Confining Strings in a Gapless Phase

TL;DR

This work analyzes confining strings embedded in a gapless 4D $\mathbb{C}\mathbb{P}^1$ nonlinear sigma model, where massless bulk pions prevent a purely 2D worldsheet description. It derives finite-tension string solutions, studies quadratic fluctuations, and computes one-loop corrections to the ground-state energy, separating infinite-volume (string-tension renormalization) from finite-volume (Luscher-like and bulk-induced) contributions. The authors find clear departures from standard EST predictions due to the gapless bulk, quantify the dependence on the transverse modulus $\lambda$, and present a stringy formalism to resum NG fluctuations to obtain the effective width, revealing regimes where EST is recovered and where bulk interactions dominate. They also discuss possible UV completions, including Abelian-Higgs theories and adjoint QCD, and argue about the (in)stability and UV-IR connections of these CP$^1$-strings, highlighting the limits of mapping to fundamental flux tubes in certain large-$N_c$ scenarios.

Abstract

We consider the dynamics of confined strings embedded in a gapless four-dimensional theory. To this end, we examine finite-tension string-like solutions to the equations of motion of the $\mathbb{C}\mathbb{P}^1$ non-linear sigma model. We present a comprehensive analysis of the quantum fluctuations around these solutions and derive the corresponding spectrum. These results allow us to determine the quantum corrections to the closed string ground state energy in both the finite- and infinite-size limits. Furthermore, we analyze quantum corrections to the string's effective width. We find that these observables generically depart from the universal predictions of standard Effective String Theory (EST), and we identify specific limits in which the bulk dynamics decouple and EST is recovered. Finally, we discuss the connection between these string configurations and stable electric and magnetic fluxes arising in certain ultraviolet completions of the $\mathbb{C}\mathbb{P}^1$ model.

Figures (7)

• Figure 1: Left: phase shifts for $l=-3,-2,0,1$. Right: Comparison with the Born approximation, which is accurate at large $\kappa$.
• Figure 2: Partial sums $\Delta_{1,l}(\kappa)$ for $l=0,1,5$ .
• Figure 3: Comparing the fit models with the numerical results for $l=0, 5, 10$.
• Figure 4: Left: the function $R_{l_{max}}(\kappa)$ for $l_{max}=1, 5, 100$. Right: the analytic function $\widetilde{\delta}_1(\kappa)$.
• Figure 5: The total phase shift as a fuction of $\kappa$ and $l_{max}=100$.