field within the five dimensional spacetime.

Fermion localization in F(R) theory
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Consider a bulk massive fermion field propagating in a background spacetime characterized by the action in eqn. (\[actionF(R)\]). The lagrangian for the Dirac fermions is given by $$\mathcal{L}_{Dirac} = \sqrt{-G} \bigg[\bar{\Psi}i\Gamma^{a}D_{a}\Psi - m_5\bar{\Psi}\Psi\bigg]
   \label{1}$$ Using the metric solution in $F(R)$ model presented in eqn.(\[grav.sol1.F(R)\_final\]), the above lagrangian can be written as,

$$\mathcal{L}_{Dirac} = e^{-4A(y)} e^{-\frac{5\kappa}{2\sqrt{3}}\big(<\Phi>+\xi(y)\big)} 
   \bigg[\bar{\Psi}i\Gamma^{a}D_{a}\Psi - m_5\bar{\Psi}\Psi\bigg]
   \nonumber$$

where $a=(x^{\mu},y)$ are the bulk coordinates, $\Psi=\Psi(x^{\mu},y)$ is the bulk fermion field and $m_5$ is its mass. $\Gamma^{a} = \bigg(e^{A(y)}e^{\frac{\kappa}{2\sqrt{3}}\big(<\Phi>+\xi(y)\big)}\gamma^{\mu}, 
  -i\gamma^{5}e^{\frac{\kappa}{2\sqrt{3}}\big(<\Phi>+\xi(y)\big)}\bigg)$ denotes the five dimensional gamma matrices where $\gamma^{\mu}$ and $\gamma^{5}$ represent 4D gamma matrices in chiral representation. Curved gamma matrices obey the Clifford algebra i.e. $[{\Gamma^{a},\Gamma^{b}] = 2G^{ab}}$. The covariant derivative $D_{a}$ can be calculated by using the metric in eqn. (\[grav.sol1.F(R)\_final\]) and is given by $$\begin{aligned}
   &D_{\mu} = \partial_{\mu} +  \frac{i}{2}\gamma_{\mu}\gamma^{5}e^{\frac{\kappa}{2\sqrt{3}}\big(<\Phi>+\xi(y)\big)}
   \bigg(A'(y)+\frac{\kappa}{2\sqrt{3}}\xi'(y)\bigg)\nonumber\\
   &D_{5} = e^{\frac{\kappa}{2\sqrt{3}}\big(<\Phi>+\xi(y)\big)}\partial_{y}
   \nonumber
  \end{aligned}$$ Using this set up, the Dirac lagrangian $\mathcal{L}_{Dirac}$ turns out to be, $$\begin{aligned}
   \mathcal{L}_{Dirac}&=&e^{-4A(y)}e^{-\frac{5\kappa}{2\sqrt{3}}\big(<\Phi>+\xi(y)\big)} \bar{\Psi}\bigg[ie^{A(y)}\nonumber\\
   &e&^{\frac{\kappa}{2\sqrt{3}}\big(<\Phi>+\xi(y)\big)}\gamma^{\mu}\partial_{\mu} 
   + \gamma^{5}\bigg[e^{\frac{\kappa}{2\sqrt{3}}\big(<\Phi>+\xi(y)\big)}\nonumber\\
   &\big(&\partial_{y}-2A'(y)-\frac{\kappa}{\sqrt{3}}\xi'(y)\big)\bigg] - m_5\bigg]\Psi
   \label{dirac lagrangian}
  \end{aligned}$$

We decompose the five dimensional spinor via Kaluza-Klein (KK) mode expansion as $\Psi(x^{\mu},y) = \sum \chi^n(x^{\mu})\zeta^n(y)$, where the superscript $n$ denotes the nth KK mode. $\chi^n(x^{\mu})$ 