Heavy Gravitino from Dynamical Generation of Right-Handed Neutrino Mass Scale, and Gravitational Waves

The absence of supersymmetric particles at the weak scale is a puzzle if supersymmetry solves the gauge hierarchy problem. We show that, if the right-handed neutrino masses arise from hidden gaugino condensation via GUT-suppressed operators, successful thermal leptogenesis leads to the lower bound on the gravitino mass, which may explain the little hierarchy problem. If domain walls associated with the gaugino condensation are formed after inflation, they annihilate when $H \sim m_{3/2}$. We address the possibility that observable gravitational wave signals might emerge from the domain wall annihilation.

. In the course of violent annihilation processes, a large amount of gravitational waves can be emitted. The frequency and abundance of the gravitational waves we observe today are respectively determined by the gravitino mass and the dynamical scale. We will address a possibility to detect the gravitational waves by the forthcoming gravitational wave experiments such as Advanced LIGO [33], Einstein Telescope [34], and an experiment using Bose-Einstein condensation (BEC) [35].
Lastly let us mention related works in the past [36,37]. In Ref. [36], they proposed a scenario where both U(1) B−L and SUSY are dynamically broken at the same time, which relates the right-handed neutrino mass to the gravitino mass, and then derived a lower-bound on the gravitino mass, m 3/2 O(1000) TeV, assuming non-thermal leptogenesis. In Ref. [37], they studied resonant leptogenesis [38][39][40][41][42][43][44] in a context of the right-handed sneutrino inflation [45], and found that the gravitino mass must be larger than O(100) TeV.
The rest of this paper is organized as follows. In Sec. 2, we give a model where the mass of the right-handed neutrino is generated by a hidden gaugino condensate. Then we show that the lower-bound on the gravitino mass is fixed by the requirement of the successful thermal leptogenesis. In Sec. 3, we discuss cosmological implications including predictions for the gravitational wave signals. Finally, Sec. 4 is devoted to conclusions.

Right-handed neutrino mass and hidden gaugino condensate
The crucial ingredient of our model is that the heavy right-handed neutrinos acquire their mass from a spontaneous breaking of the U(1) R symmetry. In the seesaw mechanism [46][47][48], the right-handed neutrino mass term is usually allowed by symmetry, and so, it can be naturally of order 10 14−15 GeV. Instead here we assume that the right-handed neutrino mass is forbidden by the U(1) R symmetry, and it is generated by the spontaneous breaking of the U(1) R symmetry such as gaugino condensation.
To be explicit, let us consider a hidden SU(N) SUSY Yang-Mills sector. The gauge interactions become strong at a dynamical scale Λ, and the hidden gaugino λ A H forms a condensate, where k = 1, · · · N labels the N distinct vacua [49]. Below the dynamical scale Λ, the effective superpotential of the gaugino condensation is given by Now let us assume that the hidden gauge sector is coupled to the right-handed neutrinos via GUT-suppressed operators, whereN R is a chiral superfield containing a right-handed neutrino, M * denotes the effective cut-off scale for the interaction between the hidden sector and the right-handed neutrino. We assume that M * is of order the GUT scale, M * ∼ 10 15 -10 16 GeV.
Here and in what follows we suppress flavor indices for simplicity. Therefore, the right-handed neutrino obtains its Majorana mass from the coupling to the hidden gaugino condensate, up to a phase factor. One example of U(1) R charge assignment is given at the end of this section.
The scale of the gaugino condensation is bounded above, and its upper bound is related to the gravitino mass. To see this, we note that the superpotential contains a constant term, W 0 , to cancel the positive contribution of the SUSY breaking to the cosmological constant, 1 where w 0 is the constant term, m 3/2 the gravitino mass, and M Pl = 2.4 × 10 18 GeV the reduced Plank mass. Here the gravitino mass is defined as a real parameter, and we have introduced a complex phase factor, e iδ . Barring cancellation among various contributions to the constant term, we obtain Combining Eqs. (2), (4), (5), and (6), we arrive at Successful thermal leptogenesis requires M N R 4 × 10 8 GeV, assuming thermal initial abundance [50]. Thus, successful thermal leptogenesis implies that the gravitino mass m 3/2 should be heavier than several TeV for the GUT-scale cut-off.
A few comments are in order. First, the assumption about the initial abundance is likely satisfied in a scenario to be discussed in the next section, where a large number of right-handed neutrinos (as well as gravitational waves) is produced by the domain wall annihilation. Secondly, if the right-handed neutrino masses are slightly degenerate, the resultant baryon asymmetry can be enhanced by the resonant leptogenesis [38][39][40][41][42][43][44]. Therefore, the lower bound on the gravitino mass can be relaxed in this case. Thirdly, the gravitino is known to cause the gravitino problem [23][24][25]: the gravitino is thermally 1 We have assumed K M 2 P , where K is a Kahler potential.
produced in the early Universe, and it later decays into the visible particles during the big bang nucleosynthesis, modifying the light element abundances in contradiction with the observation. If the gravitino is the lightest SUSY particle (LSP), its abundance may exceed the dark matter abundance. The gravitino problem is relaxed significantly if the gravitino mass is heavier than O(10) TeV, as it decays before the big bang nucleosynthesis, or if there is another dark sector (e.g. dark photon and dark photino) which contains the LSP with a mass much lighter than the gravitino mass. One can also avoid the overproduction of the LSP produced from the gravitino decay by introducing a small R-parity violating operator [51,52]. Before closing this section, let us give a concrete R-charge assignment of the matter fields which forbids the right-handed neutrino mass. Here we adopt a Z 6R symmetry with the R-charge of R(N R ) = 3. See Table 1 for the R-charges of the other SSM fields. The right-handed neutrino can couple to the SSM sector in the Z 6R symmetric manner as 2 while the bare Majorana mass is forbidden by the symmetry. Then, the Majorana mass is induced by the gaugino condensation, which spontaneously breaks Z 6R down to Z 2R , where W GC has an effective R-charge 2. Note that the µ-term is also forbidden by the symmetry. 3 In fact, one can generate the µ-term similarly by introducing non-renormalizable coupling to the hidden gaugino condensation, For Λ ∼ 10 14 GeV and M * ∼ 10 16 GeV, one can obtain µ ∼ O(1) TeV.

Cosmological implications
Next we discuss the cosmological aspects of our model. In the hidden gauge sector, the continuous R symmetry is anomalous, and it is explicitly broken to a discrete Z 2N subgroup by non-perturbative effects [54]. Furthermore, at the dynamical scale Λ, the discrete symmetry is spontaneously broken down to Z 2 symmetry due to the gaugino condensation, leading to the domain wall formation [55][56][57][58][59].
Let us suppose that domain walls are formed after inflation. This is the case if 4 is satisfied, where H inf denotes the Hubble parameter during the inflation. T max denotes the highest temperature of the background plasma given by [60,61] T where T reh is the reheating temperature. The domain walls are known to follow the scaling law, and their energy density decreases more slowly than radiation or matter. Therefore, if the domain walls were completely stable, they would dominate the Universe at the end of the day and generate intolerably large spatial inhomogeneities and anisotropies [57].
To avoid the cosmological catastrophe, one needs to introduce an energy bias between the different vacua, which makes the domain walls unstable and disappear at a later time. The domain wall annihilation is so violent that a large amount of gravitational waves can be produced. As we shall see below, the domain walls annihilate when the Hubble parameter becomes comparable to the gravitino mass, which determines the peak frequency of the gravitational waves. The domain walls are formed when either the Hubble parameter or cosmic temperature becomes equal to Λ. The tension of the domain wall between the adjacent vacua, σ DW is given by [58] Hereafter we take σ DW = 4N Λ 3 as a reference value. After formation, the network of the domain walls is known to quickly follow the scaling law. The energy density of the domain wall ρ DW in the scaling regime is estimated as where A is an O(1) parameter [62,63], and t is the cosmic time.
Here and in what follows we assume the radiation dominated Universe, H = 1/2t, unless otherwise stated. The energy density of the domain walls decreases more slowly than radiation, and so, it comes to dominate the Universe when To avoid the cosmological catastrophe, domain walls must decay before t dom . The constant term in the superpotential provides the energy bias between different vacua. 5 The relevant part of the scalar potential in supergravity reads where we have defined θ k ≡ δ − 2πk/N . The energy bias between the k-th and (k + 1)-th vacua is then given by k,k+1 In the following analysis, we take a typical energy bias bias as 6N m 3/2 Λ 3 . The domain walls annihilate when their energy density becomes comparable to the bias energy density at t = t ann . This is given by where C d is a coefficient of O(1) [66] and t ann ∼ m −1 3/2 . After the domain walls annihilate, the gaugino condensate mainly decays into a pair of right-handed neutrinos through the interaction (3). The decay rate is roughly estimated as If this is larger than the Hubble parameter at the domain wall annihilation, the gaugino condensation instantly decays into the right-handed neutrinos. Otherwise, it would take some time for the gaugino condensation to decay, and it may even dominate the Universe and produces an entropy. To see this more explicitly, let us evaluate the amount of the entropy dilution factor. The gaugino condensation after the domain wall decay is nonrelativistic, and so, its energy density decreases like matter. Therefore, it would dominate the Universe when the Hubble parameter becomes equal to where H ann = 1/2t ann . Therefore, if Γ GC > H GC,dom , there is no entropy production. On the other hand, if Γ GC < H GC,dom , the gaugino condensation dominates the Universe, and its decay increases the entropy by a factor ∆ ≡ (H GC,dom /Γ GC ) 1/2 .
The collapses of the domain walls generate the gravitational wave [67]. The peak frequency of the gravitational wave is determined by the Hubble parameter H ann which is the typical curvature radius of the domain wall: and the energy density of the gravitational wave is estimated as where G = (8πM 2 Pl ) −1 is the newton constant. The amplitudes of the gravitational wave for a frequency f can be parameterized using the following dimensionless quantity: where ρ c the critical energy density. Then, the peak amplitude is estimated as [63] Ω where˜ GW around unity is an efficiency parameter for the gravitational radiation. To estimate the property of the gravitational wave we observe today, we have to take into account the red-shift effect. The amount of the effect is given by a(t ann )/a 0 where a 0 and a(t ann ) denote the scale factors at present and at the formation of the gravitational waves, respectively. If the gravitational waves are created after the reheating, the red-shift parameter is given as where T ann and g * (t ann ) are the temperature and the relativistic degrees of freedom at H = H ann . Thus, the peak frequency and the energy fraction observed today are estimated as and On the other hand, if there is a period of the gaugino condensation domination after the domain wall annihilation, the frequency and the density parameter of the gravitational waves are reduced by a factor of ∆ 1/3 and ∆ 4/3 , respectively. In the following analysis, we take A = 0.9, C d = 5,˜ GW = 1.5, and N = 5. In Fig. 1, We show the contours of the dynamical scale Λ, right-handed neutrino mass M N R , and dilution factor ∆. The magenta regions are excluded due to t dom < t ann , namely the domain wall dominates the Universe before its annihilation. In the left(right) panel, we take M * = 10 16 GeV (M * = 10 15 GeV). Including the cosmological constraints, M N R 10 9 GeV leads to the lower-bound on m 3/2 as 10 5 GeV (10 3 GeV) for M * = 10 16 GeV (M * = 10 15 GeV). One can see that the entropy dilution factor is at most a factor of 5 or so in the viable parameter space where successful thermal leptogenesis takes place. Figure 2 shows the predicted gravitational wave spectra and the projected sensitivity reach of the current and future detectors (Advanced LIGO, Einstein Telescope (ET), and the experiment using BEC). On the top (bottom) panel, we take M * = 10 16 GeV (M * = 10 15 GeV). The peak frequency and energy fraction are determined by Eq. (24) and (25), together with the dilution factor ∆. We assume that the scale dependence of the energy fraction is Ω GW ∝ f 3 (Ω GW ∝ f −1 ) for f < f 0 (f > f 0 ) according to Ref. [62]. The sensitivity lines of Ω GW are taken from Ref. [33] for Advanced LIGO, Ref. [34] for ET, and Ref. [35] for BEC. The case of m 3/2 = 10 GeV is shown for the comparison purpose. 6 One can see that for M * = 10 15 GeV, the gravitational wave can be detected by ET and BEC in the viable region consistent with successful thermal leptogenesis.

Summary
We have shown that, if the right-handed neutrino mass arises from the hidden gaugino condensation through the GUT-suppressed interactions, successful thermal leptogenesis scenario requires the gravitino mass heavier than several TeV. Thus, the little hierarchy problem may be explained by thermal leptogenesis. Interestingly, if the gaugino conden-sation appears after inflation, domain walls may come close to dominating the Universe before they decay. As a result, a significant amount of gravitational waves is produced by the violent annihilation processes, and the peak frequency of the gravitational waves is determined by the gravitino mass. We have shown that the predicted gravitational waves are out of the Advanced LIGO, but could be within reach of Einstein Telescope and the experiment using BEC for the gravitino mass of O(1) TeV.