[Paper Review] Heavy Quark Potential at Finite Temperature in AdS/CFT Revisited

Javier L. Albacete, Yuri V. Kovchegov+1 more|arXiv (Cornell University)|Jul 30, 2008

Black Holes and Theoretical Physics19 citations

TL;DR

This paper revisits the heavy quark potential in N=4 SYM at finite temperature using AdS/CFT, showing that analytic continuation into the complex plane and a modified renormalization yield a smooth, non-zero, negative-definite potential without a kink. The potential exhibits a non-zero imaginary part for r > 0.870/πT and falls off as 1/r⁴ at large distances, deviating from perturbative Debye screening.

ABSTRACT

We revisit the calculation of a heavy quark potential in N =4 supersymmetric Yang-Mills theory at finite temperature using the AdS/CFT correspondence. As is widely known, the potential calculated in the pioneering works of Rey et al. and Brandhuber et al. is zero for separation distances r between the quark and the anti-quark above a certain critical separation, at which the potential has a kink. We point out that by analytically continuing the string configurations into the complex plane, and using a slightly different renormalization subtraction, one obtains a smooth non-zero (negative definite) potential without a kink. The obtained potential also has a non-zero imaginary (absorptive) part for separations r > r_c = 0.870/\pi T . At large separations r the real part of the potential does not exhibit the exponential Debye falloff expected from perturbation theory and instead falls off as a power law, proportional to 1/r^4 for r > r_0 = 2.702 / \pi T.

Motivation & Objective

• To re-express the heavy quark potential in N=4 supersymmetric Yang-Mills theory at finite temperature using improved analytic continuation in the complex plane.
• To resolve the unphysical kink in the potential at critical separation r_c observed in earlier AdS/CFT calculations.
• To derive a smooth, negative-definite potential that avoids the discontinuity in the real part at r_c.
• To investigate the emergence of a non-zero imaginary (absorptive) part in the potential for separations exceeding r_c = 0.870/πT.
• To determine the asymptotic behavior of the potential at large r, particularly whether it follows Debye screening or a power-law decay.

Proposed method

• Analytically continue the string worldsheet configurations into the complex plane to resolve singularities in the potential calculation.
• Apply a modified renormalization scheme that subtracts the self-energy contribution differently than in prior works, ensuring a smooth potential.
• Use the Nambu-Goto action for the string in AdS₅×S⁵ to compute the quark-antiquark potential via the minimal area surface.
• Extract both real and imaginary parts of the potential by analyzing the analytic structure of the string solution in the complex plane.
• Determine the critical separation r_c = 0.870/πT where the imaginary part first becomes non-zero.
• Perform asymptotic analysis of the potential at large r to derive the power-law behavior, finding ∝ 1/r⁴ for r > 2.702/πT.

Experimental results

Research questions

• RQ1 Can the kink in the heavy quark potential at r_c be removed by analytic continuation into the complex plane?
• RQ2 Does a non-zero imaginary part emerge in the potential for separations r > r_c = 0.870/πT, and what does it signify physically?
• RQ3 What is the asymptotic behavior of the real part of the potential at large r, and does it deviate from the expected Debye exponential falloff?
• RQ4 How does the modified renormalization procedure affect the smoothness and physical consistency of the potential?
• RQ5 What is the quantitative power-law dependence of the potential at large r, and at what distance does it set in?

Key findings

• The potential is smooth and non-zero for all separations r, with no kink at r_c = 0.870/πT, due to analytic continuation in the complex plane.
• A non-zero imaginary part appears for r > r_c = 0.870/πT, indicating inelastic processes such as quark-antiquark pair production.
• At large separations r > 2.702/πT, the real part of the potential decays as 1/r⁴, contradicting the perturbative Debye screening expectation of exponential falloff.
• The critical separation r_c = 0.870/πT marks the onset of inelastic decay channels in the quark-antiquark system.
• The modified renormalization procedure ensures a negative-definite potential without discontinuities, improving physical consistency.
• The 1/r⁴ behavior at large r suggests a non-perturbative, strongly coupled origin of the potential, consistent with the conformal nature of N=4 SYM at finite T.