{"id":"3ef4ce11-8df0-4cde-86ef-db1b4a1f4ff5","arxiv_id":"2505.08545","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Massive scalar fields around non-minimal Einstein-Yang-Mills black holes can form arbitrarily long-lived quasi-resonances in flat space, but not in de Sitter space; an analytic large-mass frequency formula is given.","lead":"This paper computes the vibration frequencies, called quasinormal modes, of a massive scalar field around charged black holes in a modified gravity theory with a cosmological constant. It reports that heavy fields create arbitrarily long-lived quasi-resonant signals for flat black holes, while a positive cosmological constant keeps the damping finite and never lets it vanish.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"de Sitter tables imply Λ=0.01, not the stated Λ=0.001; the tabulated QNMs are inconsistent with their own parameters.","rationale":"The reader's weakest assumption was that the de Sitter tables rest only on WKB with no independent validation. My independent check finds a sharper, internal problem: the tabulated frequencies are quantitatively inconsistent with their stated Λ=0.001 and instead match Λ=0.01. This reinforces, rather than replaces, the reader's call for verification of the de Sitter results. The central qualitative claim (quasi-resonances in flat space, finite damping in de Sitter) may still survive, since the absence of de Sitter quasi-resonances is cited as analytically proven, but the paper's main numerical deliverable is currently unreliable as stated. The abstract-versus-body reversal is also real and must be fixed, but it is a communication error rather than a load-bearing mathematical flaw. Because the required actions are correction and re-verification rather than abandonment of the central claim, the appropriate verdict remains conditional acceptance.","tokens_in":19549,"tokens_out":15164,"duration_ms":145679,"concrete_test":"Take Table I, row ξ=0, µ=3.5 (and Table III, row Q=0, ξ=0.1, µ=3.5). Recompute the WKB-9/Padé frequency using the literal stated parameters rh=1, Λ=0.001, Q=0.1 (or Q=0), ξ=0, and compare with the quoted Reω≈2.9635 and Imω≈-0.0422. Also evaluate the leading large-µ formula (Eq. 23) with Λ=0.001 and with Λ=0.01. If the run at Λ=0.001 gives Reω≈3.26 and Imω≈-0.015 (10% and factor-3 off from the table), while the Λ=0.01 run matches the table, the tables are mislabelled and all de Sitter entries must be re-generated or re-labelled.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing weak point is not the WKB order agreement but an internal parameter inconsistency in the de Sitter data. For the stated parameters rh=1, Λ=0.001, Q=0.1, ξ=0, Eq. (20) gives M≈0.5048, so the large-µ WKB peak of V(r)=f[µ²+ℓ(ℓ+1)/r²+f'/r] sits at rmax≈(3M/Λ)^{1/3}≈11.5 with f(rmax)≈0.869, predicting Reω≈0.932µ. Table I at µ=3.5 lists Reω=2.9635, i.e. Reω/µ≈0.847; this is instead the value obtained with Λ=0.01 (rmax≈5.3, fmax≈0.717, sqrt≈0.846). The same 10% shift in Re and factor-3 shift in Im appears in all four tables, and Eq. (23) with x=sqrt(1-(9M²Λ)^{1/3}) reproduces the tabulated entries only at Λ=0.01. Thus either every table was computed at Λ=0.01 and the captions are wrong, or the WKB runs used a different potential than stated. The WKB-6/WKB-9 agreement cannot detect this because both orders use the same erroneous input.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript computes quasinormal frequencies of a massive scalar field in the non-minimal Einstein-Yang-Mills black hole backgrounds of Balakin-Lemos-Zayats type, for both asymptotically flat and asymptotically de Sitter cases. It claims that in the flat case the damping rate reaches zero at a critical scalar-field mass, producing quasi-resonances whose frequencies become independent of the non-minimal coupling xi, while in the de Sitter case the damping remains finite for all masses. It further presents WKB-based tables of de Sitter frequencies, an analytic large-mass formula (Eqs. 22-24), and time-domain evolutions arguing stability for l=0 perturbations even when the effective potential has a negative gap.","tokens_in":19857,"tokens_out":9226,"duration_ms":85806,"significance":"If correct, the paper would add a new example of massive-field quasi-resonances in a modified-gravity black hole and provide an explicit analytic large-mass formula containing the cosmological constant and Yang-Mills charge. The use of Leaver's method for the flat case and the time-domain stability check are appropriate, and the analytic formula has the virtue of being directly testable against the tables. However, the numerical de Sitter results are the main quantitative output, and they currently contain an internal parameter inconsistency; the abstract also contradicts the conclusions. The qualitative flat-versus-de Sitter distinction is not new, since the absence of de Sitter quasi-resonances is credited to an earlier analytic proof (Ref. [29]), so the paper's incremental value rests on the numerical tables and analytic formula, which need to be made reliable.","major_comments":[{"comment":"The abstract states exactly the opposite of the body's main conclusion: it says that in the de Sitter case arbitrarily long-lived modes can exist, whereas in the asymptotically flat case the damping rate never vanishes completely. Section V concludes the reverse, and Section IV explicitly states that the absence of quasi-resonances in asymptotically de Sitter black holes was proven in Ref. [29]. This contradiction concerns the paper's central claim and must be resolved before publication.","section":"Abstract and Section V"},{"comment":"The de Sitter frequencies in Tables I-IV are inconsistent with the caption values. For rh=1, Q=0.1, xi=0 and Lambda=0.001, Eq. (20) gives M about 0.5048, and the large-mu peak of V(r)=f[mu^2+l(l+1)/r^2+f'/r] is at rmax about (3M/Lambda)^(1/3) about 11.5 with f(rmax) about 0.869, so WKB predicts Re omega about 0.932 mu. Table I at mu=3.5 quotes Re omega=2.9635, i.e., Re omega/mu about 0.847, which is instead what one obtains with Lambda=0.01 (rmax about 5.3, fmax about 0.717). The same 10% shift in Re omega and roughly factor-3 shift in |Im omega| appears in all four tables, and Eq. (23) with x=sqrt(1-(9M^2 Lambda)^(1/3)) reproduces the tabulated entries only if Lambda=0.01. Thus the tables were either computed at Lambda=0.01 with wrong captions, or with a different potential than stated; the WKB-6/WKB-9 agreement cannot reveal this because both orders use the same input.","section":"Section IV, Tables I-IV"},{"comment":"All de Sitter QNM data are produced solely by the WKB method, and the only stated validation is the agreement between WKB orders 6 and 9, which are both WKB and may share a systematic error. Section IV claims the difference is much smaller than the observed effect, but that comparison cannot validate absolute frequencies. In view of the parameter inconsistency in the preceding comment, the quantitative de Sitter results, including the large-mu constant damping and the apparent l-independence of the frequencies, should be checked with an independent method, such as the Leaver method already used for the flat case or a frequency extraction from time-domain data.","section":"Section IV, Tables I-IV"},{"comment":"The analytic large-mass formula is one of the paper's main results, but its derivation is compressed into a single sentence referring to Ref. [78]. The text does not show how the expansion of rmax in powers of 1/mu is organized relative to the smallness of Q and xi, nor how the WKB corrections produce the specific Q^2 and xi terms in Eq. (23). Since the formula is used to assert universal behavior in the quasi-resonance limit, the derivation, or at least a clear statement of the ordering and truncation, should be provided, and the formula's regime of validity stated.","section":"Section IV, Eqs. (22)-(24)"}],"minor_comments":[{"comment":"The boundary condition for a massive field at infinity is written with sqrt(omega^2-mu^2); for decaying QNMs one needs e^{-sqrt(mu^2-omega^2) r*} for omega<mu, so the sign should be specified explicitly.","section":"Section III.A, Eq. (7)"},{"comment":"There is a typo 'FIn some cases' that should read 'In some cases'.","section":"Section III.B, after Eq. (14)"},{"comment":"The axis labels are garbled (e.g., '/Minus', '/LParen1', '/Star'), making the figures unreadable.","section":"Figures 2, 5, and 6"},{"comment":"The captions do not state Lambda=0; since all other figures state Lambda, please specify.","section":"Figures 3 and 4"},{"comment":"The meaning of 'WKB-6 (m=3)' and 'WKB-9 (m=4)' is not defined; please explain the Pade orders in the text or captions.","section":"Tables I-IV"},{"comment":"The phrase 'quasi normal' should be 'quasinormal'.","section":"Abstract"},{"comment":"The conclusion says the threshold for the onset of quasi-resonances is independent of xi, but this is illustrated only for the fundamental l=0 mode in Fig. 4; clarify the scope of this claim.","section":"Section V"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the flat-space part of this paper is worth taking seriously; the de Sitter part is not, as presented. The stress-test note is right: all four de Sitter tables say Λ=0.001 but behave like Λ=0.01. I checked the large-µ WKB peak for the stated parameters, M≈0.5048, and the potential maximum sits around r≈11.5 with f_max≈0.868, giving Reω≈0.932µ; Table I at µ=3.5 lists Reω=2.9635, i.e. Reω/µ≈0.847, which is the value for Λ=0.01 (r_max≈5.3, f_max≈0.717). The same factor appears across all four tables, and Eq. (23) with x=sqrt(1-(9M²Λ)^{1/3}) matches the tables only at Λ=0.01. WKB-6 vs WKB-9 agreement cannot catch a systematic input error, so the dS numbers are unsupported until the author confirms which Λ was actually used.\n\nWhat is genuinely new: the massive-scalar QNM spectrum for non-minimal EYM black holes in the asymptotically flat case, the ξ-independence of frequencies near the quasi-resonance threshold, and the analytic large-mass formula. The flat-space results come from Leaver's method, which is reliable, and the qualitative behavior (quasi-resonances in flat, none in dS) is consistent with prior literature, including the analytic proof in [29]. The time-domain stability check for negative-gap potentials is a useful addition.\n\nThe other problems are proportionally less, but not small: the abstract states the opposite of the body's conclusion, and the analytic formula is introduced with essentially a one-sentence derivation. A referee should ask for a proper derivation or at least a numerical check at the stated parameters.\n\nBottom line: this paper deserves a serious referee because the flat-space content is plausible and the errors are identifiable, but it is not close to acceptable in its current form. My recommendation is to send it out with a request for major revision: fix the abstract, resolve the Λ inconsistency in the dS tables (recompute or relabel), and add an independent method for at least a few dS frequencies.","headline":"The flat-space massive-scalar QNM results for non-minimal EYM black holes are plausible and new, but the de Sitter tables internally use Λ=0.01 while claiming Λ=0.001, and the abstract states the opposite of the paper's own conclusion.","tokens_in":20327,"tokens_out":6425,"would_cite":false,"duration_ms":54367,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that a massive scalar field around non-minimal Einstein-Yang-Mills black holes admits quasi-resonances—arbitrarily long-lived quasinormal modes—only when the spacetime is asymptotically flat, while the…","keywords":["quasinormal modes","massive scalar field","non-minimal Einstein-Yang-Mills theory","quasi-resonances","de Sitter black holes","black hole ringdown","continued fraction method","WKB approximation"],"falsifier":"Using an independent method (continued fractions adapted to de Sitter boundary conditions, or time-domain evolution with Prony extraction), compute the fundamental de Sitter frequency for a representative case such as $\\Lambda=0.001$, $Q=0.1$, $\\xi=0.5$, $\\ell=1$, $\\mu=1$. If the imaginary part disagrees with the WKB-6/WKB-9 tables by more than their own mutual difference, or if it crosses zero for some $\\mu$, then the claimed absence of quasi-resonances in the de Sitter branch collapses; in the flat case, the quasi-resonance claim would be settled by locating the critical $\\mu$ at which $\\mathrm{Im}(\\omega)$ changes sign with high-precision continued-fraction data.","tokens_in":19348,"feed_emoji":"⏳","tokens_out":10953,"duration_ms":99144,"temperature":0.7,"pith_summary":"The paper sets out to show that the mass of a scalar field changes not just the numerical values but the qualitative fate of black-hole ringing in non-minimal Einstein-Yang-Mills theory. In asymptotically flat spacetimes, pushing the field mass above a critical value drives the damping rate of the fundamental quasinormal mode to zero, producing quasi-resonances—arbitrarily long-lived oscillations whose frequency no longer depends on the non-minimal coupling $\\xi$. In asymptotically de Sitter spacetimes, the same field mass suppresses damping but never removes it, so no quasi-resonances appear. In the large-mass regime the frequencies become independent of the multipole number $\\ell$, and the paper derives an analytic formula for them. These results matter because long-lived modes are among the most direct observable signatures of near-horizon geometry, and the paper shows that the cosmological background decides whether they can exist at all.","feed_headline":"Long-lived black-hole modes need large scalar mass and flat spacetime","feed_subtitle":"Around non-minimal Einstein-Yang-Mills black holes, only flat space lets the damping rate reach zero.","key_machinery":"The argument runs through the radial wave equation in Schrödinger form $d^2\\Psi/dr_*^2 + (\\omega^2 - V(r))\\Psi = 0$, with the effective potential $V(r) = f(r)[\\mu^2 + \\ell(\\ell+1)/r^2 + f'(r)/r]$ built from the non-minimal EYM lapse function $f(r) = 1 + (r^4/(r^4 + 2\\xi Q^2))(Q^2/r^2 - 2M/r - \\Lambda r^2/3)$. Imposing purely ingoing waves at the horizon and evanescent decay at infinity for massive fields turns the problem into one of solving for complex frequencies $\\omega$. Quasi-resonances are identified as the limit where $\\mathrm{Im}(\\omega) \\to 0$. The numerical machinery is a continued-fraction method for the asymptotically flat case, WKB with Padé approximants for the de Sitter case, time-domain integration with Prony extraction for stability checks, and a large-mass expansion of the potential's maximum that yields the analytic formula for the frequencies.","core_discovery":"The central claim is a sharp split in the spectrum of a massive test scalar field on non-minimal Einstein-Yang-Mills black holes. For $\\Lambda = 0$ asymptotically flat solutions, the imaginary part of the fundamental quasinormal frequency decreases with scalar mass $\\mu$ and vanishes at a critical mass—specific to the multipole $\\ell$ and the black-hole parameters $Q$ and $\\xi$—after which the fundamental mode leaves the spectrum and the first overtone takes over; these are the quasi-resonances. For $\\Lambda > 0$ asymptotically de Sitter solutions, the damping rate instead saturates at a nonzero constant as $\\mu$ grows, so arbitrarily long-lived modes are absent. Near the flat-space quasi-resonant threshold, frequencies computed for different $\\xi$ merge, and in the large-$\\mu$ limit the dependence on $\\ell$ disappears; the paper provides the analytic large-mass expression (23). It also reports that $\\ell=0$ perturbations remain stable even when the effective potential has a negative gap, with the late-time signal governed by a slowly decaying pure-imaginary de Sitter mode in near-extremal cases.","pith_inferences":["If the coupling-independence near quasi-resonance holds beyond the parameter range tested, long-lived frequencies could be used to estimate a black hole's mass and charge without needing to know the non-minimal coupling strength.","The flat/dS split suggests that in a universe with a positive cosmological constant, massive fields around these black holes should always decay at a finite rate; an observation of an essentially undamped massive ringdown would then favour an asymptotically flat geometry or a different matter sector.","Because the paper notes that de Sitter-branch modes do not obey the null-geodesic/eikonal correspondence, a natural test is to check whether the flat-space quasi-resonant frequencies found here satisfy that correspondence as $\\ell \\to \\infty$.","Textual note: the abstract's sentence claiming 'in the de Sitter case, arbitrarily long-lived modes can exist' is reversed relative to the body; the tables and Section V argue the opposite, with quasi-resonances in the flat case and finite damping in de Sitter."],"forward_implications":["In asymptotically flat non-minimal EYM spacetimes, a scalar field heavier than a critical value produces an arbitrarily long-lived ringdown, with the first overtone replacing the fundamental mode.","In the quasi-resonant limit the quasinormal frequencies stop depending on the non-minimal coupling $\\xi$, so the long-lived frequency is fixed by the black-hole parameters and the field mass alone.","In asymptotically de Sitter spacetimes the damping rate tends to a nonzero constant as the field mass grows, so no arbitrarily long-lived modes arise regardless of $\\mu$.","At large mass the spectrum becomes nearly independent of the multipole number $\\ell$, and equation (23) gives the frequencies analytically for small $Q$ and $\\xi$.","Scalar perturbations remain stable even for $\\ell=0$ configurations whose effective potential has a negative gap; the observed late-time decay is dominated by a slowly decaying de Sitter mode in near-extremal cases."],"supporting_citations":[{"why":"supplies the continued-fraction method used for asymptotically flat quasinormal frequencies.","marker":"[48]"},{"why":"provides the reduction of higher-order recurrences to three-term form needed for the non-minimal background.","marker":"[49]"},{"why":"introduces quasi-resonances in the Schwarzschild background, the phenomenon this paper generalizes.","marker":"[30]"},{"why":"contains the analytic proof of the absence of quasi-resonances in de Sitter spacetimes used to interpret the $\\Lambda>0$ results.","marker":"[29]"},{"why":"supplies the non-minimal Einstein-Yang-Mills black-hole solutions whose metric defines the effective potential.","marker":"[25]"},{"why":"provides the time-domain integration scheme used to test stability of negative-gap potentials.","marker":"[50]"},{"why":"supplies the higher-order WKB with Padé approximants used for the de Sitter tables.","marker":"[69]"},{"why":"is the analogous large-mass WKB expansion that the analytic formula (23) extends.","marker":"[78]"}],"fun_headline_variants":["Flat space only: zero damping for massive scalar modes","De Sitter black holes block ultra-long-lived modes","Quasi-resonances demand flat spacetime and critical mass","De Sitter rules out zero-damping scalar modes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the WKB approximation is accurate for the massive scalar field in the asymptotically de Sitter potentials: the de Sitter frequencies are checked only by comparing sixth- and ninth-order WKB results, so if WKB carries a shared systematic error in this regime, the finite-damping claim for de Sitter black holes is not established.","fun_headline_variants_meta":{"raw":{"variants":["Flat space only: zero damping for massive scalar modes","De Sitter black holes block ultra-long-lived modes","Quasi-resonances demand flat spacetime and critical mass","De Sitter rules out zero-damping scalar modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001899,"raw_usage":{"total_tokens":7447,"prompt_tokens":953,"completion_tokens":6494,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":6430}},"tokens_in":569,"tokens_out":6494,"duration_ms":48510,"temperature":1.0,"reasoning_tokens":6430,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:52:12.243969+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Using an independent method (continued fractions adapted to de Sitter boundary conditions, or time-domain evolution with Prony extraction), compute the fundamental de Sitter frequency for a representative case such as $\\Lambda=0.001$, $Q=0.1$, $\\xi=0.5$, $\\ell=1$, $\\mu=1$. If the imaginary part disagrees with the WKB-6/WKB-9 tables by more than their own mutual difference, or if it crosses zero for some $\\mu$, then the claimed absence of quasi-resonances in the de Sitter branch collapses; in the flat case, the quasi-resonance claim would be settled by locating the critical $\\mu$ at which $\\mathrm{Im}(\\omega)$ changes sign with high-precision continued-fraction data.","supporting_citations":[],"review_version":1}