Fermion masses and mixing with tri-bimaximal in SO(10) with type-I seesaw
aa r X i v : . [ h e p - ph ] J a n IFIC/11-46RM3-TH/11-12
Fermion masses and mixing with tri-bimaximal in SO (10) with type-I seesaw G. Blankenburg ∗ and S. Morisi † Dipartimento di Fisica ‘E. Amaldi’, Universit`a di Roma TreINFN, Sezione di Roma Tre, I-00146 Rome, Italy AHEP Group, Institut de F´ısica Corpuscular – C.S.I.C./Universitat de Val`enciaEdificio Institutos de Paterna, Apt 22085, E–46071 Valencia, Spain (Dated: December 6, 2018)We study a class of models for tri-bimaximal neutrino mixing in SO (10) grand unified SUSYframework. Neutrino masses arise from both type-I and type-II seesaw mechanisms. We use dimen-sion five operators in order to not spoil tri-bimaximal mixing by means of type-I contribution in theneutrino sector. We show that it is possible to fit all fermion masses and mixings including also therecent T2K result as deviation from the tri-bimaximal. PACS numbers: 11.30.Hv 14.60.-z 14.60.Pq 14.80.Cp 14.60.St 23.40.Bw
I. INTRODUCTION
Neutrino mixing leads to large atmospheric angle (maximal), large solar angle (trimaximal) and small reactor angle.In particular recently T2K collaboration has given indication of non zero reactor angle [1]. After such a result, theglobal fits of neutrino parameters give non zero reactor angle at 3 σ :sin θ = 0 . +0 . − . [2] , sin θ = 0 . +0 . − . [3] . (1)This interesting result seems in contradiction with tri-bimaximal (TBM) mixing ansatz [4] that predicts zero reactorangle. However TBM solar and atmospheric mixing angles can be used as first approximation. Deviation from zeroreactor angle can arises in grand unified theory (GUT) like SU (5) and SO (10) from the charged sector, see forinstance [5]. While neutrino mass matrix is diagonalized from TBM unitary matrix, charged leptons are not diagonalgiving deviation to the TBM. In this paper we consider such a possibility in the framework of a supersymmetric(SUSY) SO (10) model. In this scenario charged leptons and CKM mixings are strongly related, we therefore considerthe TBM as a good starting point to be corrected in general by small (CKM-like) deviations. In Ref.[6] has beenshown that in a renormalizable SO (10) model this is not possible in case of type-I seesaw. Such a difficulty arises fromthe fact that up quark and Dirac neutrino Yukawa couplings are strongly related in renormalizable SO (10) models.Some interesting attempts to obtain TBM with a flavour symmetry are developed in Ref.[7] and Ref.[8], assumingtype-II seesaw to be dominant . In Ref.[12], in the context of SUSY renormalizable SO (10) with type-II seesawdominance, a fit of all the fermion masses and mixing has been done (see also [13]). The superpotential considered isof the form w = h
16 16 10 + f
16 16 126 + h ′
16 16 120 , (2)where h is a symmetric matrix, h ′ is antisymmetric and f has the TBM structure, namely f = f f f f f + f f − f f f − f f + f , (3)where m ν = f − f , m ν = f + 2 f and m ν = f − f + 2 f . It is well know that a mass matrix with the abovestructure is diagonalized by TBM mixing matrix, see for instance [14]. No assumptions have been made taking f tobe TBM because we can always go to this basis by rotating the 16 of fermions [12]. The matrices h, h ′ are assumedto be hermitian that can correspond to an underlying parity [15]. ∗ Electronic address: blankenburg@fis.uniroma3.it † Electronic address: morisi@ific.uv.es For an incomplete list of papers with TBM in GUT see [9], for SO (10) models with discrete flavour symmetry and no TBM mixing see[10] and for a collection of general SO (10) models see at the References in Ref. [11] Another possibility to reproduce TBM mixing in the framework of SO (10) GUT models is to use non-renormalizableoperators containing a scalar field transforming as a 45 H of SO (10) [6, 16, 17]. This field allows to distinguish upquarks from neutrinos permitting TBM mixing also in the case where neutrino masses arise from type-I seesawmechanism. In particular in Ref.[6] for this purpose the dimension five operator 16 16 120 H H has been used. Thisoperator yields a contribution to the up-quark mass matrix and not to the Dirac neutrino one allowing to distinguishthe up-quark from Dirac neutrino sectors. In this way it is possible to obtain both Dirac and Majorana neutrinomasses TBM and hierarchical structure in charged fermions sector.A full fit of quark and lepton masses and mixing in models with TBM mixing from type-I seesaw in SO (10) isstill missing. In this paper we consider such a problem. We link the idea of distinguish up-quark and Dirac neutrinoby means of 16 16 120 H H operator with the result of Ref. [12] where has been shown that from the superpotential(2) it is possible to fit all the data having TBM mixing in the neutrino sector. In this paper we will translate thesuperpotential (2) in the language of dimension five operators.In the next section we will review some of the SO (10) dimension five operator that will be useful to constructan SO (10) model giving TBM mixing with type-I as well as type-II seesaw following the indication given in thesuperpotential (2) of Ref. [12]. In section III we give some examples of models and the corresponding fits, in section IVwe discuss the possibility to obtain a renormalizable model, then in section V we give our conclusions. II. DIMENSION FIVE EFFECTIVE OPERATORS
In this section we report the main ingredients that will be useful to construct our model in the next section. Itcontains some of the result of table VIII of Ref.[6] that we report in appendix A for the useful of the reader.As discussed in the introduction, one possibility to reproduce TBM mixing in SO (10) in the case of type-I seesawis by means of the dimension five operator 16 16 120 H H that allows to distinguish between up-quark and Diracneutrino sectors. In this section we remark the feature of some dimension five operators that we will use in the nextsection.In general an SO (10) dimension five operator can be written as 16 16 φ a φ b where φ a,b are scalar fields φ a,b =1 H , H , H , H , ... and so on. For simplicity we assume that SO (10) is broken through SU (5) and we describethe contribution of the dimension five operators to the fermion mass matrices in the SU (5) language. When one ofthe components of φ a and φ b take vev a i and b i respectively (where i is the SU (5) index of the component), onegenerates contributions to the quark and lepton masses. Note that a i and b i can be possibly equal if φ a = φ b .The dimension five operators that will be used are: •
16 16 16 H H This operator can be obtained, for example, integrating out a SO (10) singlet 1 χ or a 45-plet 45 χ of heavymessenger fermions: f (1616 H ) (1616 H ) , f (1616 H ) (1616 H ) . (4)¿From the table in appendix A it is possible to see that the first operator contributes to Y ν , M R and M L whilethe second one contributes to Y ν , M R , M L and Y u .We assume f to have the TBM form (3), in this way the light neutrino mass is TBM with type-I or type-IIseesaw, see also eq. (54) below. Note that f TBM is general because we can always go in this basis by rotatingthe 16 [12]. •
16 16 120 H H This operator can be obtained by integrating out a couple 16 χ − χ h (16 120 H ) (16 45 H ) . (5)It can yield a contribution to the up-quark mass matrix (and to the down-quark and charged lepton massmatrices) and not to the Dirac neutrino one, allowing to distinguish the up-quark from Dirac neutrino sectors.This can be described naively in the SU (5) language as follows. The up-type Higgs doublet in a 45 H of SU (5)(contained into the 120 H ) couples antisymmetrically to the two matter multiplets 10 that give the up-quark mass M u while it does not contribute to the Dirac neutrinos, when contracted as in eq. (5). This is a well know featureof SO (10) where the operator 16 16 120 H contributes to M u ∝ h SU (5) i and M ν ∝ h SU (5) i with antisymmetricYukawa. But with the dimension five operator the resulting mass matrix is not antisymmetric and so can givethe hierarchical structure of the quark sector. In fact with the insertion of a 24 H scalar multiplet (contained inthe 45 H of SO (10) ) that takes vev in the hypercharge direction, results that M u is not antisymmentric (but ageneric matrix) since the Clebsch-Gordan coefficients for the isospin doublet Q and the isosinglet u c are differentwith a relative factor ( −
4) that arise from the hypercharge.We can describe the property of the 16 16 120 H H operator in more detail as follow. The 45 H can take vev inits singlet 1 SU (5) component called X -direction or along the adjoint 24 SU (5) component, that is the hypercharge Y -direction (see for instance [18]). We indicate their vevs as b = h SU (5) i , b = h SU (5) i . (6)Equivalently, the 45 H can take vev along the isospin direction or the B − L direction and their correspondingvev are denoted as b and b respectively and are given by b = 15 ( b + 3 b ) , b = 15 ( − b + 2 b ) . (7)The SU (5) components of the 120 H of SO (10) that contain SU (2) doublet (giving rise to the Dirac massesterms for the fermions) are the 45 SU (5) , 45 SU (5) , 5 SU (5) and 5 SU (5) representations. We denote their vevs as a = h SU (5) i , a ¯5 = h SU (5) i , a = h SU (5) i , a ¯45 = h SU (5) i . (8)¿From the table in appendix A we have for instance that M u = h a ( b − b ) − h T a ( b + b ) , (9) M ν = 5 h a b − h T a ( − b − b ) , (10) M d = h ( a + a )( − b + 2 b ) − h T ( a + a )( b + b ) , (11) M Te = h ( a − a )( − b − b ) − h T ( a − a )( b + 6 b ) . (12)Then if a = 0 this operator contributes to Y u , Y d , Y e and not to Y ν . So h can be in a hierarchical form, with(3 ,
3) element dominant, as required by charged fermion phenomenology, without changing the TBM result.Because of the 45 H , the mass matrix that results from such an operator is not antisymmetric but general. Notethat with a 45 H in B-L direction the resulting mass matrix is symmetric, in fact using eq. (7) we have M u = h a ( b − b ) − h T a ( b ) , (13) M ν = h a ( b + 3 b ) + h T a (3 b ) , (14) M d = h ( a + a )( − b − b ) − h T ( a + a ) b , (15) M Te = h ( a − a )( − b ) − h T ( a − a )(3 b − b ) . (16)and stetting b = 0 then all the mass matrices are symmetric. •
16 16 10 H H This operator can be obtained by contracting with a couple 16 χ − χ : h ′ (16 10 H ) (16 45 H ) , (17)We denote the vev of the SU (5) components of the 10 H , containing all the possible SU (2) doublets, as a = h SU (5) i , a ¯5 = h ¯5 SU (5) i . (18)If a = 0 this operator contributes only in Y e and Y d as usual in SU (5). In fact from Appendix A we have M u = h ′ a ( b − b ) + h ′ T a ( b + b ) , (19) M ν = 5 h ′ a b + h ′ T a ( − b − b ) , (20) M d = h ′ a ( − b + 2 b ) + h ′ T a ( b + b ) , (21) M Te = h ′ a ( − b − b ) + h ′ T a ( b + 6 b ) . (22) This is the extra U (1) contained in SO (10) ⊃ SU (5) × U X (1). Again, because of the 45 H , the resulting mass matrix is not symmetric but a generic matrix and it can contributeto the down-quark and lepton masses. Note that with a 45 H in B-L direction the resulting mass matrix isantisymmetric in fact M u = h ′ a ( b − b ) + h ′ T a b , (23) M ν = 5 h ′ a ( b + 3 b ) − h ′ T a b , (24) M d = h ′ a ( − b − b ) + h ′ T a b , (25) M Te = h ′ a ( − b ) + h ′ T a ( − b + 3 b ) . (26)and putting b = 0 the mass matrices are clearly antisymmetric. • Adding an SO (10) scalar singlet 1 H we can consider also the dimension five operators16 16 126 H H ,
16 16 10 H H ,
16 16 120 H H , (27)that behave in the same way as the renormalizable ones (see appendix A) introduced in eq. (2). Also theseoperators can be obtained integrating out a 16 χ − χ of heavy messenger fermions(16 126 H ) (16 1 H ) , (16 10 H ) (16 1 H ) , (16 120 H ) (16 1 H ) . (28)We see that the key ingredient to obtain type-I seesaw and TBM mixing is that the up-type SU (2) Higgs doubletsin the 5 and 5 do not have vevs and so that they are not in linear combination of the light Higgs doublet. Thiscan be a potentially problem since ¯5 takes a vev and it is mixed with the other light Higgs doublets. However thestudy of the complete scalar potential is beyond the scope of this paper and will be studied elsewhere. III. MODELS FOR TBM AND FIT OF FERMION MASSES AND CKM
In Ref. [12] has been studied a model for TBM mixing with dominant type-II seesaw mechanism given in eq. (2). Inthis section we present some possible modifications of the model given in eq. (2). In the models we will present below,TBM arises from both type-I and type-II seesaw mechanisms differently from Ref.[7, 12] where dominant type-IIseesaw mechanism has been assumed for neutrino masses. We remark that the main problem with type-I seesaw isthat the tree-level operator 16 16 10 gives equal contribution to the up-quark and Dirac neutrino mass matrix. Butin order to fit quark masses and mixings with TBM neutrino mixing, the structure of the two mass matrices mustbe very different, namely the up quark mass matrix must be hierarchical while the Dirac neutrino mass matrix mustbe of TBM-type as in eq. (3) or the identity. So we need to disentangle the two sectors, leaving Dirac and Majorananeutrino masses of TBM-type and Dirac charged fermions masses hierachical and almost diagonal. ¿From the previoussection it is clear that one possibility is to replace the operator 16 16 10 of eq. (2) with the operator 16 16 45 120.In the following we will assume an underlying parity, like in [15], making all the mass matrices hermitian and soreducing the number of free parameters. Another way to reduce the sometimes high number of parameters is to assumethat the 45 get vev in the B-L direction. In this case the fermion mass matrices are symmetric or antisymmetric andnot arbitrary.Examples of models with TBM neutrino mixing are listed below. The details of the fit are given in appendix B andC. We fit all charged fermion masses, the two neutrino mass square differences, leptons and quarks mixings, and theCKM phase for a total of 18 observables. For the operators 16 16 120 H H and 16 16 10 H H we always take zerovev for the component 5 SU (5) of 120 H and 10 H , as described in the previous section ( a = 0). • case A : w = f
16 16 16 H H + h
16 16 45 H H + h ′
16 16 45 H H where the 45 H takes vev in a general direction, that is b and b (see eq. (6)), are both different from zero. Themass matrices are : M u = h + f, (29) M d = r [ h (2 b b −
3) + h T (2 b b + 2) + h ′ ] , (30) M e = r { c e [ h (2 b b + 7) + h T (2 b b + 2)] + h ′ ( − b b − b ) + h ′ T (4 b + b b − b
24 ) } , (31) M ν D = 12 f, (32) M R = r R f, (33) M L = r L f, (34)where h and h ′ are generic matrices, r i , c e and b i are combinations of vevs (see appendix A).Results: χ = 0 . , (35)with 26 parameters.We note that with the 45 H taking vev in B-L direction the number of parameters is considerably reduced buta good fit can not be performed. • case B : w = f
16 16 126 H H + h
16 16 45 H H + h ′
16 16 45 H H where the 45 H takes vev in B-L direction. The mass matrices are: M u = h S + r f, (36) M d = r ( h S + h A + f ) , (37) M e = r ( c e h S − h A − f ) , (38) M ν D = − r f, (39) M R = r R f, (40) M L = r L f. (41)Results: χ = 5 . , (42)with 16 parameters (2 d.o.f). • case C : w = f
16 16 126 H H + h
16 16 45 H H + h ′
16 16 120 ′ H H where the 45 H takes vev in B-L direction. The mass matrices are: M u = h S + r h A + r f, (43) M d = r ( h S + h A + f ) , (44) M e = r ( c Se h S + c Ae h A − f ) , (45) M ν D = − r f, (46) M R = r R f, (47) M L = r L f. (48)Results: χ = 0 . , (49)with 18 parameters.The last case reproduces basically the same structure of the renormalizable case (eq. (2)) with type-II seesawdominance studied for example in ref. [12], with just one more parameter c Se . We note that the analysis performed in[12] is based on a previous set of data (before the T2K and MINOS recent results). For this reason we show also anupdated fit for that case, that can be used for comparison: χ = 0 . , d F T = 461863 (50)with 17 parameters (1 d.o.f), where d F T is a parameter introduced in [12]. We note that the goodness of the fit issubstantially unchanged compared with the old analysis, showing that in this class of models it is possible to obtainthe desired (very small before T2K or more sizeable now) corrections to zero θ from the charged lepton sector,taking into account an appreciable amount of finetuning. In fact the neutrino mass matrix is of TBM-type and it isdiagonalized by TBM mixing matrix. The charged fermion mass matrices have hierarchical structure. Assuming allthe parameters to be real, the charged lepton mass matrix is diagonalized by a rotation matrix O l characterized bythree angles θ l , θ l and θ l . The lepton mixing matrix V lep is given by the product V lep = O l † · V T BM so we have( V lep ) = 1 √ s − c s ) , (51)( V lep ) = 1 √ c c + s c + s ) , (52)( V lep ) = 1 √ − c c + s ( c + s s )) , (53)where s ij = sin θ lij and c ij = cos θ lij . We can have a large value for the reactor angle in agreement with the result ofthe T2K collaboration, and at the same time ( V lep ) ≈ / √ V lep ) ≈ / √ θ lij .We observe that from type-I and type-II seesaw mechanisms we have for all the cases presented above m ν = M L − M ν D M R M Tν D = ( r L − r R ) f = r ν f , (54)where we have used the fact that f = f T . Note that only a combination of the r L,R parameters enters in the neutrinosector. So counting the number of free parameters, r L and r R are equivalent to one free parameter instead of two. IV. RENORMALIZABLE THEORY
The dimension five operators assumed in the previous section can be obtained from a renormalizable theory inte-grating out heavy messengers fields. In general the operator 16 16 φ a φ b can be obtained from w = 16 16 χ φ a + 16 16 χ φ b + M χ χ χ (55)where χ − χ is a couple of sets of fermion messengers and it gives rise to the operator 16 16 φ a φ b at a scale E ≪ M χ .Moreover it is easy to take a symmetry forbidding the direct tree level operators 16 16 φ a , for φ a = 10 H , H , H .For example we can take a Z symmetry acting as(16 , φ a,b ) → (16 , − φ a,b ) , ( χ, χ ) → − ( χ, χ ) . (56)Below we report explicit examples of renormalizable models from which the effective dimension five superpotentialsassumed in the previous section can be obtained: • case A The matter and scalar fields content of a possible renormalizable model that can give the effective superpotentialof the case A is given by: If only one messenger field is assumed the effective Yukawa mass matrix is rank one.
16 10 H H H H χ χ χ χ Z + − − − − − − − − then the renormalizable superpotential is w = 16 1 χ H + 16 45 χ H + 16 16 χ H + 16 16 χ H + 16 16 χ H + M χ χ + M χ χ + M χ χ . (57) • case B The matter and scalar field content of the model is16 1 H H H H H χ χ χ χ Z + − − − − − − − − − Z ′ + + − − − + + + − − and the superpotential is given by w = 16 16 χ H + 16 16 χ H + 16 16 χ H + 16 16 χ H + 16 16 χ H + M χ χ + M χ χ (58) • case C The matter and scalar field content of the model is16 1 H H H ′ H H χ χ χ χ Z + − − − − − − − − − Z ′ + + − − + + + + − − and the superpotential is given by w = 16 16 χ H + 16 16 χ H + 16 16 χ ′ H + 16 16 χ H + 16 16 χ H + M χ χ + M χ χ . (59) V. CONCLUSIONS
Neutrino mixing data are in well agreement with maximal atmospheric angle, tri-maximal solar angle and maybe with a non-zero and quite large (namely of order of the Cabibbo angle) reactor angle. TBM mixing gives zeroreactor angle however it can be a reasonable starting point. In fact in GUT framework large deviation of the 1 − SO (10) is still missing. In order toapproach the problem recently has been studied models where light-neutrino mass matrix arises only from type-IIseesaw mechanism. In this paper we studied the possibility that both type-I and type-II seesaw mechanisms yieldTBM neutrino mixing in a SO (10) model. We have assumed that the superpotential contains only dimension fivenon-renormalizable operators. We studied three different possible scenarios for TBM neutrino mixing In each caseproposed we make the fits of all the fermion masses and mixing angle. One case corresponds to the model studiedalready in Ref. [12] for type-II seesaw dominance, while the other two are new.We found in both cases a good fit of all the data including the recent T2K result. In particular for the first modelwe found an excellent fit ( χ = 0 . SO (10) singlets. For the second case we obtained χ = 5 . χ = 0 .
002 with 18free parameters. This case also can be considered as a good starting point for a complete flavour theory. Even if weneeded to introduce one SO (10) singlet we consider the last two cases as the most promising for the moment.For the three cases proposed, we give possible renormalizable realizations where we have introduced messengerfields and extra Abelian symmetries.We remark that in this paper we focused on the flavour secotr and we do not make a full analysis of the model.In particular we leave to a future analysis the study of the Higgs potential and related issues such as the breakingpattern of SO (10) to the SM, problems related to the doublet-triplet splitting (proton-decay) and the achieving ofexact coupling unification considering the possible breaking steps and the related threshold corrections. VI. ACKNOWLEDGMENTS
We thank Prof. G.Altarelli for the useful comments. This work was supported by the Spanish MICINN under grantsFPA2008-00319/FPA, FPA2011-22975 and MULTIDARK CAD2009-00064 (Con-solider-Ingenio 2010 Programme),by Prometeo/2009/091 (Generalitat Valenciana), by the EU Network grant UNILHC PITN-GA-2009-237920. G.B.thanks the AHEP group for partial support during his visit to Valencia for the FLASY workshop where this projecthas been started.
Appendix A
Here we report for convenience of the reader the table VIII of ref.[6]. In general an SO (10) dimension five operatorcan be written as 16 16 φ a φ b where φ a,b are scalar fields φ a,b = 1 H , H , H , H , ... and so on. For simplicity weassume that SO (10) is broken through SU (5) and we describe the contribution of the dimension five operators tothe fermion mass matrices in the SU (5) language. When one of the components of φ a and φ b take vev a i and b i re-spectively (where i is the SU (5) index of the component), one generates contributions to the quark and lepton masses. case SO (10) operator mass matricesIV (16 M H ) (16 M H ) M d = Ka b + K T a b M Te = Ka b + K T a b V (16 M H ) (16 M H ) M ν = Ka b + K T a b M L = K s a b M R = K s a b VI (16 M H ) (16 M H ) M u = 8 K s ( a b + a b ) M ν = 3( Ka b + K T a b ) M L = − K s a b M R = − K s a b VII (16 M H ) (16 M H ) M u = Ka ( b − b ) + K T a ( b + b ) M ν = 5 Ka b + K T a ( − b − b ) M d = Ka ( − b + 2 b ) + K T a ( b + b ) M Te = Ka ( − b − b ) + K T a ( b + 6 b )VIII (16 M H ) (16 M H ) M u = Ka ( b − b ) − K T a ( b + b ) M ν = 5 Ka b − K T a ( − b − b ) M d = K ( a + a )( − b + 2 b ) − K T ( a + a )( b + b ) M Te = K ( a − a )( − b − b ) − K T ( a − a )( b + 6 b )IX (16 M H ) (16 M H ) M d = K ( a b + 2 a b ) + K T ( a b + 2 a b ) M Te = K ( a b + 2 a b ) + K T ( a b + 2 a b )TABLE I: The contributions to the mass matrices from SO (10)-invariant dim-5 operators, from table VIII of ref.[6]. K is anarbitrary matrix. Below we report the contributions to the mass matrices from SO (10) invariant renormalizable Yukawa couplings.Different VEVs of the same SO (10) Higgs multiplet carry a subscript indicating the SU (5) component they belongto. case SO (10) operator mass matricesI 16 M M H M u = M ν = Y v M d = M e = Y v II 16 M M H M u = Y v M ν = Y v M d = Y ( v + v ) M Te = Y ( v − v )III 16 M M H M u = Y v M ν = − Y v M d = Y v M e = − Y v M L = Y v M R = Y v Appendix B
In this section we show the fitting procedure used in our analysis. For charged fermions and CKM mixings the fitare performed on the set of data evolved at the GUT scale showed in Tab. II. The threshold effects are not considered,because they are model dependent and we try to make a general analysis valid for the various models. In these theoriesthere are no constrains on the value of tanβ , so we use the high scale evolved data in the case of tanβ = 10.
Observables Input data m u [ MeV ] 0 . ± . m c [ MeV ] 210 ± m t [GeV] 82 . +30 . − . m d [ MeV ] 1 . ± . m s [ MeV ] 21 . ± . m b [ GeV ] 1 . +0 . − . m e [ MeV ] 0 . ± . m µ [ MeV ] 75 . ± . m τ [ GeV ] 1 . ± . V us . ± . V cb . ± . V ub . ± . J × − . ± . tgβ = 10 (ref.[20],[21]) For neutrino masses and PMNS mixings we use the results in Tab. III. These values are obtained with a global fitconsidering also the recent results from T2K and MINOS. In the models we considered we never obtain degenerateneutrino mass spectrum, so the effects of the evolution from the low energy scale to the GUT scale can be considerednegligible to a good approximation for these observables.
Observable Input data∆ m × − [ eV ] 7 . +0 . − . ∆ m × − [ eV ] 2 . +0 . − . sin θ . +0 . − . sin θ . +0 . − . sin θ . +0 . − . TABLE III: Neutrino masses and mixing in normal hierarchy (ref. [2]) Appendix C
Here we give some other details on the results of the numerical analysis. In particular for the three cases we analysedwe give the best fit parameters and the values for the observables that we obtain. • case A : h = h h e iδ h h e iδ h h e − iδ h h h e iδ h h e − iδ h h e − iδ h h (60) h ′ = h ′ h ′ e iδ h ′ h ′ e iδ h ′ h ′ e − iδ h ′ h ′ h ′ e iδ h ′ h ′ e − iδ h ′ h ′ e − iδ h ′ h ′ (61) f = f f f f f + f f − f f f − f f + f (62) (a) Observable Best fit value m u [ MeV ] 0.550 m c [ MeV ] 210 m t [ GeV ] 81.9 m d [ MeV ] 1.24 m s [ MeV ] 21.7 m b [ GeV ] 1.06 m e [ MeV ] 0.3585 m µ [ MeV ] 75.67 m τ [ GeV ] 1.292 V us V cb V ub J × − m × − [ eV ] 7.59∆ m × − [ eV ] 2.50 sin θ sin θ sin θ χ (b) Parameter Best fit value h v u [ GeV ] 1.40 h v u [ GeV ] -2.45 δ h -1.39 h v u [ GeV ] 13.1 δ h h v u [ GeV ] 5.10 h v u [ GeV ] 15.0 δ h h v u [ GeV ] 79.1 h ′ v u [ GeV ] -6.42 h ′ v u [ GeV ] -6.22 δ h ′ h ′ v u [ GeV ] 2.62 δ h ′ -0.652 h ′ v u [ GeV ] 4.55 h ′ v u [ GeV ] -31.4 δ h ′ -1.06 h ′ v u [ GeV ] 27.3 f v u [ GeV ] -3.23 f v u [ GeV ] -0.155 f v u [ GeV ] 0.987 r / tan β -0.00418 c e -0.619 b -1.12 b v ν /v u × − • case B : h S = h h h h h h h h h (63) h A = i σ σ σ σ σ − σ (64) f = f f f f f + f f − f f f − f f + f (65) (a) Observable Best fit value m u [ MeV ] 0.465 m c [ MeV ] 210 m t [ GeV ] 81.5 m d [ MeV ] 2.95 m s [ MeV ] 23.2 m b [ GeV ] 1.09 m e [ MeV ] 0.3585 m µ [ MeV ] 75.67 m τ [ GeV ] 1.292 V us V cb V ub J × − m × − [ eV ] 7.59∆ m × − [ eV ] 2.50 sin θ sin θ sin θ χ (b) Parameter Best fit value h v u [ GeV ] 0.785 h v u [ GeV ] 0.346 h v u [ GeV ] 7.83 h v u [ GeV ] 3.09 h v u [ GeV ] 4.35 h v u [ GeV ] 82.2 σ v u [ GeV ] -0.298 σ v u [ GeV ] 0.229 σ v u [ GeV ] 2.29 f v u [ GeV ] -0.935 f v u [ GeV ] 0.173 f v u [ GeV ] 0.0316 r / tan β c e r v ν /v u × − • case C : h S = h h h h h h h h h (66) h A = i σ σ σ σ σ − σ (67) f = f f f f f + f f − f f f − f f + f (68) (a) Observable Best fit value m u [ MeV ] 0.550 m c [ MeV ] 210 m t [ GeV ] 82.2 m d [ MeV ] 1.24 m s [ MeV ] 21.6 m b [ GeV ] 1.06 m e [ MeV ] 0.3585 m µ [ MeV ] 75.67 m τ [ GeV ] 1.292 V us V cb V ub J × − m × − [ eV ] 7.59∆ m × − [ eV ] 2.50 sin θ sin θ sin θ χ (b) Parameter Best fit value h v u [ GeV ] 0.584 h v u [ GeV ] -0.548 h v u [ GeV ] -5.49 h v u [ GeV ] 3.55 h v u [ GeV ] 3.99 h v u [ GeV ] 81.8 σ v u [ GeV ] -0.317 σ v u [ GeV ] 2.79 σ v u [ GeV ] -7.09 f v u [ GeV ] -0.999 f v u [ GeV ] -0.207 f v u [ GeV ] 0.0290 r / tan β r r c Se c Ae v ν /v u × − et al. [T2K Collaboration], Phys. Rev. Lett. , 041801 (2011) [arXiv:1106.2822 [hep-ex]].[2] T. Schwetz, M. Tortola, J. W. F. Valle, arXiv:1108.1376 [hep-ph].[3] G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo, A. M. Rotunno, arXiv:1106.6028 [hep-ph].[4] P. F. Harrison, D. H. Perkins, W. G. Scott, Phys. Lett. B530 , 167 (2002). hep-ph/0202074.[5] S. Antusch, V. Maurer, arXiv:1107.3728 [hep-ph]; P. S. Bhupal Dev, R. N. Mohapatra, M. Severson, arXiv:1107.2378[hep-ph], R. N. Mohapatra,M. K. Parida, arXiv:1109.2188 [hep-ph].[6] F. Bazzocchi, M. Frigerio, S. Morisi, Phys. Rev.
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