Neutrino Masses and Deviation from Tri-bimaximal mixing in Δ(27) model with Inverse Seesaw Mechanism
aa r X i v : . [ h e p - ph ] A ug Neutrino Masses and Deviation from Tri-bimaximal mixing in ∆(27) model withInverse Seesaw Mechanism
M. Abbas , , S. Khalil , , A. Rashed , and A. Sil Department of Physics, Faculty of Science, Ain Shams University, Cairo, 11566, Egypt. Center for Fundamental Physics, Zewail City of Science and Technology, Giza 12588, Egypt. Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, 11566, Egypt. Indian Institute of Technology Guwahati, 781039 Assam, India.
We propose a scheme, based on ∆(27) flavor symmetry and supplemented by other discrete symme-tries and inverse seesaw mechanism, where both the light neutrino masses and the deviation fromtri-bimaximal mixing matrix can be linked to the source of lepton number violation. The hierarchiesof the charged leptons are explained. We find that the quark masses including their hierarchies andthe mixing can also be constructed in a similar way.
The convincing evidence of small but non-vanishingneutrino masses calls for an explanation from a natu-ralness point of view. It actually points to the exis-tence of new physics beyond the electroweak scale ( v ).There exist several scenarios to explain this smallness ofneutrino masses. Among them, perhaps the most well-studied one is the conventional type-I seesaw mechanism[1]. In this mechanism, the smallness of neutrino mass( m ν ) can be obtained in an economic way at the ex-pense of introducing heavy right handed (RH) neutrinos( ν R ). For values of Yukawa couplings involved ( Y ν ) oforder unity, the mass scale of ν R ( M R ) turns out to benear the grand unified scale or so through the relation m ν = − m D M R − m TD , where m D = Y ν v . Although in-teresting, such a large scale is beyond the experimentalreach.In this regard, the inverse seesaw mechanism [2–4]offers an interesting resolution through a double sup-pression by the new physics scale M through m ν = m D M − µM T − m TD . With a small mass scale µ (of or-der KeV to few hundred MeV), a relatively low newphysics scale (accessible to LHC) associated with M re-sults. However the main caveat of this scenario is tounderstand the smallness associated with µ or in otherwords, how it is generated. Note that in case of type-I seesaw, the lepton number violation (LNV) happensthrough the majorana mass term of the RH neutrinos,which is quite large. Contrary to this, in case of inverse-seesaw, it happens via the µ term which is a tiny scalewhile compared to the electroweak scale. As the leptonnumber is only an approximate symmetry of nature, itwould be more natural to break it by a small amountrather than by a mass term like M R , which is very large.It can also be argued from the sense of ’t Hooft [5], justbecause in the limit µ tends to zero, the m ν goes to zeroand LNV vanishes so that the symmetry is enhanced.In this letter, we explain the desired smallness of µ - term in a flavor symmetric framework. We consider thepresence of a ∆(27) flavor symmetry which is supple-mented by additional Z × Z discrete groups. The struc-ture guarantees the non-appearance of the µ -term in thetree level Lagrangian. In fact, it allows the µ -term tobe generated only through a significantly higher dimen-sional operator and thereby suppressing the correspond-ing interaction by some nonzero powers of the cut-offscale (Λ) of the theory. There are flavon fields, whosevacuum expectation values (vev) would break the flavorsymmetry and thereby generates a specific structure of µ and other mass matrices like neutrino Dirac mass ma-trix ( m D ), charged leptons etc. We will elaborate moreon this as we proceed. In addition, we assume a 2-3flavor symmetry as an additional symmetry of the La-grangian (particularly for the lepton sector). The onlyplace where this 2-3 symmetry will be broken is in thevev alignment of a single flavon field ( σ ) responsible forgenerating the µ term. The vev of all other flavons re-spect the 2-3 symmetry. So, in a way our frameworksuggests a unified source (through µ term only) of break-ing the 2-3 symmetry and lepton number violation. It isknown [6] that a breaking of 2-3 symmetry may indicatea deviation from tri-bimaximal mixing in the neutrinosector. Therefore in this work, we argue that the specificstructure obtained for µ not only can explain the smallmasses of light neutrinos, but also accounts for the de-viation from an exact tri-bimaximal mixing by having anonzero θ at the same time.In realizing the above goal, the fermion field contentof the Standard Model (SM) is extended by adding threeright handed neutrinos ν R i (for i = 1 , , S i which have lepton number op-posite to that of the ν R i . In addition, the scalar sectoris extended by adding a set of flavons that break the fla-vor symmetry around few TeV scale or more. In [6], itwas emphasized that both quark and lepton masses and Fields ℓ i E Ri ν R i S i H χ σ η η ∆(27) 3 3 1 i Z − Z ω ω ω ω ω ω i 1 L / Z and Z .Here i = 1 , , L stands forthe lepton number. also their mixing angles can be simultaneously accommo-dated in a framework of ∆(27) based on the type-I see-saw mechanism. Here also we have considered the quarksector. We have constructed both the up-type and down-type quark mass matrices so that an acceptable Cabbibo-Kobayashi-Maskawa mixing matrix ( V CKM ) can be ob-tained. In realizing the V CKM , we have employed onlyone additional flavon apart from those are already in-volved in the lepton sector.It turns out that the ∆(27) symmetry alone is notenough to restrict all allowed Yukawa interactions thatcould lead to consistent mass matrices, additional Z × Z symmetries are also imposed as mentioned in Table I. χ and σ are the ∆(27) triplet flavons and η is the singletflavon field. The relevant terms in the Lagrangian forneutrino mass which are consistent with the symmetriesconsidered above are given by L = h kij Λ ¯ ℓ i ˜ Hν R j χ k + f kij Λ ¯ S ci S j σ k η + g kij ¯ ν cR i S j χ † k . (1)Here i, j are the flavor indices while k refers to the k th component of ∆(27) triplet flavon field only. Once the χ and σ fields obtain vevs, ∆(27) symmetry is broken.This breaking leads to specific flavor structures for theheavy mass matrix M = g h χ i and the µ -term µ = f h σ i h η i / Λ , where the flavor indices are suppressedfor simplicity. Each of these are 3 × µ -term in an inverse seesaw framework.The electroweak symmetry breaking thereafter results inthe Dirac neutrino mass term: m D = h h χ ih H i / Λ. Hencethe neutrino mass matrix, in the basis { ν cL , ν R , S } , isgiven by M ν = m D m TD M M T µ , (2)which is a 9 × ℓHℓH (which could contribute to the 11block of M ν ) is absent in our set-up upto dimension-9with the symmetries we considered. At this moment, a discussion about the 2-3 symmetryover the flavor or generation indices would be pertinenthere. It is well known [7], [8] that with a 2-3 symmetry,tri-bimaximal mixing can be achieved at the zeroth or-der. We have also considered this 2-3 symmetry for thelepton sector on top of the discrete symmetries listed inTable I. Therefore particles transform as f ↔ f and f ↔ f with f i stands for ℓ i , χ i , σ i , S i and ν R i . Thischoice further simplifies the structure of m D , M and µ which will be discussed later. Regarding the vevs of thetriplets involved in the model, we have chosen a specificalignment as h χ i = ( v χ , v χ , v χ ) , h σ i = (0 , v σ , . (3)Our choice is governed by the fact that it breaks boththe ∆(27)( ≡ Z ′ × Z ′′ × Z ′′′ ) and the 2 − h χ i breaks Z ′ × Z ′′ subsetof ∆(27), while h σ i breaks Z ′′′ × − µ is small) of 2-3symmetry allows to have non-zero θ in our framework.In order to analyze the vevs of the triplets σ and χ ,we consider the most general scalar potential invariantunder the symmetries considered, which is given by V = m σ ( σ † σ ) + m χ ( χ † χ ) + λ σ ( σ † σ )( σ † σ )+ λ χ ( χ † χ )( χ † χ ) + κ (cid:2) ( σ † σ )( χ † χ ) + ( σ † χ )( χ † σ ) (cid:3) , (4)where κ > − p λ σ λ χ and λ σ , λ χ ≥
0, so that the po-tential is bounded from below. For simplicity we assumeuniversal coupling for the last two terms in the abovepotential. In general this potential contains several freeparameters (masses and couplings). These plenty of freeparameters allow all type of patterns of non-zero vevsof σ and χ . Once we restrict ourselves with the partic-ular vev alignments of χ and σ as given in Eq.(3), thefollowing non-zero vevs are obtained: v χ = − (cid:2) κm σ + 3 λ σ m χ (cid:3) [4 κ + 9 λ χ λ σ ] , v σ = 12 (cid:2) κm χ − λ χ m σ (cid:3) [4 κ + 9 λ χ λ σ ] . (5)We assume that the vev’s of χ and σ are all of the sameorder and satisfy the following relation v χ Λ ∼ v σ Λ = u Λ ∼O ( λ C ) where λ C is the Cabibbo angle, i.e. λ C ∼ . M ν . From Eq. (1), one findsthat the Dirac neutrino mass matrix m D can be con-structed from the following invariants terms:1Λ h (¯ ℓ χ + ¯ ℓ χ + ¯ ℓ χ ) ν R ˜ H, h (¯ ℓ χ + ω ¯ ℓ χ + ω ¯ ℓ χ ) ν R ˜ H, h (¯ ℓ χ + ω ¯ ℓ χ + ω ¯ ℓ χ ) ν R ˜ H, where h kij is written as h k only. Thus, after the flavorsymmetry breaking, the following Dirac neutrino massmatrix is obtained m D = u h H i Λ h h h h ω h ωh h ωh ω h , (6)where h = h is considered due to the presence of 2 − M takes the form M = u g g g g ω g ωg g ωg ω g . (7)Finally, the mass matrix µ is given by µ = u Λ f f f , (8)where h η i is assumed to be of order u . Note that the2 − h σ i , which would bethe source of deviation from tribimaximal mixing patternas well in the lepton sector.The diagonalization of M ν mass matrix leads to thefollowing light and heavy neutrino masses respectively: M ν l = m D ( M µ − M T ) − m TD , (9) M ν H = M ν ′ H = q M + m D . (10) It is now clear that the light neutrino masses can be oforder eV, with a TeV scale M provided µ ≪ m D , M .We will show that with the charge assignments we haveconsidered, the charged lepton mass matrix comes outto be a diagonal one. Therefore the light neutrino massmatrix M ν must be diagonalized by the physical neu-trino mixing matrix U P MNS , i.e. , U TP MNS M ν l U P MNS =diag( m , m , m ) . Since µ matrix is generated by h σ i ,which violates the 2-3 symmetry while m D and M are 2-3symmetric, the deviation of U P MNS from tri-bimaximalmixing is proportional to the size of µ , which is quitesuppressed. A discussion regarding the lepton numberis appropriate here. We have considered the entire La-grangian to respect the lepton number. However, σ hav-ing a nonzero lepton number, while gets a vev, the lep-ton number is broken and in turn generation of neutrinomasses results.We define the parameters which characterize the de-viation of mixing angles from the tri-bimaximal valuesas D ≡ − s , D ≡ − s , D ≡ s , (11)where s ij ≡ sin θ ij . The tri-bimaximal mixing matrixcorresponds to neutrino mass matrix that satisfies the fol-lowing three conditions [9]:( M ν l ) = ( M ν l ) , ( M ν l ) =( M ν l ) , and ( M ν l ) + ( M ν l ) = ( M ν l ) + ( M ν l ) .Therefore, the deviation of tri-bimaximal mixing matrixcan be written in terms of the deviation parameters D and s as follows:∆ = ( M ν l ) − ( M ν l ) = √ (cid:2) (2 m + m ) e iδ − m (cid:3) s e − iδ + 23 ( m − m ) D , ∆ = ( M ν l ) − ( M ν l ) = 2 √
23 ( m − m ) s e iδ + 13 ( m + 2 m − m ) D , (12)where m i is the physical neutrino mass and δ is theleptonic Dirac phase. However, for simplicity, we set theDirac phase to be zero. In our model the deviations fromTBM conditions can give constraints on our parameters(couplings and VEVs) in order to get the correct mixingangles and desired scenario of mass spectra. FromEq. (9), one can write the equations above in terms ofthe model parameters ∆ = − h h H i ( − f g h + f g h + f g h + 2 f g h ) u g g Λ , ∆ = h h H i (2 f g h − f g h + f g h + 2 f g h ) u g g Λ . (13)From Eqs. (12) and (13) we can calculate the deviationfrom TBM parameters s = − h H i h [( f − f ) g h (5 m − m − m ) + g h ( f + 2 f )( m − m + 3 m )] u √ g g (2 m − m m + 2 m + 9( m + m ) m − m )Λ ,D = 3 h H i h [2( f − f ) g h ( m − m ) + ( f + 2 f ) g h ( − m + m )] u g g (2 m − m m + 2 m + 9( m + m ) m − m )Λ . (14)From these expressions, one can easily see that both thesin θ and the D are proportional to µ/u . Thereforein the limit as µ tends to zero, s , D as well as theneutrino mass vanish.As shown in Fig. 1, we can tune the involved parame-ters to get s and D within their experimental limits:0 . ≤ s ≤ .
158 and − . ≤ D ≤ .
115 [10].Here the cut-off scale is fixed at Λ = 10 GeV and the vev u ≃ O (100) TeV. The couplings involved in the m D , M ,and µ matrices, although tuned, are considered to be oforder unity. The allowed regions are consistent with theneutrino oscillation parameters [10] as well as satisfy thecosmological bound on sum of the light neutrino masses[11].The charged lepton Yukawa Lagrangian, invariant un-der the symmetries considered, is given by L l = λ Λ ¯ ℓHE R η + λ Λ ¯ ℓHE R η η † + λ Λ ¯ ℓHE R η η , (15)where the charges of η under the discrete symmetriesare specified in Table I. The Lagrangian indicates thatthe charged lepton mass matrix is diagonal. The chargedlepton masses are therefore given by, m e = λ C h H i ( λ + λ λ C + λ λ C ) ,m µ = λ C h H i ( λ + ω λ λ C + ωλ λ C ) ,m τ = λ C h H i ( λ + ωλ λ C + ω λ λ C ) , (16)where the h η i i / Λ ∼ λ C is considered. In general thecoupling constants λ i are complex, so the three leptonmasses with required hierarchy can be realized. For in-stance, with λ ≃ O (0 .
1) and λ = λ † ≃ − .
12 + 1 . i ,one finds m e = 0 . m µ = 0 . m τ = 1 . Fields ¯ Q ¯ Q ¯ Q u R c R t R d R s R b R η ∆(27) 1 Z ω ω ω ω ω ω ω ωZ − − − − − , Z and Z inthe quark sector. For completeness, we briefly discuss the quark sector.In Table II we present the charge assignments of up and s ( r a d . ) µ (GeV)-0.15-0.1-0.05 0 0.05 0.1 0.15 0 0.02 0.04 0.06 0.08 0.1 D ( r a d . ) µ (GeV) FIG. 1: The variation of the parameters s and D withrespect to the mass scale associated with the µ term are dis-played. down quarks under ∆(27), Z and Z symmetries. Wealso include an extra singlet η which is necessary forbuilding consistent quark mass matrices. The invariantLagrangian of up-quarks under the above symmetries, upto operators suppressed by 1 / Λ , are given by: L u = h u Λ ¯ Q ˜ Hu R η η † + h u Λ ¯ Q ˜ Hc R η η + h u Λ ¯ Q ˜ Ht R η η † η + h u Λ ¯ Q ˜ Hu R η η † + h u Λ ¯ Q ˜ Hc R η † + h u Λ ¯ Q ˜ Ht R η η + h u Λ ¯ Q ˜ Hu R η † + h u Λ ¯ Q ˜ Hc R η † η † η † + h u ¯ Q Ht R . (17)After the spontaneous symmetry breaking, the up-typemass matrix takes the form as m u = h H i h u aλ C λ C λ C λ C λ C λ C λ C λ C . (18)where a = h u h u ∼ O (0 . V uL = − .
998 0 .
048 00 .
048 0 .
998 0 . − . , (19)with the corresponding eigenvalues M u = diag( λ C , λ C , h H i h u GeV . (20)Therefore with h u of order unity, we can explain theorder of magnitude of the up quark masses [12].Similarly, the down mass matrix can also be obtainedfrom the following Lagrangian, L d = h d Λ ¯ Q Hd R η η + h d Λ ¯ Q Hs R η η + h d Λ ¯ Q Hb R η η † η + h d Λ ¯ Q Hd R η η η + h d Λ ¯ Q Hs R η † η + h d Λ ¯ Q Hb R η η + h d Λ ¯ Q Hd R η η † + h d Λ ¯ Q Hs R η † η † + h d Λ ¯ Q Hb R η † . (21)After the spontaneous symmetry breaking, the downmass matrix takes the form, m d = h H i h d λ C λ C λ C λ C λ C bλ C λ C λ C λ C , (22)where b = h d h d ∼ O (0 .
2) is considered so as to get theeigenvalues M d = ( λ C , λ C , λ C h H i h d GeV. Here also,with h d of order one, the order of magnitude estimate ofthe down quark mass hierarchies [12] can be explained.The left handed rotation of the matrix m d is found to be V dL = . − .
267 0 . − . − .
962 0 . .
012 0 .
047 0 . . (23) Thus, the V CKM is found out to be , V CKM = V u † L V dL = − .
975 0 .
22 0 . − . − .
974 0 . .
012 0 .
046 0 . . (24)So it is clear that a close to correct V CKM [12] can beobtained from our framework by tuning the coupling in-volved in Eq.(17) and Eq.(21).In conclusion, we have developed a flavor symmet-ric approach to realize the neutrino masses and mixingthrough the inverse seesaw mechanism. The flavor sym-metry consists of a non-abelian ∆(27) group as well astwo other discrete symmetries Z and Z . We have alsoimposed a 2-3 symmetry on the Largangian (in the lep-ton sector), the breaking of which plays instrumental rolein realizing the deviation from a TBM structure. The in-verse seesaw mechanism is characterized by a small lep-ton number violating Majorana mass term µ , while theeffective light neutrino mass is m ν ∝ µ . Therefore in the2 − σ which generates the µ term, we find thatthe neutrino mass is generated and deviation from theTBM mixing is achieved simultaneously. This is a newresult from the point of view of showing that the µ massterm which violates the lepton number has the same ori-gin as the deviation from the TBM limits. Within thismodel, the mass hierarchies in the charged lepton sectorare also obtained. Finally we are able to show that amere addition of single flavon ( η ) along with those arealready present in constructing the lepton sector can ex-plain the up and down sector mass hierarchies as well asa close to the correct V CKM for some natural values ofthe parameters involved in the model.
I. ACKNOWLEDGMENT
A.S acknowledges the hospitality of CFP, Zewail Cityof Science and Technology, Cairo during a visit when thiswork was initiated. S.K acknowledges partial supportfrom the European Union FP7 ITN INVISIBLES (MarieCurie Actions, PITN- GA-2011- 289442). The work ofM.A is partially supported by the ICTP grant AC-80. [1] P. Minkowski, Phys. Lett. B , 421 (1977); M. Gell-Mann, P. Ramond and R. Slansky, Conf. Proc.C , 315 (1979) [arXiv:1306.4669 [hep-th]];R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. , 912 (1980); T. Yanagida, Prog. Theor. Phys. , 1103(1980).[2] D. Wyler and L. Wolfenstein, Nucl. Phys. B , 205(1983).[3] R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D ,1642 (1986).[4] E. Ma, Phys. Lett. B , 287 (1987).[5] G. t Hooft, Phys. Rev. Lett. , 8 (1976);[6] M. Abbas and S. Khalil, Phys. Rev. D , no. 5, 053003(2015) [arXiv:1406.6716 [hep-ph]]. [7] V. Barger, R. Gandhi, P. Ghoshal, S. Goswami, D. Mar-fatia, S. Prakash, S. K. Raut and S. U. Sankar, Phys. Rev.Lett. , 091801 (2012) [arXiv:1203.6012 [hep-ph]].[8] C. S. Lam, Phys. Rev. D , 093001 (2005)[hep-ph/0503159].[9] M. Abbas and A. Y. Smirnov, Phys. Rev. D , 013008(2010) [arXiv:1004.0099 [hep-ph]].[10] M. C. Gonzalez-Garcia, M. Maltoni and T. Schwetz,JHEP , 052 (2014) [arXiv:1409.5439 [hep-ph]].[11] P. A. R. Ade et al.et al.