[Paper Review] Bulk Gauge Fields in the Bigravity Model

Motoi Tachibana|ArXiv.org|Jan 26, 2001

Black Holes and Theoretical Physics3 citations

TL;DR

This paper investigates bulk U(1) gauge fields in a five-dimensional bigravity model with two positive-tension AdS₄ branes in AdS₅. Unlike the Randall-Sundrum model, the gauge field zero mode is non-constant and acquires a small mass on the brane due to the warped geometry. The model constrains the parameter $kz_0 > 100$ to satisfy experimental bounds on the photon mass, ensuring the zero mode remains light enough to be phenomenologically viable.

ABSTRACT

Motivated by a recently proposed "bigravity" model with two positive tension $AdS_4$ branes in $AdS_5$ by Kogan et al.[hep-th/0011141], we study behavior of bulk gauge field in the model. In this case, the zero mode of the gauge field is not constant but depends on the fifth dimensional coordinate transverse to the brane. The zero mode then becomes massive on the brane. From the physical requirement that the mass must be small, a parameter of the model is constrained. We also discuss the Kaluza-Klein modes.

Motivation & Objective

• To analyze the behavior of bulk U(1) gauge fields in a recently proposed bigravity model with two positive-tension branes in AdS₅.
• To determine the properties of the gauge field zero mode in this background, particularly its mass on the brane.
• To constrain model parameters such that the physical mass of the zero mode remains below current experimental bounds on the photon mass.
• To examine the Kaluza-Klein (KK) mode spectrum and their wavefunction structure in the warped background.

Proposed method

• The five-dimensional action for a U(1) gauge field is derived in a warped AdS₅ background with a specific warp factor $\Omega(w)$, derived from the bigravity model.
• The gauge field is decomposed into zero mode and Kaluza-Klein modes via $A_\mu(x,w) = \sum_n a_\mu^{(n)}(x) \rho^{(n)}(w)$, with the gauge condition $A_w = 0$.
• The equations of motion are reduced to a Schrödinger-like equation: $\partial_w(\Omega \partial_w \rho^{(n)}) + m_n^2 \Omega \rho^{(n)} = 0$, with $\Omega(w)$ given by a trigonometric function of $w$.
• The zero mode solution $\rho^{(0)}(w)$ is found to be non-constant, depending on $w$ through $\sin(\tilde{k}(|w| - \theta))$, leading to a nontrivial profile.
• The effective four-dimensional action is computed by integrating over the fifth dimension, yielding a massive gauge theory with mass $M_{ZM}^2 = I_2 / I_1$, where $I_1$ and $I_2$ are $w$-integrals of $\Omega \rho^{(0)2}$ and $\Omega (\partial_w \rho^{(0)})^2$, respectively.
• For symmetric configurations, the mass is expressed in terms of $kz_0$, and the asymptotic behavior $M_{ZM}^2 \approx 8e^{-2x}/x \cdot k^2$ is derived for large $x = kz_0$.

Experimental results

Research questions

• RQ1 How does the zero mode of a bulk U(1) gauge field behave in the bigravity model with two positive-tension branes?
• RQ2 Why is the zero mode not constant in this model, unlike in the Randall-Sundrum scenario?
• RQ3 What constraints does the requirement of a small gauge boson mass impose on the model parameters, particularly $kz_0$?
• RQ4 How are the Kaluza-Klein modes structured in this warped background, and what is their wavefunction form?
• RQ5 Can the effective four-dimensional theory of the zero mode be made canonically normalized, and what is the resulting physical mass?

Key findings

• The zero mode of the gauge field is not constant in the fifth dimension but depends on $w$ through a sinusoidal function, leading to a nontrivial profile $\rho^{(0)}(w) \propto \sin(\tilde{k}(|w| - \theta))$.
• The zero mode acquires a physical mass on the brane due to the non-constant profile, with the mass squared given by $M_{ZM}^2 = I_2 / I_1$, where $I_1$ and $I_2$ are integrals over the fifth dimension.
• For the symmetric configuration with $w_L = 2\theta$, the mass is approximately $M_{ZM}^2 \approx 8e^{-2x}/x \cdot k^2$ at large $x = kz_0$, showing exponential suppression.
• To satisfy the experimental bound $M_{\text{exp}} < 2 \times 10^{-16}$ eV, the condition $kz_0 > 100$ is required, ensuring the zero mode remains light.
• The Kaluza-Klein modes satisfy a differential equation involving hypergeometric functions, with solutions $\rho^{(n)}(w)$ expressed in terms of $F(\alpha_n, \beta_n, 1/2; \sin^2(\tilde{k}(|w| - \theta)))$ and a second term with absolute value.
• The model avoids ghost fields due to the absence of negative tension branes, and the zero mode mass is naturally small due to the exponential suppression in $kz_0$, making it viable for phenomenology.