[Paper Review] KK modes of $q-$form fields on warped $p-$branes

Chun-E Fu, Yu-Xiao Liu+3 more|arXiv (Cornell University)|Dec 10, 2013

Black Holes and Theoretical Physics71 references3 citations

TL;DR

This paper investigates Kaluza-Klein (KK) modes of free massless q-form fields on warped p-branes with codimension one, focusing on thin and thick anti-de-Sitter (AdS) and de-Sitter (dS) branes. It finds that zero modes of q-form fields with q > (p+2)/2 can be localized on negative tension AdS_{p+1}^- and dS_{p+1}^- branes, while q < p/2 or q > (p+2)/2 can localize on thick AdS_{p+1} and dS_{p+1} branes, respectively, and identifies massive bound KK modes in thick brane models.

ABSTRACT

The Kaluza-Klein (KK) modes of a free massless $q-$form bulk field are investigated on warped $p-$branes with codimension one, including on thin $(p+1)-$dimensional anti-de-Sitter ($AdS_{p+1}$) and de-Sitter ($dS_{p+1}$) branes and thick ones. It is found that the zero mode of the $q-$form field with $q>(p+2)/2$ can be localized on the thin negative tension $AdS_{p+1}^-$ and $dS_{p+1}^-$ branes. For example, a localized $3-$form field will appear on the $AdS_4^-$ and $dS_4^-$ branes, which is different from that on the flat or $dS_{p+1}^+$ branes. While on the thin positive tension $dS_{p+1}^+$ brane, there is a localized zero mode for $q (p+2)/2$ and $q<p/2$ have a localized zero mode on the thick $AdS_{p+1}$ and $dS_{p+1}$ branes, respectively. Furthermore, we also find there are massive bound KK modes for a free massless $q-$form field on some warped thick $p-$branes.

Motivation & Objective

• To analyze the localization of Kaluza-Klein (KK) modes of free massless q-form fields on warped p-branes with codimension one.
• To determine under what conditions zero modes of q-form fields can be localized on thin and thick branes with anti-de-Sitter (AdS) and de-Sitter (dS) geometries.
• To investigate the existence of massive bound KK modes in the context of free massless q-form fields on warped thick p-branes.
• To compare localization behavior on positive and negative tension branes, particularly distinguishing between AdS_{p+1}^+ and AdS_{p+1}^-, and dS_{p+1}^+ and dS_{p+1}^- branes.
• To clarify the role of form rank q and spacetime dimension p in determining localization properties on different brane types.

Proposed method

• Analyzing the Kaluza-Klein decomposition of a free massless q-form field in the bulk of a warped p-brane spacetime with codimension one.
• Solving the field equations for the q-form field in the background of thin and thick branes with AdS_{p+1} and dS_{p+1} geometries.
• Applying boundary conditions and solving the resulting Schrödinger-type equation for the KK modes to determine zero mode and massive bound state solutions.
• Using the warp factor profile to determine the effective potential for the KK modes, especially focusing on the existence of zero modes and bound states.
• Classifying localization conditions based on the form rank q and the brane tension (positive or negative) in both thin and thick brane models.
• Comparing results across different brane types—thin vs. thick, positive vs. negative tension, AdS vs. dS—to identify universal and model-dependent localization behaviors.

Experimental results

Research questions

• RQ1 Under what conditions can a zero mode of a q-form field be localized on a thin negative tension AdS_{p+1}^- or dS_{p+1}^- brane?
• RQ2 How does the form rank q of the q-form field influence the localization of zero modes on thin positive tension dS_{p+1}^+ branes?
• RQ3 Can massive bound Kaluza-Klein modes exist for a free massless q-form field on warped thick p-branes?
• RQ4 What is the role of brane thickness and warp factor profile in enabling localization of q-form zero modes on thick AdS_{p+1} and dS_{p+1} branes?
• RQ5 How do the localization properties of q-form fields differ between positive and negative tension branes in AdS and dS backgrounds?

Key findings

• Zero modes of q-form fields with q > (p+2)/2 can be localized on thin negative tension AdS_{p+1}^- and dS_{p+1}^- branes, such as 3-form fields on AdS_4^- and dS_4^- branes.
• On thin positive tension dS_{p+1}^+ branes, zero modes are localized when q ≤ (p+2)/2 and q < p/2, indicating a different localization condition than on negative tension branes.
• For thick AdS_{p+1} branes, q-form fields with q > (p+2)/2 can exhibit localized zero modes, while for thick dS_{p+1} branes, q < p/2 leads to zero mode localization.
• The presence of a non-trivial warp factor in thick brane models supports the existence of massive bound Kaluza-Klein modes for free massless q-form fields.
• The localization behavior of q-form fields depends critically on the interplay between the form rank q, the brane's tension (sign), and the background geometry (AdS or dS).
• The results reveal a distinct localization pattern on negative tension branes compared to flat or positive tension dS_{p+1}^+ branes, particularly for higher-rank forms like 3-forms on 4-dimensional branes.