Gravity mediation without a Polonyi problem

Recent indications of the 125GeV Higgs at the LHC can be explained in a relatively high-scale SUSY scenario where the sparticle masses are multi-TeV as is realized in the focus-point region. However, it suffers from the notorious cosmological Polonyi problem. We argue that the Polonyi problem is solved and thermal or non-thermal leptogenesis scenario works successfully, if a certain Polonyi coupling to the inflaton is enhanced by a factor of 10-100.


I. INTRODUCTION
Recently the ATLAS [1] and CMS [2] collaborations reported event excesses, which may imply the Higgs boson with mass of about 125 GeV. While it is difficult to explain the Higgs mass in the minimal supersymmetric standard model (MSSM) as long as the sparticle masses are around 1 TeV [3], such a Higgs mass can be explained if the sparticles are heavier than multi-TeV [4][5][6][7]. One of such scenarios is the anomaly-mediated SUSY breaking (AMSB) model [8], where the sfermions and the gravitino are O(100-1000) TeV and the gaugino masses are O(100-1000) GeV, given by the AMSB relation. Phenomenological aspects of this scenario have been discussed in Refs. [6,9,10], and it was shown that it is compatible with thermal leptogenesis [11,12], which requires the reheating temperature as high as T R 10 9 GeV [13]. 1 Another attractive scenario is that all sparticles are O(10) TeV in the gravity-mediated SUSY breaking. The scenario alleviates the SUSY flavor/CP problems because of the heavy SUSY particles, while it explains the 125 GeV Higgs boson for tan β 5 [5] and the present dark matter abundance (see Ref. [7] for realization in the focus-point region [14]). However, the scenario suffers from the cosmological Polonyi problem [15,16], since there must be a singlet SUSY breaking field, called the Polonyi field, in order to generate sizable gaugino masses. Although the Polonyi may decay before the big-bang nucleosynthesis (BBN) begins for the Polonyi mass m z O(10) TeV, it releases a huge amount of entropy because it dominates the Universe before the decay. Thus the leptogenesis scenario does not work in this setup.
An interesting solution to the Polonyi problem was proposed long ago by Linde [21].
It was pointed out that, if the Polonyi field has a large Hubble-induced mass, it follows a time-dependent potential minimum adiabatically and the resultant amplitude of coherent oscillations is exponentially suppressed. Recently, two of the present authors (FT and TTY) noticed that there is an upper bound on the reheating temperature for the adiabatic solution to work [17] and also showed that such a large Hubble mass may be a consequence of the strong dynamics at the Planck scale [18] or the fundamental cut-off scale one order of magnitude lower than the Planck scale [17]. More important, the present authors found 1 In this letter T R is defined as T R ≡ (10/π 2 g * ) 1/4 Γ φ M P where Γ φ is the inflaton decay rate. Note that this definition of T R is smaller than that of Ref. [13] by a factor √ 3.
that there are generally additional contributions to the Polonyi abundance which depends on the inflation energy scale, and we showed that the Polonyi problem is still solved or greatly relaxed in high-scale inflation models [19,20]. In this solution, we do not need any additional mechanism to dilute the Polonyi abundance. Therefore, it may revive the conventional Polonyi model as a realistic SUSY breaking model, which is fully compatible with the current experiments and observations, including the 125 GeV Higgs boson.
In this letter we study the adiabatic solution in detail, considering various production processes of the Polonyi field as well as the thermal and non-thermal gravitino production.
In particular, we focus on a minimal model in which only a certain coupling of the inflaton to the Polonyi field is enhanced. We also consider explicit inflation models to see if there is an allowed parameter space where the Polonyi and gravitino problems are solved.

II. THE POLONYI MODEL FOR GRAVITY-MEDIATION
First we briefly review the cosmological Polonyi problem in the gravity mediation. Let us denote the Polonyi field by z, which makes a dominant contribution to the SUSY breaking.
Its F -term is given by F z = √ 3m 3/2 M P where m 3/2 is the gravitino mass and M P is the reduced Planck scale. It generally couples to the MSSM superfields as where Q and W a collectively denote the matter and gauge superfields, respectively, and c Q and c g are constants of order unity. Here and hereafter, c Q and c g are taken to be real and positive, for simplicity. These couplings give masses of order m 3/2 to the SUSY particles, as Note that z must be a singlet field in order to give a sizable mass to the gauginos. The following term in the Kähler potential yields the sizable µ and B terms [22], as µ = √ 3c h m 3/2 and B = m 3/2 . Thus the framework naturally solves the µ/Bµ problem. If one takes the gravitino mass to be as large as 10 TeV, the SUSY flavor and CP problems are greatly relaxed and the cosmological gravitino problem is also ameliorated. It also explains the 125 GeV Higgs boson without tuning the A-parameter [4][5][6][7]. Therefore the O(10) TeV SUSY is plausible from these phenomenological point of view. However, the model suffers from the cosmological Polonyi problem, which inevitably arises in the gravity-mediation scenario. The Polonyi abundance is estimated as where z i is the initial amplitude, which is in general of the order of M P . The reheating temperature T R is defined by where Γ tot is the inflaton decay rate, and g * counts the relativistic degrees of freedom at the reheating. Here we have assumed that the potential for z can be well approximated by a quadratic term for |z| z i , and that the z starts to oscillate before the reheating. The Polonyi abundance (4) is so large that the z dominates the energy density of the Universe soon after the reheating, and causes cosmological problems.
The Polonyi decays into gauge bosons through the interaction (1) with the decay rate given by where m z is the Polonyi mass at the zero temperature. The decay into gauginos is suppressed by (mg/m z ) 2 or (m 3/2 /m z ) 2 , where mg denotes the gaugino mass, and as we will see later, the Polonyi mass is considered to be slightly enhanced compared to mg or m 3/2 . We parametrize it as m z = c z m 3/2 with c z 1.

The interaction (3) induces the decay of the Polonyi into the Higgs boson pair [23]
, while the decay into a higgsino pair is suppressed by a factor of (m 3/2 /m z ) 2 . The Polonyi also decays into a pair of gravitinos if kinematically allowed [24]. The decay rate is given by For example, if the decay into gauge bosons is the dominant decay mode, the lifetime of the Polonyi is given by If the decay into the gravitino pair is dominant, the lifetime is given by The lifetime must be (much) shorter than 1 sec in order not to spoil the success of BBN [25].
Even if it decays before BBN, it dilutes the pre-existing baryon asymmetry of the Universe.
The dilution factor is roughly given by The dilution factor is so large that thermal and non-thermal leptogenesis scenarios do not work. Therefore some involved mechanisms to create the baryon asymmetry is required if the Polonyi problem is solved by increasing the Polonyi mass. In the next section we will consider another attractive solution to the Polonyi problem in which there is no late-time entropy production.

III. SOLUTION TO THE POLONYI PROBLEM AND IMPLICATIONS
Now we revisit the Polonyi model in light of the recent developments in the suppression mechanism for the moduli abundance [19].
Let us introduce the inflaton fields X and φ, which have R-charges of +2 and 0, respectively. The inflaton superpotential has the form where f (φ) is some function of φ. The F -term of X dominates the potential energy during inflation. Many known inflation models in supergravity fall into this category. The Polonyi field in general couples to the inflaton fields as where c X and c φ are taken to be real and positive. The adiabatic suppression mechanism works if c X ≫ 1 [18]. However, the inflaton dynamics just after the inflation induces a non-negligible amount of the coherent oscillations of the Polonyi field, which is estimated where ∆z = |z X − z φ | and H inf is the Hubble scale at the end of inflation. This expression is valid for c φ 1. For c φ ≪ 1, there remains a contribution like (13) with c φ replaced by O(1). This is much smaller than the naive estimate (4) if H inf ≫ m z , which is satisfied for the most known inflation models. From this expression, we can see that the Polonyi abundance is suppressed for c X /c φ ≫ 1 and large inflation scale. Hereafter we take c φ = 1 for simplicity. Now let us see how the present model is constrained from cosmological arguments. First, gravitinos are effectively produced at the reheating, and its abundance is proportional to the reheating temperature. If the gravitino is heavier than the lightest SUSY particle (LSP), it is unstable and decays emitting energetic particles. Such late gravitino decay changes the Helium-4 abundance [25], and produces LSPs non-thermally. The Polonyi causes similar effects: the Polonyi decay may alter the standard BBN results and yield too many LSPs.
If the Polonyi decays into the gravitino, the subsequent gravitino decay also causes similar effects. Notice that the Polonyi abundance is given by the sum of the coherent oscillation (13) and thermal production, the latter of which is comparable to the abundance of the (transverse components of) gravitino if c g ∼ 1.  Fig. 1, because it strongly depends on the mass spectrum. For instance, if it has a sizable mixing with higgsino or wino, the thermal relic abundance can be smaller than the present DM abundance. (In the latter case, we need to relax the GUT relation on the gaugino mass.) We note that, in the lower panel, the constraint comes from the LSP overproduction from the gravitino/Polonyi decay, hence all the constraints disappear if the R-parity is broken by a small amount.
It is seen that the reheating temperature of T R ≃ 10 9 GeV is allowed for H inf 10 9 -10 12 GeV. It is important that we do not need any additional late-time entropy production for solving the Polonyi problem. Thus the conventional Polonyi model for the gravity-mediation for relatively heavy SUSY scale of O(10) TeV can be compatible with leptogenesis scenario once we assume the Polonyi coupling to the inflaton is enhanced.

IV. INFLATION MODEL
Now let us see if the above solution works in some known inflation models in supergravity.
In particular, we will show that there are consistent parameter regions where thermal [11] or non-thermal [27][28][29] leptogenesis scenario works, avoiding the Polonyi and gravitino problems.

A. Hybrid inflation
First, let us consider the SUSY hybrid inflation model [30][31][32]. The superpotential is given by where W 0 = m 3/2 M 2 P . The waterfall fields, φ andφ, can be identified with the Higgs fields which break U(1) B−L gauge symmetry. This model, including the constant term W 0 , was analyzed in detail in Refs. [33][34][35]. We assume that the inflaton dominantly decays into the right-handed neutrinos N i (i = 1, 2, 3) through the interaction The inflaton decay rate into the right-handed (s)neutrino pair is given by where we have taken into account a mixing between X and φ (andφ) due to the constant term [36]. Here we consider only the decay into the lightest right-handed neutrino. On the other hand, the inflaton decays into a pair of gravitinos through the interaction in the Kähler potential [36][37][38], The decay rate into the gravitino pair is given by [38] Γ grav ≡ Γ(φ → ψ 3/2 ψ 3/2 ) = 1 32π where the mixing between X and φ (andφ) is taken into account [36]. Notice that the same interaction induces the inflaton decay into the Polonyi pair (φ → zz) with the same decay rate. Since each Polonyi field mainly decays into a pair of the gravitino, the gravitino abundance produced non-thermally by the inflaton decay is given by where the total decay rate is approximately given by Γ tot ≈ Γ(φ → NN). This imposes severe constraints on the parameter space. We have scanned parameters (κ, M), which are rewritten in terms of H inf and T R through the relation H inf = κM 2 / √ 3M P and T R = (10/π 2 g * ) 1/4 √ Γ tot M P . We have also fixed m N = 0.02m φ : the non-thermal leptogenesis works for T R 10 8 GeV in this case. Fig. 2

B. Smooth hybrid inflation
Let us consider the smooth-hybrid inflation model [39] where the inflaton superpotential is given by where m ≥ 2 is an integer. The model has a discrete symmetry Z m under which φφ has a charge +1 and X has a zero charge. This model has an advantage that it does not suffer from problematic topological defects formation since the φ andφ have nonzero VEVs during inflation and topological defects are inflated away. Hereafter we consider the case of m = 2 for simplicity. Results do not much affected by this choice. The gravitino abundance is similarly estimated by Eq. (19).
The inflaton can decay into ordinary particles through non-renormalizable interactions, for example, with cutoff parameter M c . The decay rate into the Higgs boson and higgsino pair is given by where the mixing between X and φ (andφ) is taken into account. If the right-handed neutrino mass is not much smaller than the inflaton mass, the decay rate into them through the operator K = |φ| 2 |N| 2 /M 2 c is comparable to the above expression.