No-Scale $\mu$-Term Hybrid Inflation

To solve the fine-tuning problem in $\mu$-Term Hybrid Inflation, we will realize the supersymmetry scenario with the TeV-scale supersymmetric particles and intermediate-scale gravitino from anomaly mediation, which can be consistent with the WMAP and Planck experiments. Moreover, we for the first time propose the $\mu$-term hybrid inflation in no-scale supergravity. With four Scenarios for the $SU(3)_C\times SU(2)_L\times SU(2)_R\times U(1)_{B-L}$ model, we show that the correct scalar spectral index $n_s$ can be obtained, while the tensor-to-scalar ratio $r$ is prediced to be tiny, about $10^{-10}-10^{-8}$. Also, the $SU(2)_R\times U(1)_{B-L}$ symmetry breaking scale is around $10^{14}$ GeV, and all the supersymmetric particles except gravitino are around TeV scale while gravitino mass is around $10^{9-10}$ GeV. Considering the complete potential terms linear in $S$, we for the first time show that the tadpole term, which is the key for such kind of inflationary models to be consistent with the observed scalar spectral index, vanishes after inflation. Thus, to obtain the $\mu$ term, we need to generate the supersymmetry breaking soft term $A^{S \Phi \Phi'}_{\kappa} \kappa S \Phi \Phi'$ due to $A^{S \Phi \Phi'}_{\kappa}=0 $ in no-scale supergravity, where $\Phi$ and $\Phi'$ are vector-like Higgs fields at high energy. We show that the proper $A^{S \Phi \Phi'}_{\kappa} \kappa S \Phi \Phi'$ term can be obtained in the M-theory inspired no-scale supergravity. We also point out that $A^{S \Phi \Phi'}_{\kappa}$ around 700 GeV can be generated via the renormalization group equation running from string scale.

In the original supersymmetric hybrid inflation models, the quantum corrections arising from supersymmetry breaking drive inflation, and the scalar spectral index was predicted to be n s = 1 − 1/N 0.98 [6], where N = 60 denotes the number of e-foldings necessary to resolve the horizon and flatness problems in Big Bang cosmology. Interestingly, with a class of linear supersymmetry breaking soft terms in the inflationary potential [9,10], such kind of models can be highly consistent with the observed scalar spectral index values of n s = 0.96 − 0.97 from the WMAP [11] and Planck satellite experiments [12] as well. In particular, the corresponding supersymmetry breaking A-term for the linear superpotential term can be around TeV scale [9,10].
As we know, in the Minimal SSM (MSSM), there exists a well-known µ problem. However, the µH d H u term is forbidden by U (1) R symmetry, where H u and H u are one pair of Higgs fields in the SSMs. With the linear supersymmetry breaking soft term after inflation, the inflaton field S acquires a Vacuum Expectation Value (VEV). Thus, the µ problem can be solved if there exists a superpotential term λSH d H u , as proposed by Dvali, Lazarides and Shafi (DLS) [13]. Assuming the minimal K, the magnitude of µ is typically around the gravitino mass m G [13]. Recently, such scenario has been studied in details [14]. With the reheating and cosmological gravitino constraints, it was found that a consistent inflationary scenario gives rather concrete predictions regarding supersymmetric dark matter and Large Hadron Collider (LHC) phenomenology. Especially, the gravitino must be sufficiently heavy (m G 5 × 10 7 GeV) so that it decays before the freeze out of the lightest supersymmetric particle (LSP) neutralino, which is the dark matter candidate. Moreover, the wino with mass 2 TeV becomes a compelling dark matter candidate. And the supersymmetry breaking scalar mass M 0 is expected to be of the same order as m G or larger, which can reproduce a SM-like Higgs boson mass 125 GeV for suitable tan β values, where tan β is the ratio of the VEVs for H u and H d . Depending on the underlying gauge symmetry G associated with inflationary scenario, the observed baryon asymmetry in the Universe can be explained via leptogenesis [15,16].
The compelling examples of G, in which the DLS mechanism can be successfully merged with In short, in the recent study [14], to solve the gravitino problem in the µ-term hybrid inflation, Okada [11] and Planck satellite experiments [12]. Therefore, to solve this problem, we do need the supersymmetry scenario, which can have the TeV-scale supersymmetric particles (sparticles) in the SSMs while intermediate-scale heavy gravitino. The well-known example is no-scale supergravity [21] or its generalization. In this paper, we shall realize such supersymmetry scenario via anomaly mediation [18]. In addition, we for the first time propose the µ-term hybrid inflation in no-scale supergravity 5 . We discuss it in details, and find some interesting results different from the previous study on the µ-term hybrid inflation.
First, with anomaly mediation, we will derive the supersymmetry scenario, where the sparticles are light while gravitino is heavy [18]. We consider the Kähler potential and superpotential as follows where M P l is the reduced Planck scale, z and X are respectively a hidden sector superfield and a compensator multiplet (X = 1 + F X ), Y denotes all the other superfields, is a small parameter, W 0 is a constant superpotential, and Φ and Φ are the Higgs fields which breaks the high-scale gauge symmetry in the F-term hybrid inflation [6,7]. Similar to the no-scale supergravity, the scalar potential vanishes in the limit → 0. Considering the equations of motion for the auxiliary fields, we obtain for small . Here, we define fz z ≡ ∂ 2 f (z,z)/∂z∂z, and m G is gravitino mass. So the scalar potential becomes For example, assuming fz z = (|z| 2 − 1/4) 2 − 1, we get the minimum for the scalar potential at which is an AdS vacuum. Thus, we have F X m G << m G . Because the supersymmetry breaking soft terms in the SSMs are proportional to F X via anomaly mediation, we obtain the supersymmetry breaking scenario which has TeV-scale sparticles and intermediate-scale gravitino. In particular, the supersymmetry breaking linear term for S is given by From the numerical studies in Refs. [9,10], we can still obtain the observed scalar spectral index values of n s = 0.96 − 0.97 from the WMAP [11] and Planck satellite experiments [12] as well. By the way, the AdS vacuum given by Eq. (5) can be lifted to the Minkowski vacuum by considering the F -term and D-term contributions in the anomalous U (1) theory inspired from string models [18].
In the following, we shall embed the previous µ-term hybrid inflation scenario into no-scale supergravity framework, i.e., we propose the µ-term hybrid inflation in no-scale supergravity where µ term is generated via the VEV of inflaton field after inflation. We introduce a conjugate pair of vector-like Higgs fields Φ and Φ , which breaks G down to the SM or SM-like gauge symmetry. Considering four Scenarios for the SU (3) C × SU (2) L × SU (2) R × U (1) B−L model, we show that the correct scalar spectral index n s can be obtained, while the tensor-to-scalar ratio r is prediced to be tiny, about 10 −10 − 10 −8 . Also, the SU (2) R × U (1) B−L symmetry breaking scale is around 10 14 GeV, and all the supersymmetric particles except gravitino are around TeV scale while gravitino mass is around 10 9−10 GeV. We present the complete potential terms that are linear in S, and for the first time we show that the tadpole term, which is the key for such kind of inflationary models to be consistent with the observed scalar spectral index, vanishes after inflation or say gauge symmetry G breaking. Thus, to reproduce the µ term, we need to generate the supersymmetry breaking soft term A SΦΦ κ κSΦΦ since we have A SΦΦ κ = 0 in no-scale supergravity. We show that the supersymmetry breaking soft term A SΦΦ κ κSΦΦ can be generated properly in the M-theory inspired no-scale supergravity which has no-scale supergravity at the leading or lowest order [23,24,25,26,27]. We also point out that the A SΦΦ κ κSΦΦ term with A SΦΦ κ around 700 GeV can be obtained via the renormalization group equation (RGE) running from string scale [28,29,30,31]. Therefore, we solve the finetuning problem in the previous µ-term hybrid inflation, and propose the no-scale µ-term hybrid inflation models where the sparticles in the SSMs are around TeV scale while gravitino is around 10 9−10 GeV.
Let us present our model in the following. The Kähler potential is where T is a modulus, and C i are matter/Higgs fields in the supersymmetric SMs which include Φ, Φ , H u , and H d . To simiplify the discussions, we will assume T = 1/2 in the following study.
Assuming S and superpotential have charge 2 while Φ, Φ , H u and H d are neutral under the U (1) R R-symmetry, we obtain the U (1) R invariant inflaton superpotential [13] To realize the correct symmetry breaking pattern after inflation, we require λ > κ [13]. In particular, the µH d H u term is forbidden by the U (1) R R-symmetry, and then such term can be generated only after U (1) R R-symmetry is broken down to a Z 2 symmetry, for example, by the VEV of S.
Assuming that the F-term of T breaks supersymmetry, we obtain the following scalar po- As a side remark, for Polonyi model, we will have an extra (−2) factor in the above tadpole term due to the −3|W | 2 contribution. During inflation, we have Φ = Φ = 0, as well as a tadpole term for S After inflation (or say after gauge symmetry G breaking) and neglecting the VEVs of H u and H d , we have Φ = Φ = M , and then the above tadpole term vanishes. To obtain the µ term which is forbidden by U (1) R symmetry, we need to generate the tadpole term of S, which will be discussed below.
With the supersymmetry breaking soft mass term as well as the radiative and supergravity corrections, we obtain the inflationary potential as follows where m = √ κM , φ is the real part of S, m φ is the supersymmetry breaking soft mass, M Pl is the reduced Planck scale, the renormalization scale (Q) is chosen to be equal to the initial inflaton VEV φ 0 , and the coefficient α 1 is given by In particular, the negative sign of the linear term is essential to generate the correct value for the spectral index. Without this linear term, the scalar spectral index n s is predicted to lie close to 0.98, as shown in Ref. [6]. Moreover, both φ 2 and φ 4 terms arise from the leading supergravity contribution, the quadratic supersymmetry breaking soft term can be ignored relative to the liner term in Eq. (11) [9,10], and the imaginary part of S is assumed to stay constant during inflation [For a more complete discussion of this last point, see Ref. [10].]. Thus, the inflaton potential can be simplified to In the following discussions, to be concrete, we consider the left-right model with gauge sym- Because Φ and Φ respectively have quantum numbers (1, 1, 2, 1/2) and (1, 1, 2, −1/2), we get N Φ = 2. For simplicity, we set γ ≡ λ/κ = 2, and then haveγ ≡ γ 2 + N Φ /2 = √ 5. Moreover, it seems to us that the φ 4 term can be neglected as well. Therefore, we will study the following four scenarios where the power spectrum ∆ 2 R = 2.20 × 10 −9 from the Planck 2015 results [12] has been explained simultaneously: Scenario I. The potential in Eq. (11) with m φ m G .
To obtain 0.955 ≤ n s ≤ 0.977 within about 1σ range of the Planck 2015 results [12] and the e-folding number 27 ≤ N ≤ 72, we present the numerical values of M and m G for the viable points in Fig. (1), which are normalized by the reduced Planck scale M P l =  In short, from the above numerical studies, we find that the observed scalar spectral index n s can be realized, but the tensor-to-scalar ratio r is prediced to be tiny, about 10 −10 − 10 −8 . Also, the SU (2) R × U (1) B−L symmetry breaking scale is around 10 14 GeV, and the gravitino mass is around 10 9−10 GeV. Thus, we do need the no-scale supergravity to realize the light sparticle spectrum.
Because gravitino is heavy and then unstable, we encounter the cosmological gravitino problem [32], which originates from the gravitino lifetime To avoid the constraint on the neutralino abundance from gravitino decay, we assume that the LSP neutralino is still in thermal equilibrium when gravitino decays. So the LSP neutralino abundance is not related to the gravitino yield. Using a typical value of the ratio x F ≡ mχ0/T F 20, where T F is the freeze out temperature of the LSP neutralino, this occurs for the gravitino lifetime m G 4.6 × 10 7 GeV mχ0 2 TeV Therefore, such cosmological scenario favors a gravitino mass at an intermediate scale above 10 7 GeV, and the gravitino mass in our model satisfies this bound clearly. Furthermore, after SU (2) R × U (1) B−L gauge symmetry breaking, the leading tadpole term for S in Eq. (9) vanishes. Thus, to obtain the µ term which is forbidden by U (1) R symmetry, we need to generate the supersymmetry breaking soft term A SΦΦ κ κSΦΦ . With it, we get the VEV of S as below And then the µ term is given by For λ = 2κ, we have However, in no-scale supergravity, we have A SΦΦ κ = 0. To solve this problem, first, we consider M-theory on S 1 /Z 2 [23]. For the standard Calabi-Yau compactification at the leading order or lowest order, we can realize no-scale supergravity [24], and there exists the next to leading order corrections [25,26,27]. In particular, we can have the non-zero supersymmetry breaking soft term A SΦΦ κ κSΦΦ . To compare with no-scale supergravity, we consider moduli dominant supersymmetry breaking, whose the supersymmetry breaking soft terms for universal gaugino mass, scalar mass and trilinear soft term are [27] where 0 < x < 1. For x ∼ 10 −6 − 10 −7 , we can indeed have the TeV-scale supersymmetry breaking soft terms in the SSMs while gravitino mass is around 10 9−10 GeV. Of course, there exists some fine-tuning for x.
Another way to generate the A SΦΦ κ κSΦΦ term is from the RGE running in no-scale supergravity [28,29,30,31]. Because of A = 0 from the no-scale boundary condition, we can neglect the Yukawa contributions and the RGE for A SΦΦ κ is before the SU (2) R × U (1) B−L gauge symmetry breaking, and after the SU ( Because we do not present a complete model here, let us consider the simple case. For no-scale supergravity, we should run the RGEs from the string scale, otherwise, light stau will be the LSP [28,29,30,31]. in such kind of the left-right models, we generically need to introduce more particles, and the complete RGE study is much more complicated. Note that if we have more particles above the SU (2) R × U (1) B−L symmetry breaking scale, their gauge couplings will become larger at higher scale and then the magnitude of A SΦΦ κ will be larger, which can give us larger µ term if we want. Therefore, we can indeed obtain the SSMs with TeV-scale supersymmetry and the intermediate-scale heavy gravitino.
In summary, to solve the problem in the µ-term hybrid inflation with canonical Kähler potential, we obtained the supersymmetry scenario which has the TeV-scale supersymmetric particles and intermediate-scale gravitino from anomaly mediation. Moreover, we for the first time proposed the µ-term hybrid inflation in no-scale supergravity where µ term is generated via the VEV of inflaton field after inflation. Considering four Scenarios for the SU (3) C × SU (2) L × SU (2) R × U (1) B−L model, we showed that the correct scalar spectral index n s can be obtained, while the tensor-to-scalar ratio r is prediced to be tiny, about 10 −10 − 10 −8 . Also, the SU (2) R × U (1) B−L symmetry breaking scale is around 10 14 GeV, and all the supersymmetric particles except gravitino are around TeV scale while gravitino mass is around 10 9−10 GeV.
With the complete potential terms linear in S, we for the first time showed that the tadpole term, which is the key for such kind of inflationary models to be consistent with the observed scalar spectral index, vanishes after inflation or say gauge symmetry G breaking. Thus, to obtain the µ term, we need to generate the supersymmetry breaking soft term A SΦΦ κ κSΦΦ , since we have A SΦΦ κ = 0 in no-scale supergravity. We showed that the supersymmetry breaking soft term A SΦΦ κ κSΦΦ can be realized properly in the M-theory inspired no-scale supergravity which has no-scale supergravity at the leading or lowest order. Also, we pointed out that the A SΦΦ κ κSΦΦ term with A SΦΦ κ around a few hundred GeVs can be reproduced via the RGE running from string scale. Therefore, we proposed the no-scale µ-term hybrid inflation models where the sparticles in the SSMs are around TeV scale while gravitino is around 10 9−10 GeV.