[Paper Review] Fermions on Asymmetric Bloch Branes

Zhen‐Hua Zhao, Yuxiao Liu+1 more|arXiv (Cornell University)|Nov 13, 2009

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper investigates fermion localization on asymmetric Bloch branes generated by two scalar fields, $φ$ and $χ$, using Yukawa-type couplings $\eta\bar{\Psi}F(\phi,\chi)\Psi$. It finds that only specific couplings—particularly $\bar{\Psi}(\chi - \phi)\Psi$ and $\bar{\Psi}\chi^{-2a^2/9}(\chi - \phi)\Psi$—support normalizable zero modes for left-handed fermions under certain coupling conditions, while right-handed zero modes remain non-normalizable; however, massive bound states emerge for large coupling constants.

ABSTRACT

In this paper we investigate the localization and mass spectrum of fermions on asymmetric Bloch branes generated by two scalars $\phi$ and $\chi$. The internal structure of these branes is very different from other ones. In order to trap fermions on the asymmetric Bloch branes, the couplings with background scalars $\eta\bar{\Psi}F(\phi,\chi)\Psi$ are introduced, where $\eta$ is the coupling constant and $F(\phi,\chi)$ is a function of $\phi$ and $\chi$. We find that the usual couplings $F(\phi,\chi)=\phi+\chi$ and $F(\phi,\chi)=\phi\chi$ considered in the literature do not support the localization of 4-dimensional fermions on the branes. While, for the case $\eta\bar{\Psi}(\chi-\phi)\Psi$, which is turned out to be the kink-fermion coupling corresponding to one-scalar-generated brane scenarios, the zero mode of left-handed fermions is normalizable and can be trapped on one of the branes under the condition $\eta>2 a^2/9 $ with $a$ a parameter appeared in the scalar potential, but the right-handed fermion zero mode is non-normalizable. For the coupling $\bar\Psi\chi^{-2a^2/9}(\chi-\phi)\Psi$, the zero mode of left-handed fermions is normalizable for arbitrary $\eta>0$. Again, there does not exist localized zero mode of right-handed fermions. However, discrete bound massive fermions on the branes are found for large coupling constant.

Motivation & Objective

• To explore the localization of 4D fermions on asymmetric Bloch branes generated by two scalar fields $\phi$ and $\chi$.
• To determine which fermion-scalar couplings $\eta\bar{\Psi}F(\phi,\chi)\Psi$ can trap fermions on the brane.
• To analyze the mass spectrum of fermions, particularly the existence and normalizability of zero modes and massive bound states.
• To compare the behavior of left-handed and right-handed fermions under different coupling functions.

Proposed method

• Construct asymmetric Bloch brane solutions using a two-scalar system with a potential parameterized by $a$, leading to non-symmetric brane profiles.
• Introduce fermion-scalar couplings of the form $\eta\bar{\Psi}F(\phi,\chi)\Psi$, with $F(\phi,\chi)$ chosen as $\phi + \chi$, $\phi\chi$, $\chi - \phi$, and $\chi^{-2a^2/9}(\chi - \phi)$.
• Solve the fermion zero mode equation in the background of the asymmetric brane to assess normalizability.
• Apply the standard normalization condition for zero modes and analyze the behavior of the wave function at infinity.
• Use numerical and analytical techniques to identify discrete massive bound states for large coupling constants.
• Compare results across different coupling functions to determine which support localized fermionic states.

Experimental results

Research questions

• RQ1 Which fermion-scalar coupling functions $F(\phi,\chi)$ allow for the localization of 4D fermions on asymmetric Bloch branes?
• RQ2 Under what conditions is the zero mode of left-handed fermions normalizable on these branes?
• RQ3 Why do right-handed fermion zero modes fail to be normalizable despite left-handed ones being localized?
• RQ4 Can massive bound states of fermions emerge on the brane, and under what coupling strength conditions?
• RQ5 How does the coupling $\bar{\Psi}\chi^{-2a^2/9}(\chi - \phi)\Psi$ improve localization compared to simpler couplings?

Key findings

• The coupling $\eta\bar{\Psi}(\chi - \phi)\Psi$ supports a normalizable zero mode for left-handed fermions only when $\eta > 2a^2/9$, with the right-handed zero mode remaining non-normalizable.
• The coupling $\bar{\Psi}\chi^{-2a^2/9}(\chi - \phi)\Psi$ ensures a normalizable zero mode for left-handed fermions for all $\eta > 0$, indicating improved localization behavior.
• Right-handed fermion zero modes are non-normalizable in all considered coupling scenarios, indicating a fundamental asymmetry in fermion localization.
• Discrete massive bound states of fermions are found to exist on the brane when the coupling constant $\eta$ is sufficiently large.
• The couplings $F(\phi,\chi) = \phi + \chi$ and $F(\phi,\chi) = \phi\chi$ fail to localize 4D fermions due to insufficient potential trapping.