[Paper Review] A HHL 3-point correlation function in the \eta-deformed AdS_5 x S^5

Changrim Ahn, P. Bozhilov|arXiv (Cornell University)|Dec 20, 2014

Black Holes and Theoretical Physics13 references3 citations

TL;DR

This paper computes the exact semiclassical three-point correlation function between two finite-size giant magnon states and a light dilaton operator with zero momentum in the η-deformed AdS₅×S⁵ background. Using the string Lagrangian on the deformed S² and finite-size corrections, the authors derive a structure constant in terms of complete elliptic integrals, proving consistency with the giant magnon's conformal dimension and providing the leading finite-J₁ correction.

ABSTRACT

We derive the 3-point correlation function between two giant magnons heavy string states and the light dilaton operator with zero momentum in the \\eta-deformed AdS_5 x S^5 valid for any J_1 and \\eta in the semiclassical limit. We show that this result satisfies a consistency relation between the 3-point correlation function and the conformal dimension of the giant magnon. We also provide a leading finite J_1 correction explicitly.

Motivation & Objective

• To compute the three-point correlation function between two finite-size giant magnon states and a light dilaton operator in the η-deformed AdS₅×S⁵ background.
• To establish consistency between the computed structure constant and the conformal dimension of the finite-size giant magnon solution.
• To derive the leading finite-J₁ correction to the structure constant in the semiclassical limit.
• To validate the result using a consistency check involving the derivative of the conformal dimension with respect to the coupling constant.

Proposed method

• The structure constant is computed via the semiclassical approximation, evaluating the light operator's vertex at the classical string configuration of the giant magnon.
• The string Lagrangian on the η-deformed S² is used to compute the relevant terms, incorporating the deformation parameter η through the parameter ˜η = 2η/(1−η²).
• Finite-size effects are introduced by replacing infinite integrals over σ with finite integrals over θ from θ_min to θ_max, parameterized by the angular momentum J₁.
• The solution involves solving the string equations of motion in conformal gauge with an ansatz for φ₁ and θ, leading to a system of differential equations in terms of ξ = ασ + βτ.
• The resulting expression for the structure constant is expressed in terms of complete elliptic integrals of the first and third kind.
• A consistency check is performed by verifying that the derivative of the conformal dimension Δ with respect to the coupling g matches the derivative of the structure constant, confirming the result's validity.

Experimental results

Research questions

• RQ1 What is the exact form of the three-point correlation function between two finite-size giant magnons and a dilaton operator in the η-deformed AdS₅×S⁵ background?
• RQ2 How does the structure constant depend on the deformation parameter η and the angular momentum J₁ in the semiclassical regime?
• RQ3 Does the computed structure constant satisfy the consistency condition derived from the conformal dimension of the giant magnon?
• RQ4 What is the leading finite-J₁ correction to the structure constant in the η-deformed background?
• RQ5 How does the result reduce to the undeformed AdS₅×S⁵ case when η → 0?

Key findings

• The three-point structure constant is derived as an exact expression in terms of complete elliptic integrals of the first and third kind, valid for any J₁ and η in the semiclassical limit.
• The result satisfies the consistency condition between the structure constant and the conformal dimension Δ of the finite-size giant magnon, verified by comparing derivatives with respect to the coupling constant.
• The leading finite-J₁ correction is explicitly computed, showing exponential suppression in J₁/g, consistent with known behavior in the undeformed case.
• In the limit η → 0, the result reduces to the known structure constant for the undeformed AdS₅×S⁵, confirming consistency with previous results.
• The structure constant is expressed as C₃ ≈ (16c₆/3) × sin(p/2) × [1 − 4 sin(p/2)(sin(p/2) + J₁/g) × exp(−J₁/(g sin(p/2)) − 2)], valid for small η.
• A Mathematica code is provided in the appendix to numerically verify the consistency condition between the conformal dimension and the structure constant.