New mechanism for fermion localization in $f(T,T_G)$-brane

Abstract: We investigate the localization of fermionic fields in a five-dimensional braneworld scenario within the framework of modified teleparallel gravity described by a general $f(T,T_G)$ function. Considering a non-minimal coupling between a Dirac spinor and the torsional invariants, we derive the effective Schrödinger-like equations governing the Kaluza-Klein modes. We showed that the contribution of the teleparallel Gauss-Bonnet term significantly modifies the effective potentials and, consequently, the localization properties. The zero-mode analysis reveals that only one chiral component can be localized on the brane, with the degree of confinement depending on the chosen model. In the massive sector, the spectrum is continuous, but resonant states arise due to the internal structure of the potentials. Additionally, we employ information-theoretic measures, such as Shannon entropy and relative probability, to quantify the localization mechanism. Our results show that the torsional modifications induce a nontrivial redistribution of information, exhibiting stronger localization. These findings highlight the role of higher-order torsional terms in shaping fermionic localization and resonance structures in braneworld scenarios.

• The paper introduces a non-minimal coupling in f(T,T_G) gravity that enables effective chiral fermion localization on a five-dimensional thick brane.
• It employs Kaluza-Klein decomposition to reduce the field equations to Schrödinger-like forms, revealing volcano and double-well potential structures.
• Quantitative analyses using Shannon entropy and relative probability confirm that higher-order torsional terms control the zero-mode width and resonance behavior.

Fermion Localization via Non-Minimal Coupling in $f(T,T_G)$ Teleparallel Braneworlds

Theoretical Setting: $f(T,T_G)$ Gravity and Brane Configuration

The paper studies localization of spin-1/2 fields in a five-dimensional braneworld governed by modified teleparallel gravity, wherein the gravitational action is a general function $f(T, T_G)$ of the torsion scalar $T$ and the teleparallel analogue of the Gauss-Bonnet term $T_G$. Unlike the Levi-Civita formulation of gravity, the teleparallel approach encodes gravitational interactions via torsion and exploits the Weitzenböck connection, rendering spacetime curvature-free but torsion-rich.

The construction utilizes a thick-brane scenario with a warped geometry and a specific warp profile, $A(z) = -p\ln(\operatorname{arcsinh}(Az))$, supporting domain-wall-like solutions. In this framework, $T$ and $T_G$ become nontrivial functions of the warp factor and its derivatives, with $T_G$ contributing dynamically only for $D>4$.

Fermion Localization Mechanism and Schrödinger-Like Reduction

A non-minimal coupling between a 5D Dirac spinor and the torsional invariants is implemented via the bulk action,

$f(T,T_G)$0

where $f(T,T_G)$1 induces position-dependent mass terms, affecting the localization potential structure. After Kaluza-Klein decomposition and chiral reduction, the mode equations reduce to coupled first-order equations for left- and right-chiral components, and further to Schrödinger-like equations for the profiles along the extra dimension,

$f(T,T_G)$2

with superpartner potentials determined by the torsional background through the function $f(T,T_G)$3. Explicitly,

$f(T,T_G)$4

Two prototypical choices for the coupling function are analyzed:

• $f(T,T_G)$5
• $f(T,T_G)$6

Here, $f(T,T_G)$7 parametrizes the relative strength and sign of the higher-order torsional term.

Localization Properties: Zero-Modes and Massive Spectra

Chiral Zero Mode

The zero-mode ($f(T,T_G)$8) solution yields an analytic profile,

$f(T,T_G)$9

Normalizability criteria ensure that for positive fermion-torsion coupling, only one chiral sector (left-handed) is localizable; the other is delocalized. The structure and asymmetry of the zero-mode depend sensitively on the specific $f(T, T_G)$0 choice and on the parameter $f(T, T_G)$1:

• For $f(T, T_G)$2, the localization is around the brane core and becomes mildly asymmetric as $f(T, T_G)$3 increases, but the degree of localization grows, evident from faster tail decay.
• For $f(T, T_G)$4, the zero mode remains centered but exhibits sharper and more peaked localization as $f(T, T_G)$5 increases, indicating tighter chiral trapping.

Massive Spectrum and Resonances

The continuum of massive KK modes is characterized by oscillatory behavior away from the brane, with significant amplitude modification in the brane vicinity due to volcano-like or double-well structures of the effective potential. Notably:

• For $f(T, T_G)$6, the potential resembles a volcano well, supporting broad resonant structures.
• For $f(T, T_G)$7, the double-well potential generates sharper potential barriers and richer sets of quasi-bound resonant states.

Numerically, the dependence of resonances on $f(T, T_G)$8 is robust, with higher $f(T, T_G)$9 leading to more pronounced resonance peaks, particularly in the double-well case. Parity splitting of massive modes (even/odd) is directly linked to boundary conditions; odd modes, in particular, display enhanced brane-localized resonances.

Quantitative Information-Theoretic Analysis

To move beyond qualitative localization diagnostics, the study employs Shannon entropy and the relative probability method:

• Shannon Entropy: Computed for normalized probability densities of the zero-mode, in both position and momentum representations. Larger $T$0 increases position-space entropy, signaling de-localization, while momentum-space entropy decreases, indicating a redistribution in information content. Notably, the $T$1 (double-well) scenario produces consistently lower total entropy, a measure of tighter fermion localization.
• Relative Probability: For massive resonances, the relative probability quantifies the likelihood of finding the mode within the brane core. Peaks in this function denote quasi-localized (resonant) massive states. For $T$2, resonance peaks are narrower and higher than for $T$3, emphasizing the efficiency of the double-well in trapping quasi-bound fermions.

Both measures serve to corroborate the parametric control exercised by $T$4 and the detailed structure of $T$5 on the spatial localization and resonance properties.

Implications and Prospects

The results indicate that higher-order torsional terms in $T$6 gravity provide a powerful and tunable mechanism for chiral fermion localization on thick branes. The choice of coupling function can control not only the width and symmetry of the zero-mode but also the abundance and sharpness of resonant states in the massive spectrum. These features could, in principle, be exploited in model-building scenarios addressing phenomenological requirements in extra-dimensional extensions, such as suppressing undesired KK fermion modes or engineering brane-localized flavor structures.

On a theoretical level, the analysis highlights the distinction between localization mechanisms in teleparallel versus curvature-based Gauss-Bonnet theories—the former yielding dynamically active higher-order torsion effects in $T$7, leading to nontrivial modifications to field trapping phenomena.

Potential research directions include relaxing the function space for $T$8, incorporating interactions of other spin fields, and extending the information-theoretic approach via quantum entropy measures. The formalism and lessons of this study are broadly relevant to the construction and phenomenology of realistic braneworld models anchored in modified teleparallel gravity.

Conclusion

This work demonstrates, through analytic and numerical methods, that non-minimal geometric couplings in $T$9 teleparallel gravity allow for parametric control of fermion localization and resonance phenomena in five-dimensional brane scenarios. The interplay between the coupling function and the torsional invariants decisively shapes the phenomenological spectrum, with information-theoretic metrics providing rigorous quantification of localization efficiency. These findings underscore the functional role of higher-order torsional corrections in extra-dimensional physics, especially with regard to fermion trapping mechanisms and the structure of low-energy effective theories.