A $U(1)_R$ inspired inflation model in no-scale Supergravity

We consider a cosmological inflation scenario based on a no-scale supergravity sector with $U(1)_R$ symmetry. It is shown that a tree level $U(1)_R$ symmetric superpotential alone does not lead to a slowly rolling scalar potential. A deformation of this tree level superpotential by including an explicit $R$ symmetry breaking term beyond the renomalizable level is proposed. The resulting potential is found to be similar (but not exactly the same) to the one in Starobinsky inflation model. We emphasize that for successful inflation, with the scalar spectral index $n_s \sim 0.96$ and the tensor-to-scalar ratio $r<0.08$, a correlation between the mass parameters in the superpotential and the vacuum expectation value of the modulus field $T$ in the K\"ahler potential must be adopted.


I. INTRODUCTION
Planck satellite's four years data of the cosmic microwave background radiation and the large structure in the universe support the predictions of cosmological inflation. The recent data confirmed that spectral index (scalar density fluctuations) is given by n s = 0.96 ± 0.007 and the upper bound on the tensor-to-scalar ratio is r < 0.08 [1]. These results imposed severe challenges on several inflationary models. For example, the simple chaotic and hybrid inflationary models [2] are now ruled out. On the other hand, some other models of inflation with compatible cosmological fluctuation predictions receive a growing interest. One of these models is the Starobinsky inflation [3], which is based on modified gravity.
A supergravity (SUGRA) realization of Starobinsky inflation has been studied in Ref. [4], by considering a noscale Kähler potential involving a modulus field T . It is well known that the no-scale SUGRA framework is free of the so-called η problem due to the involvement of logarithimic form in the Kähler potential. In Ref. [4], the no-scale Kähler potential of T field is combined with a Wess-Zumino superpotential consists of a quadratic and a cubic terms of the inflaton superfield S: W = µS 2 − κS 3 , with µ as a parameter of mass dimension and κ is a dimensionless parameter. It turns out that at a specific point of the parameter space, this construction becomes conformal equivalent to R+αR 2 modified gravity models similar to Starobinsky inflation model. Adding a term linear in field S to this renormalizable superpotential, the authors of [5] have shown that it is also possible to realize supersymmetry breaking at the end of inflation. It is further indicated that a successful inflation consistent with correct n s and r values may indicate an upper bound on gravitino mass, once the Starobinsky limit is implimented. Few other studies having different kinds of motivation involving Starobinsky type inflation model can be found in [6][7][8][9][10][11][12][13][14][15][16][17][18].
While the constructions in [4] and [5] are certainly elegant and minimal from their own perspectives, we notice that it is not possible to define an R charge for the su-perfield S so that superpotential W can have R charge of two units. Hence no U (1) R symmetry is prevailing in this construction. Now it is well known that R symmetry plays important roles in many supersymmetric constrctions. One such example is related to the supersymmetry breaking. According to Nelson-Seiberg theorem [19], existence of an R symmetry is a necessary condition in order to realize supersymmetry breaking. However an exact R symmetry forbids gauginos and Higgsinos to have mass. Hence it must be broken (spontaneously or explicitly). It is customary to break R symmetry spontaneously as done in many dynamical supersymmetry breaking models leading to R axions [20].
In this letter, we start with a U (1) R global symmetry. We assume that the inflaton superfield S has an R charge unity. Thus, the tree level superpotential is given by W = µ 2 S 2 , with µ as a mass scale. As we will show below, this tree level superpotential does not lead to a slowly rolling scalar potential. We propose a deformation of this tree level superpotential (having R charge 2) by including an explicit R symmetry breaking term beyond the renomalizable level. This new term is naturally expected to be suppressed by the cut-off scale M * , and hence W can be expressed as where λ is a dimensionless coupling and M * is a mass scale. Since a global symmetry is expected to be broken by gravity effects, a natural choice of M * would be the Planck scale (M P ), M * ∼ M P . A similar M P suppressed R-symmetry breaking term has been considered in supersymmetric hybrid inflation scenario [21], with minimal Kähler potential. It was emphasized there that in order to get n s within the preferred range, λ must be small enough ( < ∼ 10 −7 ). Below we study the above superpotential in Eq.
(1) and discuss how inflation can be realized in this framework. The paper is organized as follows. In section II, we study the associate inflation derived from the above mentioned U (1) R symmetric superpotential and no-scale Kähler potential. In section III, inflationary predictions are discussed, in particular the correlation between the spectral index n s and the ratio r. Finally, our concluding remarks are given in section IV.

II. THE MODEL
In addition to the superpotential W , we consider the Kähler potential (as standard in no-scale supergravity) where T is the modulus field. The Kähler potential remains invariant under U (1) R symmetry with vanishing R charge for the moduli field. The supergravity potential can be obtained using where, where i, j refer to the modulus T and inflaton S. Now using the superpotential in Eq.
(2), V F can be obtained as This is a feature of no-scale supergravity that leaves the potential V F as independent of T (apart from the dependence through the pre-factor e K/M 2 P ) and positive definite. Therefore, it can be an appropriate framework for inflationary scenarios.
Following [4], we assume here the modulus filed T is stabilized at a fixed scale such that T + T * = c. This stablization requires a non-perturbative effect at a high scale [22,23]. With this assumption, the effective Lagrangian turns out to be In order to have the kinetic term for the complex scalar field S as a canonically normalized one, following the prescription of [4], we first redefine the S field in terms of χ, With the above definition of S and considering χ = (χ 1 + iχ 2 )/ √ 2, the kinetic term (L K.E. ) becomes and the F-term scalar potential responsible for inflation will have the form From this potential, one can show that the field dependent mass squared of the imaginary component of χ, obtained by the second derivative of V F respect to χ 2 at the minimum χ 2 = 0, is much larger than the Hubble scale squared during inflation (a numerical estimate will be provided in next section). Therefore imaginary part χ 2 will be stabilized at zero during the inflation. Hence we set χ 2 to be zero from now on and identify the Fterm potential with the inflation potential, V Inf . Note that with this choice, the kinetic term of the Lagrangian (see Eq. (7)) becomes canonical. In this case, the inflation potential takes the form, where A = 3 µ 2 cM P and B = λ 3cM P µM * are two dimensionless constants. In the last expression of V Inf , we have set M P = 1 unit. Untill otherwise stated, we will use this unit for the rest of our discussion. Note that when B = 1, this potential simplifies to the form We have shown the form of this potential (normalized by A) in Fig. 1 for different choices of B. With B = 1, the shape of the V (B=1) Inf (denoted by the brown line in Fig.1) turns similar to the standard Starobinsky potential [3]. If we reduce the value of B from 1 by a tiny amount, the potential starts to become steep. On the other hand, if we enhance B from one, another distant minimum appears (other than at χ 1 = 0) at some very large value of the field χ 1 .
The inference of the above discussion is that the field χ 1 can now be identified as the inflaton in the limit (i) χ 2 → 0 and (ii) B is very close to 1 as the required flatness for inflation is obtainable from the associated potential V Inf of Eq. (9). In order to show the importance of the R symmetry breaking term λ, we include a plot of the potential against χ 1 with λ = 0 (i.e B = 0) in Fig. 1 denoted by the red curve. It is evident that such a potential can not provide sufficient inflation. Hence inclusion of an explicit R symmetry breaking term becomes instrumental in realizing inflation and that too by a restricted amount. From the nature of the plots, it is expected that for any large deviation of B from unity, the slow roll of the inflaton might be spoiled.
In order to have a better control over different values of B, we parameterize the deviation of B from 1 by ξ, B = 1 − ξ. Then the inflation potential in Eq.(9) can be expanded for small ξ as In M P = 1 unit, the slow roll parameters are given by Number of e-folds is written as where x e is the inflaton field value at which inflation ends and x * corresponds to the crossing horizon value of the inflaton. The three inflationary observables: tensor to scalar ratio (r), spectral index (n s ) and power spectrum (P s ) are provided by These observables are to be determined at x = x * .

III. INFLATIONARY PREDICTIONS
Let us now proceed to determine the inflationary observables in this scenario. The inflationary potential in Eq.(11) contains two free parameters A and ξ. Among them, ξ takes part in determining r and n s . The other parameter A will be fixed by observed value of scalar perturbation spectrum P s = 2.2 × 10 −9 . In Fig. 2 we show the Logarithmic plot of the spectral index n s versus the tensor to scalar ratio r, as predicted by our model. We Here we tabulate few reference points which provide correct values of n s and r within the allowed range of Planck limit considering N e = 55. Values of A are fixed from the value of the required power spectrum. Note that the parameters A and ξ in Table I are simply combinations of the original variables: c, µ, λ and M * . All of these variables serve significant importance from the model point of view. Therefore we should also estimate their magnitude in the set up. For the purpose we consider M * = M P , argued as the natural choice in the introduction.
Corresponding to the five reference points in Table I, below in Table II we provide values of c and µ in M P = 1 unit for λ = 1 and λ = 10 −2 .
Sl. no.  Table I. It can be noted from Table II that there exists a correlation between the two mass parameters c and µ for different values of λ. For example, in reference point I of Table II with λ = 1, the values of c and µ are found to be 4.81×10 −12 and 1.44×10 −11 respectively. For a comparatively smaller value of λ ∼ 10 −2 , magnitudes of µ and λ become O(10 −8 ). This can be interpreted by looking at the expressions of A and B which involve all the parameters c, µ, λ, M * = M P and keeping in mind that in order to achieve sucessful inflation, we have to have B = 3λc/µ value very close to unity. Therefore with a fixed choice of λ, the ratio c/µ is uniquely fixed. Then the parameter A = 3µ 2 /c will fix the value of µ from the requirement that the power spectrum P s ∼ O(10 −9 ).
For all the reference points mentioned in Table I, numerically it is found that the mass of the χ 2 field during inflation is significantly higher compared to the Hubble scale (H) during inflation given by H 2 = V Inf /3. In particular, one finds m 2 χ2 /H 2 4 at the minimum of χ 2 (χ 2 = 0). Hence χ 2 would be stabilized at origin during inflation. Furthermore we have also found numerically that the slope along the χ 2 direction is much steeper compared to the one for χ 1 . Hence χ 2 will move faster and reaches the minimum much earlier than χ 1 . This justifies our assumption χ 2 = 0 during inflation. Inflaton mass (m χ1 ) at its minimum for the above mentioned points is O(10 13 ) GeV as expected for this type of inflation scenario. We end this section by observing that even if χ 1 is super-Planckian as required by the slow-roll condition, the field S remains sub-Planckian as seen from Eq. (6). Now S being sub-Planckian, higher order U (1) R breaking terms in W are accordingly less important.

IV. CONCLUSION
In this paper, we propose a global R symmetry motivated inflation model within no-scale SUGRA. We find that the minimal U (1) R symmetric superpotential (quadratic in inflaton superfield) is unable to provide a successful inflation as the associated scalar potential turns out to be extremely steep. Then we introduce an explicit R symmetry breaking term in the superpotential at a non-renormalizable level which provides the required flatness for inflation. The introduction of such a U (1) R breaking term is motivated by the fact the any global symmetry will be broken by the gravity effect. For this reason, we associate the cut-off scale of this nonrenormalizable term with M P . The effective inflation potential resulted from our proposed set-up carries similarity with Staborinsky like inflation models in the limit, one combination of parameters of the superpotential and no-scale Kahler potential as B = 1. Varying B from unity by tiny amount leads to the predictions for the spectral index and tensor-to-scalar ratio. In order to keep these predictions within the limit allowed by Planck data, we evaluate the magnitudes of the relevant mass parameters of the model. Such a construction involving explicit R symmetry breaking term may also have some interesting consequences while supersymmetry breaking will also be involved. Since any dynamical supersymmetry breaking model requires that R symmetry should spontaneously be broken leading to the presence of R axion, such an explicit breaking term, connected with inflation, in our set-up can be helpful in providing the mass of it.

ACKNOWLEDGMENTS
The work of S.K. and A. M. is partially supported by the STDF project 18448 and the European Union