[Paper Review] On two-point correlation functions in AdS/QCD

Alexander Krikun|Jan 28, 2008

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper computes the two-point 'left-right' correlation function in the hard wall AdS/QCD model using a perturbative expansion in inverse momentum at large $Q^2$, solving the 5D equations of motion up to second order. It finds that the leading-order contribution to the correlator is independent of the 't Hooft coupling ($\lambda'$), consistent with the strong coupling regime of QCD, and confirms the model's viability for describing chiral symmetry breaking via quark condensate and quark mass effects.

ABSTRACT

In this paper we study the chiral symmetry breaking in the hard wall AdS/QCD model. We solve the equations of motion up to the second order at large momentum and obtain the first few terms in the expansion of the "left-right" correlator, which is the measure of symmetry breaking. We study the dependence on AdS radius to get the result as the series in t'Hooft constant.

Motivation & Objective

• To compute the 'left-right' two-point correlation function $\langle LR\rangle = \langle VV\rangle - \langle AA\rangle$ in the hard wall AdS/QCD model at large momentum transfer.
• To analyze the dependence of the correlator on the AdS curvature radius and the 't Hooft coupling $\lambda'$, reconstructing the power expansion in $\lambda'$.
• To investigate how chiral symmetry breaking, induced by quark mass ($m_q$) and quark condensate ($\langle \bar{q}q\rangle$), is encoded in the 5D effective action via classical solutions.
• To validate the hard wall model's ability to reproduce key features of QCD chiral symmetry breaking in the strong coupling limit.

Proposed method

• Solves the 5D equations of motion for vector and axial-vector fields perturbatively in inverse powers of momentum ($1/Q^2$) using a Green's function method derived in the appendix.
• Applies the AdS/CFT prescription: correlation functions in 4D QCD are obtained as functional derivatives of the 5D effective action evaluated on classical field configurations with fixed boundary values.
• Uses a Green function for the modified Bessel equation with specific boundary conditions: Neumann at $z = z_m$ and regularity at $z = x_0 \to 0$.
• Expands the solution in powers of $1/Q^2$ up to second order, extracting the $\mathcal{O}(1/Q^4)$ terms in the correlator.
• Relates the AdS radius $R$ to the 't Hooft coupling $\lambda' = N_c g_{YM}^2$ via the standard AdS/CFT relation $R^4/(4\pi\alpha'^2) = \lambda'$.
• Matches the parameters in the 5D action (e.g., $\Sigma$, $m$) to QCD condensates and quark masses by analyzing the boundary behavior of the scalar field $X$.

Experimental results

Research questions

• RQ1 What is the structure of the 'left-right' two-point correlation function $\langle LR\rangle$ in the hard wall AdS/QCD model at large $Q^2$?
• RQ2 How does the leading-order behavior of the correlator depend on the 't Hooft coupling $\lambda'$, and does it match expectations from QCD sum rules?
• RQ3 Can the hard wall model reproduce the correct scaling of chiral symmetry breaking effects (via $\langle \bar{q}q\rangle$ and $m_q$) in the strong coupling regime?
• RQ4 What is the role of the AdS curvature radius $R$ in determining the momentum dependence and coupling strength of the correlation functions?
• RQ5 Does the model correctly capture the $N_c$ scaling of the correlator, consistent with large-$N_c$ QCD?

Key findings

• The leading-order term in the 'left-right' correlator is independent of the 't Hooft coupling ($\lambda'^0$), contrary to the behavior in QCD sum rules which predict a linear dependence ($\lambda'^1$) at weak coupling.
• The result is consistent with the strong coupling regime of QCD, where AdS/QCD computations are expected to be valid, and explains the discrepancy with sum rules as a consequence of different coupling regimes.
• The first non-trivial correction to the correlator is proportional to the quark condensate $\sigma$ and the quark mass $m$, confirming that chiral symmetry breaking is correctly encoded in the model.
• The $\mathcal{O}(1/Q^4)$ term in the correlator is found to be proportional to $\sigma$, showing that the condensate directly contributes to the subleading behavior.
• The model correctly reproduces the $N_c$ scaling: each term in the correlator scales as $\mathcal{O}(N_c)$, as expected for current correlators in large-$N_c$ QCD.
• The Green function method successfully computes the classical solutions for the vector and axial-vector fields, enabling the evaluation of the correlation functions via the standard AdS/CFT recipe.