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REVIEW 3 major objections 5 minor 3 cited by

In any warped Randall-Sundrum braneworld, only a free scalar and the square-root NED model admit a consistent, normalizable zero mode; gauge fields, p-forms with p≠0, and Dirac fermions are excluded.

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2026-08-03 11:45 UTC pith:ZHKUQHB7

load-bearing objection Useful local consistency framework, but headline no-go claims overreach because the proofs only cover the brane-index branch. the 3 major comments →

arxiv 2601.05190 v2 pith:ZHKUQHB7 submitted 2026-01-08 gr-qc

The End of the Road for Bulk Fields in Warped Randall-Sundrum Braneworlds

G. Alencar , R. S. Almeida , R. N. Costa Filho , T. M. Crispim , Francisco S. N. Lobo This is my paper
classification gr-qc PACS 04.50.-h11.10.Kk11.25.-w
keywords braneworldsfield localizationconsistency conditionsp-form fieldsnonlinear electrodynamicsDirac fermionsRandall-Sundrumwarped geometry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper aims to close the question of which fields can propagate in the bulk of a warped Randall-Sundrum braneworld, in any number of dimensions. It derives four local consistency conditions that any bulk field must satisfy to be compatible with the full Einstein equations, and then applies them to standard separable zero modes. The result is a sweeping no-go theorem: only a free scalar field and the nonlinear electrodynamics Lagrangian L(F)=b√F survive, while Maxwell fields, all p-forms with p≠0, and Dirac fermions (with or without Yukawa couplings) are ruled out in any dimension and for any internal geometry. If correct, this sharply constrains braneworld model-building and invalidates a large class of proposed localization mechanisms.

Core claim

The paper argues that in any braneworld metric of the form ds² = e^{2σ(y)}g_μν(x)dx^μdx^ν + g_jk(y)dy^jdy^k, consistency with the full Einstein equations forces the energy-momentum tensor of any bulk field to obey four local conditions. Applying these conditions to separated zero modes, the authors show that the scalar potential must vanish, leaving the free 0-form as the only allowed p-form; that Maxwell fields fail even with scalar or geometric couplings; that among all nonlinear electrodynamics only L(F)=b√F satisfies the constraints; and that Dirac fermions are inconsistent even when Yukawa terms are added. These claims are presented as dimension-independent and independent of the warp f

What carries the argument

The central machinery is a set of four local consistency conditions (CCI–CCIV) obtained by combining the D-dimensional Einstein equations with the warped metric ansatz: (b)T_μj = 0, n (b)T^α_α − (d−2)(b)T^j_j = 0, (b)T_μν depends only on brane coordinates, and (b)T^j_j is fixed in terms of the brane Ricci scalar. The first condition forces the extra-dimensional profile of a separated zero mode to be constant; the trace condition then either kills the field's effective action or, for nonlinear electrodynamics, enforces a differential equation with unique solution L(F)=b√F.

Load-bearing premise

The no-go proofs assume that every localized zero mode decomposes with all Lorentz indices carried by the brane directions (e.g., A_μ1...μp(x) ξ(y) and Ψ(x,y)=ψ(x) ξ(y)); if a zero mode with indices along the extra dimensions (the magnetic/dual branch) is allowed, the same local conditions may admit other p-forms.

What would settle it

Construct a p-form zero mode with one index along the extra dimension, e.g., A_μ1...μ_{p−1}y(x,y)= That{A}_μ1...μ_{p−1}(x) ξ(y), in a six-dimensional warped metric of the type in Eq. (139), and verify whether it satisfies all four local consistency conditions and is normalizable; a single such solution would refute the claim that only p=0 survives.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • All existing localization mechanisms for gauge fields, p-forms, and fermions in warped braneworlds are generically excluded, including models with scalar couplings, geometric couplings, and Yukawa interactions.
  • The square-root NED model L(F)=b√F is singled out as the unique consistent and normalizable nonlinear electrodynamics, providing a concrete candidate for constructing localized black hole geometries in braneworlds.
  • Earlier global sum-rule consistency conditions are superseded by local constraints that hold pointwise in the extra dimensions, making the restrictions independent of warp factor and internal geometry.
  • Since the conditions are dimension-independent, higher-dimensional compactifications and string-inspired setups face the same restrictions on bulk fields.
  • If the paper is correct, only scalar fields remain generically available as bulk matter in Einstein gravity warped braneworlds, which would radically simplify phenomenology but also limit model-building flexibility.

Where Pith is reading between the lines

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

  • The no-go theorem rests on a separation-of-variables ansatz with all Lorentz indices along the brane; if the 'magnetic' branch—where the field carries one or more indices in the extra dimensions—were included, the conclusions for p-forms might change, as earlier Hodge-dual studies suggested.
  • The selection of L(F)=b√F follows from the tracelessness condition L=2F L_F; since this condition is conformal-like, the result might extend to other geometric settings or to trace-free formulations of gravity.
  • The consistency conditions are derived from Einstein gravity; in modified gravity (higher-curvature or scalar-tensor theories), the conditions would be altered, potentially admitting a larger class of localized fields—an avenue the paper leaves for future work.
  • A concrete test of the NED result would be to check whether the forthcoming localized black hole solutions indeed require L(F)=b√F, which would confirm the uniqueness from a different observable angle.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript generalizes the local consistency conditions derived in Ref. [1] for five-dimensional Randall-Sundrum braneworlds to arbitrary dimensions, with a d-dimensional brane and n extra dimensions (metric of Eq. (59)). The authors derive four local constraints (CCI-CCIV) on the bulk matter energy-momentum tensor from the Einstein equations, and then apply them to scalar, Maxwell, NED, p-form, and Dirac fermion fields. The advertised conclusions are that only a free scalar and the NED model L(F)=b√F admit a consistent and normalizable zero mode, that among p-forms only the 0-form is consistent, and that Dirac fermions with or without Yukawa couplings cannot be consistently localized.

Significance. If fully established, these results would constitute a strong, local, dimension-independent generalization of the Duff-Liu and Gibbons-Kallosh-Linde consistency criteria and would rule out a large class of bulk-field localization models without relying on the detailed equations of motion. The derivation of the local conditions from the Einstein equations is a useful contribution, and the paper correctly emphasizes that previously known global sum rules are weaker than local constraints. The paper also provides a broad catalog of excluded models, which is valuable. However, the advertised universality is not yet supported: the p-form no-go omits an entire branch of components, and the fermionic analysis is limited to codimension-one.

major comments (3)
  1. [Sec. IV.B, Eqs. (132)-(138)] The p-form no-go theorem is not proven for arbitrary p-forms. The proof begins by assuming 'only the components A_{\mu_1...\mu_p} are nonzero' and separating variables as A_{\mu_1...\mu_p}=\hat A_{\mu_1...\mu_p}(x)\,\xi(y). It never substitutes into CCI-CCIV the components with extra-dimensional indices, i.e., the internal-index (or Hodge-dual) branch. Thus the abstract's claim that 'among p-forms, consistency occurs solely for the free 0-form' is unsupported. This is not a pedantic point: Sec. II.B reviews Duff-Liu's result that the complementary branch is the one that can satisfy the Einstein equations. The theorem must either be extended to all component branches or restated with the additional assumption.
  2. [Sec. IV.C] The fermionic no-go is derived only for n=1 (codimension-one), using Ψ=ψ(x)⊗ξ(y) and a gamma-matrix decomposition specific to D=2k+2 or 2k+3. The abstract and introduction, however, claim that Dirac fermions are inconsistent 'within this framework' without this restriction, while the general framework of Sec. III applies to arbitrary n. The fermionic conclusion therefore needs either a genuine n>1 analysis or an explicit restriction in the abstract and conclusion. This is a mismatch between the advertised scope and the actual proof.
  3. [Secs. IV.A and IV.B (NED)] The paper asserts that the free scalar and the square-root NED have 'consistent and normalizable' zero modes, but normalizability is never checked: no extra-dimensional integral of the effective action is evaluated, and the convergence condition is not stated. The local consistency conditions CCI-CCIV do not imply finiteness of the integral over the internal space. For the positive claims, the authors must verify e.g. ∫ d^n y √{\tilde g} e^{(d-2)σ} ξ² < ∞ for the scalar, or explicitly state the class of warp factors for which such convergence holds.
minor comments (5)
  1. [Sec. IV.B, Eq. (134)] The text says 'For the condition CCIII to be satisfied' when Eq. (134) is the μ-j component and the relevant condition is CCI. Similar cross-reference mismatches occur in Sec. IV.C, where the old five-dimensional conditions (42)-(45) are cited instead of the new general conditions CCI-CCIV.
  2. [Secs. II.D, IV.C] There are many typographical and OCR-type errors: for example 'e−−−A' and 'e−333A' in Eqs. (48)-(54), and inconsistent use of σ vs. A in the fermion section (e.g., Eq. (165) uses e^{-2A} where e^{-2σ} is meant). These should be cleaned up.
  3. [Sec. II.D] The passage correcting Ref. [1] mid-review ('the terms previously highlighted in bold are in fact zero...') reads like an erratum embedded in the main text. It would be clearer placed in a footnote, an erratum, or a separate subsection.
  4. [Secs. IV.B, IV.C] The step 'CCI implies ξ=constant' in the vector, NED, and p-form proofs does not discuss exceptional configurations in which the relevant contraction (e.g., \hat F_{\mu\nu}\hat A^\nu) vanishes identically. For a fully rigorous no-go, these cases should be excluded or analyzed.
  5. [Sec. III.A] The six-dimensional background example with F_{ij}=k ε_{ij} is a gauge-field configuration with only internal components. This may appear to contradict the later statement that gauge fields are excluded; the distinction between a background field without a brane-localized zero mode and a localized zero mode should be made explicit.

Circularity Check

0 steps flagged

No significant circularity: the local consistency conditions are derived from the Einstein equations, and the field no-go statements follow algebraically from those conditions. The p-form internal-index branch is an omitted-case gap, not a circular reduction.

full rationale

The paper's core derivation is self-contained rather than circular. The four local consistency conditions (CCI-CCIV), Eqs. (119)-(122), are obtained by substituting the decomposition T_MN = (v)T_MN + (b)T_MN into the D-dimensional Einstein equations, using the background solution (95)-(96); they are not postulated or fitted. Each field application then evaluates the stress-energy tensor under an explicit separation ansatz. For scalars, CCI forces ξ constant and CCII forces V(Φ)=0. For p-forms, the proof in Eqs. (132)-(138) computes T^{μ1j}, T^{μν}, and the CCII combination; the factor "n p" in Eq. (137) makes CCII fail for p≠0, so the algebraic exclusion of the brane-index branch is a genuine consequence, not a restatement of the conclusion. For NED, the uniqueness of L(F)=b√F follows from solving the ordinary differential equation L=2F dL/dF, Eq. (146), which is derived from CCII; this is a mathematical derivation, not an input. The fermion no-go is computed from the displayed spinor energy-momentum components, and the paper explicitly corrects Ref. [1] ('contrary to the original claim, it is Eq. (48) that fails to satisfy the consistency conditions'), so it does not rely on that self-citation for its conclusion. Self-citations such as Refs. [1,17,30,80] supply standard decompositions or earlier context, but the load-bearing derivations appear in the text, so no central claim reduces to a self-citation. The genuine weakness is completeness, not circularity: the p-form no-go assumes 'only the components A_{μ1...μp} are nonzero' and never analyzes the complementary internal-index branch, which earlier Duff-Liu work suggests may behave differently. Likewise, the paper announces normalizability for L(F)=b√F without showing the extra-dimensional integral explicitly. These are important rigor gaps that could invalidate the strong 'only the 0-form' claim, but they are not cases of a prediction being equivalent to an input by construction. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work to force the conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central claims rest on the standard warped-brane metric ansatz, the split of the energy-momentum tensor into vacuum and bulk parts, constant-curvature brane, and separation of variables for zero modes. No numerical free parameters are fitted; the only undetermined constant is b in L=b√F, which is a coupling in the allowed theory rather than a fitted input.

axioms (5)
  • domain assumption The full D-dimensional metric has the warped product form (59) with no off-diagonal components g_{μj}.
    All derivations of the local conditions and all applications assume the braneworld metric ansatz (59); off-diagonal metric components would change the Einstein equations (61) and hence condition CCI.
  • domain assumption The bulk energy-momentum tensor splits as (v)T + (b)T, with the vacuum part alone sourcing the background geometry.
    Used in Sec. III.B to subtract the background equations and obtain CCII-CCIV; if the bulk field itself contributes to the background generation, the conditions change.
  • domain assumption The brane has constant curvature, so (d)G_{μν} depends only on x.
    Required for CCIII, Eq. (116); the paper assumes this in deriving the background (95).
  • standard math Spinor representations in D dimensions and the tensor-product decompositions (156) and (172) are valid.
    The fermion analysis relies on properties of gamma matrices and spinor decompositions in odd/even D, Sec. IV.C; not independently proven in the paper.
  • domain assumption A localized zero mode admits a factorization Φ(x,y)=ξ(y)ϕ(x) (separation of variables).
    Used throughout Sec. IV to compute energy-momentum tensors; if the zero mode does not factorize, the conditions must be re-evaluated.

pith-pipeline@v1.3.0-alltime-deepseek · 26904 in / 28548 out tokens · 277645 ms · 2026-08-03T11:45:14.622563+00:00 · methodology

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read the original abstract

In this manuscript we generalize Ref. [1] and derive a complete set of local consistency conditions for bulk fields in braneworld scenarios with an arbitrary number of dimensions. This provides the first fully local and dimension-independent generalization of all known criteria for bulk fields. Within this framework, we show that a free scalar field is consistent and localized, whereas minimally and non-minimally coupled Maxwell fields violate the conditions, leading to a no-go theorem valid in any dimension. For nonlinear electrodynamics, we find that only the model $L(F)=b\sqrt{F}$ admits a consistent and normalizable zero mode, and that among p-forms, consistency occurs solely for the free 0-form. We also demonstrate that Dirac fermions, with or without Yukawa terms, are inconsistent within this framework and therefore cannot propagate in the bulk. Our local approach makes explicit that these conclusions do not depend on any particular internal geometry or warp factor: previously known results arise merely as special cases of a broader and strictly local structure, highlighting the universality of the constraints derived here.

discussion (0)

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Forward citations

Cited by 3 Pith papers

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  2. Kerr-Schild Double Copy of the Randall-Sundrum Black String

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Reference graph

Works this paper leans on

96 extracted references · 86 linked inside Pith · cited by 2 Pith papers

  1. [1]

    Ψ(xν,y) =ξ(y)ψ(x ν), ¯Ψ(xν,y) =ξ †(y) ¯ψ(xν).(156) With this type of decomposition, we obtain the following spin connections

    Spinors in odd-dimensional spacetimes,D= 2k+ 3 For the odd-dimensional case, and in the absence of the Yukawa coupling, the spinor field inDdimensions can be decomposed into an effective spinor living on thed-dimensional brane and a scalar profile that encodes the dependence of the field along the extra dimension [1, 17, 80]. Ψ(xν,y) =ξ(y)ψ(x ν), ¯Ψ(xν,y)...

  2. [2]

    Spinors in even-dimensional spacetimes,D= 2k+ 2 In the even-dimensional case, we cannot use the same spinor decomposition adopted for odd values ofD. When performing the dimensional reduction tod=D−1, the effective spinor must change its number of components, since the spinorial representation ofSO(1,D−1)is reducible (it admits a chiral decomposition), wh...

    2018

  3. [3]

    Alencar, R

    G. Alencar, R. N. Costa Filho, and M. Gogberashvili, Eur. Phys. J. C85, 491 (2025), arXiv:2405.04567 [hep-ph]

    Pith/arXiv arXiv 2025

  4. [4]

    Gogberashvili, Int

    M. Gogberashvili, Int. J. Mod. Phys. D11, 1635 (2002), arXiv:hep-ph/9812296

    Pith/arXiv arXiv 2002

  5. [5]

    Gogberashvili, Mod

    M. Gogberashvili, Mod. Phys. Lett. A14, 2025 (1999), arXiv:hep-ph/9904383

    Pith/arXiv arXiv 2025

  6. [6]

    Randall and R

    L. Randall and R. Sundrum, Phys. Rev. Lett.83, 3370 (1999), arXiv:hep-ph/9905221

    Pith/arXiv arXiv 1999

  7. [7]

    Randall and R

    L. Randall and R. Sundrum, Phys. Rev. Lett.83, 4690 (1999), arXiv:hep-th/9906064

    Pith/arXiv arXiv 1999

  8. [8]

    Maartens and K

    R. Maartens and K. Koyama, Living Rev. Rel.13, 5 (2010), arXiv:1004.3962 [hep-th]

    Pith/arXiv arXiv 2010

  9. [9]

    Raychaudhuri and K

    S. Raychaudhuri and K. Sridhar,Particle Physics of Brane Worlds and Extra Dimensions, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2016)

    2016

  10. [10]

    Shiromizu, K.-i

    T. Shiromizu, K.-i. Maeda, and M. Sasaki, Phys. Rev. D62, 024012 (2000)

    2000

  11. [11]

    Fichet, JHEP04, 016 (2020), arXiv:1912.12316 [hep-th]

    S. Fichet, JHEP04, 016 (2020), arXiv:1912.12316 [hep-th]

    Pith/arXiv arXiv 2020

  12. [12]

    Davoudiasl, J

    H. Davoudiasl, J. L. Hewett, and T. G. Rizzo, Phys. Lett. B473, 43 (2000), arXiv:hep-ph/9911262

    Pith/arXiv arXiv 2000

  13. [13]

    Pomarol, Phys

    A. Pomarol, Phys. Lett. B486, 153 (2000), arXiv:hep-ph/9911294

    Pith/arXiv arXiv 2000

  14. [14]

    Kaloper, E

    N. Kaloper, E. Silverstein, and L. Susskind, JHEP05, 031 (2001), arXiv:hep-th/0006192 . 25

    Pith/arXiv arXiv 2001

  15. [15]

    M. J. Duff and J. T. Liu, Phys. Lett. B508, 381 (2001), arXiv:hep-th/0010171

    Pith/arXiv arXiv 2001

  16. [16]

    G. W. Gibbons, R. Kallosh, and A. D. Linde, JHEP01, 022 (2001), arXiv:hep-th/0011225

    Pith/arXiv arXiv 2001

  17. [17]

    W. D. Goldberger and M. B. Wise, Phys. Rev. Lett.83, 4922 (1999), arXiv:hep-ph/9907447

    Pith/arXiv arXiv 1999

  18. [18]

    Leblond, R

    F. Leblond, R. C. Myers, and D. J. Winters, JHEP07, 031 (2001), arXiv:hep-th/0106140

    Pith/arXiv arXiv 2001

  19. [19]

    L. F. F. Freitas, G. Alencar, and R. R. Landim, Eur. Phys. J. C80, 432 (2020), arXiv:2001.01267 [hep-th]

    Pith/arXiv arXiv 2020

  20. [21]

    R. R. Landim, M. O. Tahim, G. Alencar, and R. N. Costa Filho, Mod. Phys. Lett. A32, 1750193 (2017), arXiv:1505.01061 [hep-th]

    Pith/arXiv arXiv 2017

  21. [22]

    Liu, L.-D

    Y.-X. Liu, L.-D. Zhang, L.-J. Zhang, and Y.-S. Duan, Phys. Rev. D78, 065025 (2008), arXiv:0804.4553 [hep-th]

    Pith/arXiv arXiv 2008

  22. [23]

    Koley and S

    R. Koley and S. Kar, Class. Quant. Grav.22, 753 (2005), arXiv:hep-th/0407158 [hep-th]

    Pith/arXiv arXiv 2005

  23. [24]

    Zhao, Y.-X

    Z.-H. Zhao, Y.-X. Liu, H.-T. Li, and Y.-Q. Wang, Phys. Rev. D82, 084030 (2010), arXiv:1004.2181 [hep-th]

    Pith/arXiv arXiv 2010

  24. [25]

    Li, Y.-X

    H.-T. Li, Y.-X. Liu, Z.-H. Zhao, and H. Guo, Phys. Rev. D83, 045006 (2011), arXiv:1006.4240 [hep-th]

    Pith/arXiv arXiv 2011

  25. [26]

    Y.-X. Liu, J. Yang, Z.-H. Zhao, C.-E. Fu, and Y.-S. Duan, Phys. Rev. D80, 065019 (2009), arXiv:0904.1785 [hep-th]

    Pith/arXiv arXiv 2009

  26. [27]

    H. Guo, A. Herrera-Aguilar, Y.-X. Liu, D. Malagon-Morejon, and R. Mora-Luna, Phys. Rev. D87, 095011 (2013), arXiv:hep-th/1103.2430 [hep-th]

    Pith/arXiv arXiv 2013

  27. [28]

    Liu, C.-E

    Y.-X. Liu, C.-E. Fu, L. Zhao, and Y.-S. Duan, Phys. Rev. D80, 065020 (2009), arXiv:0907.0910 [hep-th]

    Pith/arXiv arXiv 2009

  28. [29]

    Brihaye and T

    Y. Brihaye and T. Delsate, Phys. Rev. D86, 024029 (2012), arXiv:1204.6110 [gr-qc]

    Pith/arXiv arXiv 2012

  29. [30]

    Oda, (2000), arXiv:hep-th/0009074

    I. Oda, (2000), arXiv:hep-th/0009074

    Pith/arXiv arXiv 2000

  30. [31]

    Ringeval, P

    C. Ringeval, P. Peter, and J.-P. Uzan, Phys. Rev. D65, 044016 (2002), arXiv:hep-th/0109194

    Pith/arXiv arXiv 2002

  31. [32]

    W. M. Mendes, G. Alencar, and R. R. Landim, JHEP1802, 018 (2018), arXiv:arXiv:1712.02590 [hep-th]

    Pith/arXiv arXiv 2018

  32. [33]

    Melfo, N

    A. Melfo, N. Pantoja, and J. D. Tempo, Phys. Rev. D73, 044033 (2006), arXiv:hep-th/0601161 [hep-th]

    Pith/arXiv arXiv 2006

  33. [34]

    L. B. Castro and L. A. Meza, EPL102, 21001 (2013), arXiv:arXiv:1011.5872 [hep-th]

    Pith/arXiv arXiv 2013

  34. [35]

    Barbosa-Cendejas, D

    N. Barbosa-Cendejas, D. Malagón-Morejón, and R. R. Mora-Luna, Gen. Rel. Grav.47, 77 (2015), arXiv:1503.07900 [hep-th]

    Pith/arXiv arXiv 2015

  35. [36]

    Kehagias and K

    A. Kehagias and K. Tamvakis, Phys. Lett. B504, 38 (2001), arXiv:hep-th/0010112 [hep-th]

    Pith/arXiv arXiv 2001

  36. [37]

    Zhao, Y.-X

    Z.-H. Zhao, Y.-X. Liu, and H.-T. Li, Class. Quant. Grav.27, 185001 (2010), arXiv:0911.2572 [hep-th]

    Pith/arXiv arXiv 2010

  37. [38]

    Farokhtabar and A

    A. Farokhtabar and A. Tofighi, Adv. High Energy Phys.2017, 3926286 (2017), arXiv:1610.00899 [hep-th]

    Pith/arXiv arXiv 2017

  38. [39]

    C. A. S. Almeida, M. M. Ferreira, Jr., A. R. Gomes, and R. Casana, Phys. Rev. D79, 125022 (2009), arXiv:0901.3543 [hep-th]

    Pith/arXiv arXiv 2009

  39. [40]

    Bazeia and A

    D. Bazeia and A. R. Gomes, JHEP05, 012 (2004), arXiv:hep-th/0403141

    Pith/arXiv arXiv 2004

  40. [41]

    Guerrero, A

    R. Guerrero, A. Melfo, N. Pantoja, and R. O. Rodriguez, Phys. Rev. D81, 086004 (2010), arXiv:0912.0463 [hep-th]

    Pith/arXiv arXiv 2010

  41. [43]

    Bajc and G

    B. Bajc and G. Gabadadze, Phys. Lett. B474, 282 (2000), arXiv:hep-th/9912232 [hep-th]

    Pith/arXiv arXiv 2000

  42. [44]

    Oda, Phys

    I. Oda, Phys. Lett. B508, 96 (2001), arXiv:hep-th/0012013 [hep-th]

    Pith/arXiv arXiv 2001

  43. [45]

    Liu, X.-H

    Y.-X. Liu, X.-H. Zhang, L.-D. Zhang, and Y.-S. Duan, JHEP0802, 067 (2008), arXiv:hep-th/0708.0065 [hep-th]

    Pith/arXiv arXiv 2008

  44. [46]

    Liu, C.-E

    Y.-X. Liu, C.-E. Fu, H. Guo, S.-W. Wei, and Z.-H. Zhao, JCAP1012, 031 (2010), arXiv:hep-th/1002.2130 [hep-th]

    Pith/arXiv arXiv 2010

  45. [47]

    Y.-X. Liu, H. Guo, C.-E. Fu, and H.-T. Li, Phys. Rev. D84, 044033 (2011), arXiv:hep-th/1101.4145 [hep-th]

    Pith/arXiv arXiv 2011

  46. [48]

    Liu, Z.-H

    Y.-X. Liu, Z.-H. Zhao, S.-W. Wei, and Y.-S. Duan, JCAP0902, 003 (2009), arXiv:hep-th/0901.0782 [hep-th]

    Pith/arXiv arXiv 2009

  47. [49]

    Oda, Phys

    I. Oda, Phys. Lett. B496, 113 (2000), arXiv:hep-th/0006203 [hep-th]

    Pith/arXiv arXiv 2000

  48. [50]

    Oda, Phys

    I. Oda, Phys. Rev. D62, 126009 (2000), arXiv:hep-th/0008012 [hep-th]

    Pith/arXiv arXiv 2000

  49. [51]

    Midodashvili, (2003), arXiv:hep-th/0308003 [hep-th]

    P. Midodashvili, (2003), arXiv:hep-th/0308003 [hep-th]

    Pith/arXiv arXiv 2003

  50. [52]

    Oda, Phys

    I. Oda, Phys. Lett. B571, 235 (2003), arXiv:hep-th/0307119 [hep-th]

    Pith/arXiv arXiv 2003

  51. [53]

    Batell and T

    B. Batell and T. Gherghetta, Phys. Rev. D75, 025022 (2007), arXiv:hep-th/0611305 [hep-th]

    Pith/arXiv arXiv 2007

  52. [54]

    Casadioa and A

    R. Casadioa and A. Gruppuso, Phys. Rev. D64, 025020 (2001), arXiv:hep-th/0103200 [hep-th]

    Pith/arXiv arXiv 2001

  53. [55]

    Alencar, R

    G. Alencar, R. R. Landim, M. O. Tahim, C. R. Muniz, and R. N. C. Filho, Phys. Lett. B693, 503 (2010), arXiv:arXiv:1008.0678 [hep-th]

    Pith/arXiv arXiv 2010

  54. [56]

    Germani, Phys

    C. Germani, Phys. Rev. D85, 055025 (2012), arXiv:arXiv:1109.3718 [hep-th]

    Pith/arXiv arXiv 2012

  55. [57]

    R. G. Landim, Eur. Phys. J. C79, 862 (2019), arXiv:arXiv:1907.10460 [hep-th]

    Pith/arXiv arXiv 2019

  56. [58]

    R. G. Landim and T. G. Rizzo, JHEP1906, 112 (2019), arXiv:arXiv:1902.08339 [hep-th]

    Pith/arXiv arXiv 2019

  57. [59]

    Oda, arXiv (2001), arXiv:hep-ph/0103052 [hep-ph]

    I. Oda, arXiv (2001), arXiv:hep-ph/0103052 [hep-ph]

    Pith/arXiv arXiv 2001

  58. [60]

    Ghoroku and A

    K. Ghoroku and A. Nakamura, Phys. Rev. D65, 084017 (2002), arXiv:hep-th/0106145 [hep-th]

    Pith/arXiv arXiv 2002

  59. [62]

    Z. H. Zhao, Q. Y. Xie, and Y. Zhong, Class. Quant. Grav.32, 035020 (2015), arXiv:arXiv:1406.3098 [hep-th]

    Pith/arXiv arXiv 2015

  60. [63]

    Fu, Y.-X

    C.-E. Fu, Y.-X. Liu, and H. Guo, Phys. Rev. D84, 044036 (2011), arXiv:arXiv:1101.0336 [hep-ph]

    Pith/arXiv arXiv 2011

  61. [65]

    Alencar, I

    G. Alencar, I. C. Jardim, R. R. Landim, C. R. Muniz, and R. N. C. Filho, Phys. Rev. D93, 124064 (2016), arXiv:arXiv:1506.00622 [hep-th]

    Pith/arXiv arXiv 2016

  62. [66]

    W. T. Cruz, M. O. Tahim, and C. A. S. Almeida, Phys. Lett. B686, 259 (2010)

    2010

  63. [67]

    W. T. Cruz, A. R. P. Lima, and C. A. S. Almeida, Phys. Rev. D87, 045018 (2012), arXiv:arXiv:1211.7355 [hep-th]

    Pith/arXiv arXiv 2012

  64. [68]

    Flachi and M

    A. Flachi and M. Minamitsuji, Phys. Rev. D79, 104021 (2009), arXiv:arXiv:0903.0133 [hep-th]

    Pith/arXiv arXiv 2009

  65. [69]

    L. F. Freitas, G. Alencar, and R. R. Landim, JHEP1902, 035 (2019), arXiv:arXiv:1809.07197 [hep-ph]

    Pith/arXiv arXiv 2019

  66. [70]

    F. W. V. Costa, J. E. G. Silva, and C. A. S. Almeida, Phys. Rev. D87, 125010 (2013), arXiv:arXiv:1304.7825 [hep-ph]

    Pith/arXiv arXiv 2013

  67. [71]

    F. W. V. Costa, J. E. G. Silva, D. F. S. Veras, and C. A. S. Almeida, Phys. Lett. B747, 517 (2015), arXiv:arXiv:1501.00632 26 [hep-ph]

    Pith/arXiv arXiv 2015

  68. [72]

    Zhao and Q.-Y

    Z.-H. Zhao and Q.-Y. Xie, JHEP1805, 072 (2018), arXiv:arXiv:1712.09843 [hep-ph]

    Pith/arXiv arXiv 2018

  69. [73]

    R. R. Landim, G. Alencar, M. O. Tahim, and R. N. C. Filho, JHEP1108, 071 (2011), arXiv:arXiv:1105.5573 [hep-ph]

    Pith/arXiv arXiv 2011

  70. [74]

    R. R. Landim, G. Alencar, M. O. Tahim, and R. N. C. Filho, JHEP1202, 073 (2012), arXiv:arXiv:1110.5855 [hep-ph]

    Pith/arXiv arXiv 2012

  71. [75]

    Alencar, R

    G. Alencar, R. R. Landim, M. O. Tahim, and R. N. C. Filho, JHEP1301, 050 (2013), arXiv:arXiv:1207.3054 [hep-ph]

    Pith/arXiv arXiv 2013

  72. [76]

    Sui, W.-D

    T.-T. Sui, W.-D. Guo, Q.-Y. Xie, and Y.-X. Liu, Phys. Rev. D101, 055031 (2020), arXiv:2001.02154 [hep-th]

    Pith/arXiv arXiv 2020

  73. [77]

    R. S. Torrealba, Phys. Rev. D82, 024034 (2010), arXiv:arXiv:1003.4199 [hep-ph]

    Pith/arXiv arXiv 2010

  74. [78]

    Giovannini, Phys

    M. Giovannini, Phys. Rev. D66, 044016 (2002), arXiv:arXiv:0205139 [hep-th]

    2002

  75. [79]

    C.-E. Fu, Y. Zhong, and Y.-X. Liu, JHEP1901, 021 (2019), arXiv:arXiv:1810.02081 [hep-th]

    Pith/arXiv arXiv 2019

  76. [80]

    Alencar, Phys

    G. Alencar, Phys. Lett. B773, 601 (2017), arXiv:arXiv:1705.09331 [hep-th]

    Pith/arXiv arXiv 2017

  77. [81]

    L. F. F. Freitas, G. Alencar, and R. R. Landim, Eur. Phys. J. C80, 1141 (2020), arXiv:2006.10640 [hep-th]

    Pith/arXiv arXiv 2020

  78. [82]

    L. F. F. Freitas, I. C. Jardim, G. Alencar, and R. R. Landim, (2022), arXiv:2212.02259 [hep-th]

    Pith/arXiv arXiv 2022

  79. [83]

    I. I. Kogan, S. Mouslopoulos, A. Papazoglou, G. G. Ross, and J. Santiago, Nucl. Phys. B584, 313 (2000), arXiv:hep- ph/9912552

    arXiv 2000

  80. [84]

    DeWolfe, D

    O. DeWolfe, D. Z. Freedman, S. S. Gubser, and A. Karch, Phys. Rev. D62, 046008 (2000), arXiv:hep-th/9909134

    Pith/arXiv arXiv 2000

Showing first 80 references.