reduction due to the running coupling is found to be roughly twice as large as the reduction due to the fixed-coupling anti-collinear resummation alone.

Significance and Claims
The paper claims to provide a consistent framework for studying high-energy evolution where both large anti-collinear logarithms and the running of the coupling are treated simultaneously. The authors emphasize that their approach captures the main numerical effects of the resummation while maintaining relative simplicity.

They note that the exact cancellation at $\gamma=1$ raises questions regarding momentum conservation in conformal theories (where $\chi(1) = \pi/\alpha_s N_c$ is expected), suggesting that the discrepancy in their result (which is slightly lower for $n_F=0$ ) may stem from treating the DGLAP cascade as point-like in rapidity rather than accounting for its finite extent. They acknowledge that their calculation of the Pomeron intercept is "semi-quantitative" due to infrared diffusion issues and the need to cut off non-perturbative contributions, but argue that the results robustly demonstrate the significant slowing of evolution caused by these combined effects.

The work serves as a necessary step toward a more complete understanding of high-energy QCD evolution, bridging the gap between fixed-coupling resummation formalisms and phenomenological requirements for running coupling effects.

The Problem: The "Too-Fast" Dance

The Big Discovery: A Surprising Magic Trick

What They Ruled Out

How Sure Are They?

The Takeaway

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