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Dark photons and tachyonic instability induced by Barbero-Immirzi parameter and axion-torsion transmutation

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Pith's one-line read The Barbero-Immirzi parameter of Holst gravity is fixed by the torsion mass through $M^4 = 1/(2\beta^2)$, giving $\beta \approx 0.775$ for $M = 0.775$ TeV and $\beta \approx 0.285$ for $M = 1.51$ TeV, with both values inside the…

desk verdict The paper's central formula is fixed by hand-setting a7=1 TeV^2, so the derived β values are artifacts, not predictions. read the letter →

arxiv 2412.16617 v1 pith:TM47RT4I submitted 2024-12-21 hep-ph gr-qc

Zhi-Fu Gao , Biaopeng Li , L.C. Garcia de Andrade This is my paper · ORCID
classification hep-phgr-qc
keywords Barbero-ImmirziparameterHolstgravitytorsiondarkphotonsaxiontachyonicinstabilitymagnetichelicityEinstein-Cartan
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the Barbero-Immirzi (BI) parameter of Holst gravity can be derived rather than assumed once torsion is coupled to dark photons. Using a coefficient-fixing technique on the Einstein-Cartan-Holst Lagrangian, the paper obtains $M^4 = 1/(2\beta^2)$, where $M$ is the typical torsion mass. This yields $\beta \approx 0.775$ for $M = 0.775$ TeV and $\beta \approx 0.285$ for $M = 1.51$ TeV, both inside the established TeV-scale range $0 \le \beta \le 1.185$. The same framework predicts an axion oscillation frequency $\omega_\phi = m_\phi/\sqrt{g}$, with $g$ depending on $\beta$ and the torsion masses, and a dark-photon frequency that is damped when magnetic helicity and the helicity parameter share a sign. The result matters because it turns a quantum-gravity ambiguity into a quantity constrained by TeV-scale torsion searches and dark-photon cosmology.

What carries the argument

The load-bearing object is the coefficient-fixing procedure applied to the Einstein-Cartan-Holst-Zanelli Lagrangian. One writes a portal Lagrangian for dark photons with generic coefficients, varies the torsion trace $T$ to obtain $T = -\frac{a_7}{\beta a_6 f^2}\,\partial a$, and then imposes constant coefficients to read off masses. The transmutation rule $S = \partial a$, in which the axial torsion is the gradient of the axion, reduces torsion dynamics to a mass spectrum. Combining $a_6 = M_T^2/f^2$, $a_5 = M_S^2/g^2$, and $a_7 = 1/(\beta M_T^2)$ with the hand-set value $a_7 = 1$ TeV$^2$ and $m_T \approx m_S$ yields the identity $M^4 = 1/(2\beta^2)$. This identity carries the numerical BI estimates and feeds the coefficient $g$ in the axion and dark-photon equations.

What would settle it

Measure the trace-torsion and axial-torsion masses in a collider search and compare $(m_S m_T)^2$ with $1/(\sqrt{2}\,\beta)$: a mismatch beyond uncertainties would falsify the relation, as would an axion oscillation frequency that disagrees with $m_\phi/\sqrt{g}$.

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Extended reading notes

Core claim

The central claim is that the Barbero-Immirzi parameter is not an independent input in this Einstein-Cartan-Holst construction; it is determined by the torsion mass spectrum through $M^4 = 1/(2\beta^2)$. Setting the free coefficient $a_7$ to $1$ TeV$^2$ and assuming the trace-torsion and axial-torsion masses are approximately equal transforms the Lagrangian coefficients into this relation. The relation is then read in both directions: $\beta \approx 1.185$ gives $M = 0.775$ TeV, while $M = 1.51$ TeV gives $\beta \approx 0.285$. The same framework yields an axion oscillation frequency $\omega_\phi = m_\phi/\sqrt{g}$ determined by $\beta$ and torsion masses, and a dark-photon dispersion relation containing a helicity-dependent damping term. The paper notes that torsion itself disappears from the final field equations, leaving only its mass spectrum and the constrained BI parameter as low-energy traces.

Load-bearing premise

The load-bearing premise is that the originally free coefficient $a_7$ may be set to $1$ TeV$^2$ by hand and that the trace-torsion and axial-torsion masses are approximately equal, since the numerical relation $M^4 = 1/(2\beta^2)$ is forced by those choices.

Editorial extensions

If this is right

  • The BI parameter becomes a measurable quantity: detecting a torsion mass near 0.775 TeV would imply $\beta \approx 1.185$, and a mass near 1.51 TeV would imply $\beta \approx 0.285$.
  • Axion oscillations in this model are constrained by torsion searches because the oscillation frequency is set by $m_\phi/\sqrt{g}$, with $g$ built from $\beta$ and the torsion masses.
  • Dark-photon production through axion-torsion transmutation inherits a helicity asymmetry: same-sign $h$ and $\lambda$ damp the dark-photon mode, while opposite signs amplify it.
  • The known black-hole-entropy value $\beta \approx 0.273$ translates into a torsion mass of 1.71 TeV, within the light-torsion territory reachable by collider searches.
  • The disappearance of an explicit torsion field from the final dark-photon equations is a decoupling: torsion leaves imprints only through $\beta$ and the mass spectrum, making the BI parameter a bridge between quantum-gravity inputs and low-energy observations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the relation $M^4 = 1/(2\beta^2)$ survives contact with data, $\beta$ and the torsion mass become interchangeable observables, so an independent measurement of either would overconstrain the other and provide a direct test of this construction.
  • The same coefficient-fixing technique could be applied to the proposed coupling between the $Z$ boson and torsion; that would yield a predicted $Z$-torsion coupling constant that collider searches could confirm or exclude.
  • The hand-set value $a_7 = 1$ TeV$^2$ carries all the numerical content of the relation; a first-principles derivation of $a_7$ from a more fundamental theory would upgrade the formula from a scaling relation to a genuine prediction.
  • The apparent decoupling of torsion from the axion field equations is a masking effect of the imposed constraint, so a version without that constraint would likely leave torsion mass terms appearing in the dark-photon equations and could produce distinct dark-photon production signatures.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The manuscript investigates Holst gravity with torsion, aiming to constrain the Barbero-Immirzi (BI) parameter using dark-photon and torsion masses. In the minimal-coupling example, the authors propose the relation M^4 = 1/(2β^2) (Eq. 21) and use it to obtain β = 1.185 for M = 0.775 TeV and β = 0.285 for M = 1.51 TeV (Table 1). In the non-minimal example, they derive dark-photon and axion field equations, and a condition on the BI parameter in terms of Planck and torsion masses (Eq. 36). The paper claims that this provides a new probe of the BI parameter from dark photons and that the resulting values fall in the range allowed by LHC-scale analyses.

Significance. If the central derivation were correct, the paper would offer a new phenomenological connection between loop quantum gravity and TeV-scale dark-photon/torsion physics, with potentially observable consequences for axion oscillation frequencies and magnetic helicity. However, the derivation of the key formula M^4 = 1/(2β^2) is not supported by the written equations, and the numerical outputs are driven by ad hoc normalizations rather than by a physically motivated calculation. The internal inconsistencies in the algebra and in the reported values make the claimed significance currently unestablished.

major comments (5)
  1. [Section 2, Eqs. (18)-(22)] The derivation of the central relation M^4 = 1/(2β^2) is not justified. Equation (19) is not obtained by inverting Eq. (18); the text simply asserts it. Taken literally, Eq. (18) with a7 = 1 gives M_T^2 = 1/β, and Eq. (19) then gives M_S^2 = 1/√2, so setting m_T ≈ m_S would force β = √2, not the values in Table 1. Moreover, Eq. (22) does not follow from Eq. (21): from M^4 = 1/(2β^2) one obtains M^2 = 1/(√2 β), not 1/(√2√(2β)). The central numerical formula is therefore unsupported by the printed equations.
  2. [Section 2, text before Eq. (18) and after Eq. (21)] The normalization a7 = 1 TeV^2 is a free input, not a derived quantity. Equation (18) states a7 = 1(TeV)/(β M_T^2), which has dimensions of inverse energy, while the later text writes a7 = 1 TeV^2; these assignments are mutually inconsistent. Since a7 is a dimensionful free parameter, any change of scale changes the predicted β for a fixed torsion mass. The values β ≈ 0.285 and β ≈ 0.775 are therefore artifacts of the chosen normalization rather than robust predictions of the model.
  3. [Table 1 and Abstract] The pairs (0.775 TeV, 1.185) and (1.51 TeV, 0.285) are internally inconsistent with Eq. (21). Evaluating M^4 = 1/(2β^2) for β = 0.285 gives M ≈ 1.57 TeV, not 1.51 TeV; conversely, M = 1.51 TeV gives β ≈ 0.310. In addition, the abstract claims that a BI parameter of about 0.775 is derived from the 1.7 TeV torsion of Barman et al., but Eq. (21) with M = 1.7 TeV yields β ≈ 0.245. These discrepancies undermine the reported numerical results.
  4. [Section 3, Eqs. (34)-(36)] The condition that the sum of the second and third terms in g vanish, together with m_T ≈ m_S, leads to β = m_P^2/(√2 m_T^2), not to √(2β) = m_P^2/m_T^2 as printed in Eq. (36). Even the corrected relation gives β ≈ 10^31 for m_T ≈ 1.7 TeV with the physical Planck scale, so the claim that this yields a BI parameter in the LQG range is numerically false. The error appears to stem from a misuse of dimensional analysis in Eq. (34), where the terms in g have mixed mass dimensions.
  5. [Section 4] The paper itself acknowledges the load-bearing nature of the constraint: 'if we do not assume the constraint chosen for the BI parameter, torsion masses would appear in the dark photon field equations.' This concession indicates that the numerical values of β are assumed rather than derived from the model. Since the stated constraint (setting g to 1) is not physically motivated and yields absurd results when combined with Eq. (36), the main conclusion of the paper is not supported.
minor comments (6)
  1. [Abstract] The phrase 'Nevertheless. this figure is still a signature of light torsion' uses a period instead of a comma and should be corrected.
  2. [Section 1] The sentence 'Throughout this paper, we use i, k, l= 0, 1, 2, 3 and Rijkl are the components of torsion' is repeated twice, once with i,k,l and once with i,j,k,l; only the latter should appear.
  3. [Section 2, Eq. (15)] The notation a2 = gaγ′/fa is undefined; presumably gaγ′ denotes a photon-axion coupling, but the subscript is not explained.
  4. [Section 2, Eq. (22)] The expression M^2 ≈ 1/(√2√(2β)) is a typographical error if derived from Eq. (21); the correct expression is M^2 = 1/(√2 β).
  5. [Section 3, Eq. (36)] The symbol 'p' in 'p(2β)' should be the square root sign: √(2β).
  6. [References] Reference [1] is dated 2017 but Phys. Rev. D 101 appeared in 2020; reference [3] lists page '01429' which should include the missing leading zeros.

Circularity Check

3 steps flagged · score 8.0 of 10

The headline BI values reduce to the hand-set normalization a7 = 1 TeV^2 and to plugging input masses into the resulting formula; the paper's own Section 4 concedes the constraint dependence.

  1. fitted input called prediction [Section 2, Eqs. (18)-(21) and the text after Eq. (21)]
    "a7 = 1(TeV) / βM_T^2 ... Inverting this equation yields M_S^2 = 1/(√2 β M_T^2) ... where we take a7 = 1 ... M^4 = 1/(2β^2) ... Using this expression, we can calculate the torsion mass from the BI parameter and vice-versa."

    The relation M^4 = 1/(2β^2) is not derived from a physical scale; it is obtained by setting the free coefficient a7 to 1 TeV^2 and assuming m_T ≈ m_S. Since a7 is an arbitrary dimensionful input, changing this scale changes the predicted β for a fixed torsion mass. The subsequent 'vice-versa' procedure therefore converts input masses into β values using a hand-set constant, so the numerical β outputs are not predictions from Holst gravity but rescalings of the inputs. In addition, the printed algebra Eqs. (18)-(20) does not by itself imply Eq. (21) without an extra identification between m_S^2 and a7/(2 M_T^2 β^2), so the central formula is imposed rather than derived.

  2. fitted input called prediction [Section 2, Table 1 and the preceding paragraph]
    "For example, consider the case by Panza et al. [6], with an upper bound of the BI parameter equal to 1.185. Substituting this into Equation (21) yields M = 0.775 TeV = 775 GeV. ... Using the result obtained by the author, we estimate 0.285 for the BI parameter, implying an average torsion mass of 1.51 TeV."

    Table 1 presents two 'derived' BI values that are actually inputs to Eq. (21). The first row inserts Panza's β = 1.185 and returns M = 0.775 TeV, so β is the input and the mass is the output, yet the text presents this as a torsion-mass result. The second row inserts the author's own previously published torsion mass of 1.51 TeV (Ref. [8]) and recovers β = 0.285; the sentence then 'implies' the same 1.51 TeV mass. Both values are therefore fitted conversions of prior numbers in a formula whose scale was set by a7 = 1, so the claim of deriving the BI parameter from dark photons reduces to plugging external and self-cited inputs into a reparametrization.

1 more flagged steps
  1. self definitional [Section 3, Eqs. (34)-(36) and Section 4]
    "From Equation (34), the BI parameter of loop quantum gravity can be obtained by setting the sum of the second and third terms in g to zero. This will reduce the coefficient g to 1, as in the regular Klein-Gordon equation. The BI parameter then becomes √(2β) = m_P^2/m_T^2, where we have used the approximation m_T ≈ m_S. ... Our findings reveal that if we do not assume the constraint chosen for the BI parameter, torsion masses would appear in the dark photon field equations."

    The β value in Section 3 is fixed by imposing a constraint — that the sum of the second and third terms in g vanishes — rather than by deriving it from the model. The authors themselves concede in Section 4 that this constraint is assumed, and that without it torsion masses would appear in the dark photon equations. Thus the BI parameter here is defined by the chosen constraint, not predicted. This is a second instance where the target quantity is manufactured by imposing a condition, and it uses the same m_T ≈ m_S approximation that underlies Eq. (21), so the numerical content remains tied to the earlier arbitrary normalization.

full rationale

The central numerical claims of the paper are not self-contained derivations. Equation (21), M^4 = 1/(2β^2), is presented as a formula from which torsion mass and BI parameter can be obtained 'vice-versa', but it follows only after setting the free coefficient a7 = 1 TeV^2 and assuming m_T ≈ m_S. Because a7 is a hand-set dimensionful constant, Eq. (21) is not a first-principles relation; it is a chosen scale that rescales input masses into β values. Table 1 then uses Eq. (21) twice: Panza's β = 1.185 is inserted to yield M = 0.775 TeV, and the author's own prior torsion mass of 1.51 TeV (Ref. [8]) is inserted to yield β = 0.285, with the text even stating that this 'implies' the same 1.51 TeV mass. This is fitted input renamed as prediction. The Section 3 derivation of β via Eq. (36) is similar, since β is fixed by imposing that the second and third terms in g sum to zero, and Section 4 concedes the constraint dependence. The later dark-photon and axion equations (Eqs. 37-48) are algebra from the stated Lagrangian and are not circular, but the headline BI values reduce by construction to the arbitrary normalization and to input masses. The abstract's 'BI parameter of approximately 0.775' also appears to confuse the mass 0.775 TeV with a dimensionless β value, a correctness issue that reinforces the impression that these numbers are not independently derived. Because the paper's central quantitative claim is forced by a hand-set scale and by plugging in external and self-cited inputs, the circularity score is 8.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central results depend on several hand-set coefficients and approximations: a7 is set to 1 TeV^2, the two torsion masses are assumed equal, and the coefficient g is forced to 1. These choices are not derived from the theory and each shifts the numerical outputs. The torsion mass inputs are drawn from collider bounds in prior work, so the paper is re-parameterizing known constraints rather than making independent predictions.

free parameters (3)
  • a7 = 1 TeV^2 (set by hand)
    The coefficient a7 is fixed to 1 TeV^2 in Section 2, which directly controls the relation between β and the torsion mass. No physical argument is given for this value.
  • m_T (torsion trace mass) = 0.775 TeV or 1.51 TeV
    The torsion mass is taken from prior literature (Panza et al., Barman et al., and the author's own 2024 result) and used as an input to compute β. This is fitting, not prediction.
  • m_S (axial torsion mass) = assumed equal to m_T
    The approximation m_S ≈ m_T is used to collapse the two masses into a single M and derive eq. (21).
assumptions (6)
  • domain assumption The flat spacetime approximation g_ij ≈ η_ij
    Used throughout Section 2 to replace the Riemannian covariant derivative with standard derivatives, simplifying the Lagrangian. This is a standard approximation but limits the validity of the results.
  • domain assumption The transmutation rule S = ∂a (or ∂φ) from Duncan et al.
    Assumed in Sections 2 and 3 to relate axial torsion to the axion field gradient. Adopted from [16] without re-derivation.
  • domain assumption The constant-coefficient constraint technique from Dombriz et al.
    The method of determining Lagrangian coefficients by requiring constant coefficients is taken from [10] and applied here, but the consistency of that method with the equations is not shown.
  • ad hoc to paper Setting a7 = 1 TeV^2
    This arbitrary normalization is the main driver of the numerical values. If a7 were different, all derived β values would shift.
  • ad hoc to paper m_T ≈ m_S
    The equality of the two torsion mass scales is assumed to simplify eq. (20) into eq. (21). No justification is provided, and it is used to produce the headline numbers.
  • ad hoc to paper g reduced to 1 by setting the sum of the second and third terms to zero
    In Section 3, the coefficient g is forced to the Klein-Gordon value of 1 by an ad hoc cancellation, which directly yields eq. (36).

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Pith. "Pith review of Dark photons and tachyonic instability induced by Barbero-Immirzi parameter and axion-torsion transmutation." pith.science (2026). https://pith.science/paper/TM47RT4I

@misc{pith2026241216617,
  author       = {Pith},
  title        = {Pith review of: Dark photons and tachyonic instability induced by Barbero-Immirzi parameter and axion-torsion transmutation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TM47RT4I}},
  note         = {Machine review of arXiv:2412.16617}
}
abstract

In this paper, we investigate Holst gravity by examining two distinct examples. The first example involves minimal coupling to torsion, while the second explores non-minimal coupling. The motivation for the first example stems from the recent work by Dombriz, which utilized a technique of imposing constraint constant coefficients to massive torsion in the model Lagrangian to determine parameters for the Einstein-Cartan-Holst gravity. We extend this methodology to investigate dark photons, where axial torsion transforms into axions.Interest in elucidating the abundance of dark photons within the framework of general relativity was sparked by Agrawal. Building on the work of Barman, who explored minimal coupling of massive torsion mediated by dark matter (DM) with light torsion on the order of 1.7 TeV, we have derived a Barbero-Immirzi (BI) parameter of approximately 0.775. This value falls within the range established by Panza et al. at TeV scales, specifically $0\le{\beta}\le{1.185}$. This seems to our knowledge the first time BI parameter is induced by dark photons on a minimal EC gravity. Very recently, implications of findings of BI parameter in cosmological bounces has appeared in the literature. For a smaller BI parameter a higher torsion mass of 1.51 TeV is obtained. Nevertheless. this figure is still a signature of light torsion which can be compatible with light dark photon masses. Magnetic helicity instability of dark photons is investigated. Axion oscillation frequency is shown to depend on the BI parameter and the BI spectra is determined by an histogram. This study not only broadens the understanding of Holst gravity but also provides crucial insights into the interplay between torsion, dark photons, and axions in the cosmological context.

Figures

Figures reproduced from arXiv: 2412.16617 by the authors.

Figure 1
Figure 1. The relation between the true BI parameter and the trace torsion [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Reheating chiral dynamos with spin-0 and massive spin-1 torsions via chiral asymmetry

    astro-ph.CO 2025-02 reject novelty 4.0 of 10

    Adding a constant torsion trace to the chiral dynamo equation produces exponential magnetic growth with a sign set by torsion chirality, but the supporting equations and headline field strengths are not reliable.

Reference graph

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