{"id":"7d1f6f0d-07b8-408f-a1b6-147794cded49","arxiv_id":"1909.02014","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A strong first-order electroweak phase transition is impossible in SO(6)/SO(5) composite Higgs models under the minimal Higgs potential hypothesis, and requires undetermined UV operator contributions, as shown in a 6+6 benchmark with LISA-detectable gravitational waves.","lead":"This paper shows that in the next-to-minimal composite Higgs model, the scalar potential generated by the lightest composite resonances cannot drive a strong first-order electroweak phase transition, for any of three viable fermion embeddings. Adding incalculable high-energy operator contributions makes the transition possible, and the resulting gravitational wave signal falls within the reach of future detectors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 6+6 and 15+15 no-go relies on the unproved inequality |alpha| >> beta f^2; explicit form factors can make this ratio O(1), so an IR-only scan is required to validate the central claim.","rationale":"The reader correctly identified the |alpha| >> beta f^2 inequality as a load-bearing assumption but framed the primary risk as heavier-resonance contributions changing the integrals. The present stress-test sharpens this: even with the paper's own N5=1, N1=2 and Nrho=Na=1 truncation, the inequality is not demonstrated and can plausibly fail when Pi_1/Pi_0 is not small. This is therefore a more direct threat to the claimed no-go for 6+6 and 15+15. The 15+6 model has a different, seemingly robust no-go based on the sign of lambda_h^2 - lambda_h lambda_eta, so the central claim's fate rests mainly on the two models with the suspect inequality. The proposed scan would settle the matter. Because the reader's CONDITIONAL verdict already flags the need for further checks, this concern does not change the recommended verdict.","tokens_in":33690,"tokens_out":9741,"duration_ms":91712,"concrete_test":"Re-run the Section 4.2 scan for the 6+6 and 15+15 models with all UV Wilson coefficients c_L,R and d_L,R set to zero, keeping the same IR parameter ranges, the same SM constraints (Mh=125.09 GeV, MZ=91.1876 GeV, Mt=172.9 GeV, f and coupling bounds), and using the full Coleman-Weinberg integrals rather than the quartic polynomial. Search for parameter points satisfying the two-step SFOEWPT conditions Eqs. (2.27)-(2.36), including v_n/T_n >= 1. If any IR-only point passes, the paper's no-go claim is falsified; if none passes, the |alpha| >> beta f^2 assumption is empirically supported for the stated ranges.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is strongest for the 6+6 and 15+15 models, where the IR-only no-go is w^2|IR = (alpha_t - s_theta^2 f^2 beta_t)/(c_{2theta}^2 beta_t) and w^2|IR = 2 alpha_q/beta_q, respectively, followed by the assertion that |alpha| >> beta f^2 because both come from expanding the same logarithm (Eqs. 3.20 and 3.35). This inequality is never proven. With the explicit N5=1, N1=2 form factors in Eq. (4.13), the ratio R(Q)=Pi_1/Pi_0 can exceed 1 at low Q^2: for M1'/M5=2.5 and M1/M5=0.5, R(0) can approach ~2.5 as couplings grow, so the integral of R^2 can exceed the integral of R, making alpha/(beta f^2) drop below 1 and allowing w < f. The numerical scan in Section 4.2 always includes the UV Wilson coefficients and does not report an IR-only run, while Appendix C only checks the polynomial approximation, not this inequality. Thus the no-go for two of the three benchmark models is not established even within the adopted lightest-resonance truncation; a heavier-resonance spectrum is a separate caveat, but the immediate gap is in the stated proof itself.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the SO(6)/SO(5) next-to-minimal composite Higgs model with one Higgs doublet and one real singlet, and asks whether a strong first-order electroweak phase transition can be generated by the minimal Higgs potential hypothesis, i.e. by the calculable IR form-factor contributions of the lightest composite resonances alone. The authors classify six fermion embeddings for qL and tR, discard three models (6+1, 6+15, 15+1) for lack of a η potential or a top mass, and derive the IR and UV contributions to the scalar potential for the remaining 6+6, 15+6 and 15+15 models. They claim that under MHP none of the three can trigger SFOEWPT: for 6+6 and 15+15 the singlet vacuum w is predicted to exceed f, while for 15+6 the condition λhη^2 > λh λη fails. They then add NDA-estimated UV local operators, scan the parameter space of the 6+6 model with MultiNest and CosmoTransitions, and show that SFOEWPT can occur with Mη around 100 GeV and f ≳ 1 TeV, producing gravitational waves visible to LISA, Tianqin, Taiji, BBO or DECIGO, and with collider phenomenology that leaves room at Mρ ≳ 3 TeV.","tokens_in":33919,"tokens_out":7574,"duration_ms":77080,"significance":"If the no-go claim is established, the paper is significant: it would rule out the commonly adopted MHP for SFOEWPT in NMCHM with embeddings up to 15 and motivate UV-dominated singlet potentials. The paper's strengths include the explicit CCWZ construction, concrete form-factor integrals with Weinberg sum rules, an appendix checking the polynomial approximation, and a complete numerical pipeline for nucleation and gravitational waves. However, the central negative claim is only as strong as the inequality |α| >> β f^2 for the 6+6 and 15+15 models, and that inequality is not proved in the manuscript.","major_comments":[{"comment":"The no-go for the 6+6 and 15+15 models rests on the claim |αt| >> βt f^2 (and analogously |αq| >> βq f^2), asserted because the coefficients come from expanding the same logarithm. This inequality is never proved. With the explicit N5=1, N1=2 form factors in Eq. (4.13), the ratio R(Q)=Π1/Π0 can exceed unity at low Q^2 for allowed mass hierarchies, so ∫R^2 can be larger than ∫R and α/(β f^2) can be O(1) or smaller, which would allow w < f. Please either prove the inequality for these form factors or perform the numerical scan of Section 4.2 with the UV Wilson coefficients set to zero and report the resulting w values.","section":"Section 3.3, Eq. (3.20); Section 3.5, Eq. (3.35)"},{"comment":"The IR form factors are saturated by one vector multiplet and one 5-plet plus two singlet partners. The no-go conclusion is derived within this truncation; a discussion of how heavier resonances would modify the ratio α/(β f^2), or an estimate based on the next resonance, is needed before the conclusion can be presented as a general statement about NMCHM under MHP. This does not require a full calculation, but the sensitivity of the central negative claim to the truncation should be quantified or at least discussed.","section":"Section 4.1, Eqs. (4.5) and (4.12)"},{"comment":"The statement that 'if only the IR contributions to the scalar potential are considered, all those three models cannot trigger SFOEWPT' is stronger than the derivation supports. Because the 6+6 and 15+15 no-go depends on the unproved inequality discussed above, the conclusion should either be restricted to the parameter ranges and truncation where the inequality is checked, or the proof should be completed. The 15+6 obstruction in Eq. (3.29) is algebraic and solid, so the issue is specific to the other two models, but the abstract does not make this distinction.","section":"Abstract and Section 5"}],"minor_comments":[{"comment":"The derivation of the chain cη/ch < μη^2/μh^2 < sqrt(λη)/sqrt(λh) < λhη/λh is compressed; please show the intermediate algebra or add a footnote explaining how each inequality follows from Eqs. (2.27)–(2.32).","section":"Section 2.2, Eq. (2.33)"},{"comment":"The phrase 'This inequality can never be achieved because η < f is given by its definition' is confusing: the issue is that the predicted minimum lies outside the allowed field range, not that the inequality is forbidden by fiat. Please rephrase.","section":"Section 3.3, after Eq. (3.20)"},{"comment":"There is a typo: 'play a important role' should be 'play an important role'; similar small typos appear elsewhere, including 'the for μ2h,η and λh,hη' in Section 4.2.","section":"Section 4.3, first paragraph"},{"comment":"It would help to state explicitly whether the plotted gravitational-wave envelope includes points that are excluded by the collider constraints shown in Fig. 4, or whether the envelope is shown before applying those constraints.","section":"Figure 3, right panel"},{"comment":"The phrase 'two degenerate vacuums' should be 'two degenerate vacua' in Eqs. (2.29)–(2.31) and the surrounding text.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is well organized and the numerical work is substantial. The main issue is the unproved α/β inequality, which is load-bearing for two of the three no-go claims; this is fixable with a dedicated IR-only numerical scan or an analytic bound. I recommend major revision rather than rejection, because if the gap is closed the paper would be a solid contribution to the composite-Higgs and electroweak-phase-transition literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a serious referee, but the headline claim is ahead of the proof. What is actually new: a systematic form-factor computation of the scalar potential for the 6+6, 15+6 and 15+15 SO(6)/SO(5) embeddings, a new 15A embedding for qL, a clean argument that 15B is killed by ZbLbar-bL, and a concrete 6+6 benchmark where UV Wilson coefficients are tuned to get SFOEWPT. The classification of which embeddings can even generate a Higgs-singlet potential (6+1, 6+15 fail) is also useful. The authors are honest that the UV part is incalculable and only NDA-sized.\n\nThe main soft spot is exactly where the stress-test puts it. For 6+6 and 15+15 the no-go reduces to w^2|IR ~ alpha/(beta f^2) >> f^2 from the unproved inequality |alpha| >> beta f^2. The paper asserts this because both come from expanding the same logarithm. But with the explicit lightest-resonance form factors in Eq. (4.13), R = Pi_1/Pi_0 can exceed 1 at low Q^2: for M1'/M5 = 2.5 and M1/M5 = 0.5, R(0) approaches about 2.5 as couplings grow. Then beta f^2 can be comparable to or larger than alpha, and w can drop below f. The numerical scan in Sec. 4.2 always includes the UV coefficients, so it never tests the IR-only claim. Appendix C checks the polynomial approximation, not this inequality. So the no-go for two of the three benchmark models is not established even within the adopted single-resonance truncation. Heavier resonances are an extra caveat, not the immediate gap.\n\nThe 15+6 no-go is on firmer ground: it follows from lambda_h-eta^2 - lambda_h lambda_eta < 0 with beta_q,t positive and does not need the alpha/beta inequality. That part survives. The positive 6+6 benchmark achieves SFOEWPT by scanning O(1) UV Wilson coefficients; the authors present it as a demonstration, not a prediction, which is fair, but it means the gravitational wave and collider numbers are conditional on those chosen coefficients.\n\nWho this is for: people building composite Higgs phase-transition and baryogenesis models. The systematic computation and benchmark will be used. Send it to a serious referee, with the request to either prove the |alpha| >> beta f^2 step under the stated truncation or show an IR-only numerical scan that confirms the no-go for 6+6 and 15+15. The paper is a useful piece of work even with that caveat, but the central negative claim needs to be marked as provisional.","headline":"Systematic form-factor computation worth taking seriously, but the no-go for 6+6 and 15+15 rests on an unproved inequality and the headline 'MHP fails' is stronger than the proof.","tokens_in":34562,"tokens_out":5807,"would_cite":true,"duration_ms":52091,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Minimal composite Higgs potentials cannot drive a strong electroweak phase transition.","keywords":["composite Higgs","next-to-minimal composite Higgs model","strong first-order electroweak phase transition","minimal Higgs potential hypothesis","gravitational waves","Weinberg sum rules","top partners","singlet scalar"],"falsifier":"Computing the scalar-potential coefficients with additional resonance states (e.g. Nρ = Na = 2 for vectors or N5 = 2, N1 = 3 for fermions) and checking whether the singlet VEV stays below f or whether λhη² > λh λη becomes possible; if either holds in the IR-only potential, the no-go conclusion would be overturned.","tokens_in":33386,"feed_emoji":"⚛️","tokens_out":2870,"duration_ms":34777,"temperature":0.7,"pith_summary":"The paper studies whether the next-to-minimal composite Higgs model, based on SO(6)/SO(5), can produce a strong first-order electroweak phase transition (SFOEWPT). It systematically builds six benchmark models by embedding the top quark in different representations of SO(6), and finds that the commonly used minimal Higgs potential hypothesis, where the scalar potential comes only from the lightest composite resonances, fails for every viable model. The paper then shows that including incalculable ultraviolet local operators can restore SFOEWPT, and works out the gravitational-wave and collider consequences for one concrete model. A sympathetic reader should care because SFOEWPT is a necessary ingredient of electroweak baryogenesis and produces observable gravitational waves.","feed_headline":"Minimal composite Higgs potentials cannot drive a strong electroweak transition","feed_subtitle":"The next-to-minimal composite Higgs model needs unknown UV operators to get SFOEWPT, yielding testable gravitational waves.","key_machinery":"The scalar potential for the Higgs doublet h and singlet η is generated by SO(6)-breaking gauge and fermion interactions, split into infrared form-factor contributions from lightest vector and fermion resonances and ultraviolet contributions from local operators. The infrared coefficients are expressed through five basic integrals αq,t, βq,t, and ε, and are made convergent by Weinberg sum rules imposed on the resonance spectrum. The argument identifies a two-step phase transition path, (0,0)→(0,w)→(v,0), as the correct way to realize SFOEWPT in this model, and derives inequalities among the potential coefficients that must hold for degenerate minima at the critical temperature.","core_discovery":"For the SO(6)/SO(5) next-to-minimal composite Higgs model, the strongly first-order electroweak phase transition cannot be triggered if only the calculable infrared contributions to the scalar potential are included. This conclusion holds for all three viable fermion embeddings considered: 6+6, 15+6, and 15+15. In the 6+6 and 15+15 cases the singlet field would need a vacuum expectation value larger than the Goldstone decay constant, breaking perturbativity, while in the 15+6 case the necessary condition λhη² > λh λη cannot be satisfied. Therefore the minimal Higgs potential hypothesis, which assumes that ultraviolet contributions are negligible, is incompatible with SFOEWPT in these models. Adding ultraviolet local operators, estimated by naive dimensional analysis, can restore SFOEWPT; the paper demonstrates this explicitly for the 6+6 model.","pith_inferences":["The no-go result plausibly extends beyond the representations studied here, because the obstruction in 6+6 and 15+15 comes from the sign and size of the β integrals, which are positive by definition; one testable extension is to check larger representations and count whether the obstruction persists.","If ultraviolet contributions are generically O(1) as naive dimensional analysis suggests, then SFOEWPT in composite Higgs models is essentially a probe of the unknown strong-sector UV completion, not of the low-energy coset structure alone.","The two-step transition required here implies that the singlet η acquires a VEV before the Higgs, which could leave a distinct imprint in gravitational-wave spectral shapes (e.g. a stronger peak from the first step) that dedicated simulations could separate from one-step transitions.","The same form-factor machinery could be applied to other cosets with extra singlets, such as SU(4)/Sp(4), to ask whether the MHP no-go is a generic feature of pNGB singlet extensions."],"forward_implications":["The minimal Higgs potential hypothesis cannot be assumed in future studies of strong first-order phase transitions in SO(6)/SO(5) composite Higgs models; ultraviolet contributions are needed.","SFOEWPT predictions in these models depend on incalculable ultraviolet operators, so statements about gravitational-wave signals carry an irreducible model-dependence from the strong-sector UV completion.","In the 6+6 model, combining IR and UV contributions produces SFOEWPT with f ≳ 1 TeV and singlet mass around 100 GeV, with gravitational-wave signals within the reach of future space-based detectors.","Composite resonances in the SFOEWPT-viable region, especially vector resonances heavier than twice the top-partner mass, may be searched for at the HL-LHC through same-sign dilepton final states.","The 15B embedding of the left-handed doublet is strongly disfavoured by ZbL bL measurements, leaving 15A as the viable 15-representation choice."],"supporting_citations":[{"why":"Defines the SO(6)/SO(5) next-to-minimal composite Higgs model, the U(1)η symmetry, and the h, η field parametrization used throughout.","marker":"[5]"},{"why":"Supplies the form-factor expressions for vector resonances and the general CCWZ framework for computing the Higgs potential.","marker":"[38]"},{"why":"Proposes the minimal Higgs potential hypothesis that the paper tests and ultimately rejects for SFOEWPT.","marker":"[42]"},{"why":"Provides the embedding of tR in the 6 representation with mixing angles θ and φ, used in the 6+6 and 15+6 models.","marker":"[44]"},{"why":"Defines the five basic integrals α, β, and ε that express the fermion-induced infrared contributions to the scalar potential.","marker":"[45]"},{"why":"Gives the high-temperature expansion and two-step phase-transition conditions used to derive the SFOEWPT inequalities.","marker":"[52]"},{"why":"Original Weinberg sum rules that make the gauge and fermion form-factor integrals convergent.","marker":"[71]"}],"fun_headline_variants":["Composite Higgs: UV operators required for strong electroweak transition","Minimal composite Higgs fails to produce strong first-order transition","UV operators mandatory for SFOEWPT in composite Higgs","SO(6)/SO(5) composite Higgs needs UV terms for SFOEWPT","No SFOEWPT in composite Higgs without UV contributions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The infrared form factors are assumed to be saturated by the lightest vector and fermion resonances, with heavier resonances contributing nothing that changes the sign or size of the computed coefficients.","fun_headline_variants_meta":{"raw":{"variants":["Composite Higgs: UV operators required for strong electroweak transition","Minimal composite Higgs fails to produce strong first-order transition","UV operators mandatory for SFOEWPT in composite Higgs","SO(6)/SO(5) composite Higgs needs UV terms for SFOEWPT","No SFOEWPT in composite Higgs without UV contributions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00102,"raw_usage":{"total_tokens":4268,"prompt_tokens":872,"completion_tokens":3396,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":3308}},"tokens_in":488,"tokens_out":3396,"duration_ms":23011,"temperature":1.0,"reasoning_tokens":3308,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:02:53.891541+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Computing the scalar-potential coefficients with additional resonance states (e.g. Nρ = Na = 2 for vectors or N5 = 2, N1 = 3 for fermions) and checking whether the singlet VEV stays below f or whether λhη² > λh λη becomes possible; if either holds in the IR-only potential, the no-go conclusion would be overturned.","supporting_citations":[{"cited_title":"General Composite Higgs Models","cited_arxiv_id":"1205.0770","evidence_quote":"Proposes the minimal Higgs potential hypothesis that the paper tests and ultimately rejects for SFOEWPT."}],"review_version":1}