Fermion localization in braneworld teleparallel f(T,B) gravity

Braneworld scenarios rs ; rs2 , show a new viewpoint of spacetime and enables a new approaches to explain a large number of outstanding issues such as the hierarchy problem rs2 , the cosmological problem cosmologicalconstant , the nature of dark matter darkmatter and dark energy. Furthermore, by assuming a warped geometry, the propagation of the gravitational field Csaki1 ; Rosa2020 and the gauge field, Kehagias , as well as fermionic fields Almeida2009 , are governed by the bulk curvature in general relativity (GR). An equivalent theory is the known teleparallel equivalent of general relativity (TEGR) Hayashi1979 ; deAndrade1997 ; andrade2000 ; ferraro2007 ; lobo2012 ; Aldrovandi ; cai2016 ; koi2020 ; olmo2020 , that is constructed in the Weitzenböck manifold, which has vanishing curvature but nonvanishing torsion. Equivalence with GR is provided by the ratio of the torsion $T$ scalar and the Ricci $R$ scalar, which is the boundary term $B$, making TEGR have the same field equations as GR.

The localization mechanism employed for matter fields living in a 5D braneworld scenarios has been the subject of many studies. The study of fermion localization on branes is rich and interesting, yet the most popular method for the localization of fermions is formulated in a rather speculative way Almeida2009 ; RandjbarDaemi2000 ; Liu2009 ; Liu2009a ; Liu2008 ; Liu2009b ; Liu2008b ; Obukhov2002 ; Ulhoa2016 . The same is true in 6D Dantas2013 ; Sousa2014 ; Dantas . This is because of the freedom one has to propose the Yukawa coupling term. The location of the fermion was studied in several modified gravity models, such as gravity $f(R)$ Mitra2017 ; Buyukdag2018 ; Wang2019 and gravity $f(T)$ Yang2012 ; Yang2017 .

A new teleparallel gravity model is the $f(T,B)$ gravity, where $B$ is the boundary term Bahamonde2015 ; Wright2016 ; Bahamonde2016 , that has attracted a lot of attention due this model features have, as well as good agreement with observational data to describe the accelerated expansion of the universe Franco2020 ; EscamillaRivera2019 , and their significant results in cosmological perturbations and thermodynamics, and dark energy, and gravitational waves Bahamonde2016a ; Caruana2020 ; Pourbagher2020 ; Bahamonde2020a ; Azhar2020 ; Bhattacharjee2020 ; Abedi2017 . Furthermore, the gravity $f(T,B)$ was studied in a brane scenario, where it was possible to observe that the additional term $B$ induces changes on the energy density causing a split in the brane, also changing the gravitational perturbations Allan .

Inspired on the results obtained in Yang2012 ; Allan , we investigate the issue of fermion localization in $f(T,B)$ gravity. In section (2) we review the main definitions of the teleparallel $f(T,B)$ theory and build the respective braneworld. In the section (3), we obtain the solutions of the system and we examined the energy density components of the brane. The section (4) deals with the fermionic sector of the model using the Yukawa coupling. Finally, additional comments and results are discussed in section (5).

In teleparallel gravity, the vielbein, $h^{\overline{M}}\ _{M}$ (rather than the metric) are the actual gravitational dynamic variables. We will use the latin letters ($M,N,Q,...=0,1,2,3,4$ ) for the indices related to the bulk, and barred indices ($\overline{M},\overline{N},\overline{Q},...=0,1,2,3,4$ ) for indices related to tangent space. We assume a mostly plus metric signature, i.e., $diag(-1,1,1,1,1)$.

The relevant connection for TEGR is the so-called Weitzenböck connection. An important feature of this connection is that the corresponding spin connection vanishes identically. Thus, the Weitzenböck connection is represented as $\widetilde{\Gamma}^{P}\ _{NM}=h_{\overline{M}}\ ^{P}\partial_{M}h^{\overline{N}}\ _{M}$, that is, $\nabla_{Q}h^{\overline{M}}\ _{M}\equiv\partial_{Q}h^{\overline{M}}\ _{M}-\widetilde{\Gamma}^{P}\ _{QM}h^{\overline{M}}\ _{P}=0$, which is called the condition of absolute parallelism Aldrovandi . The Weitzenböck and Levi-Civita connections are related by

$\displaystyle\widetilde{\Gamma}^{P}\ _{NM}=\Gamma^{P}\ _{NM}+K^{P}\ _{NM},$

(1)

where $\Gamma^{P}\ _{NM}$ is the Levi-Civita connection of general relativity, and

$\displaystyle K^{P}\ _{NM}=(T_{N}\ ^{P}\ {}_{M}+T_{M}\ ^{P}\ {}_{N}-T^{P}\ _{NM})/2$

(2)

is defined as the contortion tensor of the Weitzenböck connection Aldrovandi . We take the torsion as $T^{P}\ _{NM}=\widetilde{\Gamma}^{P}\ _{MN}-\widetilde{\Gamma}^{P}\ _{NM}$, and we also define a dual torsion tensor, known as a superpotential $S_{P}\ ^{NM}=(K^{NM}\ _{P}-\delta^{M}_{P}T^{QN}\ _{Q}+\delta^{N}_{P}T^{QM}\ _{Q})/2$ Aldrovandi . Therefore, the Lagrangian of TEGR reads

$\displaystyle\mathcal{L}=-hT/4,$

(3)

where $T=T_{PMN}S^{PMN}$ is the torsional scalar, $h=\sqrt{-g}$ and $c^{4}/4\pi G=1$ for simplicity Aldrovandi . On the other hand, the Riemann tensor in the Levi-Civita connection

$\displaystyle R^{P}\ _{MQN}=\partial_{Q}\Gamma^{P}\ _{MN}-\partial_{N}\Gamma^{P}\ _{MQ}+\Gamma^{P}\ _{SQ}\Gamma^{S}\ _{MN}-\Gamma^{P}\ _{SN}\Gamma^{S}\ _{MQ}.$

(4)

From the relation between the Weitzenböck connection and the Levi-Civita connection given by Eq.(1), one can write the Riemann tensor in the form

$\displaystyle R^{P}\ _{MQN}=\nabla_{N}K^{P}\ _{MQ}-\nabla_{Q}K^{P}\ _{MN}+K^{P}\ _{SN}K^{S}\ _{MQ}-K^{P}\ _{SQ}K^{Q}\ _{MN},$

(5)

whose associated Ricci tensor can then be written as

$\displaystyle R_{MN}=\nabla_{N}K^{P}\ _{MP}-\nabla_{P}K^{P}\ _{MN}+K^{P}\ _{SN}K^{S}\ _{MP}-K^{P}\ _{SP}K^{S}\ _{MN}.$

(6)

Using the relations $K^{(MN)S}=T^{M(NS)}=S^{M(NS)}$ and considering that $S^{M}\ _{PM}=2K^{M}\ _{PM}=-2T^{M}\ _{PM}$ along with Eq. (2), one can get BLi2010 ; BLi2011 ; Sotiriou2010

$\displaystyle R_{MN}=-\nabla^{P}S_{NPM}-g_{MN}\nabla^{P}T^{S}\ _{PS}-S^{PS}\ _{N}\ K_{SPN}.$

(7)

In turn, Ricci scalar is

$\displaystyle R=-T-2\nabla^{M}T^{N}\ _{MN}.$

(8)

Thus we can identify the boundary term

$\displaystyle B\equiv-2\nabla^{M}T^{N}\ _{MN}=\frac{2}{h}\partial_{M}(hT^{M}),$

(9)

in which $T^{M}$ is the torsion tensor that can be defined as $T_{M}=T^{N}\ _{MN}$. We can easily see that GR and TEGR will lead to exactly the same equations Aldrovandi . However, this will not be the case if one uses $f(R)$ or $f(T)$ as the Lagrangian of the theory, which therefore corresponds to different gravitational modifications Abedi2017 . However, when we consider $f(T,B)$ as the Lagrangian of the theory, we have that $f(T,B)=f(-T+B)$ is the teleparallel equivalent of $f(R)$.

We can consider a modified gravity theory where the gravitational Lagrangian depends on $T$ and $B$ Abedi2017 . Therefore, we have a gravitational action to $f(T,B)$, namely

$\displaystyle\mathcal{S}=-\frac{1}{4}\int hf(T,B)d^{5}x+\int\mathcal{L}_{m}d^{5}x,$

(10)

where $\mathcal{L}_{m}$ is the matter Lagrangian. We can get the field equations by varying the action with respect to the vielbein Abedi2017 ; Pourbagher2020

	$\displaystyle\frac{1}{h}f_{T}\Big{[}\partial_{Q}(hS_{N}\ ^{MQ})-h\widetilde{\Gamma}^{R}\ _{SN}S_{R}\ ^{MS}\Big{]}+\frac{1}{4}\Big{[}f-Bf_{B}\Big{]}\delta_{N}^{M}$
	$\displaystyle+\Big{[}(\partial_{Q}f_{T})+(\partial_{Q}f_{B})\Big{]}S_{N}\ ^{MQ}+\frac{1}{2}\Big{[}\nabla^{M}\nabla_{N}f_{B}-\delta^{M}_{N}\Box f_{B}\Big{]}$	$\displaystyle=$	$\displaystyle-\mathcal{T}_{N}\ ^{M},$		(11)

where $\Box\equiv\nabla^{M}\nabla_{M}$, $f\equiv f(T,B)$, $f_{T}\equiv\partial f(T,B)/\partial T$ e $f_{B}\equiv\partial f(T,B)/\partial B$ and $\mathcal{T}_{N}\ ^{M}$ is the stress-energy tensor, which in terms of the matter Lagrangian is given by $\mathcal{T}_{a}\ ^{M}=-\delta\mathcal{L}_{m}/\delta h^{a}\ _{M}$. The matter Lagrangian density is taken as

$\displaystyle\mathcal{L}_{m}=-h\left[\frac{1}{2}\partial^{M}\phi\partial_{M}\phi+V(\phi)\right],$

(12)

where $\phi\equiv\phi(y)$ is a background scalar field that generates the brane.

In our work, we consider the static codimension one braneworld scenario whose metric can be written as

$\displaystyle ds^{2}=e^{2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$

(13)

where $e^{A(y)}$ is the warped factor. We can choose the vielbein in the form

$\displaystyle h_{\overline{M}}\ ^{M}=diag(e^{A},e^{A},e^{A},e^{A},1).$

(14)

Using the Weitzenbock connection, the torsional scalar and the boundary term are given by

$\displaystyle T=-12A^{\prime 2}$

,

$\displaystyle B=-8(A^{\prime\prime}+4A^{\prime 2}),$

(15)

where the prime $(^{\prime})$ denotes differentiation with respect to $y$.

Thus, the gravitational field equations are given as

	$\displaystyle\phi^{\prime\prime}+4A^{\prime}\phi$	$\displaystyle=$	$\displaystyle\frac{dV}{d\phi},$		(16)
	$\displaystyle\frac{1}{4}\Big{[}f+8(A^{\prime\prime}+4A^{\prime 2})f_{B}\Big{]}+6A^{\prime 2}f_{T}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\phi^{2}-V,$		(17)
	$\displaystyle 12\Big{[}A^{\prime}(A^{\prime\prime\prime}+8A^{\prime}A^{\prime\prime})(f_{BB}+f_{TB})+3A^{\prime\prime}A^{\prime 2}(f_{TT}+f_{BT})\Big{]}$
	$\displaystyle-\frac{1}{2}(A^{\prime\prime}+4A^{\prime 2})(4f_{B}+3f_{T})+\frac{1}{4}f$	$\displaystyle=$	$\displaystyle\frac{1}{2}\phi^{2}+V.$		(18)

We can rewrite equations (17) and (2) as

	$\displaystyle 6A^{\prime 2}$	$\displaystyle=$	$\displaystyle-\frac{1}{f_{T}}\Big{(}P+P_{TB}\Big{)},$		(19)
	$\displaystyle 3A^{\prime\prime}+12A^{\prime 2}$	$\displaystyle=$	$\displaystyle-\frac{2}{f_{T}}\Big{(}\rho+\rho_{TB}\Big{)},$		(20)

where

	$\displaystyle P_{TB}$	$\displaystyle=$	$\displaystyle\frac{1}{4}\Big{[}f+8(A^{\prime\prime}+4A^{\prime 2})f_{B}\Big{]},$		(21)
	$\displaystyle\rho_{TB}$	$\displaystyle=$	$\displaystyle-12\Big{[}A^{\prime}(A^{\prime\prime\prime}+8A^{\prime}A^{\prime\prime})(f_{BB}+f_{TB})+3A^{\prime\prime}A^{\prime 2}(f_{TT}+f_{BT})\Big{]}$		(22)
			$\displaystyle+2(A^{\prime\prime}+4A^{\prime 2})f_{B}+\frac{1}{4}f.$

Note that the left side of equations (19) and (20) is equivalent to that obtained in TEGR. So we can state that modified gravity equations of the motion of the $f(T,B)$ gravities are similar to an inclusion of an additional source with $\rho_{TB}$ and $P_{TB}$.

The diagonal tetrad (14) represents a good choice among all the possible vielbeins giving metric (13). In fact, the gravitational field equations do not involve any additional constraints on the function $f(T,B)$ or the scalars $T$ and $B$. Thus, the choice in Eq. (14) can be regarded as a "good vielbein". Similarly, in the FRW cosmological models the $f(T,B)$ gravitational dynamics preserves the form of the usual Friedmann equations (two equations) Franco2020 ; Caruana2020 ; EscamillaRivera2019 ; Bahamonde2016a ; Pourbagher2020 ; Bahamonde2020a . The choice of vielbein is a rather important issue, as it fixes the number of degrees of freedom of the theory, as seen particularly in a gravitational wave analysis of $f(T,B)$ gravity Capozziello2019 .

Since the equations (17) and (2) form a second-order derivative system, it is difficult to provide an analytical solution for this case. For simplicity, we can take an ansatz Gremm1999

$\displaystyle e^{2A(y)}=\cosh^{-2p}(\lambda y),$

(23)

where the $p$ parameter modifies the warp variation within the brane core, and $\lambda$ determines the brane width.

We will then propose the cases where $f_{1}(T,B)=T+k_{1}B^{n_{1}}$ and $f_{2}(T,B)=B+k_{2}T^{n_{2}}$ , where $k_{1,2}$ and $n_{1,2}$ are parameters controlling the deviation of the usual teleparallel theory Allan .

We follow the approach carried out in Ref. Allan , where by manipulating the (17) and (2) equations, an equation relating the metric components and the scalar field is obtained. In this case, for $f_{1}(T,B)$ with equations (17),(2) and (23), we get Allan

$\displaystyle\phi^{\prime 2}(y)=\frac{3}{2}p\lambda^{2}\mathrm{sech}^{2}(\lambda y)-\frac{\alpha^{n_{1}}}{\beta^{2}}\Big{[}8^{n_{1}-1}k_{1}n_{1}(n_{1}-1)(1+4p)\sinh^{2}(\lambda y)\Big{]},$

(24)

	$\displaystyle V(\phi(y))$	$\displaystyle=$	$\displaystyle\frac{3}{4}\alpha+2^{3n_{1}-4}k_{1}(n_{1}-1)(p\lambda^{2})^{2}\alpha^{n_{1}-2}\Big{\{}4\mathrm{sech}^{4}(\lambda y)+64p^{2}\tanh^{4}(\lambda y)$		(25)
		$\displaystyle-$	$\displaystyle[3n_{1}+4(8+3n_{1})]\mathrm{sech}^{2}(\lambda y)\tanh^{2}(\lambda y)\Big{\}}.$

We can solve Eq. (24) to find a function $\phi=g(y)$ that may be inverted to give $y=g^{-1}(\phi)$, which allows us to write the potential in the usual way $V=V(\phi)$. The thick brane solution for $n_{1}=1$ is the same obtained in Refs.Gremm1999 ; Bazeia2007 , which does not depend on the $k_{1}$ parameter. For $n_{1}=2$ we get as a solution the first and second kind elliptic integrals, which depends on parameter $k_{1}$ Allan .

In Fig. 1, we plotted the $\phi(y)$ field for $f_{1}(T,B)$. The $n_{1}=1$ configuration (figure 1 $a$ ) is a kink solution. For $n_{1}=2$ configuration, for a decreasing value of $k_{1}$, the solution goes from kink to double-kink, as shown in figure 1 ($b$). This feature reflects the brane internal structure, which tends to split the brane. A similar result was obtained in Ref. Yang2012 .

For $f_{2}(T,B)$, we get

$\displaystyle\phi^{\prime 2}(y)=-2^{2n_{2}-3}(-3)^{n_{2}}p^{-1}k_{2}n_{2}(2n_{2}-1)\mathrm{csch}^{2}(\lambda y)[p\lambda\tanh(\lambda y)]^{2n_{2}},$

(26)

$\displaystyle V(\phi(y))=4^{n_{2}-2}(-3)^{n_{2}}p^{-1}k_{2}(2n_{2}-1)[4p-n_{2}\mathrm{csch}^{2}(\lambda y)][p\lambda\tanh(\lambda y)]^{2n_{2}}.$

(27)

The setting $n_{2}=4,6,...$ (even numbers) does not present a pleasant solution.

In Fig. 2, we plotted the $\phi(y)$ field for $f_{2}(T,B)$ varying the parameter $k_{2}$. For $n_{2}=1$ configuration (figure 2 $a$ ) we have a kink solution, whereas for $n_{2}=3$ configuration we have a double-kink solution (figure 2 $b$ ). Again, this feature reflects the brane internal structure, which tends to split the brane.

The energy densities for $f_{1}(T,B)$ are Allan

	$\displaystyle\rho_{1}(y)$	$\displaystyle=$	$\displaystyle\alpha\Bigg{[}2^{3n_{1}-2}(n_{1}-1)k_{1}-\frac{8^{n_{1}-1}(n_{1}-1)3n_{1}k_{1}(1+4p)\sinh^{2}(\lambda y)}{\beta^{2}}\Bigg{]}$		(28)
		$\displaystyle-$	$\displaystyle 3\Big{[}(p\lambda)^{2}-\frac{1}{2}(1+2p)p\lambda^{2}\mathrm{sech}^{2}(\lambda y)\Big{]},$

where we defined the functions $\alpha\equiv p\lambda^{2}[\mathrm{sech}^{2}(\lambda y)-4p\tanh^{2}(\lambda y)]$, and $\beta\equiv 1+2[1-\cosh(2\lambda y)]p$. In Fig. 3, we plotted the energy densities $\rho_{1}(y)$ for $f_{1}(T,B)$, with $n_{1}=1$ (figure 3 $a$ ) and $n_{1}=2$ (figure 3 $b$ ), that includes a new peak varying the parameter $k_{1}$.

The energy densities for $f_{2}(T,B)$ are Allan

$\displaystyle\rho_{2}(y)=-2^{2n_{2}-3}(-3)^{n_{2}}p^{-1}k_{2}(2n_{2}-1)\beta\mathrm{csch}^{2}(\lambda y)[p\lambda\tanh(\lambda y)]^{2n_{2}}.$

(29)

In Fig. 4, we plotted the energy densities $\rho_{2}(y)$ for $f_{2}(T,B)$ varying the parameter $k_{2}$, for $n_{2}=1$ (figure 4 $a$ ) and $n_{2}=3$ (figure 4 $b$ ), which has two peaks.

In this section, we explore the effects of the modified teleparallel $f(T,B)$ on the matter (fermionic) sector. We changed the variable from $y$ to $z$ in the metric (13), and so, $dz=e^{-A(y)}dy$ and the metric turns to $ds^{2}=e^{2A}(\eta^{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2})$. Considering a well-known Yukawa coupling between the fermion and the scalar field $\phi$, the 5-dimensional Dirac action of a spin $1/2$ fermion minimally coupled to the gravity and to the background scalar $\phi$ is

$\displaystyle\mathcal{S}_{1/2}=\int h\overline{\Psi}\Big{(}\Gamma^{M}D_{M}\Psi-\xi\phi\Psi\Big{)}d^{5}x,$

(30)

where $\Gamma^{M}=h_{\overline{M}}\ ^{M}\Gamma^{\overline{M}}$ are the Dirac curved matrices defined from the Dirac flat matrices $\Gamma^{\overline{M}}$ through the vielbeins. These matrices obey Cliford algebra $\{\Gamma^{M},\Gamma^{N}\}=2g^{MN}$. $D_{M}$ is the covariant derivative given by $D_{M}=\partial_{M}+\Omega_{M}$, where Obukhov2002 ; Ulhoa2016

$\displaystyle\Omega_{M}=\frac{1}{4}\Big{(}K_{M}\ ^{{\overline{N}}{\overline{Q}}}\Big{)}\ \Gamma_{\overline{N}}\Gamma_{\overline{Q}},$

(31)

is the spin connection, which for our case is such that $\Omega_{\mu}=\frac{1}{4}(-\partial_{z}A)\Gamma_{\mu}\Gamma^{z}$ and $\Omega_{z}=\partial_{z}A$. By choosing the spinor representation Almeida2009 ; Dantas ; Andrade2001

$\displaystyle\Psi\equiv\Psi(x,z)=\left(\begin{array}[]{cccccc}\psi\\ 0\\ \end{array}\right),\ \Gamma^{\overline{\mu}}=\left(\begin{array}[]{cccccc}0&\gamma^{\overline{\mu}}\\ \gamma^{\overline{\mu}}&0\\ \end{array}\right),\ \Gamma^{\overline{z}}=\left(\begin{array}[]{cccccc}0&\gamma^{4}\\ \gamma^{4}&0\\ \end{array}\right),$

(38)

the Dirac equation takes the form

$\displaystyle\Big{[}\gamma^{\mu}\partial_{\mu}+\gamma^{4}\partial_{z}-\xi e^{A}\phi\Big{]}\psi=0.$

(39)

We apply a decomposition to the spinor $\psi=\sum_{n}[\psi_{L,n}(x)\varphi_{L,n}(z)+\psi_{R,n}(x)\varphi_{R,n}(z)]$, being $\gamma^{4}\psi_{R,L}=\pm\psi_{R,L}$ e $\gamma^{\mu}\partial_{\mu}\psi_{R,L}=m\psi_{L,R}$. So, we have the coupled equations

	$\displaystyle\Big{[}\partial_{z}+\xi e^{A}\phi\Big{]}\varphi_{L}(z)=m\varphi_{R}(z),$
	$\displaystyle\Big{[}\partial_{z}-\xi e^{A}\phi\Big{]}\varphi_{R}(z)=m\varphi_{L}(z).$		(40)

These equations can be decoupled and reduced to Schroëdinger-like equations

	$\displaystyle\Big{[}-\partial^{2}_{z}+V_{L}(z)\Big{]}\varphi_{L}(z)=m^{2}\varphi_{L}(z),$
	$\displaystyle\Big{[}-\partial^{2}_{z}+V_{R}(z)\Big{]}\varphi_{R}(z)=m^{2}\varphi_{R}(z),$		(41)

where

	$\displaystyle V_{L}(z)=U^{2}-\partial_{z}U,$
	$\displaystyle V_{R}(z)=U^{2}+\partial_{z}U,$		(42)

and $U=\xi e^{A}\phi$ is the so-called superpotential. The supersymmetric structure of the potentials (4) leads to a massless mode in the form

$\displaystyle\varphi_{R0,L0}(z)\propto\exp{\Bigg{[}\pm\int\xi\phi e^{A}dz\Bigg{]}},$

(43)

where $\phi e^{A}|_{z\rightarrow\pm\infty}\rightarrow 0$. In other words, the zero mode for fermions can be localized on the brane for positive $\xi$ Yang2012 .

For $f_{1}(T,B)$ with $n_{1}=1$ only left-chiral fermions can be localized on the brane. For $n_{1}=2$ only right-chiral fermions can be localized on the brane. In this case note that the smaller the $k_{1}$ parameter, the more localized the mode becomes, as can be seen in the figure (5). For $f_{2}(T,B)$ with $n_{2}=1$ only left-chiral fermions can be localized on the brane. Now, the higher the parameter $k_{1}$, the more localized the mode becomes. On the other hand, for $n_{1}=3$ only right-chiral fermions can be localized on the brane, and so, the smaller the $k_{1}$ parameter, the more localized the mode becomes, as can be seen in the figure (6).

Note that the effective potentials are even functions, so the wave functions will be either even or odd. We can analyze numerically the Eq.(4). For that we impose the following conditions: $\varphi_{even}(0)=c$, $\partial_{z}\varphi_{even}(0)=0$, $\varphi_{odd}(0)=c$, and $\partial_{z}\varphi_{odd}(0)=0$, where $c$ is a constantAlmeida2009 ; Liu2009 ; Liu2009a . Here $\varphi_{even}$ and $\varphi_{odd}$ denote the even and odd parity modes of $\varphi_{R,L}(z)$, respectively.

As depicted in Fig. 7, the asymptotic divergence of the massive modes shows that they form non-localized states, which is a behavior typical of plane wave oscillations, characteristic of a free mode. This shows that these modes represent massive fermions that certainly will be leaked from the brane.

For $f_{1}(T,B)$, both for $n_{1}=1$ and $n_{1}=2$, the greater the mass, the more oscillations we obtain, as can be seen in the figure 7 ($a$ and $b$) for $n_{1}=2$. In figure 7 ($c$ and $d$), we observe that when decreasing the value of $k_{1}$, the greater the amplitude of the oscillation, mainly near the brane. For $f_{2}(T,B)$, both for $n_{2}=1$ and $n_{2}=3$, the greater the mass, the more oscillations we obtain. Increasing the value of $k_{2}$, greater the amplitude of the oscillation, mainly near the brane as we can see in the figure 7 ($e$ and $f$) for $n_{2}=3$.

In this work we considered a braneworld in the context of the $f(T,B)$ modified teleparallel gravity constructed with one scalar field. We propose two particular cases for $f(T,B)$, namely $f_{1}(T,B)=T+k_{1}B^{n_{1}}$ and $f_{2}(T,B)=B+k_{2}T^{n_{2}}$. In both cases the torsion and boundary term produce an inner brane structure tending to split the brane. We also find that the $n_{1,2}$ and $k_{1,2}$ parameters determines whether the domain wall solution is a kink or double-kink. For $f_{1}(T,B)$ where $n_{1}=2$, with the decreasing of the contribution of $k_{1}$, the configuration of the solution changes from a kink to double-kink. The same is true for $f_{2}(T,B)$ where $n_{2}=3$, when we increase the value of parameter $k_{2}$. The thick brane undergoes a phase transition evinced by the energy density components. Similar behavior was found for $f(T)$ in Ref Yang2012 .

We considered a simple Yukawa coupling between the scalar and the spinor field. We notice that potentials feel the division of brane when we vary $n_{1,2}$ and $k_{1,2}$, the same happens with the zero modes, which become more localized. We note that for $f_{1}(T,B)$ where $n_{1}=1$, only left-chiral fermions are located, the same is true for $f_{2}(T,B)$ with $n_{2}=1$. For $f_{1}(T,B)$ where $n_{1}=2$, only right-chiral fermions are located, the same is true for $f_{2}(T,B)$ with $n_{2}=3$. The massive fermionic modes are dependent on the parameters that control torsion and boundary term. This is well evidenced for $f_{1}(T,B)$ with $n_{1}=3$ since that decreasing the value of $k_{1}$, increases the amplitude of the ripples making them more intense and presenting ripples within the brane. The same goes for $f_{2}(T,B)$ when increasing the value of $k_{2}$, which is very evident for $n_{2}=3$. Therefore, the brane splitting process leads to modifications of the massive fermionic modes inside the thick brane. The interaction of the massive modes with the torsion and boundary term is more intense in the brane core where the amplitude and the rate of growth depend on the parameters $n_{1,2}$ and $k_{1,2}$.

Although only one massless chiral mode was found for each configuration $n_{1,2}$ and $k_{1,2}$, only left-handed spin $1/2$ fermions were detected so far. The configurations where the right-handed massless mode is localized on the brane are beyond the standard model states. The absence of left-handed massless mode can be used to rule out those configurations where only right-handed massless mode are trapped.

In addition, it is worthwhile to mention the role played by the parameter $k_{1,2}$ on the brane internal structure. As $k_{1,2}$ grows the brane undergoes a transition from a single into a two-brane. Therefore, the parameter $k_{1,2}$ can be regarded as phase transition parameter controlling the brane splitting process.

The authors thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), grants n^o 312356/2017-0 (JEGS) and n^o 308638/2015-8 (CASA), and Coordenação de Aperfeiçoamento do Pessoal de Nível Superior (CAPES), for financial support. The authors also thank the anonymous referee for valuable comments and suggestions.