SUSY SU(5) with singlet plus adjoint matter and A4 family symmetry

We propose a supersymmetric (SUSY) SU(5) Grand Unified Theory (GUT) including a single right-handed neutrino singlet and an adjoint matter representation below the GUT scale and extend this model to include an A4 family symmetry and a gauged anomaly-free Abelian group. In our approach hierarchical neutrino masses result from a combined type I and type III seesaw mechanism, and the A4 symmetry leads to tri-bimaximal mixing which arises indirectly. The mixing between the single right-handed neutrino and the matter in the adjoint is forbidden by excluding an adjoint Higgs, leading to a diagonal heavy Majorana sector as required by constrained sequential dominance. The model also reproduces a realistic description of quark and charged lepton masses and quark mixings, including the Georgi-Jarlskog relations and the leptonic mixing sum rules s=r*cos(delta) and a=-r^2/4 with r=theta_C/3.


Introduction
There has been much evidence to suggest that the Standard Model (SM) is not a complete description of particle physics and needs extending. Arguably one of the most important pieces of experimental evidence for physics beyond the SM is the measurement of small but non-zero neutrino mass, leading to theories of neutrino mass and mixing. In the seesaw mechanism, a natural explanation for such tiny neutrino masses is provided by the exchange of a heavy particle leading to Majorana neutrino masses suppressed by the large mass of the exchanged particle. The heavy particle in the seesaw mechanism must be a colour singlet but can be an electroweak singlet fermion with zero hypercharge, an electroweak triplet Higgs scalar with two units of hypercharge, or an electroweak triplet fermion with zero hypercharge, corresponding to the type I [1], type II [2], or type III [3] seesaw mechanisms, respectively. In this letter we shall combine the seesaw mechanism of types I and III in a new way to yield a hierarchical spectrum of neutrino masses.
However the seesaw mechanism is not by itself enough to account for the discovery that, in contrast with the smallness of the quark mixing angles, two out of the three leptonic mixing angles are large. This unexpected observation calls for a deeper theoretical understanding of the physics underlying the structure of fermion masses and mixings. It is well known that solar and atmospheric data are consistent with a simple form of the leptonic mixing matrix U, known as tri-bimaximal (TB) mixing [4]: The simple form of this matrix can be interpreted as a clue that points towards some underlying family symmetry G f , related to particular transformations which leave the mass matrix diagonalised by U T B invariant. There has been much recent work based on this idea that the postulated TB symmetry can arise from a family symmetry . The approaches taken in the literature may be separated into two distinct classes, distinguished by the breaking of the family symmetry [26]: direct models, based on A 4 , S 4 , or larger groups containing these as subgroups [27,28], have part of the family symmetry preserved at low energies and this forms some or all of the neutrino flavour symmetry; indirect models, usually based on ∆(3n 2 ) or ∆(6n 2 ) [29], have entirely broken family symmetries (in the neutrino sector), and the neutrino flavour symmetry appears accidentally.
Finding an explanation for the distinctly different mixing patterns of leptons as compared to quarks is even more important in the context of Grand Unified Theories (GUTs) [30,31], where the fermionic matter is unified at high energies into either a single representation, as in SO(10) [32] or E 6 [33], or into two representations, as in SU(5) [34] or SU(4) P S × SU(2) L × SU(2) R [35]. The minimal GUT [34] is based on the Lie group SU (5), where one family of right-handed down quarks and left-handed leptons are unified in a 5 and the rest of the family are in a 10. Three copies of each of these representations then constitutes the full fermionic matter content of minimal SU (5). It is well known that gauge coupling unification fails in this regime, however if promoted to a supersymmetric (SUSY) GUT [36] with two Higgs multiplets, H 5 and H 5 , then unification occurs at a scale of roughly 2 × 10 16 GeV [37].
It is clear that neutrino masses are zero at the renormalisable level in minimal (SUSY) SU(5) as in the minimal SM. However, in both the SM and SU (5), as pointed out by Weinberg [38], one may invoke a non-renormalisable dimension-5 operator at or above the GUT scale to generate neutrino masses. Such an operator at the GUT scale may be sufficient to describe the solar neutrino mass scale, but not the atmospheric neutrino mass scale. In order for neutrinos to obtain mass consistent with atmospheric mixing in a (SUSY) SU(5) model, the seesaw mechanism is a very attractive possibility, however this requires some extra matter or Higgs to be added below the GUT scale. The choice of additional matter or Higgs is very ad hoc since the SU(5) theory does not specify the nature of this extra matter and only requires that it be anomaly-free. A popular choice is to add three right-handed neutrinos which arise from singlet SU (5) representations. However the number of singlets is not predicted in SU (5), and it is possible to add just a single right-handed neutrino to describe the atmospheric mass scale [39]. In order to describe both atmospheric and solar neutrino masses with two large mixing angles using the type I seesaw mechanism two right-handed neutrinos are sufficient [40]. However, within SU(5) GUTs, there are other possibilities.
It has been pointed out that, in (SUSY) SU(5) GUTs, non-fundamental matter multiplets have decompositions which include both fermion singlets and fermion triplets suitable for the type I and III seesaw mechanism, the smallest such example being the adjoint 24 representation [41][42][43]. The decomposition of a matter 24 under the SM gauge group SU(3) c × SU(2) L × U(1) Y involves an SU(2) L singlet ρ 0 = (1, 1) 0 as well as a triplet ρ 3 = (1, 3) 0 , thus leading to a combination of a type I seesaw with a type III seesaw [3]. However, assuming the simplest Higgs sector, the ρ 0 and ρ 3 are constrained by SU(5) to give equal contributions to the neutrino mass matrix, up to an overall constant, resulting in a rank one neutrino mass matrix and only one non-zero neutrino mass. This problem may be addressed by allowing additional couplings to a Higgs 45 [43], but here we shall consider a different possibility.
In this letter we consider a SUSY SU(5) GUT with one single right-handed neutrino singlet superfield N plus one adjoint matter superfield ψ 24 below the GUT scale. The model combines a type I seesaw mechanism from the single right-handed neutrino N below the GUT scale [39] with a type I plus type III seesaw mechanism from the ρ 0 and ρ 3 components contained in a single adjoint matter superfield ψ 24 below the GUT scale [43]. The seesaw mechanism in our model therefore results from three distinct diagrams as shown in Fig. 1. In order to describe TB mixing we also include an Figure 1: Schematic diagrams of the type I (left) and combined type I + type III (right) seesaw mechanisms present in the model. The seesaw messenger states are N and the ρ 0 , ρ 3 components of ψ 24 . L is the SU (2) L doublet contained in the 5 of SU (5).
family symmetry, plus an anomaly-free gauged U(1) symmetry. Instead of using an adjoint Higgs representation H 24 to spontaneously break SU (5) to the SM gauge group, we shall assume implicitly that the GUT group is broken by geometrical effects in extra dimensions. However the theory here is formulated in four dimensions and we simply assume that it could subsequently be uplifted to a higher dimensional setting (as in, for example, [44]). The absence of H 24 is crucial in forbidding the mixing between the righthanded neutrino N and ψ 24 , leading to no mass mixing between N and ρ 0 and hence a diagonal heavy Majorana sector as required by constrained sequential dominance (CSD) [45]. The flavon vacuum alignments arise from the elegant D-term mechanism [46]. The model also reproduces a realistic description of quark and charged lepton masses and quark mixings, including the Georgi-Jarlskog relations [47]. The remainder of this letter is organised as follows. In Section 2 we introduce the SUSY SU(5) model with singlet plus adjoint matter below the GUT scale. Section 3 describes the full version of the model, including an A 4 family symmetry, a gauged U(1) symmetry plus two discrete Z 2 symmetries, and lists the operators allowed by these symmetries, resulting in the mass matrices for quarks, charged leptons and neutrinos. We conclude in Section 4.

SUSY SU (5) with singlet & adjoint matter
In this Section we consider a SUSY SU(5) GUT with one single right-handed neutrino arising from a singlet representation N below the GUT scale plus one extra adjoint matter representation ψ 24 with mass also below the GUT scale. The matter contained in the ψ 24 is degenerate thus avoiding problems with gauge coupling unification. The model represents a new way to achieve a hierarchical neutrino mass spectrum arising from a type I plus type III seesaw mechanism, as we now discuss.
The superpotential describing the neutrino sector takes the form with the F i denoting the three families of 5s. N is a single right-handed Majorana neutrino superfield and ψ 24 the additional adjoint matter superfield. The seesaw diagrams illustrated in Fig. 1 then yield the light neutrino mass matrix, Here v u is the vacuum expectation value (VEV) of the Minimal Supersymmetric Standard Model (MSSM) Higgs field H u which corresponds to the SU(2) L doublet within the SU (5) Higgs H 5 . The numerical factors of the two terms in parentheses are obtained by writing ψ 24 =ρ a T a , whereρ a are the 24 components of the ψ 24 and T a are the 24 appropriately normalised generators of SU(5) [30]. As can be seen from Eq. (2), the Majorana masses for the seesaw messengers ρ 0 and ρ 3 are identical, i.e. m ρ 0 = m ρ 3 = m, while N has an independent mass m N . Note that we have not introduced an adjoint Higgs H 24 which would break the degeneracy of the components in the ψ 24 and, more importantly, allow a mixing term Nψ 24 H 24 leading to a mass mixing between N and ρ 0 . Note also that c i and p i are independent dimensionless coefficients (where i and j are family indices); this independence is crucial to obtaining a rank two mass matrix and thus two non-zero neutrino masses. As c i and p i are uncorrelated parameters, Eq. (3) does not in general conform to the TB structure of the neutrino mass matrix. It is the aim of this letter to obtain TB neutrino mixing as a consequence of a discrete family symmetry in this type of model. To this end, in the next Section, we augment the adjoint SUSY SU(5) model with the tetrahedral family symmetry A 4 .

SUSY A 4 × SU (5) with singlet & adjoint matter
In this Section we uplift the model in Eq. (2) to include a tetrahedral family symmetry. We work in the basis of [15] in which two A 4 triplets a = (a 1 , a 2 , a 3 ) T and b = (b 1 , b 2 , b 3 ) T give a singlet through the combination a 1 b 1 + a 2 b 2 + a 3 b 3 . The basic idea is to unify the three families of 5s into an A 4 triplet F ∼ 3, and in order for Eq. (2) to remain invariant, to introduce extra A 4 triplets called flavons ϕ i to break the A 4 symmetry and generate the Yukawa couplings. Table 1 shows the chiral superfields present in the model. As mentioned above, the three 5s of SU(5) are embedded in a triplet of A 4 , while the three 10s are singlets. The ψ 24 is an A 4 singlet as is the right-handed neutrino N. We have fundamental Higgs fields H 5 and H 5 ; introducing another Higgs in the 45 representation, H 45 , enables the implementation of the Georgi-Jarlskog mechanism [47] to obtain the well known GUT scale mass relations m e ∼ m d 3 , m µ ∼ 3m s and m τ ∼ m b . These give phenomenologically successful predictions of down quark and charged lepton masses when evolved down to the electroweak scale.
Field ψ 24 N F T 1 T 2 T 3 H 5 H 5 H 45 ϕ 123 ϕ 23 ϕ 3 ξ ξ ′ ϕ 1 SU (5) 24 1 5 10 10 10 5 The U(1) R represents an R-symmetry. Its Z 2 subgroup gives rise to the standard R-parity which forbids unwanted operators contributing to proton decay and keeps the lightest SUSY particle a good candidate for cold dark matter. Moreover, U(1) R is essential in forbidding F -term contributions to the flavon superpotential which otherwise could dominate the relevant D-term operators used for obtaining the desired vacuum alignment (see below and the discussion in [46]). The U(1) and the two Z 2 symmetries constrain the structure of the Yukawa matrices in the quark and charged lepton sectors. The standard MSSM µ-term 1 µH u H d is forbidden by the first of the Z 2 symmetries as well as by U(1) R , allowing for a natural solution to the µ-problem of the MSSM using a GUT singlet from the hidden sector of Supergravity theories [48].
The flavon fields ϕ i , ξ and ξ ′ break the A 4 symmetry and constrain the form of the lepton and down quark Yukawa matrices. The vacuum alignments of the triplet flavon VEVs that we assume in this model are displayed in Table 2. They are achieved using the D-term vacuum alignment mechanism discussed recently in [46]. This mechanism is ideally suited for models such as this in which the flavons are used to generate the neutrino flavour symmetry as an indirect result of the A 4 symmetry as discussed in [26]. Moreover, the D-term vacuum alignment mechanism does not involve the introduction of extra "driving fields" in the superpotential and does not impose any restrictions on the model other than the requirement that higher order terms in the flavon potential do not spoil the vacuum alignment arising from the D-terms. This has been demonstrated to arise in a fairly generic way in [46] providing that the model also respects a U(1) R symmetry and involves no superfields with R = 2 which, like driving fields, could appear linearly in the superpotential and lead to large terms in the flavon potential. The present model involves only fields with R = 0, 1 and so the D-term flavon potential will not receive large corrections from the superpotential. Since the D-term vacuum alignment mechanism is generic and does not provide any other restrictions on the model than Table 2: The vacuum alignments of the triplet flavons used in the model. Without loss of generality, the alignments are given without phases; the relative sign between ϕ 23 2 and ϕ 23 3 is relevant, though the actual position of the minus sign is mere convention. those stated, in this letter we shall simply assume that this mechanism is in operation, leading to the stated alignments for ϕ 123 , ϕ 23 , ϕ 3 , ϕ 1 .
In order to avoid the massless Goldstone boson associated with the spontaneously broken U(1) symmetry, we assume it to be gauged. 2 In addition to the particle content specified in Table 1 we must then introduce extra matter to cancel the respective gauge anomalies. The cubic SU(5) anomaly requires the introduction of a Higgs field H 45 whose U(1) charge is determined by the mixed SU(5) − SU(5) − U(1) anomaly to be q(H 45 ) = − 53 24 . Finally the cubic U(1) anomaly can be removed in many ways; for example, choosing q 1 = − 126 24 we can add three extra A 4 × SU(5) singlets with U(1) charges 5 24 , 25 24 , 51 24 . Assuming that H 45 has the same Z 2 charges as H 45 while the three extra A 4 × SU(5) singlets are neutral under both Z 2 symmetries, we have checked that these additional fields lead to only negligible contributions to the fermion mass matrices discussed below, provided they get VEVs of order ǫΛ or smaller, see Eq. (8).

Allowed terms
The neutrino sector is composed of Dirac and Majorana mass terms which take the form in the superpotential: with Λ a heavy mass scale and c, p, y N , y ′ N , y dimensionless coupling constants. When the flavons get their VEVs the superpotential in Eq. (4) reproduces that in Eq. (2) but with constrained couplings c i and p i leading to TB mixing.
The superpotential terms of the down quark and charged lepton sector are given as where Λ d is the relevant messenger mass. The flavon ξ plays a role similar to a Froggatt-Nielsen field [49], except that it is not the sole contributor to the generated mass hierarchy, here combined as it is with the triplet flavons. Finally the up quark sector Yukawa superpotential terms take the form It should be mentioned that the messenger mass in this sector, Λ u , may in principle be different from that in the down quark sector. The field ξ ′ is introduced specifically to generate the T 1 T 1 term to the required order.

Fermion mass matrices
After spontaneous breakdown of the A 4 family symmetry by the flavon VEVs, the superpotential terms of Eqs. (4), (5) and (6) predict mass matrices for the respective sectors. In the following, order one coefficients in the quark and charged lepton sectors are omitted (including flavon VEV normalisation factors). Regarding the scale of the flavon VEVs we define where ϕ i = ϕ 123 , ϕ 23 , ϕ 3 , ξ or ξ ′ . In order to get the hierarchical structure of the quark and charged lepton mass matrices we assume 3 η 123 , η 23 , η ξ ′ = ǫ 2 and η 3 , η ξ = ǫ, where the numerical values for ǫ depend on the messenger scale of the relevant sector. We note that we have given the superpotential terms of the quark and charged lepton sectors up to and including O(ǫ 5 ).
In the Higgs sector, it is not the H 5 , H 5 or H 45 which get VEVs but their SM doublet components. These are the two MSSM doublets H u (corresponding to H 5 ) and H d (corresponding to a linear combination of H 5 and H 45 ); they originate below the GUT scale and remain massless down to the electroweak scale. The non-MSSM states all acquire GUT scale masses, including the linear combination of H 5 and H 45 orthogonal to H d . Electroweak symmetry is broken after the light MSSM doublets H u,d acquire VEVs v u,d and they then generate the fermion masses.
In the following all quark and charged lepton mass matrices are given in the L-R convention, i.e. the mass term for a field ψ is given in the order ψ L M LR ψ R .

Neutrino sector
In our model the light neutrino masses arise from a combination of type I and type III seesaw. Due to the absence of a H 24 the heavy seesaw messenger particles N and ρ 0 do not mix as can be seen from Eq. (4). Thus the 2 × 2 Majorana mass matrix of the heavy right-handed SU(2) L singlets is automatically diagonal. Furthermore, the seesaw messenger responsible for the type III contribution, ρ 3 , cannot mix with N as they furnish different SU(2) L representations. In CSD the (approximate) diagonal nature of the seesaw particles is usually a necessary extra assumption which often lacks a fundamental explanation. In our adjoint model, however, it is directly built into the theory by not including H 24 . Therefore our model represents a very natural realisation of CSD.
In the Dirac neutrino sector of Eq. (4), the spontaneous breaking of the A 4 family symmetry by the flavon VEVs ϕ 123 and ϕ 23 gives where the numerical factors of ρ 0 3 and ρ 0 are determined from the normalised SU(5) generators in the adjoint representation [30]. Upon application of the seesaw formula of Eq. (3) we find the effective left-handed Majorana neutrino mass matrix Since any matrix diagonalisable by Eq. (1) may be written as [26], we may readily read off the masses and state that Hence the model predicts one massless left-handed neutrino and thus a hierarchical neutrino mass spectrum.

Down quark and charged lepton sector
In the down quark and charged lepton sector, the superpotential of Eq. (5) predicts a mass matrix of the form (with messenger mass Λ d in η i ) where k f is the Georgi-Jarlskog factor (in the case that f = e, the mass matrix must also be transposed): Inserting the ǫ suppressions of the flavon VEVs from Eq. (8) the down quark mass matrix becomes whilst the charged lepton mass matrix reads Here we assume the numerical value ǫ ∼ 0.15. Upon diagonalisation, these give mass ratios of ǫ 4 : ǫ 2 : 1 for the down quarks and ǫ 4 3 : 3ǫ 2 : 1 for the charged leptons. These ratios are in good agreement with quark and lepton data and also predict GUT scale mass relations of m e ∼ m d 3 , m µ ∼ 3m s and m τ ∼ m b as desired. In the low quark angle approximation, left-handed down quark mixing angles θ d 12 ∼ ǫ, θ d 13 ∼ ǫ 3 and θ d 23 ∼ ǫ 2 are also predicted in agreement with data (assuming an approximately diagonal up sector which we obtain in the next Section). The corresponding charged lepton mixing angles are θ e 12 ∼ ǫ 3 , θ e 13 ∼ 0 and θ e 23 ∼ 0. The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix is not of exact TB form but receives small corrections from charged lepton mixing. In particular, the reactor angle deviates from zero by θ 13 ∼ 1 √ 2 ǫ 3 [50]. Furthermore, since θ e 13 ∼ θ e 23 ∼ 0, two sum rules for lepton mixing are respected [50,51]. Expressed in terms of the (r)eactor, (s)olar and (a)tmospheric deviation parameters defined as sin θ 13 = r √ 2 , sin θ 12 = 1 √ 3 (1 + s), sin θ 23 = 1 √ 2 (1 + a) [52], the sum rules read s = r cos δ and a = −r 2 /4 [18], with δ being the leptonic Dirac CP phase.

Up quark sector
Eq. (6) may be expanded after A 4 symmetry breaking and is responsible for up quark masses: Taking the VEV hierarchy as in Eq. (8), but now adopting the messenger scale Λ u ≈ 3Λ d , we obtain a mass matrix with an expansion parameter ǫ ∼ 0.05, and an up quark mass hierarchy ǫ 4 : ǫ 2 : 1. As the mass matrix of Eq. (16) is diagonal to a good approximation, the up quark mixing is negligible. An important consequence of this observation is that the Cabibbo-Kobayashi-Maskawa (CKM) mixing arises predominantly from the down quark sector, with the Cabibbo angle being θ C ∼ θ d 12 ∼ ǫ.

Conclusions
In conclusion, minimal (SUSY) SU (5) represents an attractive route to unification, but the Weinberg operator cannot account for neutrino mass and mixing, and the seesaw mechanisms all require extra matter or Higgs below the GUT scale. An appealing possibility, considered here, is to extend SUSY SU(5) by assuming a single right-handed neutrino singlet and an adjoint matter representation below the GUT scale, including an A 4 family symmetry as well as a gauged anomaly-free U(1). Hierarchical neutrino masses result from a combined type I and type III seesaw mechanism, and TB mixing arises indirectly from the A 4 family symmetry.
One attractive feature of this scheme is that the mixing between the single righthanded neutrino and the matter in the adjoint can be forbidden by not including the H 24 , leading to a diagonal heavy Majorana sector as required by CSD. The flavon vacuum alignments arise from the elegant SUSY D-term mechanism. The model also reproduces a realistic description of quark and charged lepton masses and quark mixings, including the Georgi-Jarlskog relations.
Corrections to TB mixing in the lepton sector come solely from the 1-2 mixing of the left-handed charged leptons, resulting in a PMNS matrix which is within the experimentally allowed limits. In particular the model respects the sum rules s = r cos δ and a = −r 2 /4 with r = θ C /3.