Shifted $\mu$-hybrid inflation, gravitino dark matter, and observable gravity waves

We investigate supersymmetric hybrid inflation in a realistic model based on the gauge symmetry $SU(4)_c \times SU(2)_L \times SU(2)_R$. The minimal supersymmetric standard model (MSSM) $\mu$ term arises, following Dvali, Lazarides, and Shafi, from the coupling of the MSSM electroweak doublets to a gauge singlet superfield which plays an essential role in inflation. The primordial monopoles are inflated away by arranging that the $SU(4)_c \times SU(2)_L \times SU(2)_R$ symmetry is broken along the inflationary trajectory. The interplay between the (above) $\mu$ coupling, the gravitino mass, and the reheating following inflation is discussed in detail. We explore regions of the parameter space that yield gravitino dark matter and observable gravity waves with the tensor-to-scalar ratio $r \sim 10^{-4}-10^{-3}$.


I. INTRODUCTION
In its simplest form supersymmetric (SUSY) hybrid inflation [1,2] is associated with a gauge symmetry breaking G → H, and it employs a minimal renormalizable superpotential W and a canonical Kähler potential K. Radiative corrections and soft SUSY breaking terms together play an essential role [3][4][5][6] in the inflationary potential that yields a scalar spectral index in full agreement with the Planck data [7]. In this minimal model the symmetry breaking G → H occurs at the end of inflation, and the symmetry breaking scale M is predicted to be of the order of (2 − 3) × 10 15 GeV [1, [3][4][5][6]. One simple extension of this minimal model retains a minimal W but invokes a nonminimal K [8], such that the correct scalar spectral index is obtained without invoking the soft SUSY breaking terms. Nonminimal Kähler potentials are also used to realize symmetry breaking scales comparable to the grand unified symmetry (GUT) scale M GUT (∼ 2 × 10 16 GeV) [9], and to predict possibly observable gravity waves [10,11].
If the symmetry breaking G → H produces topological defects such as magnetic monopoles, a more careful approach is required in order to circumvent the primordial monopole problem. The first such example is provided by the so-called 'shifted-hybrid inflation' [12,13], in which the monopole producing Higgs field actively participates in inflation such that, during inflation, G is broken to H and the monopoles are inflated away.
In this paper we explore inflation and reheating in the framework of the gauge symmetry SU (4) c × SU (2) L × SU (2) R (G 4-2-2 ) [14]. A SUSY model based on this symmetry including hybrid inflation was first explored in Ref. [15]. However, the primordial monopole problem was not resolved, but it was subsequently addressed and successfully rectified in Ref. [12] based on shifted hybrid inflation. In the model proposed here, we employ the mechanism invented in Refs. [15,16] for generating the MSSM µ term, and we exploit shifted hybrid inflation to overcome the monopole problem. We implement this scenario using both minimal and nonminimal Kähler potentials, and address in both cases important issues related to the gravitino problem [17]. For a discussion of leptogenesis via right-handed neutrinos in models where the dominant inflaton decay channel yields Higgsinos, see Ref. [18].
The plan of the paper is as follows: In Sec. II, we present the SUSY G 4-2-2 model including the superfields, their charge assignments, and the superpotential which respects a U (1) R symmetry. In Sec. III, the inflationary setup is described. This includes the scalar potential for global SUSY as well as the one including supergravity (SUGRA). The shifted µ-hybrid inflation (µHI) scenario with minimal Kähler potential and its compatibility with the gravitino constraint [19] is studied in Sec. IV. The analysis is extended by employing a nonminimal Kähler potential in Sec. V, discussing again the gravitino problem and the bounds it imposes on reheat temperature, and focusing on solutions with observable gravity waves. Our conclusions are summarized in Sec. VI.
The matter superfields F i and F c i belong in the following representations of G 4-2-2 , where the index i (= 1, 2, 3) denotes the three families of quarks and leptons, and the subscripts r, g, b, l are the four colors in the model, namely red, green, blue, and lilac. The and acquire nonzero vacuum expectation values (vevs) along the right-handed sneutrino directions, that is | ν c Hl | = | ν c Hl | = v = 0, to break the G 4-2-2 gauge symmetry to the standard model (SM) gauge symmetry G SM = SU (3) c ×SU (2) L ×U (1) Y , around the GUT scale (∼ 2 × 10 16 GeV), while preserving low scale SUSY [20]. The electroweak breaking is triggered by the electroweak Higgs doublets, h u and h d , which reside in the bidoublet Higgs superfield h represented as follows: Note that such doublets can remain light because of appropriate discrete symmetries [21].
A gauge singlet chiral superfield S = (1, 1, 1) is introduced, which triggers the breaking of G 4-2-2 and whose scalar component plays the role of the inflaton. A sextet Higgs The first three terms in the superpotential are of the standard µHI case as discussed in Refs. [19,22]. The first two and the fourth term characterize the 'shifted case' by providing additional inflationary tracks to avoid the monopole problem. The third term λSh u h d yields the effective µ term. Indeed assuming gravity-mediated SUSY breaking [23,24], the scalar component of S acquires a nonzero vev proportional to the gravitino mass m 3/2 and generates a µ term with µ = −λm 3/2 /κ, thereby resolving the MSSM µ problem [16]. The λ ij -terms contain the Yukawa couplings, and hence provides masses for fermions. The γ ij -terms yield large right-handed neutrino masses, needed for the see-saw mechanism. The other possible couplings similar to γ ij -terms which are allowed by the symmetries are F F H c H c , F F H c H c , and F c F c H c H c . The last two terms in the superpotential involving the sextuplet superfield G are included to provide superheavy masses to d c H and d c H . It is important to mention here that the matter-parity symmetry Z mp 2 , which is usually invoked to forbid rapid proton decay operators at renormalizable level, is contained in U (1) R as a subgroup. The superpotential W is invariant under Z mp 2 and this symmetry remains unbroken. There is no domain wall problem and the lightest SUSY particle (LSP) is stable and consequently a plausible candidate for dark matter (DM).
The relevant part of the superpotential for shifted µHI contains the terms where ξ = β 1 M 2 /κΛ 2 is a dimensionless parameter. We ignore the β 2,3 -terms in our future discussions as they become irrelevant in the D-flat direction, that is the direction where the D-term contributions vanish (i.e. with |ν c H | = |ν c H | and all other components zero). For simplicity, the superfields and their scalar components will be denoted by the same notation.
The global SUSY minimum obtained from Eq. (5) is given as which requires that ξ ≤ 1/4 for real values of v.
The global SUSY scalar potential obtained from the superpotential in Eq. (5) is

+D-terms. (7)
Taking the D-flat direction the scalar potential takes the form: Rotating the complex field S to the real axis by suitable transformations, we can identify the normalized real scalar field σ = √ 2S with the inflaton. Introducing the dimensionless the normalized potential V takes the form The extrema of the above potential with respect to u are given as: From now on we assume the system to be stabilized along a particular direction with z = 0.
These extrema can be visualized with the help of the potential V (u, w, z = 0), plotted in In Fig. 1, the standard µHI case with ξ = 0 is reproduced in plot (a). In this case, u = 0, w > 1 is the only inflationary valley available. It evolves at w = 0 into a single pair of global SUSY minima with vev v = ±M . For ξ = 0, in addition to the standard track at u = u 1 , two shifted local minima appear at u = u 2 for w > 1/8ξ − 1/2. In plot (b) for ξ < 1/8, the shifted tracks lie higher than the standard track. Following Ref. [12], in order to have suitable initial conditions for realizing inflation along the shifted tracks, we assume ξ ≥ 1/8. The normalized scalar potential V is shown in plots (c)-(e) for some realistic values of ξ, namely for ξ = 1/8, ξ = 1/6, and ξ = 1/4. In the last plot (f) with ξ > 1/4, we obtain V min = 0, and therefore SUSY will be broken at high scale after inflation. So for our analysis, it is appropriate to keep ξ within the interval [1/8, 1/4].
As the inflaton slowly rolls down the inflationary valley and enters the waterfall regime at w = 1/8ξ − 1/2, inflation ends due to fast rolling and the system starts oscillating about a vacuum at w = 0. Note that in the H c direction there are actually two pairs of vacua at However, the path leading to v − appears before the one leading to v + , as explained in Ref. [12]. The necessary slope for realizing inflation in the valley with w > 1/8ξ − 1/2, u = u 2 , z = 0 is generated by the inclusion of the one-loop radiative corrections, the SUGRA corrections, and the soft SUSY breaking terms. The one-loop radiative corrections V loop , arising as a result of SUSY breaking on the inflationary path, are calculated using the Coleman-Weinberg formula [25]: where F i and M 2 i are the fermion number and squared mass of the ith state. The function F (x) is given by y = γ/2 x with γ = λ/κ, and x is defined in terms of the canonically normalized real inflaton field σ as x = σ/m with m 2 = M 2 (1/4ξ − 1). The function F (y) exhibits the contribution of the µ term in the superpotential W , and for γ 1, is expected to play an important role in the predictions of inflationary observables. The renormalization scale Q is set equal to σ 0 , the field value at the pivot scale k 0 = 0.05 Mpc −1 [7].
The soft SUSY breaking terms are added in the inflationary potential as: where A is the complex coefficient of the trilinear soft-SUSY-breaking terms.
The F -term SUGRA scalar potential is evaluated using where The Kähler potential K is expanded in inverse powers of m P : where the minimal canonical Kähler potential K c is given by The inflationary potential along the D-flat direction with |H c | = |H c |, stabilized along the h = 0 direction, and incorporating the SUGRA corrections [24], the radiative corrections [1], and the soft-SUSY-breaking terms [3,4], is given by Here A, B, and C are the coefficients of the constant, quadratic, and quartic SUGRA terms, respectively, and are defined in terms of H P = (M/m P )/ √ 2ξ as where γ S = 1 + 2κ 2 S − 3κ SS − 7κ S /2 [26]. The independently varying parameters c 0 , c 1 , and c 2 for the nonminimal case are similar to the ones given in Ref. [26]. Our choice for these parameters will be shown in the relevant sections. The parameter a depends on arg S as follows: Assuming negligible variation in arg S, with a = −1, the scalar spectral index n s is expected to lie within the experimental range [4,26]. This could also be achieved by taking an intermediate-scale, negative soft mass-squared term for the inflaton [27]. But with the nonminimal terms in the Kähler potential, one can also obtain the central value of n s with TeV-scale soft masses even for a = 1 [8,9]. The variation in arg S with general initial condition has been studied in Refs. [3,6,9].
The slow-roll parameters are defined by where the primes denote derivatives with respect to x. The scalar spectral index n s , the tensor-to-scalar ratio r, the running of the scalar spectral index dn s /d ln k, and the scalar power spectrum amplitude A s , to leading order in the slow-roll approximation, are as follows: where A s (k 0 ) = 2.196×10 −9 and x 0 denotes the value of x at the pivot scale k 0 = 0.05 Mpc −1 [7]. For the numerical estimation of the inflationary predictions, these relations are used up to second order in the slow-roll parameters.
Assuming a standard thermal history, the number of e-folds N 0 between the horizon exit of the pivot scale and the end of inflation is The reheat temperature T r is approximated by where g * = 228.75 for MSSM and Γ S is the inflaton decay width. From the µ-term coupling λSh 2 in Eq. (5), we see that the inflaton can decay into a pair of Higgsinos h u , h d with a decay width where is the inflaton mass [12]. The reheat temperature, the inflaton decay width, and the inflaton mass defined above in Eqs. (28)- (30) are used together with Eq. (27) in order to derive the numerical predictions for the present inflationary scenario.
First we consider the possibility that the LSP is the gravitino. In Fig. 2, the upper thick solid-magenta, dashed-blue, dot-dashed-green curves show the variation of the gravitino mass constrained by inflation, for ξ = 0.125, 0.148, 0.245 respectively. The lower bound on the reheat temperature T r 2 × 10 10 GeV is obtained for a gravitino mass m 3/2 1 TeV. If the gravitino is the LSP and hence constitutes DM, the DM relic abundance Ω DM h 2 = 0.12 [7] can be used to obtain the variation of the gluino mass mg with T r : This variation is depicted by the lower-thin-faded curves in Fig. 2 (again for ξ = 0.125, 0.148, 0.245), which are cutoff at 2 TeV, thus complying with the LHC bound on the gluino mass. The shaded region is excluded by the LHC lower bound on gluino mass (mg 2 TeV). Since the thick curves representing the gravitino mass are above the gluino mass depicted by thin-faded curves in the region above the LHC cutoff, the gravitino cannot be the LSP.
The bounds on the reheat temperature from the inflationary constraints for gravitino masses 1 and 10 TeV are T r 2.2×10 10 GeV and 7.5×10 10 GeV respectively (see Fig. 2). These are clearly inconsistent with the above mentioned BBN bounds, and so the unstable long-lived gravitino scenario is not viable.
Lastly, for the unstable short-lived gravitino case, we compute the LSP lightest neutralino (χ 0 1 ) density produced by the gravitino decay and constrain it to be smaller than the observed DM relic density. For reheat temperature T r 10 11 GeV with m 3/2 > 25 TeV (see Fig. 2), the resulting bound on the neutralino mass mχ0 1 comes out to be inconsistent with the lower limit set on this mass mχ0 1 18 GeV in Ref. [31]. To circumvent this, the LSP neutralino is assumed to be in thermal equilibrium during gravitino decay, whereby the neutralino abundance is independent of the gravitino yield. For an unstable gravitino, the lifetime is (see Fig. 1 Now for a typical value of the neutralino freeze-out temperature, T F 0.05 mχ0 1 , the gravitino lifetime is estimated to be τ 3/2 10 −11 1 TeV GeV. Thus, minimal shifted µHI conforms with the conclusion of the standard case [19,22] by requiring split-SUSY with an intermediate-scale gravitino mass and reheat temperature T r 10 13 GeV (see Fig. 2). To check whether the shifted µHI scenario is also compatible with low reheat temperature (i.e. T r 10 12 − 10 8 GeV [33]) and TeV-scale soft SUSY breaking, we employ nonminimal Kähler potential in the next section.

V. µ-HYBRID INFLATION WITH NONMINIMAL KÄHLER POTENTIAL
The nonminimal Kähler potential used in the following analysis is which includes only the nonminimal couplings of interest κ S and κ SS . (For a somewhat different approach to µ-hybrid inflation with nonminimal K, see Ref. [34]). Thus, for the nonminimal scenario we take c 0 = c 1 = 1 and c 2 = 1 − κ S in Eq. (23) [26]. Using these values the potential of the system can easily be read off from Eq. (22).
It is worth noting that with the nonminimal Kähler potential we can realize the central value of n s with TeV-scale soft masses even for a = 1 [8,9]. Our study is conducted in two parts, described separately in the following subsections, first with κ SS = 0 and then by allowing κ SS to be nonzero. The appearance of a negative mass term with a single nonminimal coupling κ S in the potential in Eq. (22) is expected to lead to red-tilted inflation with low reheat temperature, as for standard µHI (see Ref. [22]). Furthermore for nonzero κ SS , the possible larger r solutions leading to observable gravity waves are also anticipated.
These expectations along with the impact of an additional parameter ξ on inflationary predictions are discussed below. the shifted µHI as compared to the standard µHI (see Fig. 2 of Ref. [22]), as can be seen from Fig. 3. Also, it is not surprising that around κ ∼ 10 −3 the system is oblivious to the gravitino mass, since the contribution of the linear term becomes less important compared with the SUGRA or radiative corrections [8]. The interesting new feature is due to the presence of another parameter ξ , whose effect is to increase the range of symmetry breaking scale M . For a particular value of κ, say κ ∼ 10 −6 , and m 3/2 = 1 TeV, a wider range to the tensor-to-scalar ratio r with r 10 −9 , which is experimentally inaccessible in the foreseeable future [35][36][37][38].
As Fig. 5 shows, the running of the scalar spectral index dn s /d ln k also turns out to be small in the present scenario, namely 10 −10 −dn s /d ln k 10 −4 , which is a common feature of small field models. The nonminimal Kähler coupling κ S remains constant in the low reheat temperature range as can be seen from the lower plot of Fig. 5, since the radiative and the quartic-SUGRA corrections can be neglected in this regime. The scalar spectral index n s in the low reheat temperature region is n s 1 − 2κ S [15], and so for the central value of the scalar spectral index n s = 0.9655, one obtains κ S = 0.0173, as exemplified by Proceeding next to the role of the gravitino in cosmology, one can read off the lower bounds on the reheat temperature T r from Fig. 3. Since, at low reheat temperatures, inflation occurs near the waterfall region (with x 0 close to 1), we devised a criterion by allowing only 0.01% fine-tuning on the difference x 0 − 1. This yields T r 2 × 10 6 , 7 × 10 5 , or 2 × 10 5 GeV for m 3/2 = 1, 10, or 100 TeV.
For the first scenario with the gravitino being the LSP in shifted µHI with nonminimal Kähler potential, the upper bounds on the reheat temperature obtained in Ref. [22] (see Fig. 3 and For the third scenario of a short-lived gravitino (for instance with mass m 3/2 = 100 TeV), the gravitino decays before BBN, and so the BBN bounds on the reheat temperature no longer apply. The gravitino decays into the LSP neutralinoχ 0 1 . We find that the resulting neutralino abundance is given by where the gravitino yield Y 3/2 2.3 × 10 −12 T r 10 10 GeV (40) is acceptable over the range T r ∼ 10 5 GeV − 10 12 GeV [32]. The LSP (lightest neutralino) density produced by the gravitino decay should not exceed the observed DM relic density Ω DM h 2 0.12 [7]. The resulting bound on the lightest neutralino mass The canonical measure of primordial gravity waves is the tensor-to-scalar ratio r and the next-generation experiments are gearing up to measure it. One of the highlights of PRISM [35] is to detect inflationary gravity waves with r as low as 5 × 10 −4 and a major goal of LiteBIRD [36] is to attain a measurement of r within an uncertainty of δr = 0.001. Future missions include PIXIE [37], which aims to measure r < 10 −3 at 5 standard deviations, and CORE [38], which forecasts to lower the detection limit for the tensor-to-scalar ratio down to the 10 −3 level. As seen in previous sections, with κ SS = 0, the tensor-to-scalar ratio remains in the undetectable range r 10 −6 . It is therefore instructive to explore our model further to look for large-r solutions, which, as it turns out, yield r's in the 10 −4 − 10 −3 range.
To achieve this, we employ nonzero κ SS in addition to a nonzero κ S , and the results are The curves corresponding to field values S 0 close to m P are terminated since, at some point, either the nonminimal coupling κ SS takes unnatural values ≈ 10 (see Fig. 9) or M reaches m P . Indeed, for ξ = 0.125, the coupling κ SS can exceed the bound of 10 on curves at T r = 10 12 GeV and going down to values as low as 10 4 − 10 5 GeV.
The upper bound on r as can be read off from Fig. 6 is r 0.001 for the choice S 0 = m P and r 10 −5 for S 0 ∼ 0.1 m P . The Fig. 6 also shows that r 10 −6 − 10 −3 from the requirement that T r 10 11 GeV for circumventing the gravitino problem. The running of the scalar spectral index dn s /d ln k remains small namely 10 −7 −dn s /d ln k 4 × 10 −3 , as shown in Fig. 7. The variation of the symmetry breaking scale M with κ is shown in Fig. 8, where we find values of κ up to 5 × 10 −4 for large values of M (∼ 10 17 − 10 18 GeV). The respective variation in the couplings κ S and κ SS is shown in Fig. 9. They remain acceptably small and well within the bound κ S , κ SS 1, for natural values of S 0 = 0.5 m P or less.
Although the plots in Figs. 6-9 are for gravitino mass m 3/2 = 1 TeV, the curves, for these larger r solutions, are independent of the gravitino mass and are valid for a gravitino mass range m 3/2 = 1 − 100 TeV.

VI. CONCLUSION
We have implemented a version of SUSY hybrid inflation in SU (4) c × SU ( . This points towards split SUSY. In the nonminimal Kähler case, we have realized successful inflation with reheat temperatures as low as 10 5 GeV. This is favorable for the resolution of the gravitino problem and compatible with a stable LSP and low-scale (∼TeV) SUSY. Compared with standard µ hybrid inflation [22], the reheat temperature is lowered by half an order of magnitude and, due to the additional parameter ξ, an order of magnitude increase in the spread of M is seen.
We have discussed how primordial monopoles are inflated away and provided a framework that predicts the presence of primordial gravity waves with the tensor-to-scalar ratio r in the observable range (∼ 10 −4 − 10 −3 ). This is realized with the G 4-2-2 symmetry breaking scale approaching values that are comparable to the string scale (∼ 5 × 10 17 GeV) and a gravitino mass lying in the 1 − 100 TeV range. It is worth noting that the inflaton field values do not exceed the Planck scale, which may be an additional desirable feature in view of the swampland conjectures [39,40]. For a recent discussion and additional references see Ref. [41].