FCNC-free multi-Higgs-doublet models from broken family symmetries

Ivo de Medeiros Varzielas; Jim Talbert

FCNC-free multi-Higgs-doublet models from broken family symmetries

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Residual flavour symmetries can forbid tree-level FCNCs in multi-Higgs-doublet models.

desk verdict A short, honest paper that correctly derives generalized Yukawa Alignment from residual flavour symmetries, but the 'model-independent' framing overreaches beyond its own stated assumption that all Higgs doublets share one residual symmetry. read the letter →

arxiv 1908.10979 v2 pith:NNJSYGSL submitted 2019-08-28 hep-ph

Ivo de Medeiros Varzielas , Jim Talbert This is my paper · ORCID

classification hep-ph

keywords multi-Higgs-doubletmodelsflavour-changingneutralcurrentsresidualflavoursymmetriesfamilysymmetrybreakingYukawaalignmentA4modelCKMandPMNSmixingtree-levelFCNCfreedom

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 establishes a model-independent way to keep multi-Higgs-doublet models safe from tree-level flavour-changing neutral currents (FCNC) without appealing to mass hierarchies or hand-picked Yukawa textures. It shows that residual flavour symmetries—the Abelian subgroups left over after a family symmetry breaks—naturally force every Higgs doublet's Yukawa matrix to be diagonal in the fermion mass basis, so long as the same residual generator applies to all of them. In that strict limit the Yukawa couplings take the generalised Yukawa Alignment form, and in weaker limits FCNC are confined to the same flavour sectors that the symmetry cannot use to predict quark or lepton mixing. The paper also sketches an explicit $A_4$ toy model that realises the mechanism with multiple Higgses singlet under the family symmetry and produces a viable PMNS matrix.

What carries the argument

The carrying object is the residual symmetry generator $T_A$, a diagonal $U(1)^3$ rephasing matrix that survives in a given fermion sector after the family group breaks. The paper's key step is to demand, for each Higgs doublet $H_k$, the invariance relation $Y^A_k \stackrel{!}{=} T_A Y^A_k T_A^\dagger$ in the fermion mass basis. Expanding this equation shows that the $(i,j)$ entry is multiplied by $e^{i(\psi^i_A-\psi^j_A)}$, so entries with unequal phases must vanish; equal phases leave the entry unconstrained. This single constraint simultaneously gives the strict diagonal limit (Eq. (20)) and the weak limit in which FCNC are restricted to specific flavour sectors. The paper also handles product-group residual symmetries of the form $G_A \simeq G^1_A \times G^2_A$, which can forbid different sets of entries and relax mixing ambiguities.

What would settle it

Construct or identify a complete flavour model with two or more Higgs doublets in which different flavons, aligned along different residual subgroups of a family group, generate Yukawa couplings for different Higgs doublets; if the resulting Yukawa matrices are not simultaneously diagonal, the strict claim fails. A sharper check: in the weak RFS limit, scan residual groups with a phase degeneracy in the $(1,2)$ sector and compute the tree-level contribution to $K^0$–$\overline{K}{}^0$ mixing; if the induced FCNC exceeds measured neutral-meson mixing limits for all allowed parameter choices, the paper's flavour-specific protection is experimentally excluded.

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Extended reading notes

Core claim

The paper's central claim is that residual flavour symmetries—the Abelian groups left invariant after a family symmetry breaks—can be imposed on the Yukawa sector of any multi-Higgs-doublet model, and in the strict limit force every extra-Higgs Yukawa matrix to be diagonal in the same basis as the fermion masses. Starting from the invariance condition $Y^A_k = T_A Y^A_k T_A^\dagger$ in the mass-eigenstate basis, with $T_A = \mathrm{diag}(e^{i\alpha_A}, e^{i\beta_A}, e^{i\gamma_A})$, the paper shows that any unequal pair of phases forbids the corresponding off-diagonal entry; if all phases are distinct, each $Y^A_k$ is diagonal and tree-level FCNCs vanish. The surviving couplings take the form $Y^A_k = \zeta^A_k Y^A_1$ with diagonal $\zeta^A_k$, which is the generalised Yukawa Alignment ansatz. When phase degeneracies exist, FCNC are allowed only in the same flavour sectors that the residual symmetry fails to control in CKM/PMNS mixing, so predictivity and FCNC protection are tied together. An $A_4$ toy model with family-symmetry singlet Higgses realises the strict limit and yields a viable lepton mixing matrix.

Load-bearing premise

The load-bearing premise is that the same residual symmetry generator applies to every Higgs doublet's Yukawa operator; if different Higgses are governed by different residual subgroups, the mechanism's strict FCNC-free diagonal form no longer follows.

Editorial extensions

If this is right

Strictly RFS-controlled multi-Higgs models are automatically tree-level FCNC-free, with no discrete symmetry or mass hierarchy needed beyond the family-symmetry breaking pattern.
In the weak limit, dangerous FCNC are restricted to specific flavour sectors, so a model can, for example, have the residual symmetry explain Cabibbo mixing while protecting the light-quark sector from the strongest bounds.
Generalised Yukawa Alignment ceases to be an ansatz: it is the low-energy form imposed by broken family symmetries, which connects flavour-model predictivity directly to Higgs phenomenology.
The leading loop-level FCNC in the aligned limit are dimension-seven operators suppressed by three mass insertions, two CKM factors, and alignment factors, and are consistent with current bounds when alignment is imposed at high scales.

Reading between the lines

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

The same residual-symmetry logic should transfer to other scalar extensions, such as models with multiple leptoquarks, where an analogous constraint would sequester flavour-changing leptoquark couplings; the paper already notes a correspondence to independent leptoquark analyses.
FCNC bounds can be repurposed as model-selection criteria: residual groups that predict viable CKM/PMNS but permit phase degeneracies in experimentally tight sectors would be disfavoured, turning the paper's link between mixing and FCNC into a tool for scanning flavour groups.
A testable pattern emerges for future precision: if the $\zeta^A_k$ alignment parameters are measured through Higgs couplings, they should obey the residual-symmetry relations sector by sector, and any violation would signal different residual subgroups for different Higgses.

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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

The useful thing here is the derivation: if the same residual symmetry generator T_A is imposed on every Higgs-doublet Yukawa operator, then in the strict limit where no phases coincide, each Y_k is forced diagonal in the same basis. That is short, correct, and I believe genuinely new as a stated result. The identification of this outcome with generalized Yukawa Alignment (A6) is also new and useful. The paper is honest about its main caveats in A3, A5, and A7, which is more than many short theory notes manage.

The soft spot is the recurring 'model-independent' claim in the abstract and conclusion. The strict FCNC-free result is conditional on assumption A5: that the same residual Abelian subgroup is preserved by the flavon insertions feeding every doublet. If H_1 is dressed by a flavon preserving one subgroup and H_2 by another, you get different T_A^{(k)} and the simultaneous diagonalization fails; tree-level FCNC can come back. The stress-test note is right about this. The A4 toy model is an existence proof of the shared-symmetry case, not a theorem for all RFS models. The conclusion should say 'models in which the same residual symmetry is preserved in every Yukawa operator' rather than implying RFS generally forbid FCNC. The toy model also fits one angle to the reactor measurement, so it is illustrative, not a parameter-free prediction — the paper says as much, but the abstract could be clearer.

The RGE discussion is responsible: they note that alignment is not automatically stable and cite the conditions under which it is. The citation pattern is fair; the self-citations are to closely related prior work and are not padding.

Who is this for? People building MHDM with flavour symmetries, and anyone who wants a quick algebraic criterion to check whether a symmetric model is tree-level FCNC-free. It deserves a serious referee. I would recommend acceptance after asking the authors to temper the model-independence language and to state A5 more prominently in the abstract.

Referee Report

0 major / 4 minor

Summary. The paper argues that residual flavour symmetries (RFS)—Abelian subgroups of a family symmetry left unbroken by flavon VEVs—can be used to control the Yukawa sector of multi-Higgs-doublet models (MHDMs), forbidding or limiting tree-level FCNCs. Section II reviews how RFS constrain CKM/PMNS mixing. Section III derives the invariance condition Eq. (13) for each Yukawa matrix in the mass basis, notes that absent phase degeneracies Eq. (14) forces each matrix diagonal, and concludes that in the strict limit all Y_k are simultaneously diagonal, giving a generalized Yukawa alignment of the form Eq. (20). The weak limit with phase degeneracies allows flavour-specific off-diagonal entries. Section IV constructs an A4 toy model with Higgs doublets that are A4 singlets and a single flavon φ_l, yielding diagonal charged-lepton Yukawas of the form (24) and a PMNS matrix with one free angle θ fit to the reactor angle. Section V discusses RGE stability and subtleties (A1–A7).

Significance. The central observation—that RFS give a symmetry rationale for generalized Yukawa alignment—is interesting and likely useful for model building. The derivation of Eq. (20) from Eq. (13) is explicit and does not rely on fitted inputs; the discussion of the weak limit, product-group RFS, and RGE effects is balanced. The paper is honest about the main assumption (A5) that all Higgs doublets share the same residual generator. If the result holds, it provides a model-independent organizing principle that connects flavour predictivity with FCNC suppression, although the toy model is illustrative rather than fully predictive since θ is fit.

minor comments (4)

[Abstract and Conclusion] The phrases 'naturally forbid' in the abstract and 'in the limit where the RFS controls all parameters' in the conclusion should be qualified by the A5 assumption that a single residual generator T_A acts on all Yukawa operators; as written, the abstract overstates the domain, because if different H_k are dressed by flavons preserving different subgroups, simultaneous diagonality and Eq. (20) are lost.
[III(A5)] The sentence 'In this more baroque scenario FCNC-inducing elements can then be in different matrix sectors' is the right caveat, but I suggest adding one explicit sentence stating that in that case the Y_k are diagonal in different bases and Eq. (20) no longer holds, so that the consequence of relaxing A5 is unambiguous.
[Eqs. (9), (10), (27)] Equations (9), (10), and (27) are garbled by brace artifacts such as '/bracehtipupleft' in the manuscript text; these should be typeset cleanly in the published version.
[Section IV and Conclusion] The parameter θ is introduced as a free angle and fit to θ13 via Eq. (28), so the Conclusion's phrase 'predicts a viable PMNS mixing matrix' should be softened to 'accommodates' or 'can reproduce', since the example contains a fitted parameter.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the strict FCNC-free limit follows by direct algebra from the stated RFS invariance constraint, and the key assumption is explicitly flagged.

full rationale

The central derivation is self-contained. Section III imposes Eq. (13), Y_k^A = T_A Y_k^A T_A^dagger, on every FCNC-inducing Yukawa; expanding in the mass-eigenstate basis gives Eq. (14), where each off-diagonal entry obeys Y_ij = exp[i(psi_i-psi_j)] Y_ij. If no two RFS phases coincide, each such entry must vanish, so every Y_k is diagonal in the same basis. Eq. (20) is then just the statement that a diagonal Y_k is a diagonal rescaling of the diagonal Y_1 (zeta_k = diag(Y_k,ii/Y_1,ii)); the identification with the generalized Yukawa Alignment of Refs. [13,14] is an external comparison, not an input used to derive the result. The paper explicitly flags the load-bearing shared-generator condition in Section III(A5): 'Applying the same T_A representation in each Yukawa operator for all k embeds an inherent (albeit natural) assumption,' and it notes the alternative in which different Higgs operators preserve different residual subgroups. That is an honest domain restriction, not a circular step. The conclusion likewise qualifies its model-independence with 'so long as one assumes family symmetry breaking patterns similar to those in (6).' Self-citations [52,53,59,60] appear only in illustrative or contextual roles (leptoquark correspondence, UV-complete/sketch model) and do not carry the FCNC theorem. The toy model's angle theta is explicitly fitted to sin(theta_13) in Eq. (28) and is presented as a fit, not as the model-independent claim. No fitted parameter is renamed as a prediction in the central argument, and no uniqueness theorem is imported from the authors' prior work.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central argument rests on the standard RFS framework and one explicit modeling assumption (A5): all Higgs doublets share the same residual symmetry. The toy model additionally imports the A4 vacuum alignments of [60] and fits one angle θ to the reactor data. No new particles or forces are introduced; the multi-Higgs and flavon content is standard.

free parameters (1)

θ (free rotation angle in A4 toy model) = determined by the measured reactor angle via |sin θ13| = (2/√6)|sin θ|
Introduced in Section IV as the angle in the R13 rotation that defines U_TM = U_TBM R13; the PMNS reactor angle is then set by this fit, so the toy model accommodates rather than predicts θ13.

assumptions (5)

domain assumption Residual symmetries T_A are exact symmetries of the SM fermion mass matrices in the mass basis, with T_fL = T_fR for charged fermions.
Standard RFS framework reviewed in Section II, Eqs. (2)-(4); assumed to descend from a spontaneously broken family symmetry G_F.
ad hoc to paper Every Higgs doublet's Yukawa matrix Y_k^A is invariant under the same RFS generator T_A, i.e. Eq. (13) holds for all k.
The paper's key modeling assumption, flagged in observation (A5), Section III. It is natural when all H_k are G_F-singlets and all operators are generated by the same flavon alignments, but it need not hold in general; if it fails, the strict FCNC-free conclusion is lost.
domain assumption In the strict limit, the RFS phases are pairwise distinct (ψ_i ≠ ψ_j for i≠j), so Eq. (14) forces all off-diagonal Yukawa entries to vanish.
Needed for complete control of three-generation mixing; Eqs. (15)-(16) show phase equalities leave a continuous 2x2 rotation ambiguity. Product-group RFS (A4) can partially relax this.
domain assumption The Altarelli-Feruglio flavon VEV alignments are taken as given: ⟨φ_l⟩ = (v_l,0,0), ⟨φ_ν⟩ = (v_ν,v_ν,v_ν).
Imported from [23,24,60]; the toy model does not re-derive vacuum alignment, only the Yukawa consequences.
domain assumption The extra Higgs doublets are A4 singlets in the toy model.
Design choice stated at the end of Section IV, used to avoid the no-go theorem of [61].

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Cite this review

Pith. "Pith review of FCNC-free multi-Higgs-doublet models from broken family symmetries." pith.science (2026). https://pith.science/paper/NNJSYGSL

@misc{pith2026190810979,
  author       = {Pith},
  title        = {Pith review of: FCNC-free multi-Higgs-doublet models from broken family symmetries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NNJSYGSL}},
  note         = {Machine review of arXiv:1908.10979}
}

abstract

We demonstrate how residual flavour symmetries, infrared signatures of symmetry breaking in complete models of flavour, can naturally forbid (or limit in a flavour specific way) flavour-changing neutral currents (FCNC) in multi-Higgs-doublet models (MHDM) without using mass hierarchies. We first review how this model-independent mechanism can control the fermionic mixing patterns of the Standard Model, and then implement the symmetries in the Yukawa sector of MHDM, which allows us to intimately connect the predictivity of a given flavour model with its ability to sequester FCNC. Finally, after discussing various subtleties of the approach, we sketch an $A_4$ toy model that realises an explicit example of these simplified constructions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

83 extracted references · 56 canonical work pages

[1]

While invariance under (

is invariant under the action of (2) when the transformations on left- and right-handed fermions are equivalent, as they must be if fermion mass terms are generated. While invariance under (
[2]

(4) We consider scenarios where (

may seem trivial in the mass basis, when the same relation is rotated to the ﬂavour basis, the transformations become functions ofUA, the (highly non-trivial) fermionic mixing matrices of the SM: TA →TAU =UATAU † A, (3) which are appropriately interpreted as symmetries of the mass matrix (and not of the full LSM , as weak inter- actions do not respect the...
[3]

G. C. Branco, L. Lavoura and J. P. Silva, Int. Ser. Monogr. Phys. 103 (1999) 1

work page 1999
[4]

This can occur, for example, in mod- els where the family-symmetry breaking occurs through the VEV of scalar ﬂavons {φ}

is not accidental, but is instead the remnant signature of an ultra-violet (UV) ﬂavour theory controlled by a family symmetry GF that commutes with the SM gauge group (or indeed any other BSM symmetry). This can occur, for example, in mod- els where the family-symmetry breaking occurs through the VEV of scalar ﬂavons {φ}. Regardless of the dy- namics that...
[5]

A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32 [JETP Lett. 5 (1967) 24] [Sov. Phys. Usp. 34 (1991) no.5, 392] [Usp. Fiz. Nauk 161 (1991) no.5, 61]

1967
[6]

(or relaxed versions of it) have been well studied in the context of fermionic mixing in the SM (see e.g. [24, 26, 27] for reviews), with the literature including complete models realising the RFS as well as both analytic and computa- tional approaches to cataloguing the types of GF that can break to phenomenologically viable CKM/PMNS matri- ces [28–51]. ...

work page
[7]

is then expanded as LY = − ( 1 + S0 1 v ) (¯dLmddR + ¯uLmuuR + ¯ lLmeeR ) − 1 v N∑ k=2 ( S0 k +iP 0 k )( ¯dLY d kdR + ¯uLY u kuR + ¯ lLY e keR ) − √ 2 v N∑ k=2 S+ k ( ¯uLVY d kdR − ¯uRY u,† k Vd L + ¯νLY e keR ) +h.c., (12) 5 W e will adopt the notation of [14] in the following discussio n. 4 where V ≡ UCKM , the second line is clearly capable of generati...

work page
[8]

Weinberg, Phys

S. Weinberg, Phys. Rev. Lett. 37 (1976) 657

work page 1976

Show all 83 references

[9]

For the k ⁄= 1 terms, expanding the implicit ﬂavour indices in (

are to remain RFS invariant, the asso- ciated Yukawa couplings must respect the following RFS constraint: Y A k ! =TAY A k T † A, (13) where the ! notation implies ‘must be equal to’ . For the k ⁄= 1 terms, expanding the implicit ﬂavour indices in (
[10]

G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Re- belo, M. Sher and J. P. Silva, Phys. Rept. 516 (2012) 1 [arXiv:1106.0034 [hep-ph]]

2012 arXiv
[11]

are 3 × 3 matrices in ﬂavour space, and after EWSB the associated mass matrixM ′ k will receive contributions from the VEV of all H ′ k. After EWSB we go, without loss of generality, to the Higgs basis where all the VEV is in H1 (this is done by a Higgs-basis transformation, a...
[12]

is subject to constraints (for each k). Moving line-by-line we ﬁrst observe that the mass term RFS analysis proceeds precisely as in the SM; as we are in the fermion mass-eigenstate basis we immedi- ately conclude that non-degenerate fermion masses re- quire TA to be in a diag...
[13]

yields   Y11 ei(αl−βl)Y12 ei(αl−γl)Y13 ei(βl−αl)Y21 Y22 ei(βl−γl)Y23 ei(γl−αl)Y31 ei(γl−βl)Y32 Y33   ! =   Y11 Y12 Y13 Y21 Y22 Y23 Y31 Y32 Y33   (14) for each k. Denoting a generic phase in TA by ψ A ∈ {α A,β A,γ A}, we now make a few important observa- tions: (A1) FCN...
[14]

However, since we take Λ F > Λ EW, the imprint of GF through GA-symmetricY A k occurs without reference to the isospin decompo- sition that yields the diﬀerent terms in (

are not RFS invariant despite the fact that, unlike in the SM, these interactions are sourced from the same ﬂavon-enhanced operators as the mass terms, and so one might naively expect the RFS to hold. However, since we take Λ F > Λ EW, the imprint of GF through GA-symmetricY A...
[15]

( 4)) which controls Y A k in the CC, but not the over- all term

after 5 EWSB.6 Hence the RFS is again properly under- stood as a symmetry of mass/Yukawa matrices (cf. ( 4)) which controls Y A k in the CC, but not the over- all term. Regardless, using that Y d k is forced into a speciﬁc shape through ( 14), we observe that one can artiﬁ- ci...
[16]

Serôdio, Phys

H. Serôdio, Phys. Lett. B 700 (2011) 133 [arXiv:1104.2545 [hep-ph]]

2011 arXiv
[17]

Hence GF ,Q must be Abelian if it is closed via this generator alone, which is less phenomenologically interesting to engineer in ﬂavon models

implies TuU =TdU (18) in the ﬂavour basis. Hence GF ,Q must be Abelian if it is closed via this generator alone, which is less phenomenologically interesting to engineer in ﬂavon models. If TeU is also part of the overall generating set then GF can again be non-Abelian, and an...
[18]

(A4) Product Group RFS: We have implicitly as- sumed that GA are generated by one matrix TA

could appear if neutrino masses and non-trivial UP M N S are considered. (A4) Product Group RFS: We have implicitly as- sumed that GA are generated by one matrix TA. If the RFS are instead described by product groups of the form GA ∼ = G1 A × G 2 A ×... (19) then each RFS gene...
[19]

can also alleviate ambiguities in mix- ing predictions entering through ( 15). (A5) Generator Representation: Applying the same TA representation in each Yukawa operator for all k embeds an inherent (albeit natural) assumption about the structure of the family symmetry break- ...
[20]

A similar expression holds for the up quarks, while the RGE for charged leptons does not misalign

is proportional to (in the Higgs and neutral scalar mass-eigenstate basis) ˜Θ d k = −V † N∑ l=1 ζu,† l mum† uζu k V ζd l +ζd kV † N∑ l=1 ζu,† l mum† uV ζd l + 1 4 [ V † ( N∑ l=1 ζu,† l mum† uζu l ) V, ζ d k ] , (22) where the last term is a commutator. A similar expression hol...
[21]

corresponds to the Generalised Yukawa Alignment of (A6). The Dirac neutrino masses in the MHDM framework are analogous to those in ( 7), H ′ 1y1 ν [ ¯LLνR ] +H ′ 2y2 ν [ ¯LLνR ] , (25) and the Majorana masses given by (xAξ +x′ Aξ′) [νRνR] +xB [φ ννRνR] (26) similarly mimic ( 7...
[22]

That is, the leading FCNC contributions are di- mension seven and are suppressed by two CKM fac- tors, three alignment factors ζA, and three mass in- sertions

couples to neutral Higgs ﬁelds, left- and right- chiral down quarks, and an additional power of md. That is, the leading FCNC contributions are di- mension seven and are suppressed by two CKM fac- tors, three alignment factors ζA, and three mass in- sertions. This structure ca...
[23]

T. D. Lee, Phys. Rev. D 8 (1973) 1226

1973
[24]

We have explicitly given the assignments in [24]

LL νR eR µ R τR Hu,d φ l φ ν ξ ˜ξ SU(2)L 2 1 1 1 1 2 1 1 1 1 A4 3 3 1 1′′ 1′ 1 3 3 1 1 TABLE I: The relevant ﬁeld and symmetry content of the A4 Altarelli-Feruglio model. We have explicitly given the assignments in [24]. in a given fermion sector (and in the appropriate basis)...
[25]

Unlike the µ −τ operator generating Z′ 2, this remaining Z2 is an A4 subgroup, and therefore a proper RFS

only changes the overall normalization of the low-energy Majorana mass matrix from [60], and so the RFS analysis presented there still holds — the ξ′ breaks the original Altarelli-Feruglio Z2 × Z′ 2 RFS pre- served by ⟨φ ν ⟩, with Z′ 2 an accidental µ −τ operator enforcing θl ...

2015
[26]

J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, Front. Phys. 80 (2000) 1

2000
[27]

R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38 (1977) 1440

1977
[28]

S. L. Glashow and S. Weinberg, Phys. Rev. D 15 (1977) 1958

1977
[29]

E. A. Paschos, Phys. Rev. D 15 (1977) 1966

1977
[30]

Cree and H

G. Cree and H. E. Logan, Phys. Rev. D 84 (2011) 055021 [arXiv:1106.4039 [hep-ph]]

2011 arXiv
[31]

T. P. Cheng and M. Sher, Phys. Rev. D 35 (1987) 3484

1987
[32]

J. L. Diaz-Cruz, R. Noriega-Papaqui and A. Rosado, Phys. Rev. D 69 (2004) 095002 [hep-ph/0401194]

2004 arXiv
[33]

Pich and P

A. Pich and P. Tuzón, Phys. Rev. D 80 (2009) 091702 [arXiv:0908.1554 [hep-ph]]

2009 arXiv
[34]

Peñuelas and A

A. Peñuelas and A. Pich, JHEP 1712 (2017) 084 [arXiv:1710.02040 [hep-ph]]

2017 arXiv
[35]

Rodejohann and U

W. Rodejohann and U. Saldaña-Salazar, JHEP 1907 (2019) 036 [arXiv:1903.00983 [hep-ph]]

2019 arXiv
[36]

de Medeiros Varzielas, Phys

I. de Medeiros Varzielas, Phys. Lett. B 701 (2011) 597 [arXiv:1104.2601 [hep-ph]]. 8

2011 arXiv
[37]

G. C. Branco, W. Grimus and L. Lavoura, Phys. Lett. B 380 (1996) 119 [hep-ph/9601383]

1996 arXiv
[38]

P. M. Ferreira and J. P. Silva, Phys. Rev. D 83 (2011) 065026 [arXiv:1012.2874 [hep-ph]]

2011 arXiv
[39]

I. P. Ivanov, V. Keus and E. Vdovin, J. Phys. A 45 (2012) 215201 [arXiv:1112.1660 [math-ph]]

2012 arXiv
[40]

Serôdio, Phys

H. Serôdio, Phys. Rev. D 88 (2013) no.5, 056015 [arXiv:1307.4773 [hep-ph]]

2013 arXiv
[41]

I. P. Ivanov and C. C. Nishi, JHEP 1311 (2013) 069 [arXiv:1309.3682 [hep-ph]]

2013 arXiv
[42]

Altarelli and F

G. Altarelli and F. Feruglio, Nucl. Phys. B 741 (2006) 215 [hep-ph/0512103]

2006 arXiv
[43]

Altarelli and F

G. Altarelli and F. Feruglio, Rev. Mod. Phys. 82 (2010) 2701 [arXiv:1002.0211 [hep-ph]]

2010 arXiv
[44]

P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B 530 (2002) 167 [hep-ph/0202074]

2002 arXiv
[45]

S. F. King and C. Luhn, Rept. Prog. Phys. 76 (2013) 056201 [arXiv:1301.1340 [hep-ph]]

2013 arXiv
[46]

Grimus and P

W. Grimus and P. O. Ludl, J. Phys. A 45 (2012) 233001 [arXiv:1110.6376 [hep-ph]]

2012 arXiv
[47]

C. S. Lam, Phys. Lett. B 656 (2007) 193 [arXiv:0708.3665 [hep-ph]]

2007 arXiv
[48]

S. F. Ge, D. A. Dicus and W. W. Repko, Phys. Lett. B 702 (2011) 220 [arXiv:1104.0602 [hep-ph]]

2011 arXiv
[49]

S. F. Ge, D. A. Dicus and W. W. Repko, Phys. Rev. Lett. 108 (2012) 041801 [arXiv:1108.0964 [hep-ph]]

2012 arXiv
[50]

de Adelhart Toorop, F

R. de Adelhart Toorop, F. Feruglio and C. Hagedorn, Nucl. Phys. B 858 (2012) 437 [arXiv:1112.1340 [hep-ph]]

2012 arXiv
[51]

Hernandez and A

D. Hernandez and A. Y. Smirnov, Phys. Rev. D 86 (2012) 053014 [arXiv:1204.0445 [hep-ph]]

2012 arXiv
[52]

C. S. Lam, Phys. Rev. D 87 (2013) no.1, 013001 [arXiv:1208.5527 [hep-ph]]

2013 arXiv
[53]

Holthausen, K

M. Holthausen, K. S. Lim and M. Lindner, Phys. Lett. B 721 (2013) 61 [arXiv:1212.2411 [hep-ph]]

2013 arXiv
[54]

Holthausen and K

M. Holthausen and K. S. Lim, Phys. Rev. D 88 (2013) 033018 [arXiv:1306.4356 [hep-ph]]

2013 arXiv
[55]

S. F. King, T. Neder and A. J. Stuart, Phys. Lett. B 726 (2013) 312 [arXiv:1305.3200 [hep-ph]]

2013 arXiv
[56]

Lavoura and P

L. Lavoura and P. O. Ludl, Phys. Lett. B 731 (2014) 331 [arXiv:1401.5036 [hep-ph]]

2014 arXiv
[57]

R. M. Fonseca and W. Grimus, JHEP 1409 (2014) 033 [arXiv:1405.3678 [hep-ph]]

2014 arXiv
[58]

Hu, Phys

B. Hu, Phys. Rev. D 90 (2014) no.7, 073012 [arXiv:1407.4722 [hep-ph]]

2014 arXiv
[59]

A. S. Joshipura and K. M. Patel, JHEP 1404 (2014) 009 [arXiv:1401.6397 [hep-ph]]

2014 arXiv
[60]

A. S. Joshipura and K. M. Patel, Phys. Rev. D 90 (2014) no.3, 036005 [arXiv:1405.6106 [hep-ph]]

2014 arXiv
[61]

Talbert, JHEP 1412 (2014) 058 [arXiv:1409.7310 [hep- ph]]

J. Talbert, JHEP 1412 (2014) 058 [arXiv:1409.7310 [hep- ph]]

2014 arXiv
[62]

C. Y. Yao and G. J. Ding, Phys. Rev. D 92 (2015) no.9, 096010 [arXiv:1505.03798 [hep-ph]]

2015 arXiv
[63]

S. F. King and P. O. Ludl, JHEP 1606 (2016) 147 [arXiv:1605.01683 [hep-ph]]

2016 arXiv
[64]

de Medeiros Varzielas, R

I. de Medeiros Varzielas, R. W. Rasmussen and J. Tal- bert, Int. J. Mod. Phys. A 32 (2017) no.06n07, 1750047 [arXiv:1605.03581 [hep-ph]]

2017 arXiv
[65]

C. Y. Yao and G. J. Ding, Phys. Rev. D 94 (2016) no.7, 073006 [arXiv:1606.05610 [hep-ph]]

2016 arXiv
[66]

J. N. Lu and G. J. Ding, Phys. Rev. D 95 (2017) no.1, 015012 [arXiv:1610.05682 [hep-ph]]

2017 arXiv
[67]

C. C. Li, J. N. Lu and G. J. Ding, JHEP 1802 (2018) 038 [arXiv:1706.04576 [hep-ph]]

2018 arXiv
[68]

J. N. Lu and G. J. Ding, Phys. Rev. D 98 (2018) no.5, 055011 [arXiv:1806.02301 [hep-ph]]

2018 arXiv
[69]

Hagedorn and J

C. Hagedorn and J. König, arXiv:1811.07750 [hep-ph]

arXiv
[70]

J. N. Lu and G. J. Ding, JHEP 1903 (2019) 056 [arXiv:1901.07414 [hep-ph]]

2019 arXiv
[71]

de Medeiros Varzielas and J

I. de Medeiros Varzielas and J. Talbert, Eur. Phys. J. C 79 (2019) no.6, 536 [arXiv:1901.10484 [hep-ph]]

2019 arXiv
[72]

Bernigaud, I

J. Bernigaud, I. de Medeiros Varzielas and J. Talbert, arXiv:1906.11270 [hep-ph]

1906 arXiv
[73]

Tanabashi et al

M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98 (2018) no.3, 030001

2018
[74]

P. M. Ferreira, L. Lavoura and J. P. Silva, Phys. Lett. B 688 (2010) 341 [arXiv:1001.2561 [hep-ph]]

2010 arXiv
[75]

F. J. Botella, G. C. Branco, A. M. Coutinho, M. N. Re- belo and J. I. Silva-Marcos, Eur. Phys. J. C 75 (2015) 286 [arXiv:1501.07435 [hep-ph]]

2015 arXiv
[76]

G. C. Branco, J. I. Silva-Marcos and M. N. Rebelo, Phys. Lett. B 237 (1990) 446

1990
[77]

Fritzsch, Nucl

H. Fritzsch, Nucl. Phys. Proc. Suppl. 40 (1995) 121 [hep- ph/9411419]

1995
[78]

de Medeiros Varzielas and L

I. de Medeiros Varzielas and L. Merlo, JHEP 1102 (2011) 062 [arXiv:1011.6662 [hep-ph]]

2011 arXiv
[79]

de Medeiros Varzielas and D

I. de Medeiros Varzielas and D. Pidt, JHEP 1303 (2013) 065 [arXiv:1211.5370 [hep-ph]]

2013 arXiv
[80]

González Felipe, I

R. González Felipe, I. P. Ivanov, C. C. Nishi, H. Serôdio and J. P. Silva, Eur. Phys. J. C 74 (2014) no.7, 2953 [arXiv:1401.5807 [hep-ph]]

2014 arXiv
[81]

González Felipe, H

R. González Felipe, H. Serôdio and J. P. Silva, Phys. Rev. D 87 (2013) no.5, 055010 [arXiv:1302.0861 [hep-ph]]

2013 arXiv
[82]

González Felipe, H

R. González Felipe, H. Serôdio and J. P. Silva, Phys. Rev. D 88 (2013) no.1, 015015 [arXiv:1304.3468 [hep-ph]]

2013 arXiv
[83]

Pramanick and A

S. Pramanick and A. Raychaudhuri, JHEP 1801 (2018) 011 [arXiv:1710.04433 [hep-ph]]

2018 arXiv

Pith tools

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

[1] [1]

While invariance under (

is invariant under the action of (2) when the transformations on left- and right-handed fermions are equivalent, as they must be if fermion mass terms are generated. While invariance under (

[2] [2]

(4) We consider scenarios where (

may seem trivial in the mass basis, when the same relation is rotated to the ﬂavour basis, the transformations become functions ofUA, the (highly non-trivial) fermionic mixing matrices of the SM: TA →TAU =UATAU † A, (3) which are appropriately interpreted as symmetries of the mass matrix (and not of the full LSM , as weak inter- actions do not respect the...

[3] [3]

G. C. Branco, L. Lavoura and J. P. Silva, Int. Ser. Monogr. Phys. 103 (1999) 1

work page 1999

[4] [4]

This can occur, for example, in mod- els where the family-symmetry breaking occurs through the VEV of scalar ﬂavons {φ}

is not accidental, but is instead the remnant signature of an ultra-violet (UV) ﬂavour theory controlled by a family symmetry GF that commutes with the SM gauge group (or indeed any other BSM symmetry). This can occur, for example, in mod- els where the family-symmetry breaking occurs through the VEV of scalar ﬂavons {φ}. Regardless of the dy- namics that...

[5] [5]

A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32 [JETP Lett. 5 (1967) 24] [Sov. Phys. Usp. 34 (1991) no.5, 392] [Usp. Fiz. Nauk 161 (1991) no.5, 61]

1967

[6] [6]

(or relaxed versions of it) have been well studied in the context of fermionic mixing in the SM (see e.g. [24, 26, 27] for reviews), with the literature including complete models realising the RFS as well as both analytic and computa- tional approaches to cataloguing the types of GF that can break to phenomenologically viable CKM/PMNS matri- ces [28–51]. ...

work page

[7] [7]

is then expanded as LY = − ( 1 + S0 1 v ) (¯dLmddR + ¯uLmuuR + ¯ lLmeeR ) − 1 v N∑ k=2 ( S0 k +iP 0 k )( ¯dLY d kdR + ¯uLY u kuR + ¯ lLY e keR ) − √ 2 v N∑ k=2 S+ k ( ¯uLVY d kdR − ¯uRY u,† k Vd L + ¯νLY e keR ) +h.c., (12) 5 W e will adopt the notation of [14] in the following discussio n. 4 where V ≡ UCKM , the second line is clearly capable of generati...

work page

[8] [8]

Weinberg, Phys

S. Weinberg, Phys. Rev. Lett. 37 (1976) 657

work page 1976