Inflation from a No-scale supersymmetric $SU(4)_{c}\times{SU(2)_{L}\times{SU(2)_{R}}}$ model

We study inflation in a supersymmetric Pati-Salam model driven by a potential generated in the context of no-scale supergravity. The Pati-Salam gauge group $SU(4)_{c}\times{SU(2)_{L}\times{SU(2)_{R}}}$ , is supplemented with a $Z_{2} $ symmetry. Spontaneous breaking via the $SU(4)$ adjoint leads to the left-right symmetric group. Then the $SU(2)_{R}$ breaks at an intermediate scale and the inflaton is a combination of the neutral components of the $SU(2)_{R}$ doublets. We discuss various limits of the parameter space and we show that consistent solutions with the cosmological data for the a spectral index $n_{s}$ and the tensor-to-scalar ratio r are found for a wide range of the parameter space of the model. Regarding the latter, which is a canonical measure of primordial gravity waves, we find that $r\sim{10^{-3}-10^{-2}}$. An alternative possibility where the adjoint scalar field $S$ has the role of the inflaton is also discussed.


INTRODUCTION
In cosmological models inflation is realized by a slowly rolling scalar field, the so called inflaton, whose energy density dominates the early history Universe [1,2,3,4]. Among several suggestions regarding its origin, the economical scenario that this field can be identified with the Standard Model (SM) Higgs state h, has received considerable attention [5]. In this approach, the Higgs field drives inflation through its strong coupling, ξ h 2 R, where R is the Ricci scalar and ξ is a dimensionless parameter that acquires a large value, ξ 10 4 .
In modern particle physics theories, cosmological inflation is usually described within the framework of supergravity or superstring grand unified theories (GUTs). In these theories the SM is embedded in a higher gauge symmetry and the field content including the Higgses are incorporated in representations of the higher symmetry which includes the SM gauge group. In this context, several new facts and constraints should be taken into account. For instance, since new symmetry breaking stages are involved, the Higgs sector is usually extented and alternative possibilities for identifying the inflaton emerge. In addition, the effective potential has a specific structure constrained from fundamental principles of the theory. In string theory effective models, for example, in a wide class of compactifications the scalar potential appears with a no-scale structure as in standard supergravity theories [6,7]. In general, the scalar potential is a function of the various fields which enter in a complicated manner through the superpotential W and the Kähler potential K. Thus, a rather detailed investigation is required to determine the conditions for slow roll inflation and ensure a stable inflationary trajectory in such models. Modifications of the basic no-scale Kähler potential and various choices for the superpotential have been studied leading to a number of different inflationary cases [8]- [14], while studies of inflation within supergravity in a model independent way can be found in [15,16].
In the present work we implement the scenario of Higgs inflation in a model based on the Pati-Salam gauge symmetry SU (4) C × SU (2) L × SU (2) R [17] (denoted for brevity with 4-2-2). This model has well known attractive features (see for example the recent review [18]) and has been successfully rederived in superstring and D-brane theories [19,20,21,22]. Early universe cosmology and inflationary predictions of the model (or its extensions) have been discussed previously in several works [23,24,25]. Here we consider a supersymmetric version of the 4-2-2 model where the breaking down to the SM gauge group takes place in two steps. First SU (4) breaks spontaneously at the usual supesymmetric GUT scale M GU T 10 16 GeV, down to the left-right group 1 via the adjoint representation. Then, depending on the specific structure of the Higgs sector, the SU (2) R scale can break either at the GUT scale, i.e., simultaneously with SU (4), or at some lower, intermediate energy scale. The variety of possibilities are reflected back to the effective field theory model implying various interesting phenomenological consequences. Regarding the Higgs inflation scenario, in particular, the inflaton field can be identified with the neutral components of the SU (2) R doublet fields associated with the intermediate scale symmetry breaking. In this work we will explore alternative possibilities to realise inflation where the inflaton is identified with the SU (2) R doublets. We also examine the case of inflation in the presence of the adjoint representation.
The layout of the paper is as follows. In section 2, we present a brief description of the 4-2-2 model, focusing in its particle content and the symmetry breaking pattern. In sections 3 we present the superpotential and the emergent no-scale supergavity Kähler potential of the effective model. We derive the effective potential and analyse the predictions on inflation when either the SU (2) R doublets or the adjoint play the rôle of the inflaton. We present our conclusions in section 4.

DESCRIPTION OF THE MODEL
In this section we highlight the basic ingredients of the model with gauge symmetry, This model unifies each family of quarks and leptons into two irreducible representations, F i andF i transforming as [28] F i = (4, 2, 1) i andF i = (4, 1, 2) i , under the corresponding factors of the gauge group (2.1). Here the subscript i (i = 1, 2, 3) denotes family index. Note that F +F comprise the 16 of SO(10), 16 → (4, 2, 1) + (4, 1, 2). The explicit embedding of the SM matter fields, including the right-handed neutrino is as follows: where the subscript (r, g, b) are color indices.
Collectively we have the following SM assignments: (2.9) Fermions receive Dirac type masses from a common tree-level invariant term, FF h, whilst right-handed (RH) neutrinos receive heavy Majorana contributions from nonrenormalisable terms, to be discussed in the next sections. In addition, the colour triplets d c H and d c H are combined with the D 3 and D 3 states via the trilinear operators HHD 6 +HHD 6 and get masses near the GUT scale.
After the short description of the basic features of the model, in the following sections we investigate various inflationary scenarios in the context of no-scale supergravity, by applying the techniques presented in [29,30].

INFLATION IN NO SCALE SUPERGRAVITY
In this section we consider the 4-2-2 model as an effective string theory model and study the implications of Higgs inflation. The 'light' spectrum in these constructions contains the MSSM states in representations transforming non-trivially under the gauge group and a number of moduli fields associated with the particular compactification. We will focus on the superpotential and the Kähler potential which are essential for the study of inflation.
The superpotential is a holomorphic function of the fields. Ignoring Yukawa interaction terms, the most general superpotential up to dimension four which is relevant to our discussion is where from now on we set the reduced Planck mass to unity, M P l = 1. We focus on the dynamics of inflation during the first symmetry breaking stages at high energy scales. For this reason we ignore all the terms involving the bi-doubled since this state mostly contribute in low energies by ginving mass to the MSSM particles and do not play an important rôle during inflation. In addition we impose a Z 2 symmetry, under which Σ is odd and all the other fields are even. As a result the trilinear termsHΣH and tr (Σ 3 ) are eliminated from the superpotential in (3.1). The elimination of these trilinear terms of the superpotential is important, since if we useHΣH and tr (Σ 3 ) instead ofH tr(Σ 2 )H and tr (Σ 4 ), the shape of the resulting potential is not appropriate and it leads to inconsistent results with respect to the cosmological bounds while at the same time returns a low scale value for the parameter M in the superpotential, which usually expected to be close to the GUT scale. Then, using (2.5) and (2.6) the superpotential takes the following form: whereλ = 3λ 4 andκ = 7κ 12 . From the phenomenological point of view we expect S = v to be at the GUT scale. By assuming v 3 × 10 16 GeV and using the minimization condition ∂W/∂S = 0, we estimate that m 2κv 2 which, forκ = 1/2, gives m ∼ 10 14 GeV.
In the two step breaking pattern that we consider here, L H and L H must remain massless at this scale in order to break the SU (2) R symmetry at a lower scale. The SU (2) R breaking scale should not be much lower than the GUT scale in order to have a realistic heavy Majorana neutrino scenario. In addition we have to ensure that the coloured triplets Q H and Q H will be heavy. In order to keep the L H , L H doublets at a lower scale, and at the same time the coloured fields Q H and Q H to be heavy, we assume that M ≈λ S 2 =λυ 2 . In this case Q H , Q H acquire GUT scale masses M Q H ≈ 8λ 9 S 2 . During inflation the colored triplets Q H , Q H and the charged components of the RH doublets, L H and L H , do not play an important rôle. The SU (2) R symmetry breaks via the neutral components 2 ν H and ν H . In terms of these states the superpotential reads: where we have made use of the relation M λ υ 2 .
The Kähler potential has a no-scale structure and is a hermitian function of the fields and their conjugates. For the present analysis, we will consider the dependence of the Higgs fields of the 4-2-2 gauge group and the 'volume' modulus T . Therefore, assuming the fields φ i = (S, T, H, h) and their complex conjugates, we write where ξ is a dimensionless parameter. In the expression (3.4), we can ignore the last term which involves the bidoublet and in terms of ν H , ν H and S, the Kähler potential reads: In order to determine the effective potential we define the function Then the effective potential is given by where G i (G j * ) is the derivative with respect to the field φ i (φ * j ) and the indices i, j run over the various fields. V D stands for the D-term contribution.
Computing the derivatives and substituting in (3.6) the potential takes the form where we have ignored the D-term contribution and we have assumed that the value of the T modulus field is stabilized at T = T * = 1/2, see [31,32]. Notice that in the absence of the Higgs contributions in the Kähler potential, the effective potential is exactly zero, V = 0 due to the well known property of the no-scale structure.
We are going now to investigate two different inflationary cases: firstly, along Hdirection and secondly along S-direction.

INFLATION ALONG H-DIRECTION
We proceed by parametrizing the neutral components of the L H and L H fields as ν H = 1 2 (X + Y ) e iθ and ν H = 1 2 (X − Y ) e iϕ , respectively. These yield Assuming θ = 0 and ϕ = 0, along the D-flat direction, Y = 0, and the combination X is identified with the inflaton. The shape of the potential, as a function of the fields S and X, is presented in Figure 1. In order to avoid singularities from the denominator we have assume a condition which is described in the following.
The potential along the S = 0 direction is: (3.9) The shape of the V (X, S) scalar potential presented in Figure 1 along with the inflaton trajectory description and the simplified form in (3.9) is similar with the one presented in [29,30]. As it is usually the case in no-scale supergravity, the effective potential displays a singularity when the denominator vanishes. The presence of these singularities lead to an exponentially steep potential which can cause violation of the basic slow-roll conditions (i.e. ε 1, |η| 1). Consequently, these singularities must be removed. In our specific model described by the potential (3.9), we first notice that for the special value ξ = 1 the potential is free from singularities. For generic values of ξ however, i.e. ξ = 1, the potential displays a singularity for X = 6 1−ξ . In order to remove the zeros of the denominator in (3.9), we assume the following condition [29], This is a strong assumption which relates parameters with different origins. Indeed, α is a superpotential parameter while ξ descents from the Kahler potential. Since in our specific model the condition (3.10) lacks an explanation from first principles, it will be reasonable in the subsequent analysis to study the effects of a slightly relaxed version of (3.10). This can be achieved by introducing a small parameter δ (with δ 1) and modifying the condition as follows, In the remaining of this section, we are going to study the potential for special ξ values using the conditions (3.10) and (3.11).
We will start by analysing some special cases first. By imposing (3.10), which means δ = 0 the scalar potential simplifies to a quadratic monomial, something that can be also seen from the plots in Figure 1, where for small values of S (along the S = 0 direction) the potential receives a quadratic shape form. The equation (3.12) shows the potential of a chaotic inflation scenario. However, at this stage, the inflaton field X is not canonically normalized since its kinetic energy terms take the following form We introduce a canonically normalized field χ satisfying (3.14) After integrating, we obtain the canonically normalized field χ as a function of X Next, we investigate the implications of equation (3.15) by considering two different cases, for ξ = 0 and ξ = 0.
• For ξ = 0 we have X = √ 6 tanh χ √ 6 and the potential becomes, which is analogous to the conformal chaotic inflation model (or T-Model) [33]. In these particular type of models the potential has the general form: As we can see, for n = 1 we receive our result in (3.16) with λ = 3λ 2 υ 4 . This potential can be further reduced to subcases depending upon the value of χ. For χ 1 the potential in equation (3.16) reduces to Starobinsky model [34]. In this case the inflationary observables have values (n s , r) ≈ (0.967, 0.003) and the tree level prediction for ξ = 0 is consistent with the latest Planck bounds [35]. This type of models will be further analysed in the next section where inflation along the S-direction is discussed.
• The particular case of ξ = 1 implies a quadratic chaotic inflation and the tree-level inflationary prediction (n s , r) ≈ (0.967, 0.130) is ruled out according to the latest Planck 2015 results. For 0 < ξ < 1 , the prediction for (n s , r), can be worked out numerically.
After this analysis we turn our attention to a numerical calculation. In our numerical analysis we imply the modified condition (3.11) were as mentioned previously a small varying parameter δ has been introduced in order to soften the strict assumption (3.10). By substitute the relaxed condition (3.11) in (3.9) and neglecting O(δ 2 ), the potential receives the following form: (3.18) As we observe the first term in the above relation is the quadratic potential (3.12), while the second term encodes the effects of the small parameter δ. In addition, we note that the order of the singularity enhancement have been improved in comparison with the initial potential (3.9). Next we present our numerical results where the rôle of the parameter δ is also discussed.

NUMERICAL ANALYSIS
Before presenting numerical predictions of the model it is useful to briefly review here the basic results of the slow roll assumption. The inflationary slow roll parameters are given by [36,37]: The third slow-roll parameter is, 4 (3.20) where a prime denotes a derivative with respect to X. The slow-roll approximation is valid as long as the conditions 1,| η | 1 and ς 2 1 hold true. In this scenario the tensor-to-scalar ratio r, the scalar spectral index n s and the running of the spectral index The number of e-folds is given by, where l is the comoving scale after crossing the horizon, X l is the field value at the comoving scale and X e is the field when inflation ends, i.e max ( (X e ) , η (X e ) , ς (X e )) = 1. Finally, the amplitude of the curvature perturbation ∆ R is given by: .
Focusing now on the numerical analysis, we see that we have to deal with three parameters: ξ, δ andλ. We took the number of e-folds (N ) to be 60, and in Figure 2 we present two different cases in the n s − r plane, along with the Planck measurements (Planck TT,TE,EE+lowP) [35]. Specifically, in Figure 1(a), we fixed ξ and varyλ and δ. The various colored (dashed) lines corresponds to different fixed ξ-values. The green line corresponds to the limiting case with ξ = 1 and as we observe the results are more consistent with the Plank bounds (black solid contours) as the value of ξ decreases. Similar, in Figure 1(b) we treat δ as a fixed parameter while we vary ξ andλ. Also, in this case, we observe that for a significant region of the parameter space the solutions are in good agreement with the observed cosmological bounds. The green curve here corresponds to δ = 10 −6 . The special case with δ = 10 −6 ∼ 0 and ξ = 1 is represented by the black dot and as we discussed earlier is ruled out from the recent cosmological bounds. We observe from the plot that, as ξ approaches to unity the splitting between the curves due to different values of δ is small and the solution converges to δ ∼ 0 case. However, as we decrease the values of ξ we have splitting of the curves and better agreement with the cosmological bounds. Finally in plots 1(c) and 1(d) we present values of the running of the spectral index with respect to n s . We observe that the running of the spectral index, approximately receives values in the range −5 × 10 −4 < dn S d ln k < 5 × 10 −4 . Next we present additional plots to better clarify the rôle of the various parameters involved in the analysis.
Firstly, we study the spectral index n s as a function of the various parameters. The results are presented in Figure 3. In plots (a) and (b) we consider the cases with fixed values for ξ and δ respectively, and we take variations forλ. We vary the parameter ξ in the range ξ ∼ [0.92, 1] with the most preferable solutions for ξ [0.96, 1]. In addition the two plots suggest that acceptable solutions are found in the rangeλ ∼ [10 −2 , 10 −1 ]. In plots (c) and (d) n s is depicted in terms of δ and ξ respectively. As we expected the dependence on δ is negligible when it receives very small values, since we observe from plot 3(c) that the various curves are almost constant for very small δ values. The results are become more sensitive on δ as we decrease the value of ξ. This behaviour can also be confirmed from the potential (3.18). As we can see for ξ ∼ 1 the second term is simplified and the potential receives a chaotic like form. In this case the effects of small δ in the observables are almost negligible (green line). However as we decrease the value of ξ and we increase the values of δ the second term becomes important and contributes to the results.
Next, in Figure 4 we consider various cases for the tensor to scalar ratio, r. The description of the plots follows the spirit of those presented in Figure 3 for the spectral index n S . In particular, by comparing the plots 4(c) and 3(c) we notice that the dependence of r on δ is weaker in comparison with n S . Thus the relaxation parameter δ strongly affects the spectral index n S while for δ < 10 −4 and fixed ξ the tensor-scalar ratio r remains almost constant. In summary from the various figures presented so far we observe that consistent solutions can be found in a wide range of the parameter space. We also note that the model predicts solutions with r ≤ 0.02, which is a prediction that can be tested with the discovery of primordial gravity waves and with bounds of future experiments.
Regarding the superpotential parameterλ, we can see from the various plots that and we evaluate it at the pivot scale. In Figure 5 we show the values of the Hubble parameter in the (H inf − n s ) plane. We observe that the values of the Hubble parameter with respect to n s bounds are of order 10 13 GeV.

REHEATING
As  Majorana mass for the RH-neutrinos can be realized from the following non-renormalisable term where we have suppressed generation indices for simplicity, γ is a coupling constant and M * represents a high cut-off scale (for example the compactification scale in a string model or the Planck scale M P l ). In terms of SO(10) GUTs this operator descent from the following invariant operator and as described in [38] can be used to explain the reheating process of the universe after the end of inflation. In our case the 4-2-2 symmetry breaking occur in two steps: We can see that a heavy Majorana scale scenario implies that the SU (2) R breaking scale should not be much lower than the SU (4) scale and also γ should not be too small. Another important role of the higher dimensional operators is that after inflation the inflaton X decays into RH neutrinos through them to reheat the Universe. In addition the subsequent decay of these neutrinos can explain the baryon asymmetry via leptogenesis [39,40] . For the reheating temperature, we estimate [38] (see also [41]) : where the total decay width of the inflaton is given by with M ν c = γ ν H 2 M P l the mass of the RH neutrinos and M X the mass of the inflaton. The later is calculated from the effective mass matrix at the local minimum and approximately  is M X = 2M 2λυ 2 . Since M 10 13 GeV, the decay condition M X > M ν c it is always satisfied for appropriate choices of the parameters ν H and γ. In Figures 6 we present solutions in n s − T RH and r − T RH plane with respect to the various parameters of the model. For the computation of T RH we assume that ν H = M λ v 2 and we present the results for γ = 0.1 (solid), γ = 0.5 (dashed) and γ = 1 (dotted). In this range of γ values we have a Majorana mass, M ν c ∼ 10 6 − 10 7 GeV, which decreases as we decrease the value of γ. In addition, gravitino constraints implies a bound for the reheating temperature with T RH < 10 6 − 10 9 GeV and as we observe from the plots there are acceptable solutions in this range of values. More precisely, from plots (a) and (c) we see that for ξ > 0.97 and γ > 0.5 most of the results predict T RH > 10 9 GeV. However, it is clear that the consistency with the gravitino constraints strongly improves as we decrease γ, since all  the curves with γ = 0.1 (solid lines) predicts T RH 10 9 GeV. Similar conclusions can be derived from plots (b) and (d). In addition, from the r − T RH plots (c) and (d) we observe that for T RH < 10 6 − 10 9 there are regions in the parameter space with r ∼ 10 −2 − 10 −3 . Furthermore, we observe from plot 6(c) that the tensor-scalar ratio and the reheating temperature are decreased as we decrease the value of ξ since the curves are shift to the left and down regions of the plot.
A sample of the results have been discussed so far is presented in Table 1. The table is organized in horizontal blocks and each block contains three sets of values. For each set in a block we change only the coupling constant γ (γ = 1, 0.5, 0.1) while we keepλ, ξ and δ constant. We observe that as we decrease the values ofλ and ξ the values of the tensor to scalar ratio (r) and the reheating temperature (T RH ) also decreased.

INFLATION ALONG S DIRECTION
Here we briefly discussed the case where the S field has the rôle of the inflaton. In the potential (3.7) we put ν H = 0 and ν H = 0 so we have: In order to remove the singularity of the denominator, we take m = 6κ. In this case we get the following simple form which is of the form of a chaotic-potential.
Now the kinetic energy is defined as, then the potential in (3.30) becomes, V = 72κ 2 X 2 , and from the coefficient of the kinetic energy term we can find X in terms of a canonical normalized field χ: The potential in terms of the canonical normalized field reads as which is analogous to the conformal chaotic inflation model or T-Model inflation already mentioned before. Potentials for the T-Model inflation are given in Equation (3.17). For n = 1 the potential become, V (χ) = λ tanh 2 χ √ 6 , which is similar to our potential in (3.33) for λ = 432κ 2 . We can understand the inflationary behaviour in these type of models, by considering two cases.
First for χ 1, by writing the potential in exponential form we have Using the relation above we have for the slow-roll parameter that, (3.38) Similarly the second slow-roll parameter η is found to be, Finally, the predictions for the tensor-to-scalar ratio r and the natural-spectral index n s are, r = 12 N 2 , n s = 1 + 2η − 6 = 1 − 2 N − 9 4N 2 (3.40) and for N = 60 e-foldings we get n s 0.9673 and r 0.0032.
Regarding the case with χ 1, we can see from the expression (3.33) that the potential reduces to a quadratic chaotic form. The tree-level inflationary predictions in this case are (n s , r) ≈ (0.967, 0.130), which are ruled out with the latest Planck 2015 results.
The discussion above strongly depends on the assumption m = 6κ that we imposed on the potential in order to simplify it. If we consider small variations of this assumption similar to (3.11) and modify the condition as, m = 6κ + δ, we will see that the parameter δ contributes only to n S while the tensor-to-scalar ratio r remains constant.

CONCLUSIONS
In the present work we have studied ways to realise the inflationary scenario in a no-scale supersymmetric model based on the Pati-Salam gauge group SU (4) × SU (2) L × SU (2) R , supplemented with a Z 2 discrete symmetry. The spontaneous breaking of the group factor SU (4) → SU (3) × U (1) B−L is realised via the SU (4) adjoint Σ = (15, 1, 1) and the breaking of the SU (2) R symmetry is achieved by non-zero vevs of the neutral components ν H , ν H of the Higgs fields (4, 1, 2) H and (4, 1, 2)H.
We have considered a no-scale structure Kähler potential and assumed that the Inflaton field is a combination of ν H , ν H and find that the resulting potential is similar with the one presented in [29,30] but our parameter space differs substantially. Consequently, there are qualitatively different solutions which are presented and analysed in the present work. The results strongly depend on the parameter ξ and for various characteristic values of the latter we obtain different types of inflation models. In particular, for ξ = 0 and canonical normalized field χ ≥ 1, the potential reduces to Starobinsky model and for ξ = 1 the model receives a chaotic inflation profile. The results for 0 < ξ < 1 have been analysed in detail while reheating via the decay of the inflaton in right-handed neutrinos is discussed.
We also briefly discussed the alternative possibility where the S field has the rôle of the inflaton. In this case, the potential is exponentially flat for χ ≥ 1. Similar conclusions can be drawn for the Starobinsky model. On the other hand for small χ it reduces to a quadratic potential.
In conclusion, the SU (4)×SU (2) L ×SU (2) R model described in this paper can provide inflationary predictions consistent with the observations. Performing a detailed analysis we have shown that consistent solutions with the Planck data are found for a wide range of the parameter space of the model. In addition the inflaton can provide masses to the right-handed neutrinos and depending on the value of reheating temperature and the right-handed neutrino mass spectrum thermal or non-thermal leptogenesis is a natural outcome. Finally we mention that, in several cases the tensor-to-scalar ratio r, a canonical measure of primordial gravity waves, is close to∼ 10 −2 − 10 −3 and can be tested in future experiments.