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arxiv: 2605.19221 · v1 · pith:TY7BP6BQnew · submitted 2026-05-19 · ✦ hep-th · gr-qc

Thick branes and fermion localization in five-dimensional f(T,T_G) gravity

A. R. P. Moreira , F. M. Belchior , Shi-Hai Dong , E. N. Saridakis This is my paper

Pith reviewed 2026-05-20 05:10 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords thick branesf(T,T_G) gravityteleparallel gravityfermion localizationwarped geometrybrane splittingKaluza-Klein modes
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0 comments X

The pith

In five-dimensional f(T,T_G) gravity the torsional Gauss-Bonnet term deforms thick branes and produces a normalizable chiral fermion zero mode together with resonant massive states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs explicit thick-brane solutions inside a warped five-dimensional geometry supported by a scalar field. It demonstrates that the torsional Gauss-Bonnet invariant T_G, which enters the gravitational action at linear order in five dimensions, controls the shape of the warp factor and the energy-density profile. Varying the coupling strength produces brane splitting and a nontrivial internal structure. For spin-1/2 fermions coupled through a Yukawa term the model yields a normalizable left-handed zero mode localized on the brane while the right-handed mode stays delocalized; the massive Kaluza-Klein tower experiences modified effective potentials that generate resonant quasi-localized states.

Core claim

Within a warped geometry supported by a scalar field, we construct explicit solutions and show that the T_G sector significantly modifies the brane structure. In particular, the coupling parameter controls the deformation of the warp factor and energy density, allowing for the emergence of brane splitting and nontrivial internal structure. We further analyze the localization of spin-1/2 fermions via a Yukawa coupling. The system admits a normalizable chiral zero mode, while the opposite chirality remains delocalized. The massive Kaluza-Klein spectrum is strongly affected by the torsional Gauss-Bonnet term, which modifies the effective potentials and leads to the appearance of resonant quasi-

What carries the argument

The torsional Gauss-Bonnet invariant T_G appearing in the five-dimensional f(T,T_G) action, which supplies an additional dynamical contribution to the warped metric and to the effective potentials felt by fermions.

If this is right

  • The coupling parameter in f(T,T_G) can be tuned to produce brane splitting and a rich internal energy-density profile.
  • A single chiral zero mode for fermions is normalizable and trapped on the brane while the opposite chirality is not.
  • Modified effective potentials for massive Kaluza-Klein fermions produce resonant quasi-localized states.
  • f(T,T_G) gravity supplies a broader class of braneworld geometries than standard teleparallel or Einstein gravity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same T_G-induced deformation mechanism could be examined in other extra-dimension scenarios to see whether splitting persists when the scalar field potential is changed.
  • Resonant massive modes might leave observable signatures in collider or cosmological data if the extra dimension is not too far above the electroweak scale.
  • Analogous torsional corrections could be studied in six- or higher-dimensional braneworld models to test the robustness of the localization results.

Load-bearing premise

The warped geometry ansatz supported by a scalar field admits consistent explicit solutions for the metric, warp factor, and energy density under the f(T,T_G) field equations.

What would settle it

Substitute the reported warp-factor and scalar-field profiles into the full set of f(T,T_G) field equations and verify that they are satisfied; any mismatch would show that the claimed brane solutions and the resulting localization properties do not hold.

Figures

Figures reproduced from arXiv: 2605.19221 by A. R. P. Moreira, E. N. Saridakis, F. M. Belchior, Shi-Hai Dong.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: illustrates the effective potentials VL(z) and VR(z), together with the corresponding zero-mode profiles χ (0) L (z) and χ (0) R (z), for the case ξ = λ = p = 1 and β = 1. As we observe, the left-handed potential VL(z) exhibits a characteristic volcano-like structure, with a central well surrounded by potential barriers. This structure supports a normalizable zero mode, whose wave function is sharply peake… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: presents the corresponding results for β = 2. As in the previous case, the modes display oscillatory profiles, but with noticeably enhanced amplitudes. This behavior arises from the stronger Yukawa coupling, which intensifies the interaction between the fermionic field and the scalar background. Moreover, the impact of the torsional Gauss￾Bonnet term becomes more pronounced. Negative values of α lead to st… view at source ↗
Figure 8
Figure 8. Figure 8: shows the behavior of P(m) as a function of m2 , for both even and odd modes, considering β = 1 (left panel) and β = 2 (right panel), with fixed parameters ξ = λ = p = 1 and α = −0.01. The presence of pronounced peaks at low masses signals the existence of resonant states, corresponding to quasi-localized fermions with enhanced probability density near the brane. As shown, for β = 1, the odd sector exhibit… view at source ↗
read the original abstract

We investigate thick-brane configurations in five-dimensional $f(T,T_G)$ modified teleparallel gravity. In five dimensions, the torsional Gauss-Bonnet invariant $T_G$ contributes dynamically, leading to genuinely new effects even at linear order. Within a warped geometry supported by a scalar field, we construct explicit solutions and show that the $T_G$ sector significantly modifies the brane structure. In particular, the coupling parameter controls the deformation of the warp factor and energy density, allowing for the emergence of brane splitting and nontrivial internal structure. We further analyze the localization of spin-$1/2$ fermions via a Yukawa coupling. The system admits a normalizable chiral zero mode, while the opposite chirality remains delocalized. The massive Kaluza-Klein spectrum is strongly affected by the torsional Gauss-Bonnet term, which modifies the effective potentials and leads to the appearance of resonant quasi-localized states.Our results show that $f(T,T_G)$ gravity provides a richer framework for braneworld models, where torsional higher-order corrections play a key role in shaping both geometry and field localization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper investigates thick brane configurations in five-dimensional f(T, T_G) modified teleparallel gravity. Within a warped geometry supported by a scalar field, explicit solutions are constructed showing that the T_G sector modifies the brane structure, with a coupling parameter controlling deformation of the warp factor and energy density to produce brane splitting and nontrivial internal structure. Fermion localization is studied via Yukawa coupling, yielding a normalizable chiral zero mode while the opposite chirality is delocalized; the massive Kaluza-Klein spectrum exhibits resonant quasi-localized states due to modifications of the effective potentials by the torsional Gauss-Bonnet term.

Significance. If the reported explicit solutions satisfy the full set of f(T, T_G) field equations, the work would demonstrate how torsional higher-order corrections enrich braneworld models by enabling controlled brane splitting and affecting fermion resonances, extending beyond standard teleparallel or Einstein gravity frameworks. The construction of explicit solutions and the identification of normalizable zero modes represent concrete strengths that could be falsifiable through the derived effective potentials.

major comments (2)
  1. [Section 3 (explicit solutions)] The central claim rests on explicit solutions for the warp factor A(y), scalar profile, and energy density within the warped ansatz. These must be shown to satisfy the complete f(T, T_G) field equations obtained by varying the action, including all linear-order contributions from the torsional Gauss-Bonnet invariant T_G; if the solutions are obtained only from a reduced system that neglects higher-order T_G terms, the reported deformation, splitting, and resonance effects would not follow from the theory.
  2. [Section 4 (fermion localization)] The effective potentials for the massive Kaluza-Klein fermions are stated to be strongly modified by the T_G term, leading to resonant states. The derivation of these potentials from the five-dimensional Dirac equation in the warped background must explicitly incorporate the torsional contributions at the level of the spin connection and metric functions to confirm that the resonances are not an artifact of the auxiliary assumptions used for the background.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should specify the explicit functional form chosen for f(T, T_G) at the outset, as the linear-order T_G effects depend on this choice.
  2. [Figures 1-2] In the plots of warp factor and energy density versus the extra dimension, the specific numerical values of the coupling parameter corresponding to the splitting cases should be indicated directly on the figures or in the captions for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each of the major comments in detail below and have made revisions to the manuscript to enhance clarity and rigor.

read point-by-point responses

  1. Referee: [Section 3 (explicit solutions)] The central claim rests on explicit solutions for the warp factor A(y), scalar profile, and energy density within the warped ansatz. These must be shown to satisfy the complete f(T, T_G) field equations obtained by varying the action, including all linear-order contributions from the torsional Gauss-Bonnet invariant T_G; if the solutions are obtained only from a reduced system that neglects higher-order T_G terms, the reported deformation, splitting, and resonance effects would not follow from the theory.

    Authors: We appreciate the referee pointing out the necessity of verifying the solutions against the full field equations. In our work, the solutions were derived by substituting the warped metric ansatz into the complete set of field equations from the variation of the f(T, T_G) action. Since in five dimensions T_G contributes at linear order in the perturbations around the background, the equations are solved including these terms without reduction or neglect of higher-order contributions. The specific choice of f(T, T_G) allows analytic solutions that satisfy all components of the equations. To make this explicit, we have added a new subsection in Section 3 detailing the substitution and verification that the solutions fulfill the full field equations derived from the action. revision: yes

  2. Referee: [Section 4 (fermion localization)] The effective potentials for the massive Kaluza-Klein fermions are stated to be strongly modified by the T_G term, leading to resonant states. The derivation of these potentials from the five-dimensional Dirac equation in the warped background must explicitly incorporate the torsional contributions at the level of the spin connection and metric functions to confirm that the resonances are not an artifact of the auxiliary assumptions used for the background.

    Authors: We thank the referee for this comment, which helps strengthen the presentation. The effective potentials are derived from the five-dimensional Dirac equation, where the spin connection in the teleparallel formulation includes the torsional contributions. The warp factor A(y) and its derivatives, modified by the T_G term, enter the potentials through both the metric and the connection terms. We have revised Section 4 to include the explicit steps: starting from the Dirac equation with the full spin connection incorporating torsion, reducing to the effective 4D Schrödinger equation, and showing how the T_G-induced modifications to A(y) affect the potential shape leading to resonances. This confirms the effects are genuine consequences of the theory. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives explicit solutions for the warp factor, scalar profile, and energy density by assuming a standard five-dimensional warped metric ansatz supported by a scalar field and substituting into the field equations obtained from varying the f(T, T_G) action. The T_G contributions enter the equations at linear order and modify the resulting profiles for chosen values of the coupling parameter, leading to brane splitting and altered fermion potentials. These steps are independent: the solutions satisfy the modified teleparallel equations by construction of the ansatz, the localization analysis follows from the resulting effective potentials without renaming prior results or relying on load-bearing self-citations, and no fitted subset is relabeled as a prediction. The framework is self-contained against the stated assumptions and does not reduce the central claims to inputs by definition.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only view limits detail, but the model rests on an assumed warped metric ansatz, a scalar-field source, and an unspecified functional form for f(T, T_G) whose parameters are tuned to produce splitting.

free parameters (1)
  • coupling parameter
    Described as controlling warp-factor deformation and energy-density profile to produce brane splitting.
axioms (1)
  • domain assumption Warped geometry ansatz supported by a scalar field in five dimensions
    Invoked to construct explicit solutions for the brane and to analyze fermion localization.

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Reference graph

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