Emilian Dudas
email: emilian.dudas@cpht.polytechnique.fr

Lucien Heurtier
email: lucien.heurtier@cpht.polytechnique.fr

Yann Mambrini
email: yann.mambrini@th.u-psud.fr

Generating X-ray lines from annihilating dark matter

We propose different scenarios where a keV dark matter annihilates to produce a monochromatic signal. The process is generated through the exchange of a light scalar of mass of order 300 keV -50 MeV coupling to photon through loops or higher dimensional operators. For natural values of the couplings and scales, the model can generate a gamma-ray line which can fit with the recently identified 3.5 keV X-ray line.

I. INTRODUCTION

Physics community is still in the quest of the dominant matter component in the Universe. Even if we know quite well the cosmological abundance of this dark matter [1,2], little is known about its mass and couplings. In view of recent results from both direct and indirect detection experiments, the Weakly Interacting Massive Particle (WIMP) paradigm, corresponding to a ∼ 100 GeV particle interacting weakly with the visible sector begins to be severely constrained by XENON [3], LUX [4] or FERMI satellite [5]. On the other hand, several other scenarios offer much lighter [9] or heavier [10] candidates with feeble [6,7] or very feeble [8] couplings. Their thermal histories are relatively different (but not less motivated) from the standard freeze out one. The cases of FIMP (for Freeze In Massive Particle or Feebly Interacting Massive Particle) or WISP (for Weakly Interacting Slim Particle) are typical cases where the coupling is too weak to reach the thermal equilibrium with the standard model bath [6]. The dark matter candidate can be so weakly coupled that it decoupled from the bath at the reheating epoch, like the gravitino or candidates motivated by SO (10) schemes. Other scenario proposed an even more weakly interacting particle, so weakly interacting that the dark matter is stable at the scale of the age of the universe: the decaying dark matter (see [START_REF] Dugger | [END_REF] for a review on the subject).

At present, clues for the presence of an interacting dark matter are rare. However, recently a 3.55 keV X-ray line has been reported in the stacks analysis of 73 galaxy clusters from the XMM-Newton telescope [12] with a significance larger than 3σ. A similar analysis finds evidence at the 4.4 σ level for a 3.52 keV line from their analysis of the X-ray spectrum of the Andromeda galaxy (M31) and the Perseus Cluster [13]. In both analysis, the unidentified line was interpreted as a possible signal of sterile neutrino dark matter ν s [14] decaying through a loop ν s → γν.

While more conventional explanations in term of atomic physics effects are currently lacking, several works have been released in the following weeks, all focussing on a decaying dark matter candidate. Extensions with a sterile neutrino as dark matter candidate [15], axions or ALPs [16], axinos [17], pseudo-Nambu-Goldstone bosons [START_REF] Nakayama | X-ray line signal[END_REF] or supersymmetric models (gravitino [START_REF] Kolda | [END_REF], sgoldstino [START_REF] Demidov | [END_REF] or low scale supersymmetry breaking [START_REF] Ko | [END_REF]) were proposed, all relying on processes ensuring a fine tuned lifetime τ 10 28 seconds to fit the observed line. Other more exotic candidates like decaying moduli [START_REF] Krall | [END_REF], millicharged dark matter [START_REF] Aisati | [END_REF], dark atoms 1 [START_REF] Cline | [END_REF], magnetic dark matter [26], majoron decay [27] or multicomponent [28] have been proposed. Another original model was studied in [29,30] with an eXciting dark matter, where the photons come from the transition from an excited state down to the ground state for the dark matter particle, which in this case can be significantly heavier than 3.5 keV. Moreover, it is well known that for a warm dark matter candidate of mass ∼keV, free streaming produces a cutoff in the linear fluctuation power spectrum at a scale corresponding to dwarf galaxies and can fit observations for m s 1.5 keV [31].

All of these scenarios seem to exclude (maybe too early) an annihilating dark matter scenario [32]. Indeed, the cross section needed to fit the excess measured in [12,13] is σv 10 -33 cm 3 s -1 which corresponds to an effective scale Λ of Λ 10 GeV for a 3.5 keV dark matter, in the classical effective operators approach. This scale is too low to be due to heavy particles charged under the electromagnetic field.

In this work however, we show that such cross section arises naturally when we go beyond the naive effective operators approach and consider microscopic constructions with the presence of a light hidden mediator coupling to the standard model through effective operators. Such approach had been explored in the case of a vector boson or a heavy fermionic mediator [36] but did not consider the possibility of a scalar one. Furthermore these studies had worked out a lower bound on the dark matter mass of order O (MeV) in order to fit both ray fluxes and relic density measurements [37]. We will show that this bound can actually be overcome if one considers the dark sector to be living in a thermal bath of temperature differing from the one of the visible sector.

The paper is organized as follows. After a summary of the effective operators approach in the next section, we describe our model in Section III and extract the parameter space allowed to fit the observed X-ray line. We then compute the relic abundance predicted in this restricted parameter space by adding an additional coupling of the mediator to a light sterile right-handed neutrino in Section IV. We show that the model can fit WMAP/PLANCK data taking into account that the bath of the hidden sector is at a different temperature compared to the standard model one. We then conclude in section V with an explicit example of an UV model, before giving the relevant technical formulae in the appendix in the case of a fermionic dark matter.

II. THE FAILURE OF A NAIVE EFFECTIVE OPERATORS APPROACH

A. A priori Since Fermi's time the classical way to explain a signal or to work out predictions at a scale beyond the standard model one2 (which is the one that one can reach experimentally at a given time) is to use the effective operators approach. Indeed, it seems natural to imagine that the physics beyond a measurable scale of energy is represented by heavy particles not yet produced in accelerators, but coupling to the observable sector. This is frequently used in LHC studies [38] or dark matter searches [39]. However, it is obvious that this effective operators approach has its limitations. First of all, the coupling to the visible sector generated by loops of heavy states can be highly dependent on the hidden microscopic physics (gauge or Yukawa-like couplings) but, more dramatically, the presence of light states modifies considerably the predictions, as was shown in [40]. Supersymmetric or grand unified models do not escape this rule: light stau or Z for instance generate new processes observable at LHC and not predicted by a naive effective operators approach. This is exactly what is happening in the case of a cosmological monochromatic signal as one discusses below.

If the signal analysed in [12,13] is generated by dark matter annihilation to two photons ss → γγ (with a dark matter mass of 3.5 keV) then one should fit the annihilation cross section σv γγ with the flux measured in the vicinity of the sun. A naive estimate of the total luminosity of Perseus can be computed using

L = R P e 0 4πr 2 n 2 DM σv γγ = R P e 0 4πr 2 ρ(r) m s 2 σv γγ (1)
with R pe is the Perseus radius. As a first approximation, one can consider like in [33] a mean density of dark matter in the cluster. The Perseus observation involved a mass3 of M P e = 1.49 × 10 14 M in a region of R P e = 0.25 Mpc at a distance of D P e = 78 Mpc from the solar system. One can then estimate

n DM 1.49 × 10 14 M m s 3 4πR 3 P e = 2.0 × 10 -37 GeV 3 = 2.6 × 10 4 cm -3 . (2)
Combining Eq.( 1) and ( 2), one can compute the luminosity in the Perseus cluster, L 1.2×10 55 One can then deduce the flux φ γγ = L/(4πD 2 P e ) that one should observe on earth

φ γγ = 1.7×10 -5 3.5 keV m s 2 σv γγ 10 -32 cm 3 s -1 cm -2 s -1 (4)
The cumulative flux of φ γγ ∼ 4 × 10 -6 cm -2 s -1 from [12] is hard to interpret in the dark matter framework as it arises from a combination of clusters at a variety of distances. However, according to the authors of [13,29], one can identify a monochromatic signal arising from M31 or Perseus cluster with a flux φ γγ = 5.2 +3.70 -2.13 × 10 -5 photons cm -2 per seconds at 3.56 keV with the cluster core. One can make a more refined analysis, considering for instance a more complex halo structure like Einasto or NFW profile as in [29] but the result will not change 4 . One could also look at the Centaurus observation like in [33] with M Ce = 6.3 × 10 13 M and a radius of R Ce = 0.17 Mpc.

Finally, taking into account also other observations like M31, we will impose in our analysis a conservative required annihilation cross section estimated as

σv γγ (2 × 10 -33 -4 × 10 -32 ) cm 3 s -1 . (5)
However, for such a light dark matter particle annihilating into photons, it is important to check the consequences of injecting secondary particles on the recombination, leaving an imprint on Cosmic Microwave Background (CMB) anisotropies. The authors of [34] show that the corresponding condition is given by

σv CM B γγ 2.42 × 10 -27 m s 1 GeV cm 3 s -1 , (6)
which for a 3.5 keV dark matter is σv γγ < 8.5 × 10 -33 cm 3 s -1 . In f ine one will then restrict ourself to the parameter space allowing a monochromatic signal and respecting the CMB constraints:

2 × 10 -33 cm 3 s -1 < σv γγ < 8.5 × 10 -33 cm 3 s -1 . (7)

C. A posteriori

In the case of a scalar particle annihilating into two photons, the CP-even effective lagrangian can be written 5

L ef f = S 2 Λ 2 F µν F µν , (8)
with

F µν = ∂ µ A ν -∂ ν A µ being the electromagnetic field strength.
The scale Λ is related to the mass of the particles running in the loops (see Fig.( 1)) which , being charged under U (1) em , should be heavier or at least of the order of TeV. A list of generic couplings of this type can be found in [41]. We will write in the appendix the results we obtained in other cases.

It is important to notice that such a light dark matter can contribute to the effective number of neutrinos N ef f . However, it has been shown recently that a dark matter annihilating into photons in sub-MeV masses is possible only in the case of a scalar dark matter [42]. Such processes have already been computed in [43] and one obtains

σv ef f γγ = 2m 2 s πΛ 4 . (9)
Applying the constraints (7) to the annihilation cross section ( 9) one obtains

10 GeV < Λ < 15 GeV . ( 10
)
This value is obviously far below any accelerator limit on charged particles. It seems then impossible to UV complete this operator and achieve a large enough rate. However, as we will see below, the effective operators approach cannot be applied anymore when the UV sector contains light states.

III. A NATURAL MICROSCOPIC APPROACH

It is then natural to build a microscopic model and to see how observables are modified. But natural in which sense? Natural in the sense that the presence of a keV-MeV dark matter particle naturally leads to a keV scale dynamics, as the presence of GeV particles in the standard model naturally leads to GeV scale dynamics in the Higgs sector. We then can suppose the presence of a (pseudo)scalar coupling to the dark matter candidate, and generating the keV dynamics. The simplest way to generate such dynamics is through a "higgs-like" portal. We will consider by simplicity a scalar dark matter; other dark matter spin or couplings do not change our conclusions and are treated in the appendix.

A. The scalar model

We will work in the framework of a scalar portal φ, coupling directly at tree-level to dark matter, but indirectly to the standard model through loops. This is a typical secluded dark matter type of model [44]. The lagrangian can then be written for a scalar dark matter

L ef f ⊃ - m 2 s 2 S 2 - m 2 φ 2 φ 2 -mφS 2 + φ Λ F µν F µν . ( 11
)
We assume the parameter m to be a free mass scale parameter. However such coupling can be explicitely generated by symmetry breaking in renormalizable models, as illustrated in section V. In the latter case, m is expected to be at most of the same order of magnitude than m φ since it gets its value from the vev of a field Φ = v φ + φ after spontaneous symmetry breaking. Furthermore this is what would be more generally expected if m is generated by whatever dynamical mechanism involving only φ and the light field S. The mass scale Λ is related to the mass of heavy particles integrated in the loop. In a perturbative set up with N charged fermions running in the loop Λ ∼ 4π N h φ α M ψ , where h φ is the Yukawa coupling of φ to the charged fermions of mass M ψ . Using the constraint M ψ 500 GeV from collider searches and perturbativity one finds that the minimum natural values for Λ are Λ ∼ 50 -500 TeV, whereas Λ ∼ 5 TeV can only be obtained in a strongly coupled hidden sector. Such a lagrangian gives for the annihilation cross section (process depicted in Fig. (2) )

σv micro γγ = 4m 2 s m2 πΛ 2 (4m 2 s -m 2 φ ) 2 . (12)

B. X-ray line

Depending on the hierarchy between the masses of the mediator φ and the dark matter particle S, the condition (7) leads to two kinds of constraints :

Both cases give at first sight viable results. One can understand easily why it is so in the microscopic approach compared to the effective operators approach of Eq.( 9). Indeed, as recently emphasized by the authors of [40] for the LHC analysis of mono jet events, the effective operators approach ceases to be valid once the ultraviolet (microscopic) theory contains some light mediators, which is exactly our case. This comes from two powers less in Λ in the computation of observables: heavier states become now reasonably heavy compared to the result Eq.( 10).

We will see however that experimental bounds on light scalar particle interactions with the electromagnetic sector are strongly restrictive.

C. Experimental Bounds

As we just mentioned above, interactions of a light scalar, or axion-like particle (ALP) with the visible sector is very much constrained by collider data (LEP) and astrophysics. Indeed bounds on pseudoscalar particles interacting with photons (see [46]) have been studied, using LEP data from ALEPH, OPAL, L3 and DELPHI, and shown that the coupling of the pseudoscalar with photons cannot exceed a value of 2.6 × 10 -4 GeV -1 for a mediator of mass m φ 50 MeV, which means, in terms of our mass scale

Λ 3 TeV [m φ 50 MeV] . (15)
Furthermore, one of the most restrictive constraints on ALPs comes from the non-observation of anomalous energy loss of horizontal branch (HB) stars via a too important ALP production [47]. Indeed those contraints impose

Λ 10 10 GeV [m φ 30 keV] , (16)
for a mediator mass up to m φ 30 keV. At higher masses arise constraints coming from the CMB and BBN studies, setting lower limits on the coupling with photons. A nice review on the subject can be found in [48,49]. Various astrophysical constraints on ALP mass and coupling to photons are summarized in, e.g. [50].

These constraints on our model essentially put lower bounds on Λ. Indeed, for a light mediator (Case B) HB experiments impose that the mass scale Λ takes very high values ( 10 10 GeV). In this case, as indicated by Eq.( 14), one would need the tri-linear coupling to be of order m 10 -3 GeV. However, in this case, since m φ is assumed to be smaller than the keV scale, one would conclude that m/m φ 10 3 which is, as mentioned in the previous section, quite unnatural. We will then concentrate our study on Case A, where the mediator φ is assumed to be heavier than the dark matter field S.

In Case A, the discussion is a bit more subtle, as far as the experimental constraints are concerned. For mediator masses lower than a hundred keV, the mass scale Λ must reach very high values ( 10 16 GeV) to escape experimental exclusion bounds. Still such region of the parameter space is not acceptable since it would lead to a very heavy parameter m. Then for higher masses of the mediator (m φ 300 keV) more reasonable values of Λ are allowed, and we are left with lower bounds coming from LEP (mentioned above) and upper bounds on Λ arising from CMB dilution and BBN perturbations. Different choices of Λ will then lead to different pairs of (m φ , m), as depicted in Fig. In order to fix ideas, and anticipating results of section V, we indicated in red in the figure the region where m m φ . This shows clearly, that imposing m φ 300 keV sets an upper limit for Λ, giving approximately Λ 1000 TeV .

(17)

Furthermore, the lower limit Λ 5 TeV mentioned in section III A -still acceptable if there is some strongly coupled hidden sector generating the effective mass scale Λ -imposes an upper limit on the mediator mass, m φ 50 MeV 6 . One would thus expect from this model that the mediator mass lies in the region 300 keV m φ 50 MeV .

6 As mentioned in previous sections, assuming that the effective coupling between the mediator φ and the photons comes from some perutrbative heavy physics sets a stronger limit on Λ leading to masses of the mediator m φ 5 MeV.

IV. RELIC ABUNDANCE A. State of the art

Computing the relic abundance in models with a very weak annihilation cross section and a keV dark matter particle is highly non-standard. Indeed, it is well known from the standard lore that a hot dark matter candidate leads to a relic density

Ωh 2 9.6 × 10 -2 g ef f g s (x f ) m s 1 eV , (19)
where g ef f is the effective number of degrees of freedom of the dark matter candidate and g s the effective number of degrees of freedom for the entropy. Eq.( 19) gives m s 5 eV if one wants to respect PLANCK [2] limit Ω DM h 2 = 0.1199 ± 0.0027. However, this condition is valid only under the hypothesis that the dark matter is in thermal equilibrium with a common temperature T with the thermal bath. In the case of the line signal observed in the clusters, the cross section necessary to fit the result is far below the classical thermic one σv therm = 3×10 -26 cm 3 s -1 . This idea had led previous studies to rule out scalar dark matter candidates lighter than O (MeV) [37]. In fact, the dark bath, composed of the light mediator φ and the dark matter S cannot be in equilibrium with the standard plasma.

There are several ways to address this issue. A first possible attempt to solve the problem, proposed in [6] and [7], is to suppose that the dark matter is produced through the freeze in mechanism: the interacting photons annihilate to produce the dark matter in the inverse process of Fig.( 2). Yet it is not possible to get the right relic density in this way since, solving the Boltzmann equation in this case would produce too much dark matter. Indeed equilibrium dark matter density would reach quickly a value that would overclose the Universe.

Another way to solve the problem was proposed in [54,55] where the authors noticed that the condition ( 19) is not valid anymore if the temperature of the hidden sector T h is different from the one of the thermal bath T . In this case, one can compute the temperature T h needed to obtain a 3.56 keV particle respecting the relic abundance constraint. Yet, as we will see in what follows, we still need the hidden sector content to be richer in order to provide new dark matter annihilation channels leading to the right relic abundance. This will be done adding to the model a right-handed sterile neutrino.

B. Dark matter annihilation into sterile neutrinos

One way of solving the lack of annihilation of dark matter described above is to assume that a right-handed sterile neutrino couples directly to the mediator scalar particle previously introduced. This would provide another channel for annihilating dark matter which would boost the relic density to its experimental value. In a similar fashion than in section III A, one can add to the usual neutrino interaction terms a Yukawa coupling between the field φ and the sterile neutrino ν R

7 -L ν = M 2 ν R ν R + m D ν L ν R + λ ν φν R ν R + h.c. , (20)
After diagonalization of the mass matrix, the sterile neutrino gets mass m st M while the active one obtains a mass of m act m 2 D /M via the seesaw mechanism. Such interactions give the following cross section for dark matter annihilation into a pair of sterile neutrinos

σv νν = λ 2 ν m2 8π 2 (m 2 s -M 2 ) m 2 s (4m 2 s -m 2 φ ) 2 1 - M 2 m 2 s . (21)
Assuming that M is negligible compared to m s leads to a cross section depending only on m s , m φ and m, λ ν :

σv νν ∼ λ 2 ν 8π 2 m m s 2 m 2 s (4m 2 s -m 2 φ ) 2 . (22)

C. Cosmological constraints on a hidden sector

In such a framework, dark matter and sterile neutrinos can be seen as living together in a hidden thermal bath decoupled from the visible sector. Its temperature can be denoted by

T h ≡ ξ(t)T , (23)
where in what follows ξ is assumed to be a constant parameter8 . As described in [54,55], such a point of view can have important consequences on the thermal dynamics of the hidden sector, since ξ enters into the Boltzmann equation. Indeed dark matter can decouple while still relativistic or semi-relativistic and lead to different relic densities, as the parameter ξ takes different values (See Fig.( 6)). Such freedom in the temperature of the hidden sector is yet constrained by astrophysical considerations. In particular, a hidden sector dark matter can freeze out while still being relativistic. It is then important to check that its free streaming length is smaller than typical galactic length scales so that it does not destroy the matter power spectrum. Another constraint, introduced by Tremaine and Gunn in [56], comes from bounding the phase-space density of structures like small dwarf galaxies by statistical quantum mechanics considerations. Both constraints are computed and summarized in the case of a ∼ 3 keV dark matter in [54] and lead naturally to a relic density value different from the one of the visible thermal bath ( σv 0 ∼ 10 -9 GeV -2 ), depending on how cold the hidden sector is. We thus get 0.015 σv 0 σv 0.045 σv 0 ,

the upper limit corresponding to the free streaming constraint whereas the lower one to a strict Tremaine-Gunn bound. One should also remark that the dark matter is relatively warm (x f = m S /T f 2 -4 for points respecting WMAP/PLANCK) in this case. As a warm candidate, it can elude the classical issues of cold dark matter i.e. the few number of galaxy mergers ( 10%) or the core observed profiles compared to the cusp ones predicted by N-body simulations.

D. Results

In the light of section III C and equation [START_REF] Krall | [END_REF] one can now constrain couplings between the mediator φ to both the dark and the visible sector m and λ ν , taking into account that relic density can lie in the region [START_REF] Cline | [END_REF] exhibited in the previous section, as well as imposing constraints on the photons flux measurement. Results are presented in Fig. (5) where we show the allowed parameter space imposing cosmological bounds [54] superimposed within the regions fitting the 3.5 keV excess, for m φ = 500 keV. As one can see, a relatively large region respects the cosmological bounds and the monochromatic excess. The values of λ ν are also quite constrained (∼ 10 -7 -10 -5 , depending on the mass scale Λ), leading to different values of the hidden sector temperature (as represented in Fig.( 6)). Furthermore it could be interesting in a future work to combine our analysis with neutrino bounds.

V. AN EXPLICIT UV MODEL

We give here an explicit example realizing the case A of Section III. The model contains a scalar σ and a pseudoscalar S, together with a (set of) fermion(s) ψ, with the lagrangian ψ(h

L = L kin + µ 2 2 (σ 2 + S 2 ) - λ 4 (σ 2 + S 2 ) 2
1 σ + ih 2 Sγ 5 )ψ + σ Λ F µν F µν . ( 25
)
The set of fermions ψ are part of the hidden sector, in particular they are singlets with respect to the Standard Model. By defining the complex field Φ = 1 √ 2 (σ + iS), we can split the lagrangian (25) into two parts:

L = L 0 + L 1 , where L 0 = L kin + µ 2 Φ † Φ -λ(Φ † Φ) 2 -(h 1 ψL Φψ R + h.c.) , L 1 = i(h 1 -h 2 ) S ψγ 5 ψ + σ Λ F µν F µν . ( 26
)
Notice that the whole lagrangian is invariant under the vectorial transformation

U (1) V : ψ → e iα ψ , Φ → Φ , (27)
whereas L 0 is also invariant under the axial transformation

U (1) A : ψ → e iβγ5 ψ , Φ → e -2iβ Φ . ( 28
)
The axial transformation is broken by L 1 and is therefore an exact symmetry of the hidden sector lagrangian in the limit h 1 = h 2 . The symmetry is broken additionally by the coupling to the photons. At tree-level, S is massless since it is the (pseudo)Goldstone boson of the axial U (1) A symmetry. For µ 2 > 0 there is a symmetry breaking vacuum, with σ = v and v 2 = µ 2 λ . By expanding around the minimum one also finds at tree-level

m 2 φ = 2µ 2 , m = λ 2 m φ . ( 29
)
At the one-loop level, there is a quantum correction to the potential

V 1 = V 1 (h 2 1 σ 2 + h 2 2 S 2
), which generates a mass for the pseudo-goldstone boson S, proportional to

m 2 S ∼ O( h 2 2 -h 2 1 16π 2 )m 2 φ
, which has a one-loop suppression with respect to the mediator mass φ. This model is an example of a UV embedding of case A of Section III, in which one naturally expects m ≤ m φ and selects therefore natural regions in the parameter space in Section III.

CONCLUSIONS

In this work, we have shown that a keV scalar dark matter can be the main constituent of the matter of the universe, producing monochromatic X-ray signals that can be fitted with the recently claimed events of a 3.5 keV line in nearby clusters of galaxies. Moreover, we know that for a warm dark matter mass of order of a keV, free streaming produces a cutoff in the linear fluctuation power spectrum at a scale corresponding to dwarf galaxies and can fit observations for m s 1.5 keV [31]. We have shown that astrophysical, collider and relic density constraints are more difficult to accomodate. They are however possible to satisfy for certain values of the mediator mass m φ and scale Λ of the couplings between the mediator and the photon 300 keV m φ 50 MeV and 5 TeV Λ 1000 TeV. These values can conversely be considered as a prediction of our setup of keV dark matter models leading to X-ray monochromatic lines. We also showed that as pointed out recently by [40], the study of pure effective models -as it is often done in the literature -misses important quantitative effects. Indeed, we could reach a cross section in agreement with the analysis of XMM Newton data only through the building of a microscopic model and the exchange of a FIG. 7. Effective diagram for dark matter annihilation in the fermionic case light scalar. Finally, it should be interesting to look at the observability of such a light meson. This task is far beyond the scope of our paper but is under investigation.

APPENDIX

For completeness, we present in this appendix the main alternatives to the Lagrangian [START_REF] Dugger | [END_REF]. They do not affect however our conclusions.

Fermionic dark matter

Effective operator approach

With the same philosophy used in section II, we could consider a fermionic dark matter χ interacting with photons through effective operators of the form χχ

Λ 3 F µν F µν , χγ 5 χ Λ 3 F µν F µν , (30)
where F µν = 1 2 µνρσ F ρσ . The obtained cross sections are, respectively

σv ef f γγ = 8m 4 χ v 2 πΛ 6 , 8m 4 χ πΛ 6 . (31)
The constraints (7) give then 0.061 < Λ < 0.078 , 0.091 < Λ < 0.12 (GeV) (32) It is clear then that in the fermionic case these values are even more incompatible with experimental constraints coming from colliders.

Microscopic Model

Scalar Mediator. Following the discussion of section III but working in the alternative frame of a fermionic dark matter (see Fig.( 8)) particle interacting with the standard model via a scalar portal, one can write the following lagrangian

L ef f = g χ φ χχ + φ Λ F µν F µν .
One then gets the annihilation cross section

σv micro γγ = 2g 2 χ m 4 χ v 2 πΛ 2 (4m 2 χ -m 2 φ ) 2 . ( 33
)
which in what follows is fixed in order to fit the X-ray monochromatic signal.

Furthermore, one can still differentiate between the two cases where the mass of the mediator is heavier or lighter than the dark matter one.

g 2 χ v 2 Λ 2 ∼ 4.2 × 10 -14 GeV -2 (35)
The cross section is velocity suppressed here and leads, for a given value of the mediator mass and of the coupling g χ , to smaller values of the mass scale Λ. For example, for g χ = 1 in case B, one gets Λ ∼ 1.4 × 10 6 GeV.

Pseudo-Scalar Mediator. Another option is to make the dark sector communicate with the standard model through the exchange of a pseudo-scalar particle. The lagrangian is given by

L ef f = g χ φ χγ 5 χ + φ Λ F µν F µν ,
and the annihilation cross section is

σv micro γγ = 2g 2 χ m 4 χ πΛ 2 (4m 2 χ -m 2 φ ) 2 , (36)
which is now not velocity suppressed anymore.

In case B fitting the X-ray signal requires therefore Λ ∼ 5 × 10 6 GeV.

In both the cases of scalar and pseudo-scalar mediators, as is the case in the rest of the paper, case B is always excluded, regarding to the HB experimental bounds. As far as the Case A is concerned, the discussion is similar and g χ plays the same role as m. However the needed values of Λ are different compared to the scalar dark matter : Λ 10 TeV for Eq.( 34) and Λ 100 TeV for Eq.( 37).

3 .5 keV m s 2 σv γγ 10 -

 3210 26 cm 3 s -1 ph/s .(3)

FIG. 1 .

 1 FIG. 1. Effective diagram for dark matter annihilation

  dramatically

FIG. 2 .

 2 FIG. 2. Microscopic diagram for dark matter annihilation

  Case A : m φ m s (Heavy Mediator), m φ (12.3 -17.6) m s 3.5 keV m Λ GeV (13) Case B : m φ m s (Light Mediator), m Λ ∼ (1.63 -3.36) × 10 -13 .

( 3 )FIG. 3 .

 33 FIG. 3. (m φ , m) parameter space allowed by the γ flux measurements in the case of a heavy mediator (Case A), for different values of Λ. The red shaded region indicates where m is higher than m φ .

FIG. 4 .

 4 FIG. 4.Microscopic model of dark matter decaying into right-handed sterile neutrinos.

FIG. 5 .

 5 FIG. 5.Constraints on the parameter space ( m, λν ), for m φ = 500 keV; considering cosmological constraints (relic abundance, free streaming and Tremaine-Gunn bounds) in the red/dark band and a 3.5 keV excess (blue/green region) for different values of the BSM scale Λ (see the text for details).

FIG. 6 .

 6 FIG. 6. Hidden temperature factor ξ allowed by the flux measurement limits, as a function of the relic density value, for different values of the mass scale Λ.

FIG. 8 .

 8 FIG. 8. Microscopic diagram for dark matter annihilation in the fermionic case

Case A : m φ m χ (

  Case B : m φ m χ (Light Mediator)

2 χΛ 2 ∼ 4 . 2 ×

 2242 Case A : m φ m χ (Heavy Mediator) m φ (0.97 -1Case B : m φ m χ (Light Mediator) g 10 -14 GeV -2

The "standard model scale" varying with time: MeV in 1897, GeV in 1932, 100 GeV in 1983, TeV scale nowadays.B. The monochromatic signal

See table 2 of[12]

We use two-component spinor notation in this equation.

See ref.[55] for a discussion on the validity of this approximation.

Acknowledgements. The authors would like to thank M. Goodsell and A. Ringwald for very useful discussions. E.D. and L.H. would like to thank the Alexander von Humboldt foundation and DESY-Hamburg for support and hospitality during the final stage of the project and Y.M. would like to thank K. Tarachenko for constant support. This work was supported by the French ANR TAPDMS ANR-09-JCJC-0146, the Spanish MICINN's Consolider-Ingenio 2010 Programme under grant Multi-Dark CSD2009-00064, the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442) and the ERC advanced grants MassTeV and Higgs@LHC.


Emilian Dudas
email: emilian.dudas@cpht.polytechnique.fr

Tony Gherghetta

Yann Mambrini
email: yann.mambrini@th.u-psud.fr

Keith A Olive
email: olive@physics.umn.edu

Inflation and High-Scale Supersymmetry with an EeV Gravitino

We consider inflation and supersymmetry breaking in the context of a minimal model of supersymmetry in which the only "low" energy remnant of supersymmetry is the gravitino with a mass of order an EeV. In this theory, the supersymmetry breaking scale is above the inflaton mass, m 3 × 10 13 GeV, as are all sfermion and gaugino masses. In particular, for a no-scale formulation of Starobinsky-like inflation using the volume modulus T , we show that inflation can be accommodated even when the supersymmetry breaking scale is very large. Reheating is driven through a gravitational coupling to the two Higgs doublets and is enhanced by the large µ-parameter. This leads to gravitino cold dark matter where the mass is constrained to be in the range 0.1 EeV m 3/2 1000 EeV.

I. INTRODUCTION

While the Higgs boson was discovered at the LHC [1,2] and is consistent with predictions of low energy supersymmetry (SUSY), [3][4][5][6][7], so far supersymmetry has not been seen experimentally [8]. Whether supersymmetry is waiting around the corner, or is broken at some high scale (intermediate or above) is currently unknown. If indeed supersymmetry is broken above the inflationary scale, it may well be that the only remnant of supersymmetry at low energies is the gravitino which may yet play the role of dark matter [9,10] 1 .

Below the Planck and grand unified theory (GUT) scales, it would appear that there is an intermediate scale (between the GUT scale and electroweak scale) associated with inflation. For the sake of definiteness, let us consider the Starobinsky model of inflation as an example [12][13][14]. The inflaton potential can be written as

V (t) = 3 4 m 2 1 -e - √ 2 3 t 2 , ( 1
)
where t is the canonically normalized inflaton field. The inflaton mass scale, m, can be determined by the amplitude of density fluctuations [15],

A s = 3m 2 8π 2 sinh 4 (t * / √ 6) = 2.1 × 10 -9 , (2)
where t * ≈ 5.35 corresponds to 55 efolds of inflation. Solving for m in (2), we have m = 1.2 × 10 -5 M P ≈ 3 × 10 13 GeV, where M P = 1/ √ 8πG N 2.4 × 10 18 GeV.

Interestingly, the mass scale around 10 13 GeV, may also correspond to an intermediate scale gauge group whose breaking may yield a large Majorana mass for right-handed neutrinos, M R m, appropriate for the see-saw mechanism [16].

Here, we consider the possibility that the supersymmetry breaking scale is also of order the inflaton mass, m. An example is provided by supersymmetry breaking via a Polonyi sector which is achieved with a superpotential of the form [17]

W P = m 2 (Z + b) , (3)
where Z is the chiral superfield responsible for breaking supersymmetry with auxiliary field component F ≡ m 2 . Of course if m in (3) is of order the inflaton mass, m, then the masses of the entire supersymmetric spectrum would be of order the intermediate scale and clearly out of reach of any accelerator search. However, the gravitino mass,

m 3/2 = m 2 √ 3M P , (4)
would be significantly lighter and could still provide for the dark matter in the universe [9,10]. In fact, to avoid over production of gravitinos through the decay of the inflaton to R-parity = -1 matter fields (which subsequently decay to gravitinos), it was argued [10], that the sparticle spectrum should lie above the inflaton mass, thus providing a lower limit to the supersymmetry breaking scale and hence a lower limit to the gravitino mass of m 3/2 > 0.2 EeV. Only an EeV gravitino mass is left behind.

The EeV gravitino as a dark matter candidate is produced after inflation in the process of reheating. The common mechanism [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], for producing a single gravitino in thermal scattering processes has a cross section which is temperature independent and scales as m2 SUSY /M 2 P m 2 3/2 , where m SUSY is a typical sparticle mass. The rate therefore is roughly Γ ∼ T 3 m 2 SUSY /M 2 P m 2 3/2 , where we have assumed predominantly Goldstino production in the limit m 3/2 m SUSY . If sparticle production is kinematically forbidden, single gravitino production (which must be accompanied by a massive sparticle (gluino), is not operative. Instead, the rate for gravitino production during reheating is suppressed, as only processes which produce two gravitinos are allowed. This cross section is temperature dependent and scales as σv ∝ T 6 /F 4 , so that the rate is roughly Γ ∼ T 9 /F 4 . In this case, the final gravitino abundance scales as n 3/2 /n γ ∼ Γ/H ∼ T 7 M P /F 4 evaluated at the reheating temperature, in contrast to the abundance for single gravitino production,

n 3/2 /n γ ∼ Γ/H ∼ T m 2 SUSY /M P m 2 3/2 .
For reheating temperatures of order 10 10 GeV, the gravitino abundance matches the CMB determined cold dark matter density [15]. It is also possible that the inflaton can decay to two gravitinos, but this is more model dependent and we return to this possibility in section III.D.

The phenomenology of this high scale supersymmetric model is simple. The (not so) low energy spectrum consists of the gravitino and perhaps the scalars associated with the chiral superfield, Z. However, as we discuss in section III.A, we expect that these are also hierarchically more massive than the gravitino. As has been shown recently [33,34], even with a spectrum as massive as discussed here, a 125 GeV Higgs mass can still be attained if tan β is either small (close to 1) or large (above 60)

The clear drawback of such a model is its testability. In fact, the model in its simplest and most minimal form predicts no signatures in either accelerator searches, or direct and indirect searches for dark matter. Of course if supersymmetry is actually discovered at the LHC, then this model can be ruled out. In a modest extension of the model with R-parity violation in the lepton-Higgs sector, the gravitino becomes unstable, though still suitably long lived. The detection of very high energy neutrinos or photons at HAWC or the Pierre Auger Observatory would be a signature of this model. Another possible signature may come from the observation of non-gaussianities in the CMB due to scalars with masses near the Hubble scale during inflation [35].

The paper is organized as follows. In section II, we discuss exemplary inflation models with high scale supersymmetry breaking. We focus on models based on no-scale supergravity [36,37] which lead to Starobinskylike potentials [38][39][40][41][42][43][44][45][46][47][48][49][50][51]. In particular, models which allow for high scale supersymmetry breaking with stabilized fields [38,39,41,43,45,47] without spoiling the inflationary properties of the potential. In section III, we discuss the phenomenological aspects of the model 2 . We begin in section III.A with a model for gaugino and scalar masses. The model utilizes the strong stabilization of the Polonyi field [45,[54][55][56][57][58][59][60][61] used to generate large gaugino masses. Scalar masses are then obtained through threshold corrections as in gaugino mediation [62] or a more strongly-coupled mediation mechanism [63,64]. The requirement that all sparticle masses lie above the inflaton mass will set a constraint on the stabilization scale. The effects of supersymmetry breaking on inflation is then discussed in Section III.B. In section III.C we determine the conditions under which we can obtain a Higgs mass of 125 GeV, as well as preserving the stability of the Higgs vacuum. The requirements for gravitino dark matter are outlined in section III.D. Our concluding remarks are given in section IV.

II. INFLATION AND SUPERSYMMETRY BREAKING

There are many ways to proceed in constructing a model of inflation which incorporates supersymmetry breaking. While it would be an overstatement to say that recent Planck results [15] on the CMB spectrum parameters, n s and r, corresponding to the tilt of the scalar perturbation spectrum and the scalar to tensor ratio, respectively predict Starobinsky-like inflation models, it is clear that these models are for now in very good agreement with Planck results. In particular, we will use formulations of the Starobinsky model based on no-scale supergravity.

A. No-scale supergravity and Starobinsky-like Inflation

The Kähler potential in the context of no-scale supergravity can be written in the form [36,37],

K = -3 ln T + T - 1 3 i |φ i | 2 , ( 5
)
where T is a volume modulus and the φ i include all matter fields and possibly a supersymmetry breaking Polonyi-like field, Z [17]. The inflaton may be identified with either T or a matter-like field, φ. Equivalently, we may use a set of field redefinitions [65] and write

K = -3 ln 1 - i |y i | 2 3 , (6)
where the y i include all matter fields, moduli and the inflaton. For now, let us ignore the matter fields, and concentrate on a two-field model. There are at least two independent families [41] of superpotentials which lead to Starobinsky-like inflation. In the first, T is a modulus and φ is the inflaton with a Wess-Zumino (WZ) superpotential written as [38]

W = m φ 2 2 - φ 3 3 √ 3 WZ , (7)
or in the symmetric basis

W = m y 2 1 2 1 + y 2 √ 3 - y 3 1 3 √ 3 WZ , (8)
which is a WZ model for the inflaton y 1 with an interaction term y 2 1 y 2 . In both bases, when φ (y 1 ) is redefined to a field x with a canonical kinetic term, the potential is exactly of the form of the Starobinsky potential (1) (with t identified as x), assuming that some dynamics stabilizes and fixes T (y 2 ): T = T * = 1/2 ( y 2 = 0). One way to accomplish this is by adding a quartic term in the Kähler potential [41,66].

The second family of models first formulated as an R 2 extension to supergravity by Cecotti [67] can be written as

W = √ 3mφ(T -1/2) C , (9)
or

W = my 1 y 2 (1 + y 2 / √ 3) C , (10)
In this case, the inflaton is associated with T (y 2 ) and it must be assumed that φ (y 1 ) is stabilized at the origin [39,41]. Again, when T (y 2 ) is normalized to give a proper kinetic term, we get the Starobinsky potential shown in Eq. (1) [39].

In either case (WZ or C), the mass parameter m is related to the inflaton mass, and is set by the amplitude of density fluctuations measured in the CMB through Eq. (2).

B. Effects of supersymmetry breaking on Inflation

Supersymmetry breaking can be accomplished in various ways, but in many of these, there are constraints on the SUSY breaking scale due to its effect on the inflationary potential. In general, SUSY breaking perturbs the potential, but these effects may be small, if the supersymmetry breaking scale, m m. Indeed, this was one of the initial motivations behind supersymmetric formulations of inflation [68].

The simplest possibility we can consider is adding a constant, w 0 to the superpotential. In most low energy models of SUSY phenomenology, we would relate w 0 to the weak scale through w 0 = mM 2 P and the gravitino mass is just

m 3/2 = w 0 (T + T ) 3/2 = m , (11)
with T + T = 1, in Planck units. However, in this case low energy SUSY breaking parameters such as soft scalar masses, m 0 , trilinear A-terms and the bi-linear B 0 are all proportional to m 3/2 (m 0 = 0 for untwisted matter fields) [45,69]. The gaugino mass in this case is

M 1/2 = 1 2 e G/2 fT Ref (G -1 ) T T G T = 1 2 w 0 fT Re f , (12)
where f αβ = f δ αβ is the gauge kinetic function. For T + T = 1, it is unlikely that we get a hierarchy m 3/2 M 1/2 . Moreover, we would like to relate the supersymmetry breaking scale to the inflationary scale m = m. However, a priori, we can set w 0 to be either mM 2 P , or m2 M P , or m3 , giving m 3/2 = m, m 2 /M P ≈ 0.4 EeV, or m 3 /M 2 P ≈ 5 TeV (though the latter may be of phenomenological interest at the LHC).

In [50], a linear term a 2 φ for the inflaton in the WZ model given in (7) was proposed, making the association between the inflaton and Polonyi field. For small a, the theory works quite well, and thus predicts a small (weak scale) gravitino mass. The inflationary capability of the theory breaks down when a 5 × 10 -5 corresponding to an upper limit on the gravitino mass of m 3/2 = a 4 /2m 10 6 GeV.

Next we can consider adding a strongly stabilized (twisted) Polonyi field to the WZ model,

w 0 → m2 (z + b)(T + 1/2) p , (13)
with

K ⊃ z z - (z z) 2 Λ 2 z . (14)
The factor (T + 1/2) p is needed to avoid a deSitter vacuum with weak scale energy density and we will take p = 3 as an example here [45]. Choosing b 1/ √ 3 gives a minimum with zero vacuum energy at z Λ 2 z / √ 12. The mass of the Polonyi field is now hierarchically larger than the gravitino mass

m 2 z = 12m 2 3/2 M 2 P Λ 2 z . ( 15
)
Once again, for m2 mM P , this works quite well so long as Λ z is not too small (Λ z 2( m2 /mM P ) .3 ). Increasing m, leads to the formation of a new minimum at large field values (of the canonically normalized inflaton), which quickly becomes the global minimum. In Fig. 1, we show the potential for fixed p = 3, Λ z = 10 -2 , and several values of m. For m2 /m < 10 -8 M P , the potential is indistinguishable from that shown as 10 -8 . For m ≈ 10 -5 M P , we see that this model works fine for weak scale supersymmetry breaking, and for scales as large as m2 /M P 10 -12 M P ∼ 3 PeV, corresponding to a gravitino mass of m 3/2 m2 / √ 3M P 1.7 PeV. In particular, m 3/2 = 0.2 EeV would correspond to m2 /m ≈ 10 -5 M P which would badly spoil the inflationary potential.

While the WZ models are perfectly acceptable for low scale supersymmetry breaking, our objective here is high scale breaking and thus we turn our attention to the case C, for the superpotential given by ( 9). To achieve supersymmetry breaking and generate a finite gravitino mass, we can again add a constant, w 0 to the superpotential. In this case, if w 0 m, the minimum is shifted slightly to [45]

T = 1 2 - w 2 0 m 2 , φ = √ 3 w 0 m , (16)
but the vacuum energy density is necessarily negative, V 0 = -3 e G = -3m 2 w 2 0 /(m 2 -3w 2 0 ) < 0. However, adding an untwisted Polonyi field, so that the Kähler potential becomes

K = -3 ln T + T - 1 3 i |φ i | 2 - 1 3 |z| 2 + |z| 4 Λ 2 z , (17)
with the superpotential given in (13) with p = 0 leaves the Starobinsky potential (now a function of T ) unchanged, save for a shift in the minimum to

T 1 2 + 1 3 m2 mM P 2 , φ m2 m , z Λ 2 z 6 √ 3 , b 1 √ 3 1 - 1 6 m2 mM P 2 , (18)
when m2 /(mM P ), Λ z /M P 1. The mass of z is √ 3 times larger than the twisted Polonyi mass given in Eq. ( 15).

Alternatively, one can add a twisted Polonyi field with

Kähler potential

K = -3 ln T + T - 1 3 i |φ i | 2 + |z| 2 - |z| 4 Λ 2 z , (19)
and the same superpotential (9). This also leaves the Starobinsky potential unchanged, with a similar shift in the minimum to [45]

T 1 2 + 2 3 m2 mM P 2 , φ m2 m , z Λ 2 z 2 √ 3 , b 1 √ 3 1 - 1 2 m2 mM P 2 , (20)
Unlike the WZ case discussed above, the inflationary potential maintains its form even for large m, and arbitrarily small Λ z . For example, in Fig. 2, we show the inflationary potential with m2 = 0.9mM P and Λ z = 10 -3 M P both the twisted (solid) and untwisted (dashed) Polonyi models. Here t = 3/2 ln(2T ) is the canonically normalized inflaton. In the figure, z and b have been fixed at the approximate values given in ( 18) and (20), respectively (higher order terms in ( 18) cannot be neglected). The gravitino mass for small Λ z can be written as

m 3/2 = m 5 m6 + 6 m2 m 2 M 2 P 2(3m 2 M 2 P + m4 ) 3/2 untwisted , ( 21
)
m 3/2 = m 4 m6 + 2 m2 m 3 M P 2 √ 3(m 2 M 2 P + m4 ) 3/2 twisted , (22)
which in the limit of small m2 /mM P for both cases gives the expected result m 3/2 = m2 / √ 3M P . Indeed, for m2 = 0.9mM P , we obtain m 3/2 ∼ 10 13 GeV. It is indeed rather surprising that even for a large Polonyi mass scale, the inflationary dynamics are little affected. This is only true for case C given by Eq. ( 9). Thus we are free to make the simple choice of m = m ≈ 10 -5 M P . In this case, the gravitino mass is

m 3/2 = m 2 √ 3M P ≈ 0.2 EeV. (23)
Furthermore, it was shown in the first reference of [47] that one also needs to impose m 3/2 < H in order to keep perturbative control of the Kähler potential. In what follows, we will restrict our attention to the twisted Polonyi model.

To avoid the production of the Polonyi field during reheating, we can derive an upper limit on Λ z from the requirement that m z = √ 12m 3/2 M P /Λ z > m. This limit is shown by the blue dashed line in Fig. 3. Another upper limit on Λ z is obtained by requiring that the branching ratio of inflaton decays to gravitinos does not lead to an excess abundance of gravitinos (discussed in more detail in section III.D). This constraint is shown by the negatively sloped pink dot-dashed line. Acceptable parameters lie below both lines (blue dashed and pink dot-dashed) Finally, we also have a lower bound on Λ z , stemming from our effective correction to the Kähler potential which imposes F < z < Λ 2 z or 1/2 log(m 3/2 /M P ) < log(Λ z /M P ). This lower bound is shown by the green dotted line in Fig. 3, and all values of 10 -5 < m 3/2 /m < 10 -1 are allowed so long as Λ z lies in the pale shaded region.

III. THE PARTICLE SPECTRUM

A. Gaugino and Scalar masses

We next discuss ways to generate gaugino and scalar masses via perturbative mediations of supersymmetry breaking, and then we will turn to more strongly-coupled mediation mechanisms.

One perturbative possibility for mediation of supersymmetry breaking is via gaugino mediation [62]. New physics at a messenger mass scale M generates a coupling of the supersymmetry breaking field, Z = z + θ 2 F (ignoring fermionic components) to the gauge fields

d 2 θ f 0 + f 1 Z M trW α W α , (24)
where at leading order f 0 = 1/(4g 2 ). This will give rise to gaugino masses of order

M 1/2 ∼ g 2 f 1 F M . (25)
Scalar masses are then generated by SM loop corrections and will be of order 3. Bounds on the stabilization parameter Λz as a function of the gravitino mass. The blue dashed line shows the upper limit on Λz from the requirement that the mass of the Polonyi field lies above the inflaton mass. The pink dot-dashed line (negatively sloped) is an upper limit on Λz derived from an upper limit on the branching ratio of inflaton decays to gravitinos. The green dotted line shows the lower limit on Λz, assuming F < Λ 2 z . The shaded region is allowed by all constraints. Note, however, that there are lower bounds on the gravitino mass given in Eqs. ( 27) and (30).

m 2 0 ∼ g 2 16π 2 M 2 1/2 , (26) -7 -6 -5 -4 -3 -2
where g is a SM gauge coupling. Usually there is a log enhancement of the scalar masses due to the running between the mediation scale M and the scale of the superpartners. In our case however, since the superpartners are very heavy there is not much running, so numerically we can use as an order of magnitude estimate m 0 ∼ (α/g) M 1/2 ∼ 0.06M 1/2 . In order for the effective field theory to be well-defined (higher-dimensional operators are negligible), one needs to impose F < M 2 . We also forbid inflaton decays into superpartners, which roughly requires m 0 > 3×10 13 GeV. Under such assumptions with √ F ∼ M , one obtains √ F > 5 × 10 14 GeV and therefore the minimum value of the gravitino mass in such a scenario is

m 3/2 > 6 × 10 10 GeV = 60 EeV . ( 27
)
It seems difficult to decrease the gravitino mass much below 100 EeV with known perturbative mediations of supersymmetry breaking. For example, in a model based on gauge mediation, scalar masses would be expected to be of order the gaugino masses such that m 0 ∼ M 1/2 ∼ (g 2 /16π 2 )F/M (hence there is no relative loop suppression). The lower limit on m 3/2 is then increased to m 3/2 > 2 × 10 13 GeV , which is essentially incompatible with the required upper limit, m 3/2 < m.

In order to decrease the viable values of the gravitino mass, some strong coupling effects seem to be needed, like for example in holographic models of supersymmetry breaking of the type described in [63], or general gauge mediation [64]. In more generic terms, this means a mediation mechanism with no loop suppression in the generation of visible sector soft masses, so that we obtain m 0 ∼ M 1/2 ∼ F/M . Thus we can start with the same gauge kinetic function as in (24), namely

f αβ = f 0 + f 1 Z M δ αβ , (28)
and generate squark/slepton/Higgs soft masses through operators of the type

d 4 θ Z † Z M 2 Q † Q , (29)
where Q denotes a generic MSSM chiral superfield. In this case, we recover the limit

m 3/2 > 0.2 EeV , (30)
and corresponds to the bound derived in [10]. Perturbativity of the correction to gauge couplings f 1 z M < f 0 , together with the requirement, M 1/2 > m leads to the new lower limit

m 3/2 > 4 z √ 3 m M P . (31)
However, this constraint is easily satisfied, once the other constraints, such as F ≤ M 2 are taken into account.

Note that at the lower gravitino mass limit (30) we obtain the bound g 2 f 1 1. Since mediation is strongly coupled, this is not really surprising. In both holographic models and general gauge mediation setups, additional states of mass, M and heavier are expected. In order to not perturb our single-field inflation framework, the masses of these states should be above the Hubble scale H during inflation3 , which implies generically M > H. This condition is satisfied by the range of gravitino masses in Fig. 3 and is saturated at the lower bound.

Finally, we comment on the partial wave unitarity limit arising from the scattering of two gluons into two gravitinos [70]. For gaugino mediation, tree-level unitarity is violated at a scale 17/(g 2 f 1 )M , which for g 2 f 1 17 is above the messenger scale M (where new degrees of freedom should appear), and therefore compatible with the constraint arising from the gravitino mass limit (30).

predominantly SM particles (the abundance of gravitinos is discussed in section III.D). Radiative corrections with MSSM fields in loops correct the inflaton potential. In low-energy supersymmetry such corrections are tiny, since they are proportional to the scale of supersymmetry breaking [68]. In our case with high-scale supersymmetry breaking, there may be large radiative corrections that can spoil flatness of the inflaton potential. Such constraints can put upper limits on the superpartner masses and therefore on the gravitino mass.

For example, a direct coupling (through the gauge kinetic function) of the inflaton, t to gauge fields, f h 1 t/M P , would induce quadratic and quartic corrections of magnitude

δm 2 ∼ h 2 1 16π 2 M 2 1/2 , ( 32
)
δλ ∼ h 4 1 16π 2 M 2 1/2 M 2 P , (33)
which both place non-trivial bounds on the coupling h 1 .

For reheating dominated decays to gauge bosons, this can be translated into a limit on the reheating temperature and eventually the gravitino abundance.

As we discuss in more detail in section III.D, reheating in this model proceeds via the gravitational coupling of the inflaton to two Higgs bosons. The coupling of the inflaton field, t to MSSM fields was derived in [45], and the relevant bosonic coupling is

L eff ReT √ 3 (n I + n L -3)W IL WLJ Φ I ΦJ , ∼ µ 2 e √ 2 3 t (|h u | 2 + |h d | 2 ) , (34)
where n I,J are modular weights4 of the superfields Φ I,J and should be taken to be equal to one for untwisted Higgs fields. The coupling for the Higgs fields is then µ 2 / √ 3M P , where µ is the MSSM Higgs mixing mass, which is now expected to be of order the scalar masses. The quadratic and quartic corrections in Eqs. ( 32)- (33) are found with the replacement h 1 → µ 2 /mM P . Requiring δm 2 m 2 and δλ 10 -14 sets a rough bound on µ/m 10 2 which we will see below is satisfied when µ is adjusted to give the correct gravitino relic density. However a more model-independent statement is that reheating sets constraints on inflaton couplings to MSSM fields. Once we fix such couplings, the requirement that quantum corrections do not spoil flatness of the inflaton potential generically set upper bounds on superpartner masses. Such bounds are dependent on the inflationary model and details of reheating, but they typically indicate that the scale of supersymmetry breaking should not be too much higher than the inflationary mass scale.

C. The Higgs mass and vacuum stability

The fact that the sfermion, gaugino and Higgsino masses are above 3 × 10 13 GeV, leads to important implications for the Higgs boson mass and vacuum stability. It is well-known that to obtain the 125 GeV Higgs mass for tan β 50, the maximum supersymmetry breaking scale is approximately 10 10 GeV. However in Ref. [33], it was noted that the supersymmetry breaking scale can be increased to the GUT scale (∼ 10 Instead, Ref. [34] considered non-degenerate superpartner masses, and showed that GUT scale masses can in fact be accommodated for much smaller values of tan β. In particular, assuming m = m QL ,3 = m tR = 10 16 GeV and tan β = 1, a scan of gaugino and first/second generation sfermion masses in the range [ m, 100 m], and Higgsino masses in the range [ m 100 , m] gives rise to the required threshold corrections of the Higgs quartic coupling.

The analysis in Ref. [34] suggests that our superpartner spectrum can give rise to similar threshold corrections needed for a 125 GeV Higgs mass. For concreteness we will consider a gaugino-mediated spectrum, and expect similar qualitative features for a spectrum generated by a more strongly-coupled mediation mechanism. Identifying the right-handed stau mass with the scale m = m τR ∼ 5× 10 13 GeV, will then determine the size of the remaining sfermion and gaugino masses. Assuming that the gaugino masses are generated at a scale m < µ 0 ≤ 10 16 GeV, the sfermion masses are then approximately given by

m 2 Q 1 16π 2 32 3 g 2 3 + 6g 2 2 + 2 15 g 2 1 M 2 1/2 log µ 0 m , (35) m 2 ũ 1 16π 2 32 3 g 2 3 + 32 15 g 2 1 M 2 1/2 log µ 0 m , (36)
m 2 d 1 16π 2 32 3 g 2 3 + 8 15 g 2 1 M 2 1/2 log µ 0 m , (37)
m 2 L 1 16π 2 6g 2 2 + 6 5 g 2 1 M 2 1/2 log µ 0 m , (38)
m 2 ẽ 1 16π 2 24 5 g 2 1 M 2 1/2 log µ 0 m . ( 39
)
Other soft parameters such as A t , Bµ and m 2 H u,d are also generated radiatively. Using this approximate spectrum we can then compute the one-loop threshold corrections as given in Ref. [34,71]. The result in shown in Figure 4. The contribution to the Higgs quartic coupling from our superpartner spectrum overlaps with the ±3σ band of the λ H coupling provided that 0.75 tan β 1.34.

However, note that the one-loop threshold correction matches at a negative value of the Higgs quartic coupling. This occurs because of our somewhat compressed spectrum, m Q/ m ∼ 1.9, M 1/2 / m ∼ 2, µ/ m ∼ 1.5, and the fact that the gaugino/Higgsinos contribute negatively, while the positive stop contribution is suppressed by mixing near tan β ∼ 1. A more detailed determination of the superpartner spectrum that gives different mass ratios may lead to a positive correction and then the matching could occur for positive values of λ H .

Nevertheless even with a positive one-loop threshold correction from our superpartner spectrum, the value of the top quark Yukawa coupling, y t (m t ) would need to be near its -3σ extreme value that is allowed by the large uncertainty in the top-quark mass measurement. Otherwise one needs to rely on other threshold effects to stabilize the Higgs potential below the scale m. For instance this could be due to the inflaton coupling to the Higgs,

y I ∼ 1 √ 3 µ 2
mMp ∼ 10 -3 . However, assuming that the supersymmetric and soft contributions to the inflaton mass are of the same order, this coupling causes a shift [73] in the Higgs quartic coupling by an amount given by δλ H y 2 I sin 2 2β ≤ y 2 I ∼ 10 -6 , which is negligibly small. Alternatively, a heavy scalar singlet with a quartic coupling to the Higgs, could be introduced that is related to the generation of the neutrino masses [74]. 5 If this correction causes the Higgs quartic to be large and positive at the SUSY scale m, then a negative one-loop threshold correction would not be a problem, since it could then be absorbed by the SUSY tree-level contribution for suf-ficiently large tan β. Nonetheless this does require an extra tuning in the model in order that this scalar singlet remains light (at an intermediate scale).

Another potential concern is that the required tan β values in Figure 4 are near one, and normally in the MSSM this would cause a Landau pole in the top quark Yukawa coupling to appear below the GUT scale. However since our sparticle spectrum is quite heavy, the top quark Yukawa coupling is reduced by a factor of two at the scale m 5 × 10 13 GeV. The matching condition for the top Yukawa coupling y t = y t sin β, where y t is the superpotential Yukawa coupling, then allows for a lower value of tan β, with a corresponding larger value of y t . Since there is relatively little running in our high-scale SUSY model above the scale m, the larger y t value can remain perturbative below the GUT scale. Furthermore, to radiatively break electroweak symmetry requires that at some scale,

dm 2 Hu dt = 0, or y t m Q ∼ g 2 M 2 .
In the gaugino-mediated model this condition occurs when tan β ∼ 0.5, which is incompatible with the range required in Figure 4 6 . However if there were a new positive contribution to the Higgs quartic coupling then it may be possible to also achieve radiative electroweak symmetry breaking. For instance, starting the running above the GUT scale 10 16 GeV would give different sfermion mass ratios to make δλ H positive and/or allow m 2 Hu to run negative before m. The details of these possibilities are beyond the scope of this paper and will be left for future work.

D. Dark Matter

One of the main motivations of our high-scale SUSY model is its ability to account for the dark matter in the form of gravitinos with masses m 3/2 0.2 EeV. Because the supersymmetric particle spectrum lies above the inflaton mass, the dominant mechanism for gravitino production becomes SM + SM → 2 gravitinos with longitudinal polarizations [9,10] or the decay of the inflaton directly to gravitinos depending on the reheating temperature.

The gravitino production rate was derived in [9]

R = n 2 σv 21.65 × T 12 F 4 , ( 40
)
where n is the number density of incoming states. This temperature dependence can be understood as follows: one uses n ∝ T 3 , and we expect the gravitino production cross section to scale as σv ∝ T 6 /F 4 . From the rate R(T ), we can determine that Γ ∼ R/n ∼ T 9 /M

assuming instantaneous decay and thermalization. Thus, thermal production of gravitinos with m 3/2 > 0.2 EeV would require T RH > 3 × 10 10 GeV.

It is known however, that the reheating process is not instantaneous, and that the temperature of the Universe during inflaton decay can exceed T RH by orders of magnitude [32,76] up to a value T max . Due to the strong temperature dependence of the gravitino production cross section, there will be significant production of gravitinos at T max , which is not fully diluted by the entropy produced in subsequent decays. The final gravitino abundance in this case (with σ ∝ T 6 ) relative to the instantaneous approximation is [77]

r 3/2 = 56 5 ln T max T RH , (42)
where

T max 0.5 m Γ T 1/4 T RH , (43)
for inflationary models of this type, where Γ T is the total inflaton decay rate.

For the inflationary model discussed above, there are many possible decay channels all of which are Planck suppressed. The decay channel given in Eq. ( 34) is ordinarily (with weak scale supersymmetry breaking) negligible as µ 2 /(mM P )

1. However, in our case, since µ > m, this is actually the dominant inflaton decay mode t → H u,d H * d,u which ultimately corresponds to a decay of t → hh where h is the SM Higgs boson. The decay rate to two Higgs bosons is [45]

Γ 2h = µ 4 384πmM 2 P sin 2 2β , (44)
where an additional factor of 1/16 has been included in writing

H 0 u = h/ √ 2
where we have included the enhancement factor r 3/2 from Eq. ( 42). The enhancement factor depends on ln µ, and for the range of µ values considered here (roughly 10 14 -10 15 GeV), r 3/2 varies very little and we take it as a constant r 3/2 = 25.

The value of µ needed to obtain the correct relic density of gravitinos is shown by the solid line in Fig. 5 using Eq. (46). It is rather amazing that independent of the supersymmetric particle spectrum discussed above, the value of µ needed for the correct abundance of gravitinos is in the range of roughly 3-30 times the inflaton mass. This is exactly where one might expect the Higgsino mass to lie given our spectrum of heavy scalars and gauginos.

It is also possible that µ takes values below the solid line in Fig. 5. In that case, the abundance of gravitinos is below the needed relic density of dark matter (by the same token, values of µ above the solid line are excluded as they yield a relic density in excess of the observed one). Nevertheless, it is still possible to recover the correct relic density through inflaton decay to gravitinos. The gravitino abundance produced by inflaton decay for a given branching fraction to gravitinos, B 3/2 = Γ 3/2 /Γ 2h , was computed in [10] Ω decay 3/2 h 2 = 0.11

B 3/2 1.3 × 10 -13 y I 2.9 × 10 -5 (47)
× m 3/2 0.1 EeV 3 × 10 13 GeV m 1/2
.

in [78] are not effective in this model. The adiabatic condition ṁh /m 2 h > 1 is at best satisfied during the first inflaton oscillation, where m h is the t-dependent Higgs mass. One can easily check that ṁh /m 2 h ∼ Am/µM P , where A M P is the amplitude of inflaton oscillations and we require m < µ. The decay of the inflaton to two gravitinos was computed in [45] with

Γ 3/2 = Λ z M P 4 3m 2 3/2 m 256πM 2 P , (48)
so that

B 3/2 = 9 2 Λ z M P 4 m 3/2 m 2 m µ 4 . (49)
Using Eq. ( 48), inputting B 3/2 and y I , we obtain the correct relic density of gravitinos along the sloped dashed and dotted lines for different values of Λ z /M P as labelled.

As one can see from Fig. 5, the value of the abundance of gravitinos from the decay of the inflaton is strongly dependent on the value of Λ z as this scale controls the branching ratio B 3/2 . As a consequence, we can derive an upper limit to Λ z

Λ z M P ≤ 2.4 × 10 -4 m m 3/2 9/14 , (50)
which is shown in Fig. 3 by the negatively sloped pink dot-dashed line. This upper limit in turn imposes an upper limit to the gravitino mass m 3/2 0.1m.

IV. CONCLUSIONS

It may be that supersymmetry is not physically realized at energy scales accessible to the LHC. The hierarchy problem and naturalness biased our expectations that the supersymmetry mass scale was at or near the weak scale making experimental discovery all but inevitable. Such is not (yet) the case, and the mass scale of the supersymmetric spectrum remains unknown. It is therefore plausible to consider the possibility that nearly the entire supersymmetric spectrum lies at very high energies. If it is above the inflationary scale it is quite possible that the rich spectrum of supersymmetric partners were never produced in the early universe after inflationary reheating.

The exception could be the gravitino whose mass may remain below the inflaton mass. In this case, gravitinos could be produced (in pairs) during reheating [9,10] and because of the strong sensitivity to temperature, have enhanced production at the start of reheating when the temperature of the radiation plasma is above the reheating scale [77]. For a sufficiently high reheating temperature (T RH 10 10 GeV), thermally produced gravitinos would have the correct relic density to account for the observed cold dark matter in the Universe.

In this paper, we have constructed a working model incorporating both inflation and supersymmetry breaking which leads to the heavy supersymmetric spectrum with gravitino dark matter. Our starting point is no-scale supergravity [36,37]. In the family of models formulated in Ref. [67], the inflaton is associated with the volume modulus, T , and the scalar potential is identical to that derived in the Starobinsky model [12]. Supersymmetry breaking is achieved by adding a strongly stabilized Polonyi field which preserves the potential for inflation. Reheating occurs through the gravitational coupling of the inflaton to the Standard Model Higgs scalars. Because supersymmetry breaking occurs at a high scale, the µ parameter is large (larger than the inflaton mass) and the dominant decay channel is to two Higgs bosons. Depending on the value of µ, gravitinos may be produced through reheating with the correct relic density over a wide range of gravitino masses. For smaller values of µ (but still larger than the inflaton mass), the correct relic density may be obtained via the decays of the inflaton directly to gravitinos. All constraints are satisfied for gravitino masses in the range 0.1 EeV m 3/2 1000 EeV.

While we have shown that the Higgs mass in this class of models can be compatible with the experimental value when tan β ≈ 1, we leave open for future study the questions of vacuum stability, and radiative electroweak symmetry breaking -two attractive features normally associated with supersymmetric models. Both stability and symmetry breaking can be achieved without supersymmetry via an intermediate scale such as in SO (10) grand unification [79], and therefore suggests that in models of high scale supersymmetry breaking, grand unification plays a crucial role in determining the Higgs mass.

Finally, the minimal setup of our model predicts no signatures in either collider or direct/indirect dark matter searches. Instead, scalars with masses near the Hubble scale during inflation could lead to non-gaussianities in the CMB that may eventually be observed. Alternatively, by introducing R-parity violation in the lepton-Higgs sector, the gravitino can become unstable with a suitably long-lived decay. The detection of the decay products, such as very high energy neutrinos or photons at HAWC or the Pierre Auger Observatory, would then be a possible sign of high scale supersymmetry with an EeV gravitino, and thus these types of signatures are worthy of further study.

V/m 2 xFIG. 1 .√ 12 and b 1/ √ 3 ,

 21123 FIG. 1.Projections of the effective inflationary potential for the model(7) with the Polonyi sector ((13) and (14)), for p = 3. Here T = 1/2, z Λ 2 z / √ 12 and b 1/ √ 3, and we use the nominal value Λz = 10 -2 . Shown is the potential for different choices of m2 /m = 10 -8 , 10 -7 , 2 × 10 -7 , 5 × 10 -7 (in Planck units) in black, blue, green, and red.

FIG. 2 .

 2 FIG. 2.Projections of the effective inflationary potential for the model(9) with the Polonyi sector ((13) and (17)), with p = 0. We use the nominal values Λz = 10 -3 with m2 /m = 0.9. The values z , b, and φ are given approximately by(18) (shown by the dashed curve), and by(20) (shown by the solid curve).

  FIG.3. Bounds on the stabilization parameter Λz as a function of the gravitino mass. The blue dashed line shows the upper limit on Λz from the requirement that the mass of the Polonyi field lies above the inflaton mass. The pink dot-dashed line (negatively sloped) is an upper limit on Λz derived from an upper limit on the branching ratio of inflaton decays to gravitinos. The green dotted line shows the lower limit on Λz, assuming F < Λ 2 z . The shaded region is allowed by all constraints. Note, however, that there are lower bounds on the gravitino mass given in Eqs. (27) and(30).

FIG. 4 .

 4 FIG.4. The Higgs quartic coupling, λH at the matching scale m = 5 × 10 13 GeV, as a function of tan β. The solid blue line is the difference between the SM and tree-level SUSY values, with the dashed lines indicating the ±3σ contours due to the error in the top-quark Yukawa coupling yt(mt), where mt = 173.1 ± 0.7 GeV[72]. The green line shows the one-loop threshold correction arising from our superpartner spectrum.

5 FIG. 5 .

 55 FIG.5. The value of µ relative to the inflaton mass needed to obtain the correct relic density of gravitinos thermally through reheating (solid line) as a function of the gravitino mass. Also shown (dashed lines) are the values of µ needed to obtain the correct relic density of gravitinos through inflaton decays for a given value of log Λz/MP as labelled.

  sin α, and H 0 d = h/ √ 2 cos α, and noting that α = β in the high scale SUSY limit 7 .

	If we define an effective Yukawa-like coupling, y I = µ 2 /(4 √ 3mM P ), such that Γ 2h = y 2 I 8π m, we can express the reheating temperature in terms of y I [10, 30, 32]
	T RH =	10 g s	1/4	2Γ 2h M P π c	1/2	= 0.5	y I 2π	(m M P )	1/2 ,
										(45)
	where g s is the effective number of light degrees of free-
	dom, in this case set by the Standard Model, g s = 427/4
	and c ≈ 1.2 is a constant. We can then re-express the
	relic abundance (41) as
	Ω 3/2 h 2 0.11 r 3/2	m 3/2 0.1 EeV	3		3 × 10 13 GeV m	7/2
				×	y I 2.9 × 10 -5	7	,
		= 0.11 r 3/2	m 3/2 0.1 EeV	3		m 3 × 10 13 GeV	7/2
				×	µ 1.2 × 10 14 GeV

14

,

For alternative approaches to inflation in high scale supersymmetric models see[52,53].

It is also possible that all additional scalars obtain Hubble scale masses during inflation, therefore avoiding this condition.

Note that the definition of modular weights in our paper is opposite in sign with respect to the standard convention.

In[75], an additional SU(2) L triplet was introduced to insure stability and aid in keeping the Higgs mass at 125 GeV.

In addition, requiring that there is no color-breaking minimum deeper than the electroweak minimum[71], leads to tan β 0.6 for our spectrum.

We note that parametric resonance effects such as those studied

ACKNOWLEDGMENTS

We would like to thank Sebastian Ellis and Marco Peloso for helpful discussions. This work was supported by the France-US PICS no. 06482. YM.M acknowledges partial support from the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie: RISE InvisiblesPlus (grant agreement No 690575), the ITN Elusives (grant agreement No 674896), and the ERC advanced grants Higgs@LHC. E.D. acknowledges partial support from the ANR Black-dS-String. The work of T.G. and K.A.O. was supported in part by the DOE grant DE-SC0011842 at the University of Minnesota.


Emilian Dudas
email: emilian.dudas@cpht.polytechnique.fr

Yann Mambrini
email: yann.mambrini@th.u-psud.fr

Keith A Olive
email: olive@physics.umn.edu

The Case for an EeV Gravitino

We consider the possibility that supersymmetry is broken above the inflationary mass scale and that the only "low" energy remnant of supersymmetry is the gravitino with mass of order the EeV scale. The gravitino in this class of models becomes a candidate for the dark matter of the Universe. To avoid the over-production of gravitinos from the decays of the next-to-lightest supersymmetric particle we argue that the supersymmetric spectrum must lie above the inflationary mass scale (MSUSY > 10 -5 MP ∼ 10 13 GeV). Since m 3/2 M 2 SUSY /MP, we expect m 3/2 0.2 EeV. Cosmological constraints then predict a relatively large reheating temperature between 10 10 and 10 12 GeV.

Introduction.-To date, there is no significant experimental signal for weak scale (TeV) supersymmetry at the LHC [1]. In parallel, direct detection experiments such as XENON100 [2], LUX [3] or PandaX [START_REF] Fu | PandaX-II Collaboration[END_REF] have set strong limits on the elastic scattering cross section of neutralinos on nucleons exceeding common pre-run I LHC predictions [5,6]. This may indicate one of the following: 1) low energy supersymmetry is still around the corner waiting to be discovered at a slightly higher energy scale [6,7]; 2) part of the supersymmetric spectrum lies at very high energy as in split supersymmetry [8]; 3) essentially the entire supersymmetric spectrum lies at very high energy as in supersplit supersymmetry [9] (aka the Standard Model). Here, we consider the possibility that the only remnant of supersymmetry surviving down to energies significantly below the Planck scale 1 is the gravitino.

The gravitino may either be an excellent dark matter candidate [10][11][12][13][14][15][16][17][START_REF] Bailly | [END_REF][19] or a severe cosmological problem [START_REF] Fayet | Talk at the XVIIth Rencontre de Moriond[END_REF]21]. If the gravitino is the lightest supersymmetric particle (LSP) and therefore a dark matter candidate, there is the risk of overproduction from the decay of the next-to-lightest supersymmetric particle (NLSP) [21][22][23].

In fact, as we discuss below, a combination of limits from big bang nucleosynthesis (BBN) [24] and the relic density allows us to place an upper limit of 4 TeV on the gravitino mass. However, if the sparticle spectrum lies above the inflationary mass scale, and none of the superpartners are ever produced after inflationary reheating, we derive a lower limit on the gravitino mass of 0.2 EeV and the gravitino may once again become a dark matter candidate. Note that such a spectrum implies that supersymmetry is nonlinearly realized [START_REF] Volkov | For applications of nonlinear supersymmetry to inflation[END_REF]. Finally the thermal production of gravitinos allows us to bound reheat temperature after inflation between a few ×10 10 and 10 12 GeV.

The letter is organized as follows. We first discuss limits on the gravitino mass in typical supersymmetric models. We discuss both the limits from BBN and from NLSP decay. We then consider a high scale supersymmetric model where only the gravitino lies below the inflationary mass scale. We derive a new lower limit to the gravitino mass in this case. Assuming that the gravitino is the dark matter, we consider general consequences for inflationary models, particularly aspects of reheating.

Upper limits to the gravitino mass in typical SUSY scenarios.-The physics behind the limits on the gravitino mass can be very different depending on the specific mass range under consideration. With the exception of the cases of light (MeV, keV, or sub-keV) masses, typical gravitino masses discussed in the literature are in the 10-1000 GeV range similar to the masses expected for MSSM superpartners if the SUSY scale is related to the hierarchy problem. However, it is well known that a gravitino with O(100) GeV mass is potentially problematic [START_REF] Fayet | Talk at the XVIIth Rencontre de Moriond[END_REF]21]. On the one hand, if it is not the LSP, it will decay to lighter sparticles, and if it is the LSP, the NLSP would decay to the gravitino. In either case, the lifetime may easily fall within the range of 100 -10 8 s and be subject to constraints from BBN [24,[26][27][28][29][30][31]. For example, the decay rate of a neutralino NLSP to a gravitino and photon is given by [14,15,31]

Γ decay C 2 16π m 5 χ m 2 3/2 M 2 P ( 1
)
where C depends on the neutralino diagonalization matrix and we have ignored phase space factors (and other factors of O(1)). In the case of a gravitino LSP, there are typically strong constraints on the SUSY parameter space forcing one into regions where the NLSP is the tau slepton [28,29].

The BBN constraints begin to be relaxed when the lifetime of the NLSP becomes less than O(100) s [26,27], and for a neutralino NLSP, we can use Eq.( 1) to obtain a relation between the neutralino and gravitino masses,

τ χ 100 s. ⇒ m χ > 300 GeV m 3/2 GeV 2/5 (2)
for C ∼ 1. Thus avoiding the limits from BBN will require a rather heavy SUSY spectrum for TeV scale (and above) gravitino masses. We note that the relaxation of the BBN bound at 100 s requires satisfying the upper bound on the density of decaying particles of roughly [26],

m χ n χ /n γ 7 × 10 -9
GeV. If we exceed this density, we must use the more strict BBN bound of τ χ 0.1s. In this case, the lower limit on m χ in Eq.( 2) is increased by a factor of ∼ 4.

In addition to the BBN constraints, there is an additional constraint coming from the relic density of the NLSP whose decay contributes to the relic density of gravitinos [21][22][23]. The gravitino relic density from NLSP decays can be written simply as

Ω 3/2 h 2 = m 3/2 m χ Ω χ h 2 (3)
and thus the NLSP relic density is limited by

Ω χ h 2 0.12 m χ m 3/2 (4)
where 0.12 is the approximate upper limit on the cold dark matter density from PLANCK experiment [32]. As long as m χ is not much greater than m 3/2 , the NLSP density is constrained to be near the cold dark matter density. Even in the event that m χ m 3/2 , the relic density of the NSLP is still constrained by the BBN unless its lifetime is very short (< 0.1 s) as noted above.

Thus as we attempt to increase the mass of a gravitino LSP, we are forced to higher NLSP masses to insure both a relatively short lifetime and low relic density. For example, for m 3/2 = 2 TeV, we must require m χ 6 TeV (20 TeV) to obtain τ χ < 100 s (< 0.1s). Generally, it is very difficult to obtain an acceptable neutralino relic density when the neutralino masses surpass the TeV scale [6,7]. In particular, the neutralino relic density in the TeV regime must be regulated by either some strong resonant process or co-annihilation. Indeed, the strongest such process involves the co-annihilation with the gluino [33][34][35][36]. Pushing the mass scales to their limit (when the neutralino and gluino masses are degenerate), an upper limit to the neutralino mass of roughly 8 TeV was found [34][35][36]. This translates (using Eq. 2) to an upper bound on the gravitino mass of roughly m 3/2 < 4 TeV.

High scale SUSY breaking and Inflation -EeV scale gravitinos-High scale SUSY-In order to go beyond the derived upper limit on the gravitino mass of 4 TeV, we must make a more substantial departure from the common paradigm of weak scale supersymmetry. To this end, we consider the possibility for a higher gravitino masses along with a very high SUSY breaking scale, leaving only the gravitino surviving at low energies as a dark matter candidate.

As we demonstrated in the previous section, a gravitino mass in excess of 4 TeV, would require a SUSY spectrum in excess of 8 TeV in order to obtain NLSP lifetimes short enough to be compatible with constraints from BBN. However, even in the limit of degenerate neutralinos and gluinos, strong co-annihilations are insufficient to lower the NLSP relic density to acceptable levels. Further increasing the SUSY mass scale, weakens the interaction strengths, lowering the annihilation (and co-annihilation) cross sections, leading to an overabundance. Without resorting to some unknown form of dilution, one possibility for larger gravitino masses is to move the SUSY matter spectrum to such high scales, so that SUSY particles were never part of the thermal bath after inflation.

To completely remove the supersymmetric particle spectrum from the thermal history, we must assume that the SUSY mass spectrum is larger than both the inflationary reheating temperature, T R , and the inflaton mass, m φ , so as to prevent SUSY particles from being produced by either thermal processes during reheating or by the decay of the inflaton. Here, we will not tie ourselves to a particular inflationary model, but note that in many models considered, the inflaton mass is set by amplitude of density perturbations seen in the microwave background, and yields a value of roughly 3 × 10 13 GeV. When we need to refer to a specific example, we consider a no-scale supergravity model of inflation [37] which leads to Starobinsky-like inflation [38].

If we denote as F the order parameter for supersymmetry breaking, then typical soft SUSY masses will be proportional to F ,

M SU SY = F Λ mess (5)
where Λ mess is the mass scale associated with the mediators of supersymmetry breaking2 . We expect Λ mess ≥ M SU SY . Thus M SU SY > m φ translates to F > m 2 φ . The gravitino mass is also determined by F [39],

m 3/2 = F √ 3M P (6)
And hence we have a lower bound on the gravitino mass given by

m 3/2 > m 2 φ √ 3M P 0.2 EeV (7)
Thus we have a gravitino mass gap between 4 TeV and 0.2 EeV which remains cosmologically problematic. As a toy model, one could consider supersymmetry breaking using a Polonyi-like field, z [44] with superpotential W = F (z + ∆) where ∆ is a constant. In this case, the goldstino is the fermionic component of z eaten by the gravitino. However, the transmission of supersymmetry breaking to the observable sector cannot be of standard gravity type. One option is (general) gauge mediation, with a high scale for messenger masses and supersymmetry breaking.

Gravitino Production-Clearly the LHC bounds can be satisfied if the sparticle mass spectrum lies above a few TeV. The direct detection limits can also be satisfied as the spectrum approaches its upper limit [7]. It is also possible that the dark matter lies beyond the MSSM and has weaker couplings to matter, e.g. through a t-channel exchange of a massive Z' or Higgs as shown in [45] or invoking a pseudoscalar or pure axial mediator to velocity suppress σ scat N [46,47]. Furthermore, if the dark matter couples too weakly with the standard model, it will never reach thermal equilibrium as its production rate is dn dt = n 2 γ σv . The particle is frozen in during the process of thermalization. The weak coupling of the dark sector with the standard model can be due to either an effectively small coupling (of the order of 10 -10 ) [48] or because the mass of the mediator between the two sectors is very large, as in the case of Non-Equilibrium Thermal Dark Matter (NETDM) models [51].

By increasing the SUSY mass scale, we have also removed most of the standard gravitino production mechanisms. Namely both NSLP decay, and the thermal production from standard model annihilations such as gluon, gluon → gluino, gravitino are no longer kinematically allowed. The rate for the latter is well known [40,41] and scales as Γ ∼ T 3 M 2 SU SY /M 2 P m 2 3/2 , where we have assumed predominantly goldstino production in the limit m 3/2 M SU SY . In this case, the gravitino abundance is approximately n 3/2 /n γ ∼ Γ/H ∼ T M 2 SU SY /M P m 2 3/2 , where we have simply taken the Hubble parameter as T 2 /M P .

In the limit that the SUSY mass scale is above the inflationary scale, there remains, however, (at least) two sources of gravitino production. Inflaton decay to gravitinos [41,42], and thermal production of two gravitinos from the thermal bath (gluon, gluon → gravitino, gravitino) [43] as this is only kinematically allowed channel. A careful computation of the gravitino production rate was derived in [43]

R = n 2 σv 21.65 × T 12 F 4 ( 8
)
where n is the number density of incoming states and we see that the rate has a strong dependence on temperature and is even stronger than the NETDM case [51] where the dependence is R(T ) ∝ T 8 . This dependence can be easily ascertained on dimensional grounds. Recall that n ∝ T 3 , and for gravitino production, we expect σv ∝ T 6 /F 4 . The consequences of such a high temperature dependence are important: we expect that all gravitino production will occur early and rapidly in the reheating process. This differs from the feably coupled case [48] where the smallness of the dark matter coupling to the standard model bath renders the production rate slower.

From the rate R(T ), we can determine that Γ ∼ R/n ∼ T 9 /M

In the absence of direct inflaton decays, a gravitino at the lower mass limit (7) would require a reheating temperature of roughly 3 × 10 10 GeV, above the upper limit allowed by the relic abundance constraint (T R 10 7 GeV) in the more common thermal scenario [40], thus favoring thermal leptogenesis [49,50].

Consequences for inflationary models-The reheating temperature appearing in Eq.( 9) is generated by the decay of an inflaton field φ of mass m φ and width Γ φ . We assume that the decay and thermalization occur instantaneously at the time t φ , Γ φ t φ = 2Γ φ /3H = c, where c ≈ 1.2 is a constant. In this case, the reheating temperature is given by [41,52]

T RH = 10 g s 1/4 2Γ φ M P π c 1/2 = 0.55 y φ 2π m φ M P c 1/2
(10) where we have defined a standard "yukawa"-like coupling y φ of the inflaton field to the thermal bath, Γ φ = y 2 φ 8π m φ and g s is the effective number of light degrees of freedom in this case set by the Standard Model, g s = 427/4. We can then re-express the relic abundance (9) as function of y φ :

Ω 3/2 h 2 0.11 0.1 EeV m 3/2 3 m φ 3 × 10 13 GeV 7/2 y φ 2.9 × 10 -5 7 (11)
where we have set c = 1.2. The cosmological constraint is plotted in Fig. (1) in the (m 3/2 , y φ ) plane, where we show the region allowed by PLANCK [32]. The black (solid) line represents the PLANCK constraint Ωh 2 = 0.11. One immediately sees the linear increase in the Yukawa coupling y φ with increasing gravitino mass in order to counterbalance the weakening of the effective coupling 1/F responsible for its production in the thermal bath.

A large inflaton-matter coupling produces a high reheating temperature, which in turn increases the gravitino abundance. Then, as one can see from Eq.( 11), the solid curve in Fig. 1 is an upper bound on y φ to avoid an overabundant gravitino. In fact, one can extract an

m 3/2 [EeV] y Φ � � /� = � � � / � = � � -� � � � / � = � � -� � � � / � = � � - � � ����������� ������� � �/� > � �� FIG. 1.
Region of the parameter space allowed by PLANCK constraints [32] in the plane (m 3/2 , y φ ) for different values of the branching ratio B 3/2 and m φ = 3 × 10 13 GeV (see the text for upper bound on y φ independent of m 3/2 simply requiring m 3/2 < T RH , a necessary condition for the gravitino to be thermally produced. The condition m 3/2 < T RH implemented in Eq.( 11) with the expression (10) gives

y φ 1.6 × 10 -3 3 × 10 13 GeV m φ 1/2 , (12)
shown as the horizontal dashed line in the Figure 1. We can then extract the maximum reheating temperature T RH 1.1 × 10 12 GeV. Combined with the condition (7) m 3/2 > 0.2 EeV, the relic abundance constraint (9) gives 2.7 × 10 10 GeV T RH 1.1 × 10 12 GeV (13) which is a strong prediction of our model.

Gravitino production by inflaton decay-It is also possible to produce gravitinos through the direct decay of the inflaton. For example, in no-scale supergravity models of inflation, the decay of the inflaton to gravitinos is highly suppressed. In simple models, there is no coupling at the tree-level [53]. However, it is possible to couple the inflaton to moduli without spoiling the inflationary potential [41,42]. We can parameterize the decay to a pair of gravitinos as Γ 3/2 = m φ y 2 3/2 72π . The branching ratio of decays to gravitinos is then

B 3/2 = Γ 3/2 /Γ φ = |y 3/2 | 2 9y 2 φ . (14)
Using the result from [41] for the gravitino abundance produced by inflaton decay at the epoch of reheating, we get

n 3/2 n γ ≈ 3.6B 3/2 (Γ φ M P ) 1/2 m φ ≈ 0.7B 3/2 y φ M P m φ 1/2 (15)
corresponding to Ω decay 3/2 h 2 = 0.11

B 3/2 1.3 × 10 -13 y φ 2.9 × 10 -5 (16) × m 3/2 0.1 EeV 3 × 10 13 GeV m φ 1/2 .
today.

The condition ( 7) is then translated into

B 3/2 y φ = |y 3/2 | 2 9|y φ | 1.9 × 10 -18 0.1 EeV m 3/2 (17)
for m φ = 3 × 10 13 GeV. Contrary to the case of thermal gravitino production, our limit to the coupling y φ is strengthened as m 3/2 is increased when gravitino production occurs through inflaton decay. Since the density through the decay of the inflaton is proportional to n φ B 3/2 m 3/2 , where m φ n φ is the inflaton energy density, the limit on the coupling is improved when either the branching ratio or the gravitino mass is increased.

This result is also shown in Fig. (1) where we clearly see the changing in the slope for larger value of B 3/2 > 10 -19 where the direct production from inflaton decay may dominate over the thermal production. We note that the constraints obtained on the inflaton coupling to gravitinos are strong. We recall, however, that in no-scale models of inflation [41,42,53] and in classes of inflationary models with so-called stabilized field [54,55], this coupling is naturally very small. Finally, we point out that in the case of the direct production of the gravitino through inflaton decay, both the ±3/2 and the ±1/2 components of the gravitino populate the Universe, whereas in the case of thermal production (Eq.9) only the longitudinal goldstino component contributes to the relic abundance.

Perspectives and Conclusions-In many ways, it seems quite natural that a particle with only gravitational interactions should make up the dark matter of the Universe. We have seen that in the generic context with gravitino dark matter where the supersymmetric particle spectrum thermalizes with standard model bath, an upper limit to the mass of the gravitino of 4 TeV is obtained. However, if one makes the minimal hypothesis that the supersymmetric spectrum lies above the inflaton mass, a new cosmologically allowed window opens for gravitino mass above 0.2 EeV. Indeed, despite the weakness of its coupling, the gravitino can be produced directly from the thermal bath by the exchange of virtual heavy superpartners (or equivalently by higher dimensional operators). It can also be produced directly from the inflaton decay. In order to obtain gravitino dark matter from the thermal bath, we predict a relatively large reheating temperature 10 10 GeV, compatible with the thermal leptogenesis scenario. If stable, this gravitino is virtually undetectable as it is the only R-parity odd state ever present in the Universe after inflation. If unstable through an R-parity violating coupling, the decay of the gravitino would produce EeV-like monochromatic photons or neutrinos, which are not yet observable by present experiments.

10 - 1

 1 10 0 10 1 10 2 10 3 10 4 10 5

These messengers could in principle also play a role in restoring unification at high scale.

Acknowledgements.

The authors acknowledge Karim Benakli and Yifan Chen for useful discussions. This work was supported by the France-US PICS no. 06482. Y. Mambrini acknowledges partial support from the European Union's Horizon 2020 research and innovation programme (under the Marie Sklodowska-Curie grant agreements No 690575 and No 674896), and the ERC advanced grants Higgs@LHC. E.D. acknowledges partial support from the ANR Black-dS-String. The work of K.A.O. was supported in part by DOE grant DE-SC0011842 at the University of Minnesota.