Real and Complex Fundamental Partial Compositeness

Alessandro Agugliaro; Francesco Sannino

REVIEW 3 major objections 4 minor 1 cited by

Real and Complex Fundamental Partial Compositeness

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 1908.09312 v1 pith:YJNYINYT submitted 2019-08-25 hep-ph

Alessandro Agugliaro , Francesco Sannino This is my paper · ORCID

classification hep-ph

keywords fundamentalpartialcompositenesscompositeHiggstechnicolorvacuummisalignmentelectroweakprecisionrhoparametercustodialsymmetryfour-topproduction

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

The 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.

Minor 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.

Verdict: 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Sec. 3.5, Eq. (3.34)] 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.
[Sec. 3.7, Eq. (3.44)] 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.
[Sec. 3.4, Eqs. (3.31)-(3.33)] 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.

minor comments (4)

[Sec. 1] There are spelling errors: 'hyerarchies' should be 'hierarchies' and 'trulys' should be 'truly'.
[Sec. 3.5, Eq. (3.34) and Sec. 3.4, Eqs. (3.32)-(3.33)] 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.
[Appendix A.3, text after Eq. (A.22)] 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.
[Sec. 3.7, Eq. (3.44)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

There is no significant circularity: the central triplet-VEV claims follow from spurion and CP-parity analyses, with the dependence on uncomputed strong-dynamics coefficients explicitly acknowledged rather than fitted.

full rationale

The paper's new claims are the real-case SU(5)/SO(5) eta_3^0 tadpole and the resulting rho-parameter contribution, and the complex-case absence of triplet VEVs. None of these is assumed as an input. The real-case tadpole coefficient in Eq. (3.34) is obtained by spurion expansion of the three Yukawa-generated operators O1-3_Vf listed in Eqs. (3.28)-(3.30), and the complex-case argument is a CP-parity statement about the analogous operators in Section 4.4. The paper does not fit the coefficients C_i_Vf to the desired VEV; instead it leaves them as free strong-dynamics coefficients and explicitly states in the Introduction that 'there could be an unforeseen cancellation emerging once the coefficients of these operators will be fully determined from the fundamental theory.' This makes the real-case result conditional on unknown coefficients, which is a correctness risk, not circularity. The use of the measured mb/mt ratio in Eq. (3.35) and external constraints from ATLAS, TopFitter, and Zbb measurements is not used to infer the triplet VEV; those inputs constrain other combinations of parameters. Citations to Refs. [18,24,25,30] provide the FPC framework and pion-matrix conventions, but the specific SU(5)/SO(5) and SU(4)xSU(4)/SU(4) operator constructions, vacuum potentials, and VEV statements are derived in the present paper rather than imported as conclusions. No equation in the derivation reduces to its own input, and no fitted parameter is renamed as a prediction. The derivation is therefore self-contained with respect to circularity, though its phenomenological reach depends on uncomputed strong-dynamics coefficients.

Assumptions & free parameters 6 free parameters · 6 assumptions · 2 invented entities

The central claims rest on a standard EFT parameterization of strong dynamics plus model-specific fields from the FPC framework. There are no machine-checked proofs or shipped code; the quantitative constraints are parametric in the effective coefficients.

free parameters (6)

Strong-dynamics potential coefficients C_g, C_m
Enter the pNGB potential from gauge and fermion-mass loops (eqs. 3.23, 3.24, 4.17, 4.18); signs assumed positive, values not computed from the UV theory.
Vacuum-potential Yukawa coefficients C1_Vf, C2_Vf, C3_Vf (real), C1_Vf, C2_Vf (complex)
Coefficients of the spurion potential operators (eqs. 3.28-3.30, 4.19); they set A and B (eqs. 3.32, 3.33) and the triplet tadpole (eq. 3.34), so they control the main real-case result.
Yukawa operator normalization C_Yuk
Overall coefficient of the effective Yukawa operator (eqs. 3.15, 4.14); determines masses (eqs. 3.18, 4.16) and enters all derived constraints; value not derived.
NLO operator coefficients C_Pi_f, C_i_4f, C_i_yPiD, C_i_PiD
Set the size of Zbb, rho parameter, four-top and dipole constraints (eqs. 3.40-3.46, 4.22-4.33, appendices A and B); unknown strong-dynamics form factors.
TC fermion mass parameters mu_d, mu_s (real), mu_L, mu_R (complex)
Explicit symmetry-breaking masses in eqs. (3.4) and (4.9); appear in the vacuum potential and triplet mass (eqs. 3.24, 3.39, 4.18); inputs from the underlying theory.
Fundamental Yukawa couplings y_Q, y_tilde_Q, y_t, y_b (real); y_Q, y_t, y_b (complex)
Inputs encoding the SM mass hierarchy (eqs. 3.14, 4.12-4.13); the paper only constrains combinations of them, never predicts their values.

assumptions (6)

domain assumption The strong dynamics realizes the assumed condensates leading to SU(5)/SO(5) and SU(4)xSU(4)/SU(4)_D cosets.
Used throughout Sections 3.1 and 4.1; not derived from gauge dynamics, standard for composite Higgs model building.
domain assumption The low-energy dynamics is described by a nonlinear sigma model with scale f and spurion formalism for Yukawa operators.
Eqs. (3.7), (3.11), (3.15), (4.14); the whole EFT machinery presumes this effective description.
domain assumption The fundamental Yukawa couplings y_Q, y_t, y_b encode the SM fermion mass hierarchy; their values are not explained.
Eqs. (3.14), (4.12)-(4.13); the paper uses these as inputs, only deriving constraints on combinations.
ad hoc to paper The strong-dynamics coefficients C_i are assumed to have signs and magnitudes such that EWSB occurs and no tachyons appear.
Eq. (3.31) and text 'for the electroweak symmetry to be spontaneously broken we need A,B>0'; eq. (4.20) and 'misaligned for C_tilde_t > C_tilde_g'; eq. (3.39) tachyon constraint. Without this assumption the central vacuum-alignment statements do not follow.
domain assumption The underlying theory is CP even, and CP violation does not generate tadpole coefficients.
Sec. 4.4: 'Because the underlying theory is CP even, the operators O1-2 generate potential terms proportional to absolute values of Yukawa couplings, de facto forbidding tadpole operators'; needed for the claim that triplet VEVs vanish in the complex case.
domain assumption TC scalars have common masses that are light compared with the dynamical scale.
Sec. 3.2: 'we will assume all the scalars to have common masses that are light with respect to the dynamical scale'; this fixes the composite spectrum and the omega_ij replacement if scalars are heavy, eq. (3.19).

invented entities (2)

Colored TC scalars S_t and S_b (real case), S_q (complex case)
purpose: Enable renormalizable Yukawa terms that mix SM fermions with TC fermions, realizing partial compositeness (eqs. 3.12, 4.5).
These scalars are model content inherited from the FPC framework, not introduced to explain a specific datum; no masses or couplings are predicted, only generic colored-scalar signatures (Sec. 5, [39]).
Composite vector-like partner fermions B_ia (Table 2)
purpose: Heavy partners that linearly mix with SM quarks and generate the top and bottom masses (eq. 3.21).
Composite states of one TC fermion and one TC scalar; their masses and mixings are parameterized but not predicted, so there is no falsifiable handle outside the model.

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Cite this review

Pith. "Pith review of Real and Complex Fundamental Partial Compositeness." pith.science (2026). https://pith.science/paper/YJNYINYT

@misc{pith2026190809312,
  author       = {Pith},
  title        = {Pith review of: Real and Complex Fundamental Partial Compositeness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJNYINYT}},
  note         = {Machine review of arXiv:1908.09312}
}

read the original abstract

We complete the analysis of the effective field theory at the electroweak scale for minimal models of fundamental partial compositeness. Specifically, we consider fermions in the complex and real representation of the gauge group underlying the composite Higgs dynamics, since the pseudo real representation was investigated earlier. The minimal models feature the cosets SU(4)xSU(4)/SU(4)_D and SU(5)/SO(5) respectively for the complex and real representations. We determine the vacuum alignment, the electroweak precision constraints, additional collider constraints. We finally discuss the main differences among the different models of minimal partial compositeness.

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Forward citations

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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