SO(10)× S 4 Grand Unified Theory of Flavour and Leptogenesis
SSO ( ) × S Grand Unified Theory of Flavourand Leptogenesis
Francisco J. de Anda † , Stephen F. King (cid:63) , Elena Perdomo (cid:63) (cid:63) School of Physics and Astronomy, University of Southampton,SO17 1BJ Southampton, United Kingdom † Tepatitl´an’s Institute for Theoretical Studies, C.P. 47600, Jalisco, M´exico
Abstract
We propose a Grand Unified Theory of Flavour, based on SO (10) together with anon-Abelian discrete group S , under which the unified three quark and lepton 16-plets are unified into a single triplet 3 (cid:48) . The model involves a further discrete group Z R × Z which controls the Higgs and flavon symmetry breaking sectors. The CSD2flavon vacuum alignment is discussed, along with the GUT breaking potential andthe doublet-triplet splitting, and proton decay is shown to be under control. TheYukawa matrices are derived in detail, from renormalisable diagrams, and neutrinomasses emerge from the type I seesaw mechanism. A full numerical fit is performedwith 15 input parameters generating 19 presently constrained observables, takinginto account supersymmetry threshold corrections. The model predicts a normalneutrino mass ordering with a CP oscillation phase of 260 ◦ , an atmospheric anglein the first octant and neutrinoless double beta decay with m ββ = 11 meV. Wediscuss N leptogenesis, which fixes the second right-handed neutrino mass to be M (cid:39) × GeV, in the natural range predicted by the model. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] a r X i v : . [ h e p - ph ] D ec ntroduction The Standard Model (SM) [1], though highly successful, does not address the origin ofneutrino mass and lepton mixing [2]. One attractive possibility is the type I seesawmechanism, which can account for the smallness of neutrino masses by introducing threeright-handed neutrinos with very large Majorana masses [3]. Such right-handed neutrinosarise very naturally from SO (10) Grand Unified Theories (GUTs) [4] in which a singlefamily of quarks and leptons, together with a right-handed neutrino, is unified into asingle 16-plet. Supersymmetry (SUSY) is then naturally suggested for gauge couplingunification and to ameliorate the gauge hierarchy problem. However the origin of thethree families, and their hierarchical masses are not explained by traditional SO (10)SUSY GUTs.The almost tri-bimaximal lepton mixing observed over recent years [5], combined with areactor angle of order 8 . ◦ [6], suggests that some sort of non-Abelian family symmetrymay be at work in the lepton sector [7]. The first models to consider a non-Abelian SU (3) symmetry as an explanation of bi-large lepton mixing were put forward in [8].Models based on SO (10) with a non-Abelian discrete symmetry were first proposed in[9, 10], and further flavoured GUTs were considered in [11]. A more general study offlavour symmetries in SO (10) can be found in [12]. Here we shall be interested in aSUSY GUT theory of flavour in which all quarks and leptons are fitted into a single(3 ,
16) representation of S × SO (10) [13, 14]. While the former model predicted a zeroreactor angle [13], the latter model [14] was based on CSD3 flavon vacuum alignment[16], leading to approximate tri-bimaximal mixing with the correct value of the reactorangle. However the latter model is so far incomplete since it did not include any explicitdiscussion of the flavon vacuum alignment, or GUT breaking potential, and also did notinclude any discussion of leptogenesis.In the present paper we consider a more complete S × SO (10) SUSY GUT of flavour,which also involves a further discrete group Z R × Z which controls the Higgs and flavonsymmetry breaking sectors. In the model here, we prefer the simpler CSD2 [18] vacuumalignment, which, in conjunction with small charged lepton corrections arising from the SO (10) structure of Yukawa matrices, is capable of yielding the desired reactor angle. Italso allows successful leptogenesis, as discussed below. Here the flavon vacuum alignmentpotential is discussed, along with the GUT breaking potential and the doublet-tripletsplitting, and proton decay are shown to be under control. The Yukawa matrices arederived in detail, from renormalisable diagrams, and neutrino masses emerge from thetype I seesaw mechanism. A full numerical fit is performed with 15 input parametersdescribing 19 observables, taking into account supersymmetry threshold corrections. Themodel predicts a normal neutrino mass ordering with a CP oscillation phase of 260 ◦ , anatmospheric angle in the first octant and neutrinoless double beta decay with m ββ =11 meV. We also discuss N leptogenesis [19, 20], which fixes the second right-handedneutrino mass M (cid:39) × GeV, in the natural range predicted by the model . CSD refers to “constrained sequential dominance” first introduced in [15]. In this paper CSD issimply used as a label which refers to a particular flavon vacuum alignment as discussed later. Suchvacuum alignments motivates the choice of S as the family symmetry, as discussed by Luhn et al [16]. Interestingly we find that N leptogenesis is not consistent with the earlier model based on CSD3vacuum alignment [14], which is a significant motivation for considering the new model based on CSD2. N leptogenesis. Section 10 lists our conclusions. The symmetry of the model is SO (10) × S × Z R × Z . The model has a gauge symmetry SO (10) which is the GUT symmetry. The symmetry S is the flavour symmetry whichgives the specific CSD2 structure to the fermion mass matrices. The Z R is an R symmetrywhile the other three Z ’s are shaping symmetries. Furthermore, we assume that theGUT theory is invariant under trivial CP symmetry, which is spontaneously broken bythe complex VEVs of the flavons.Field Representation S SO (10) Z R Z Z Z ψ (cid:48)
16 1 0 0 0 H u H d H H H X,Y H W,Z H B − L ζ (a) Matter, and Higgs superfields. Field Representation S SO (10) Z R Z Z Z φ (cid:48) φ (cid:48) φ (cid:48) φ S,U (cid:48) φ T ξ φ t (b) Flavon superfields. Table 1: Field content of the model that relates directly to the low energy fields.
In the table 1 we present the fields that contain the Higgs, flavons and matter fields,which are relevant to Yukawa sector. The field ψ contains the full SM fermion content.The fields H u,d contain the MSSM Higgs doublets h u,d respectively. The H breaks SO (10) → SU (5) and gives masses to the right handed neutrinos (RHN). The H ’s break SU (5) → SM and introduce the necessary Clebsch-Gordan (CG) relations to generatecorrect charged lepton and down quark masses. The flavon fields φ i , with i = 1 , , completely with the CSD2 vacuum alignment [18], (cid:104) φ (cid:105) = v , (cid:104) φ (cid:105) = v − , (cid:104) φ (cid:105) = v , (1)with | v | (cid:28) | v | (cid:28) | v | . This CSD2 flavon alignment is fixed by a superpotential asdiscussed in Sec. 3.With these fields, a very specific mass structure for the SM fermion fields is generated.For the up-type quark and the neutrino sectors, the Yukawa terms look like H u ( ψφ )( ψφ ) + H u ( ψφ )( ψφ ) + H u ( ψφ )( ψφ ) , (2)where the brackets denote S singlet contractions. Each of these terms generates a rank-1 matrix. The hierarchy between the flavon VEVs, shown in Sec 4, gives a naturalexplanation of the hierarchical Yukawa couplings y u ∼ v /M , y c ∼ v /M , y t ∼ v /M . The RHN Majorana masses are similar to Eq. 2 replacing H u by H H . Thefact that the RHN masses have the same structure as the Dirac neutrino masses generateexactly the same structure for the left handed neutrino Majorana masses, as shown inSec. 7.3, after the seesaw mechanism.For the down-type quark and the charged lepton sectors, the Yukawa terms look like H d ( ψφ )( ψφ ) + H d ( ψφ )( ψφ ) + H d ( ψφ )( ψφ ) + H d ( ψψ ) (cid:48) ( φ ) , (3)where the brackets denote S singlet contractions apart from the 3 (cid:48) contraction whichis necessary to combine with φ ∼ (cid:48) into a singlet. They have a different structurecompared to the up sector, due to a mixing term between the flavons φ and φ , whichexplains why there is a milder hierarchy in the down and charged lepton sectors comparedto the up one. It also introduces a texture zero in the (1,1) element of the down Yukawamatrix, reproducing the GST relation [21], i.e. the Cabibbo angle is predicted to be θ q (cid:39) (cid:112) y d /y s . With this setup the full SM fermion masses are generated in a veryspecific and predictive way, this being the main aim of the paper.After GUT symmetry breaking, all the messenger fields and adjoints obtain a GUT scalemass. Furthermore, the triplets inside the H u,d also get a GUT scale mass through theDimopoulos-Wiclzeck mechanism [22], as shown in the Sec. 5. This way we make surethat at low energies, only the MSSM remains. We now present the effective Yukawa terms in more detail than in the previous section.With the fields in the table 1 we may write the superpotential relevant to the Yukawa3ield Representation S SO (10) Z R Z Z Z ¯ χ χ χ χ χ χ χ d χ d χ u χ u χ d χ d ζ ζ (a) Messenger superfields. Field Representation S SO (10) Z R Z Z Z X (cid:48) (cid:48) X X X X X (cid:48) (cid:48) Z (cid:48) (cid:48) Z (cid:48) (cid:48) Z Z (b) Alignment superfields. Table 2: Fields that appear only at high energies. Together with the ones in Table 1 they listthe complete field content of the model. terms, including terms O (1 /M P ), as W Y ∼ H u ( ψφ )( ψφ ) (cid:104) H W,Z (cid:105) + H u ( ψφ )( ψφ ) (cid:104) H W,Z (cid:105) + H u ( ψφ )( ψφ ) (cid:104) H W,Z (cid:105) + H d ( ψφ )( ψφ ) (cid:104) H W,Z (cid:105) + H d ( ψφ )( ψφ ) (cid:104) H X,Y (cid:105) + H d ( ψφ )( ψφ ) (cid:104) H X,Y (cid:105) + H H ( ψφ )( ψφ ) M P (cid:104) H W,Z (cid:105) + H H ( ψφ )( ψφ ) M P (cid:104) H W,Z (cid:105) + H H ( ψφ )( ψφ ) M P (cid:104) H W,Z (cid:105) + H d ( ψψ ) (cid:48) ( φ ) M P (4)where ( ) (cid:48) means a 3 (cid:48) contraction, while ( ) without any subscript means the singletcontraction of S . There are plenty of terms supressed by M P and they are expectedto make small mass contributions of O ( M GUT /M P ) < − , and therefore negligible .We have ignored all the O (1) dimensionless couplings for simplicity. The diagrams thatgenerate these terms are shown in Figs. 1-3, where they include the messengers χ , listed inTable 2. In the Sec. 7 we present them in full detail together with the specific messengerstructure.The full field content of the model is listed in Tables 1-2. Even though the list of fieldsseems large, it is substantially smaller than previous flavoured GUT models that attempt The most important correction, of O (10 − ), is made to the up-quark Yukawa coupling. From table5, we see that it is of comparable magnitude. We performed the fit ignoring these corrections. If theywere included, they would shift the fit parameters. The largest contribution to the electron Yukawacoupling is of O (10 − ) and therefore negligible. H W,Z i h H W,Z i H u φ φ ψ ψ ¯ χ χ χ ¯ χ h H W,Z i h H W,Z i H u φ φ ψ ψ ¯ χ χ χ ¯ χ h H W,Z i h H W,Z i H u φ φ ψ ψ ¯ χ χ χ ¯ χ Figure 1: Diagrams coupling ψ to H u . When flavons acquire VEVs, these give the up-typequark and Dirac neutrino Yukawa matrices. h H W,Z i h H W,Z i H d φ φ ψ ψ ¯ χ χ χ ¯ χ h H X,Y i h H X,Y i H d φ φ ψ ψ ¯ χ χ ′ χ ′ ¯ χ h H X,Y i h H X,Y i H d φ φ ψ ψ ¯ χ χ ′ χ ′ ¯ χ Figure 2: Diagrams coupling ψ to H d . These generate the down-type quark and chargedlepton Yukawa matrices. to be complete [10]. The flavon superpotential fixes the symmetry breaking flavon VEVs in Eq. 1. To derivethis alignment we use a set of driving fields, listed in Table 2, coupled to the flavon fieldsin Table 1. We follow a sequence of steps using supersymmetric F-terms equations toalign all the flavons. The letter subscript in the flavons refers to the symmetry preservinggenerator. The alignments depend on the S representation of the alignment field, denotedby its index. The superpotential is given by W φ ∼ X (cid:48) ( φ S,U ) + X ( φ T ) + X ( φ t ) + ˜ X φ T φ t + X (cid:48) φ T φ + ˜ X φ t φ + Z (cid:48) ( φ S,U φ T + ξφ ) + ˜ Z (cid:48) ξ (cid:18) φ φ M P − φ (cid:19) , (5)where we have ignored dimensionless O (1) parameters since they are not relevant. Solvingthe F-term equations from the alignment fields fixes the flavon VEV alignment, while theF-term equations from flavons forbid the alignment fields from getting a VEV.The three S generators, working in the T diagonal basis, are S = 13 − − − , T = ω
00 0 ω for or (cid:48) , (6)and U = ∓ , SU = U S = ∓ − − − , for , (cid:48) respectively . (7)5 H W,Z i h H W,Z i H φ φ ψ ψ ¯ χ a χ a χ a ¯ χ a H M P h H W,Z i h H W,Z i H φ φ ψ ψ ¯ χ a χ a χ a ¯ χ a H M P h H W,Z i h H W,Z i H φ φ ψ ψ ¯ χ a χ a χ a ¯ χ a H M P Figure 3: Diagrams coupling ψ to H . These give the RH neutrino mass matrix. The first 3 terms in the superpotential in Eq. 5 fix the alignments X (cid:48) ( φ S,U ) −→ ω n ω n , (8) X ( φ T ) −→ , − ω n − ω n , (9) X ( φ t ) −→ , , x − /x , (10)up to an integer ( n ∈ Z ) or continuos ( x ∈ R ) parameter, with ω = e πi/ . We may noticethat the three solutions for (cid:104) φ S,U (cid:105) are related one to another by a T transformation. Wemay choose it to be (1 , , T without loss of generality.The (cid:104) φ T (cid:105) has four different solutions. The last three solutions are related by a T trans-formation. From these three, the one without any ω is related to the first solution byan S transformation. Since they are all related, we may choose (1 , , T without loss ofgenerality.The (cid:104) φ t (cid:105) has three different solutions. The third solution is not related to the first two byany symmetry transformations. The fourth term in the superpotential fixes the solutionto be either (0 , , T or (0 , , T , which are related by an U transformation and wechoose the former without loss of generality.The fifth and sixth terms fix φ to be orthogonal to φ t and φ T so that it is fixed to be(0 , , T .The first term from the second line in Eq. 5 involves( (cid:104) φ S,U (cid:105) · (cid:104) φ T (cid:105) ) (cid:48) ∝ − , (11)and together with the fifth one fixes (cid:104) φ (cid:105) into this direction. The third term in the secondline involves ( (cid:104) φ (cid:105) · (cid:104) φ (cid:105) ) (cid:48) ∝ , (12)and together with the seventh term we fix (cid:104) φ (cid:105) into this direction. Furthermore the ξ field that does not add anything to the alignment but it plays a role in the VEV drivingas explained below. 6he F-term equations from the X, Z fix the alignments to be (cid:104) φ S,U (cid:105) = v , (cid:104) φ T (cid:105) = v , (cid:104) φ t (cid:105) = v t (cid:104) φ (cid:105) = v , (cid:104) φ (cid:105) = v − , (cid:104) φ (cid:105) = v , (13)where the last three flavons couple to the matter superfield ψ and determine the fermionmass matrix structure. The flavon VEVs v i are, in general, complex, and spontaneouslybreak the assumed CP symmetry of the high energy theory. The model gives a natural understanding of the SM fermion masses through the hierarchybetween the flavon VEVs | v | (cid:28) | v | (cid:28) | v | . Here, we show the symmetry breakingsuperpotential that produces such hierarchy between the VEVs, W DV ∼ ˜ Z ξ (cid:18) φ − φ φ M P (cid:19) + ˜ Z φ T M P (cid:18) φ φ − φ (cid:80) i φ i M P + O (1 /M P ) (cid:19) + Z (cid:32) M GUT + (cid:88) i φ i + ( H W,Z ) + ( H B − L ) + ζ + Z + O (1 /M P ) (cid:33) + H B − L (cid:18) ( H X,Y ) + ζM P (cid:0) ( H W,Z ) + ( H B − L ) (cid:1) + H X,Y H H M P + DT + O (1 /M P ) (cid:19) , (14)where we have ignored dimensionless couplings for simplicity.The first term of Eq. 14 also appears in the alignment potential in Eq. 5 and fixes | ˜ κ v | = (cid:12)(cid:12)(cid:12)(cid:12) v v M P (cid:12)(cid:12)(cid:12)(cid:12) , (15)where ˜ κ denotes an effective dimensionless coupling coming from the ones in the su-perpotential. Note that we have written this equation as only fixing the modulus. Thishappens due to the appearance of the field ξ ; We assume there are two copies of that field,which get a VEV with an arbitrary phase. This phase, together with the dimensionlesscouplings for each term, does not allow to relate the phases of the v i .The second term of Eq. 14 fixes the VEVs˜ κ v v = v M P (cid:88) i v i , (16)where ˜ κ denotes an effective dimensionless coupling coming from the ones in the super-potential. This equation, together with the previous one, require a hierarchy in the v i ’s.Specifically it requires v , (cid:29) v . 7he field ˜ Z does not obtain a VEV to comply with the F-term equations from the flavons.The second line of Eq. 14 drives the linear combination M GUT ∼ (cid:88) i v i + (cid:104) H W,Z (cid:105) + (cid:104) H B − L (cid:105) + (cid:104) ζ (cid:105) + (cid:104) Z (cid:105) , (17)where we assume that the sum of v i and the all adjoints get a GUT scale VEV. Thefield Z does not get a VEV due to the F-term equations coming from the adjoints. Thisequation does not fix the phases of the VEVs. We assume that the (cid:104) H W,Z (cid:105) are real whilethe phase of the sum of flavon VEVs is unconstrained (only related to the one of (cid:104) ζ (cid:105) which does not appear at low energies). We assume that the flavons obtain a VEV thatbreak the CP symmetry with an arbitrary phase.The third line of Eq. 14 drives (cid:104) ζ (cid:105) M P (cid:0) (cid:104) H W,Z (cid:105) + (cid:104) H B − L (cid:105) (cid:1) ∼ (cid:104) H X,Y (cid:105) + (cid:104) H X,Y (cid:105) (cid:104) H H (cid:105) M P , (18)where we assume that the (cid:104) H X,Y (cid:105) is real. The DT represent all the terms involved in theD-T splitting (shown in Sec.5) that do not contribute to the F-term equation, but theyare there nonetheless. The F-term equations coming from the H X,Y force the χ u,d to alsoget a VEV and does not change any low energy phenomenology.The F-term equations previously discussed can give a VEV to the adjoint fields but donot fix their direction. The adjoint fields can get a VEV in any SM preserving direction.In general they can be written as a linear combination of the U (1) X,Y directions. We donot assume any specific direction for the VEVs (cid:104) H W,X,Y,Z , ζ (cid:105) . We assume that (cid:104) H B − L (cid:105) lies in the U (1) B − L direction. We assume that the (cid:104) H , (cid:105) lie in the right handedneutrino direction.Using the first three equations 15- 17, we may find that the flavon VEVs v = ˜ κ M GUT √ ˜ κ ˜ κ M P v , v = √ ˜ κ ˜ κ M GUT √ ˜ κ , v = ˜ κ M GUT , (19)in this way, if we assume that ˜ κ (cid:39) . , ˜ κ (cid:39) , ˜ κ (cid:39)
1, we have v (cid:39) M GUT , v (cid:39) . M GUT , v (cid:39) . M GUT , (20)which generates the hierarchy between the fermion families. We note that the hierachybetween v and v is given by the structure of the F-term equations. The hierarchybetween v and v is assumed and realized by a much milder hierarchy between thecouplings in the superpotential.Using Eq. 16 and knowing that v (cid:29) v , , we approximately get˜ κ v v (cid:39) v M P , (21) It can be written as the linear combination B − L = ( − X + 4 Y ) / v + arg v (cid:39) v , (22)that in terms of the physical phases is η (cid:39) η (cid:48) − γ, (23)this way there are only 2 free physical phases. The Higgs fields H u,d and H , contain SU (2) doublets and SU (3) triplets. We needthe triplets to be heavy since they mediate proton decay, while two of the doubletsremain light so they can be associated to the MSSM Higgs doublets. This is known asthe doublet-triplet splitting problem and can be solved using the Dimopolous-Wilczekmechanism [22]. In our case this mechanism is in place since we assume that (cid:104) H B − L (cid:105) liesin the U (1) B − L direction. Furthermore, there are extra pairs of doublets, and they arerequired to be heavy to preserve gauge coupling unification. Using the fields in Tables1-2, we may write the superpotential involving the Higgs fields (ignoring dimensionlessparameters) W H = H B − L (cid:0) H u H d + ζ ζ + H χ u + H χ d (cid:1) + H H u χ u + H H d χ d + H H ζ + ζ ( ζ ζ + χ u χ u + χ d χ d )+ H B − L (cid:32) H H H d M P + H H H u M P + H u H d ( H X,Y,W,Z ) M P (cid:33) . (24)After integrating out the messengers ζ i , χ j , the superpotential becomes W H = H B − L (cid:32) κ H u H d + κ ( H H ) (cid:104) ζ (cid:105) + κ H u H d ( H X,Y,W,Z ) M P + κ H H H u (cid:104) ζ (cid:105) + κ H H H d M P + κ H H H u M P + κ H H H d (cid:104) ζ (cid:105) (cid:19) . (25)We remember that the magnitude of the VEVs is (cid:104) H (cid:105) (cid:39) (cid:104) H (cid:105) (cid:39) (cid:104) H (cid:105) = M GUT , (26)and we define z = M GUT / (cid:104) ζ (cid:105) , y = M GUT /M P . (27)Denoting the up (down)-type doublet inside each H as u ( d ) ( H u, ( d )10 ), and similarly forthe triplets, the mass matrix for the triplets becomes M T ∼ u ( H u ) u ( H d ) u ( H ) d ( H d ) κ κ y d ( H u ) 0 − κ κ z d ( H ) κ y κ z κ z M GUT , (28)9hat has as approximate eigenvalues m T ∼ κ M GUT , κ M GUT , κ z M GUT , (29)so that it requires κ ∼ κ z ∼ , to get the triplets at the GUT scale. The doubletsmass matrix is M D ∼ u ( H u ) u ( H d ) u ( H ) d ( H d ) − κ y κ y d ( H u ) 0 κ y κ z d ( H ) κ y κ z κ z M GUT , (30)that has as eigenvalues m D ∼ − y M GUT , κ κ z M GUT , κ z M GUT , (31)so that we must have κ κ z ∼ κ z ∼ , to get two doublet pairs at the GUT scale.Furthermore, there is a µ term generated by µ ∼ y M GUT ∼ T eV, (32)which happens at the correct order.The light MSSM doublets are h u (cid:39) u ( H u ) + κ yκ z u ( H d ) , h d (cid:39) d ( H d ) + κ yκ z u ( H d ) , (33)so that the second term is suppressed to be < − and we may safely assume that h u ( d ) lies only inside H u ( d )10 . One of the characteristic features of GUTs is the prediction of proton decay. It has notbeen observed and the proton lifetime is constrained to be τ p > years [1].Proton decay can be mediated by the extra gauge bosons of the GUT and by the tripletsaccompanying the Higgs doublets. In SUSY SO (10) GUTs, the main source for protondecay comes from the triplet Higgsinos. The decay width is dependent on SUSY breakingand the specific coupling texture of the triplets and determining it exactly lies beyondthe scope of this paper. In general the constraints are barely met when the triplets havea mass at the GUT scale [23], and in Sec. 5 we have shown this is our case.The existence of additional fields in the model may allow proton decay to arise fromeffective terms of the type gQQQL (cid:104) X (cid:105) M P . (34)10uch terms must obey the constraint g (cid:104) X (cid:105) < × GeV [23]. In our model, the largestcontribution of this type comes from the term ψψψψ (cid:104) H B − L ( H X,Y ) (cid:105) M P ⇒ (cid:104) X (cid:105) = ( M GUT ) M P ∼ GeV . (35)The constraints are met when g < .
3. With an O (1) g parameter, the contributionscoming from these terms are the same order as the ones coming from the Higgs triplets.In this model, proton decay complies with experimental constraints but lies fairly closeto detection. Now that we have given VEVs to the fields in a specific direction, we may write the fullydetailed Yukawa structure.With the fields in the Table 1, together with the messenger fields in Table 2 we may writethe superpotential relevant to the Yukawa terms, up to O (1 /M P ), W Y = (cid:88) a =1 , , (cid:18) λ φa ( ψφ a ) ¯ χ a + ( λ Wa H W + λ Za H Z ) χ a ¯ χ a + λ ua χ a χ a H u + λ Na χ a χ a H H M P (cid:19) + (cid:88) b =2 , (cid:16) χ b χ db ( λ Xa H X + λ Ya H Y ) + λ db χ db χ db H d (cid:17) + λ d χ χ H d + λ dt ( ψψ ) (cid:48) φ H d M P , (36)where ( ) , ( ) (cid:48) means an S singlet or 3 (cid:48) contraction respectively. The λ ’s are dimen-sionless and real coupling constants (due to CP conservation) and are all expected to be O (1).After integrating the messengers χ , we obtain the superpotential W Y = (cid:88) a =1 , , (cid:18) ( λ φa ) ( ψ (cid:104) φ a (cid:105) ) ( ψ (cid:104) φ a (cid:105) )( λ Wa (cid:104) H W (cid:105) + λ Za (cid:104) H Z (cid:105) ) λ uχ H u + ( λ φa ) ( ψ (cid:104) φ a (cid:105) ) ( ψ (cid:104) φ a (cid:105) )( λ Wa (cid:104) H W (cid:105) + λ Za (cid:104) H Z (cid:105) ) λ Na M P (cid:104) H (cid:105) (cid:104) H (cid:105) (cid:19) + (cid:32) (cid:88) b =2 , λ ua ( λ φb ) ( ψ (cid:104) φ b (cid:105) ) ( ψ (cid:104) φ b (cid:105) )( λ Xb (cid:104) H X (cid:105) + λ Yb (cid:104) H Y (cid:105) ) + λ d λ φ λ φ ( ψ (cid:104) φ (cid:105) ) ( ψ (cid:104) φ (cid:105) )( λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) )( λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) )+ λ dt ( ψψ ) (cid:48) (cid:104) φ (cid:105) M P (cid:19) H d . (37)This superpotential generates all the SM fermion masses. In Eq. 37, all the terms suppressed by (cid:104) H X,Y,W,Z (cid:105) involve integrating out the messengersby assuming M GUT (cid:29) v i . This naive integration is not possible for the third flavon since11t has a much larger VEV v ∼ M GUT . Let us single out the terms in W Y involving thesefields. Ignoring O (1) couplings, and after the fields get their VEV, the relevant terms are W (3) Y ∼ v ψ χ + (cid:104) H W,Z (cid:105) χ χ . (38)Naively, one would interpret ψ as the set of third-family particles, but the first term inEq. 38 generates mixing with χ . To obtain the physical (massless) states, which we label t , we rotate into a physical basis ( ψ , χ ) → ( t, χ ) ψ = (cid:104) H W,Z (cid:105) t + v χr , χ = − v t + (cid:104) H W,Z (cid:105) χr ; r = (cid:113) v + (cid:104) H W,Z (cid:105) . (39)Physically, it may be interpreted as follows: inside the original superpotential W Y lie theterms W Y ⊃ χ χ H u,d ⊃ v v + (cid:104) H W,Z (cid:105) t t H u,d , (40)which generate renormalisable mass terms for the third family at the electroweak scale. The superpotential in Eq. 37 generates all the SM fermion mass matrices. The structureof the mass matrices is fixed by the flavon VEV structure shown in Eq. 1. We mayredefine the dimensionless couplings to obtain the mass structure of the SM fermions at12ow energies y ua =1 , = λ ua ( λ φa ) | v a | [ λ Wa (cid:104) H W (cid:105) + λ Za (cid:104) H Z (cid:105) ] Q [ λ Wa (cid:104) H W (cid:105) + λ Za (cid:104) H Z (cid:105) ] u c ,y u = ( λ φ ) | v | ( λ φ ) v + [ λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) ] Q [ λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) ] u c ,y νa =1 , = λ ua ( λ φa ) | v a | [ λ Wa (cid:104) H W (cid:105) + λ Za (cid:104) H Z (cid:105) ] L [ λ Wa (cid:104) H W (cid:105) + λ Za (cid:104) H Z (cid:105) ] ν c ,y ν = ( λ φ ) | v | ( λ φ ) v + ( λ χ ) [ λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) ] L [ λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) ] ν c ,y e = λ d ( λ φ ) | v | [ λ X (cid:104) H X (cid:105) + λ Y (cid:104) H Y (cid:105) ] L [ λ X (cid:104) H X (cid:105) + λ Y (cid:104) H Y (cid:105) ] e c ,y e = λ d ( λ φ ) | v | ( λ φ ) v + [ λ X (cid:104) H X (cid:105) + λ Y (cid:104) H Y (cid:105) ] L [ λ X (cid:104) H X (cid:105) + λ Y (cid:104) H Y (cid:105) ] e c ,y d = λ d ( λ φ ) | v | [ λ X (cid:104) H X (cid:105) + λ Y (cid:104) H Y (cid:105) ] Q [ λ X (cid:104) H X (cid:105) + λ Y (cid:104) H Y (cid:105) ] d c ,y d = λ d ( λ φ ) | v | ( λ φ ) v + [ λ X (cid:104) H X (cid:105) + λ Y (cid:104) H Y (cid:105) ] Q [ λ X (cid:104) H X (cid:105) + λ Y (cid:104) H Y (cid:105) ] d c ,y e = λ d λ φ λ φ | v v | [ λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) ] L + e c [ λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) ] L + e c ,y d = λ d λ φ λ φ | v v | [ λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) ] Q + d c [ λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) ] Q + d c ,M R a =1 , = λ Na v M P ( λ φa ) | v a | [ λ Wa (cid:104) H W (cid:105) + λ Za (cid:104) H Z (cid:105) ] ν c ,M R3 = λ N v M P ( λ φ ) | v | ( λ φ ) v + [ λ W (cid:104) H W (cid:105) + λ Z (cid:104) H Z (cid:105) ] ν c ,y P = λ dt Y P v M P , (41)where (cid:104) H X,Y,W,Z (cid:105) f denotes the adjoint VEV with the corresponding CG coefficients foreach SM fermion f . This allows for each y, M parameter in Eq. 41 to be independent.The VEVs (cid:104) H X,Y (cid:105) obtain a VEV in an arbitrary SO (10) breaking direction. They needto be different from one another.For a better understanding we can show an explicit example. Let us assume that (cid:104) H X,Y (cid:105) is aligned in the U (1) X,Y direction respectively with an M GUT magnitude. In this casethe effective Yukawa couplings y e,d would be y e = λ d ( λ φ ) | v | [3 λ X − λ Y / − λ X + λ Y ] M GUT , y d = λ d ( λ φ ) | v | [ − λ X + λ Y / λ X + λ Y / M GUT , (42)where the coefficients multiplying each λ X,Y are the U (1) X,Y charges of the correspondingSM field. Since the λ X,Y appear with different coefficients in y e,d , we can use them toobtain a arbitrarily different effective Yukawa coupling for charged leptons and down typequarks. 13ssuming the adjoints have all real VEVs, the physical phases are η = 2 arg v − v η (cid:48) = 2 arg v − v γ = arg v − v , (43)while all the y (cid:48) s and M (cid:48) s are real.With these definitions we may write the fermion mass matrices M e /v d = y e e iη/ + y e + y e e iη (cid:48) + y P e iγ ,M d /v d = y d e iη/ + y d + y d e iη (cid:48) + y P e iγ ,M u /v u = y u e iη + y u + y u e iη (cid:48) ,M νD /v u = y ν e iη + y ν + y ν e iη (cid:48) ,M R = M R e iη + M R + M R e iη (cid:48) . (44)We note the remarkable universal structure of the matrices in the up and neutrino sectors,which differ from the down and charged lepton sectors.The y and M parameters are all free and independent while there is a constraint in thephases η (cid:39) η (cid:48) − γ, (45)as shown in the Sec. 4. We have in total 18 free parameters that fix the whole spectrumof fermion masses and mixing angles, as discussed in Sec. 8.2.As shown in the Sec. 4, the flavons and adjoint fields get a VEV M GUT ∼ v ∼ v ∼ v ,M GUT ∼ v X,Y,W,Z ∼ M ρ ∼ v . (46)Assuming all the parameters in the superpotential we have ignored are O (1), and tan β ∼
20, the mass matrix parameters are expected to be y u ∼ y ν ∼ v /v ∼ − , y u ∼ y ν ∼ v /v ∼ − ,y u ∼ y ν ∼ v /v ∼ , y d ∼ y e ∼ cos β v v /v ∼ − ,y d ∼ y e ∼ cos β v /v ∼ − , y d ∼ y e ∼ cos β v /v ∼ . ,y P ∼ cos β v /M P ∼ − , M R ∼ v v /v M P ∼ GeV,M R ∼ v v /v M P ∼ GeV, M R ∼ v v /v M P ∼ GeV. (47)14hese values denote only an approximate order of magnitude for each parameter and areexpected to be different due to the appearance of dimensionless couplings. This appliesspecially to the last 4 parameters that come from unknown Planck suppressed physicsand may deviate significantly from our naive expectation.
Since we have very heavy RHN Majorana masses, the left handed neutrinos get a verysmall Majorana mass through type I seesaw m νL = M νD ( M R ) − ( M νD ) T . (48)As we see in Eq. 44, the Dirac neutrino masses M νD and RHN Majorana masses M R havethe same matrix structure. These are rank one matrices so that we may write them as M νD /v u = y ν e iη ϕ ϕ T + y ν ϕ ϕ T + y ν e iη (cid:48) ϕ ϕ T ,M R = M R e iη ϕ ϕ T + M R ϕ ϕ T + M R e iη (cid:48) ϕ ϕ T , (49)with ϕ T = (1 , , , ϕ T = (0 , , , ϕ T = (0 , , . (50)We may always find vectors ˜ ϕ a such that˜ ϕ Ti ϕ j = δ ij , (51)this way we may write the inverse matrix as( M R ) − = e − iη M R ˜ ϕ ˜ ϕ T + 1 M R ˜ ϕ ˜ ϕ T + e − iη (cid:48) M R ˜ ϕ ˜ ϕ T . (52)Plugging this into the seesaw mechanism we obtain the light effective left-handed Majo-rana neutrino mass matrix m νL , m νL = µ e iη ϕ ϕ T + µ ϕ ϕ T + µ e iη (cid:48) ϕ ϕ T , with µ a = ( y νa v u ) M Ra (53)so that we may conclude that the small left handed neutrino mass matrix has the sameuniversal structure m νL = µ e iη + µ + µ e iη (cid:48) , (54)after the seesaw mechanism. To test our model we perform a numerical fit using a χ test function. We have a set ofinput parameters x = { y ui , y di , y ei , y P , µ i , η (cid:48) , γ } , from which we obtain a set of observables15 n ( x ). We minimize the function defined as χ = (cid:88) n (cid:18) P n ( x ) − P obs n σ n (cid:19) , (55)where the 19 observables are given by P obs n ∈ { θ qij , δ q , y u,c,t , y d,s,b , θ (cid:96)ij , δ l , y e,µ,τ , ∆ m ij } withstatistical errors σ n . This test assumes data is normally (Gaussian) distributed, which istrue for most of the observables except for θ (cid:96) . The atmospheric mixing angle octant, i.e. θ (cid:96) < ◦ or θ (cid:96) > ◦ , has not been determined yet. Current data favours θ (cid:96) = 41 . Observable Data ModelCentral value 1 σ range Best fit θ (cid:96) / ◦ → θ (cid:96) / ◦ → θ (cid:96) / ◦ → δ (cid:96) / ◦ → y e / − → y µ / − → y τ → m / (10 − eV ) 7.510 7.330 → m / (10 − eV ) 2.524 2.484 → m /meV 10.94 m /meV 13.95 m /meV 51.42 (cid:80) m i /meV <
230 76.31 α / ◦ α / ◦ m ββ /meV < β = 20, M SUSY = 1 TeV and ¯ η b = − . χ is 1.2. Theneutrino masses m i as well as the Majorana phases are pure predictions of our model. Thebound on (cid:80) m i is taken from [24]. The bound on m ββ is taken from [25]. We need to run up all the measured Yukawa couplings and mixing angles up to the GUTscale in order to compare it with the predictions of our model. In doing so, we needto match the SM to the MSSM at the SUSY scale, M SUSY , which involves adding thesupersymmetric radiative threshold corrections. This has been done in [27]. At the GUTscale, the values depend to a good approximation only on ¯ η b and tan β . A good fit is foundfor large ¯ η b , which can be explained if tan β (cid:38)
10, as shown in the Sec. 8.1. We also needtan β <
30 to keep Yukawa couplings perturbative. The best fit is found for ¯ η b = − . β = 20. The SUSY scale does not affect the fit and we choose M SUSY = 1 TeV.The fit has been performed using the Mixing Parameter Tools (MPT) package [28].The best fit found has a χ = 11 .
9. Table 3 shows the best fit to the charged leptons andneutrinos observables. Neutrino data is taken from the Nufit global fit [26]. Only the Note that we are performing the numerical fit in terms of the effective neutrino mass parameters µ i defined in Eq. 54. We are ignoring any renormalisation group running corrections in the neutrino sector. m ββ . All the lepton sectoris fitted to within 1 σ except for the leptonic CP phase. δ (cid:96) is not yet well measured,although a negative CP phase is preferred [29]. Observable Data ModelCentral value 1 σ range Best fit θ q / ◦ → θ q / ◦ → θ q / ◦ → δ q / ◦ → y u / − → y c / − → y t → y d / − → y s / − → y b → β = 20, M SUSY = 1 TeV and ¯ η b = − . χ is 10.7. In table 4, we have all the quark Yukawa couplings and mixing parameters for the min-imum χ . The biggest contribution to the χ is coming from this sector, as shown inFig. 4. This figure shows the corresponding pulls for lepton (light orange) and quark(blue) observables. As we can see, all parameters lie inside the 2 σ region and the biggestpulls are in the quark Yukawa couplings.Table 5 shows the input parameter values. There are 13 real parameters plus twoadditional phases, a total of 15 input parameters to fit 19 data points. Naively, we canmeasure the goodness of the fit computing the reduced χ , i.e. the χ per degree offreedom χ ν = χ /ν . The number of degrees of freedom is given by ν = n − n i , where n = 19 is the number of measured observables, while n i = 15 is the number of inputparameters. A good fit is expected to have χ ν ∼
1. We have 4 degrees of freedom andthe best fit has a reduced χ ν (cid:39)
3. We view this as a good fit and it also remarks thepredictivity of the model, not only fitting to all available quark and lepton data but alsofixing the neutrino masses and Majorana phases. Assuming the Dirac neutrino Yukawa parameters y νi in Eq. 47, we can compute the RHN masses,using the Seesaw formula in Eq. 53 and taking the µ i values from the fit, such that M R ∼ GeV, M R ∼ GeV and M R ∼ GeV. Only M has the expected natural value given in Eq. 47. Weremark that RHN Majorana masses come from unknown Planck suppressed physics, which is presumablyresponsible for the mismatch. l θ l θ l δ l Δ m Δ m y e y μ y τ θ q θ q θ q δ q y u y c y t y d y s y b - - - - Figure 4: Pulls for the best fit of model to data, as shown in Tables 3-4, for lepton (lightorange) and quark (blue) parameters.Parameter Value y u / − y u / − y u − . y d / − − . y d / − − . y d − .
238 Parameter Value y e / − . y e / − . y e . µ /meV 6.845 µ /meV 27.18 µ /meV 42.17 Parameter Value y P / − . γ . πη (cid:48) . π Table 5: Best fit input parameter values. The model has 13 real parameters: y ui , y di , y ei , µ i and y P and two additional free phases: η (cid:48) and γ . The total χ is 11.9. The running of the MSSM Yukawa couplings to the GUT scale, M GUT , was performedin [27]. Here, the threshold corrections at the SUSY scale, M SUSY , are parametrized by y MSSM u,c,t (cid:39) y SM u,c,t csc ¯ β,y MSSM d,s (cid:39) (1 + ¯ η q ) − y SM d,s sec ¯ β,y MSSM b (cid:39) (1 + ¯ η b ) − y SM b sec ¯ β,y MSSM e,µ (cid:39) (1 + ¯ η (cid:96) ) − y SM e,µ sec ¯ β,y MSSM τ (cid:39) y MSSM τ sec ¯ β. (56)The CKM parameters become θ q, MSSM i (cid:39) η b η q θ q, SM i , θ q, MSSM12 (cid:39) θ q, SM12 , δ q, MSSM (cid:39) δ q, SM . (57)When running between M SUSY and M GUT , the most relevant parameters are ¯ η b and tan ¯ β .Due to their small contribtutions, we assume ¯ η q = ¯ η (cid:96) = 0 and β = β . These assumptionsdo not affect the quality of the fit. Similarly, we fix M SUSY = 1 TeV. The effect on thefit, of having it up to O (10) TeV, is minor.18pecifically, the parameter ¯ η b is required to be somewhat large ¯ η b = − . χ ∼ η b (cid:39) tan β π (cid:18) g m ˜ g µ m + λ t µA t m (cid:19) , (58)where m represents the squark masses, g the strong coupling, m ˜ g the gluino mass and A t the SUSY softly breaking trilinear coupling involving the stops. We see that a largecontribution can be achieved when m ˜ g , µ, A t > m , tan β (cid:38) . (59)Since SUSY breaking lies beyond the scope of our paper, it is sufficient for us to showthat there is a parameter space in the softly broken SUSY that generates the necessarycorrections. In this section we explain and clarify the number of parameters in our model. Clearly atthe high energy scale there are many parameters associated with the undetermined O (1)Yukawa couplings of the 43 superfields of the model. For example the renormalisableYukawa superpotential in Eq. 36 contains 23 parameters alone. Then we must add tothis all the O (1) Yukawa couplings associated with vacuum alignment, GUT symmetrybreaking and doublet-triplet splitting, many of which we have not defined explicitly.Despite this, we are claiming that our model is predictive at low energies. How can thisbe? The short answer is that most of these parameters are irrelevant for physics belowthe GUT scale, as discussed in detail below.The effective fermion mass matrices generated below the GUT scale are given in Eq. 44 asfunction of 18 free effective parameters (remembering the constraint on η ) that will fix allthe fermion masses and mixing angles, including RHN Majorana masses and Majoranaphases. This compares favourably to the 31 parameters of a general high energy model,comprising 21 parameters in the lepton sector of a general 3 right-handed neutrino seesawmodel [31], plus the 6 quark masses and 4 CKM parameters. However, below the seesawscale of right-handed neutrino masses, the effective parameter counting is different againand requires further discussion below.In order to perform the fit and compare our model with available data, we apply the see-saw mechanism to write the light effective left-handed Majorana neutrino mass matrixas a function of the new parameters µ i in Eq 53. Therefore, we have 15 effective pa-rameters at low energy (shown in Table 5) that fit the 19 so far measured or constrainedobservables in Fig. 4 . After the fit is performed, the model predicts all the three lightneutrino masses with a normal ordering, a CP oscillation phase of 260 ◦ and both theMajorana phases, corresponding to a total of 22 low energy observables which will be We need to run up to the GUT scale these observables and, therefore, we need to include SUSYthreshold corrections. The fit is therefore also dependent on η b and tan β . As shown earlier, we find agood fit for η b = − . β = 20. N Leptogenesis
The source of the Baryon Asymmetry of the Universe (BAU) η CMBB = (6 . ± . × − , (60)remains unexplained in the SM. One of the most convincing sources for it is Leptogenesis,where the asymmetry is generated through CP breaking decays of heavy RHNs intoleptons, then converted into baryons through sphalerons [32].The simplest mechanism to generate the correct BAU, happens when the lightest RHNshas CP breaking decays and has a mass of about ∼ GeV . In our model, accordingto Eq. 47, it is the second RHN the one that is expected to be at that scale. Whenleptogenesis is generated mainly by the decays of the second RHN it is called N lepto-genesis. This has already been calculated in [20] and we will apply such calculations toour specific model. N leptogenesis. Leptogenesis calculations are done in the so called Flavor Basis, where the charged leptonand RHN mass matrices are diagonal and we work with the Dirac neutrino mass matrix m D = V eL M νD U TN ,V eL M e † M e V † eL = diag ( y e , y µ , y τ ) v d , U N M R U TN = diag ( M , M , M ) . (61)The total and flavoured decay parameters, K i and K iα respectively, can be written as K iα = | m Dαi | m MSSM(cid:63) M i and K i = (cid:88) α K iα = ( m † D m D ) ii m MSSM(cid:63) M i , (62)where the equilibrium neutrino mass is given by m MSSM(cid:63) (cid:39) . × − eV sin β. (63)The wash-out at the production is described by the efficiency factor κ ( K α ) that for aninitial thermal N abundance can be calculated as κ ( K α ) = 2 z B ( K α ) K α (cid:16) − e − K α zB ( K α )2 (cid:17) , z B ( K α ) (cid:39) K . α e − . K α . (64)20n the hierarchical RH neutrino mass limit, as our model is, the CP asymmetries can beapproximated to ε = (cid:88) α ε α , ε α (cid:39) π M v Im (cid:104)(cid:0) m † D (cid:1) iα (cid:0) m D (cid:1) α (cid:0) m † D m D (cid:1) i (cid:105) M M (cid:101) m , (65)where (cid:101) m ≡ ( m † D m D ) /M .In the regime where 5 × GeV (1 + tan β ) (cid:29) M (cid:29) × GeV (1 + tan β ), the final B − L asymmetry can be calculated using N f B − L (cid:39) (cid:34) K e K τ ⊥ ε τ ⊥ κ ( K τ ⊥ ) + (cid:32) ε e − K e K τ ⊥ ε τ ⊥ (cid:33) κ ( K τ ⊥ / (cid:35) e − π K e ++ (cid:34) K µ K τ ⊥ ε τ ⊥ κ ( K τ ⊥ ) + (cid:32) ε µ − K µ K τ ⊥ ε τ ⊥ (cid:33) κ ( K τ ⊥ / (cid:35) e − π K µ ++ ε τ κ ( K τ ) e − π K τ , (66)where we indicated with τ ⊥ the electron plus muon component of the quantum flavourstates produced by the N -decays defining K τ ⊥ ≡ K e + K µ , ε τ ⊥ ≡ ε e + ε µ . The finalasymmetry, in terms of the baryon to photon number ratio is η B (cid:39) a sph N B − L N recγ , (67)where α sph = 8 /
23 is the fraction of B − L asymmetry converted into baryon asymmetryby sphalerons. The photon asymmetry at recombination is ( N recγ ) MSSM (cid:39)
78. The factorof 2 accounts for the asymmetry generated by the RH neutrinos and sneutrinos.
Using the matrices in Eq. 44 and the fit in Table 5, we may calculate the BAU generatedthrough N Leptogenesis in our model. The first thing to note is that the parametersare quite hierarchical so that the rotation angles of the diagonalizing matrices can beneglected since they only give 1% contributions V eL (cid:39) , U N (cid:39) diag ( e − iη/ , , e − iη (cid:48) / ) , (68)and the neutrino mass matrix in the Flavor Basis becomes m Dij (cid:39) y ν e iη/ y ν e iη y ν e iη/ y ν y ν e − iη (cid:48) / y ν y ν e iη (cid:48) / v u . (69)Also, due to the hierarchical nature of the couplings we may safely assume that the RHNmass parameter are equal to their mass eigenvalues M Ra (cid:39) M a .21ne of the features of the matrix structure is that K τ vanishes, due to the approximatezero in the (3,1) entry of the Dirac mass matrix , so that the last term in Eq. 66 isgreatly enhanced since it overcomes the exponential suppression. With these approxima-tions, the baryon asymmetry becomes η B (cid:39) α sph N recγ κ ( K τ ) ε τ ,K τ = ( y ν ) v u m MSSM(cid:63) M , (cid:15) τ = sin η (cid:48) π M M ( y ν ) β. (70)We note that η (cid:48) is identified with the leptogenesis phase. With use of Eq. 53, we maywrite the neutrino Yukawa couplings as y νa = (cid:112) µ a M Ra /v u so that η B (cid:39) sin η (cid:48) π α sph N recγ κ (cid:18) µ m MSSM(cid:63) (cid:19) µ M v , (71)where we note that the only free parameter is M . Using the parameters from the fit, inTable 5, the correct BAU is generated when M (cid:39) . × GeV. (72)From Eq. 47 we see that this is the natural value for the second RHN mass, so thatthe model naturally explains the origin of the BAU through N leptogenesis without anyneed for tuning.
10 Conclusion
We have constructed a SUSY GUT of flavour based on the symmetry S × SO (10) × Z × Z R that is relatively simple, predictive and fairly complete. The Higgs sector of the modelinvolves two SO (10) 10-plets, a 16-plet and its conjugate representation, and three 45-plets. These low dimensional Higgs representations are all that is required to break theGUT symmetry, yield the Clebsch relations responsible for the difference of the chargedfermion masses, and account for heavy Majorana right-handed neutrino masses. In orderto account for the hierarchical mixing structure of the Yukawa matrices, we also need aparticular set of S triplet flavons with hierarchical VEVs and particular CSD2 vacuumalignments, where both features are fully discussed here. To complete the model we alsorequire a rather rich spectrum of high energy messenger and alignment superfields, which,like most of the Higgs fields, do not appear in the low energy effective theory.We highlight and summarise the main successes and features of the model as follows: • The model is succesfully built with an SO (10) gauge symmetry where all of thefields belong to the small “named” representations: fundamental, spinorial andadjoints; this could be helpful for a possible future string embedding. The zero is a consequence of the CSD2 vaccum alignment; it would not be zero for CSD3 vacuumalignment. M has been computed numerically, including the rotation angles of the diagonalizing matrices inEq. 69. It contains a superpotential that spontaneously breaks the original symmetry: S × SO (10) × Z × Z R → SU (3) C × SU (2) L × U (1) Y × Z R . The model alsospontaneously breaks CP . • The S breaking superpotential that yields the CSD2 vacuum alignment is fairlysimple. • All the GUT scale parameters are natural and ∼ O (1), explaining the hierarchyof the low energy parameters, where the family mass hierarchy is due the derivedhierarchy of flavon VEVs | v | (cid:28) | v | (cid:28) | v | , rather than by Froggatt-Nielsen. • The model contains a working doublet-triplet mechanism, that yields exactly twolight Higgs doublets from two SO (10) Higgs multiplets, respectively and withoutmixing, apart from the µ term which is generated at the correct scale. It also haswell behaved proton decay. • The model naturally generates sufficient BAU through N Leptogenesis, which fixesthe second right-handed neutrino mass M (cid:39) × GeV, in the natural rangepredicted by the model. • At low energies, the model contains 15 free parameters that generate 19 presentlyconstrained observables so that it is quite predictive. The model achieves an excel-lent fit of the SM fermion masses and mixing angles, with χ = 11 . O (1 /M P ) terms for the right-handedneutrino masses. Indeed M and M apparently do not have such natural values as M ,and we are forced to explain this away by appealing to the unknown physics at the Planckscale. The symmetry breaking superpotential gives VEVs to most of the GUT breakingfields but it does not drive all of them. Also we do not address the strong CP problem,inflation or Dark Matter (DM) (which may in principle be due to the lightest SUSYparticle, stabilised by the R-parity). Indeed we have not considered the SUSY spectrumat all. Such issues are beyond the stated aims of the present paper, which is to propose acomplete grand unified theory of flavour and leptogenesis, consistent with the latest dataon quark and lepton masses and mixing parameters, in which the three families of quarksand leptons are unified into a single (3 (cid:48) ,
16) representation of S × SO (10).Importantly, the model can be tested due to its robust predictions of a normal neutrinomass ordering, a CP oscillation phase of 260 ◦ , an atmospheric angle of 42 ◦ in the firstoctant and a neutrinoless double beta decay parameter m ββ = 11 meV, with the sum ofneutrino masses being 76 meV. These predictions, together with the other lepton mixingangles given earlier, will enable the model to be tested by the forthcoming neutrinoexperiments. 23 cknowledgements We thank Fredrik Bj¨orkeroth for discussions. S. F. K. acknowledges the STFC Con-solidated Grant ST/L000296/1. This project has received funding from the EuropeanUnion’s Horizon 2020 research and innovation programme under the Marie Sk(cid:32)lodowska-Curie grant agreements Elusives ITN No. 674896 and InvisiblesPlus RISE No. 690575.
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