SU(4)_C X SU(2)_L X SU(2)_R Models With C-parity

We construct supersymmetric SU(4)_C X SU(2)_L X SU(2)_R models with spontaneously broken left-right symmetry (C-parity). The minimal supersymmetric standard model (MSSM) can be recovered at low scales by exploiting the missing partner mechanism. The field content is compatible with realistic fermion masses and mixings, proton lifetime is close to or exceeds the current experimental bounds, and supersymmetric hybrid inflation can be implemented to take care of C-parity domain walls as well as magnetic monopoles, and to realize the observed baryon asymmetry via non-thermal leptogenesis.

In a previous article [1] we constructed supersymmetric SU(3) C × SU(2) L × SU(2) R × U(1) B−L models [2,3] in which the scalar (Higgs) sector respects a spontaneously broken discrete left-right symmetry (C-parity) [4,5]. A variety of symmetry breaking scales were discussed, and it was shown that for TeV scale breaking, a large number of new particles potentially much lighter than the SU(2) R charged gauge boson could be found at the LHC.
This current article is a continuation of [1] and here we will explore supersymmetric models based on the well known gauge symmetry G 422 ≡ SU(4) C × SU(2) L × SU(2) R [2]. Being a maximal subgroup of Spin(10) (also known as SO (10)), G 422 captures many of its most salient features. For instance, G 422 gives rise to electric charge quantization, explains the standard model quantum numbers of each family, and predicts the existence of right handed neutrinos. However, there are also some important differences between SO(10) and G 422 which can be experimentally tested. For instance, in G 422 the lightest magnetic monopole carries two quanta of Dirac magnetic charge [6]. (In SO(10) the lightest monopole carries one quantum of Dirac magnetic charge, unless SO(10) breaks via G 422 .) By the same token in the absence of SO(10), G 422 predicts the existence of SU(3) color singlet states carrying electric charges ±e/2 [6,7,8]. Finally, gauge coupling unification and gauge boson mediated proton decay are a characteristic feature of SO(10) (C-parity reduces from three to two the number of independent gauge couplings in G 422 ). Following [1], we construct G 422 based supersymmetric models supplemented by C-parity. Ref. [9] considered similar models but there are some important differences. For instance, we exploit a missing partner mechanism [10] to realize MSSM at low energies without fine tuning. In our models, among other things, we also can realize supersymmetric inflation to avoid the monopole problem and C-parity domain walls.
In the simplest G 422 models, the MSSM electroweak doublets come from a bidoublet H(1, 2, 2), the matter fields are unified into three generations of Ψ(4, 2, 1), the antimatter fields into three generations of Ψ c (4, 1, 2), and the Yukawa couplings for matter come from HΨ c Ψ. In the simplest such mechanism, we are left with the unwanted relation Y U = Y D = Y E = Y Dirac between the Yukawa couplings, which can at best match experimental data for the third generation. We will discuss later how to get around this.
Two of the simplest ways of breaking G 422 down to MSSM is to either use (4, 1, 2)/(4, 1, 2), which we will call Φ c R and Φ R respectively and/or (10, 1, 3)/(10, 1, 3) superfields which we will call ∆ c R and ∆ R respectively. (This is analogous to the Φ's and ∆'s of Ref. [1], except that these fields also contain color triplets). By invoking C-parity, we also ensure that these chiral superfields will come with their C-conjugates, namely Φ L (4, 2, 1), Φ c L (4, 2, 1), ∆ L (10, 3, 1) and ∆ c L (10, 3, 1). We begin by analyzing a model with only Φ's and no ∆'s. Unlike the SU(2) L × SU(2) R × U(1) B−L models we considered in [1], the Φ's here decompose into SU(3) C singlets as well as triplets. The mechanisms presented there can serve to pair up the color singlet components but will fail to pair up the color triplet components. Let us see why this is the case. We double the number of bidoublets and consider the following terms in the superpotential W: . This is the so-called missing partner mechanism [10] because what remains at low energies of the bidoublets is the down-type component of A solution to this proposed in [7,11] is to introduce a (6, 1, 1) Higgs field and the Here, however, we also have Φ L and Φ c L Higgs fields containing (3, 2) 1 3 and (3, 2) − 1 3 components. Thus, instead of (6, 1, 1), we introduce a (15, 1, 1) Higgs superfield and add the following terms to W: This induces a nonzero VEV along the MSSM singlet direction for H 15 .
By varying with respect to Φ R and Φ c R , we find that the 44-component of κS½ 4×4 + αH 15 has to be zero. This means that the color triplet components of Φ L and Φ R get paired up but the color singlet components do not. The non-zero VEVs are as follows: The (1, 1) ±2 components become the Goldstone and the sgoldstones. S pairs up with a linear combination of (1, 1) 0 and the orthogonal combination form the goldstone multiplet.
which has a zero determinant. The direction with the zero eigenvalue is both the goldstone and sgoldstone direction.
Note that the unpaired set of weak doublets from Φ L and Φ c L have a mass term which goes as κ S ½ + αH 15 . This is where the missing partner mechanism involving two Higgs bidoublets comes in handy, as explained earlier. Because of the SU(4) C symmetry, the model as it stands still suffers from the unwanted relation Y D = Y E . Now it turns out that there are a number of ways around this problem but we will only deal with the two most common strategies here; one of them is to introduce the nonrenormalizable coupling H 1 H 15 Ψ c Ψ/Λ and the other is to introduce a (15, 2, 2) Higgs field [12]. Eq. 3b tells us that the contribution of H 1 H 15 Ψ c Ψ/Λ to Y D is − 1 3 that of the contribution to Y E . If Λ corresponds to some value an order of magnitude or so larger than the GUT scale, and if the H 1 Ψ c 3 Ψ 3 coupling for the 3rd generation turns out to be of order unity, then the latter coupling will dominate and we will have the approximate relation Y b = Y τ . For the 1st and 2nd generations, the contributions from both couplings can be comparable. To get something more predictive, we may insist upon using certain texture ansatzes. For instance, the ansatzes in [13] which lead to realistic fermion masses and mixings can be realized with the field content listed in Table I, possibly supplemented by (15,2,2). [The model needs to be augmented with the Majorana coupling (Φ c R Ψ c ) 2 and its C-conjugate to give superheavy masses to the right-handed neutrinos.] We would like to make a remark on the MSSM µ term. The required coupling is µH 1 H 2 , with µ on the order of the electroweak scale. In the present scheme, the Giudice-Masiero mechanism [14] is a plausible way to accomplish this.
Next we will look at the issue of proton decay in these G 422 models. In order to get proton decay, we need SU(3) C operators like 333 or 333. This would correspond to SU(4) C operators like 4 444, 4 444, 644 or 644. So far, we have not included any couplings of this nature. (Even the Majorana term Φ c R Φ c R Ψ c Ψ c /Λ is not of this form). In principle, these couplings can be forbidden by introducing a global symmetry U(1) X which commutes with all the gauge symmetries. The X-charge assignments are given in Table III. The proton in this case turns out to be essentially stable.
In the absence of U(1) X , proton decay can occur via dimension five operators along the lines discussed in [13] and according to which, the dominant decay modes are νK + and νπ + with lifetime ∼ 10 34−35 yrs.
Next, we consider an alternative model without the missing partner mechanism. Recall that the missing partner mechanism is needed for two things; to give masses to the color singlet components of Φ L and to break the Y U = Y D relation. The former can be taken care of by the nonrenormalizable coupling Φ c L Φ L Φ c R Φ R /Λ once the Φ R 's get a nonzero VEV. Introducing an SU(2) R Higgs triplet T R (1, 1, 3) together with its C-conjugate, we can arrange for T R to get a nonzero VEV by employing the couplings T 2 R and T R Φ c R Φ R as well as their C-conjugates. Without the missing partner mechanism, we only have one bidoublet H which contains both H u and H d . The up-down relation between the Yukawa couplings can be broken by the nonrenormalizable coupling H T R Ψ c Ψ/Λ. However, as the cutoff scale Λ will typically be an order of magnitude or so larger than the SU(2) R breaking scale M, the splitting between the Yukawa couplings will only be significant for the first and second generations and not for the third. Thus, we will still have the approximate relation Y t ≃ Y b .
Let us note that we may also break the G 422 symmetry with ∆ L (10, 3, 1), ∆ R (10, 1, 3), ∆ c L (10, 3, 1) and ∆ c R (10, 1, 3) instead of the Φ's [15]. (We note that the Z 2 matter parity of MSSM is automatically embedded within G 422 in the absence of the Φ fields.) The Majorana coupling will now be ∆ c R Ψ c Ψ c , and its C-conjugate. Under the decomposition from SU(4) C to SU(3) C , 10 → 6 ⊕ 3 ⊕ 1 and 10 → 6 ⊕ 3 ⊕ 1. If we only have a (15, 1, 1) Higgs field and a singlet S and we use the same mechanism as our previous model, we find that the color sextet and triplet components of the ∆'s will get nonzero masses because the Clebsch-Gordon contributions from S and H 15 do not cancel but the mass contribution to the color singlet components cancel (This is a consequence of setting the F -terms to zero). This includes the (1, 3) ±2 components of ∆ L and ∆ c L . Since the ∆ R 's are SU(2) R triplets, the color singlet components decompose into three once G 422 is broken. One The up-down Yukawa relation is broken not by the missing partner mechanism but by the nonrenormalizable coupling H (1, 1, 3) Ψ c Ψ/Λ. (The C-conjugate of this coupling is also included.) For the same reason as before, this difference is suppressed by (1, 1, 3) /Λ and so, Y t ≃ Y b for the third generation. Let us summarize by putting together all the terms in the superpotential: One consequence of C-parity is the existence of Z 2 domain walls once it is spontaneously broken. The scale at which this occurs happens to be same as the G 422 breaking scale.
Such domain walls can give rise to cosmological problems unless they are inflated away. In addition, when G 422 breaks down to MSSM, magnetic monopoles carrying two quanta of Dirac magnetic charge [6] can be generated. Astrophysical and cosmological bounds on such monopoles are fairly stringent and the standard solution is to inflate them away. One way to do this in our case is to invoke shifted hybrid inflation [16] where a nonrenormalizable ) is added to the superpotential. With such a suitably altered model, it is possible to start inflation with a trajectory where C-parity as well as G 422 are already spontaneously broken. In such a scenario, domains do not form once inflation ends and neither do monopoles. During inflation itself, the inflationary trajectory is identical to that analyzed in [16] with Φ L = Φ c L = 0 throughout. Although the postinflationary trajectory will be different, the end result is that both C-domain walls and monopoles are eliminated.
The end of inflation is followed by the decay of the inflaton fields Φ R , Φ c R and S into right handed neutrinos (ν c ) and sneutrinos ( ν c ). As discussed in [17,18], following [19], the subsequent out of equilibrium decay of ν c and ν c generates lepton asymmetry, which is then partially converted to the observed baryon asymmetry by the electroweak sphalerons.
Note that hybrid inflation requires that the G 422 breaking scale is comparable to M GUT ∼ 10 16 GeV. If G 422 breaks at significantly lower (such as intermediate) scales, an alternative scenario for suppressing monopoles should be employed (see, for instance, [20]).
In conclusion, we have constructed realistic supersymmetric G 422 models in which the scalar (Higgs) sector respects a discrete left-right symmetry (C-parity) which is spontaneously broken at the same scale as the SU(2) R gauge symmetry. We have shown how the MSSM is recovered without fine tuning at low scales. The scalar fields we employ enable us to reproduce, following [13], the observed fermion masses and mixings, implement a missing partner mechanism [1], and realize inflation followed by non-thermal leptogenesis.