Effective magnetic $SU(N)$ gauge theory with classical $Z_N$ flux tubes of intrinsic width $\frac{1}{M}$ is an effective field theory of the long-distance quark-antiquark interaction in $SU(N)$ Yang-Mills theory. Long-wavelength fluctuations of the $Z_N$ vortices of this theory lead to an effective string theory. In this paper, we clarify the connection between effective field theory and effective string theory, and we propose a new constraint on these vortices. We first examine the impact of string fluctuations on the classical dual superconductor description of confinement. At interquark distances $R\sim \frac{1}{M}$, the classical action for a straight flux tube determines the heavy quark potentials. At distances $R\gg \frac{1}{M}$, fluctuations of the flux tube axis $\tilde{x}$ give rise to an effective string theory with an action $S_{\text{eff}}(\tilde{x})$, the classical action for a curved flux tube, evaluated in the limit $\frac{1}{M}\to 0$. This action is equal to the Nambu-Goto action. These conclusions are independent of the details of the $Z_N$ flux tube. Further, we assume the QCD flux tube satisfies the additional constraint, ${\int }_0^{\infty }rdr\frac{T_{\theta \theta }(r)}{r^2}=0,$ where $\frac{T_{\theta \theta }(r)}{r^2}$ is the value of the $\theta \theta$ component of the stress tensor at a distance $r$ from the axis of an infinite flux tube. Under this constraint, the string tension $\sigma$ equals the force on a quark in the chromoelectric field $\overrightarrow{E}$ of an infinite straight flux tube, and the Nambu-Goto action can be represented in terms of the chromodynamic fields of effective magnetic $SU(N)$ gauge theory, yielding a field theory interpretation of effective string theory.

• Received 10 December 2015

DOI:https://doi.org/10.1103/PhysRevD.93.054012