#  Information measures for fermion localization in $f(T,B)$ gravity with non-
minimal couplings

A. R. P. Moreira<EMAIL_ADDRESS>Research Center for Quantum
Physics, Huzhou University, Huzhou, 313000, P. R. China.  Secretaria da
Educação do Ceará (SEDUC), Coordenadoria Regional de Desenvolvimento da
Educação (CREDE 9), Horizonte, Ceará, 62880-384, Brazil.  Shi-Hai Dong
<EMAIL_ADDRESS>Research Center for Quantum Physics, Huzhou University,
Huzhou, 313000, P. R. China.  Centro de Investigación en Computación,
Instituto Politécnico Nacional, UPALM, CDMX 07700, Mexico.  Emmanuel N.
Saridakis<EMAIL_ADDRESS>National Observatory of Athens, Lofos Nymfon, 11852
Athens, Greece  CAS Key Laboratory for Researches in Galaxies and Cosmology,
School of Astronomy and Space Science, University of Science and Technology of
China, Hefei, Anhui 230026, China  Departamento de Matemáticas, Universidad
Católica del Norte, Avda. Angamos 0610, Casilla 1280 Antofagasta, Chile

###### Abstract

We investigate the dynamics of fermion localization within the framework of
$f(T,B)$ gravity featuring non-minimal couplings. Starting from the Dirac
action for a spin-$1/2$ fermion in a five-dimensional spacetime governed by
torsional $f(T,B)$ gravity, we derive the Dirac equation and we explore its
solutions under various non-minimal coupling functions. We examine two
realistic forms of the torsional non-minimal coupling and reveal distinct
behaviors that impact the localization of both massless and massive fermionic
modes on the brane. Additionally, we employ probabilistic measurements,
including Shannon entropy theory, Fisher information theory, and relative
probability, to analyze the localization of these fermionic modes. The
observed effects offer potential insights into probing torsional
modifications.

###### pacs:

04.50.h, 11.10.Kk, 98.80.Cq

##  I Introduction

The investigation of extra-dimensional theories traces its origins back to the
pioneering works of Kaluza Kaluza1921  and Klein Klein1926  in the early 20th
century. Initially motivated by the unification of Einstein’s general
relativity (GR) and Maxwell’s electromagnetism, these theories opened new
directions in theoretical physics. Much later, the possibility of non-compact
extra-dimensional theories Arkani-Hamed:1998jmv ; rs ; rs2 ; dsh  broadened
these directions further. In particular, the Randall-Sundrum model paved the
way for braneworld scenarios, while the Gregory-Rubakov-Sibiryakov model
Gregory:2000jc , the universal extra dimension model Appelquist:2000nn , the
Dvali-Gabadadze-Porrati model Dvali:2000hr , and others, explored these
possibilities even further, including the construction of thick branes using
scalar, vector, and spinor fields DeWolfe:1999cp ; Gremm1999 ; Csakil ;
Gremm2000 ; Dzhunushaliev:2009 ; Dzhunushaliev:2010 ; Herrera-Aguilar:2009 ;
Dzhunushaliev:2007 ; Gogberashvili:2003a ; Gogberashvili:2003b ;
Goldberger1999 ; Bazeia2008 ; Geng:2015kvs ; Dzhunushaliev:2011mm .

On the other hand, theoretical motivations such as the non-renormalizability
of GR, as well as cosmological requirements like solving the cosmological
constant problem or addressing observational tensions (e.g., the $H_{0}$ and
the $\sigma_{8}$ tensions) Abdalla:2022yfr , have led to extensive research in
the direction of modified gravity CANTATA:2021ktz ; Capozziello:2011et ;
Cai:2015emx . In this framework, various extensions and modifications of GR
have been constructed to address these issues by introducing a richer
structure. The simplest approach involves extending the Einstein-Hilbert
action, resulting in $f(R)$ gravity Starobinsky:1980te ; Capozziello:2002rd ;
DeFelice:2010aj , $f(G)$ gravity Nojiri:2005jg ; DeFelice:2008wz , cubic
gravity Asimakis:2022mbe , Lovelock gravity Lovelock:1971yv ; Deruelle:1989fj
, or scalar-tensor theories such as Horndeski gravity Horndeski:1974wa  and
generalized Galileon theory DeFelice:2010nf ; Deffayet:2011gz ;
Kobayashi:2010cm ; DeFelice:2011bh . Alternatively, one can start with the
Teleparallel Equivalent of General Relativity (TEGR), where gravity is
described by the torsion tensor, and extend it in various ways. This includes
$f(T)$ gravity Cai:2015emx ; Linder:2010py ; Chen:2010va , the Teleparallel
Equivalent of Gauss-Bonnet and $f(T,T_{G})$ gravity Kofinas:2014owa ;
Kofinas:2014daa , and $f(T,B)$ gravity, where $B$ is the boundary term
connecting the torsion and curvature scalars Bahamonde:2015zma ;
Bahamonde:2016grb . Inspired by scalar-tensor theories Geng:2011aj , one can
also construct scalar-torsion theories by introducing a new scalar field
coupled to torsion terms, resulting in the Teleparallel Equivalent of
Horndeski gravity Bahamonde:2019shr ; Bahamonde:2020cfv ; Capozziello:2023foy
; Aldrovandi . Torsional modified gravities, particularly $f(T,B)$ theories,
have been shown to have interesting cosmological and black hole phenomenology,
thus attracting significant research interest Franco2020 ; EscamillaRivera2019
; Bahamonde2016a ; Abedi2017 ; Farrugia:2018gyz ; Soudi:2018dhv ;
Gakis:2019rdd ; Caruana2020 ; Pourbagher2020 ; Bahamonde2020a ; Azhar2020 ;
Bahamonde:2021dqn ; Bhattacharjee2020 ; Mylova:2022ljr .

Due to the success of braneworld constructions and modified theories of
gravity, a new direction has emerged: constructing braneworld scenarios within
the framework of modified gravity. Among other topics, one can study fermion
localization in teleparallel gravity models, such as considering fermionic
Yukawa couplings in $f(T)$ gravity Yang2012  or $f(T,B)$ gravity Moreira20211
. However, although the Yukawa-type coupling is the most commonly used, it may
not sufficiently show the influence of gravitational changes on the fermion’s
location in the brane. A potential solution is to propose new non-minimal
coupling mechanisms between the fermion and the geometrical structure.

In this work, we investigate fermion localization in $f(T,B)$ gravity,
allowing for the possibility of non-minimal couplings, which are known to play
an important role in teleparallel Horndeski theories. The structure of this
paper is as follows: In Section II, we study the influence of geometry on the
location of fermions through a non-minimal coupling between the spinor field
and torsional geometry. In Section III, we apply probabilistic measurements as
tools to further analyze how geometry affects the localization of the
fermionic field on the brane. Finally, additional conclusions and comments are
presented in Section IV.

##  II Fermion localization in $f(T,B)$ gravity

In this section, we investigate fermion localization within the framework of
$f(T,B)$ gravity, incorporating non-minimal couplings between the fermionic
field and the geometry. To begin, we briefly review the fundamentals of
$f(T,B)$ gravity, working in $D$ dimensions for generality. Throughout this
work, we set $\hbar=1$.

###  II.1 Modified teleparallel gravity

In torsional gravity, the dynamical fields are the vielbeins, which are
related to the metric through the following relationship:

$\displaystyle g_{MN}=\eta_{ab}h^{a}\ _{M}h^{b}\ _{N},$ (1)

with $\eta_{ab}$ a $D$-dimensional Minkowski metric, and where capital Latin
indices $M={0,...,D-1}$ denote the coordinates in the bulk, while small Latin
indices $a={0,...,D-1}$ denote the Lorentz, tangent space. Furthermore, one
introduces the Weitzenböck connection Aldrovandi

$\displaystyle\widetilde{\Gamma}^{P}\ _{NM}=h_{a}\ ^{P}\partial_{M}h^{a}\
_{N},$ (2)

which corresponds to zero curvature, and therefore the corresponding torsion
tensor is

$\displaystyle T^{P}\ _{MN}=\widetilde{\Gamma}^{P}\
_{NM}-\widetilde{\Gamma}^{P}\ _{MN},$ (3)

which quantifies all information about the gravitational field. The
relationship between the Weitzenböck and the Levi-Civita connections is
$\widetilde{\Gamma}^{P}\ _{NM}=\Gamma^{P}\ _{NM}+K^{P}\ _{NM}$, where

$\displaystyle K^{P}\ _{NM}=\frac{1}{2}\Big{(}T_{N}\ ^{P}\ {}_{M}+T_{M}\ ^{P}\
{}_{N}-T^{P}\ _{NM}\Big{)},$ (4)

is the contortion tensor. Furthermore, it proves convenient to define the
tensor

$\displaystyle S_{P}\ ^{MN}=\frac{1}{2}\Big{(}K^{MN}\
_{P}-\delta^{N}_{P}T^{QM}\ _{Q}+\delta^{M}_{P}T^{QN}\ _{Q}\Big{)}.$ (5)

Finally, we can define the torsion scalar as Aldrovandi

$\displaystyle T=T_{PMN}S^{PMN}.$ (6)

If one uses $T$ as a Lagrangian, namely consider the action

$S=\int d^{D}x\,h\,T,$ (7)

where $h=det(h^{A}{}_{M})$ is the vielbein determinant. By performing
variation with respect to the vielbein, one obtains exactly the same equations
as in GR, and this equivalence is why the theory in four dimensions is named
the TEGR. The underlying reason for this equivalence is that the torsion
scalar corresponding to the Weitzenböck connection is related to the Ricci
scalar $R$ corresponding to the Levi-Civita connection through

$\displaystyle-T+B=R,$ (8)

where

$\displaystyle B=-2\nabla^{M}T^{N}\ _{MN},$ (9)

is a boundary term. However, considering functions of the above quantities
does not lead to equivalent theories anymore, because functions of total
derivatives are not total derivatives themselves. Therefore, a more general
theory is $f(T,B)$ gravity, which is characterized by the action

$S=\int d^{D}x\,h\,f(T,B).$ (10)

When $f(T,B)=f(T)$, the theory above corresponds to $f(T)$ gravity. When
$f(T,B)=f(-T+B)$, it corresponds to $f(R)$ gravity. Finally, when
$f(T,B)=-T+const$, the theory reproduces the TEGR, which includes GR along
with a cosmological constant.

###  II.2 Braneworld scenario and fermion localization

Let us now proceed to the fermion localization in a 5-dimensional braneworld
scenario embedded in a bulk governed by $f(T,B)$ gravity. We consider the
metric ansatz that describes such a braneworld scenario as follows

$ds^{2}=e^{2A}(\eta^{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}),$ (11)

where $\eta_{\mu\nu}=(-1,1,1,1)$ represents the four-dimensional Minkowski
metric and $e^{A(r)}$ is the warp factor that controls the thickness of the
brane. Specifically, we consider the warp factor in the form obtained in
Gremm1999 , namely

$\displaystyle A(z)=-p\ln\Big{[}\mathrm{arcsinh}(\lambda z)\Big{]},$ (12)

where $p$ and $\lambda$ are parameters that modify the warp variation and
determine the width within the brane core, respectively.

Now, apart from the bulk $f(T,B)$ gravity action (10), we consider that the
fermionic matter field $\Psi$ with spin $1/2$ is coupled to the torsional
geometry in a non-minimal way. In particular, we consider the action

$\displaystyle\mathcal{S}_{1/2}=\int
h\overline{\Psi}\Big{(}\Gamma^{M}D_{M}\Psi-\xi f(T,B)\Psi\Big{)}d^{5}x,$ (13)

where we use the gravitational model $f(T,B)$ as a non-minimal coupling
function. Here $\Gamma^{M}=h_{a}\ ^{M}\Gamma^{a}$ represents the curved Dirac
matrices ($\Gamma^{a}$ being the flat Dirac matrices), and
$D_{M}=\partial_{M}+\Omega_{M}$ denotes the covariant derivative. Moreover,
the spin connection $\Omega_{M}$ is expressed as Obukhov2002 ; Ulhoa2016 ;
Moreira20211

$\displaystyle\Omega_{M}=\frac{1}{4}\Big{(}K_{M}\ ^{bc}\Big{)}\
\Gamma_{b}\Gamma_{c}.$ (14)

For our specific scenario, the spinor representation is given by Almeida2009 ;
Dantas

$\displaystyle\Psi\equiv\Psi(x,z)=\left(\begin{array}[]{cccccc}\psi\\\
0\end{array}\right)\ \mathrm{and}\ \
\Gamma^{a}=\left(\begin{array}[]{cccccc}0&\gamma^{a}\\\
\gamma^{a}&0\end{array}\right).$ (19)

Consequently, the Dirac equation takes the form:

$\displaystyle\Big{[}\gamma^{\mu}\partial_{\mu}+\gamma^{4}\partial_{z}-\xi
e^{A}f(T,B)\Big{]}\psi=0.$ (20)

Decomposing Eq.(20) using
$\psi=\sum_{n}[\psi_{L,n}(x)\varphi_{L,n}(z)+\psi_{R,n}(x)\varphi_{R,n}(z)]$,
where $\gamma^{4}\psi_{R,L}=\pm\psi_{R,L}$ and
$\gamma^{\mu}\partial_{\mu}\psi_{R,L}=m\psi_{L,R}$, yields two coupled
equations, namely

$\displaystyle\Big{[}\partial_{z}+\xi
e^{A}f(T,B)\Big{]}\varphi_{L}(z)=m\varphi_{R}(z),$
$\displaystyle\Big{[}\partial_{z}-\xi
e^{A}f(T,B)\Big{]}\varphi_{R}(z)=m\varphi_{L}(z).$ (21)

These equations can be decoupled, leading to Schrödinger-like equations

$\displaystyle\Big{[}-\partial^{2}_{z}+V_{L}(z)\Big{]}\varphi_{L}(z)=m^{2}\varphi_{L}(z),$
$\displaystyle\Big{[}-\partial^{2}_{z}+V_{R}(z)\Big{]}\varphi_{R}(z)=m^{2}\varphi_{R}(z),$
(22)

with the potential functions $V_{L}(z)$ and $V_{R}(z)$ defined as

$\displaystyle V_{L}(z)=U^{2}-\partial_{z}U,$ $\displaystyle
V_{R}(z)=U^{2}+\partial_{z}U,$ (23)

and $U=\xi e^{A}f(T,B)$ representing the superpotential.

In order to ensure the localization on the brane and self-consistency of our
considerations, certain conditions on the form of $f(T,B)$ must be imposed.
Drawing inspiration from the Yukawa-type coupling, which is prevalent in the
literature, we propose the following conditions for $f(T,B)$:

  1. 1.

The function $f(T(z),B(z))$ should exhibit an isometric behavior, undergoing a
phase transition at the origin ($z=0$).

  2. 2.

The $f(T,B)|_{z\rightarrow\pm\infty}$ should approach a constant value $c$
asymptotically.

  3. 3.

For our model to retain physical validity, the so-called superpotential
$U|_{z\rightarrow\pm\infty}$ must tend to zero in vacuum.

Obeying these conditions, we define two forms for the geometric coupling
function, namely,

$\displaystyle f_{1}(T,B)=\sqrt{-T-\alpha(B)}$ (24)

and

$\displaystyle f_{2}(T,B)=\sqrt{-T}-\alpha B,$ (25)

where the parameter $\alpha$ is controlling the effect of the boundary term on
the location of the fermion.

Figure 1:  The effective potential for $\lambda=p=\xi=1$, and various values
of the modified gravity parameter $\alpha$, for two cases of the $f(T,B)$
function, i.e., $f_{1}$ of (24), and $f_{2}$ of (25).

For the first choice of $f_{1}$, the effective potential (II.2) exhibits a
standard behavior commonly discussed in the literature, portraying a perfectly
symmetric brane (see $f_{1}$ of Fig. 1). This outcome is significant as it
ensures the localization of fermionic modes, thereby validating the physical
viability of our model. Moving forward, we observe how variations in the
geometry of the model influence the potential’s shape. Increasing the
parameter $\alpha$ enhances both the potential barriers and the depth of the
potential well. This feature suggests that a greater contribution from the
boundary term in our model enhances the localization of fermionic modes on the
brane.

Our second choice, $f_{2}$, represents a novel aspect of this work,
illustrating how the underlying geometry induces defects in brane symmetry. As
depicted in $f_{2}$ of Fig. 1, when $\alpha\rightarrow 0$, the potential
exhibits a perfect and symmetric solution. However, as the influence of the
boundary term on the model increases, the potential solution deforms, showing
slight asymmetry. Importantly, the likelihood of localizing fermionic modes is
higher on the right side of the brane. This asymmetry signifies a distorted
brane solution, demonstrating that model geometry can indeed induce
deformations on the brane. Notably, similar results have been observed in
other studies Battisti:2008am ; Perez-Payan:2011cvf ; Sabido:2015xfk ;
Lopez:2017xaz ; Lopez-Picon:2023rhc .

In the following subsections we investigate the massless and massive fermionic
modes separately.

####  II.2.1 Massless modes

Similarly to the Yukawa-type coupling scenario, the specific form of the non-
minimal coupling $f(T,B)$ plays a crucial role in governing the dynamics of
the Kaluza-Klein states. Massless modes, also known as zero modes, exhibit the
following form

$\displaystyle\varphi_{R0,L0}(z)\propto\exp{\Bigg{[}\pm\int\xi
f(T,B)e^{A}dz\Bigg{]}},$ (26)

owing to the supersymmetric structure of the potentials (II.2).

To ascertain whether the zero modes can indeed be localized on the brane, we
need to verify whether the normalization condition for the zero modes is
satisfied, namely whether we have that

$\displaystyle\int\varphi_{R0,L0}(z)^{2}dz<\infty.$ (27)

Given that $f(T,B)e^{A}|_{z\rightarrow\pm\infty}\rightarrow 0$, only the left-
handed massless mode is confined to the brane (for positive $\xi$), a feature
shared by Yukawa coupling models Yang2012 . Therefore, defining $p=1$, we can
derive a straightforward expression for the warp factor in conformal
coordinates, namely $A(z)=-\ln\sqrt{1+\lambda^{2}z^{2}}$. As $z$ tends to
infinity, $e^{A}\rightarrow 1/\lambda|z|$, consequently, one has

$\displaystyle\varphi_{L0}(z\rightarrow\pm\infty)\rightarrow|z|^{-\frac{\xi}{\lambda}f(T,B)|_{\infty}},$
(28)

where $f(T,B)|_{\infty}=c$. If the normalization condition holds, we obtain
the following equivalent condition

$\displaystyle\int|z|^{-2\frac{\xi}{\lambda}f(T,B)|_{\infty}}dz<\infty.$ (29)

The above integral converges only when $\xi>\lambda/2f(T,B)|_{\infty}$,
implying that the left-chirality zero modes can be localized on the brane
under this condition.

Figure 2:  The massless modes for $\lambda=p=\xi=1$, and various values of the
modified gravity parameter $\alpha$, for two cases of the $g(T,B)$ function,
namely, $g_{1}$ of (24), and $g_{2}$ of (25).

In Fig. 2 we present the massless modes for the two choices of the $f(T,B)$
function. As we observe, the massless fermionic modes are well localized in
our model. We mention that only the left-chirality fermions are localized in
the brane. For $f_{1}$ case, the massless fermions feel the changes in the
effective potential as $\alpha$ increases, and thus the massless modes become
more localized in the brane (see $f_{1}$). Concerning the second case $f_{2}$,
note that when $\alpha\rightarrow 0$ the massless mode appears symmetrically
at the origin of the brane. However, when we increase the effect of the
boundary term, the zero-mode solutions become deformed and more localized on
the right side of the brane (see $f_{2}$). This is an important feature, as
these fermions, although localized, are located outside the core of the brane.
Thus, this indicates that theoretically these fermions could only be detected
on one (right) side of the brane. This is one of the important results of the
present work, since it may correspond to a possible observable of these
fermionic particles and thus an indication for extra dimensions.

####  II.2.2 Massive modes

Let us now proceed to the determination of the massive fermionic modes. To
achieve this, considering the even symmetry of the effective potentials
illustrated in Fig. 1, we will numerically solve equations (II.2). It is
crucial to note that the resulting wave functions will exhibit either even or
odd symmetry. Consequently, the boundary conditions are set as follows

$\displaystyle\varphi_{even}(0)$ $\displaystyle=$ $\displaystyle c,\ \
\partial_{z}\varphi_{even}(0)=0,$ $\displaystyle\varphi_{odd}(0)$
$\displaystyle=$ $\displaystyle 0,\ \ \partial_{z}\varphi_{odd}(0)=c,$ (30)

where $c$ is a constant Almeida2009 ; Liu2009 ; Liu2009a . Here,
$\varphi_{even}$ and $\varphi_{odd}$ denote the even and odd parity modes of
$\varphi_{R,L}(z)$, respectively.

Figure 3:  The massive modes for $\lambda=p=\xi=1$, for the $f_{1}$ case of
(24) and various values of the modified gravity parameter $\alpha$, for even
case (a), and odd case (b).

Figure 4:  Same as Fig. 3 but for the $f_{2}$ case of (25).

As observed, massive fermionic modes typically resemble free wave solutions.
This resemblance stems from the substantial energy that these massive fermions
possess, ensuring that very massive fermions (high energy) tend to escape into
the extra dimension. However, it is important to note that despite their
tendency to escape into the bulk, these fermions can still sense geometric
modifications near the brane core. In Fig. 3, we depict the solutions of the
massive fermionic modes for the case $f_{1}$ of (24). It is evident that as
the parameter $\alpha$ increases, the oscillations in the brane core undergo
significant modifications. Thus, even though these modes are not localized on
the brane, their influence can be detected there, potentially affecting
measurements such as gravitational wave detections, leading to noise effects.

Moreover, the massive fermionic modes for $f_{2}$ exhibit asymmetric behavior,
as illustrated in Fig. 4. Increasing the influence of the boundary term in the
model results in heightened oscillations on the right side of the brane.
Noticeably, in this scenario, the brane behaves akin to a barrier, decreasing
the likelihood of fermions crossing to the left side.

Figure 5:  The entropy densities for $\lambda=p=\xi=1$. The left panels
correspond to position space (32), while the right panels, correspond to
momentum space (33). Additionally, the upper panels, correspond to the case
$f_{1}$ of (24), while the lower panels correspond to the case $f_{2}$ of
(25). Lighter regions correspond to higher probability of appearance than the
darker ones.

##  III Information measures

In this section, we apply information measures as a tool to further analyze
how geometry influences the localization of fermions on the brane.
Specifically, we utilize Shannon information entropy theory and Fisher
information measurement theory, focusing on the massless case. Additionally,
for the massive modes, we employ the relative probability formalism.

###  III.1 Shannon theory

Shannon’s entropy theory is basic in information theory Shannon . Its power
lies in offering insights into particle localization via the system
probability density $|\varphi_{L0,R0}(z)|^{2}$, and thus it can explore the
loss of information or noise between a transmitting and receiving source. We
start by applying the Fourier transformation to the massless mode function,
expressed as

$|\varphi_{L0,R0}(p_{z})|^{2}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}|\varphi_{L0,R0}(z)|^{2},\text{e}^{-ipz}dz,$
(31)

where $p_{z}$ is the coordinate of momentum space or reciprocal space. We
proceed to define Shannon entropy for both position and momentum spaces, as

$\displaystyle S_{z}$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{\infty}\rho_{s}(z)dz,$ (32) $\displaystyle
S_{p_{z}}$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{\infty}\rho_{s}(p_{z})dp_{z},$ (33)

where

$\displaystyle\rho_{s}(z)$ $\displaystyle=$
$\displaystyle|\varphi_{L0,R0}(z)|^{2}\ln|\varphi_{L0,R0}(z)|^{2},$ (34)
$\displaystyle\rho_{s}(p_{z})$ $\displaystyle=$
$\displaystyle|\varphi_{L0,R0}(p_{z})|^{2}\ln|\varphi_{L0,R0}(p_{z})|^{2},$
(35)

are the entropy densities of the system.

Relations (32), (33) yield an uncertainty relation, known as the Beckner,
Bialynicki-Birula, Mycielski (BBM) relation Beckner ; Bialy . The important
aspect of this entropy uncertainty relation is its effectiveness as an
alternative to the Heisenberg uncertainty principle. The BBM uncertainty
relation is mathematically expressed as

$S_{z}+S_{p_{z}}\geq D(1+\text{ln}\pi),$ (36)

where $D$ denotes the dimensions capable of discerning changes in the system
information. In our model, only the extra dimension perceives entropy
modifications within the system, and therefore $D=1$.

Figure 6:  The information densities for $\lambda=p=\xi=1$. The left panels
correspond to position space (37), while the right panels correspond to
momentum space (38). Additionally, the upper panels correspond to the case
$f_{1}$ of (24), while the lower panels correspond to the case $f_{2}$ of
(25). Lighter regions correspond to higher probability of appearance than the
darker ones.

The entropy densities can provide information on the behavior of the fermions
and their possible location. In Fig. 5, we present the shape of the entropy
density for our two choices of the $f(T,B)$ modified gravity function of (24)
and (25). Lighter regions correspond to a higher probability of appearance
than darker ones. For the $f_{1}$ case, the entropy density in position space
appears linearly at the origin of the brane, while in momentum space, the
density appears thicker. In contrast, for the $f_{2}$ case, the entropy
density deviates from the origin as we increase the value of $\alpha$.

Lastly, Shannon entropy provides additional information about the possible
location of the massless fermions. We numerically compute the equations and
present the results in Table 1. Increasing the effect of the boundary term
decreases Shannon entropy in position space, indicating reduced information
loss and increased certainty about the fermion’s location. However, Shannon
entropy in momentum space increases, reflecting greater uncertainty about the
particle momentum with larger information losses, consistent with the
uncertainty principle and satisfying the BBM relation. Finally, the total
entropy sum increases with larger $\alpha$ values, indicating increased total
information loss about the fermion.

Table 1: Shannon entropy values in position and momentum space for the two cases of the $f(T,B)$ function, with $\xi=p=\lambda=1$. $n$ | $\alpha$ | $S_{z}$ | $S_{p_{z}}$ | $S_{z}+S_{p_{z}}$
---|---|---|---|---
$g_{1}$ | 0.01 | 0.21998 | 2.28577 | 2.50575
| 0.10 | 0.20899 | 2.30903 | 2.51802
| 0.20 | 0.16477 | 2.40538 | 2.57015
| 0.25 | 0.15473 | 2.42789 | 2.58262
$g_{2}$ | 0.01 | 0.22122 | 2.28318 | 2.50441
| 0.10 | 0.22023 | 2.28722 | 2.50745
| 0.20 | 0.19858 | 2.37979 | 2.57837
| 0.25 | 0.19099 | 2.41879 | 2.60978

###  III.2 Fisher theory

Fisher information theory is a useful tool of quantifying the information
content carried by an observable random variable about an unknown parameter
Fisher . One key aspect of this theory is its connection to the probability
density of an information system, which can also be correlated with the
probability density observed in quantum systems. The informational density
associated with Fisher theory is expressed as

$\displaystyle\rho_{F}(z)$ $\displaystyle=$
$\displaystyle|\varphi_{L0,R0}(z)|^{2}\bigg{[}\frac{d}{dz}\ln|\varphi_{L0,R0}(z)|^{2}\bigg{]}^{2},$
(37) $\displaystyle\rho_{F}(p_{z})$ $\displaystyle=$
$\displaystyle|\varphi_{L0,R0}(p_{z})|^{2}\bigg{[}\frac{d}{dp_{z}}\ln|\varphi_{L0,R0}(p_{z})|^{2}\bigg{]}^{2}.$
(38)

Consequently, for the probability density of a continuous system, Fisher
information can be computed for position space $F_{z}$ and momentum space
$F_{p_{z}}$, namely

$\displaystyle F_{z}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\rho_{F}(z)dz,$ (39) $\displaystyle
F_{p_{z}}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\rho_{F}(p_{z})dp_{z}.$ (40)

We mention that these probabilistic measures of information also propose an
alternative to the Heisenberg uncertainty principle, namely

$\sigma_{z}\sigma_{p_{z}}\geq\frac{1}{(F_{z}F_{p_{z}})^{\frac{1}{2}}}\geq\frac{1}{2},$
(41)

leading to

$F_{z}F_{p_{z}}\geq 4,$ (42)

where $\sigma_{z}^{2}=\langle z^{2}\rangle-\langle z\rangle^{2}$ and
$\sigma_{p_{z}}^{2}=\langle p_{z}^{2}\rangle-\langle p_{z}\rangle^{2}$.

Information densities, akin to information entropy, reveal the possible
location of the fermions. In Fig. 6, we present the behavior of information
density for our two choices of the $f(T,B)$ modified gravity function of (24)
and (25). In the $f_{1}$ case, we observe that regions with low information
density separate regions with higher information density. Additionally, in
position space, there are two peaks in information density, while in momentum
space, there is only one. For the $f_{2}$ case, the information density
illustrates the deformation of the brane in both position and momentum spaces.

We numerically compute the equations and present the results in Table 2. As
observed, increasing the effect of the boundary term enhances the information
measures in position space. However, in momentum space, the opposite occurs
such that the total sum of information measures decreases as $\alpha$
increases. Larger Fisher information measures correspond to higher
probabilities of receiving information from the fermion (indicating less
information loss and noise), thereby increasing certainty about the fermion’s
location.

Table 2: Fisher information values in position and momentum space for the two cases of the $f(T,B)$ function, with $\xi=p=\lambda=1$.  $n$ | $k$ | $F_{z}$ | $F_{p_{z}}$ | $F_{z}F_{p_{z}}$
---|---|---|---|---
$g_{1}$ | 0.01 | 2.81545 | 7.74121 | 21.7949
| 0.10 | 2.87569 | 7.39004 | 21.2514
| 0.20 | 3.13204 | 6.02243 | 18.8623
| 0.25 | 3.19351 | 5.72364 | 18.2785
$g_{2}$ | 0.01 | 2.80880 | 7.91084 | 22.2199
| 0.10 | 2.81871 | 7.22346 | 20.3608
| 0.20 | 3.05356 | 5.75099 | 17.5609
| 0.25 | 3.15738 | 5.29128 | 16.7065

###  III.3 Relative probability

Let us now focus on the massive fermionic modes. Although they do not reside
on the brane, certain heavy states may exhibit significant presence in close
proximity to it Almeida2009 . These substantial states manifest when the
potentials near the brane form a potential well, accommodating masses $m^{2}$
within the highest value of the potential barrier Liu2009 ; Liu2009a . We
refer to these occurrences as resonant modes, which arise from the analogous
quantum mechanical structure of heavy modes Almeida2009 ; Liu2009 ; Liu2009a .

Figure 7:  The relative probability (43) for $\xi=p=\lambda=1$ and
$\alpha=0.1$, for two cases of the $g(T,B)$ function, namely: $g_{1}$ of (24),
and $g_{2}$ of (25).

To pinpoint the solutions of the Schrödinger-like equation (II.2) with
significant amplitudes near the brane, we apply the resonance method. The
relative probability $P(m)$ of detecting a particle with mass $m$ within a
narrow band $2z_{b}$ is given by Almeida2009 ; Liu2009 ; Liu2009a

$\displaystyle
P_{R,L}(m)=\frac{\int_{-z_{b}}^{z_{b}}|\varphi_{R,L}(z)|^{2}dz}{\int_{-z_{\text{max}}}^{z_{\text{max}}}|\varphi_{R,L}(z)|^{2}dz},$
(43)

where $z_{\text{max}}$ represents the domain boundaries. While larger values
of the parameter $z_{b}$ maintain consistency in locating the resonance peaks,
smaller $z_{b}$ values prove more adept at pinpointing these peaks.

We numerically compute the equations, and in Fig. 7, we illustrate the
relative probabilities for our model. The peaks in relative density delineate
resonant modes with the highest likelihood of locating massive fermions on the
brane. For the $f_{1}$ case, the peak occurs around $12<m^{2}<14$ (odd), while
for the $f_{2}$ case, the peak is around $4.5<m^{2}<6.0$ (odd). This indicates
that in this model, massive fermions with these mass spectra have a greater
chance of being detected on the brane.

##  IV Conclusions

In this work, we investigated the dynamics of fermion localization within the
framework of $f(T,B)$ gravity. Starting from the Dirac action for a fermion
with spin $1/2$ embedded in a 5-dimensional spacetime governed by torsional
$f(T,B)$ gravity, we formulated the Dirac equation and explored its solutions
under various non-minimal coupling functions.

We examined two realistic forms of the non-minimal coupling function $f(T,B)$,
i.e., $f_{1}(T,B)=\sqrt{-T-\alpha(B)}$ and $f_{2}(T,B)=\sqrt{-T}-\alpha B$.
Our analysis revealed distinct behaviors, highlighting implications for the
localization of both massless and massive fermionic modes on the brane.
Particularly, we observed how geometric deformations induced in the brane can
influence the localization of fermionic modes, potentially providing avenues
for experimental detection of these particles.

Furthermore, we employed probabilistic measures-Shannon entropy theory, Fisher
information theory, and relative probability-to investigate the localization
of massless and massive fermionic modes. Our analysis provided insights into
the information content and uncertainty associated with fermion localization
on the brane, offering a nuanced understanding of the interplay between
position and momentum spaces.

In summary, this study represents a significant step toward comprehending the
impact of modified gravity on fermion localization in brane world scenarios.
The observed effects could potentially serve as probes for torsional
modifications and may open new avenues for exploring holography and brane
cosmography.

## Acknowledgments

SHD would like to thank the partial support of project 20240220-SIP-IPN,
Mexico and started this work on the research stay in China. ENS acknowledges
the contribution of the LISA CosWG, and of COST Actions CA18108 ”Quantum
Gravity Phenomenology in the multi-messenger approach” and CA21136” Addressing
observational tensions in cosmology with systematics and fundamental physics
(CosmoVerse)”.

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