Topological pseudodefects of a supersymmetric $SO(10)$ model and cosmology

Obtaining realistic supersymmetry preserving vacua in the minimal renormalizable supersymmetric $Spin(10)$ GUT model introduces considerations of the non-trivial topology of the vacuum manifold. The $D$-parity of low energy unification schemes gets lifted to a one-parameter subgroup $U(1)_D$ of $Spin(10)$. Yet, the choice of the fields signaling spontaneous symmetry breaking leads to disconnected subsets in the vacuum manifold related by the $D$-parity. The resulting domain walls, existing due to topological reasons but not stable, are identified as topological pseudodefects. We obtain a class of one-parameter paths connecting $D$-parity flipped vacua and compute the energy barrier height along the same. We consider the various patterns of symmetry breaking which can result in either intermediate scale gauge groups or a supersymmetric extension of the Standard Model. If the onset of inflation is subsequent to GUT breaking, as could happen also if inflation is naturally explained by the same GUT, the existence of such pseudodefects can leave signatures in the CMB. Specifically, this could have an impact on the scale invariance of the CMB fluctuations and LSS data at the largest scale.


I. INTRODUCTION
There are several indications for physics beyond the Standard Model (SM) which demands the need to connect with the high energy scales unlikely to be accessible to accelerators. One of them is the minuscule masses of neutrinos [1,2] which through see-saw mechanism [3][4][5][6] suggest the existence of a high mass scale. Further the precarious hierarchy of the Higgs mass with respect to the Planck scale is conceptually unnatural and can be easily ameliorated by new physics beyond the SM. Finally, unification of couplings remains ever a desirable feature and can be accomplished by grand unification in supersymmetric or non supersymmetric SO(10), or larger gauge groups. The scale of such models is beyond the reach of accelerators but the early cosmology and its imprints on the CMB data and Large Scale Structure data can be important checks on this model. Other than the CMB, consistent Big Bang Nucleosynthesis and successful inflation remain important requirements on any model with new physics at high energies. Indeed for the class of models that unifies, the high scale physics natural to them is constrained by inflation, by the need to generate baryon asymmetry, and by exotic relics such as cosmic strings and domain walls (DW) that survive the homogenizing effects of the high temperature.
An early study [7] considered the consequence of unstable domain wall formation in Spin (10), which can decay due to the formation of cosmic strings as punctures or boundaries. Several other works have considered these issues, notably [8][9][10][11][12] for the context of topological defects and others [13][14][15][16][17][18][19] which have utilized the group theoretic constraints arising from such considerations for unification proposals. The present investigation is concerned with studying the interplay of symmetry breaking * ila.garg@iitb.ac.in † yajnik@iitb.ac.in patterns with cosmology in the context of supersymmetric Spin (10). There are two broad directions that have been pursued along these ones. One, that is motivated by superstring unification, as in [20][21][22][23], and the other class of models relies on the minimal representations of the Higgs and has been explored in [24][25][26][27][28][29]. It has been advanced as a renormalizable minimal supersymmetric SO(10) grand unified theory (MSGUT). In the present work we shall restrict to the latter class of models due to their rich topological structure. The model utilizes Higgs supermultiplets 10, 210, 126 (126) required to break the symmetry and provide fermion masses. Among these the 210 and 126 (126) are responsible for breaking of SO(10) symmetry down to MSSM, and 10 and 126 give masses to the fermions. The 16-dim Spinor representation contains one generation of SM fermions and a right handed neutrino. The GUT model considered has many phenomenological as well as cosmological virtues. It can provide inflaton candidates [30,31], mechanism for baryogenesis through leptogenesis and the lightest supersymmetric particle (LSP) of the model can serve as weakly interacting cold dark matter candidate. A hallmark of this class of models is the occurrence of D-parity, a discrete symmetry that can exchange the chiral matter fields with their charge conjugates and also appropriately the corresponding Higgs bosons which gives them masses. While this is a discrete symmetry of the partial unification model, viz., the left-right symmetric model, when lifted to SO(10) it gets embedded in a one parameter U (1) D subgroup of the covering group Spin (10). Since the parent group is simply connected there are no stable domain walls. However we argue here that the topology of the vacuum manifold can be non-trivial, and can give rise to defects that are best dubbed topological psuedodefects (TPD). In this paper we want to study the implications to cosmology of TPD's arising in a SUSY SO(10) GUT model. In [26][27][28][29] the implicit assumption is that there is a one step breaking from SUSY SO(10) to MSSM. However the physics of Big Bang implies the existence of many causally dis-connected regions in space and the non-trivial vacuum structure would give rise to domain walls [7].
Generically, the presence of domain walls (DW) in a model has important interplay with inflation. One of the successes of inflationary proposal is the removal unpleasant relics of GUTs such as monopoles. The same applies to cosmic strings, whose density can be easily diluted by cosmological inflation. The same is however not true of DW, as these may form a mutually locked structure which may not be blown apart by inflation easily. This is a relevant possibility if inflation has a preceding hot period allowing DW forming phase transition. The walls if stable would conflict with standard cosmology due to the inhomogeneities they would introduce in the CMB. On the other hand, for a network of walls which is unstable, the question shifts to the time scale of the decay and disappearance of the walls. The unified model in this case can be constrained by requiring that the inhomogeneities generated by their early presence should not affect the successful outcomes of inflation, specifically the nearly scale invariant CMB spectrum and the observed Gaussian nature of the density perturbations.
The plan of the paper is as follows. In section II we review the topological role of D-parity as first considered in [7]. In III, we treat a warm up example of the energy barrier of DW in the minimal supersymmetric left-right where D parity occurs as a discrete symmetry, avoiding the subtleties of the large group SO(10). In section IV, we give a brief introduction to the GUT model of [28] and calculate the energy barrier associated with the Dparity breaking. The height of such a barrier estimates the energy per unit area of the TPD walls that may form. In sec. V we discuss the implications to cosmological inflation which in turn may imply constraints on the scales the possible symmetry breaking schemes. The conclusions are in Sec. VI.

II. METASTABLE DOMAIN WALLS
Here we briefly review the topological issues relevant to the domain walls, paraphrased from [7]. Spin (10) can be broken to its subgroup H 0 = Spin(6) ⊗ Spin(4), where the first factor contains the color SU (3) c and the second factor contains the SM SU (2) L and a potential SU (2) R . This breaking can be achieved by using the 54 dimensional scalar χ which takes on a vacuum expectation value (VEV) However, the stability group of this VEV contains a discrete set of additional elements. Consider the oneparameter curve in Spin(10) of the form U J67 (θ) = exp(iθJ 67 ). The 6-7 submatrix of the VEV of χ transforms under this as Thus χ is left invariant by U J67 (θ = nπ) with n ∈ Z. It can be seen that all such choices derived from mixed a and α indices, with a ∈ 1, 2, ...6 and α ∈ 7, 8, 9, 10, have the same property, and are indeed equivalent to each other under a suitable transformation by an element from H 0 . Thus the full stability group of χ is H 0 ⊕ H 0 consisting of two disconnected continuous subsets, where H 0 are all the elements of the form h(iJ 67 ) with h ∈ H 0 . It may be noted that iJ 67 also enters the D-parity defined as D ≡ (iJ 67 )(iJ 23 ), the effective charge conjugation operator due to the charges of the fermions assigned to 16.
In the model of [7] the sequence of breaking is Stable cosmic strings arise at the first phase transition, due to Π 0 (H 0 ⊕ H 0 ) = Z 2 . At the next stage of breaking when the 126 acquires a VEV, domain walls appear due to the breaking of this Z 2 . But the Z 2 comes embedded in a continuous loop U (1) D the one-parameter subgroup generated by the D-parity generator. Such loops continuously connect a VEV to its D-parity conjugate. Specifically such walls separate vacua with 126 = (10, 1, 3) ( written in its components with Pati-Salam quantum numbers) from its charge conjugate VEV (10, 3, 1). Thus the walls are not stable, and decay due to tension of the string boundaries which are liable to shrink. The walls can also disintegrate due to creation of holes formed in them due to quantum tunnelling assisted by thermal fluctuations. If the second phase transition is first order, there is a phase of wall domination and the possibility of wall decay only through large black hole formation. This is certainly ruled out by the CMB inputs into primordial fluctuations. On the other hand, a second order phase transition at second stage of breaking creates a short period of wall persistence though the walls do not come to dominate over radiation. These considerations are a warm up for the study of topology of the vaccum manifold arising in [26][27][28][29] which study the breaking of SUSY SO(10) to MSSM. We argue that D-parity TPD walls are a necessary consequence in such a breaking and expect that an epoch similar to the second order phase transition at second stage as reviewed in this section may unfold. A short period of substantial wall presence can have definite consequences to CMB data. We shall discuss this in Sec.s IV and V.

III. D-PARITY SYMMETRIC VACUA IN LEFT-RIGHT SYMMETRIC MODEL
Before proceeding to the SUSY SO(10) case, to illustrate the procedure we start with a warm up exercise for a related system, the minimal supersymmetric left-right symmetric model (MSLRM) considered in [32], based on the group While unlikely to have implications for inflation, the model is interesting in its own right as an intermediate scale group. The walls were studied from the point of view of cosmology and leptogenesis earlier in [33,34].
The Higgs superfields proposed for breaking the G LR symmetry to SM are ∆ = (1, 3, 1, 2);∆ = (1, 3, 1, −2); These fields transform under D-parity as The renormalizable superpotential corresponding to these Higgs superfields is given as, The vacua are sought assuming the supersymmetry to be unbroken and remaining so till the electroweak scale (∼ O(TeV)). These can be obtained by imposing F-flatness and D-flatness conditions given in [32]. The set of vacuum expectation values (VEV's) for the Higgs fields required to obtain the MSSM is, The required minimum is obtained at w = −m∆ a and d = ( 2m∆mΩ a 2 ) 1 2 [32]. Here, w and d set two mass scales in the problem. At first step, Ω c acquires VEV at scale M R and SU (2) R is broken to U (1) R , and then the VEV's of Thus, at this scale we get the minimal supersymmetric standard model (MSSM). However the D and F -flatness conditions [32] also give another set of possibility of vacuum which is degenerate to the one given by Eq. (7) which preserves the SU (2) R × U (1) L × U (1) B−L symmetry. The alternative set of VEV's is given by Due to the left-right symmetry, numerically d =d, d c = d c and w = w c . It is the breaking of this symmetry that leads to the formations of domain walls. Now we have two degenerate vacua separated by a domain wall. Since in this case the D-parity is a discrete symmetry, the walls are topologically stable. Here we consider an ansatz for a trajectory in the group space which connects the two vacua. We parameterize the VEV's as follows with a parameter θ, When θ=0, we have left like vacuum and for θ = π, right like. On substituting these paramterised VEV's in the superpotential we obtain, We can then compute θ derivative of the scalar potential as This gives, using the numerical equality of the VEV's noted below Eq. (8), It is easy to see that the two expressions with the braces mutually cancel at the symmetric point θ= π 2 . The value of the energy at this point is given by, The two set of vacua considered above, Eqs. (7), (8) are degenerate and related by D-parity, which is a discrete symmetry of the group G LR . The domain walls are therefore topologically stable. The solitonic domain walls thus arising were obtained as solutions of this theory in [35]. The motivation here is different. The considerations of this section illustrate how one rotates from one vacuum to another, not necessarily along energetically optimal path, but in order to estimate the height of the barrier. The same strategy will be utilised even for the more general case when the parent group is simply connected. The main point is that degeneracies that might occur in single field minimization are lifted due to the presence of several mutually coupled fields providing a general quartic polynomial. The two minima of Eqs. (7) and (8) are two of the solutions of the extremization condition. Such extrema are necessarily isolated points, being the zeros of a generic polynomial. Further, supersymmetry ensures that the supersymmetry preserving minima are absolute minima. Thus the third extremum, the intermediate point, is a local maximum determined along the parameterized curve. While this is not guaranteed to be the lowest energy peak separating the two minima, it provides an upper bound on the height of the saddle point lying on the barrier.

IV. DOMAIN WALLS IN MINIMAL SUPERSYMMETRIC SO(10) GUT
We now turn to the main problem of the minimal SUSY GUT model (MSGUT) [24][25][26]28]. The wall ansatz for MSGUT turns out to be complicated due to the presence of several Higgs fields in different representations. At first we proceed to establish the presence of a variety of walls that may appear in this MSGUT model, which may also involve intermediate phases of smaller gauge groups, if we allow some variation in the parameters. In each case we focus on D-parity walls which are unstable yet have non-trivial consequences.
The MSGUT can be broken down to MSSM directly or through intermediate symmetries depending upon the choice of Higgs multiplet getting VEV [27]. During these symmetry breakings, the D-parity is also broken and leads to the formation of TPD domain walls. The Higgs content of this model is 210 (Φ ijlk , four index totally antisymmetric), 126(126) (Σ ijklm (Σ ijklm ), five index totally anti-symmetric self-dual (anti-self-dual) representation) and the vector representation 10 (H i ). Here i, j, k, l, m = 1, 2...10 run over the vector representation of SO(10). The 126(126) and 210 break the SO(10) gauge symmetry to MSSM; the 10 breaks the electroweak symmetry, while the 10 and 126 give masses to the fermions.
Here a, b, c, d, e take values 0 or 1.
The superpotential in terms of these VEVs is given by The SUSY preserving minima using the F -term and Dterms vanishing conditions are given by [26], where x is the solution of following cubic equation Individual components of the fields transform differently under this generalized U D -rotation, as follows, (with s θ , c θ standing for sin θ and cos θ respectively) Similarly one can write out for the other field components of Σ − R and Σ + R given in A1. Now we calculate the θ dependent potential from the corresponding superpotential as, Here i runs over number of field components given in Eq. (A1). The form of potential for different values of x assuming |η| = |λ| is The variation of the potential (in units of |mΦ| 4 |λ| 2 ) as a function of the U (1) D -rotation angle θ is shown in Fig. 1. One can see from Fig. 1 that at θ = 0 and π, the potential energy is zero and the VEVs satisfy the relations in Eq. (19) which are x dependent. For intermediate values of θ the potential is not symmetric under θ → π − θ and its overall magnitude is also strongly x dependent. Our main motivation in performing this detailed calculation is to establish that TPD domain walls indeed form even if the group contains no discrete symmetry. In the last para of Sec. III, it was pointed out that our strategy at least yields an upper bound on the energy barrier separating vacua which are two distinct points. Unlike in that example, the group is simply connected here. However, it may be observed that there is an inadvertent (not accidental) discrete symmetry of the D and F flatness conditions. The flatness conditions cannot single out a unique vacuum but signal two for a given set of parameters, related by the D-parity. We now need to argue that this vacuum pair related by the flip symmetry are necessarily separated by an energy barrier. It is sufficient to focus on the 210 whose three independent sets of MSSM singlet components are assigned different VEV's p, a and w. The parameters in the superpotential are tuned according to Eq.s (19) (20) and the five possi-bilities listed below them. Then the value x = 1/4 is one zero of a cubic polynomial, which is necessarily isolated. Further, any values of the component fields P , A and Ω accessed by small variations of p, a and w are necessarily of higher energy. Thus the preferred vacua with unbroken MSSM are also isolated points at best connected by discrete transformations. This ends the existence proof of isolated vacua. While it is convenient to build low energy phenomenology based on the preferred vacuum, the conditions in the early universe allow domains of both types to form. Eventually the unstable TPD walls must disintegrate or have unfavorable consequences as discussed in II, based on [7]. In the next section we turn to cosmological consequences for MSGUT. Case A is generic to chaotic inflation [36] where inflation originates close to the Planck scale. In this case after reheating, the temperature could be less than or more than M GU T . In the former case the thermal state should be directly in the required MSSM phase. In the latter case however, after the symmetry breaking phase transition, TPD walls would emerge. Due to their unstable nature they eventually disintegrate. The resulting epoch of wall domination would end with entropy dumping with return to pure radiation dominated universe. It has been pointed out [37][38] [39] that there are models of inflation in which the duration of the reheating phase and the effective equation of state during that phase can be correlated with other inflation observables and is being pursued in [40]. For such cases the presence of TPD walls during reheating could have important consequences.
In Case B, the TPD walls would be copiously present when inflation commences. This would produce signatures similar in nature to but more pronounced than in the case C to be discussed below. The epoch over which the power law inflation caused by the walls would compete with the scale invariant inflation would be determined by the ratio M GU T /M inf . In late time observables, this would reflect in deviations from scale invariance at the largest scales. Since there are no strong indications to this effect we do not analyse this further, however the framework would be similar to that we pursue for the case C.
Case C could be accidental, but more interestingly, also occurs if inflationary physics emerges from the same Grand unified theory. In this case the formation of TPD walls could occur before the inflaton potential energy dominates, giving rise to the signatures in the primordial fluctuations as encoded in the CMB data [41,42]. Recently this question has been addressed in [43] and it is shown that the presence of frustrated domain walls can alleviate the quadrupole anomaly of the CMB fit occurring in the Lambda-CDM model.
Consider the presence of domain walls which are conformally stretched, [7] and the wall complex as a whole obeys the coarse grained equation of state [44], and corresponding dependence on the Friedmann scale factor, where ρ 1 ≡ M 4 GU T , and in the latter equation the numerator sets the initial conditions on its value. The inflaton has a comparable energy density, V 0 ≡ M 4 inf so that H 2 0 = (8π/3)GV 0 would be the Hubble parameter if only the inflaton were present. The combined Friedmann equation, has the solution with In the regime where ρ 1 > V 0 , and for H 0 (t − t 1 ) < 1 one gets the behaviour characteristic of the p = −2ρ/3 equation of state. At late times of course the vacuum energy dominates. But a brief period of wall domination would still have the behavior similar to inflation, in which physical scales like those of the scalar field perturbations would be growing faster than the Hubble horizon. The amplitude of the perturbations would be imprinted on the earliest of the scales to leave the horizon. The parameter of inflation calculated in the present case gives, At t = t 1 this gives = 1/2 as expected for a pure power law expansion with domain walls. But it soon turns over to approaching the value 0 of the vacuum energy dominated phase. Thus the early modes to leave the horizon would be far from scale invariant, whereas within a few e-foldings of the time scale H −1 0 the slow roll condition is satisfied [40]. The departure from approximate scale invariance could therefore be detected. Further, the presence of domain walls would introduce non-gaussianities. Since these would affect inflation only in its earliest stages of slow roll, they may not have entered our horizon yet. But in principle these could be detected. While cosmic strings have been studied for their effect on CMB data extensively [45] [46][47] [48], the presence of such primordial domain walls is also warranted, as a countercheck on models of unification as well as inflation.

VI. CONCLUSIONS
We have studied the example of a unification group wherein domain walls can form although the group is simply connected, with no discrete symmetries that break spontaneously. But inadvertent symmetries of the minimization conditions imply the possibility of a discrete set of vacua. Such vacua turn out to be related by discrete symmetries in the parent group. The case in point is the well known D-parity of Spin(10) and its descendants. We have explicitly computed value of the energy along one-parameter paths connecting two possible subgroups to which the symmetry breaking of Spin(10) could have occurred. We thus show that the vacua are indeed separated by an energy barrier. Then the causal structure of the early Universe creates the interesting possibility of topological pseudodefects, dubbed TPD walls, separating regions of such vacua. Even though manifestly unstable, the walls may live long enough to leave imprints on the observables. Such signatures in the CMB signals create the exciting possibility of accessing grand unification in current observations.
We have shown that in the context of inflation with a preceding epoch of radiation domination, (cases B and C), the formation of domain walls would leave scale dependent imprints on the very long wavelengths which leave the horizon at the onset of inflation. At current state of knowledge we do not know if these are indeed the scales being seen in the lowest multipoles. Likewise it is important to study non-gaussianities resulting from such objects in the phase at the onset of inflation.