Gravity Waves and Proton Decay in Flipped SU(5) Hybrid Inflation Model

We revisit supersymmetric hybrid inflation in the context of flipped $SU(5)$ model. With minimal superpotential and minimal K\"ahler potential, and soft SUSY masses of order $(1 - 100)$ TeV, compatibility with the Planck data yields a symmetry breaking scale $M$ of flipped $SU(5)$ close to $(2 - 4) \times 10^{15}$ GeV. This disagrees with the lower limit $M \gtrsim 7 \times 10^{15}$ GeV set from proton decay searches by the Super-Kamiokande collaboration. We show how $M$ close to the unification scale $2\times 10^{16}$ GeV can be reconciled with SUSY hybrid inflation by employing a non-minimal K\"ahler potential. Proton decays into $e^+ \pi^0$ with an estimated lifetime of order $10^{36}$ years. The tensor to scalar ratio $r$ in this case can approach observable values $\sim 10^{-4} - 10^{-3}$.


Introduction
Supersymmetric (SUSY) hybrid inflation model [1,2,3,4,5] has attracted a fair amount of attention due to its simplicity and eleganance in realizing the grand unified theory (GUT) models of inflation [5]. In models with minimal Kähler potential, the soft linear and mass squared terms play important role in attaining the scalar spectral index compatible with the current experimental observations [6,7]. The next important task is to explore the possibility of realizing the gauge symmetry breaking scale M close to a typical GUT scale ∼ 2 × 10 16 GeV. This can, in turn, adequetely suppress the proton decay rate from dimension six operators usually present in GUT models . Achieving M ∼ 2 × 10 16 GeV was one of the main prediction of the original SUSY hybrid inflation model where only radiative correction was included in otherwise a flat potential [1]. We, therefore, investigate the possibility of realizing large enough M in SUSY hybrid inflation model with minimal 1 Email: mansoor@qau.edu.pk 2 Email: shafi@bartol.udel.edu 3 Email: umer@udel.edu Kähler potential, including various important corrections [1,3,5,6,7]. Specifically, we update the status of SUSY flipped SU(5) hybrid inflation model [8,7] with minimal Kähler potential and soft SUSY masses ∼ 1 − 100 TeV. For other hybrid models of inflation in flipped SU(5) gauge group see [9] where each of two hybrid fields is shown to realize inflation. For no-scale SUSY flipped SU(5) models of inflation see [10,11].
The flipped SU(5) ≡ SU(5) × U(1) X model [12,13] exhibits many remarkable features and constitutes an attractive choice as a grand unified gauge group. In flipped SU(5) model, the doublet-triplet splitting problem is elegantly solved due to the missing partner mechanism [13]. The proton decay occurs via dimension six operators and is naturally long lived with M around GUT scale. Moreover, it lacks the monopole problem that appears in the spontaneous breaking of other GUT gauge groups (i.e. SU(5), SU(4) C × SU(2) L × SU(2) R or SO (10)). This property also makes the flipped SU(5) model appropriate choice for the standard version of SUSY hybrid inflation where gauge symmetry is broken after the end of inflation. Finally, flipped SU (5) is also regarded as a natural GUT model due to its connection with F-theory [14].
The outline of the paper is as follows. In the section-2 we briefly introduce the SUSY hybrid model of flipped SU(5) which was first proposed in [8]. We update the status of this model with minimal Kähler potential in section-3 and check its compatibility with the proton lifetime constraint. The minimal model with ∼ 1 − 100 TeV scale soft SUSY masses is shown to predict fast proton decay. However, with the help of leading order non-minimal terms in the Kähler potential we overcome this problem and the predictions of inflationary parameters are found to be in accordance with the latest Planck data. This is discussed in detail in section-4. The dominant proton decay mode is p → e + π 0 with a lifetime estimated to be of order 10 36 years. Finally, we provide a brief summary of our findings in section-5.

SUSY FSU(5) hybrid inflation
The minimal Higgs sector of Flipped SU(5) ≡ F SU(5) ≡ SU(5) × U(1) X consists of a pair of Higgs superfields (10 H , 10 H ), and a second pair of 5-plet Higgs superfields (5 h , 5 h ), which are decomposed under the SM gauge group as where N c is the right handed neutrnio superfield. Assuming the following R-charge assignment of the superfields the superpotential of the model is given by [8], where the scalar component of the gauge singlet superfield S acts as the inflaton. The first line in Eq. (4) is relevant for inflation and is also responsible for the gauge symmetry breaking of F SU(5) into MSSM as the 10-plet Higgs pair attains non-zero vev in the The second line in Eq. (4) contains the terms that are involved in the solution of doublettriplet splitting problem. The U(1) R symmetry plays a key role here. This symmetry not only eliminates the S 2 and S 3 terms to realize successful inflation, it also forbids the bilinear term 5 h 5 h to avoid GUT scale masses of the MSSM Higgs doublets H u and H d .
The MSSM µ problem is assumed to be solved by the Giudice-Masiero mechanism [15]. ij denote the corresponding Yukawa couplings. For a discussion of light neutrino masses in this model see [8]. Another possibility to realize light neutrino masses by assuming R-breaking at non-renormaizable level is discussed in [16]. As all matter superfields are neutral under U(1) R symmetry, an additional Z 2 symmetry (or matter parity) is assumed [8]. This symmetry not only realizes the possibility of LSP as a cold dark matter candidate but also avoids some unwanted terms in the superpotential.
In the D-flat direction, the relevant part of the global SUSY potential may be written as Along the inflationary valley (|10 H | = |10 H | = 0), SUSY is temporarily broken by the vacuum energy density V 0 = κ 2 M 4 , and is restored later at the global minimum (| 10 H | = | 10 H | = M, | S | = 0). In the inflationary trajectory, the effective contributions of 1-loop radiative correction and soft SUSY breaking terms can be written as with and Here, N = 10 is the dimensionality of the 10-plet Higgs conjugate pair, Q is the renormalization scale and we have defined x ≡ |S|/M. The a and M S are the coefficients of soft SUSY-breaking linear and mass terms for S, respectively and m 3/2 is the gravitino mass.

Minimal Kähler potential
In order to include the supergravity (SUGRA) correction we first consider the minimal canonical Kähler potential, The F-term SUGRA scalar potential is given by with z i being the bosonic components of the superfields z i ∈ {S, 10 H , 10 H , · · · }, and we have defined and Putting all these corrections together, we obtain the following form of inflationary potential The prediction of various inflationary parameters can now be estimated using standard slow-roll definitions described below. The leading order slow roll parameters are defined as, where m P = 2.4 × 10 18 GeV is the reduced Planck mass. In the leading order slow-roll approximation, the scalar spectral index n s , the tensor-to-scalar ratio r and the running of the scalar spectral index dn s /d ln k are given by For negligibly small values of r and dns d ln k , the relevant Planck constraint on the scalar spectral index n s in the base ΛCDM model is [17] n s = 0.9677 ± 0.0060 (68%CL, P lanck T T + lowP + lensing).
The amplitude of the primordial spectrum is given by, and has been measured by Planck to be A s = 2.137 × 10 −9 at k 0 = 0.05 Mpc −1 [17]. The last N 0 number of e-folds before the end of inflation is, where x 0 is the field value at the pivot scale k 0 , and x e is the field value at the end of inflation. The value of x e is fixed either by the breakdown of the slow roll approximation, or by a 'waterfall' destabilization occurring at the value x c = 1 if the slow roll approximation holds.
The results of our numerical calculations are depicted in Figs. 1 and 2. Following [7], we have taken a = −1 assuming appropriate initial condition for arg S [18]. In addition, we set the number of e-folds N 0 = 50 and the scalar spectral index n s is fixed at the central value (0.968) of Planck data bounds. The left panel of Fig. 1 shows the behavior of κ with respect to M S , while the behavior of GUT symmetry breaking scale M with respect to M S is shown in the right panel and is of particular importance because of proton decay considerations. The curves are drawn for different values of m 3/2 . The solid curves are drawn for M 2 S < 0, while the dashed curves are drawn for M 2 S > 0. To make our discussion relevant for the current experiments we restrict the soft masses to be ∼ 1 − 100 TeV. This automatically includes the split-susy scenario where soft scalar masses can take values upto 100 TeV [19]. In our region of interest, the radiative correction provides the dominant contribution while the sugra correction is mostly negligible. The suppression of sugra correction is supported by the tiny values of S 0 /m p 2 × 10 −3 as shown in the left panel of Fig. 2. This approximation simplifies the expressions of the amplitude of curvature perturbation and the scalar spectral index as Next we estimate the values of κ, M and M S such that the contributions of the soft linear and mass squared terms are comparable in Eq. (23). Assuming x 0 ∼ 1 we obtain the following expressions using the above equations, For M S 10 4 TeV, the soft mass squared term begins to take over which drives the curve upward for M 2 S < 0, and downward for M 2 S > 0. Furthermore, the tensor to scalar ratio r turns out to be extremely small, taking on values r ∼ 1.5 × 10 −13 − 6.3 × 10 −11 . This can be seen in the right panel of Fig. 2 where r is plotted against M S for different values of m 3/2 . Our findings in this section are compatible with the results of [20] where SUSY hybrid inflation with minimal Kähler and soft SUSY masses of order ∼ (0.1 − 10) TeV is considered for U(1) B−L gauge group. Finally, it is important to note that with a minimal Kähler potential and soft SUSY masses of order ∼ (1 − 100) TeV, the symmetry breaking scale turns out to be relatively small, M ∼ (1.7 − 3.8) × 10 15 GeV. This leads to fast proton decay rate as briefly discussed below.

Proton Decay in FSU(5)
In the flipped SU(5) model, the dangerous dimension five proton decay operators are highly suppressed due to R-symmetry. For example, even though the operators 10 F (5) gauge invariant, they are not invariant under R-symmetry. Further, consider the following R-symmetric F SU(5) gauge invariant operators [8], With softly broken SUSY, the superfield S is expected to attain non-zero vacuum expectation value S ≃ −m 3/2 /κ [21,22]. This makes the above operators heavily suppressed. There are additional R-symmetric F SU(5) gauge invariant operators which lead to proton decay. These include, Although heavily suppressed to have any observable signature for proton decay, these operators are not allowed due to the additional Z 2 matter parity imposed on the superpotential (Eq. (4)). The matter superfields 10 F i , 5 f j and 1 e l are odd under this Z 2 matter parity whereas all other superfields are even. This matter parity not only forbids many unwanted couplings as mentioned in [8], but it also makes the neutral LSP a suitable dark matter candidate. Therefore, proton decay occurs via dimension six operator from the superheavy gauge boson exchange, and the lifetime for the channel p → e + π 0 is given by [23,24,25],

Non-minimal Kähler potential
The non-minimal Kähler potential may be expanded as, Using Eqs. (4), (12) and (33) along with the radiative correction in Eq. (7) and soft mass terms in Eq. (8), we obtain the following inflationary potential, where γ S = 1 − 7κ S 2 + 2κ 2 S − 3κ SS , and we have retained terms up to O ((|S|/m P ) 4 ) from SUGRA corrections (Recall that during inflation both |10 H | and |10 H | are zero). We further assume soft masses, am 3/2 and M S to be ∼ 1 − 100 TeV, with a and M 2 S either positive or negative.
The parameter space consistent with the Planck data bounds is enlarged with the addition of two non-minimal parameters κ S and κ SS . However, to make our discussion interesting for near future experiments [28,29], we restrict ourselves to the parameter region with the largest possible values of the tensor to scalar ratio r with M ∼ 2 × 10 16 GeV. As previously discussed in [30], the possibility of larger r solutions restricts the non-minimal parameters, namely κ S < 0 and κ SS > 0 with the quartic coupling γ S < 0. Therefore, large r solutions are obtained mainly with a potential of the form The linear and soft mass squared terms with am 3/2 and M S ∼ 1 − 100 TeV are suppressed, while the radiative and SUGRA corrections parametrized by κ S and κ SS play the dominant role. To keep the SUGRA expansion under control we further limit S 0 ≤ m P . We also require that the non-minimal couplings |κ S | ≤ 1 and |κ SS | ≤ 1. Using next to leading order slow roll approximation, the results of our numerical calculations are displayed in Figs. 3 -6. The lighter (darker) region represents the Planck 2-σ (1-σ) bounds on r and n s . The upper and lower cutoffs correspond to the S 0 = m P and κ SS = 1 constraints. We have also included a brown shaded region for |S 0 | 0.5 m P . This is the ultraviolet sensitivity region where higher order Planck suppressed terms in the SUGRA expansion become important. Outside this region with |S 0 | 0.5 m P we not only obtain a natural suppression of higher order terms but also ensure the boundness of the potential, a problem arising due to γ S < 0 [31]. Fig. 3 depicts the behavior of κ and the tensor to scalar ratio r with respect to the non-minimal coupling κ S . These plots resemble one another and their behavior can be understood from the following approximate relation between r and κ (from Eqs. (18) and (21)), (36)  Fig. 3). In the leading order slow roll approximation, the spectral index n s and tensor to scalar ratio r are given by Solving these two equations simultaneously for S 0 ≃ m P , r ≃ 5 × 10 −3 and n s ≃ 0.968 (central value) we obtain κ S ≃ −0.03 and γ S ≃ −0.014. Further, with |S 0 | ∼ 0.5 m P and r ∼ 10 −3 , we get κ ∼ 0.09, κ S ≃ −0.03 and γ S ≃ −0.06. These estimates are in good agreement with our numerical results displayed in Figs. 3−6. The behavior of κ SS and γ S with respect to κ S is presented in Fig. 4, while Fig. 5 depicts the behavior of S 0 /m P with respect to κ S and γ S . To facilitate this discussion further, we have also provided a plot of r versus γ S in the left panel of Fig. 6 where it can be seen that larger values of r (∼ 10 −3 ) are obtained for smaller values of γ S (∼ −0.06). It is important to note that large tensor modes can be obtained for any value of scalar spectral index n s within Planck bounds. In short, for non-minimal couplings −0.034 ≤ κ S ≤ −0.027 and 0.37 ≤ κ SS ≤ 1, we obtain the scalar spectral index n s within the Planck 2 − σ bounds and tensor to scalar ratio r in the range (3 × 10 −5 − 5 × 10 −3 ). Moreover, with the symmetry breaking scale fixed at (∼ 2 × 10 16 GeV), the proton is naturally stable with a lifetime of ∼ 2 × 10 36 years.  The right panel of Fig. 6 shows the dependence of the spectral running dn s /d ln k on tensor-to-scalar ratio r. It can be seen that the spectral running does not vary appreciably with r and takes on roughly the same values for large and small r values. The scalar spectral running dn s /d ln k varies in the range (0.0024 − 0.0034) and this justifies the use of latest Planck's data of ΛCDM+tensors with no running for the purpose of comparing the predictions of this model. Finally, smaller r values (∼ 10 −5 ) are obtained for S 0 0.1 m P and large κ SS ≃ 1 for which γ S is negative and rather large (∼ −2), as depicted in Figs. 4 -6. Since both the quadratic and quartic couplings (κ S , γ S ) are negative in this region, the form of potential remains the same as in Eq. (35). For κ S > 0 and γ S < 0 with soft masses ∼ 1 − 100 TeV, only tiny values of r 10 −12 are obtained as discussed recently in [32].

Summary
To summarize, we have revisited supersymmetric hybrid inflation in the framework of flipped SU(5) model. We have shown that with a minimal Kähler potential and soft SUSY masses of order (1 − 100) TeV, this model predicts a symmetry breaking scale M ∼ (2 − 4) × 10 15 GeV, for the central value n s = 0.968. This value of M is significantly below the GUT unification scale 2×10 16 GeV and leads to proton lifetime τ ∼ 10 32−33 years as compared to the current lower limit τ p→e + π 0 1.6 ×10 34 years determined by the Super-Kamiokande collaboration. The tensor-to-scalar ratio also turns out to be extremely small, taking on values r ∼ (10 −13 − 7 × 10 −11 ). By employing non-minimal Kähler potential, with soft SUSY masses of order 1 − 100 TeV, the symmetry breaking scale M ∼ 2 × 10 16 GeV is easily achieved within the Planck σ-bounds on n s and the proton is naturally of order ∼ 10 36 years. Moreover, larger tensor modes with observable values (∼ 10 −4 − 10 −3 ) are obtained with non-minimal couplings −0.034 ≤ κ S ≤ −0.027 and 0.37 ≤ κ SS ≤ 1.