Dirac neutrinos in the 2HDM with restrictive Abelian symmetries

F. R. Joaquim; R. G. Felipe; S. S. Correia

REVIEW 2 major objections 3 minor 132 references

Dirac neutrinos in the 2HDM with restrictive Abelian symmetries

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that, in a two-Higgs-doublet extension of the Standard Model with Dirac neutrinos, only 5 of the 28 maximally restrictive texture-zero mass-matrix pairs compatible with oscillation data can be realized by Abelian flavor…

desk verdict A genuinely useful classification with explicit charge assignments, but the 'only 5 of 28' count leans on an appendix that needs to be opened up before I'd fully trust the negative claims. read the letter →

arxiv 1909.00833 v1 pith:Z5G4O4BD submitted 2019-09-02 hep-ph

S. S. Correia , R. G. Felipe , F. R. Joaquim This is my paper · ORCID

classification hep-ph

keywords Diracneutrinostwo-Higgs-doubletmodeltexturezerosAbelianflavorsymmetriesU(1)andZ_Nchargesneutrinomassmixingleptonviolationuniversality

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 tries to determine how far texture zeros can go in a two-Higgs-doublet model (2HDM) with Dirac neutrinos: it asks which maximally restrictive patterns of zeros in the charged-lepton and neutrino mass matrices are both compatible with current neutrino oscillation data and obtainable from an Abelian (U(1) or Z_N) flavor symmetry. The answer is five patterns out of 28 candidates, and for those five the symmetry is essentially unique: a U(1) whose minimal discrete charges form a Z5. Why it matters: these are the most economical models in which the whole lepton-flavor structure is fixed by symmetry, giving one-to-one relations between Yukawa couplings and observable masses, mixing angles, and CP phases. The paper then shows how these five patterns translate into concrete predictions for leptonic CP violation, lepton universality in tau decays, and rare lepton-flavor-violating processes.

What carries the argument

The argument runs on two complementary classification tools. The first is the canonical decomposition of a mass-matrix texture into the two Yukawa matrices Y1 and Y2: it determines, for a given texture, which pairs of zero/nonzero patterns can coexist under a phase symmetry, using the realizable-texture chains of a (3,3) degenerate-charge class. The second is the Smith normal form (SNF) method, which takes the integer charge vectors of all allowed Yukawa interactions and diagonalizes them into a canonical diagonal matrix whose entries encode the residual rephasing symmetry group as a product of Z_d and U(1) factors. Together they decide realizability: a texture pair is realizable exactly when some decomposition exists whose SNF yields a symmetry that imposes all required zeros and only those zeros. The two tools are applied in sequence—first the canonical method to reject 23 of the 28 candidate pairs, then the SNF method to identify the minimal U(1) (with Z5 representative charges) for the five survivors.

What would settle it

Find any assignment of U(1) charges (or any sequence of Z_N transformations) in the 2HDM that realizes any of the 23 excluded texture pairs of Table II, for instance by solving the linear phase equations of the canonical method symbolically and exhibiting a nonzero solution; one such solution would invalidate the 'only five' claim. Conversely, a direct symbolic proof that the phase equations for a specific excluded pair, such as (4ℓ3, 6ν2), have no solution in any charge normalization would confirm the classification for that case.

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Extended reading notes

Core claim

The paper establishes that, in a 2HDM extended by three right-handed neutrinos with conserved lepton number, exactly five maximally restrictive texture pairs (Mℓ, Mν) simultaneously satisfy two conditions: they are consistent with neutrino oscillation data (mostly at 1σ; one pair only at 3σ and for inverted ordering), and they can be realized by a continuous U(1) or a discrete Z_N Abelian symmetry. The five surviving pairs are (4ℓ3, 6ν1,3,7,9) and (5ℓ1, 5ν8), and each admits exactly one decomposition into the two Yukawa matrices Y1 and Y2, enforced by a single U(1) flavor symmetry; the minimal set of discrete charges corresponds to a Z5 symmetry. For these pairs, after exhausting field-rephasing freedom, only one complex phase α remains in the mass matrices, and this phase is correlated one-to-one with the Dirac CP phase δ of the lepton mixing matrix. The paper further derives explicit relations expressing all mass-matrix entries in terms of the charged-lepton and neutrino masses plus the mixing angles, and shows that for the (5e1, 5ν8) pattern the μ→eγ decay is naturally suppressed by the flavor structure, whereas the (4ℓ3, 6νk) patterns require the CP-odd and CP-even neutral scalar masses to be nearly degenerate to evade the same bound.

Load-bearing premise

The completeness of the canonical exclusion procedure: the claim that only five pairs are realizable assumes that every possible decomposition of every texture pair into the two Yukawa matrices, and every combination of consecutive Abelian transformations, has been considered and that none of the 23 rejected pairs has a hidden solution.

Editorial extensions

If this is right

Only five of the 28 maximally restrictive texture pairs can be realized by Abelian symmetries in the 2HDM with Dirac neutrinos; every other pair would require more than two Higgs doublets or a non-Abelian symmetry.
Each of the five viable pairs has a unique Yukawa decomposition, and the minimal discrete implementation is a Z5 symmetry, making the flavor structure fully determined once the texture is chosen.
For all five pairs, the single physical phase α left in the mass matrices is directly related to the Dirac CP phase δ (for example δ ≃ −α in the (5e1, 5ν8) case), so leptonic CP violation is predicted once masses and mixing angles are fixed.
All nine lepton-sector parameters can be reconstructed from the observables; for (5e1, 5ν8) the authors give explicit analytic formulas expressing every mass-matrix entry in terms of lepton masses, mixing angles, and α.
Phenomenologically, only the (5e1, 5ν8) pattern avoids the μ→eγ bound without tuning the neutral scalar masses, while the (4ℓ3, 6νk) patterns require m_R ≈ m_I; sizable deviations from tau lepton universality appear only for a light charged Higgs, m_H± ≲ 300 GeV.

Reading between the lines

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

If the same maximally restrictive scan were applied to the quark sector, which the authors state is in preparation, the analogous classification could produce a similarly small list of realizable quark texture pairs, with parameter-free predictions for flavor-changing processes such as B→Xsγ and meson decays.
The count 'five' depends on the equivalence-class reduction borrowed from the earlier texture classification; using different equivalence conventions, such as including generalized CP transformations, could merge or split some of the 28 candidates and change the final number even if each individual realizability statement is unchanged.
As neutrino oscillation data sharpen the values of θ23 and δ, some of the five surviving patterns may become disfavored; re-running the same compatibility scan on an updated global fit would provide a direct test of their viability.
The Z5 nature of the minimal discrete charges suggests these models could be embedded in a Z5-symmetric ultraviolet completion, which would predict additional structure such as domain walls or cosmic strings if the U(1) is spontaneously broken with a Z5 remnant.

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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

Dear colleague,

Here's the take: the paper delivers a genuinely new result — a systematic enumeration of which maximally-restrictive texture pairs in a 2HDM with Dirac neutrinos can be realized by Abelian symmetries, ending with five survivors and explicit U(1) charge assignments, Yukawa reconstructions, and phenomenology. The identification of (5e1,5ν8) as the only pair with naturally suppressed LFV (no mR≈mI tuning) is a genuinely useful, concrete conclusion for model builders.

The positive side is explicit and checkable. The negative side — 23 of 28 pairs excluded — rests on Appendix B, and that appendix is the weakest part. B.1 and B.2 are fine: the pigeonhole arguments about full rows/columns forcing identical rows/columns are solid. B.3 also holds up; the ±(θ1−θ2) incompatibility is real. But B.4 is too compressed. The lemma is plausible, but the applications to (4_l^3,6_ν^{4,6}) and (5_l^1,5_ν^4) are asserted with phrases like "we can identify the indices" rather than demonstrated. Since 23 of 28 exclusions are decided by that case analysis, the headline count is only as trustworthy as the completeness of those handwritten arguments. I'm not claiming the count is wrong; I'm saying the paper doesn't yet let the reader verify it.

Two smaller issues. First, the mass-ordering statement for (6_l^1,4_ν^17) is internally inconsistent: Table II says IO, the main text says NO. It's a genuine error, though minor because that pair is excluded regardless. Second, the "minimal Z5" claim is asserted without proof. A short argument showing why no smaller N works would settle it.

The data fitting is standard, not circular — the symmetry classification is a mathematical statement made before the parameter reconstruction. The citation pattern is fine; the chain classification of Ref. [68] is genuine prior work, not a citation for its own sake.

Recommendation: send it to peer review, but tell the referee to insist on an expanded Appendix B — either full charge equations for the excluded pairs or a more explicit proof of the B.4 lemma — and on fixing the ordering typo. With those, this is a solid contribution to the texture-zero literature.

Referee Report

2 major / 3 minor

Summary. This paper considers a 2HDM extended by three right-handed neutrinos with conserved lepton number, so that neutrinos are Dirac particles, and asks which maximally-restrictive texture-zero pairs (M_l, M_nu) can be realized by Abelian U(1) or Z_N flavor symmetries. The authors first identify 28 such pairs compatible with neutrino oscillation data, then apply the canonical method and the Smith normal form (SNF) method to decide realizability. The central claim is that only 5 of the 28 pairs survive, each with a single decomposition into Yukawa matrices and a U(1) flavor symmetry (with minimal discrete charges corresponding to Z5). For the surviving pairs, the paper reconstructs all Yukawa parameters in terms of lepton masses and mixing parameters, studies the correlation between the single physical phase alpha and the Dirac CP phase delta, and analyzes constraints from lepton universality in tau decays and from rare lepton-flavor-violating processes.

Significance. If the classification is correct, the paper provides a useful and sharply stated result: the space of viable maximally-restrictive texture pairs in the 2HDM with Dirac neutrinos reduces to five explicit cases. The positive side of the claim is well documented through the explicit charge assignments in Table IX, the decomposition lists in Table V, and the SNF-based realizability analysis. The phenomenology section gives concrete, falsifiable predictions, such as the allowed tan-beta and m_H+ ranges for (5_e^1, 5_nu^8) and the m_R ~ m_I tuning required for the (4_l^3, 6_nu^k) cases. The approximate analytic formulas in Section VI, supported by the numerical scans, are another strength, as is the one-to-one reconstruction of Lagrangian parameters from observables.

major comments (2)

[Section V, Appendix B (B.3 and B.4)] The negative half of the headline result, namely that 23 of the 28 pairs in Table II cannot be realized, is not demonstrated at the level of detail needed to verify exhaustiveness. In Appendix B.3 the exclusions of (4_l^3, 6_nu^{2,5,8}) and (5_l^1, 5_nu^5) are justified by the statement that the constituent textures 'can only be realized by transformations which obey non-compatible versions of Eq. (B3)', but the full systems of charge equations and the enumeration of decompositions are not shown. In Appendix B.4 the exclusions of (4_l^3, 6_nu^{4,6}) and (5_l^1, 5_nu^4) rely on 'we can identify the indices i,j,k and easily arrive at the conclusion', without displaying those indices or the resulting contradiction. Because 23 of the 28 pairs are rejected, even one missed decomposition or ordering would change the count from 'only 5' to 'at least 6'. Please provide, for each excluded pair, the explicit systems arising from Eq. (20) and their solution sets, or a reproducible computer enumeration.
[Section V.A, Tables V-VII] The decomposition information needed to check the negative claim is not self-contained. Table V lists decompositions only for the five surviving pairs, while the chain data of Ref. [68] that underlies the exclusions is not reproduced. A reader cannot check whether an alternative assignment of the nonzero entries of a texture to Y^l_{1,2} and Y^nu_{1,2} (including both orderings, or a decomposition not listed in Table V) evades the incompatibility arguments used in Appendix B. The final sentence of Appendix B, stating that 'no other decompositions ... arise from the application of such consecutive transformations', is asserted without proof. I request either a complete decomposition table for every texture appearing in the excluded pairs, or an explicit algorithmic/code-based verification of the 23 exclusions.

minor comments (3)

[Section IV vs Table II] Section IV states that the pair (6_l^1, 4_nu^17) is compatible with data only at 3 sigma and for a NO mass spectrum, while the caption of Table II says this pair is consistent only at 3 sigma and for IO; these two statements cannot both be correct and must be reconciled.
[Fig. 5 caption] The caption of Fig. 5 refers to the 'MEG bound given in (77)', but the upper limit on Br(mu -> e gamma) appears in Eq. (81); Eq. (77) lists the normalization branching ratios, not the bound.
[Table X caption] The caption of Table X repeats the same expression twice ('Allowed l_alpha -> l_beta gamma and l_alpha -> l_beta gamma'); the second expression should presumably be the three-body decay l^-_alpha -> l^-_beta l^+_gamma l^-_delta.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the symmetry-realization classification is independent of the data fit, and the cited classification theorems are external to the authors.

full rationale

I walked the derivation chain and found no step in which a claimed result reduces by construction to its own inputs. The paper first uses a standard chi-square fit to identify the 28 maximally-restrictive texture pairs in Table II; this is data fitting, not circular reasoning. It then applies the canonical and Smith-normal-form methods to decide which of those pairs admit Abelian realizations. That decision depends on solving the phase-equation systems in Eqs. (16)-(20) and on the Yukawa-texture classification of Ref. [68], which is an external mathematical classification rather than a result derived from the present paper or from the same fitted data. The later reconstruction of Yukawa parameters in terms of masses and mixing angles is also standard model fitting, and the LFV/universality analysis uses experimental bounds that were not used in the texture selection. The negative claim that 23 pairs are non-realizable rests on the completeness of the case analysis in Appendix B; that is a correctness or completeness risk, not a circularity, because the exclusions are mathematical solubility arguments, not restatements of the input data. No load-bearing self-citation chain is present: the methods of Refs. [68-70] are cited as independent technical tools, and the authors do not invoke a private uniqueness theorem to forbid alternatives. Accordingly, the appropriate finding is no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The model introduces no new particles beyond the three right-handed neutrinos and the second Higgs doublet that are standard in the literature. The U(1) charges in Table IX are symmetry assignments, not new entities. The free scalar-sector parameters are scanned in the phenomenology rather than fitted, and the lepton-sector mass-matrix parameters are eventually fixed by the observable masses, angles, and phase.

free parameters (4)

tanβ = scanned 0.01 to 100
Ratio of the two Higgs doublet VEVs. It is scanned in Section VI B to test lepton universality and LFV bounds; it does not affect the texture realization count.
mH± (charged scalar mass) = scanned 80 GeV to 1 TeV
Mass of the charged Higgs, scanned to compute |gµ/ge| and LFV rates.
mR (CP-even neutral scalar mass) = scanned 100 GeV to 10 TeV
Mass of the heavy CP-even scalar; LFV three-body and radiative decays depend on it.
mI (CP-odd neutral scalar mass) = scanned 100 GeV to 10 TeV
Mass of the U(1)-Goldstone-like scalar after soft breaking; the ratio mR/mI controls the cancellation in µ→eγ.

assumptions (5)

domain assumption The 2HDM extended by three right-handed neutrino singlets, with conserved lepton number so neutrinos are Dirac particles.
This is the model setup of Section II, Eqs. (2)-(7).
domain assumption The complete classification of texture-zero pairs from Ludl and Grimus (Ref. [78]) is exhaustive.
Section IV relies on this to enumerate the 28 maximally-restrictive pairs; if other equivalence classes exist, the starting set changes.
domain assumption The canonical method and Smith normal form procedure of Refs. [68-70] correctly identify all U(1) and Z_N realizable textures in multi-Higgs models.
Section V and Appendix B use these methods to exclude 23 pairs and realize the remaining 5 pairs.
domain assumption The scalar potential has no CP violation, contains a soft m12² Φ1†Φ2 term, and H0 is identified with the observed 125 GeV SM-like Higgs.
Assumed in Section VI B and used for all LFV and lepton-universality computations.
domain assumption Best-fit oscillation parameters and 3σ ranges from Ref. [72] (Table I) are the correct current experimental inputs.
The compatibility scan in Section IV A and the numerical scans in Section VI use these values.

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Pith. "Pith review of Dirac neutrinos in the 2HDM with restrictive Abelian symmetries." pith.science (2026). https://pith.science/paper/Z5G4O4BD

@misc{pith2026190900833,
  author       = {Pith},
  title        = {Pith review of: Dirac neutrinos in the 2HDM with restrictive Abelian symmetries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z5G4O4BD}},
  note         = {Machine review of arXiv:1909.00833}
}

abstract

Recently, there has been a growing interest in extensions of the Standard Model in which naturally small Dirac neutrino masses arise due to existence of a symmetry which protects neutrino's Diracness. Motivated by this, we consider an extension of the Standard Model with a second Higgs doublet (2HDM) and three right-handed neutrinos where lepton number is conserved and, thus, neutrinos are Dirac particles. In this framework, we identify the most restrictive texture-zero combinations for the Dirac-neutrino and charged-lepton mass matrices that lead to masses and mixings compatible with current experimental data. We then investigate, in a systematic way, which of these combinations can be realized by Abelian continuous U(1) or discrete $\mathbb{Z}_N$ symmetries. We conclude that, from the 28 initially possible sets of maximally-restricted lepton mass matrices, only 5 have a symmetry realization in the 2HDM. For these cases, one-to-one relations among the Yukawa couplings and the neutrino mass and mixing parameters are established, and the fermion interactions with the neutral and charged scalars of the 2HDM are also determined. Consequences for lepton universality in $\tau$ decays and rare lepton-flavor-violating processes are also discussed.

Figures

Figures reproduced from arXiv: 1909.00833 by the authors.

**Figure 2.** FIG. 2. Correlation between the phase [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 2.** In conclusion, all parameters in the mass matrices [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

Figures from the paper (4 more)

**Figure 3.** Figure 3: The natural suppression of LFV decays does not however occur when M` ∼ 4 ` 3 . As can be seen from (83), in these cases the couplings Ne do not exhibit any decoupling behavior and, thus, the decay rates are not naturally suppressed. In the particular case of µ → eγ,…

**Figure 3.** Figure 3: FIG. 3. Results for the (5 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png]

**Figure 4.** Figure 4: FIG. 4. Results for the (4 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: FIG. 5. Dependence of Br( [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

132 extracted references · 65 canonical work pages

[68]

R. Foot, H. Lew, X. G. He and G. C. Joshi, Z. Phys. C 44, 441 (1989)

1989
[1]

we obtainδ≃α (δ≃π + α), which also agrees with the numerical output shown in Fig. 2. In conclusion, all parameters in the mass matrices M𝓁 and Mν can be determined in terms of the charged- lepton and neutrino masses and mixing angles through Eqs. (51)-(58) and (62). B. Lepton universality and rare LFV decays In the 2HDM, Yukawa interactions may induce ﬂav...
[2]

Also notice that, with the exception of texture 4 ν 17, any representative of Mν given in Table IV features a mass- less neutrino, since it contains a full column of zeros

which is consistent with data only at 3 σ CL and for a NO mass spectrum. Also notice that, with the exception of texture 4 ν 17, any representative of Mν given in Table IV features a mass- less neutrino, since it contains a full column of zeros. We emphasize that these maximally-restrictive texture pairs cannot be implemented in the SM by imposing Abelian...
[3]

Since none of the textures belonging to the pairs (4 𝓁 3, 6ν 1,3,7,9) and (5𝓁 1, 5ν
[4]

(B3) for the same rowsi and j

T exture pairs (4𝓁 3, 6ν 4,6) and (5𝓁 1, 5ν 4) From the application of the canonical method one can conclude that if a mass matrix textureT1 is characterized by two non-identical columns c1 and c2 with non-zero entries in some row i and zero entries in some row j, then it cannot be realized through Abelian symmetries in the 2HDM, together with a texture T...
[5]

The associated chains and their corresponding building matrices are shown in Ta- bles VI and VII, for the charged-lepton and neutrino tex- tures, respectively

pairs (see Table V). The associated chains and their corresponding building matrices are shown in Ta- bles VI and VII, for the charged-lepton and neutrino tex- tures, respectively. Based on Table V, we generate all possible pairs of (4𝓁 3, 6ν 1,3,7,9) and (5𝓁 1, 5ν
[6]

decompositions. Notice that one must consider both Yukawa matrix orderings, since swap- ping Yν 1 and Yν 2, while maintaining the charged-lepton 8 Charged leptons Texture decomposition Y𝓁 1 Y𝓁 2 4𝓁 3,I   0 0 × 0 × 0 × 0 0     0 0 0 0 0 × 0 × 0   5𝓁 1,I   0 0 × 0 × 0 × 0 0     0 0 0 0 0 0 0 0 ×   5𝓁 1,II   0 0 × 0 0 0 × 0 0     0 0 0 ...
[7]

decomposition pairs in Table V, considering both orderings of Yν 1,2. We ﬁnd that DSNF can only take one of the following two forms: DSNF =   1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0   , g= 0, 2, (31)...

Show all 132 references

[8]

has identical columns, and neither 4 𝓁 3 nor 5𝓁 1 has identical rows, the implementation of any of these tex- tures require non-degenerate transformations, i.e three distinctαi, three distinctβi and three distinctγi phases. Thus, the search for decompositions is limited to the...
[9]

that are exactly realized by the symme- tries (34) (see Table VIII). We ﬁnd that all 5 mass matrix pairs have one decomposition which can be implemented imposing a GF = U(1)× U(1)νRi symmetry, being U(1) alone able to reproduce the entire corresponding texture structure. This ...
[10]

The results shown in the left (right) panel correspond to a NH (IH) neutrino mass spectrum

texture pairs. The results shown in the left (right) panel correspond to a NH (IH) neutrino mass spectrum. In all points, the mixing angles θij lie within the 3σ ranges given in Table I. M𝓁 Mν 4𝓁 3,I :   0 0 a1 0 a2 b1 a3 b2 0   6ν 1,I :   0 0 0 0 0 x2 0 x1 yeiα   6ν 3...
[11]

In all cases, both the NH and IH mass spectra are considered

pairs, respectively. In all cases, both the NH and IH mass spectra are considered. By looking at Fig. 1 we see that the results are simi- lar for diﬀerent pairs of ( M𝓁, Mν) textures. This could suggest that those pairs are equivalent in the sense that they can be transformed ...
[12]

Implementation of the texture pairs from Ta- ble VIII in a 2HDM

( γ′ 1,γ′ 2,γ′ 3) (4𝓁 3,I, 6ν 1,I) (0 ,θ, 2θ) (2 θ,θ, 0) ( η, 3θ, 2θ) (0, 1, 2) (2 , 1, 0) (4 , 3, 2) (4𝓁 3,I, 6ν 3,I) (0 ,θ, 2θ) (2 θ,θ, 0) ( η, 3θ,θ ) (0, 1, 2) (2 , 1, 0) (4 , 3, 1) (4𝓁 3,I, 6ν 7,I) (0 ,θ, 2θ) (2 θ,θ, 0) ( η, 2θ, 0) (0, 1, 2) (2 , 1, 0) (4 , 2, 0) (4𝓁 3,I, ...
[13]

of Section IV)

= (0 , 1) and ϕ = 2π/5. of Section IV). However, we advocated that only non- equivalent pairs of textures were kept. The similarity between some of the results have to do with the fact that up to a very small parameter, which does not have much impact in the results for the mi...
[14]

By look- ing at Table VIII we notice that 6ν 3 =P13 6ν 7P23 , 4𝓁 3 = a3→0 P13 4𝓁 3P23

and (4𝓁 3, 6ν 7). By look- ing at Table VIII we notice that 6ν 3 =P13 6ν 7P23 , 4𝓁 3 = a3→0 P13 4𝓁 3P23. (35) The last relation simply indicates that 4 𝓁 3 =P13 4𝓁 3P23 13 in the limit a3→ 0 and, thus, the two pairs would be equivalent in this case. 6 Obviously, a3 = 0 would l...
[15]

From Eqs

pair. From Eqs. (10), (37), (41), (C1) and (C2), the lep- ton mixing matrix U is computed and the mixing angles and the phase δ are extracted. Notice that, for the case (5e 1, 5ν 8), one has U1j = ( Uν L)2j. Therefore, given the parametrization (11), x2 and y2 in Mν depend onl...
[16]

+s2 13 , y2 2 = ∆m2 31(s2 13 +rc2 13s2 12), (51) IH :x2 2 = ∆m2 31(1 +r)(1 +rs2 12)s2 13 rs2 12(r− 2s2 13−rc2 13s2
[17]

was found to be consistent with experimental data only at 3σ and for IO. 3𝓁 2∼   0 × × 0 × × × 0 ×   4𝓁 1∼   0 0 × 0 × 0 × × ×   4𝓁 2∼   0 0 × 0 × × × 0 ×   4𝓁 3∼   0 0 × 0 × × × × 0   5𝓁 1∼   0 0 × 0 × 0 × 0 ×   6𝓁 1∼   0 0 × 0 × 0 × 0 0   TABLE III. R...
[18]

(52) It now remains to expressθL (orb2) appearing in Eq

+s2 13 , y2 2 = ∆m2 31c2 13(1 +rs2 12). (52) It now remains to expressθL (orb2) appearing in Eq. (37) and the phase α in terms of the measurable neutrino parameters. Including the charged-lepton corrections to the mixing we have tan2θ23≃ [rcαtL sin(2θ12)− 2s13]2 +r2t2 L sin2(2...
[19]

In the left (middle) column we plot |gµ/ge|−1 as a function ofmH ± (tanβ), while in the right column the same points are shown in the (mR,mI)-plane

texture pairs in the two upper and lower pan- els, respectively. In the left (middle) column we plot |gµ/ge|−1 as a function ofmH ± (tanβ), while in the right column the same points are shown in the (mR,mI)-plane. We conclude that the (5 µ,τ 1 , 5ν
[20]

Instead, for (5 e 1, 5ν

cases are disfavored by the|gµ/ge|− 1 constraint (73) (indicated by the hori- zontal gray bands in the plots). Instead, for (5 e 1, 5ν
[21]

(73) for 80 GeV ≲ mH ± ≲ 200 GeV and tan β ≲ 0.03 or tanβ & 30, for both NH and IH

the deviation from universality is in agreement with Eq. (73) for 80 GeV ≲ mH ± ≲ 200 GeV and tan β ≲ 0.03 or tanβ & 30, for both NH and IH. Notice that for large (small) tanβ the Yukawa couplings in Y𝓁 1 (Y𝓁
[22]

Similar results are presented in Fig

are en- hanced, leading to an enhancement of |gµ/ge|− 1. Similar results are presented in Fig. 4 for the (4 𝓁 3, 6ν k) texture pairs given in Table VIII. We do not present the results in terms of tan β since the behavior is similar to that of the (5𝓁 1, 5ν
[23]

The main diﬀerence between the results in Figs

cases i.e., in general, there is a small and large tanβ region. The main diﬀerence between the results in Figs. 3 and 4 is evident from the comparison of the (mR,mI) plots. While for the (5𝓁 1, 5ν
[24]

3), for the texture sets Decay 5 e 1 5µ 1 5τ 1 𝓁α→𝓁βγ (τ,µ ) ( τ,e ) ( µ,e ) 𝓁− α→𝓁− β 𝓁+ γ 𝓁− δ (τ,µµµ ) ( τ,eee ) ( µ,eee ) (τ,eeµ ) ( τ,µµe ) TABLE X

texture pair all constraints are veriﬁed for non-correlatedmR,I masses (see the left column plots in Fig. 3), for the texture sets Decay 5 e 1 5µ 1 5τ 1 𝓁α→𝓁βγ (τ,µ ) ( τ,e ) ( µ,e ) 𝓁− α→𝓁− β 𝓁+ γ 𝓁− δ (τ,µµµ ) ( τ,eee ) ( µ,eee ) (τ,eeµ ) ( τ,µµe ) TABLE X. Allowed 𝓁α → 𝓁βγ ...
[25]

On the other hand, τ radiative decays are forbidden in that case since the τ is decoupled

the electron (muon) is de- coupled and, thus, µ−e transitions are not allowed. On the other hand, τ radiative decays are forbidden in that case since the τ is decoupled. Applying the same rea- soning to the 3-body decays 𝓁− α→𝓁− β 𝓁+ γ 𝓁− δ , we conclude that, at most, only tw...
[26]

In the particular case of µ → eγ, the terms enhanced by mτ/mµ are potentially large and the experimental bound on that decay is respected only 17 FIG

As can be seen from (83), in these cases the couplings Ne do not exhibit any de- coupling behavior and, thus, the decay rates are not nat- urally suppressed. In the particular case of µ → eγ, the terms enhanced by mτ/mµ are potentially large and the experimental bound on that ...
[27]

In the left (middle) columns we plot|gµ/ge|− 1 as a function of mH± (tanβ)

texture pairs (upper and lower rows, respectively). In the left (middle) columns we plot|gµ/ge|− 1 as a function of mH± (tanβ). The horizontal grey bands correspond to the constraint (73). In the right column, the same points as in the corresponding |gµ/ge|− 1 plots are shown ...
[28]

In all points the mixing angles θij lie within the 3σ ranges given in Table I and |gµ/ge|− 1≥ 10−4

texture pair, for the remaining (4𝓁 3, 6ν k) cases shown in Table VIII the results are similar. In all points the mixing angles θij lie within the 3σ ranges given in Table I and |gµ/ge|− 1≥ 10−4. The red points are excluded by theµ→eγ MEG bound given in (77). two Higgs doublet...

2019
[29]

Hereafter, we shall refer to such structures as identical columns (rows)

T extures 4𝓁 1, 4𝓁 2, 4ν 17 In the context of a 2HDM, a mass matrix texture with a row (column) full of non-zero entries can only be re- alized through an Abelian symmetry if it has at least two columns (rows) with an identical texture structure. Hereafter, we shall refer to s...
[30]

T exture pairs (3𝓁 2, 7ν 1,3), (6𝓁 1, 4ν
[31]

and (5𝓁 1, 5ν 1,6) First we note that, for each of the texture pairs (3𝓁 2, 7ν 1,3), (6𝓁 1, 4ν
[32]

Since αi are common to the charged-lepton and Dirac-type neutrino sectors, Eq

and (5 𝓁 1, 5ν 1,6), one of the textures is characterized by a column full of non-zero entries and it has two identical texture rows (generated by two identical left-handed continuous phasesαi). Since αi are common to the charged-lepton and Dirac-type neutrino sectors, Eq. (17...
[33]

Consequently, another 5 maximally- restrictive pairs of leptonic mass matrix textures can be eliminated from Table II for the purpose of our model implementation

and (5𝓁 1, 5ν 1,6) cannot be real- ized through any continuous or discrete Abelian symme- tries in the 2HDM. Consequently, another 5 maximally- restrictive pairs of leptonic mass matrix textures can be eliminated from Table II for the purpose of our model implementation
[34]

(B3) In each of the pairs (4 𝓁 3, 6ν 2,5,8) and (5 𝓁 1, 5ν 5), the con- stituent textures can only be realized by transformations which obey non-compatible versions of Eq

T exture pairs (4𝓁 3, 6ν 2,5,8) and (5𝓁 1, 5ν 5) From the application of the canonical method one can conclude that, in the 2HDM, a mass matrix texture in- cluding a column with two non-zero entries associated to non-identical rows i and j can only be realized by a symmetry tr...
[35]

As a result, we ex- clude another 4 maximally-restrictive pairs of leptonic mass matrix textures from Table II

cannot be realized through any continuous or dis- crete Abelian symmetries, in the context of the 2HDM, regardless of the number of consecutive symmetry trans- formations of type (15) imposed. As a result, we ex- clude another 4 maximally-restrictive pairs of leptonic mass mat...
[36]

y2 √ (∆21−x2 2)(∆31−y2 2) ∆21∆−(y2 2−x2
[37]

y2 √ (y2 2− ∆21)(∆31−x2 2) ∆31∆−(y2 2−x2 2) y2 √ (x2 2− ∆21)(∆31−y2 2) ∆21∆31(y2 2−x2
[38]

−x2 √ (y2 2− ∆21)(∆31−x2 2) ∆21∆−(y2 2−x2
[39]

Deﬁning conditions for the charged-lepton and neutrino Yukawa textures given in Table VIII

x2 √ (∆21−x2 2)(∆31−y2 2) ∆31∆−(y2 2−x2 2)   , (C1) 23 Texture Deﬁning conditions 4𝓁 3 :   0 0 a1 0 a2 b1 a3 b2 0   a2 1 = b2 2 ∆ ∆− Σ , a 2 2 = ∆− Σ a2 3b2 2 , b 2 1 =a2 3 + a4 3 b2 2 + χ b2 2 − a2 3 T b2 2 − b2 2 ∆ ∆− Σ > 0 Σ =a2 3 [ χ + (a2 3 +b2 2)(a2 3 +b2 ...
[40]

in terms of a2 3 andb2 2 (a2
[41]

In 5 𝓁 1, the state 𝓁1 can be identiﬁed with e,µ orτ leading, respectively, to the cases 5e 1, 5µ 1 and 5τ 1 discussed in Section VI

and the charged-lepton masses. In 5 𝓁 1, the state 𝓁1 can be identiﬁed with e,µ orτ leading, respectively, to the cases 5e 1, 5µ 1 and 5τ 1 discussed in Section VI. For the 6ν k (5ν
[42]

textures we write x2 2 and y (x2 1 and y2
[43]

in terms of x2 1 (x2 2 and y2
[44]

The Yukawa couplings in Y𝓁,ν 1 (Y𝓁,ν 2 ) entering Eq

and the neutrino mass-squared diﬀerences ∆ 21 and ∆ 31. The Yukawa couplings in Y𝓁,ν 1 (Y𝓁,ν 2 ) entering Eq. (3) are determined by dividing ai and xi (bi and yi) by v cosβ (v sinβ). For simplicity, we use the notation ∆ 21≡ ∆m2 21 and ∆31≡ ∆m2 31 > 0 (∆m2 31 corresponds to|∆m...
[45]

y2 √ (y2 2− ∆31)(∆+−x2 2) ∆21∆+(y2 2−x2
[46]

x2 √ (y2 2− ∆31)(∆+−y2 2) ∆31∆+(y2 2−x2 2) −x2 √ (y2 2− ∆31)(∆+−x2 2) ∆21∆+(y2 2−x2
[47]

In the above equations and in Table XI, we have used the notation ∆± = ∆31± ∆21

x2 √ (∆31−x2 2)(∆+−y2 2) y2 2−x2 2 y2 √ (∆31−x2 2)(∆+−x2 2) ∆21∆+(y2 2−x2 2)   , (C2) for NH and IH neutrino mass spectra, respectively. In the above equations and in Table XI, we have used the notation ∆± = ∆31± ∆21. Moreover, the −(+) sign in U′ ν corresponds to t...
[48]

Kayser and R

B. Kayser and R. E. Shrock, Phys. Lett. 112B, 137 (1982)

1982
[49]

Yoshimura, Phys

M. Yoshimura, Phys. Rev. D 75, 113007 (2007)

2007
[50]

Yoshimura, Phys

M. Yoshimura, Phys. Lett. B 699, 123 (2011)

2011
[51]

Fukumi et al., PTEP 2012, 04D002 (2012)

A. Fukumi et al., PTEP 2012, 04D002 (2012)

2012
[52]

D. N. Dinh, S. T. Petcov, N. Sasao, M. Tanaka and M. Yoshimura, Phys. Lett. B 719, 154 (2013)

2013
[53]

J. M. Berryman, A. de Gouvˆ ea, K. J. Kelly and M. Schmitt, Phys. Rev. D 98, no. 1, 016009 (2018)

2018
[54]

A. B. Balantekin, A. de Gouvˆ ea and B. Kayser, Phys. Lett. B 789, 488 (2019)

2019
[55]

Schechter and J

J. Schechter and J. W. F. Valle, Phys. Rev. D 25, 2951 (1982)

1982
[56]

Weinberg, Phys

S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979)

1979
[57]

Minkowski, Phys

P. Minkowski, Phys. Lett. 67B (1977) 421

1977
[58]

Gell-Mann, P

M. Gell-Mann, P. Ramond and R. Slansky, Conf. Proc. C 790927 (1979) 315

1979
[59]

Yanagida, Conf

T. Yanagida, Conf. Proc. C 7902131 (1979) 95

1979
[60]

Schechter and J

J. Schechter and J. W. F. Valle, Phys. Rev. D 22 (1980) 2227

1980
[61]

S. L. Glashow, NATO Sci. Ser. B 61 (1980) 687

1980
[62]

R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980)

1980
[63]

Barbieri, D

R. Barbieri, D. V. Nanopoulos, G. Morchio and F. Stroc- chi, Phys. Lett. 90B, 91 (1980)

1980
[64]

T. P. Cheng and L. F. Li, Phys. Rev. D 22, 2860 (1980)

1980
[65]

Magg and C

M. Magg and C. Wetterich, Phys. Lett. 94B, 61 (1980)

1980
[66]

Lazarides, Q

G. Lazarides, Q. Shaﬁ and C. Wetterich, Nucl. Phys. B 181, 287 (1981)

1981
[67]

R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23, 165 (1981)

1981
[69]

Ma, Phys

E. Ma, Phys. Rev. Lett. 81, 1171 (1998)

1998
[70]

Brahmachari, E

B. Brahmachari, E. Ma and U. Sarkar, Phys. Lett. B 520, 152 (2001)

2001
[71]

Ma and O

E. Ma and O. Popov, Phys. Lett. B 764, 142 (2017)

2017
[72]

Arkani-Hamed, S

N. Arkani-Hamed, S. Dimopoulos, G. R. Dvali and J. March-Russell, Phys. Rev. D 65, 024032 (2001)

2001
[73]

G. R. Dvali and A. Y. Smirnov, Nucl. Phys. B 563, 63 (1999)

1999
[74]

Centelles Chuli, R

S. Centelles Chuli, R. Srivastava and J. W. F. Valle, Phys. Rev. D 98, no. 3, 035009 (2018)

2018
[75]

Calle, D

J. Calle, D. Restrepo, C. E. Yaguna and . Zapata, Phys. Rev. D 99, no. 7, 075008 (2019)

2019
[76]

Enomoto, S

K. Enomoto, S. Kanemura, K. Sakurai and H. Sugiyama, Phys. Rev. D 100, no. 1, 015044 (2019)

2019
[77]

Roncadelli and D

M. Roncadelli and D. Wyler, Phys. Lett. 133B, 325 (1983)

1983
[78]

Roy and O

P. Roy and O. U. Shanker, Phys. Rev. Lett. 52, 713 (1984) Erratum: [Phys. Rev. Lett. 52, 2190 (1984)]

1984
[79]

C. Y. Yao and G. J. Ding, Phys. Rev. D 97, no. 9, 095042 (2018)

2018
[80]

E. Ma, N. Pollard, R. Srivastava and M. Zakeri, Phys. Lett. B 750, 135 (2015)

2015
[81]

Bonilla, E

C. Bonilla, E. Ma, E. Peinado and J. W. F. Valle, Phys. Lett. B 762, 214 (2016)

2016
[82]

Ma and R

E. Ma and R. Srivastava, Phys. Lett. B 741, 217 (2015)

2015
[83]

Centelles Chuli, R

S. Centelles Chuli, R. Srivastava and J. W. F. Valle, Phys. Lett. B 781, 122 (2018)

2018
[84]

C. Y. Yao and G. J. Ding, Phys. Rev. D 96, no. 9, 095004 (2017). Erratum: [Phys. Rev. D 98, no. 3, 039901 (2018)]

2017
[85]

Centelles Chuli, R

S. Centelles Chuli, R. Cepedello, E. Peinado and R. Sri- vastava, arXiv:1907.08630 [hep-ph]

1907 arXiv
[86]

Farzan and E

Y. Farzan and E. Ma, Phys. Rev. D 86, 033007 (2012)

2012
[87]

Okada, arXiv:1404.0280 [hep-ph]

H. Okada, arXiv:1404.0280 [hep-ph]

arXiv
[88]

Centelles Chuli´ a, E

S. Centelles Chuli´ a, E. Ma, R. Srivastava and J. W. F. Valle, Phys. Lett. B 767, 209 (2017)

2017
[89]

M. Reig, D. Restrepo, J. W. F. Valle and O. Zapata, Phys. Rev. D 97, no. 11, 115032 (2018)

2018
[90]

Bonilla, S

C. Bonilla, S. Centelles-Chuli´ a, R. Cepedello, E. Peinado and R. Srivastava, arXiv:1812.01599 [hep-ph]

arXiv
[91]

Dasgupta, S.K

A. Dasgupta, S.K. Kang and O. Popov, arXiv:1903.12558 [hep-ph]. 25

1903 arXiv
[92]

Ma, arXiv:1907.04665 [hep-ph]

E. Ma, arXiv:1907.04665 [hep-ph]

1907 arXiv
[93]

K. Dick, M. Lindner, M. Ratz and D. Wright, Phys. Rev. Lett. 84, 4039 (2000)

2000
[94]

Murayama and A

H. Murayama and A. Pierce, Phys. Rev. Lett. 89, 271601 (2002)

2002
[95]

Borah and A

D. Borah and A. Dasgupta, JCAP 1612, no. 12, 034 (2016)

2016
[96]

P. H. Gu, arXiv:1711.11333 [hep-ph]

arXiv
[97]

Narendra, N

N. Narendra, N. Sahoo and N. Sahu, Nucl. Phys. B 936, 76 (2018)

2018
[98]

Grimus, A

W. Grimus, A. S. Joshipura, L. Lavoura and M. Tani- moto, Eur. Phys. J. C 36 (2004) 227

2004
[99]

Dighe and N

A. Dighe and N. Sahu, arXiv:0812.0695 [hep-ph]

arXiv
[100]

Adhikary, A

B. Adhikary, A. Ghosal and P. Roy, JHEP 0910, 040 (2009)

2009
[101]

S. Dev, S. Gupta and R. R. Gautam, Phys. Lett. B 701, 605 (2011)

2011
[102]

Gonz´ alez Felipe and H

R. Gonz´ alez Felipe and H. Serˆ odio, Nucl. Phys. B886, 75 (2014)

2014
[103]

L. M. Cebola, D. Emmanuel-Costa and R. G. Felipe, Phys. Rev. D 92, 025005 (2015)

2015
[104]

Samanta and A

R. Samanta and A. Ghosal, Nucl. Phys. B 911, 846 (2016)

2016
[105]

Kobayashi, T

T. Kobayashi, T. Nomura and H. Okada, Phys. Rev. D 98, 055025 (2018)

2018
[106]

M. H. Rahat, P. Ramond and B. Xu, Phys. Rev. D , 055030 (2018)

2018
[107]

Nath, arXiv:1808.05062 [hep-ph]

N. Nath, arXiv:1808.05062 [hep-ph]

arXiv
[108]

D. M. Barreiros, R. G. Felipe and F. R. Joaquim, Phys. Rev. D 97, no. 11, 115016 (2018)

2018
[109]

D. M. Barreiros, R. G. Felipe and F. R. Joaquim, JHEP 1901, 223 (2019)

2019
[110]

G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and J. P. Silva, Phys. Rept. 516, 1 (2012)

2012
[111]

G. C. Branco, W. Grimus and L. Lavoura, Phys. Lett. B 380, 119 (1996)

1996
[112]

F. J. Botella, G. C. Branco and M. N. Rebelo, Phys. Lett. B 687, 194 (2010)

2010
[113]

F. J. Botella, G. C. Branco and M. N. Rebelo, Phys. Lett. B 722, 76 (2013)

2013
[114]

Nebot, F

M. Nebot, F. J. Botella and G. C. Branco, arXiv:1808.00493 [hep-ph]
[115]

Serˆ odio, Phys

H. Serˆ odio, Phys. Rev. D88, 056015 (2013)

2013
[116]

Petersen, M

B. Petersen, M. Ratz and R. Schieren, JHEP 0908, 111 (2009)

2009
[117]

I. P. Ivanov and C. C. Nishi, JHEP 1311, 069 (2013)

2013
[118]

Tanabashi et al

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

2018
[119]

P. F. de Salas, D. V. Forero, C. A. Ternes, M. Tortola and J. W. F. Valle, arXiv:1708.01186 [hep-ph]

arXiv
[120]

Esteban, M

I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler and T. Schwetz, JHEP 1701, 087 (2017)

2017
[121]

Capozzi, E

F. Capozzi, E. Di Valentino, E. Lisi, A. Marrone, A. Mel- chiorri and A. Palazzo, Phys. Rev. D 95, no. 9, 096014 (2017)

2017
[122]

Palanque-Delabrouille et al., JCAP 1511, 011 (2015)

N. Palanque-Delabrouille et al., JCAP 1511, 011 (2015)

2015
[123]

C. I. Low, Phys. Rev. D 71, 073007 (2005)

2005
[124]

P. M. Ferreira, J. P. Silva, Phys. Rev. D 83, 065026 (2011)

2011
[125]

P. O. Ludl and W. Grimus, JHEP 1407, 090 (2014) Er- ratum: [JHEP 1410, 126 (2014)]

2014
[126]

Gonzalez Felipe and H

R. Gonzalez Felipe and H. Serodio, J. Phys. G 44, no. 6, 065002 (2017)

2017
[127]

I. P. Ivanov, C. C. Nishi, JHEP 1311, 069 (2013)

2013
[128]

F. J. Botella, G. C. Branco, A. Carmona, M. Nebot, L. Pedro and M. N. Rebelo, JHEP 1407, 078 (2014)

2014
[129]

Amhis et al

Y. Amhis et al. [HFLAV Collaboration], Eur. Phys. J. C 77, no. 12, 895 (2017)

2017
[130]

Abbiendi et al

G. Abbiendi et al. [ALEPH and DELPHI and L3 and OPAL and LEP Collaborations], Eur. Phys. J. C73, 2463 (2013)

2013
[131]

C. I. Low and R. R. Volkas, Phys. Rev. D 68, 033007 (2003)

2003
[132]

S. S. Correia, R. G. Felipe, F. R. Joaquim and M. Nebot, in preparation

Pith tools

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

[1] [68]

R. Foot, H. Lew, X. G. He and G. C. Joshi, Z. Phys. C 44, 441 (1989)

1989

[2] [1]

we obtainδ≃α (δ≃π + α), which also agrees with the numerical output shown in Fig. 2. In conclusion, all parameters in the mass matrices M𝓁 and Mν can be determined in terms of the charged- lepton and neutrino masses and mixing angles through Eqs. (51)-(58) and (62). B. Lepton universality and rare LFV decays In the 2HDM, Yukawa interactions may induce ﬂav...

[3] [2]

Also notice that, with the exception of texture 4 ν 17, any representative of Mν given in Table IV features a mass- less neutrino, since it contains a full column of zeros

which is consistent with data only at 3 σ CL and for a NO mass spectrum. Also notice that, with the exception of texture 4 ν 17, any representative of Mν given in Table IV features a mass- less neutrino, since it contains a full column of zeros. We emphasize that these maximally-restrictive texture pairs cannot be implemented in the SM by imposing Abelian...

[4] [3]

Since none of the textures belonging to the pairs (4 𝓁 3, 6ν 1,3,7,9) and (5𝓁 1, 5ν

[5] [4]

(B3) for the same rowsi and j

T exture pairs (4𝓁 3, 6ν 4,6) and (5𝓁 1, 5ν 4) From the application of the canonical method one can conclude that if a mass matrix textureT1 is characterized by two non-identical columns c1 and c2 with non-zero entries in some row i and zero entries in some row j, then it cannot be realized through Abelian symmetries in the 2HDM, together with a texture T...

[6] [5]

The associated chains and their corresponding building matrices are shown in Ta- bles VI and VII, for the charged-lepton and neutrino tex- tures, respectively

pairs (see Table V). The associated chains and their corresponding building matrices are shown in Ta- bles VI and VII, for the charged-lepton and neutrino tex- tures, respectively. Based on Table V, we generate all possible pairs of (4𝓁 3, 6ν 1,3,7,9) and (5𝓁 1, 5ν

[7] [6]

decompositions. Notice that one must consider both Yukawa matrix orderings, since swap- ping Yν 1 and Yν 2, while maintaining the charged-lepton 8 Charged leptons Texture decomposition Y𝓁 1 Y𝓁 2 4𝓁 3,I   0 0 × 0 × 0 × 0 0     0 0 0 0 0 × 0 × 0   5𝓁 1,I   0 0 × 0 × 0 × 0 0     0 0 0 0 0 0 0 0 ×   5𝓁 1,II   0 0 × 0 0 0 × 0 0     0 0 0 ...

[8] [7]

decomposition pairs in Table V, considering both orderings of Yν 1,2. We ﬁnd that DSNF can only take one of the following two forms: DSNF =   1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0   , g= 0, 2, (31)...