{"id":"5300118b-7f4a-42be-9fbf-7058eb5f69d6","arxiv_id":"2502.09699","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Custodial Naturalness uses classical scale invariance plus a custodial SO(6) symmetry to make the Higgs a naturally light pseudo-Goldstone boson, with testable new particle predictions.","lead":"This paper develops a symmetry-based mechanism, Custodial Naturalness, that explains why the Higgs boson is so much lighter than the scale of new physics without fine-tuning. It predicts a new Z' boson and a dilaton-like scalar, and connects the mechanism to neutrino masses and dark matter.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The custodial-boundary condition at MPl is imposed, not protected, and the claimed robustness scan may not probe the dangerous direction: a Planck-scale-symmetric SO(6) potential plus a small explicit λH−λΦ breaking.","rationale":"The reader's weakest_assumption is precisely the exact SO(6) boundary condition at MPl, and the paper itself flags this as an assumption (Sec. 3.1: 'The equality of quartic couplings by custodial symmetry is an assumption that we expect to be explained by some mechanism at the high scale'). The stress-test pass finds that the numerical stability argument does not cover the most direct violation of that assumption. The scan varies g12|MPl and yψ, which are additional, loop-level custodial-breaking parameters; a tree-level splitting of λH vs. λp vs. λΦ at MPl is not scanned. Since Eq. (2.24) shows the low-scale Higgs mass is generated by λp−λΦ, any boundary splitting of these quartics propagates directly into the Higgs mass and could reintroduce tuning. The correct response is not to reject: the paper is honest about the assumption, the mechanism is plausible, and the analytical argument is internally consistent. But the central claim of robustness under high-scale boundary variations is broader than what the numerics demonstrate. A targeted numerical scan varying δλ/λ at MPl would settle the question. If the allowed splitting turns out to be tiny, the model still works as an existence proof but the word 'natural' carries less force. Hence CONDITIONAL is appropriate, matching the reader's verdict and reasoning.","tokens_in":31896,"tokens_out":2275,"duration_ms":23108,"concrete_test":"Re-run the minimal-model scan with the SO(6) boundary condition relaxed to λH(MPl)=λ+δλ, λp(MPl)=λ−δλ', λΦ(MPl)=λ, varying δλ/λ and δλ'/λ over, say, [0,10⁻⁴] (and also [0,10⁻²]) while keeping g12|MPl=0 and all other inputs as in Sec. 3.1. Then plot Δ and ⟨H⟩/⟨Φ⟩ as functions of δλ/λ. If Δ≲10 survives for δλ/λ up to ~10⁻³, the robustness claim is supported; if the correct EW scale requires δλ/λ≲10⁻⁵, the central naturalness claim is quantitatively contingent on an unexplained exact boundary condition.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The core argument rests on Eq. (2.4): at µ=MPl the potential is exactly SO(6)-symmetric, V=λ(|H|²+|Φ|²)². The paper asserts (Sec. 3.1) that 'none of our mathematical results... are affected by small variations in the boundary conditions even if they slightly break custodial symmetry,' and the scan varies g12|MPl and the fermion Yukawas as proxies. But those are not the dangerous variations. The dangerous variation is a Planck-suppressed operator that splits the boundary quartics, e.g. δλ|H|⁴ at MPl with δλ/λ ~ O(1)×(v/MPl)² or even smaller, because the whole pNGB protection is the equality λp(MPl)=λΦ(MPl). The reader's weakest_assumption identifies exactly this: the exact quartic equality is imposed, not derived. The paper's own Eq. (2.24) shows that the Higgs mass term is ≈2(λp−λΦ)Φ0² at the low scale; a boundary splitting δλ runs down and directly feeds this difference. The scan only varies g12 and yψ, which are additional custodial-breaking sources that are loop-suppressed and can plausibly be small. It does not scan a tree-level custodial-breaking quartic at MPl, nor does it quantify how large such a splitting can be before Δ≫10 or before ⟨H⟩/⟨Φ⟩~10⁻³ is lost. Without that check, the claim 'the hierarchy is stable under variations of high-scale boundary conditions' is not actually demonstrated for the most direct boundary perturbation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops 'Custodial Naturalness,' a mechanism in which classical scale invariance at the Planck scale is combined with an exact SO(6) custodial symmetry of the scalar potential, V = λ(|H|²+|Φ|²)², for the SM Higgs doublet H and a complex singlet Φ. RG running generates an intermediate scale by dimensional transmutation, while the SM-like Higgs emerges as a pseudo-Nambu-Goldstone boson of the spontaneous SO(6) → SO(5) breaking, with a mass controlled by the custodial-violating difference of quartic couplings. The manuscript analyzes the minimal model and two extensions (a neutrino-portal model and a two-component dark-matter model), provides approximate analytic formulas for the effective potential and scalar masses, reports numerical scans of the parameter space of each model using one- and two-loop RGEs, and assesses experimental constraints and future reach for the predicted Z′ and dilaton. The central claim is that the hierarchy ⟨H⟩/⟨Φ⟩ ≈ 10⁻³ is achieved with a fine-tuning measure Δ ≲ 10 and is stable under variations of high-scale boundary conditions and under new sources of custodial symmetry violation.","tokens_in":32445,"tokens_out":17039,"duration_ms":196604,"significance":"If the high-scale boundary assumption is granted, the mechanism is a genuinely new way to address the little hierarchy problem without top partners: the Higgs mass is an output of the RG flow, not an input, and the paper exhibits a concrete, falsifiable correlation between the top quark mass and the Higgs mass (Fig. 9). The numerical work is largely reproducible in structure: scans are described in enough detail, public tools (PyR@TE, MadGraph, micrOMEGAs, SARAH) are used, and LHC dilepton limits are recast. The extensions to neutrino masses and dark matter broaden the phenomenological relevance, and the predicted Z′ and dilaton provide concrete collider targets. The main weakness is that the robustness of the mechanism under the most direct boundary perturbation—a tree-level split among the quartic couplings at MPl—is asserted rather than quantitatively demonstrated; this is the load-bearing gap in the paper's naturalness claim.","major_comments":[{"comment":"The robustness claim that 'none of our mathematical results... are affected by small variations in the boundary conditions even if they slightly break custodial symmetry' is not demonstrated for the most direct perturbation of the boundary condition. The mechanism rests on Eq. (2.4), i.e. λH(MPl) = λp(MPl) = λΦ(MPl), and Eq. (2.24) shows that the low-scale Higgs mass term is set by (λp − λΦ)Φ₀² (in the limit of small g12 and yψ). A scale-invariant but SO(6)-violating marginal operator such as δλ|H|⁴ at MPl runs directly into this difference, and the search for an explanation of the mechanism is exactly the equality of the quartic couplings. The scans in Secs. 3.1–3.3 vary g12|MPl and yψ as proxies, but these are not the same perturbation: g12 enters through the gauge-kinetic RGEs, and the scan definition in Sec. 3.1 formally replaces λH, λp|MPl by λΦ|MPl, thereby removing the quartic splitting from the parameter set over which the fine-tuning measure in Eq. (3.1) is evaluated. I request a dedicated scan, or an analytic bound, that independently adds δλ|H|⁴ at MPl (or varies λp(MPl) − λΦ(MPl) and λH(MPl) − λΦ(MPl)) and reports the maximal |δλ/λ| for which Δ ≲ 10 and vH/vΦ ≈ 10⁻³ survive. Without this, the stability claim covers only a subset of boundary perturbations, not the perturbation that directly controls the pNGB mass formula.","section":"Sec. 3.1 and Eq. (2.24)"},{"comment":"The Conclusions state that the mechanism 'naturally explains' the EW suppression, but the defining condition Eq. (2.4) is an exact SO(6) constraint whose origin is delegated to an unspecified high-scale mechanism. Since SO(6) is explicitly broken by the gauge and Yukawa interactions of the full Lagrangian, there is no symmetry-based reason for the quartic couplings to be exactly equal at MPl; a marginal SO(6)-violating operator is not forbidden by any principle used in the paper. I am not asking for a UV completion, but the conclusions should distinguish between (i) the radiative stability of the hierarchy once the boundary is imposed and (ii) the unexplained choice of the boundary itself. The present text conflates these two statements, and the phrase 'small variations' in Sec. 3.1 is never quantified for the quartic-splitting direction.","section":"Sec. 7 and Sec. 3.1"}],"minor_comments":[{"comment":"The heading 'V ariations and embeddings of Custodial Naturalness' contains a typo and should read 'Variations and embeddings of Custodial Naturalness'.","section":"Sec. 6 heading"},{"comment":"The caption lists the neutrino-portal panels as 'yψ ≠ 0, g12|MPl = 0 (top right)' and 'with yψ ≠ 0, g12|MPl = 0 (bottom left)'; the bottom-left panel appears to correspond to g12|MPl ≠ 0, ȳψ = 0, matching the layout of the other figures.","section":"Fig. 9 caption"},{"comment":"The abstract and Sec. 5 state that the cosmological evolution features a strongly supercooled phase transition testable by gravitational-wave observatories, but no finite-temperature computation is performed for the models of this paper; the statements are based on analogous conformal B−L models. The text does say 'future work should investigate finite temperature effects,' but the abstract should be qualified so that the gravitational-wave statement is not read as a new result of this work.","section":"Sec. 5 and Abstract"},{"comment":"In the definition of the fine-tuning measure, the set of parameters {g_i} over which the maximum is taken is not specified. For reproducibility, please state explicitly which couplings at MPl are varied and whether the derivatives are computed after the custodial replacement λH, λp → λΦ.","section":"Eq. (3.1)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript substantially extends the authors' earlier letter (Ref. [32]) and contains enough new material for a full-length JHEP paper. The central mechanism is defensible, and the numerical implementation is standard. The main obstacle is not a technical error but a robustness gap: the paper's strongest claimed virtue—stability under variations of high-scale boundary conditions—is not tested for the tree-level quartic splitting that directly feeds the Higgs mass formula. The requested boundary-splitting scan is straightforward and should be feasible within a revision. If the authors add that scan and temper the wording of the robustness and cosmological claims, the paper would be acceptable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious, readable follow-up to the authors' own Custodial Naturalness proposal, not a radical new idea. What it adds is a systematic account of what breaks the SO(6) custodial symmetry and how each source feeds into the Higgs mass, plus two concrete extensions: a neutrino portal that predicts two massive Dirac neutrinos and one exactly massless active neutrino, and a two-component fermion dark matter model. The numerics use standard tools (PyR@TE, micrOMEGAs, MadGraph) and the scans are described in enough detail that they look reproducible. The Mt–mh correlation and the Z'/dilaton mass targets are concrete and worth testing. I would not call the central mechanism wrong: within the zero-temperature effective potential/RGE setup, the pNGB suppression of the EW scale is real and the small fine-tuning measure is credible.\n\nThe soft spots, in rough order. (1) The high-scale boundary condition is the load-bearing assumption: V = λ(|H|²+|Φ|²)² at M_Pl. The paper asserts robustness against 'small variations' and scans g12 and new Yukawas at M_Pl, but those are not the dangerous directions. A tree-level Planck-suppressed operator like δλ |H|⁴ at M_Pl splits λH from λp at the boundary and feeds directly into the low-scale Higgs mass term, approximately 2(λp−λΦ)Φ0². The paper does not scan such a quartic splitting or bound how large it can be before Δ≫10 or the hierarchy is lost. This is not fatal—many scale-invariant constructions start from an assumed exact symmetry, and the authors are transparent that the SO(6) point is imposed, not derived. But the robustness claim is currently stronger than what the scan demonstrates. (2) The cosmological section borrows the strongly supercooled phase transition from conformal B−L papers; the abstract states it more crisply than the body, which defers a quantitative analysis. (3) The DM model needs m_ψ ≈ m_Z'/2 and yψ ≈ yψ′ within a few percent; the viable relic region is small and much of it is already excluded by dilepton searches. The authors say this. Minor: the fine-tuning measure is measure-dependent, as they admit.\n\nWho it is for: BSM phenomenologists working on pNGB Higgs models, conformal B−L, and future collider and DM probes. It deserves a serious referee. My advice: send it out; ask the referee to check whether the boundary-condition robustness claim can be supported against a tree-level δλ at M_Pl, and to make the abstract match the caveated cosmology. Those are revision-level issues, not grounds for rejection.","headline":"A solid systematic follow-up on the pNGB Higgs idea, with a real but checkable soft spot: the M_Pl SO(6) boundary condition is imposed, and the robustness scan does not probe tree-level quartic splittings.","tokens_in":32848,"tokens_out":2870,"would_cite":true,"duration_ms":32001,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The Higgs could be a pseudo-Goldstone boson whose small mass is set by an SO(6) custodial symmetry at the Planck scale, with no fine-tuning.","keywords":["Custodial Naturalness","pseudo-Nambu-Goldstone boson","classical scale invariance","Coleman-Weinberg mechanism","little hierarchy problem","SO(6) custodial symmetry","U(1)X gauge boson","dilaton"],"falsifier":"A precise measurement of the top quark pole mass and Higgs mass could falsify the mechanism: the paper finds no viable parameter points with $M_t \\lesssim 171.5$ GeV and predicts an approximately linear $M_t$--$m_h$ correlation in the minimal model, so a top mass below that bound or a Higgs mass outside the predicted band would rule it out. A future 100 TeV collider search for a $Z'$ decaying to dileptons across the full predicted $m_{Z'}\\sim 4$--$100$ TeV range, with no signal, would similarly exclude the minimal realization.","tokens_in":31652,"feed_emoji":"⚛️","tokens_out":7657,"duration_ms":74729,"temperature":0.7,"pith_summary":"Custodial Naturalness is a proposed explanation for why the electroweak scale is so much smaller than the scale of new physics. The paper argues that if the scalar sector has an SO(6) custodial symmetry at the Planck scale, with the Higgs doublet and a new complex singlet treated on equal footing, then radiative symmetry breaking at an intermediate scale makes the Higgs a pseudo-Nambu-Goldstone boson. Its mass then comes only from explicit custodial-symmetry violation, approximately $m_h^2 \\approx 2(\\lambda_\\Phi-\\lambda_p)\\,v_\\Phi^2$, instead of from quadratically divergent corrections. The authors show numerically that the resulting hierarchy $\\langle H\\rangle \\sim 10^{-3}\\langle\\Phi\\rangle$ requires little fine-tuning, and that the mechanism survives new sources of symmetry breaking, making the idea testable through a heavy $Z'$, a light dilaton, and a correlation between the top and Higgs masses.","feed_headline":"A Planck-scale symmetry keeps the Higgs mass naturally tiny","feed_subtitle":"The Higgs as a pseudo-Goldstone of SO(6) would explain the weak scale and predict a Z' boson and a dilaton at future colliders.","key_machinery":"The load-bearing object is the SO(6)-symmetric, classically scale-invariant scalar potential $V=\\lambda(|H|^2+|\\Phi|^2)^2$ imposed at the Planck scale, together with a gauged U(1)X under which $H$ and $\\Phi$ have equal charges. The radiative breaking follows the Gildener-Weinberg flat-direction approximation: at the scale where one quartic coupling turns negative, the vacuum direction is mostly $\\Phi$, so $\\langle\\Phi\\rangle$ (the intermediate scale) is generated by dimensional transmutation while $\\langle H\\rangle$ is suppressed. Expanding the one-loop effective potential in $H_b/\\Phi_0$ yields the Higgs mass-squared term $\\approx 2\\lambda_p\\Phi_0^2 H_b^2$, and the physical Higgs mass $m_h^2 \\approx 2(\\lambda_\\Phi-\\lambda_p)v_\\Phi^2$ is set by the custodial-symmetry-violating splitting between the quartic couplings, not by the large $\\lambda_H$ driven by the top Yukawa. The other emerging field, the dilaton, is the pseudo-Goldstone boson of broken scale invariance with mass set by the $\\beta$ function.","core_discovery":"The central claim is that the large hierarchy between the electroweak scale and the scale of ultraviolet completion can be generated from just two ingredients: classical scale invariance and an SO(6) custodial symmetry of the scalar potential at the Planck scale. The potential $V=\\lambda(|H|^2+|\\Phi|^2)^2$ is symmetric under rotations between the SM Higgs doublet $H$ and a complex singlet $\\Phi$, and neither field has a tree-level mass. Quantum corrections drive the couplings so that the potential develops a flat direction, and the Coleman-Weinberg mechanism spontaneously breaks both scale and custodial symmetry at an intermediate scale, SO(6) $\\to$ SO(5). Four of the five resulting Goldstone bosons are eaten, while the fifth is the physical Higgs, whose small mass is fixed by the amount of custodial symmetry violation: $m_h^2 \\approx 2(\\lambda_\\Phi-\\lambda_p)v_\\Phi^2$ in the limit of small kinetic mixing and new Yukawas. Because the top Yukawa and electroweak gauge couplings break the custodial symmetry only in subleading order, the Higgs mass is protected without introducing top partners. The paper demonstrates stability of this picture under variations of high-scale boundary conditions and under new sources of custodial symmetry violation, and constructs minimal, neutrino-portal, and dark-matter realizations.","pith_inferences":["Taken at face value, the mechanism suggests that many existing scale-invariant models with a singlet scalar may be fine-tuned versions of a more symmetric setup; the SO(6) symmetry is the ingredient that removes the need to adjust $\\lambda_p-\\lambda_\\Phi$.","The Planck-scale boundary condition is the least protected part of the construction; a future theory of quantum gravity that produces small explicit violations of scale or custodial symmetry at $M_{\\rm Pl}$ would feed directly into the Higgs mass, and quantifying that sensitivity is a natural next step.","Non-universal flavor charge assignments are an obvious extension suggested by the paper's structure; if custodial symmetry survives such assignments, the same mechanism could tie the electroweak hierarchy to the flavor structure and the muon $g-2$ anomaly.","One can test the mechanism indirectly now by using the top-mass measurement to improve predictions for $m_{Z'}$ and the dilaton mixing angle, since the paper shows the top mass is the dominant source of theoretical uncertainty."],"forward_implications":["Electroweak symmetry breaking is naturally hierarchical: the scan yields $\\langle H\\rangle/\\langle\\Phi\\rangle \\sim 10^{-3}$ with fine-tuning measure $\\Delta\\lesssim 10$, so no top-partner mechanism is needed.","A heavy $Z'$ with mass roughly 4--100 TeV and a dilaton-like scalar with mass roughly 30--1000 GeV are generic predictions; future colliders and Higgs factories can probe them, and tiny Higgs-dilaton mixing opens displaced-vertex signatures.","The model predicts a correlation between the top quark and Higgs masses; measuring $M_t$ to about 0.1 GeV precision would sharpen the prediction of $m_{Z'}$.","The neutrino-portal extension predicts two massive Dirac neutrinos and one exactly massless active neutrino, while the dark-matter extension provides two stable WIMP candidates whose relic density can match observation near the $Z'$ resonance.","The cosmological history contains a strongly supercooled first-order phase transition, which could produce gravitational waves observable by future detectors."],"supporting_citations":[{"why":"Supplies the Coleman-Weinberg mechanism by which radiative corrections generate a scale from a classically scale-invariant potential.","marker":"[2]"},{"why":"Provides the Gildener-Weinberg flat-direction approximation used to analyse the vacuum and identify the pseudo-Goldstone Higgs.","marker":"[4]"},{"why":"Introduced the Custodial Naturalness mechanism that this paper extends to new sources of symmetry violation and new models.","marker":"[32]"},{"why":"Gives the two-loop Standard Model matching and running formulas used to translate measured top and Higgs parameters to the high scale.","marker":"[59]"},{"why":"Supplies the renormalization-group equations used in the numerical scans of the parameter space.","marker":"[60]"},{"why":"Defines the fine-tuning measure used to quantify how much adjustment the hierarchy requires.","marker":"[61]"},{"why":"Provides the dilepton resonance search limits that currently bound the predicted Z' mass range.","marker":"[66]"}],"fun_headline_variants":["Custodial symmetry makes the Higgs naturally light","Scale invariance plus SO(6) yields tiny Higgs mass","Higgs as pseudo-Goldstone solves hierarchy naturally","New mechanism predicts Z' and dilaton from Higgs mass","Symmetry-based naturalness: no hierarchy problem"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"At the Planck scale the scalar potential must be exactly scale-invariant and SO(6)-symmetric, meaning three quartic couplings are exactly equal; this equality is imposed as a boundary condition rather than derived, and even a small Planck-suppressed breaking of scale invariance or custodial symmetry would change the predicted hierarchy.","fun_headline_variants_meta":{"raw":{"variants":["Custodial symmetry makes the Higgs naturally light","Scale invariance plus SO(6) yields tiny Higgs mass","Higgs as pseudo-Goldstone solves hierarchy naturally","New mechanism predicts Z' and dilaton from Higgs mass","Symmetry-based naturalness: no hierarchy problem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000284,"raw_usage":{"total_tokens":1732,"prompt_tokens":1062,"completion_tokens":670,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":678,"completion_tokens_details":{"reasoning_tokens":593}},"tokens_in":678,"tokens_out":670,"duration_ms":6794,"temperature":1.0,"reasoning_tokens":593,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T20:47:11.400180+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A precise measurement of the top quark pole mass and Higgs mass could falsify the mechanism: the paper finds no viable parameter points with $M_t \\lesssim 171.5$ GeV and predicts an approximately linear $M_t$--$m_h$ correlation in the minimal model, so a top mass below that bound or a Higgs mass outside the predicted band would rule it out. A future 100 TeV collider search for a $Z'$ decaying to dileptons across the full predicted $m_{Z'}\\sim 4$--$100$ TeV range, with no signal, would similarly exclude the minimal realization.","supporting_citations":[],"review_version":1}