Soft breaking of the μ ↔ τ symmetry by 𝐒 4 ⊗ 𝐙 2

García-Aguilar, J. D.; Pozas Ramírez, A. E.; Suárez Castañeda, M. M.; Gómez-Izquierdo, J. C.; García-Aguilar, J. D.; Pozas Ramírez, A. E.; Suárez Castañeda, M. M.; Gómez-Izquierdo, J. C.

doi:10.31349/revmexfis.69.030802

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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.69 no.3 México may./jun. 2023 Epub 06-Sep-2024

https://doi.org/10.31349/revmexfis.69.030802

High Energy Physics

Soft breaking of the μ ↔ τ symmetry by 𝐒 ₄ ⊗ 𝐙 ₂

J. D. García-Aguilar^a

A. E. Pozas Ramírez^b

M. M. Suárez Castañeda^b

J. C. Gómez-Izquierdo^a

^{^a} Centro de Estudios Científicos y Tecnológicos No 16, Instituto Politécnico Nacional, Kilómetro 1.500, Actopan-Pachuca, Distrito de Educación, Salud, Ciencia, Tecnología e Innovación, San Agustín Tlaxiaca, Hidalgo; México. E-mail: jdgarcia@ipn.mx; cizquierdo@ipn.mx

^{^b} Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Av. Instituto Politécnico Nacional s/n Edificio 9 Unidad Profesional “Adolfo López Mateos” Col. San Pedro Zacatenco, Alcaldía Gustavo A. Madero, 07738, Ciudad de México; México.

Abstract

The μ ↔ τ symmetry has been ruled out by its predictions on the reactor and atmospheric angles, nevertheless, a breaking of this symmetry might provide correct values. For that reason, we build a non-renormalizable lepton model where the mixings arise from the spontaneous breaking of the 𝐒 ₄ ⊗ 𝐙 ₂ discrete group, subsequently the μ ↔ τ symmetry is broken in the effective neutrino mass matrix, that comes from the type II see-saw mechanism. As main result, the reactor and atmospheric angles are corrected and their values are in good agreement with the experimental data for the inverted hierarchy. Furthermore, we point out a link between the atmospheric angle and reactor one. In the quark sector, under certain assumptions, the generalized Fritzsch textures shape to the quark mass matrices so that the CKM matrix values are guaranteed.

Keywords: μ ↔ τ symmetry; CKM matrix

1. Introduction

The confirmed (g-2)μ anomaly by the Fermilab [¹] and the W boson mass new measurement [²], which are not consistent with the theoretical prediction, have shown again that the Standard Model (SM) is incomplete. Along with this, the flavor puzzle remains to be solved so that there is a need to enlarge the SM.

The neutrino oscillation opened the window to search physics beyond the SM, as it is well known, these established that neutrinos have mass so they mix. Although there are many mechanism [³] to get tiny neutrino masses, so far there is no a convincing theory that explains the origin of such mass and the peculiar pattern which is completely different from the quark sector. In the last years, several experiments have measured the neutrino mixing angles with great accuracy, also the masses seem to obey two orderings (normal and inverted hierarchy) due to the lacking of information on the absolute neutrino mass. Certainly, the normal ordering is preferred by the available data [⁴, ⁵] but the inverted hierarchy is not completely discarded [⁶]. Another important point is that, conforming to the experimental data, the PMNS mixing matrix exhibits large values in its entries, in addition, the second and third rows satisfy the relation |Uμi|=|Uτi| (i = 1,2,3) in good approximation for the normal and inverted hierarchy. The aforementioned facts might be understood by means of a symmetry in the effective neutrino mass matrix, then the concept of flavor symmetry turn out being crucial to explain the mixings, and a variety of discrete symmetries [⁷, ⁸, ⁸-¹⁴] have been applied to the lepton sector. In particular, the neutrino data seem to obey an approximated μ ↔ τ symmetry (for a complete review see [¹⁵]), that consists in the exchange label μ ↔ τ in the effective neutrino mass matrix when the charged lepton mass one is diagonal. Speaking of exact μ ↔ τ symmetry, which is is outdated currently due to its predictions, would imply to obtain 0∘ and 45∘ for the reactor and atmospheric angles, respectively. Besides that, the solar angle and the Dirac CP-violating phase keep as a free and unknown parameters. Despite this, from the model building point of view, the well studied μ ↔ τ symmetry has been a guide to construct lepton models [¹⁶-²⁴] and there is a possibility that a soft breaking [²¹-²³, ²⁵-²⁹] of this symmetry can accommodate the experimental results so that there is still strong motivation to study on the μ ↔ τ symmetry. Apart from this, elaborated flavored models have been proposed to face the lepton mixings and related issues as leptogenesis, dark matter, and so forth [¹⁴, ³⁰, ³¹].

On the other hand, in the quark sector, according to the available data [32] the CKM matrix is close to the identity one, this pattern might be explained by the notable hierarchy among the quark masses. In addition, this feature is exhibited by some matrices like the nearest neighbor interactions (NNI) [³³-³⁶] and the generalized Fritzsch [³⁷-³⁹] mass textures which can be obtained by means the flavor symmetries [⁷-¹⁰]. The contrasting behavior between the PMNS and CKM mixing matrices is undoubtedly a puzzling problem, so far one of the main task for model builders is to match simultaneously the fermion mixings by the same flavor symmetry in the suitable framework.

In order to address the masses and mixing problem, a phenomenological scalar extension of the SM is realized where the type II see-saw mechanism is responsible to obtain small neutrino masses and special emphasis is put on the lepton sector under a soft breaking of the μ ↔ τ symmetry scheme. To do so, we use the 𝐒₄ [⁴⁰-⁴⁴, ⁴⁶-⁴⁸] non-abelian discrete group to handle the Yukawa couplings, at the same time, this symmetry allows to treat the quark, lepton and scalar sector in different manner. Additionally, we include a 𝐙₂ symmetry, to have a non-renormalizable Yukawa mass term for neutrinos. On the other hand, the inclusion of three Higgs doublets are required to obtain the quark and charged lepton masses and mixings, this latter comes out being diagonal as result of the matter assignation under the flavor symmetry. Then, an enriched scalar (flavons) sector is included to provide desirable mass textures. In consequence, the mixings arises from the spontaneous breaking of the S4⊗Z2 discrete group and the μ ↔ τ symmetry is broken in the effective neutrino mass matrix. Eventually, the reactor and atmospheric angles come out being different of 0° and 45° respectively. CP parities phases in the neutrino masses play an important role to get sizable values for θ ₁₃ and the deviation of θ ₂₃ from maximality which turn out being consistent with neutrino data for the inverted hierarchy. In the quark sector, under certain assumptions, the generalized Fritzsch textures shape to the quark mass matrices so that the CKM matrix values are guaranteed.

It is worthy mentioned that a similar study was carried out [²⁴], nonetheless there are clear differences namely. The first one is scalar matter and the flavor symmetry, the second one is related with the mechanism to generate small neutrino masses and the corresponding predictions: in the aforemen- tioned paper, they got exact μ ↔ τ symmetry. Lastly, the NNI textures, in the quark mass matrices, appeared in a natural way so that they obtained correct values for the mixings. Although our model has some limitations like the flavon alignments and one benchmark (in the quark sector), the main purpose of this work was to show that a simple soft breaking of the μ ↔ τ symmetry is enough to correct the lepton mixing angles.

The layout of the paper is as follows. In Sec. 2, we describe the general framework to explore the 𝐒₄ discrete symmetry, the full assignation for the matter content is shown and the mass matrices and the corresponding mixing matrix are obtained. In addition, a brief analytical study is carried out to fix some free parameters in the model. Main results are presented in scattered plots where the set of free parameters values, that fit the mixing angles, are shown. All of this is included in Sec. 3. We give some conclusions in Sec. 4.

2. Flavored model

2.1. General framework

Although, there are fascinating theoretical frameworks that can be good candidates to replace the SM, conforming to our interest, a scalar extension of the SM will be considered. Thus, apart from the SM matter content a Higgs triplet (Δ) is required to generate tiny neutrino masses by means the type II see-saw mechanism. Furthermore, extra Higgs doublets and flavon gauge singlets (ϕ, φ and ξ) will be added to provide the CKM and PMNS matrices, respectively. In Table I, we can see the rest of the matter fields.

Table I Matter content.

Matter	QL=udL	d _R	u _R	L=vl	e _R	H=H+H0	ΔL=δ+2δ++δ0-δ+2
SU(3)_c	3	3	3	1	1	1	1
SU(2)_L	2	1	1	2	1	2	3
U(1)_Υ	1/3	-2/3	4/3	-1	-2	1	2

The relevant gauge invariant Lagrangian is given by

-L=ydQ-LHdR+yuQ-LH~uR+yeL-HeR+12yνL-(iσ2)ΔLc+V(H,Δ,ϕ,φ,ξ)+h.c. , (1)

with H~=iσ2H* and the scalar potential

V(H,Δ,ϕ,φ,ξ)=mH2H†H+12λHH†H2+mΔ2Tr(Δ†Δ)+12λΔTr(Δ†Δ)2+λHΔH†HTrΔ†Δ+λHΔ'HTΔ†H+mϕ2|ϕ|2+12λϕ|ϕ|4+λHϕH†H|ϕ|2+λΔϕTrΔ†Δ|ϕ|2+mφ2|φ|2+12λφ|φ|4+λHφH†H|φ|2+λΔφTrΔ†Δ|φ|2+mξ2|ξ|2+12λξ|ξ|4+λHξH†H|ξ|2+λΔξTrΔ†Δ|ξ|2+λφϕ|φ|2|ϕ|2+λξϕ|ξ|2|ϕ|2+λξφ|ξ|2|φ|2. (2)

In flavored models, the scalar potential turns out being important to get a viable model, in here, a detailed study on the scalar potential is not the purpose of this paper however we add a comment about it. The 𝐒₄ discrete symmetry [⁷-¹⁰] was selected to control the flavor mixings since it has singlet, doublet and triplet irreducible representations (see the Appendix A for more details), this feature represents an advantage for us because the quark and Higgs sector will be assigned in doublets and singlets whereas the lepton sector in triplets. The main achievement to do that is to get desirable mass textures in both sectors.

Along with this, we wish to highlight the scalar potential, where the three Higgs doublets are only involved, has been study exhaustively [⁴⁹-⁵¹]. In the aforementioned paper, three Higgs doublets were assigned under the 𝐒₃ group as follows: the first and second family were put in a 𝟐 whereas the third one in 𝟏₁. In these circumstances the scalar potential was minimized and the 〈H2〉=〈H1〉 alignment is allowed by the flavor symmetry. Having commented that, we go back to our work where the 𝐒₄ flavor symmetry drives the Yukawa couplings as well as the scalar potential. It is worthy mentioned that the non-abelian groups 𝐒₄ and 𝐒₃ are completely different from each other (𝐒₃ is a subgroup of the 𝐒₄), however, the 𝐒₄ scalar potential with three Higgs doublets can be mimicked from the previous study [⁴⁹-⁵¹]. This asseveration is supported due to the 𝐒₄ representation can be decomposed in the 𝐒₃ ones [⁷]. To be more explicit, as we can see in the Appendix B, the irreducible representations 𝟏_𝟏, 𝟏_𝟐 and 𝟐 of both groups coincide so that the tensor product respects the same rules among them as can be verified. Then, in this sense, similar results are expected for the Higgs alignments because we are using the same assignation for the three Higgs families under the 𝐒₄, as one can see in Table II. On the other hand, a complete analysis of the scalar potential is beyond the scope of this work so that the flavor alignments will be considered as a matter of fact.

Table II Assignment under 𝐒₄ flavor group. Here, I = 1, 2 and i = 1, 2, 3.

Matter	Q _IL	Q _3L	d _IR	d _3R	u _IR	u _3R	L _i	e _iR	H _I	H ₃	𝚫	𝛟	𝛗 _I	𝛏 _i
𝐒₄	2	1₁	2	1₁	2	1₁	3₁	3₁	2	1₁	1₁	1₁	2	3₁
𝐙₂	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1

Further to our previous comments, the full symmetry breaks down as follows: SU(3)C⊗SU(2)L⊗U(1)Y⊗S4⊗Z2→SU(3)C⊗SU(2)L⊗U(1)Y→SU(3)C⊗U(1)Q, where the Λ scale of the spontaneous breaking of the S4⊗Z2 group is larger than the 𝑣 = 246 GeV electroweak one.

2.2. The model

Having commented briefly the theoretical framework, we put now attention to the matter field assignation under the 𝐒₄ flavor symmetry. Hence, those are assigned as follows: the first and second family of quark and Higgs are put in 𝟐 doublet; the third family is assigned to the 𝟏₁ singlet. This choice has been exploited in many 𝐒₃ models with three Higgs doublets (see for instance [⁵²]) and interesting mass textures can be obtained, for this reason, the same assignation is used in our work. On the other hand, the lepton sector is treated in different way since the three families of L(e _R)left-handed (right-handed) doublets (singlets) are put in the 𝟑₁ triplet irreducible representations. This allows to obtain a diagonal charged lepton mass matrix so that the mixings will arise from the neutrino sector where an enriched scalar one is needed as can be seen in Table II. Let us add a comment on the role 𝐙₂ symmetry, the main purpose is to prohibit the renormalizable neutrino mass term, L-(iσ2)ΔLC.

Consequently, the most relevant terms which are flavor and gauge invariant are written as^ⁱ

L=y1dQ-1LH1d2R+H2d1R+Q-2LH1d1R-H2d2R+y2dQ-1LH3d1R+Q-2LH3d2R+y3dQ-1LH1+Q-2LH2d3R+y4dQ-3LH1d1R+H2d2R+y5dQ-3LH3d3R+y1uQ-1LH~1u2R+H~2u1R+Q-2LH~1u1R-H~2u2R+y2uQ-1LH~3u1R+Q-2LH~3u2R+y3uQ-1LH~1+Q-2LH~2u3R+y4uQ-3LH~1u1R+H~2u2R+y5uQ-3LH~3u3R+y1eL-1H3e1R+L-2H3e2R+L-3H3e3R+y2eL-1H2e1R-12L-23H1+H2e2R+12L-33H1-H2e3R+y1NL-1(iσ2)ΔϕL1C+L-2(iσ2)ΔϕL2C+L-3(iσ2)ΔϕL3C1Λ+y2NL-1(iσ2)Δφ2L1C-12L-2(iσ2)Δ3φ1+φ2L2C+12L-3(iσ2)Δ3φ1-φ2L3C1Λ+y3NL-1(iσ2)Δξ2L3C+ξ3L2C+L-2(iσ2)Δξ1L3C+ξ3L1C+L-3(iσ2)Δξ1L2C+ξ2L1C1Λ+h.c. (3)

Once the scalar fields get their vev’s, the fermion masses are written as

Mq=y2q〈H3〉+y1q〈H2〉y1q〈H1〉y3q〈H1〉y1q〈H1〉y2q〈H3〉-y1q〈H2〉y3q〈H2〉y4q〈H1〉y4q〈H2〉y5q〈H3〉,Me=y1e〈H3〉+y2e〈H2〉000y1e〈H3〉-12y2e3〈H1〉+〈H2〉000y1e〈H3〉+12y2e3〈H1〉-〈H2〉,Mν=y1N〈ϕ〉+y2N〈φ2〉y3N〈ξ3〉y3N〈ξ2〉y3N〈ξ3〉y1N〈ϕ〉-12y2N3〈φ1〉+〈φ2〉y3N〈ξ1〉y3N〈ξ2〉y3N〈ξ1〉y1N〈ϕ〉+12y2N3〈φ1〉-〈φ2〉〈Δ〉Λ. (4)

Evidently, there are too many free parameters in the fermion mass matrices however some ones can be reduce notably by making an alignment in the vev’s of the scalar fields. In particular, 〈H1〉=〈H2〉 will be assumed in the Higgs sector as we already commented. Also, Higgs vev’s have to satisfy the relation 〈H1〉2+〈H2〉2+〈H3〉2=v=246 GeV. For the flavon sector, the alignment that provides a good phenomenology in the neutrino mass matrix is given by

〈ξ〉=vξ1,vξ,vξ, 〈ϕ〉=vϕ1,1,1, 〈φ〉=vφ1,0. (5)

As it is usual, each vev’s of the flavons are set to be proportional to λΛ where λ (0.225) is the Wolfenstein parameter and the cutoff scale of the model.

2.3. Fermion masses and mixings

2.3.1. Lepton sector

As was already commented, we put special emphasis on the lepton sector. To start with, let us focus in the charged lepton sector which is diagonal and the physical masses can be obtained straightforwardly. Nonetheless, a particular alignment was assumed, this is, 〈H1〉=〈H2〉 [⁴⁹-⁵¹] and the principal motivation has to do with the quark sector where outstanding mass textures appear.

As consequence of the mentioned choice in the Higgs sector, the y2e Yukawa coupling has to be negative and an extra rotations in the fields are necessary to obtain M^e=Diag.me,mμ,mτ=UeL†MeUeR with Ue(L,R)=S23ue(L,R), therefore M^e=ueL†meueR with

me=y1e〈H3〉+y2e〈H2〉000y1e〈H3〉+12y2e3-1〈H2〉000y1e〈H3〉-12y2e3+1〈H2〉,S23=100001010. (6)

From Eq. (6), one can identify the charged lepton masses

me=|y1e〈H3〉+y2e〈H2〉|, mμ=|y1e〈H3〉+12y2e3-1〈H2〉|, mτ=|y1e〈H3〉-12y2e3+1〈H2〉|. (7)

We stress that there are few parameters to adjust the three charged lepton masses and this can be a weak point. This can be solved by including extra flavons however we want to keep the model simple so that this will not be carried out.

In the neutrino sector, on the other hand, due to phenomenological implications in the mass matrix we assume the alignments given in Eq. (5). Along with this, in the standard basis, ℳ_v is diagonalized by the 𝐔_v matrix such that M^ν=Diag.m1,m2,m3=Uν†MνUν* with Uν=S23uν, then M^ν=uν†mνuν* where 𝐒₂₃ has been shown before and 𝐦_v is given by

mν=y1Nvϕy3Nvξy3Nvξy3Nvξy1Nvϕ+32y2Nvφy3Nvξ1y3Nvξy3Nvξ1y1Nvϕ-32y2Nvφ〈Δ〉Λ=meemeμmeμmeμmμμmμτmeμmμτmττ. (8)

Due to the charged lepton mass matrix is diagonal, one can identify clearly the physical masses, see Eq. (7). Therefore, in the effective mass matrix, 𝐦_v , the μ↔τ symmetry is broken because of the difference mμμ≠mττ as one can notice. As result of this, the reactor and atmospheric angles will be deviated from 0° and 45°, respectively. As it is usual, in the context of μ↔τ, the solar angle is a free parameter which can be fixed to the current experimental values but this will get correction since that mμμ≠mττ.

In order to diagonalize the neutrino mass matrix, a perturbative analysis will be done in such a way that the matrix can be written as

mν=meemeμmeμmeμmμμmμτmeμmμτmμμ⏞mν0+00000000mμμϵ⏞mνϵ, (9)

where the former matrix possesses exact μ↔τ symmetry and it is broken in the latter one where the dimensionless parameter ϵ≡mττ-mμμ/mμμ has been defined and this quantify the breaking. As we observe, this can be written as ϵ∼y2N/(y1N-y2N) (vev’s of the flavons are proportional to λΛ). So that, if y2N was zero, the μ↔τ symmetry would be exact, then we assume that 𝜖 is small such that this parameter will be treated as a perturbation, thus, a pertubative study at first order in 𝜖 will be carried out. Therefore, we demand that |ϵ|≤0.3 as consequence quadratic (|ϵ|2) terms will be neglected.

As a result of having a diagonal charged lepton mass matrix, there is no contribution to the mixings, then the neutrino sector will provide it. To see this, we go back to the 𝐦_v mass matrix where mν0 is diagonalized by the following mixing matrix^ⁱⁱ

Uν0=cosθsinθ0-sinθ2cosθ2-12-sinθ2cosθ212, (10)

Hereafter, the superscripted in Uν0 and the matrix elements mαβ0 (α, β = e, μ, τ), denotes quantities when the μ ↔ τ symmetry is exact.

Going back to the expression M^ν=uν†mνuν*, then uν≈Uν0Uνϵ which implies

M^ν=Uνϵ†Uν0†mν0Uν0*+Uν0†mνϵUν0*Uνϵ*, with Uν0†mν0Uν0*=Diag.m10,m20,m30. (11)

In addition, we have

Uν0†mνϵUν0*=ϵ mμμ02sin2θ-sin2θ2-sinθ-sin2θ2cos2θcosθ-sinθcosθ1, mμμ0=12m10sin2θ+m20cos2θ-m30. (12)

As we already commented, the parameter 𝜖 is considered as a perturbation so that the mixing matrix Uνϵ is obtained by using perturbation theory^ⁱⁱⁱ at first order in |ϵ|. Consequently, we obtain

Uνϵ≈N1-mμμ0m20-m10sin2θ4ϵN2mμμ0m10-m30sinθ2ϵN3mμμ0m20-m10sin2θ4ϵN1N2-mμμ0m20-m30cosθ2ϵN3-mμμ0m10-m30sinθ2ϵN1mμμ0m20-m30cosθ2ϵN2N3, (13)

where the normalization factors are written as

N1=1+|ϵ|2sin2θ4mμμ0m10-m302+cos2θmμμ0m20-m102-1/2,N2=1+|ϵ|2cos2θ4mμμ0m20-m302+sin2θmμμ0m20-m102-1/2,N3=1+|ϵ|24sin2θmμμ0m10-m302+cos2θmμμ0m20-m302-1/2. (14)

At last, the theoretical formulas for the mixing angles are obtained by comparing our PMNS mixing matrix, U≈Ue†Uν=Uν0Uνϵ, with the standard parametrization, then we finally get

sinθ13=U13=N34mμμ0m20-m10ϵm20-m30m10-m30sin2θ,sinθ12=|U12|1-sin2θ13=N2sinθ1-ϵ2mμμ0m20-m10cos2θ1-sin2θ13,sinθ23=U231-sin2θ13=N321+ϵ2mμμ0m20sin2θ+m10cos2θ-m30m20-m30m10-m301-sin2θ13. (15)

As one can realize if ϵ goes to zero, one would obtain the well known predictions: θ13=0, θ12=θ and θ ₂₃= π/4.

In order to figure out the set of free parameter values, an analytical study on the theoretical formulas is carried out. It is important to note that the reactor angle depends strongly on the breaking parameter and the ratio among complex masses, mi0=|mi0|eiαi. In the former factor, the associated phase ϵ=|ϵ|eiαϵ is irrelevant however the difference m20-m10 and m10-m30 are crucial to enhance the reactor angle value, then CP parities values turn out being relevant to accommodate the reactor angle. As result of this, we choose the following CP parities values m20=-m20, m10=m10 and m30=|m30|. To add to it, the solar and atmospheric angles are sensitive to the associated phase α _𝜖 and the CP parities values of the neutrino masses.

In the current analysis, the normal hierarchy is not favored as one can check straightforward, then we just focus in the inverted ordering. Due to the CP parities in the neutrino masses, we obtain

sinθ13=U13=N38m20+m10cos2θ-m10-m30m20+m10m20+m30m10-m30ϵsin2θ,sinθ12=U121-sin2θ13=N2sinθ1-ϵ4cos2θ-m10-m30m20+m10cos2θ1-sin2θ13,sinθ23=U231-sin2θ13=N321-ϵ4m20+m102cos2θsin2θ-m20+m10m10-m30+m10-m302m20+m30m10-m301-sin2θ13. (16)

Let us consider two extreme cases where the lightest neutrino mass takes part. According to the squared mass scales Δm212=|m20|2-|m10|2 and Δm132=|m10|2-|m30|2, two neutrino masses might write as |m20|=|m10|2+Δm212 and |m10|=|m30|2+Δm132.

Strict inverted hierarchy (|m30|=0): In this case, we have |m10|=Δm132 and

|m20|≈|m10|1+12Δm212|m10|2. (17)

Then, one can obtain a precise values for the mixing angles

sinθ13≈N34|ϵ|sin2θ2(1+rA)cos2θ-11-rA=218|ϵ|1-12rA,sinθ12≈N2sinθ1-ϵ4cos2θ-121-rAcos2θ1-sin2θ13=131-ϵ361-sin2θ13,sinθ23≈N321-ϵ44sin2θcos2θ-11-sin2θ13=121+ϵ361-sin2θ13, (18)

where rA=Δm212/2Δm132.

In the above expressions, we have considered sinθ=1/3 which is a good approximation to the tribimaximal scenario [⁵³-⁵⁶], |ϵ|=0.3 and αϵ=0 we obtain sinθ13≈0.0234, sinθ12≈0.587 and sinθ23≈0.713. In the case where αϵ=π, the solar and atmospheric angles have similar values in comparison to above case.

Almost degenerate |m30|≫Δm132: In this limit, we obtain the following masses

|m10|≈|m30|1+rB, |m20|≈|m30|1+rB+rC, (19)

with rB≈Δm132/2|m30|2 and rC≈Δm212/2|m30|2.

For this reason, the mixing angles formulas are written as

sin θ13≈N32sin θ cos3θ|ϵ|rB=29|ϵ|rB,sin θ12≈N2 sin θ1-ϵ4cos4 θ1-sin2 θ13=131-ϵ91-sin2θ13,sin θ23≈N321-ϵ2rBsin2 θcos2θ1-sin2 θ13=121-ϵ9rB1-sin2θ13. (20)

Remarkably, in this scheme the three angles can be accommodated with great accuracy according to the experimental data as we will see later.

Before finishing this section, it is worthy of mentioning the relation among the reactor angle and the deviation of the solar and atmospheric angles, respectively. To do so, in the strict hierarchy case we have

ϵ≈182 sin θ13, (21)

then

sinθ12≈131±sinθ1381-sin2θ13, sinθ23≈121±sinθ1381-sin2θ13. (22)

In the almost degenerate case, one can write

ϵ≈92rB sin θ13, (23)

subsequently

sinθ12≈13|1±rB2sinθ13|1-sin2θ13, sinθ23≈12|1±sinθ132|1-sin2θ13, (24)

where the ± represents the π and 0 values for the αϵ phase.

2.3.2. Quark sector

As we already commented, the lepton sector was studied mainly in this paper. Then, we want to address briefly the quark sector within a particular benchmark as follows. We adopted the following alignments 〈H1〉=〈H2〉 which is consistent with the minimization of the scalar potential [⁴⁹-⁵¹]. Hence, one gets

Mq=BqbqCqbqAqCqDqDqEq, (25)

where q=u,d and the defined parameters can be read of Eq. (4). Let us remark that ℳ_q has bee studied exhaustively in [⁵²] and significant results were released. Nonetheless, we want to address the quark mass matrices in different way so that some assumption will be done. To do so, notice that ℳ_q is diagonalized^iv by Uq(L,R) such that Mq=UqL†MqUqR with Mq=Diag.mq1,mq2,mq3 denoting the quark physical masses. Then, the following rotations is realized Uq(L,R)=Uπ/4uq(L,R) so that Mq=uqL†mquqR. Notice that

mq=Aqbq0bqBq2Cq02DqEq, Uπ/4=12120-12120000. (26)

At this stage, two assumptions are made A _q = 0 and C _q = D _q . To be honest, we could not eliminate the former entry by means the S4⊗Z2 discrete symmetry and the latter assumption might be realized within the left-right theory[⁵⁷-⁶⁰] by invoking parity symmetry. Also, as was shown in [³⁷], the second assumption can be realized by a suitable transformation in the right-handed quarks fields (there are no right-handed currents in the model), which are SU(2)L singlets, such that the resultant quark mass matrix turns out being hermitian. Due to this fact, we could have assumed that ℳ_q is hermitian but only the aforementioned simplification was carried out. In this benchmark the quark mass matrix has the generalized Fritzsch textures [37-39] which fit with great accuracy the CKM mixing matrix.

As a result, the CKM mixing matrix is given by V=Uu†Ud=OuTP-qOd where P-q=Pu†Pd and the 𝐎_q orthogonal matrix has the following form

Oq=mq3|mq2||Eq|-mq1Rq1-mq1mq3|Eq|+|mq2|Rq2mq1|mq2|mq3-|Eq|Rq3mq1|Eq|-mq1|Eq|Rq1|mq2||Eq|+|mq2||Eq|Rq2mq3mq3-|Eq||Eq|Rq3-mq1|Eq|+|mq2|mq3-|Eq|Rq1-|mq2||Eq|-mq1mq3-|Eq|Rq2mq3|Eq|-mq1|Eq|+|mq2|Rq3 (27)

where q=u,d. In addition,

Rq1=mq3-mq1|mq2|+mq1|Eq|,Rq2=mq3+|mq2||mq2|+mq1|Eq|,Rq3=mq3+|mq2|mq3-mq1|Eq|. (28)

As we can show in the Appendix B, in the CKM matrix there are four parameters namely |Eq| (q = u, d) and two effective CP-violating phases (α and β) so that a numerical study will be realized to fix them.

3. Results

3.1. Lepton sector

We have shown that our theoretical formulas on the mixing angles can accommodate the experimental data where the inverted hierarchy is favored. In order to get a full set of free parameter values that fit the mixing angles, then some scattered plot will be elaborated as follows.

The mixing angles depend on three free parameters, explicitly

sinθ13=sinθ13ϵ,θ,|m30|,sinθ12=sinθ13ϵ,θ,|m30|,sinθ23=sinθ13ϵ,θ,|m30|. (29)

Hence, from the previous analytical study the three free parameters let vary on the following ranges: ϵ∈-0.3,0, θ∈0,π/3 and |m30|∈0,0.09 eV. Therefore, we demand our theoretical formulas satisfy (at 3σ) the following values [⁴]

sin2θ12/10-1∈2.71-3.69,sin2θ13/10-2∈2.018-2.424,sin2θ23/10-1∈4.33-6.08, (30)

for the inverted hierarchy. Additionally,

Δm21210-5 eV2∈6.94-8.14,Δm13210-3 eV2∈2.37-2.53. (31)

Having included the experimental data, the scattered plots are constructed by using the theoretical formulas given in Eq. (16) which have to satisfy the experimental values. As a result, the mixing angles as function of the lightest neutrino mass are displayed in Fig. 1. The allowed region of values for the |m30| is consistent with the previous analytical study.

Figure 1 From left to right: the reactor, solar and atmospheric angles versus the |m30| lightest neutrino mass. The thick line stands for 3σ of C. L.

As it was already commented, the θ parameter is identified with the solar angle in the limit of μ↔τ exact. Then, the following scattered plots exhibit the region where θ parameter lies around the experimental value of the solar angle. In fact, this value is close to tribimaximal prediction since the solar angle receives a small correction from 𝜖.

In the previous analytical study, we showed the 𝜖 parameter must be negative and this may vary in the interval 0,-0.3. The numerical study shows the favored region where the mixing angles are fitted at 3σ, see Fig. 3. Evidently, the case with |ϵ|=0 is excluded due to this stands for the limit of exact μ↔τ symmetry.

Figure 2 From left to right: the reactor, solar and atmospheric angles versus the θ parameter. The thick line stands for 3σ of C. L.

Figure 3 From left to right: the reactor, solar and atmospheric angles versus the |ϵ| parameter. The thick line stands for 3σ of C. L.

As model prediction, we have calculated numerical the effective Majorana mass of electron neutrino which is defined as follows

|mee|=∑i=13‍miUei2, (32)

with m _i represents the physical neutrino mass and Uei PMNS matrix elements. This effective mass has been measured by GERDA phase I[⁶¹] and II [⁶²], and the lowest upper bound is |mee|<0.22 eV.

In our model, CP parities have been used in the neutrino masses. In particular, we utilized m20=-m20, m10=|m10| and m30=|m30| since this fit quite well the mixing angles. Consequently, the predicted region for the effective Majorana mass of electron neutrino is shown in the following scattered plots.

3.2. Quark sector

Our numerical study consists in making scattering plots. To do so, we compare our CKM theoretical expression with the standard parametrization one. In particular, we consider the entries Vuith (i=d,s,b) and Vcbth that depend on the free parameters

|Vuith|=|Vuith|Eu|,|Ed|,α,β|,|Vcbth|=|Vuith|Eu|,|Ed|,α,β|. (33)

Therefore, we demand the magnitude of mentioned entries must satisfy the following experimental values [³²]

|Vudex|=0.97401±0.0001,|Vusex|=0.22650±0.0004,|Vubex|=0.00361-0.00009+0.00011,|Vcbex|=0.04053-0.00061+0.00083. (34)

In the current study, the physical quark masses are considered as input values. To be more precise, the normalized quark masses (mqi/mq3) will be used due to their ratios do not change drastically at different energy scales as one can verify directly from [⁶³]. So that, at the top quark mass scale we have [²²]

m~u=(1.33±0.73)×10-5,m~c=(3.91±0.42)×10-3,m~d=(1.49±0.39)×10-3,m~s=(2.19±0.53)×10-2. (35)

In addition, for simplicity, two dimensionless parameters have been defined yq≡|Eq|/mq3 (q = u, d), then we now have the constraint 1>yq>m~q2≡|mq2|/mq3>m~q1. Explicitly, for the up and down sector 1>yu>m~c≡|mc|/mt>m~u and 1>yd>m~s≡|ms|/mb>m~d.

With all the above information, we calculate the allowed regions for the four CKM entries and constrain the free parameter set of values. However, let us show you only the scattered plots for |Vub| and |Vcb| since that these entries usually are complicated to fit. As we already commented the theoretical expression are required to satisfy the experimental data up to 3σ. Moreover, the normalized quark masses let vary up to 2σ and the two CP-violating phases are in the range 0,2π. Then, as one notices, in Fig. 5, there is a set of values in which |Vub| is fitted with great accuracy.

Figure 4 The effective Majorana mass of neutrino electron versus the fitted parameters.

Figure 5 |V _ub | as function of the four free parameters. The thick line stands for 3σ of C. L.

Focusing in the dimensionless parameters yu and yd, the favorable regions lie in [0.5,1) approximately. Additionally, there are two regions of values for the CP phases, α and β, where the magnitude of |Vub| is accommodated. In the Fig. 6, we see |Vcb| as function of the four free parameters and these have the same allowed region as the above case.

Figure 6 |V _cb | as function of the four free parameters. The thick line stands for 3σ of C. L.

To finish this section, we want to comment our naive analysis showed a large region of values for the free parameters where the theoretical CKM entries are in good agreement with the experimental data up to 3σ. A strict study, as for example an χ ² fit, must determine better the space of values however the principal aim of this numerical study was shown the generalized Fritzsch mass textures fit the CKM matrix as it is well known.

4. Summary and conclusions

To sum up, we have built a non-renormalizable model where the fermion mixing is driven by the spontaneous breaking of the S4⊗Z2 discrete group. An appropriated alignment of the scalar vev’s allows to break the μ↔τ symmetry in the effective neutrino mass matrix. Therefore, under a perturbative study, we were able to correct the wrong predictions on the reactor and atmospheric angles, and a set of values for the free parameters was found such that the mixing angles are consistent with the latest neutrino data. Due to the lack of extra symmetries, in the quark sector, a benchmark allows to get consistent mass textures that accommodate the CKM mixing matrix.

We have learned that the flavor symmetries have been useful to eliminate spurious parameters in the Yukawa sector. At the same time, those shape the fermion mass matrices, consequently the mixing pattern can be obtained straightforwardly. Ambitious flavored models have gone beyond of fitting the mixings and prediction on some free parameters (Majorana phases, Dirac CP phase for instance) have been done. In conclusion, despite the μ↔τ is outdated, in this constrained model, we wanted to show you that a simple soft breaking is enough to correct the mixing angles. Although the model predictions are so limited and the favored inverted hierarchy goes against the data, a soft breaking of μ↔τ is still alive from theoretical point of view nevertheless the experiments have the verdict.

Acknowledgements

This work was partially supported by Secretaria de Investigación y Posgrado del Instituto Politécnico Nacional under Projects 20221285 and 20220411 and the program PAPIIT IN109321.

References

1. B. Abi, et al. [Muon g-2], Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm, Phys. Rev. Lett. 126 (2021) 141801, https://doi.org/10.1103/PhysRevLett.126.141801. [arXiv:2104.03281 [hep-ex]]. [ Links ]

2. T. Aaltonen, et al. [CDF], High-precision measurement of the W boson mass with the CDF II detector, Science 376 (2022) 170, https://doi.org/10.1126/science.abk1781. [ Links ]

3. J. C. Romao, Supersymmetric Models for Neutrino Mass, 2007 [arXiv:0710.5730 [hep-ph]]. [ Links ]

4. P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C. A. Ternes, M. Tórtola and J. W. F. Valle, 2020 global reassessment of the neutrino oscillation picture, JHEP 02 (2021) 071, https://doi.org/10.1007/JHEP02(2021)071. [arXiv:2006.11237 [hep-ph]]. [ Links ]

5. I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, The fate of hints: updated global analysis of three-flavor neutrino oscillations, JHEP 09 (2020) 178, https://doi.org/10.1007/JHEP09(2020)178. [arXiv:2007.14792 [hep-ph]]. [ Links ]

6. S. Gariazzo, M. Gerbino, T. Brinckmann, M. Lattanzi, O. Mena, T. Schwetz, S. Choudhury Roy, K. Freese, S. Hannestad and C. A. Ternes, et al. Neutrino mass and mass ordering: no conclusive evidence for normal ordering, JCAP 10 (2022) 010, https://doi.org/10.1088/1475-7516/2022/10/010. [arXiv:2205.02195 [hep-ph]]. [ Links ]

7. H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, Non-Abelian Discrete Symmetries in Particle Physics, Prog. Theor. Phys. Suppl. 183 (2010) 1, https://doi.org/10.1143/PTPS.183.1. [arXiv:1003.3552 [hep-th]]. [ Links ]

8. W. Grimus and P. O. Ludl, Finite flavour groups of fermions, J. Phys. A 45 (2012) 233001, https://doi.org/10.1088/1751-8113/45/23/233001. [arXiv:1110.6376 [hep-ph]]. [ Links ]

9. H. Ishimori , T. Kobayashi , H. Ohki, H. Okada , Y. Shimizu and M. Tanimoto, An introduction to non-Abelian discrete symmetries for particle physicists, Lect. Notes Phys. 858 (2012) 1, https://doi.org/10.1007/978-3-642-30805-5. [ Links ]

10. S. F. King and C. Luhn, Neutrino Mass and Mixing with Discrete Symmetry, Rept. Prog. Phys. 76 (2013) 056201, https://doi.org/10.1088/0034-4885/76/5/056201. [arXiv:1301.1340 [hep-ph]]. [ Links ]

11. S. F. King, Models of Neutrino Mass, Mixing and CP Violation, J. Phys. G 42 (2015) 123001, https://doi.org/10.1088/0954-3899/42/12/123001. [arXiv:1510.02091 [hep-ph]]. [ Links ]

12. F. Feruglio and A. Romanino, Lepton flavor symmetries, Rev. Mod. Phys. 93 (2021) 015007, https://doi.org/10.1103/RevModPhys.93.015007. [arXiv:1912.06028 [hep-ph]]. [ Links ]

13. Z. z. Xing, Flavor structures of charged fermions and massive neutrinos, Phys. Rept. 854 (2020) 1-147, https://doi.org/10.1016/j.physrep.2020.02.001. [arXiv:1909.09610 [hep-ph]]. [ Links ]

14. G. Chauhan, et al., Discrete Flavor Symmetries and Lepton Masses and Mixings, 2022 [arXiv:2203.08105 [hep-ph]]. [ Links ]

15. Z. z. Xing and Z. h. Zhao, A review of μ-τ flavor symmetry in neutrino physics, Rept. Prog. Phys. 79 (2016) 076201, https://doi.org/10.1088/0034-4885/79/7/076201. [arXiv:1512.04207 [hep-ph]]. [ Links ]

16. W. Grimus and L. Lavoura, A Discrete symmetry group for maximal atmospheric neutrino mixing, Phys. Lett. B 572 (2003) 189, https://doi.org/10.1016/j.physletb.2003.08.032. [arXiv:hep-ph/0305046 [hep-ph]]. [ Links ]

17. W. Grimus, A. S. Joshipura, S. Kaneko, L. Lavoura and M. Tanimoto , Lepton mixing angle θ ₁₃ = 0 with a horizontal symmetry D ₄, JHEP 07 (2004) 078, https://doi.org/10.1088/1126-6708/2004/07/078. [arXiv:hep-ph/0407112 [hep-ph]]. [ Links ]

18. A. Adulpravitchai, A. Blum and C. Hagedorn, A Supersymmetric D4 Model for mu-tau Symmetry, JHEP 03 (2009) 046, https://doi.org/10.1088/1126-6708/2009/03/046. [arXiv:0812.3799 [hep-ph]]. [ Links ]

19. H. Ishimori, T. Kobayashi, H. Ohki, Y. Omura, R. Takahashi and M. Tanimoto, D(4) Flavor Symmetry for Neutrino Masses and Mixing, Phys. Lett. B 662 (2008) 178, https://doi.org/10.1016/j.physletb.2008.03.007. [arXiv:0802.2310 [hep-ph]]. [ Links ]

20. C. Hagedorn and R. Ziegler, μ - τ Symmetry and Charged Lepton Mass Hierarchy in a Supersymmetric D ₄ Model, Phys. Rev. D 82 (2010) 053011, https://doi.org/10.1103/PhysRevD.82.053011. [arXiv:1007.1888 [hep-ph]]. [ Links ]

21. J. C. Gómez-Izquierdo, Non-minimal flavored S3⊗Z2 left-right symmetric model, Eur. Phys. J. C 77 (2017) 551, https://doi.org/10.1140/epjc/s10052-017-5094-0. [arXiv:1701.01747 [hep-ph]]. [ Links ]

22. E. A. Garcés, J. C. Gómez-Izquierdo and F. Gonzalez-Canales, Flavored non-minimal left-right symmetric model fermion masses and mixings, Eur. Phys. J. C 78 (2018) 812, https://doi.org/10.1140/epjc/s10052-018-6271-5. [arXiv:1807.02727 [hep-ph]]. [ Links ]

23. J. C. Gómez-Izquierdo and M. Mondragón, B-L Model with S3 symmetry: Nearest Neighbor Interaction Textures and Broken μ↔τ Symmetry, Eur. Phys. J. C 79 (2019) 285, https://doi.org/10.1140/epjc/s10052-019-6785-5. [arXiv:1804.08746 [hep-ph]]. [ Links ]

24. S. Morisi and E. Peinado, An S4 model for quarks and leptons with maximal atmospheric angle, Phys. Rev. D 81 (2010) 085015, https://doi.org/10.1103/PhysRevD.81.085015. [arXiv:1001.2265 [hep-ph]]. [ Links ]

25. S. Morisi, K. M. Patel and E. Peinado , Model for T2K indication with maximal atmospheric angle and tri-maximal solar angle, Phys. Rev. D 84 (2011) 053002, https://doi.org/10.1103/PhysRevD.84.053002. [arXiv:1107.0696 [hep-ph]]. [ Links ]

26. S. Gupta, A. S. Joshipura and K. M. Patel, Minimal extension of tri-bimaximal mixing and generalized Z ₂ → Z₂ symmetries, Phys. Rev. D 85 (2012) 031903, https://doi.org/10.1103/PhysRevD.85.031903. [arXiv:1112.6113 [hep-ph]]. [ Links ]

27. S. Gupta, A. S. Joshipura and K. M. Patel, How good is μ-τ symmetry after results on non-zero θ ₁₃?, JHEP 09 (2013) 035, https://doi.org/10.1007/JHEP09(2013)035. [arXiv:1301.7130 [hep-ph]]. [ Links ]

28. Z. h. Zhao, On the breaking of mu-tau flavor symmetry, 2017 https://doi.org/10.1142/9789813145504_0017. [arXiv:1605.04498 [hep-ph]]. [ Links ]

29. E. Becerra-García and A. Pérez-Lorenzana, Are neutrino oscillation mixings linked to the smallness of solar neutrino scale?, 2022 [arXiv:2202.00864 [hep-ph]]. [ Links ]

30. Z. z. Xing and Z. h. Zhao, The minimal seesaw and leptogenesis models, Rept. Prog. Phys. 84 (2021) 066201, https://doi.org/10.1088/1361-6633/abf086. [arXiv:2008.12090 [hep-ph]]. [ Links ]

31. L. Calibbi, M. L. López-Ibáñez, A. Melis and O. Vives, Implications of the Muon g-2 result on the flavour structure of the lepton mass matrix, Eur. Phys. J. C 81 (2021) 929, https://doi.org/10.1140/epjc/s10052-021-09741-1. [arXiv:2104.03296 [hep-ph]]. [ Links ]

32. R. L. Workman, et al. [Particle Data Group], Review of Particle Physics, PTEP 2022 (2022) 083C01, https://doi.org/10.1093/ptep/ptac097. [ Links ]

33. G. C. Branco, L. Lavoura and F. Mota, Nearest Neighbor Interactions and the Physical Content of Fritzsch Mass Matrices, Phys. Rev. D 39 (1989) 3443, https://doi.org/10.1103/PhysRevD.39.3443. [ Links ]

34. G. C. Branco and J. I. Silva-Marcos, NonHermitian Yukawa couplings?, Phys. Lett. B 331 (1994) 390, https://doi.org/10.1016/0370-2693(94)91069-3. [ Links ]

35. K. Harayama and N. Okamura, Exact parametrization of the mass matrices and the KM matrix, Phys. Lett. B 387 (1996) 614, https://doi.org/10.1016/0370-2693(96)01079-9. [arXiv:hep-ph/9605215 [hep-ph]]. [ Links ]

36. K. Harayama, N. Okamura, A. I. Sanda and Z. Z. Xing, Getting at the quark mass matrices, Prog. Theor. Phys. 97 (1997) 781, https://doi.org/10.1143/PTP.97.781. [arXiv:hep-ph/9607461 [hep-ph]]. [ Links ]

37. H. Fritzsch and Z. z. Xing, Mass and flavor mixing schemes of quarks and leptons, Prog. Part. Nucl. Phys. 45 (2000) 1, https://doi.org/10.1016/S0146-6410(00)00102-2. [arXiv:hep-ph/9912358 [hep-ph]]. [ Links ]

38. J. Barranco, F. Gonzalez Canales and A. Mondragon, Universal Mass Texture, CP violation and Quark-Lepton Complementarity, Phys. Rev. D 82 (2010) 073010, https://doi.org/10.1103/PhysRevD.82.073010. [arXiv:1004.3781 [hep-ph]]. [ Links ]

39. H. Fritzsch, Neutrino Masses and Flavor Mixing, Mod. Phys. Lett. A 30 (2015) 1530012, https://doi.org/10.1142/S0217732315300128. [arXiv:1503.01857 [hep-ph]]. [ Links ]

40. K. M. Patel, An SO(10)XS4 Model of Quark-Lepton Complementarity, Phys. Lett. B 695 (2011) 225, https://doi.org/10.1016/j.physletb.2010.11.024. [arXiv:1008.5061 [hep-ph]]. [ Links ]

41. P. V. Dong, H. N. Long, D. V. Soa and V. V. Vien, The 3-3-1 model with S ₄ flavor symmetry, Eur. Phys. J. C 71 (2011) 1544, https://doi.org/10.1140/epjc/s10052-011-1544-2. [arXiv:1009.2328 [hep-ph]]. [ Links ]

42. H. Ishimori , Y. Shimizu , M. Tanimoto and A. Watanabe, Neutrino masses and mixing from S ₄ flavor twisting, Phys. Rev. D 83 (2011) 033004, https://doi.org/10.1103/PhysRevD.83.033004. [arXiv:1010.3805 [hep-ph]]. [ Links ]

43. R. N. Mohapatra and C. C. Nishi, S₄ Flavored CP Symmetry for Neutrinos, Phys. Rev. D 86 (2012) 073007, https://doi.org/10.1103/PhysRevD.86.073007. [arXiv:1208.2875 [hep-ph]]. [ Links ]

44. P. S. Bhupal Dev, B. Dutta, R. N. Mohapatra and M. Severson, θ₁₃ and Proton Decay in a Minimal SO(10)×S4 model of Flavor, Phys. Rev. D 86 (2012) 035002, https://doi.org/10.1103/PhysRevD.86.035002. [arXiv:1202.4012 [hep-ph]]. [ Links ]

45. V. V. Vien, H. N. Long and D. P. Khoi, Neutrino Mixing with Non-Zero θ ₁₃ and CP Violation in the 3-3-1 Model Based on S ₄ Flavor Symmetry, Int. J. Mod. Phys. A 30 (2015) 1550102, https://doi.org/10.1142/S0217751X1550102X. [arXiv:1506.06063 [hep-ph]]. [ Links ]

46. V. V. Vien, H. N. Long and A. E. Cárcamo Hernández, Fermion Mass and Mixing in a Low-Scale Seesaw Model based on the S ₄ Flavor Symmetry, PTEP 2019 (2019) 113B04, https://doi.org/10.1093/ptep/ptz119. [arXiv:1909.09532 [hep-ph]]. [ Links ]

47. V. V. Vien and H. N. Long, Multiscalar B - L extension based on S ₄ flavor symmetry for neutrino masses and mixing, Chin. Phys. C 45 (2021) 043112, https://doi.org/10.1088/1674-1137/abe1c7. [arXiv:2012.01715 [hep-ph]]. [ Links ]

48. V. V. Vien, H. N. Long and A. E. Cárcamo Hernández, Lepton masses and mixings, and muon anomalous magnetic moment in an extended B - L model with the type-I seesaw mechanism, PTEP 2022 (2022) 093B11, https://doi.org/10.1093/ptep/ptac116. [arXiv:2206.06564 [hep-ph]]. [ Links ]

49. S. Pakvasa and H. Sugawara, Discrete Symmetry and Cabibbo Angle, Phys. Lett. B 73 (1978) 61, https://doi.org/10.1016/0370-2693(78)90172-7. [ Links ]

50. O. F Beltran, M. Mondragon and E. Rodriguez-Jauregui, Conditions for vacuum stability in an S(3) extension of the standard model, J. Phys. Conf. Ser. 171 (2009) 012028, https://doi.org/10.1088/1742-6596/171/1/012028. [ Links ]

51. D. Das and U. K Dey, Analysis of an extended scalar sector with S ₃ symmetry, Phys. Rev. D 89 (2014) 095025, [erratum: Phys. Rev. D 91 (2015) 039905] https://doi.org/10.1103/PhysRevD.89.095025. [arXiv:1404.2491 [hep-ph]]. [ Links ]

52. F. González Canales, A. Mondragón, M. Mondragón, U. J. SaldañaSalazar and L. Velasco-Sevilla, Quark sector of S3 models: classification and comparison with experimental data, Phys. Rev. D 88 (2013) 096004, https://doi.org/10.1103/PhysRevD.88.096004. [arXiv:1304.6644 [hep-ph]]. [ Links ]

53. P. F. Harrison and W. G. Scott, Permutation symmetry, tri - bimaximal neutrino mixing and the S ₃ group characters, Phys. Lett. B 557 (2003) 76, https://doi.org/10.1016/S0370-2693(03)00183-7. [arXiv:hep-ph/0302025 [hep-ph]]. [ Links ]

54. P. F. Harrison, D. H. Perkins and W. G. Scott, Tri-bimaximal mixing and the neutrino oscillation data, Phys. Lett. B 530 (2002) 167 https://doi.org/10.1016/S0370-2693(02)01336-9. [arXiv:hep-ph/0202074 [hep-ph]]. [ Links ]

55. Z. z. Xing, Nearly tri bimaximal neutrino mixing and CP violation, Phys. Lett. B 533 (2002) 85, https://doi.org/10.1016/S0370-2693(02)01649-0. [arXiv:hep-ph/0204049 [hep-ph]]. [ Links ]

56. G. Altarelli, F. Feruglio and L. Merlo, Tri-Bimaximal Neutrino Mixing and Discrete Flavour Symmetries, Fortsch. Phys. 61 (2013) 507, https://doi.org/10.1002/prop.201200117. [arXiv:1205.5133 [hep-ph]]. [ Links ]

57. J. C. Pati and A. Salam, Lepton Number as the Fourth Color, Phys. Rev. D 10 (1974) 275, [erratum: Phys. Rev. D 11 (1975) 703, ] https://doi.org/10.1103/PhysRevD.10.275. [ Links ]

58. R. N Mohapatra and J. C Pati, A Natural Left-Right Symmetry, Phys. Rev. D 11 (1975) 2558, https://doi.org/10.1103/PhysRevD.11.2558. [ Links ]

59. G. Senjanovic and R. N. Mohapatra, Exact Left-Right Symmetry and Spontaneous Violation of Parity, Phys. Rev. D 12 (1975) 1502, https://doi.org/10.1103/PhysRevD.12.1502. [ Links ]

60. G. Senjanovic , Spontaneous Breakdown of Parity in a Class of Gauge Theories, Nucl. Phys. B 153 (1979) 334, https://doi.org/10.1016/0550-3213(79)90604-7. [ Links ]

61. M. Agostini, et al. [GERDA], Results on Neutrinoless Double-β Decay of ⁷⁶ Ge from Phase I of the GERDA Experiment, Phys. Rev. Lett. 111 (2013) 122503, https://doi.org/10.1103/PhysRevLett.111.122503. [arXiv:1307.4720 [nucl-ex]]. [ Links ]

62. M. Agostini, et al. [GERDA], Improved Limit on Neutrinoless Double-β Decay of ⁷⁶ Ge from GERDA Phase II, Phys. Rev. Lett. 120 (2018) 132503, https://doi.org/10.1103/PhysRevLett.120.132503. [arXiv:1803.11100 [nucl-ex]]. [ Links ]

63. Z. z. Xing, H. Zhang and S. Zhou, Updated Values of Running Quark and Lepton Masses, Phys. Rev. D 77 (2008) 113016, https://doi.org/10.1103/PhysRevD.77.113016. [arXiv:0712.1419 [hep-ph]]. [ Links ]

64. R. N Mohapatra and S. Nussinov, Bimaximal neutrino mixing and neutrino mass matrix, Phys. Rev. D 60 (1999) 013002, https://doi.org/10.1103/PhysRevD.60.013002. [arXiv:hep-ph/9809415 [hep-ph]]. [ Links ]

65. C. S. Lam, A 2-3 symmetry in neutrino oscillations, Phys. Lett. B 507 (2001) 214, https://doi.org/10.1016/S0370-2693(01)00465-8. [arXiv:hep-ph/0104116 [hep-ph]]. [ Links ]

66. T. Kitabayashi and M. Yasue, S_2L permutation symmetry for left-handed μ and τ families and neutrino oscillations in an SU(3)L×SU(1)N gauge model, Phys. Rev. D 67 (2003) 015006, https://doi.org/10.1103/PhysRevD.67.015006. [arXiv:hep-ph/0209294 [hep-ph]]. [ Links ]

67. Y. Koide, Universal texture of quark and lepton mass matrices with an extended flavor 2 ↔ 3 symmetry, Phys. Rev. D 69, (2004) 093001, https://doi.org/10.1103/PhysRevD.69.093001. [arXiv:hep-ph/0312207 [hep-ph]]. [ Links ]

68. T. Fukuyama and H. Nishiura, Mass matrix of Majorana neutrinos, 1997 [arXiv:hep-ph/9702253 [hep-ph]]. [ Links ]

69. B. D Claude Cohen-Tannoudji and F. Laloe, Quantum Mechanics , volume II, pages 1093 - 1104, Wiley (1991) [ Links ]

70. F. F. González-Canales, Simetría Permutacional S ₃ : Sabor y Ceros de Textura, Ph.D. thesis, Universidad Autónoma de México, Posgrado en Ciencias Físicas (2011). [ Links ]

i We ought to comment there are terms as for example (L-ILIC)11(H~I2)11/Λ2, (L-ILIC)2(H~I2)2/Λ2 and (L-ILIC)11(H~32)11/Λ2 which are invariant under all symmetries but these contributions are very subleading due to the hierarchy 〈Δ〉≪v≪Λ.

iiSee Appendix B for a brief overview on μ ↔ τ symmetry.

iiiIn Appendix C, we detail the process to figure out Uνϵ.

Appendix

A. 𝐒₄ flavour symmetry

𝐒₄ is the group of permutations of four objects and this is the smallest non abelian group having doublet, triplet and singlet irreducible representations [⁷]. Geometrically, 𝐒₄ is the symmetry of a cube. This discrete group contains five irreducible representations, this is, 11,12,2,31,32 which has the following tensor product rules [⁷]:

a1a22⊗b1b22=(a1b1+a2b2)11⊕(-a1b2+a2b1)12⊕a1b2+a2b1a1b1-a2b22 , (A.1)

a1a22⊗b1b2b331=a2b1-12(3a1b2+a2b2)12(3a1b3-a2b3)31⊕a1b112(3a2b2-a1b2)-12(3a2b3+a1b3)32 , (A.2)

a1a22⊗b1b2b332=a1b112(3a2b2-a1b2)-12(3a2b3+a1b3)31⊕a2b1-12(3a1b2+a2b2)12(3a1b3-a2b3)32 , (A.3)

a1a2a331⊗b1b2b331=(a1b1+a2b2+a3b3)11⊕12a2b2-a3b316(-2a1b1+a2b2+a3b3)2⊕a2b3+a3b2a1b3+a3b1a1b2+a2b131⊕a3b2-a2b3a1b3-a3b1a2b1-a1b232 , (A.4)

a1a2a332⊗b1b2b332=(a1b1+a2b2+a3b3)11⊕12(a2b2-a3b3)16(-2a1b1+a2b2+a3b3)2⊕a2b3+a3b2a1b3+a3b1a1b2+a2b131⊕a3b2-a2b3a1b3-a3b1a2b1-a1b232 , (A.5)

a1a2a331⊗b1b2b332=(a1b1+a2b2+a3b3)12⊕16(2a1b1-a2b2-a3b3)12(a2b2-a3b3)2 ⊕a3b2-a2b3a1b3-a3b1a2b1-a1b231⊕a2b3+a3b2a1b3+a3b1a1b2+a2b132 . (A.6)

In this section, we remark an interesting feature between the 𝐒₃ [7] and 𝐒₄ non-abelian groups. As it is well known, these are different, the former one has three irreducible representations namely two singlets, 𝟏_𝟏 and 𝟏₂, and one doublet 𝟐. This group is smaller than 𝐒₄ as one can see in Ref. [⁷]. In addition, each representation of 𝐒₄ can be decomposed in representation of 𝐒₃ as follows: 11→11, 12→12,2→2, 31→11⊕2 and 32→12⊕2.

B. Symmetry μ ↔ τ

In the basis where the charged lepton mass matrix is diagonal, the effective neutrino mass term is given by

L=ν-lL(Mν)ll'(νl'L)C+h.c, (B.1)

where l,l'=e,μ,τ. If the neutrino mass matrix is invariant under the interchange label μ↔τ, one would have

Mν=meemeμmeμmeμmμμmμτmeμmμτmμμ. (B.2)

As one can notice, in the previous mass matrix the entries 12 (22) and 13 (33) are equals, then that matrix possesses the μ↔τ symmetry [⁶⁴-⁶⁸] and its prediction on the mixing angles are obtained as follows. In the mentioned basis, the neutrino mass matrix is diagonalized by 𝐔_v , this means, Uν†MνUν*=M^ where the latter matrix stands for the neutrino masses, M^=Diag.(m1,m2,m3), which can be complex.

As it is well known, Uν=Uπ/4Uθ so that Uν†MνUν*=Uθ†mνUθ* where

mν=mee2meμ02meμmμμ+mμτ000mμμ-mμτ, Uπ/4=100012-1201212. (B.3)

In addition, 𝐦_v is diagonalized by Uθν whose form is given as

Uθ=cosθsinθ0-sinθcosθ0001, (B.4)

and one condition should be satisfied

tan 2θ=8meμmμμ+mμτ-mee. (B.5)

Therefore, one can write the full matrix

Uν=cosθsinθ0-sinθ2cosθ2-12-sinθ2cosθ212. (B.6)

Comparing the above mixing matrix with the standard parametrization of the PMNS matrix, one gets the reactor and atmospheric angles are θ13=0 and θ23=π/4, respectively. Speaking about the solar angle, this is free parameter and can be identified by θ12=θ.

At the same time, the matrix elements can be written in terms of the physical neutrino mass as follows

mee=m1cos2θ+m2sin2θ,meμ=sin2θ8m2-m1,mμτ=12m1sin2θ+m2cos2θ+m3,mμμ=12m1sin2θ+m2cos2θ-m3. (B.7)

C. Perturbative study to obtain U^ϵ _v

To start with, we have to remember the stationary perturbation theory from our lectures on quantum mechanics [⁶⁹]. This method is applied to a system whose Hamiltonian is given by H=H0+W where the eigenstates (ϕn0) and eigenvalues (En0) of H ₀ are known, also W (known as a perturbation) is smaller than H ₀, besides that, H ₀ and W are time independent. With W=λ W~ and λ≪1, at first order in the λ perturbative parameter, one can perform the correction to the eigenstates and eigenvalues of H(λ) (H(λ)|Φn(λ)〉=En(λ)|Φn(λ)〉) which are given respectively by

|Φn(λ)〉=|ϕn0〉+∑k≠n‍〈ϕk0|W|ϕn0〉En0-Ek0|ϕk0〉, En(λ)=En0+〈ϕn0|W|ϕn0〉. (C.1)

Then, we adapt the above results to diagonalization problem given in the neutrino sector. As it was shown, ℳ_v is diagonalized by the Uν matrix such that M^ν=Diag.m1,m2,m3=Uν†MνUν* with Uν=S23uν, then M^ν=uν†mνuν*. 𝐦_v can be written as

mν=meemeμmeμmeμmμμmμτmeμmμτmμμ⏞mν0+00000000mμμϵ⏞mνϵ, (C.2)

where the former matrix possesses exact μ↔τ symmetry and it is broken in the latter one where the dimensionless parameter ϵ≡mττ-mμμ/mμμ has been defined and this quantify the breaking. In addition, this parameter will be treated as a perturbation, thus, a pertubative study at first order in 𝜖 will be carried out. It was shown in the above Appendix, mν0 is diagonalized by Uν0, then, M^ν=uν†mνuν* with uν≈Uν0Uνϵ implies

M^ν=Uνϵ†Uν0†mν0Uν0*+Uν0†mνϵUν0*Uνϵ*, with Uν0†mν0Uν0*=Diag.m10,m20,m30. (C.3)

In addition,

Uν0†mνϵUν0*⏞m~νϵ=ϵ mμμ02sin2 θ-sin 2θ2-sin θ-sin 2θ2cos2 θcos θ-sin θcos θ1, mμμ0=12m10sin2θ+m20cos2θ-m30. (C.4)

In here, we make contact with the perturbation theory, instead of having a Hamiltonian, we have mν=mν0+mνϵ where the eigenstates (νi0) and eigenvalues (mi0) of mν0 are well known. In consequence, Uνϵ is built by means the adapted eigenvector, this is, Uνϵ=|Φ1(ϵ)〉,|Φ2(ϵ)〉,|Φ3(ϵ)〉 where

|Φn(ϵ)〉=|νn0〉+∑k≠n3‍〈νk0|m~νϵ|νn0〉mn0-mk0|νk0〉, (C.5)

where n = 1, 2, 3. In order to apply correctly the formula, we have to observe that matrix mνϵ has been rotated by Uν0 so that the new perturbative matrix is denoted by m~νϵ. Additionally, each eigenvector |Φn(ϵ)〉 must be normalized. Finally, one gets

Uνϵ≈N1-mμμ0m20-m10sin2θ4ϵN2mμμ0m10-m30sinθ2ϵN3mμμ0m20-m10sin2θ4ϵN1N2-mμμ0m20-m30cosθ2ϵN3-mμμ0m10-m30sinθ2ϵN1mμμ0m20-m30cosθ2ϵN2N3, (C.6)

where the normalization factors are written as

N1=1+|ϵ|2sin2θ4mμμ0m10-m302+cos2θmμμ0m20-m102-1/2,N2=1+|ϵ|2cos2θ4mμμ0m20-m302+sin2θmμμ0m20-m102-1/2,N3=1+|ϵ|24sin2θmμμ0m10-m302+cos2θmμμ0m20-m302-1/2. (C.7)

To sum up, Mν is diagonalized approximately by Uν≈S23uν with uν≈Uν0Uνϵ. As a result of this, the PMNS mixing matrix, 𝐔, is defined as U=UeL†Uν where UeL=S23ueL was performed in the lepton section. Therefore, U≈ueL†Uν0Uνϵ, as one realizes, ueL† contains unphysical phases which are irrelevant to the magnitude for PMNS matrix elements that are written explicitly as

U11=N1cosθ1+ϵ2sin2θmμμ0m20-m10,U12=N2sinθ1-ϵ2cos2θmμμ0m20-m10,U13=N34sin2θ ϵ mμμ0m20-m10m20-m30m10-m30,U21=-N12sinθ1-ϵ2mμμ01m10-m30+cos2θm20-m10,U22=N22cosθ1-ϵ2mμμ01m20-m30-sin2θm20-m10,U23=-N321-ϵ2mμμ0cos2θm20-m30+sin2θm10-m30,U31=-N12sinθ1+ϵ2mμμ01m10-m30-cos2θm20-m10,U32=N22cosθ1+ϵ2mμμ01m20-m30+sin2θm20-m10,U33=N321-ϵ2mμμ0cos2θm20-m30+sin2θm10-m30. (C.8)

In the limit ϵ→0, one recovers the μ↔τ scenario.

D. Diagonalizing 𝓜_q

D. Diagonalizing 𝓜 _q

After the spontaneous symmetry breaking and with the appropriated alignment in the vev’s, the quark mass term is given by

q-LMqqR+h.c., (D.1)

with q = u, d. Explicit, we have

Mq=BqbqCqbqAqCqDqDqEq. (D.2)

In order to diagonalize the above mass matrix v, a transformation in the left and right-handed quarks is made such that qL=UqL q~L and qR=UqR q~R where q~(L,R) are the quark fields in the mass basis. Then, Mq=UqL†MqUqR where Mq=Diag.(mq1,mq2,mq3) stands for the quark physical masses. With UqL=Uπ/4uq(L,R), one obtains Mq=uqL†mquqR, in this case, we have

mq=Aqbq0bqBq2Cq02DqEq, Uπ/4=12120-12120000. (D.3)

As it was discussed, a benchmark was considered such that Aq=0 and Cq=Dq so that we end up having a complex symmetric mass matrix

mq=0bq0bqBq2Cq02CqEq. (D.4)

Given mq, this can be written in the polar form, this means, bq=|bq|eiαbq, Bq=|Bq|eiαBq and so forth. The phases can be absorbed in the quark fields, to do so let us write mq=PqmPq with Pq=Diag.eiηq1,eiηq2,eiηq3 where the following condition must be satisfied

ηq1=2arg(bq)-arg(Bq)2, ηq2=arg(Bq)2, ηq3=arg(Eq)2, arg(Bq)+arg(Eq)=2arg(2 Cq), (D.5)

and

m-q=0|bq|0|bq||Bq||2Cq|0|2Cq||Eq|. (D.6)

As a result of this, uqL=PqOq and uqR=Pq†Oq, 𝐎q being the orthogonal matrix that diagonalizes to m _q . Therefore, Mq=uqL†mquqR=OqTmqOq, this last expression is useful to fix three free parameters, in terms of the quark physical masses and one unfixed parameter (|Eq|), through the following invariants

trM^q, trM^q2, detM^q, (D.7)

where tr and det stand for the trace and determinant. In consequence,

|bq|=mq3|mq2|mq1|Eq|, |Bq|=mq3-|mq2|+mq1-|Eq|, 2|Cq|=mq3|mq2|mq1|Eq|. (D.8)

In the above parameters, mq2=-|mq2| has been chosen in order to have real parameters.

Once many free parameters have been fixed, the 𝐎_q orthogonal matrix is built by means the Xqi eigenvectors, Oq=Xq1,-Xq2,Xq3, which are given by

Xqi=1Nqi|bq||2Cq|mqi|2Cq|mqimqi-|Bq|-|bq|2. (D.9)

Here, Nqi stands for the normalization factors whose explicit form is determined by the condition XqiTXqi=1. Finally, one obtains

Oq=mq3|mq2||Eq|-mq1Rq1-mq1mq3|Eq|+|mq2|Rq2mq1|mq2|mq3-|Eq|Rq3mq1|Eq|-mq1|Eq|Rq1|mq2||Eq|+|mq2||Eq|Rq2mq3mq3-|Eq||Eq|Rq3-mq1|Eq|+|mq2|mq3-|Eq|Rq1-|mq2||Eq|-mq1mq3-|Eq|Rq2mq3|Eq|-mq1|Eq|+|mq2|Rq3. (D.10)

In addition

Rq1=mq3-mq1|mq2|+mq1|Eq|,Rq2=mq3+|mq2||mq2|+mq1|Eq|,Rq3=mq3+|mq2|mq3-mq1|Eq|. (D.11)

Let us point out the unfixed parameter, |Eq|, has to satisfy the constraint mq3>|Eq|>|mq2|>mq1 in order to get a real orthogonal matrix, 𝐎_q .

Having calculated the ingredients that take places in the CKM matrix, we have UqL=Uπ/4uqL=Uπ/4PqOq. Consequently, V=UuL†UdL=OuTP-qOd with P-q=Pu†Pd, Here, let us emphasize an important point, this has to do with the CP phases that enter in the CKM matrix. Notice that P-q contains three phases but two of them only play an important role to fit the mixings. At the end, 𝐕 has four parameters namely |Eq| (q = u, d) and two effective CP-violating phases (α and β).

In the CKM matrix, the involved matrices are written explicitly

Ou=mt|mc||Eu|-muRu-mumt|Eu|+|mc|Rcmu|mc|mt-|Eu|Rtmu|Eu|-mu|Eu|Ru|mc||Eu|+|mc||Eu|Rcmtmt-|Eu||Eu|Rt-mu|Eu|+|mc|mt-|Eu|Ru-|mc||Eu|-mumt-|Eu|Rcmt|Eu|-mu|Eu|+|mc|Rt,Od=mb|ms||Ed|-mdRd-mdmb|Ed|+|ms|Rsmd|ms|mb-|Ed|Rbmd|Ed|-md|Ed|Rd|ms||Ed|+|ms||Ed|Rsmbmb-|Ed||Ed|Rb-md|Ed|+|ms|mb-|Ed|Rd-|ms||Ed|-mdmb-|Ed|Rsmb|Ed|-md|Ed|+|ms|Rb,P-q=Diag eiη-q1,eiη-q2,eiη-q3, (D.12)

with η-q1=ηd-ηu,η-q2=ηs-ηc and η-q3=ηt-ηb. In addition,

Ru=mt-mu|mc|+mu|Eu|,Rc=mt+|mc||mc|+mu|Eu|,Rt=mt+|mc|mt-mu|Eu|;Rd=mb-md|ms|+md|Ed|,Rs=mb+|ms||ms|+md|Ed|,Rb=mb+|ms|mb-md|Ed|. (D.13)

On the other hand, for the up and down quark sector we have the following constraints on the free parameters mt>|Eu|>|mc|>mu and mb>|Ed|>|ms|>md. Having written the main ingredients that take place in the quark mixing, the CKM matrix elements are

VCKMud=Ou11Od11eiη-q1+Ou21Od21eiη-q2+Ou31Od31eiη-q3,VCKMus=Ou11Od12eiη-q1+Ou21Od22eiη-q2+Ou31Od32eiη-q3,VCKMub=Ou11Od13eiη-q1+Ou21Od23eiη-q2+Ou31Od33eiη-q3,VCKMcd=Ou12Od11eiη-q1+Ou22Od21eiη-q2+Ou32Od31eiη-q3,VCKMcs=Ou12Od12eiη-q1+Ou22Od22eiη-q2+Ou32Od32eiη-q3,VCKMcb=Ou12Od13eiη-q1+Ou22Od23eiη-q2+Ou32Od33eiη-q3,VCKMtd=Ou13Od11eiη-q1+Ou23Od21eiη-q2+Ou33Od31eiη-q3,VCKMts=Ou13Od12eiη-q1+Ou23Od22eiη-q2+Ou33Od32eiη-q3,VCKMtb=Ou13Od13eiη-q1+Ou23Od23eiη-q2+Ou33Od33eiη-q3. (D.14)

Remarkable, for each entry its magnitude only depends on two effective phases, this is, αq≡η-q2-η-q1 and βq≡η-q3-η-q1.

Received: August 30, 2022; Accepted: November 16, 2022

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