Localization mechanism of the Kalb-Ramond field on brane with codimension-two

TL;DR

This work studies the localization of a six-dimensional Kalb-Ramond (KR) field on codimension-2 branes by performing a general Kaluza-Klein decomposition into one 4D KR field, two 4D vector fields, and one 4D scalar field. The 4D masses are determined by the two extra dimensions, with $m_{ ext{KR}}^2=m_1^2+m_2^2$, $m_{ ext{C}}^2=-oldsymbol{\lambda}_7$, and $m_{ ext{D}}^2=-oldsymbol{\lambda}_{12}$, while the scalar remains massless, and the five interaction sectors among the 4D fields are identified. In the minimal coupling scenarios, only the 4D scalar can be localized for $ ext{R}_1 imes ext{R}_1$, and the vector from the compact dimension along with the scalar can localize for $ ext{R}_1 imes ext{S}_1$. Introducing a background scalar coupling $F(oldsymbol{\phi})$ in a concrete thick-brane model, the authors show that localization of all four 4D components occurs when $t>rac{v^2}{12}$, with the zero mode $w_1^{(0,0)}(z) o F(oldsymbol{\phi})^{1/2}$ and volcano-like potentials enabling resonant KK modes near the brane, whose number grows with $t$. These results illuminate how higher-codimension brane setups and scalar couplings control KR field localization and yield observable resonant spectra.

Abstract

The $2$-form Kalb-Ramond (KR) field, together with the metric tensor and dilaton, arises as one of the massless excitation mode of a closed string. Subsequently, this field plays an important role in both string theory and field theory. In this paper, we investigate the localization of the KR field on the brane with codimension-2. A general Kaluza-Klein (KK) decomposition is adopted, wherein the six-dimensional KR field is expanded into one four-dimensional (4D) KR field, two 4D vector fields, and one 4D scalar field. Then, for the case of the extra dimensions $\mathcal{R}_1\times\mathcal{R}_1$, only the 4D scalar field can be localized on the brane. In contrast, for the case of extra dimensions $\mathcal{R}_1\times\mathcal{S}_1$, one 4D vector field and the 4D scalar field can be localized on the brane at the same time. In both cases, the mass of the 4D scalar field remains zero. Next, we examine the localization of the KR field within a specific six-dimensional brane model with extra dimensions $\mathcal{R}_1\times\mathcal{S}_1$. By introducing the background scalar coupling, we show that the 4D KR field, along with the other three 4D fields, can be localized on the brane under the condition of the coupling parameter $t>v^2/12$. Additionally in this case, for both the 4D KR field and the one 4D vector field which acquires its mass from the non-compact extra dimension, the resonant KK modes could exist near the origin of this extra dimension.

Figures (3)

• Figure 1: The effective potential $V(z)$ and the zero mode $w_1^{(0,0)}(z)$ with parameters $v=2$, $k=1$ and $t=10,15,20$.
• Figure 2: The mass spectra, the effective potential $V(z)$, and corresponding relative probability $P$ with the coupling parameter $t=10,15,20$. The potential $V(z)$ for the black line, the zero mode for the grey line, the even parity resonant KK modes for the blue lines, and the odd parity resonant KK modes for the red lines. The parameters are set as $v=2$ and $k=1$.
• Figure 3: The shapes of the zero mode and the resonance modes with different $m_1^{2}$. The parameters are set as $v=2,k=1$, and $t=15$.