On the matching method and the Goldstone theorem in holography

TL;DR

This paper analyzes scalar field transitions in AdS space, deriving boundary propagators and confirming the Goldstone theorem through higher-order matching methods in holography.

Contribution

It extends the matching method to higher orders and demonstrates the Goldstone theorem in the context of holographic scalar fields.

Findings

Derived the boundary propagator for scalar fields in AdS.

Confirmed the Goldstone theorem at next-to-leading order.

Showed a simple pole at zero momentum in the propagator.

Abstract

We study the transition of a scalar field in a fixed $AdS_{d+1}$ background between an extremum and a minimum of a potential. We compute analytically the solution to the perturbation equation for the vev deformation case by generalizing the usual matching method to higher orders and find the propagator of the boundary theory operator defined through the AdS-CFT correspondence. We show that, contrary to what happens at the leading order of the matching method, the next-to-leading order presents a simple pole at $q^2=0$ in accordance with the Goldstone theorem applied to a spontaneously broken dilatation invariance.