⚛️ phenomenology

Scheme Transformations in Non-Linear QCD Evolution: Towards a Stable NLO BK Equation

This paper introduces a systematic scheme-transformation framework based on the Doi-Peliti formalism to cancel large double collinear logarithms in the Next-to-Leading Order Balitsky-Kovchegov equation, thereby stabilizing high-energy QCD evolution up to rapidity $Y=10$ for various initial conditions.

Original authors: Renaud Boussarie, Paul Caucal, Piotr Korcyl, Yacine Mehtar-Tani

Published 2026-09-29

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CC BY 4.0

Original authors: Renaud Boussarie, Paul Caucal, Piotr Korcyl, Yacine Mehtar-Tani

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

In the subatomic world, matter is not made of solid, unchanging bricks, but of a seething, dynamic soup of particles called quarks and gluons. These particles are bound together by the strong nuclear force, the most powerful force in nature, which operates according to the rules of a theory known as Quantum Chromodynamics. When scientists smash particles together at incredibly high speeds, such as in the experiments at the Large Hadron Collider, they are trying to see how this soup behaves when it is squeezed and stretched to its limits. A key part of understanding these collisions involves looking at what happens when particles move at speeds close to the speed of light, a regime where the number of gluons inside a proton can multiply rapidly. To predict the outcome of these high-speed collisions, physicists use mathematical tools that describe how the density of these gluons changes as the energy increases. However, when they tried to make these predictions more precise by adding the next level of detail to their calculations, the equations began to break down, producing results that were physically impossible, such as negative probabilities.

A team of researchers has now developed a new way to fix these broken equations, offering a stable path forward for understanding high-energy collisions. The problem they tackled arises when physicists try to calculate the behavior of gluons using a method called the Balitsky-Kovchegov equation, which tracks how the density of gluons evolves as the energy of the collision increases. When the scientists added the next layer of complexity to this equation, known as the next-to-leading order, they found that the calculations were being overwhelmed by massive mathematical terms that grew too large to ignore. These terms, which arise from the way the particles are organized in space and time, caused the predictions to become unstable and nonsensical, effectively preventing the theory from describing reality accurately. The researchers realized that this instability was not a fundamental flaw in the laws of physics, but rather an artifact of how the equations were being set up, specifically related to how the scientists chose to divide the problem into different parts for calculation.

To solve this, the team introduced a systematic method for reshuffling the mathematical framework, a process they call a scheme transformation. Imagine trying to describe the motion of a complex machine; if you choose to measure the speed of one part relative to the ground, the math might become messy and confusing, but if you choose to measure it relative to a moving part of the machine itself, the description might become much cleaner. The researchers applied this kind of shift to their equations, moving from a perspective that focused on the incoming projectile to one that focused on the target being hit. By doing this, they were able to cancel out the troublesome, oversized terms that were causing the instability. The result is a new version of the evolution equation that remains mathematically sound and physically sensible, even when the calculations are pushed to higher levels of precision.

The team tested this new, stable equation by running detailed computer simulations to see how the gluon density would evolve over time. They started with two different realistic models for the initial state of the particles, representing different ways the gluons might be arranged before the collision begins. As they let the simulation run forward, tracking the changes over a vast range of energy levels, the equation held steady. The researchers observed that the solutions remained stable and physically meaningful, reaching a point corresponding to a rapidity value of ten without breaking down or producing negative results. This stability was maintained even when they included the effects of the strong force changing strength at different distances, a feature known as the running coupling. The simulations showed that the new equation successfully tamed the wild mathematical behavior that had plagued previous attempts, allowing for a consistent description of the gluon cloud as it grows denser.

This work provides a crucial foundation for future experiments, particularly those planned at the future Electron-Ion Collider, where scientists hope to take detailed 3D images of the proton's interior. By ensuring that the theoretical tools used to interpret these images are stable and reliable, the researchers have cleared a major hurdle in the field. The new framework allows physicists to combine these stable evolution equations with other advanced calculations, known as impact factors, which describe how the particles interact at the moment of collision. While the immediate results are confined to computer simulations and theoretical checks, the method offers a consistent and robust way to move forward, ensuring that the next generation of high-energy physics experiments can be understood with greater clarity and confidence.

Technical Summary: Scheme Transformations in Non-Linear QCD Evolution

Problem Statement
High-energy Quantum Chromodynamics (QCD) evolution at Next-to-Leading Order (NLO) faces significant stability issues, particularly within the Balitsky-Kovchegov (BK) equation. The perturbative expansion of the NLO BK equation is destabilized by large collinear logarithms. Specifically, double collinear logarithms arise because the phase-space integration over emitted gluons extends into a kinematically forbidden region ( $k^- > P^-$ ), corresponding to gluon fluctuations with formation times shorter than the longitudinal extent of the target shock wave. This region lies outside the validity of the high-energy operator product expansion. While previous remedies, such as kinematic constraints or explicit resummations, have improved numerical behavior, they introduce modifications to the leading-order kernel that raise conceptual questions regarding the underlying factorization scheme, the universality of high-energy operators, and the consistency of matching Color Glass Condensate (CGC) evolution to hard coefficient functions.

Methodology
To address these issues without altering the operator content of the high-energy operator product expansion, the authors develop a systematic framework based on scheme transformations implemented at the level of physical observables. The core methodology relies on the Doi-Peliti formalism, which allows non-linear evolution and scheme transformations to be formulated as linear transformations in a Fock space spanned by dipole-chain states.

The authors define a similarity transformation generated by a $\mu$ -dependent operator $L(\bar{\alpha}_s, \mu^2)$ to relate different factorization schemes:

Transformation Definition: They transform the target operator vector $|S\rangle$ (dependent on projectile rapidity $k^+$ ) to a new vector $|\bar{S}\rangle$ (dependent on target rapidity $k^-$ ) via $|\bar{S}\rangle = \exp(-\bar{\alpha}_s L)|S\rangle$ .
Kernel Modification: This transformation induces a change in the evolution kernel $\bar{K}$ , calculated via the Campbell identity: $\bar{K} = e^{-\bar{\alpha}_s L} K e^{\bar{\alpha}_s L}$ .
Cancellation Mechanism: By choosing a specific scheme transformation (target rapidity factorization with $\sigma=2$ ), the commutator term $[\bar{\alpha}_s K_{LO}, L_{LO}]$ generates NLO terms that exactly cancel the problematic double collinear logarithms present in the original $k^-$ -ordered NLO BK equation.
Physical Consistency: The transformation is accompanied by a compensating redefinition of the hard coefficient functions (impact factors) to ensure the physical cross-section remains scheme-independent.

Key Contributions

Formalism: The paper establishes a rigorous framework for scheme transformations in the non-linear saturation regime using the Doi-Peliti formalism, treating the evolution of CGC operators as vectors in a Fock space.
Stabilization: The authors derive a modified NLO BK equation where the double collinear logarithmic instability is removed. The resulting equation retains only single collinear logarithms (associated with DGLAP dynamics) and double anti-collinear enhancements.
Explicit Kernel: The paper provides the explicit form of the transformed NLO kernel, identifying the specific commutator terms responsible for the cancellation of the instability.

Numerical Results
The authors performed numerical solutions of the modified NLO BK equation (Eq. 10) using a fixed running coupling ( $\alpha_s = 0.3$ at NLO, with the LO scale set via the BLM prescription to remove $\beta$ -terms).

Initial Conditions: Two distinct initial conditions were tested: the Golec-Biernat-Wusthoff (GBW) model and the McLerran-Venugopalan (MV) model.
Stability: The solutions were evolved up to rapidity $Y=10$ . The results demonstrate that the modified equation yields stable evolution for both initial conditions.
Observations: While the solutions remain stable, the authors note the onset of possible single collinear logarithmic instabilities at larger rapidities, visible as small negative structures for dipole sizes $|x_{12}| \approx 0.05$ fm. However, the primary double-logarithmic instability is absent.

Significance and Claims
The paper claims to provide a consistent path toward combining stable NLO evolution with rotated NLO impact factors. By systematically incorporating collinear logarithms through scheme transformations rather than ad-hoc kinematic constraints, the framework preserves the standard factorization structure while improving perturbative stability order by order. The authors emphasize that this approach resolves the conceptual ambiguity regarding the definition and universality of high-energy operators in the presence of large logarithms. The work is presented as a necessary step toward a fully consistent NLO description of high-energy QCD processes, specifically preparing the ground for future applications involving "rotated" NLO impact factors.

Technical Summary: Scheme Transformations in Non-Linear QCD Evolution

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