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REVIEW 2 major objections 5 minor 39 references

The lack of influence of the scalar hair on the DC conductivity

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read In charged hairy black holes with axionic momentum dissipation, the DC conductivity carries no explicit dependence on the primary scalar hair; the hair only shifts the horizon location.

desk verdict The explicit hairy axion conductivity computation is fine; the general hair-independence claim is an unproven conjecture as it stands. read the letter →

arxiv 2412.19388 v2 pith:4IXTQACD submitted 2024-12-27 hep-th

Ulises Hernandez-Vera This is my paper · ORCID
classification hep-th MSC 83C5783D05 PACS 04.70.-s11.25.Tq
keywords primaryscalarhairbeyond-HorndeskitheoriesaxionicblackholesDCconductivityholographictransportmomentumdissipationHorndeskigravityGauss-Bonnetcoupling
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

The paper asks whether the primary scalar hair of black holes in beyond-Horndeski gravity can be seen in holographic transport. It constructs charged planar black holes with two axionic fields that dissipate momentum, extending previously found hairy solutions, and computes the DC conductivity from horizon data. The central result is a single formula that depends on the electric charge, the axionic charge, the k-essence exponent, and the horizon radius, but never on the hair parameter. The calculation is repeated in a broad shift-invariant beyond-Horndeski class and in two Gauss-Bonnet branches, where the same cancellation occurs. The sympathetic reading is that hair would only matter by resizing the horizon, not through any direct coupling in transport.

What carries the argument

The machinery is horizon evaluation of the linearized equations. A conserved radial current $J_{x^i}=-\sqrt{hf}\,a'_{x^i}-Q_e H_{t x^i}$ follows from Maxwell's equations, and the perturbed metric equation $\varepsilon_{r x^i}$ is solved at the horizon for $H_{t x^i}$, which combines with the current to give the conductivity. All coupling-function dependence is contained in the combination $W$ defined in Eq. (14), and the argument hinges on $W$ and its derivatives dropping out at $r_h$ when the theory is shift-invariant and regular there.

What would settle it

Construct an explicit solution of the general beyond-Horndeski action within ansatz (3) whose $W$, defined in Eq. (14), is nonzero at the horizon; evaluating Eq. (17) there would then produce hair-dependent terms in $H_{t x^i}$ and hence in the conductivity, directly contradicting the claim.

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Extended reading notes

Core claim

The paper claims that the primary hair $q$, though present in the metric and in the scalar ansatz $\varphi(t,r)=qt+\psi(r)$, does not appear in the DC conductivity of these axionic black hole solutions. Evaluating the conserved current $J_{x^i}$ and the metric fluctuation $H_{t x^i}$ at the horizon yields $\sigma_{\mathrm{DC}}^{x^i x^i}=1+\frac{2^{k-1}Q_e^2}{k\,\omega^{2k}r_h^{2(2-k)}}$, where $r_h$ is the horizon radius. The same expression emerges for the logarithmic $k=3/2$ solution, for a general shift-invariant beyond-Horndeski action where all hair-dependent terms assemble into a function $W$ that vanishes at the horizon, and for two Gauss-Bonnet-coupled branches where the Gauss-Bonnet coupling also drops out. The hair thus enters only indirectly, by determining where the horizon sits.

Load-bearing premise

The general argument assumes the theory is shift-invariant and that the combination $W$ of coupling functions is regular at the horizon, so all hair-dependent terms vanish when the perturbed equation is evaluated there.

Editorial extensions

If this is right

  • For the standard axionic coupling $k=1$, the conductivity reduces to the familiar minimally-coupled result, so momentum dissipation behaves as in simpler holographic models.
  • For the conformal axionic coupling $k=2$, the horizon-radius dependence cancels and the conductivity depends only on the ratio $Q_e^2/\omega^4$, removing the horizon scale from transport.
  • Within the general shift-invariant beyond-Horndeski class, the same conductivity formula holds for every choice of the coupling functions, so no member of that family can make hair visible in DC transport.
  • In the Gauss-Bonnet-coupled branches, the additional coupling constant $\alpha$ also drops out, extending the cancellation beyond scalar hair to another theory parameter.
  • If the claim is correct, any observable signature of primary hair in DC transport must come through the horizon location $r_h$, not through direct hair couplings.

Reading between the lines

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

  • An obvious testable extension is to search for a non-shift-invariant solution within the same ansatz: if such a solution exists with $W(r_h)\neq 0$, the perturbed equation (17) would generate hair-dependent conductivity terms, delimiting the claim to the shift-invariant class.
  • The same horizon-cancellation mechanism likely applies to other transport coefficients, such as the thermal or Hall conductivities, since they are built from the same conserved currents and horizon data.
  • The $k=2$ conformal case offers a cleaner observational diagnostic: a measured conductivity independent of temperature or horizon radius would point to conformal axion dynamics together with a fixed charge-to-axion ratio.
  • The pattern suggests a transport no-hair statement: whenever hair enters only through a function that vanishes at the horizon, the DC conductivity cannot see the hair at all.
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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

2 major / 5 minor

Summary. The manuscript studies DC conductivity of planar black holes in beyond-Horndeski theories with primary scalar hair and axionic fields that break translational invariance. In Section II the author constructs a charged solution of a specific beyond-Horndeski model with a k-essence axionic term and computes the DC conductivity using the Donos–Gauntlett horizon technique. The result is \(\sigma = 1 + 2^{k-1} Q_e^2/(k \omega^{2k} r_h^{4-2k})\), which contains the hair parameter only through the horizon radius \(r_h\). In Section III the calculation is repeated for a broader beyond-Horndeski Lagrangian; the author argues that a combination \(W\) of coupling functions drops out at the horizon and that the same formula results. The paper concludes that scalar hair has no direct effect on DC conductivity.

Significance. If the Section II calculation is correct, it provides a concrete new example of hair-independent transport in a higher-order scalar-tensor theory and extends the known axionic conductivity formula to k-essence axions. The Section III argument, however, is not a complete proof for the general class: it relies on existence, regularity, and shift-invariance assumptions that are not derived. The paper is explicit about some of these limitations and frames the general proof as future work, which is commendable, but the abstract and conclusions overstate what has been shown.

major comments (2)
  1. [III, Eq. (17)] The horizon evaluation that eliminates \(W\) is not derived. The term multiplying \(H'_{t x_i}\) in Eq. (17) is \(-\frac{r^2}{2}\left[\frac{h}{2}(f/h)'W + f(4W/r+W') + \sqrt{f/h}\,\dot{W}\right]\). Its vanishing at \(r=r_h\) requires that \(f/h\), \(W\), and \(W'\) be regular and that \(\dot{W}=0\). The last condition is not "without loss of generality": for the general couplings \(G_i(\phi,X)\) and \(F_4(\phi,X)\) in Eq. (12), \(W\) depends on \(v\) through \(\phi\), and \(\dot{W}\) need not vanish. If \(f\neq h\), \(X\) can diverge at the horizon, so regularity of \(W\) is not automatic. Because no explicit solution is exhibited for this class, the step from Eq. (17) to Eq. (18) is conditional. The author should either prove these assumptions from the field equations or explicitly restrict the claim to the shift-invariant, regular-horizon case.
  2. [Abstract and Section IV] The abstract and conclusions state that the paper "shows" that scalar hair has no direct impact on DC conductivity for a general class of beyond-Horndeski theories, but Section III only provides a formal calculation conditional on an assumed solution and on the regularity and shift-invariance assumptions discussed above. The final paragraph of Section IV explicitly defers the formal proof to future work. The wording should be aligned with the actual result: a proof for the explicit Section II model, together with a conditional argument and a conjecture for the general case.
minor comments (5)
  1. [II, Eq. (10) and III, Eq. (19)] The notation \(r_h\) appears to be rendered as a separate \(h\) in the denominator, making the formula look singular since \(h(r_h)=0\). Please typeset the horizon radius explicitly as \(r_h^{2(2-k)}\) and \(r_h^{4-2k}\).
  2. [II, Eq. (8) vs III, Eq. (17)] The sign of the \(E_{x_i}Q_e\) term differs between Eq. (8) (plus) and Eq. (17) (minus); please verify that this is intended and explain the sign convention.
  3. [II, text near Eq. (3)] The solution is said to lie "within the ansatz (3)", but the ansatz is Eq. (2); the cross-reference should be corrected.
  4. [III, Eq. (14)] The definition of \(W\) is typographically ambiguous: the grouping of \(\sqrt{q^2-2hX}\), \(f/r\), and \(G_{5\phi}\) should be made explicit with parentheses or by matching the notation of Refs. [14,15].
  5. [Abstract] The phrase "and seem to confirm" has a subject-verb agreement problem; it should read "and seems to confirm". In addition, some references lack publication years or journal volume/page information, and the bibliography should be completed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the explicit conductivity derivation is self-contained; the general beyond-Horndeski argument has an unproven horizon cancellation that is a mathematical gap, not a circular reduction.

full rationale

The explicit Section II computation is self-contained: the solution (3) is substituted into the linearized equations (7)-(9) and the DC conductivity (10) is read off from horizon data, with no fitted parameter and no appeal to the target conclusion. The hair q enters only through the metric and hence the horizon radius r_h, an indirect effect the paper explicitly acknowledges. Section III is where the argument becomes delicate. The paper states that, when evaluated at the horizon, the W-dependent terms in Eq. (17) vanish, giving Eq. (18). Shift-invariance removes the sqrt(f/h) dot W term and f(r_h)=0 kills f(4W/r+W'), but the remaining coefficient (h/2)(f/h)' W is generically nonzero at the horizon unless f=h or W(r_h)=0; neither condition is proven for the general class, and in the explicit example W=1 at the horizon with f=h. The paper itself ends by deferring "a formal proof" of the general intuition. This is an omitted proof/correctness gap, not a circularity: nothing is fitted, renamed, or reduced to a self-citation. The cited prior hairy solutions in [14,15] (one co-authored by the author) are explicit parameter-free constructions whose assumptions do not include the conductivity result, so the self-citation is not load-bearing in a circular sense.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data; M, Q_e, omega and q are integration or background parameters, and the result is independent of the hair q. The derivation relies on the standard holographic membrane paradigm and on two unproven structural assumptions for the general class, namely shift-invariance and W regularity at the horizon. No new entities are introduced.

assumptions (5)
  • standard math The Donos-Gauntlett membrane paradigm maps DC conductivities to horizon data without solving the full bulk.
    Used in Section II.A and III; it is an established result in holography, treated as a theorem.
  • domain assumption The planar ansatz (2) supports regular black hole solutions for the chosen beyond-Horndeski actions.
    Solutions (3)-(4) are stated; in Section III the author notes no explicit solutions exist for the general class.
  • ad hoc to paper Shift-invariant couplings can be assumed without loss of generality, so the dot-derivative of W vanishes.
    Section III, before Eq. (17); restricts the theory class without proof.
  • ad hoc to paper The function W in Eq. (14) is regular at the horizon so that h W, f(4W/r + W') and sqrt(f/h) dot-W vanish there.
    Needed for the horizon evaluation to leave only axion and Maxwell contributions; not demonstrated for general G5 theories.
  • domain assumption The holographic dictionary with homogeneous axion fields gives momentum dissipation.
    Standard framework described in the Introduction and used throughout.

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Cite this review

Pith. "Pith review of The lack of influence of the scalar hair on the DC conductivity." pith.science (2026). https://pith.science/paper/4IXTQACD

@misc{pith2026241219388,
  author       = {Pith},
  title        = {Pith review of: The lack of influence of the scalar hair on the DC conductivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4IXTQACD}},
  note         = {Machine review of arXiv:2412.19388}
}
read the original abstract

Recently obtained black hole solutions within the framework of beyond-Horndeski theories, which have the advantage of featuring primary hair, are generalized in the presence of two axionic fields. In order to induce a momentum dissipation, the axionic field solutions are homogeneously distributed along the horizon coordinates of the planar base manifold. We show that, despite the explicit dependence of the scalar field and the metric on the primary hair, this latter does not directly affect the calculation of transport properties. Its influence is indirect, modifying the horizon location, but the transport properties themselves do not explicitly depend on the hair parameter. We take a step further and show that even within a more general class of beyond-Horndeski theories, where the scalar field depends linearly on the hair parameter, the scalar hair still has no direct impact on the DC conductivity. This result underscores the robustness of our earlier findings, and seem to confirm that the transport properties remain unaffected by the explicit presence of the hair parameter.

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