Critical coupling with zero-mode corrections in discretized light-cone quantized $\phi^4$ theory

Technical Summary: Critical Coupling with Zero-Mode Corrections in Discretized Light-Cone Quantized $\phi^4$ Theory

Problem Statement
The Discretized Light-Cone Quantization (DLCQ) method is a primary non-perturbative tool for solving relativistic quantum field theories, particularly for bound-state phenomena. However, in the context of 2D interacting scalar $\phi^4$ theory with Periodic Boundary Conditions (PBCs), DLCQ faces a significant convergence challenge. Conventional implementations omit the constrained zero-momentum mode ($\phi_0$) to avoid divergences associated with its nonlinear operator constraint equation and zero-mode kinetic energy. This omission introduces systematic truncation errors that scale as $O(1/K)$, where $K$ is the longitudinal momentum resolution. Consequently, achieving a precise estimate of the continuum limit for the critical coupling ($\lambda_c$) requires extremely high resolutions (e.g., $K=88$), leading to Hamiltonian matrices with dimensions exceeding 22 million states per sector. This imposes prohibitive computational costs on high-performance resources.

Methodology
The authors address the slow convergence by incorporating perturbative zero-mode contributions into the effective DLCQ Hamiltonian. Rather than explicitly including the $K^+=0$ mode in the basis space—which introduces intractable divergences—the authors derive an effective interaction term, $H_{\text{zero mode}}$, representing the zero-mode field exchange among dynamical modes.

1. Effective Hamiltonian Construction: The zero-mode correction is derived as a normal-ordered expression involving creation and annihilation operators for dynamical modes ($n \ge 1$). This term is added perturbatively to the original DLCQ Hamiltonian.
2. Perturbative Correction: The correction is applied to the eigenvalues of the uncorrected Hamiltonian. The corrected mass-squared eigenvalue at finite resolution $K$ is calculated as:
   $$M^2_{\text{corrected}} = M^2 + \langle \Phi_0 | K H_{\text{zero mode}} | \Phi_0 \rangle$$
   where $|\Phi_0\rangle$ is the lowest eigenvector of the uncorrected Hamiltonian.
3. Continuum Extrapolation: To determine the critical coupling in the continuum limit ($K \to \infty$), the authors employ Gaussian Process Regression (GPR) using a Matérn kernel ($\nu=1.5$). This non-parametric Bayesian approach is chosen to minimize model dependence and avoid artifacts associated with high-degree polynomial fits, particularly given the sharp convergence behaviors near the vanishing mass gap.

Key Results
The application of the zero-mode correction yields two primary outcomes:

• Accelerated Convergence: The corrected critical coupling values converge significantly faster toward the continuum limit compared to uncorrected data. At a resolution of $K=60$, the uncorrected coupling ($\lambda_c/\mu^2 \approx 33.7$) remains far from the continuum limit, whereas the corrected coupling ($\lambda_c/\mu^2 \approx 22.5$) is already very close.
• Extrapolated Critical Coupling: Using GPR on the corrected data, the authors determine an extrapolated critical coupling of $\lambda_c = 23.10 \pm 0.25$. This is compared to the uncorrected GPR re-analysis of previous data, which yielded $\lambda_c = 23.58 \pm 0.60$ (originally published as $22.64 \pm 0.17$ via polynomial fit).
• Computational Efficiency: The method allows for comparable accuracy to be achieved at a maximum resolution of $K=70$ instead of $K=88$. Because the basis space dimension is governed by the partition function, this reduction in $K$ results in an order-of-magnitude decrease in the Hamiltonian matrix dimension:
  • $K=88$ (uncorrected): $\approx 22$ million states per sector.
  • $K=70$ (corrected): $\approx 2$ million states per sector.
    This translates to an approximate 100-fold reduction in memory and CPU time requirements.

Significance and Claims
The paper claims that the inclusion of perturbative zero-mode corrections is a highly efficient mechanism for advancing DLCQ and related Hamiltonian Light-Front techniques, such as Basis Light-Front Quantization (BLFQ). The authors assert that:

1. Validation of Approach: The rapid convergence of corrected data confirms the physical consistency of the DLCQ approach when zero-mode effects are systematically included, effectively suppressing the dominant $O(1/K)$ truncation errors.
2. Resource Optimization: The method provides a practical pathway to attain high-precision regimes required for comparison with modern experimental data without the exponential growth in computational resources typically associated with higher resolutions.
3. Generalizability: While demonstrated in 2D $\phi^4$ theory, the approach suggests a viable strategy for higher-dimensional gauge theories (e.g., 3+1 dimensional QCD). Specifically, it offers a way to handle the $K^+=0$ mode, which often introduces intractable divergences in gauge theories, by treating it as an effective interaction within a conventional DLCQ framework rather than requiring explicit inclusion in the basis.

The authors emphasize that their results validate the use of effective Hamiltonians with zero-mode contributions as a suitable substitute for explicit $K^+=0$ mode inclusion, retaining the ability to work within standard DLCQ frameworks while capturing vacuum effects usually lost.