Localization and quasilocalization of a spin-$1/2$ fermion field on a two-field thick braneworld

Localization of a spin-$1/2$ fermion on the braneworld is an important and interesting problem. It is well known that a five-dimensional free massless fermion $\mathrm{\Psi }$ minimally coupled to gravity cannot be localized on the Randall-Sundrum braneworld. In order to trap such a fermion, the coupling between the fermion and bulk scalar fields should be introduced. In this paper, localization and quasilocalization of a bulk fermion on the thick braneworld generated by two scalar fields (a kink scalar $\varphi$ and a dilaton scalar $\pi$) are investigated. Two types of couplings between the fermion and two scalars are considered. One coupling is the usual Yukawa coupling $-\eta \overline{\mathrm{\Psi }}\varphi \mathrm{\Psi }$ between the fermion and kink scalar, another one is $\lambda \overline{\mathrm{\Psi }}{\mathrm{\Gamma }}^M{\partial }_M\pi {\gamma }^5\mathrm{\Psi }$ between the fermion and dilaton scalar. The left-chiral fermion zero mode can be localized on the brane, and both the left- and right-chiral fermion massive Kaluza-Klein modes may be localized or quasilocalized. Hence the four-dimensional massless left-chiral fermion and massive Dirac fermions, whose lifetime is infinite or finite, can be obtained on the brane.

Figure 1

The shapes of the effective potentials of the left- and right-chiral fermions $V_{L,R}(z)$ for different values of the parameter $\eta$. The parameter $\eta$ is set to $\eta =0$ for the thick line, $\eta =5$ for the thin line, and $\eta =10$ for the dashed line. Other parameters are set to $v=1$, $a=1$, and $\lambda =5$. (a) $V_L(z),\lambda =5$ and (b) $V_R(z),\lambda =5$.

Figure 2

The shapes of the effective potentials of the left- and right-chiral fermions $V_{L,R}(z)$ for different values of the parameter $\lambda$. The parameter $\lambda$ is set to $\lambda =0$ for the thick line, $\lambda =5$ for the thin line, and $\lambda =10$ for the dashed line. Other parameters are set to $v=1$, $a=1$, and $\eta =5$. (a) $V_L(z),\eta =5$ and (b) $V_R(z),\eta =5$.

Figure 3

The shapes of the zero mode of the left-chiral fermion $L_0(z)$ for different values of the parameter $\eta$. The coupling coefficient $\eta$ is set to $\eta =0$, $\eta =5$, and $\eta =10$ for thick, thin, and dashed lines, respectively. Other parameters are set to $v=1$, $a=1$, $C_L=1$ and $\lambda =5$.

Figure 4

The shapes of the zero mode of the left-chiral fermion $L_0(z)$ for different values of the parameter $\lambda$. The coupling coefficient $\lambda$ is set to $\lambda =0$, $\lambda =5$, and $\lambda =10$ for thick, thin, and dashing lines, respectively. Other parameters are set to $v=1$, $a=1$, $C_L=1$, and $\eta =5$. (a) $-3\le z\le 3$ and (b) $10\le z\le 30$.

Figure 5

The shapes of the relative probability $P_{L,R}(m^2)$ for the (a) left- and (b) right-chiral fermion resonant KK modes, with the parameters $a=1$, $v=1$, $\lambda =5$, and $\eta =5$.

Figure 6

The shapes of the wave functions for the left-chiral fermion KK modes $L_n$, with the parameters $a=1$, $v=1$, $\lambda =5$, and $\eta =5$. (a) $n=0$, bound state; (b) $n=1$, resonant state; (c) $n=2$, resonant state; and (d) $n=3$, resonant state.

Figure 7

The shapes of the wave functions for the right-chiral fermion KK modes $R_n$, with the parameters $a=1$, $v=1$, $\lambda =5$, and $\eta =5$. (a) $n=1$, resonant state; (b) $n=2$, resonant state; and (c) $n=3$, resonant state.

Figure 8

The shapes of the wave functions for the left-chiral fermion KK modes $L_n$, with the parameters $a=1$, $v=1$, $\lambda =15$, and $\eta =5$. (a) $n=0$, bound state; (b) $n=1$, bound state; (c) $n=2$, resonant state; (d) $n=3$, resonant state; (e) $n=4$, resonant state; and (f) $n=5$, resonant state.

Figure 9

The mass spectra and effective potentials for the (a) left- and (b) right-chiral fermion KK modes. The blue lines represent the potentials $V_L$ and $V_R$, the red lines represent the square of the mass for bound KK modes, and the red dashed lines represent the square of the mass for resonant KK modes. The parameters are set to $a=1$, $v=1$, $\lambda =15$, and $\eta =5$.