[Paper Review] Linear Confinement of Quarks from Supergravity

Girma Hailu|arXiv (Cornell University)|Jul 28, 2011

Black Holes and Theoretical Physics4 references4 citations

TL;DR

This paper proposes a supergravity background in type IIB string theory that realizes linear confinement of quarks in four dimensions, using a warped geometry with D7-branes at the infrared boundary. The string connecting a quark-antiquark pair stretches toward the IR, yielding a potential energy that increases linearly with separation, reproducing Regge-like mass spectra observed in QCD.

ABSTRACT

A supergravity background that produces linear confinement of quarks in four dimensions is presented.

Motivation & Objective

• To construct a supergravity background that reproduces linear confinement of quarks in four-dimensional gauge theories.
• To realize the Regge-type mass spectra observed in QCD within a string-theoretic framework.
• To demonstrate that the energy of a quark-antiquark pair increases linearly with separation in a consistent supergravity background.
• To establish a gravity dual for pure N=1 supersymmetric SU(N) gauge theory with confinement, chiral symmetry breaking, and running coupling.
• To provide a mechanism for tuning the slope of Regge trajectories via the IR regularization parameter ρ₀.

Proposed method

• Utilizes type IIB supergravity on a background with radial coordinate r and compact internal spaces: C¹/Z₂ × T²/Z₂ × T²/Z₂.
• Imposes D7-branes at r=1 (IR boundary) to source color charge, while quarks and antiquarks are placed at the UV boundary (r = exp(2π/(3gₛN))).
• Applies the Nambu-Goto action for the string worldsheet to compute the energy of a quark-antiquark pair in curved spacetime.
• Solves the string worldsheet equation of motion with boundary conditions at the UV boundary (x = ±L/2) and a cutoff at ρ₀ > 0 to regularize divergence.
• Uses the warp factor sech(u) with u defined via ρ = 3 ln r and sech(u) = √(1 - (gₛN/(2π)ρ)²), which governs the geometry and string tension.
• Derives the string profile x(ρ) analytically, showing logarithmic stretching toward the IR (ρ → 0), with energy proportional to L for large separation.

Experimental results

Research questions

• RQ1 Can a supergravity background in type IIB string theory produce linear confinement of quarks in four dimensions?
• RQ2 Does the string configuration between quark-antiquark pairs in this background yield a linear potential energy with separation?
• RQ3 How does the geometry of the background, particularly the warp factor and radial dependence, lead to confinement?
• RQ4 Can the slope of the Regge trajectories in this model be tuned by adjusting the IR cutoff ρ₀?
• RQ5 Is this background a consistent gravity dual for N=1 supersymmetric Yang-Mills theory with nonperturbative features like confinement and chiral symmetry breaking?

Key findings

• The string connecting a quark and antiquark in the background stretches toward the infrared (IR) boundary, minimizing the Wilson loop area.
• The energy of the quark-antiquark pair increases linearly with separation L, confirming linear confinement.
• The analytic solution for the string profile x(ρ) shows logarithmic divergence as ρ → 0, indicating infinite stretching in the IR, which is regularized by ρ₀ > 0.
• For large L, the energy scales linearly with L, with the slope determined by the 't Hooft coupling gₛN and the IR cutoff ρ₀.
• The potential energy is proportional to L, with the proportionality constant depending on gₛN and ρ₀, allowing tuning of the Regge trajectory slope.
• The background realizes a consistent gravity dual for N=1 supersymmetric SU(N) gauge theory, including confinement, chiral symmetry breaking, and running coupling, with features matching QCD phenomenology.