Homotopy approach for scattering amplitude for running QCD coupling

Technical Summary: Homotopy Approach for Scattering Amplitude with Running QCD Coupling

Problem Statement
The paper addresses the challenge of solving the non-linear Balitsky-Kovchegov (BK) evolution equation for the scattering amplitude in the saturation region of Quantum Chromodynamics (QCD), specifically incorporating the effects of a running QCD coupling ($\alpha_S$). While the BK equation describes the high-energy evolution of the dipole scattering amplitude, the inclusion of a running coupling introduces significant complexity, particularly regarding the violation of geometric scaling behavior and the difficulty of finding analytic solutions. Previous work by the authors established a homotopy approach for fixed coupling; this paper extends that methodology to the running coupling case, aiming to develop a regular iteration procedure that yields analytic or semi-analytic solutions with controlled accuracy.

Methodology
The authors employ a homotopy approach, a perturbative technique that constructs a solution to a non-linear equation by continuously deforming a solvable linear (or simplified non-linear) equation into the target equation. The methodology is structured in two distinct stages:

1. Analytic Solution to a Simplified Model: The first stage involves finding an analytic or "almost analytic" solution to the non-linear equation using a simplified "leading twist" BFKL kernel. In this approximation, the kernel retains only the dominant logarithmic contributions relevant to the saturation region. The authors transform the variables to $l = \int \bar{\alpha}_S(r'^2) dr'^2/r'^2$ and $\Omega$, reducing the equation to a form similar to the fixed-coupling case but with a modified variable structure. They explore various solution types, including asymptotic solutions for large $\Omega$, traveling wave solutions, and self-similar solutions dependent on the variable $\zeta = Y(l - l_s)$.
2. Iterative Correction Procedure: The second stage utilizes the homotopy method to calculate corrections to the initial analytic solution. By expanding the solution in powers of a homotopy parameter $p$, the authors derive a sequence of linear differential equations for successive iterations ($\Omega^{(1)}, \Omega^{(2)}, \dots$). These iterations account for the full non-linear terms and the differences between the simplified leading-twist kernel and the general BFKL kernel.

The analysis distinguishes between two scenarios:

• Leading Twist Kernel: A model where the kernel is simplified to isolate the running coupling effects.
• General BFKL Kernel: The full kernel including all twist contributions, treated via an alternative approach that leverages the asymptotic behavior at high energies ($Y \gg 1$).

Key Contributions and Results

• Leading Twist Kernel Analysis:

  • The authors demonstrate that the first iteration of the homotopy approach, based on the self-similar solution of the simplified equation, provides a solution with approximately 5% accuracy for small rapidities ($Y \leq 5$) and better than 2% accuracy for $Y > 5$.
  • A second iteration is shown to improve the accuracy to $\leq 1\%$ across the entire region of $Y$ and the variable $l - l_s$.
  • The study confirms that the solution exhibits $\zeta$-scaling ($\zeta = Y(l - l_s)$) at large values of $\zeta$ and $l - l_s$, but this scaling is violated at low values, necessitating higher-order corrections.
  • Geometric scaling (dependence on $z = \xi_s + \xi$) is found to hold only in the vicinity of the saturation scale and for small $l - l_s$, consistent with the expectation that running coupling effects modify the scaling behavior compared to the fixed coupling case.

• General BFKL Kernel Analysis:

  • For the general kernel, the authors propose an alternative first iteration based on the high-energy asymptotic solution ($N \to 1$), which yields a simple analytic form.
  • The second iteration in this framework is calculated analytically and is shown to reduce the error to $\leq 1\%$ for $l - l_s > 1$.
  • For small values of $l - l_s$ (near the saturation scale), higher iterations (third and fourth) are required to achieve similar accuracy, as the corrections are more significant in this region.
  • The authors observe that the general BFKL kernel solution does not strictly reproduce the $\zeta$-scaling inherent to the leading-twist model, highlighting the impact of non-leading twist contributions when running coupling is included.

• Infrared Cutoff:

  • The paper addresses the Landau pole singularity in the running coupling by introducing an infrared cutoff ($r_{max} = 1/\Lambda_{QCD}$). The authors demonstrate that contributions from distances of the order of the cutoff are negligible in their equations, validating the perturbative treatment within the defined kinematic regions.

Significance and Claims
The paper claims that the homotopy approach provides a robust and systematic method for solving the non-linear BK equation with running QCD coupling. The primary significance lies in the ability to:

1. Obtain analytic or semi-analytic solutions that satisfy the necessary initial and boundary conditions, avoiding purely numerical black-box approaches.
2. Achieve high precision (better than 1%) with a manageable number of iterations (typically two or three), making the method computationally efficient.
3. Clarify the kinematic regions where geometric scaling and $\zeta$-scaling hold or break down due to running coupling effects.

The authors emphasize that while the leading-twist kernel serves as a useful model to isolate running coupling effects, it is insufficient for a complete description of the general BFKL kernel, particularly near the saturation scale where higher-twist contributions become relevant. The work concludes that the homotopy approach is a viable tool for treating non-linear QCD dynamics and accounting for running coupling effects, though it requires careful handling of initial conditions and iteration depth depending on the specific kinematic regime.