Finite-length flux tube in the dual Ginzburg-Landau theory on the dual lattice

TL;DR

This work develops a finite-length flux-tube analysis within the dual Ginzburg-Landau framework by formulating the theory on a dual lattice and employing a Hodge decomposition to separate Coulombic and solenoidal contributions. The authors implement a robust Newton-Raphson solver on the lattice to obtain flux-tube solutions with endpoints, enabling precise computation of interquark potentials, flux-tube widths, and interactions in single, two-, and multiflux configurations across type-I, Bogomolnyi, and type-II regimes characterized by the GL parameter $\kappa=m_{\chi}/m_{B}$. The results show that the interquark potential comprises a short-range Yukawa-like term $-A_{y}e^{-m_{y}r}/r$ plus a linear term $\sigma_{y}r$, with the effective mass $m_{y}(\kappa)$ and string tension $\sigma_{y}(\kappa)$ revealing rich dependence on $\kappa$; flux tubes interact in geometry-dependent ways, including string bending and flips, and high-density configurations can trigger a transition toward a normal phase. Overall, the dual-lattice DGL approach provides a quantitative framework to interpret nonperturbative QCD vacuum structure, enabling direct comparisons with lattice QCD flux-tube profiles and extensions to baryonic and multiquark flux-tube networks.

Abstract

The dual Ginzburg-Landau (DGL) theory is one of the nonperturbative effective field theories of quantum chromodynamics (QCD). The DGL theory describes the QCD vacuum as a dual superconductor and possesses electric flux-tube solutions via the dual Meissner effect, which applies to the quark confinement mechanism. We demonstrate a powerful numerical method for solving the field equations in the DGL theory with U(1) dual gauge symmetry. An essential aspect of our method is to formulate the DGL theory on the dual lattice, which enables us to investigate any system composed of finite-length flux tubes in a systematic manner. Taking full advantage of the dual lattice formulation, we investigate the finite-length flux-tube solution corresponding to the quark-antiquark system in detail, which exposes the significant terminal effects absent in the infinitely long flux-tube solution. We also study the flux-tube interaction in the two-flux-tube and multiflux-tube systems, providing new insights into the nonperturbative properties of QCD.

Figures (35)

• Figure 1: Histories of the maximum violation of the field equations $\max (|X \!(n)| )$ and $\max ( |Y \!(n)| )$ for $\kappa =1.0$ as a function of iteration step.
• Figure 2: History of the value of $\sigma$ in Eq. \ref{['eqn:dgl-cylind']} for $\kappa =1.0$ as a function of iteration step. The dotted line denotes the analytical continuum value $\sigma/m_{B}^{2}=\pi$ in Eq. \ref{['eqn:stringtension-bb']}.
• Figure 3: The field profiles for $\kappa =1.0$ as a function of the radial coordinate $\rho$. The circles with dotted lines are for $\hat{m}_{B}=1.0$ (coarse), and the solid lines are for $\hat{m}_{B}=0.10$ (fine).
• Figure 4: The ratio of the monopole supercurrent to the electric field with the normalization $\hat{m}_{B}$ for $\kappa =1.0$ as a function of the radial coordinate $\rho$. The ratio is compared to that of the modified Bessel functions $K_{1}/K_{0}$.
• Figure 5: How to put the nonzero $\Sigma_{\mu\nu}$ for the quark-antiquark ($q\bar{q}$) system in $D=3$ dimensions. This example is for the $R=3$ case, where the $q\bar{q}$ axis is taken along the $x_{3}$-axis. The shaded part of $\Sigma_{12}$ is set to be $-N_{q}$.