{"id":"199d91b7-5435-472c-92f9-639ba0b7c014","arxiv_id":"1908.09312","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"The paper completes the minimal fundamental partial compositeness EFT for real and complex TC representations, finding a custodial-triplet VEV with a rho-parameter contribution in SU(5)/SO(5) and no triplet VEV in SU(4)xSU(4)/SU(4).","lead":"Two new composite-Higgs models are analyzed, in which quarks get mass by mixing with composite partners and the new technicolor fermions come in real or complex representations. The real case develops an extra Higgs-triplet vacuum value that degrades electroweak precision, while the complex case avoids it.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Real-case triplet VEV is conditional on uncomputed coefficients C1-3_Vf in eq. (3.34); the paper itself concedes that a cancellation is possible.","rationale":"The reader's weakest assumption is the correct one. The headline comparison between real and complex FPC models is the paper's main contribution, and its real-case branch is entirely controlled by uncomputed strong-dynamics coefficients C_i_Vf. The paper itself concedes the cancellation possibility, so this is not a hidden flaw; it is a self-flagged limitation. My stress test does not find an internal inconsistency: the EFT construction, spurion assignments, operator counting, and the CP argument for the complex-case triplets appear coherent. The issue is that the central positive claim is not determined by symmetries alone; it depends on dynamical coefficients whose signs and magnitudes are not calculated. In the natural simplifying limit y_Q = y_Qtilde, the whole claim reduces to C2_Vf being nonzero, making that coefficient decisive. A lattice or perturbative computation of the operator coefficients would settle it. Until then CONDITIONAL is the right verdict, and the reader's conditionality should be retained without change.","tokens_in":23690,"tokens_out":8094,"duration_ms":85958,"concrete_test":"Evaluate the tadpole combination T_eta = 4 Lambda^2/(16 pi^2)[-C1_Vf(...) + C2_Vf(...) + C3_Vf(...)] from eq. (3.34) with the Wilson coefficients computed at one loop in a perturbative scalar-extended version of the UV Lagrangian (2.1), or extracted from a lattice simulation of the SU(N_TC) theory with fundamental fermions and scalars. A minimal decisive check: set y_Q = y_Qtilde (left-handed custodial limit) and compute C2_Vf; if C2_Vf = 0 or has the wrong sign, the inevitable tadpole is absent and the real-case VEV and rho contribution (3.37)-(3.38) do not follow. Also verify that the same coefficients give B>0 in eq. (3.31); if B <= 0, the vacuum is not electroweak-misaligned and the analysis is moot.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that SU(5)/SO(5) necessarily develops a custodial-triplet VEV while SU(4)xSU(4)/SU(4) does not rests on the bracket in eq. (3.34). In the simplest way to accommodate mb/mt, with y_Q = y_Qtilde, the bracket reduces to C2_Vf(|yt|^2-|yb|^2) times the remaining loop factor; if C2_Vf is small, zero, or of opposite sign relative to the other operators in the general case, the tadpole vanishes and the delta-rho contribution in eq. (3.38) disappears. No computation of C1_Vf, C2_Vf, C3_Vf from the underlying TC-plus-scalar dynamics is provided, and the Introduction explicitly flags an unforeseen cancellation at O(p^4). The same coefficients also control B in eq. (3.31), so the assumption A,B>0 needed for electroweak misalignment is itself parametric. The complex-case no-VEV argument is more robust because it relies on CP parity; the real-case positive result is a conditional statement, not a prediction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes two minimal models of fundamental partial compositeness at the electroweak scale, one with TC fermions in a real representation (SU(5)/SO(5)) and one with TC fermions in a complex representation (SU(4)xSU(4)/SU(4)_D). For each case the authors construct the effective Lagrangian, derive the effective potential and vacuum alignment, compute corrections to the rho parameter and to Zbb, and list the relevant operator bases in the appendices. The central claim is that the custodial-triplet pNGB in the SU(5)/SO(5) model acquires a VEV, contributing to the rho parameter, while the corresponding triplet VEVs in the SU(4)xSU(4)/SU(4)_D model vanish because the would-be tadpole coefficients are imaginary and the underlying theory is CP even. The paper explicitly notes in the Introduction that the real-case triplet VEV is generated at O(p^4) and could be subject to an unforeseen cancellation once the strong-dynamics coefficients are determined.","tokens_in":24101,"tokens_out":3280,"duration_ms":35812,"significance":"If the central claim is established, the paper provides a useful classification of minimal FPC models: the real-representation model deviates from custodial symmetry through a triplet VEV, while the complex-representation model avoids this effect for symmetry reasons. The complex-case argument based on CP parity is robust and internally consistent, and the systematic operator counting in Sections 3, 4, and the appendices is a valuable technical contribution. The paper also identifies concrete phenomenological constraints from Zbb, four-top production, and dipole operators. However, the headline real-case result is contingent on uncomputed strong-dynamics coefficients that the authors themselves flag in the Introduction, and the rho parameter estimate is labeled 'rough' in Sec. 3.7. The significance would be enhanced by a clear statement of the parametric assumptions under which the real-case prediction holds, or by a computation or estimate of the relevant coefficients.","major_comments":[{"comment":"The statement that 'requiring physical masses inevitably induces a tadpole for the eta_3^0' is not established by the argument given. The tadpole is a linear combination of the strong-dynamics coefficients C1_Vf, C2_Vf, C3_Vf times Yukawa combinations. While the combination vanishes in the custodial limit, it can also vanish for non-custodial values of the Yukawa couplings if the coefficients take appropriate signs and sizes, and the paper itself acknowledges in the Introduction that an unforeseen cancellation at O(p^4) is possible. Since the abstract and conclusions present the triplet VEV as a definite finding, the claim should be reframed as a scenario under explicit assumptions about C1_Vf, C2_Vf, C3_Vf, or supported by a computation or estimate of these coefficients from the underlying TC-plus-scalar dynamics.","section":"Sec. 3.5, Eq. (3.34)"},{"comment":"The central quantitative result, the contribution to delta-rho from the triplet VEV, is called 'a rough estimate' in the text and depends on the same uncomputed strong-dynamics coefficients C1_Vf, C2_Vf, C3_Vf through T_eta in Eq. (3.37). The authors also note that additional contributions from gauge-boson vacuum polarization with vector-like partners running in the loop are not included. As written, this is not a prediction from the model but a conditional estimate. The paper should consistently present it as such, and the abstract's claim that the triplet VEV 'is indeed the case' for SU(5)/SO(5) should be conditioned on the absence of the cancellations mentioned in the Introduction.","section":"Sec. 3.7, Eq. (3.44)"},{"comment":"The misalignment of the electroweak vacuum in the real case requires A,B>0 in Eq. (3.31). These conditions depend on the signs and magnitudes of the uncomputed coefficients C_g, C_m, C1_Vf, C2_Vf, C3_Vf. The paper does not demonstrate that there exists a region of parameter space satisfying A,B>0 together with the no-tachyon requirement around Eq. (3.39). While this is not an internal inconsistency, it means that even the real-case vacuum alignment is parametric. This point is load-bearing because the triplet tadpole and the resulting rho contribution are evaluated at the misaligned vacuum. The authors should either provide an explicit parameter scan or an existence argument, or state this as an assumption in the abstract and conclusions.","section":"Sec. 3.4, Eqs. (3.31)-(3.33)"}],"minor_comments":[{"comment":"There are spelling errors: 'hyerarchies' should be 'hierarchies' and 'trulys' should be 'truly'.","section":"Sec. 1"},{"comment":"The relative signs of C2_Vf in Eqs. (3.32)-(3.33) and in Eq. (3.34) matter for the cancellation argument, but the definitions are spread over two subsections. It would help to state explicitly that the same operator coefficient C2_Vf appears with sign changes dictated by the contractions, rather than leaving the reader to compare the expressions.","section":"Sec. 3.5, Eq. (3.34) and Sec. 3.4, Eqs. (3.32)-(3.33)"},{"comment":"The completeness claim in Section 5 ('we provided in the appendices a complete list of the effective operators') should be qualified by the fact that for each template in Eqs. (A.17)-(A.22) only one scalar-index contraction is shown and two additional contractions analogous to Eqs. (3.28)-(3.30) are said to be 'understood'. The actual, fully expanded list should either be given or its derivation indicated.","section":"Appendix A.3, text after Eq. (A.22)"},{"comment":"The combination of coefficients C_yPiD and C_PiD in Eq. (A.24) is presented without derivation. A brief statement of how the operator templates in Eqs. (A.17)-(A.22) reduce to these combinations would make the result easier to check.","section":"Sec. 3.7, Eq. (3.44)"}],"recommendation":"major_revision","confidential_remarks":"The paper is suitable in scope for JHEP. The complex-case symmetry argument is solid, and the operator counting is systematic. The main issue is that the headline real-case triplet VEV and the derived rho contribution are conditional on uncomputed strong-dynamics coefficients, and the paper's own Introduction admits this possibility. This is a framing and support problem rather than a mathematical error, so I recommend major revision with a request to either compute or bound the coefficients, or to clearly reframe the real-case results as a scenario under explicit parameter assumptions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. This is the real- and complex-representation companion to the earlier pseudo-real FPC analysis, and as an EFT classification exercise it is careful and useful. The central claim—that SU(5)/SO(5) develops a custodial-triplet VEV while SU(4)xSU(4)/SU(4) does not—is only half robust. The complex-case no-VEV result follows from CP parity and is solid. The real-case triplet VEV depends on uncomputed coefficients that the authors themselves concede could cancel.\n\nWhat is actually new: the spurion construction for real and complex TC fermions, the full operator bases in the appendices, the identification of the different CP properties of the triplets, and the first pass at Zbb, rho, and LHC constraints for these two cosets. The cosets were studied earlier, and the paper cites that work honestly. The internal symmetry logic checks out—the CP-odd triplet argument in the complex case is clean and correct.\n\nThe soft spot is real and load-bearing. The real-case positive result rests on eq. (3.34). The bracket contains three uncomputed strong-dynamics coefficients C1_Vf, C2_Vf, C3_Vf. In the simple y_Q = y_Qtilde limit, the bracket reduces essentially to C2_Vf(|yt|^2-|yb|^2); if C2_Vf is small, zero, or opposite sign relative to the other operators, the tadpole vanishes and the delta-rho estimate disappears. The paper flags this in the Introduction (an 'unforeseen cancellation' at O(p^4)) and labels its rho estimate 'a rough estimate' in Sec. 3.7. So the headline real-case result is a conditional statement, not a prediction. The same coefficients enter B in the potential, so the A,B>0 misalignment assumption is also parametric. This is not a fatal flaw—it is honest model building in the EFT framework—but it should be presented as a scan over coefficients, not a determination.\n\nMinor soft spots: the extensive four-fermion and dipole operator lists are not matched to concrete lattice inputs, and the collider constraints translated into bounds on fundamental Yukawas assume NDA for coefficient sizes. Reasonable, but worth remembering.\n\nVerdict: a solid EFT model-building paper. It deserves a serious referee, and the right referee will want the real-case triplet discussion reframed as parameter-dependent rather than a unique prediction. The complex-case section is the more robust part and is the part I would cite. I would bring it to the reading group, but I would not cite the triplet-VEV claim without a caveat.","headline":"Extends fundamental partial compositeness to real and complex TC representations with a careful operator classification, but the headline real-case triplet VEV is conditional on uncomputed strong-dynamics coefficients, not a prediction.","tokens_in":24606,"tokens_out":1948,"would_cite":false,"duration_ms":20311,"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":"In the minimal real-representation model of fundamental partial compositeness, physical quark masses force a custodial-symmetry-breaking triplet vacuum value that shifts the rho parameter; in the complex-representation model all triplet…","keywords":["fundamental partial compositeness","composite Higgs","technicolor","vacuum misalignment","electroweak precision","rho parameter","custodial symmetry","four-top production"],"falsifier":"A lattice or other first-principles determination of the real-case coefficients $C_g$, $C_m$, $C_1^{Vf}$, $C_2^{Vf}$, $C_3^{Vf}$ that yields $A \\le 0$ or $B \\le 0$, or an exact cancellation in the tadpole combination of eq. (3.34), would falsify the claim that physical quark masses inevitably induce a triplet VEV and the associated rho contribution.","tokens_in":23485,"feed_emoji":"⚛️","tokens_out":8596,"duration_ms":79299,"temperature":0.7,"pith_summary":"This paper completes the effective field theory analysis of minimal fundamental partial compositeness by treating the two remaining possibilities for the technicolor fermion representation: real and complex. It establishes that in the real case, realised by the coset SU(5)/SO(5), giving physical masses to the top and bottom quarks inevitably generates a tadpole for the neutral custodial triplet pNGB, giving it a vacuum expectation value and hence a contribution to the rho parameter. In the complex case, realised by SU(4)xSU(4)/SU(4), the same triplet fields are CP-odd, so the CP-even underlying dynamics cannot generate tadpoles and all triplet VEVs vanish. The paper also derives the low-energy Yukawa sector, the vacuum alignment conditions, corrections to Z to b bbar, and collider constraints on four-top and top-dipole operators, making the two minimal models phenomenologically distinguishable.","feed_headline":"Real composite Higgs breaks custodial symmetry; complex twin does not","feed_subtitle":"In SU(5)/SO(5) the custodial triplet shifts the rho parameter; in SU(4)xSU(4)/SU(4) it cannot.","key_machinery":"The load-bearing machinery is the spurion formalism for fundamental partial compositeness: the SM fermions are coupled to composite partners through spurion fields $\\psi$ that carry one index under the scalar flavour symmetry, which contains QCD colour, and one under the fermionic global symmetry, so that invariants built from $\\psi$, the nonlinear $\\sigma$ field $\\Sigma$, and the $\\omega$ tensor encode all Yukawa, potential, and precision operators. Within this formalism, the decisive objects are the triplet tadpole operator of eq. (3.34) and the effective potential $V = f^4(-A\\cos 2\\theta + B\\cos 4\\theta)$: the condition $A,B > 0$ selects the misaligned electroweak vacuum, and the combination of strong-dynamics coefficients $C_i^{Vf}$ in the tadpole controls whether the custodial triplet develops a VEV. The CP parity of the triplet pNGBs is the mechanism that separates the two cases.","core_discovery":"The central discovery is a CP-parity selection rule for triplet vacuum values. For SO(N_TC) with real TC fermions, the minimal coset SU(5)/SO(5) contains a CP-even neutral custodial triplet $\\eta_3^0$; the Yukawa spurion potential of eq. (3.34) contains a tadpole proportional to $s_\\theta^2$ and to combinations of fundamental Yukawa couplings that vanish only in the custodial limit $y_t = y_b$, $y_Q = \\tilde{y}_Q$. Since realistic top and bottom masses require these couplings to differ, the triplet necessarily acquires a vacuum value $\\langle\\eta_3^0\\rangle = -f^3 T_\\eta s_\\theta^2/(2m_{\\eta_3^0}^2)$, which produces $\\delta\\rho = -2f^4 T_\\eta^2 s_\\theta^2/m_{\\eta_3^0}^4$, an order-$p^4$ effect. For SU(N_TC) with complex TC fermions, coset SU(4) x SU(4)/SU(4)_D, the two triplets $N^0$ and $\\Delta^0$ are CP-odd; because the underlying theory is CP-even, the effective potential depends on absolute values of the Yukawa couplings and no tadpole operator is allowed, so the triplet VEVs vanish. The paper further provides complete operator bases at NLO, including four-fermion, dipole, and kinetic operators, and uses them to extract Zbb, four-top, and top-dipole constraints in both models.","pith_inferences":["The same CP-parity argument suggests a model-building rule: among minimal cosets, only those whose custodial triplet is CP-even are exposed to triplet-VEV contributions to the T parameter; cosets with CP-odd triplets are automatically protected, assuming the underlying theory preserves CP.","A first-principles lattice determination of the uncomputed coefficients $C_g$, $C_m$, $C_i^{Vf}$ and their complex-case analogues could decide whether the real-model vacuum is actually misaligned; without those coefficients the sign and size of $A$ and $B$ in eq. (3.31) remain unknown.","The accidental cancellation of the triplet tadpole mentioned by the authors would erase the delta-rho shift from the VEV but would not remove the order-$p^4$ nature of the effect, so precision electroweak data at future colliders could discriminate between the tuned and untuned regimes."],"forward_implications":["If the real-representation model is correct, the neutral custodial triplet necessarily develops a small VEV once top and bottom masses are realistic, and its contribution to the rho parameter must be included in any electroweak precision fit.","The complex-representation model is protected from this particular source of custodial breaking, but it contains two Higgs doublets and a neutral CP-odd triplet sector, so its leading constraint is the Z b_L b_L correction rather than the rho parameter.","The extracted operator bases allow the four-top and top-dipole LHC bounds to be translated directly into bounds on the fundamental Yukawa couplings and the condensation scale $\\Lambda$ in both models.","The real case requires two distinct left-handed partial-composite couplings for top and bottom, so the top-bottom mass hierarchy is not solely a right-handed Yukawa effect; this distinguishes it from the pseudo-real and complex cases.","The requirement that the pNGB spectrum be tachyon-free restricts the strong-dynamics coefficients, since the neutral triplet mass can turn negative when fermionic-loop effects overcome electroweak gauge-loop contributions."],"supporting_citations":[{"why":"introduces the fundamental partial compositeness framework with fundamental techniscalars that this paper extends to real and complex TC representations.","marker":"[24]"},{"why":"establishes the minimal FPC model with pseudo-real fermions and supplies the spurion and operator methods, and the SU(4)/Sp(4) baseline, used throughout.","marker":"[18]"},{"why":"provides the SU(4)/Sp(4) effective Lagrangian analysis and lattice connection on which the present real and complex operator classification builds.","marker":"[29]"},{"why":"supplies the SU(5)/SO(5) vacuum misalignment analysis, the pion matrix, and the triplet tadpole formalism adopted in the real-case section.","marker":"[30]"},{"why":"supplies the SU(4)xSU(4)/SU(4) coset conventions, the Goldstone decomposition, and the heavy-quark-loop potential used in the complex-case section.","marker":"[37]"},{"why":"provides gauge-theory UV completions of partial compositeness, including the SU(5)/SO(5) structure, that motivate the real-representation model.","marker":"[11]"},{"why":"gives a UV-complete SU(5)/SO(5) composite Higgs realisation whose low-energy coset structure the real case extends with FPC dynamics.","marker":"[23]"},{"why":"supplies the experimental best-fit value for the Zbb coupling used to derive the bound in eq. (3.42).","marker":"[32]"},{"why":"is the LHC four-top search whose 95% C.L. limit is translated into the constraint on the right-handed top Yukawa combination in eq. (3.46).","marker":"[34]"},{"why":"is a global fit of top-quark effective interactions whose bounds on anomalous top couplings produce the constraints in eqs. (3.51), (3.52), (4.32), and (4.33).","marker":"[36]"}],"fun_headline_variants":["Real composite Higgs breaks rho; complex twin stays protected","CP-odd triplets keep complex twin custodial; real breaks","Triplet VEV only in real coset: rho shifts, complex safe","Custodial rule: real rep triggers triplet VEV; complex forbids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusions depend on the uncomputed strong-force coefficients having signs and sizes that tilt the vacuum in the electroweak-breaking direction and that do not cancel the triplet tadpole; if those coefficients take different values, the real case would not develop the triplet vacuum value and the rho shift would vanish.","fun_headline_variants_meta":{"raw":{"variants":["Real composite Higgs breaks rho; complex twin stays protected","CP-odd triplets keep complex twin custodial; real breaks","Triplet VEV only in real coset: rho shifts, complex safe","Custodial rule: real rep triggers triplet VEV; complex forbids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000903,"raw_usage":{"total_tokens":3902,"prompt_tokens":978,"completion_tokens":2924,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":2845}},"tokens_in":594,"tokens_out":2924,"duration_ms":21558,"temperature":1.0,"reasoning_tokens":2845,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:16:08.980360+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A lattice or other first-principles determination of the real-case coefficients $C_g$, $C_m$, $C_1^{Vf}$, $C_2^{Vf}$, $C_3^{Vf}$ that yields $A \\le 0$ or $B \\le 0$, or an exact cancellation in the tadpole combination of eq. (3.34), would falsify the claim that physical quark masses inevitably induce a triplet VEV and the associated rho contribution.","supporting_citations":[],"review_version":1}