Sampling the lattice Nambu-Goto string using Continuous Normalizing Flows

TL;DR

This work demonstrates that Continuous Normalizing Flows can efficiently sample the lattice-regularized Nambu-Goto string, enabling precise computation of the partition function and the flux-tube width within EST. By benchmarking against analytic zeta-function results in LT and HT regimes, the CNF approach reproduces universal terms and extracts next-to-leading corrections to the flux-tube width, including previously unexplored NL corrections. The method also compares favorably with Hybrid Monte Carlo, offering reduced autocorrelation and scalable GPU-based performance. Together, these results show CNFs as a powerful tool for probing beyond-Nambu-Goto corrections in EST and for exploring complex observables where traditional regularization fails.

Abstract

Effective String Theory (EST) represents a powerful non-perturbative approach to describe confinement in Yang-Mills theory that models the confining flux tube as a thin vibrating string. EST calculations are usually performed using the zeta-function regularization: however there are situations (for instance the study of the shape of the flux tube or of the higher order corrections beyond the Nambu-Goto EST) which involve observables that are too complex to be addressed in this way. In this paper we propose a numerical approach based on recent advances in machine learning methods to circumvent this problem. Using as a laboratory the Nambu-Goto string, we show that by using a new class of deep generative models called Continuous Normalizing Flows it is possible to obtain reliable numerical estimates of EST predictions.

Figures (12)

• Figure 1: Schematic representation of the lattice configurations $\phi$ with volume $L\times R=10\times 10$. The green sites represent the Dirichlet boundary and they are fixed to 0, blue and red sites represent the "active" volume seen by the CNFs. The width $\sigma w^2$ of one configuration $\langle \phi^2(x^0,R/2)\rangle_{x^0}$ is computed by averaging over $x^0$ the square of the red sites.
• Figure 2: Effective Sample Size as a function of $R$ for fixed $L=10$ and various $\sigma$. Error bars are not visible due to the very small statistical errors.
• Figure 3: Effective Sample Size as a function of $R$ for $L=90$ and different $\sigma$.
• Figure 4: Plot of $a^{(0)}_{LT}(L)-A^{(0)}_{LT}R-C^{(0)}_{LT}$ of eq. (\ref{['eq:logZLT2']}) as a function of $R$ compared to the Lüscher term $-\frac{\pi}{24R}$ (solid line).
• Figure 5: Plot of $a^{(0)}_{HT}(L)-A^{(0)}_{HT}L$ of eq. (\ref{['eq:logZHT2']}) as a function of $L$ compared to the Dedekind function prediction $-\frac{\pi}{6L}$ (solid line).