The Static Heavy Quark-Antiquark Potential within String Theory in Arbitrary Stationary Backgrounds

TL;DR

This paper derives a general Nambu–Goto framework for the static heavy quark-antiquark potential in arbitrary stationary spacetimes, capturing both holographic and effective-string perspectives. It identifies a parity-violating geometric factor $h_{pr}$ that generally renders the string profile asymmetric about its turning point, while showing that symmetry and a clean linear-in-$L$ term emerge for backgrounds with $h_{pr}=0$. The authors provide a universal, renormalized expression for $V(L)$ and demonstrate, via the Rindler-AdS example, how acceleration modulates the interquark distance and potential, and raises the deconfinement temperature, with a scale-invariant structure under acceleration rescaling. The results illuminate how rotation/acceleration and off-diagonal metric components influence quark confinement in holographic settings and suggest observable parity-violating signatures in interquark interactions.

Abstract

We analyze a static open string in a general stationary spacetime, which can represent a heavy quark-antiquark pair within the holographic framework or effective theory. We establish that for a simple U-shaped string with only radial dependence on the space string coordinate, $x_r'(σ) \neq 0$, the string is generally not symmetric about its turning point, and the symmetry restores only for backgrounds with $h_{pr} = G_{00} G_{pr} - G_{0p} G_{0r} = 0$. Consequently, such asymmetric strings directly probe a possibility of the parity violation in the quark-antiquark interaction. Nevertheless, we identify a wide family of metrics for which the symmetry is preserved, enabling a direct isolation of the linear-in-distance term in the static interquark potential for simple symmetric string configurations, even in non-diagonal backgrounds. Applying the holographic framework, we further study the Rindler-AdS spacetime dual to an accelerated $\mathcal{N}=4$ super Yang-Mills plasma. We show that the distance between quarks decreases, the static potential between them increases, and the deconfinement phase transition temperature, $T_{\rm dec} = (π/3) T_H = a_c/6$, increases with an acceleration. However, we observe that an acceleration-scaled potential as a function of the acceleration-scaled distance does not depend on the certain value of the acceleration This result, reflecting the scale invariance and self-similarity of the holographic setup, can be also obtained in the dimensionless metric after scaling of the coordinates onto the acceleration, $\tilde{x}_i = a_c x_i$, for which one obtains an universal value of the phase transition temperature, $\tilde{T}_{\rm dec} = (π/3) \tilde{T}_H = 1/6$.

Figures (4)

• Figure 1: Numerical calculations in the Rindler-AdS background \ref{['eq:rindler-ads:metric']} for (1) the quark-antiquark potential $V$\ref{['eq:result:V:special']} depending on the distance $L$\ref{['eq:result:L:special']} between quarks; (2) the potential $V$ as a function of the absolute value of constant $\mathcal{C}$; (3) the distance $L$ as a function of the absolute value of constant $\mathcal{C}$. Different line colors correspond to different values of the acceleration: blue for $a=0.4/\ell$, red for $a=0.6/\ell$, black for $a=1/\ell$, and green for $a=2/\ell$.
• Figure 2: Contours plots of the distance $L$ (first pane) and potential $V$ (second pane) between quarks as functions of the dimensionless acceleration parameter $a_0 = a_c \ell$ and of the dimensionless temperature at the string turning point $\ell T = \ell/(2\pi\xi_m)$. Different values of $V$ or $L$, corresponding certain contours, are depicted by colors and labels. The critical temperature $T_{\text{dec}} = (\pi/3) T_\text{H} = a_c/6$ of the phase transition is shown by a red dashed line, while the Hawking temperature $T_\text{H}$ is drawn by a rest dashed line.
• Figure 3: 3-dimensional plots of the distance $L$ (first pane) and potential $V$ (second pane) between quarks as functions of the dimensionless acceleration parameter $a_0 = a_c \ell$ and of the dimensionless temperature at the string turning point $\ell T = \ell/(2\pi\xi_m)$; the potential $V$ as a function of the distance $L$ and acceleration parameter $a_0$ (third pane).
• Figure 4: The scaled onto acceleration potential $\tilde{V} = a_c V$ as a function of the acceleration-scaled distance $\tilde{L} = a_c L$.