Stochastic normalizing flows for Effective String Theory

TL;DR

Effective String Theory (EST) provides a non-perturbative framework for confinement by modeling the confining flux tube as a string, with observables tied to the EST action $S_{EST}$. The authors adopt Stochastic Normalizing Flows (SNFs) to overcome sampling challenges on the lattice, combining flow-based layers with non-equilibrium updates and an NE-iMH equilibrium step, and they train the model via a KL-divergence objective linked to Crooks' fluctuation theorem. They compute the Binder cumulant $U$, defined as $U = 1 - \frac{\langle \phi^4(\tau,R/2)\rangle_{\tau}}{3 \langle \phi^2(\tau,R/2)\rangle_{\tau}^2}$, to probe non-Gaussianity of the flux density and find $U$ effectively zero across NG and BN G sectors, with small deviations attributed to finite-size and boundary effects. The results indicate EST yields near-Gaussian flux tubes, in contrast to LGT where larger non-Gaussian effects and intrinsic width may be required; the SNF approach provides a scalable path to explore EST observables and paves the way for extensions to higher dimensions and more intricate EST corrections.

Abstract

Effective String Theory (EST) is a powerful tool used to study confinement in pure gauge theories by modeling the confining flux tube connecting a static quark-anti-quark pair as a thin vibrating string. Recently, flow-based samplers have been applied as an efficient numerical method to study EST regularized on the lattice, opening the route to study observables previously inaccessible to standard analytical methods. Flow-based samplers are a class of algorithms based on Normalizing Flows (NFs), deep generative models recently proposed as a promising alternative to traditional Markov Chain Monte Carlo methods in lattice field theory calculations. By combining NF layers with out-of-equilibrium stochastic updates, we obtain Stochastic Normalizing Flows (SNFs), a scalable class of machine learning algorithms that can be explained in terms of stochastic thermodynamics. In this contribution, we outline EST and SNFs, and report some numerical results for the shape of the flux tube.

Figures (2)

• Figure 1: Binder cumulant $U$ as a function of $\sqrt{\sigma}R$ in the high-temperature limit ($R\gg L=8$, left plot) and in the low-temperature limit ($L=80\gg R$, right plot). Different values of the string tension $\sigma$ are listed in different colors.
• Figure 2: Binder cumulant $U$ for $S^1_{\hbox{\tiny{BNG}}}$ (\ref{['eq:NGK2action']}) at fixed (left plot) and for $S^2_{\hbox{\tiny{BNG}}}$ (\ref{['eq:NGK4action']}) at fixed $\gamma_2=0.02$ and $\gamma_3=0.02$. Both theories are studied at $\sigma=100$.