Dipole-dipole scattering at high energy in the Pomeron field theory with Braun Hamiltonian and beyond

Technical Summary: Dipole-Dipole Scattering at High Energy in the Pomeron Field Theory with Braun Hamiltonian and Beyond

Problem Statement
The paper addresses the determination of the asymptotic behavior of the scattering amplitude for dipole-dipole interactions at high energies within the framework of Quantum Chromodynamics (QCD) at large $N_c$. Specifically, it investigates the scattering matrix ($S$-matrix) deep within the saturation region. The study highlights a significant discrepancy between existing theoretical approaches:

1. The "rare" fluctuation approach and the summation of large Pomeron loops predict a specific asymptotic form ($S_{d}^{d} \propto \sqrt{S_{BK}}$).
2. The Balitsky-Kovchegov (BK) equation, describing dipole-nucleus scattering, yields a different asymptotic form ($S_{BK}$).
3. The current work utilizes the Braun Hamiltonian, which incorporates Pomeron vertices in the leading $1/N_c$ approximation, to derive the dipole-dipole scattering amplitude. The author notes that both the Braun and BK Hamiltonians suffer from violations of $s$-channel unitarity, yet they aim to determine the asymptotic behavior within this shared theoretical framework.

Methodology
The author employs the BFKL Pomeron calculus in the leading order of $N_c$, formulated as a path integral with the Braun Hamiltonian ($H_B$). The methodology proceeds through the following steps:

1. Equations of Motion: The study analyzes the classical equations of motion derived from the action $S = S_0 + S_I + S_E$, where $S_0$ describes free Pomerons and $S_I$ their mutual interaction. The fields $\Phi$ and $\bar{\Phi}$ represent the scattering amplitudes of external dipoles on the target and projectile, respectively.
2. Fixed Point Analysis: The author examines the solutions to the equations of motion in the vicinities of the system's fixed points: $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$.
   • They analyze the perturbative QCD region (near $(0,0)$) where geometric scaling holds.
   • They focus on the saturation regions, specifically the fixed point $(1,1)$, where both $\Phi$ and $\bar{\Phi}$ approach unity (black disk limit).
3. Linearization and Solution: In the saturation region, the author introduces small deviations $\Delta = 1 - \Phi$ and $\bar{\Delta} = 1 - \bar{\Phi}$. By neglecting higher-order terms ($\Delta^2, \bar{\Delta}^2, \Delta\bar{\Delta}$), they derive linear equations for these deviations. Using Mellin transforms and the method of steepest descent, they solve these equations for large values of the kinematic variable $z$ (related to rapidity $Y$ and dipole sizes).
4. Scattering Matrix Calculation: The $S$-matrix is calculated using a path integral formulation. The author evaluates the contribution of the action $S_0 + S_I$ at the saddle point (solution to the equations of motion) and determine the dominant behavior of the integral.
5. Validation via One-Dimensional Model: To address the reliability of summing large Pomeron loops and the issue of $s$-channel unitarity, the author utilizes a one-dimensional "Unitarity Toy Model" (UTM). This model possesses a Hamiltonian that satisfies both $s$- and $t$-channel unitarity. They compare the results of summing large Pomeron loops against the exact solution of this model to establish the kinematic range of validity.

Key Contributions and Results

• Asymptotic Form of Dipole-Dipole Scattering: The primary result is the derivation of the scattering matrix for dipole-dipole interaction deep in the saturation region ($z \gg 1$). The author finds:
  $$S_{d}^{d} \xrightarrow{z \gg 1} \exp\left(-C_d \frac{z^2}{\kappa}\right) \propto (S_{BK})^4 = \exp\left(-4 C_{BK} \frac{z^2}{\kappa}\right)$$
  where $z = \bar{\alpha}_S \kappa Y + \ln(r^2/r'^2)$. This result contrasts sharply with the predictions from the "rare" fluctuation approach and the summation of large Pomeron loops, which yield $S_{d}^{d} \propto \sqrt{S_{BK}}$ (i.e., an exponent of $-z^2/4\kappa$).

• Stability of Fixed Points: The analysis confirms that the fixed point $(1,1)$ is stable within the Braun Hamiltonian framework. The solutions for the deviations $\Delta$ and $\bar{\Delta}$ decay exponentially as $\exp(-z^2/\kappa)$, ensuring the stability of the black disk limit.

• Validity of Summing Large Pomeron Loops: Using the one-dimensional model, the author demonstrates that the summation of large Pomeron loops reproduces the correct scattering amplitude only within a limited energy range. They conclude that this approach is trustworthy for rapidities:
  $$Y \leq \frac{1}{\alpha_S^4}$$
  Beyond this range, small Pomeron loops (size $\delta Y \sim \Delta$) become significant, altering the intercepts of multi-Pomeron exchanges and invalidating the large-loop approximation.

• Unitarity and the Hamiltonian Structure: The paper highlights that the Braun Hamiltonian, like the BK Hamiltonian, violates $s$-channel unitarity. In the one-dimensional model, the conservation of the Hamiltonian ($\bar{P}P = \text{const}$) shifts the effective fixed points from $(1,1)$ to $(1, \alpha)$ and $(\alpha, 1)$. Despite this, the final scattering amplitude in the UTM coincides with the form derived in the main analysis (Eq. 41) if the Hamiltonian possessed a $(1,1)$ fixed point. This suggests that while the specific structure of fixed points depends on the Hamiltonian, the asymptotic form of the amplitude may be robust under certain conditions.

Significance and Claims
The paper claims to resolve the asymptotic behavior of dipole-dipole scattering within the specific theoretical framework of the Braun Hamiltonian, providing a result that contradicts the "rare" fluctuation and large-loop summation approaches. The author expresses "mixed feelings" regarding the robustness of their result:

• Pessimism: The specific form of the Hamiltonian crucially determines the fixed points of the equations of motion. Since the Braun Hamiltonian violates $s$-channel unitarity, the derived asymptotic behavior might be an artifact of this violation.
• Optimism: The fact that the exact solution in the unitary one-dimensional model yields an amplitude consistent with their derived form (Eq. 41) suggests that the result might hold even if the Hamiltonian is corrected to satisfy unitarity.

The author concludes that a definitive resolution requires the construction of a QCD Hamiltonian for Pomeron interactions that satisfies both $t$- and $s$-channel unitarity. Without such a Hamiltonian, even the asymptotic behavior of the Balitsky-Kovchegov amplitude remains uncertain. The paper serves as a continuation of previous work (Ref. [21]), emphasizing the need for further effort in formulating a unitary Pomeron field theory.