1 Introduction

# Making the Sneutrino a Higgs with a $U(1)_R$ Lepton Number

Making the Sneutrino a Higgs with a Lepton Number

Claudia Frugiuele and Thomas Grégoire

Ottawa-Carleton Institue for Physics , Department of Physics, Carleton University,

1125 Colonel By Drive, Ottawa, Canada, K1S 5B6

## 1 Introduction

Supersymmetry (SUSY) at the weak scale remains one of the favorite paradigm for physics at the terascale. In the minimal supersymmetric version of the Standard Model (MSSM) and many extensions thereof, the weak scale is protected against large quadratically divergent radiative corrections, there exist a natural dark matter candidate and gauge couplings unify at a high scale. Unfortunately the fact that LEP, Tevatron and now the first data from LHC did not find any superpartners or the Higgs makes the realization of this scenario difficult without a fair amount of fine-tuning. This motivates the exploration of a larger portion of the weak scale supersymmetry landscape. For example, one can consider models where the gaugino soft masses are Dirac instead of Majorana [1, 2, 3]. This requires the introduction of new superfields and a different couplings of the susy breaking sector to the Standard Model gauge sector. These soft Dirac gaugino masses do not contribute to the running of scalar soft masses, and are therefore dubbed ’supersoft’ [3]. This could help create a small hierarchy between gaugino and scalar masses which might be an interesting starting point in order to improve on the fine tuning issues of the MSSM. It also allow the possibility of writing models that are invariant under a full instead of the usual -parity. The flavour constraints on such models are relaxed and supersymmetry breaking can be transmitted to the visible sector through gravitational interactions [4] . Because symmetry forbid a term, the Higgs sector of these models need to be different than the MSSM. One option [5] is to enlarge the field content and include two new doublets and . The other option [6] is to give masses to the down-type quarks and to the leptons through a SUSY-breaking term .

In this work we examine the possibility of instead giving masses to the down-type quarks and to the leptons through the vev of a sneutrino (the idea of giving the sneutrino a vev has a long history, see [7] for examples). In the MSSM, the lepton doublet has the same quantum number as the down type Higgs, it can therefore serve this purpose. However, in the MSSM such a vev is very strongly constrained, mainly due to the fact that it breaks lepton number and induce neutrino masses that are too large. In models with a symmetry however, the can be identified with a lepton number (see [8] for an early implementation of this idea), and the sneutrino can acquire a relatively large vev [9]. The goal of this paper is to explore the main features of such a scenario. In section 2 we present the particle content of the model and the Lagrangian. Because one of the lepton number is a symmetry, the gauginos carry a lepton number and mix the corresponding lepton and neutrino. Constraints on such mixing from electroweak precision measurement are presented in section 3. In the same section we present constraints that arise from gravitino decay, and also from the generation of neutrino masses through unavoidable -symmetry breaking. In section 5 we discuss possibilities for mediating SUSY breaking in such a model and the related problem. Finally in section 6 we discuss the main features of the collider phenomenology.

## 2 The model

### 2.1 Particle content and Lagrangian

The particle content of our model consists of the usual particle content of the MSSM to which we add an adjoint chiral superfield for each SM gauge group . This is necessary to give Dirac mass to the gauginos and is the minimal particle content needed to accommodate a symmetry in a supersymmetric extension of the Standard Model. In fact, this particle content is more minimal than the minimal R-symmetric supersymmetric extension of the Standard Model (MRSSM) presented in [5], as the latter includes two additional weak doublets in order to give mass to the gauginos as the standard term is forbidden by -symmetry. We therefore refer to our model as the MMRSSM. Table 1 shows the MMRSSM superfields and their quantum numbers; the charge assignments is chosen such that we can use the -symmetry as the lepton number of type , where or . Indeed all the Standard Model particles, except the charged lepton and the neutrino carry -charge zero. The situation with the SUSY partners is reversed: the charged slepton and the sneutrino of flavour do not carry any lepton number while all other have lepton number. This means in particular that a sneutrino vev does not break the lepton number, and this is crucial for making the sneutrino the down type Higgs. The squarks are leptoquarks because they carry both the baryon number and the lepton number As we will show in a following section, this feature characterizes and distinguishes the phenomenology of the model. Moreover, the higgsinos, the wino, the bino together with their adjoint partners carry -charge this means they can mix with the ordinary leptons of flavor In the MMRSSM the lightest chargino and the lightest neutralino coincide with the charged lepton and the neutrino

The up-type Higgs has charge 0, and it acquires a vev. Instead, is an inert doublet1, which is introduced to cancel the anomalies, and to give mass to the higgsinos. It is the sneutrino of flavor that acquires a vev and gives mass to the down-type fermions.

With this particle content the MMRSSM superpotential is then:

 W= yuUcQHu−ydDcQLa−ybEcbLbLa−ycEccLcLa+μHuRd. (1)

where and are matrices in family space, while and

As usual, the up-type fermions acquire mass through while the down type Yukawa couplings involve the leptonic superfield which then plays the role of the down-type Higgs. However, it is important to note that the superpotential in equation (1) does not contain the Yukawa coupling for the lepton of flavor as the term is null, while the term is forbidden by the -symmetry. Therefore, this coupling needs to be generated in the SUSY breaking sector as we will discuss in a following section. The down-type Yukawa couplings of equation (1) violate the conventional parity as well as the standard lepton number. Indeed, here these couplings correspond to the trilinear violating coupling and often discussed in the literature [10]. These couplings have very stringent bounds in conventional -parity breaking models that come from the Majorana neutrino masses they induce. In our model however, there is a conserved lepton number which forbids such masses. In fact in the limit of massless neutrinos, we impose three separate lepton numbers, one for each flavour: which is the -symmetry as well as and which are not symmetries. As a consequence, the bounds on those coupling are in the MMRSSM much less stringent than in conventional -parity violating models, and come mainly from electroweak precision measurements. This, as we will see, has interesting phenomenological consequences.

The inert doublet does not interact with the SM fermions as the trilinear couplings and are forbidden by the -symmetry. As we have already commented, is necessary to give mass to the higgsinos. Indeed, a bilinear term is forbidden by the -symmetry, and the higgsinos acquire mass through the -symmetric term

Finally, the soft supersymmetry breaking terms allowed by both gauge symmetries and by the -symmetry are:

 Lsoft =Lfmass+Lsmass−Bμ(Hu~la+cc), (2)

where the gaugino masses are given by:

 Lfmass=M~Bλ~Bψ~B+M~Wλ~Wψ~W+M~gλ~gψ~g, (3)

and the soft scalar masses by:

 Lsmass =m2~q~q†~q+m2~l~l†~l+m2~uc~uc†~uc+m2~dc~dc†~dc (4) +m2e~ec†~ec+m2HuH†uHu+m2RdR†dRd+m2Φ~BΦ†~BΦ~B+ +m2Φ~WΦ†~WΦ~W+m2Φ~gΦ†~gΦ~g+M2Φ~B(Φ2~B+cc)+M2Φ~W(Φ2~W+cc)+M2Φ~g(Φ2~g+cc).

We notice that equation (2) contains a -term that mixes the sneutrino with but not a mixing term for This ensures that will not get a vev as long it does not acquire a negative mass while the sneutrino will. Moreover, we note that the soft SUSY lagrangian of equation (2) does not contain scalar trilinear coupling nor Majorana mass terms for the gauginos.

As we have observed in the introduction, -symmetric models can be generated through the supersoft SUSY breaking mechanism [3]. In this scenario, supersymmetry breaking is parametrized by a non-dynamical vector superfield with a non-zero D-term. The Dirac masses for the gauginos in equation (3) are then generated by the following operator:

 ∫d2θciMW′αWαiΦi, (5)

where is the field strength superfield of the SUSY breaking spurion: . This is known as a supersoft operator as the Dirac gaugino masses it produces will not lead to log-divergent susy breaking scalar masses. In principle it allows that gauginos to be parametrically heavier than the sfermions. The adjoint scalars can also get a mass from an operator involving :

 ∫d2θW′αW′αM2Φ2i. (6)

This operator gives rise to the term proportional to in equation (4). It gives a positive mass square to the real part of , but a negative mass square which is potentially dangerous to the imaginary part of . In theories of R-symmetric gauge mediation which we will consider for this model one can also generate an operator of the form [11, 12]:

 ∫d4θW′αDαV′Φ†iΦiM2 (7)

which gives a common mass squared to the real and imaginary parts of the adjoint scalar. With appropriate choice of couplings for the messengers [13] it is possible in R-symmetric gauge mediation to avoid having tachyonic adjoint scalars. Moreover, they are the heaviest particle of the spectrum with their masses parametrically the square root of a loop factor above the gauginos masses. The sfermions are the lighest superpartners with soft masses squared that come from the following finite one loop contribution [3, 13]:

 m2s=3∑b=1CbsαbM2bπlogM2Φ2RbM2b, (8)

where is the quadratic Casimir of the scalar under the group which is equal to for and for

Finally, as we have already anticipated, the SUSY breaking lagrangian should contain the Yukawa coupling . This term needs to come from the mechanism of SUSY breaking mediation and we will discuss it’s origin in section 5

### 2.2 Electroweak symmetry breaking

In the present section we will study how electroweak symmetry breaking is realized in our model. Such an analysis was also done for a quite general model in [14] . The part of the potential that is relevant for electroweak symmetry breaking contains only , as well as the adjoint scalars and as they can acquire a non-zero vev. All other fields do not get a vev and are set to in what follows. The potential consists of three terms:

 VEW=VD+VF+Vsoft. (9)

The first is the contribution from the and D-term and is given by:

 VD =12(√2M2~B(~ϕ~B+~ϕ†~B)+g′2(|H0u|2−|~νa|2))2+12(√2M2~W(~ϕ0~W+~ϕ0∗~W)+g2(|H0u|2−|~νa|2))2, (10)

The second contribution comes, instead, from the superpotential, and it only contains a mass term for the up-type Higgs:

 VF=μ2|H0u|2. (11)

Finally, the third contribution contains the following soft SUSY breaking terms:

 Vsoft =m2~ϕ~B~ϕ†~B~ϕ~B+m2~ϕ~W~ϕ†~W~ϕ~W+M2~ϕ~B(~ϕ2~B+cc)+M2~ϕ~W(~ϕ2~W+cc)+ (12) m2Hu|H0u|2+m2La|~ν2a|−Bμ(H0u~νa+h.c.).

The scalar potential is then:

 VEW= (μ2+m2Hu)|H0u|2+m2~νa|~νa|2−Bμ(H0u~νa+h.c.)+g′+g8(|Hu|0−|~νa|2)2+ (13) +12(m2~ϕ~B+M2~ϕ~B+4M2~B)~ϕR2~B+g′M2~B~ϕR~B(|Hu|0−|~νa|2)+gM2~W~ϕR~W(|H0u|2−|~νa|2).

with denoting the real part of .

As we have already noticed, in gauge mediation models, the adjoint scalars are the heaviest particle of the spectrum [12] and can be integrated out of the potential. This has two effects: first it lowers the Higgs quartic and second it shift the mass of the boson, creating a contribution to the parameter:

 Δρ=v2M4ΦRWg2M~Wcos(2β), (14)

where is the mass of the real part of the adjoint scalar and is the ratio of the vev of the up-type Higgs and the vev of the sneutrino: . With larger than a few TeV, the above contribution to is within the experimental bound, and we can neglect the correction to the Higgs potential and minimize the following potential:

 VEW=(μ2+m2Hu)|H0u|2+m2~νa|~νa|2−Bμ(H0u~νa+h.c.)+g2+g′28(|H0u|2−|~νa|2)2. (15)

This is exactly the scalar potential of the MSSM with except that here we do not have the contribution to the sneutrino mass, as the invariant term contains only Therefore, in order for the potential to be bounded from below the quadratic part should be positive along the flat directions:

 2Bμ<μ2+m2Hu+m2~νa. (16)

Furthermore, the condition for electroweak symmetry breaking is:

 Bμ>(μ2+m2Hu)m2~νa. (17)
 sinβ =2Bμm2Hu+μ2+m2La, (18) M2Z =|μ2+m2Hu−m2La|√1−sin2β−m2Hu−m2La−μ2. (19)

The spectrum of the Higgs sector of the model contains the usual CP odd neutral particle the two CP even and the charged Higgs. Their masses are:

 m2A0 =2bsin2β=m2Hu+μ2+m2La, (20) m2H± =m2A0+m2W, (21) m2h0,H0 =12Ê(m2A0+m2Z0∓+√(m2A0−m2Z0)2+m2A0m2Z0sin22β. (22)

This is identical to the case of the MSSM and we therefore, we inherit also the MSSM little hierarchy problem. In a -symmetric model this problem could be even more severe. The -symmetry forbids the left/right stop mixing, and this reduces the contribution of the stop radiative corrections to the SM Higgs mass. Indeed, the full one loop contribution of the stop sector to the Higgs mass is [15]:

 δm2h0 =34π2sin2βy2t[m2tln(m~t1m~t2m2t)+c2~ts2~t(m2~t2−m2~t1)ln(m2~t1m2~t2)+ (23) +c4~ts4~t((m2~t2−m2~t1)2−12(m4~t2−m4~t1)ln(m2~t1m2~t2))/m2t],

where and are the cosine and the sine of the stop mixing angle, and the mass eigenstate. From equation (2.2) we see how the absence of left/right mixing considerably reduces the radiative contribution from the stop sector forcing the mass of the stop to increase in order to make the Higgs sufficiently heavy.

On the other hand, the supersoft SUSY breaking mechanism ameliorates the fine tuning problem, because now the radiative contribution to is:

 ΔM2Hu=3ytm2~t4π2lnm~tΛ, (24)

where the cutoff scale is the mass of the real adjoint scalars, and not the messenger scale as in the typical gauge mediation scenarios.

As in the MSSM, one might wonder how to increase the Higgs quartic coupling, and reduce in this way the fine tuning. The only symmetric dimension five operator that gives a contribution to the Higgs quartic coupling is:

 ∫d2θM(HuHd)(HuLa). (25)

A possible way to generate this operator is to introduce a singlets which couples to the Higgs superfields in the following way:

 mSS¯S+k1HuHdS+k2HuLa¯S. (26)

This is a possible solution to the little hierarchy problem in our model inspired by the NMSSM. Alternatively, if we consider a very low SUSY breaking scale we might be able to increase the Higgs quartic coupling through the following operator:

 ∫d4θX†XM4(H†uHu)2. (27)

We plan to explore in more detail the fine tuning problems of the model in future work.

### 2.3 Lepton mixing

In the MMRSSM all the sparticles are leptons, except for the sneutrino and the slepton of flavour In particular, the new fermions (gauginos, adjoints, higgsinos), and the neutrino as well as the charged lepton carry charge and therefore they can all mix.

In the gauge eigenstate basis with and the chargino mass term is given by:

 LC=ΨT−MCΨ+, (28)

where:

 MC=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0M~W−gvu√20M~W00000μ0−gva√200ma\par⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

The smallest eigenvalue corresponds to the mass of the charged lepton and is give by to first order in . The left-handed component of the charged lepton mixes with the charged components of the adjoint triplet that is:

 a′−=cosϕ a−+sinϕ ψ−~W, (29)

where the mixing angles are:

 cosϕ =−√2M~W√(2M2~W+g2v2a)∼−1+g2v2aM2~W+O(v2aM2~W), (30) sinϕ =g va√(2M2~W+g2v2a)∼vaM~W+O(v2aM2~W). (31)

In the same way the neutrino corresponds to the lightest neutralino. In the gauge-eigenstates basis and the neutralinos mass term has the form:

 LN=−12(Ψ0−1)TMNΨ01+c.c., (32)

where the mass matrix is:

 MN=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝\parg′vu√2−gvu√2−μg′va√2−gva√20M~B000M~W0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (33)

Then, the physical neutrino corresponds to the following mixture:

 ν′a=cννa+c~Bψ~B+c~Wψ~W, (34)

where the mixing angle:

 cν =−1√12(gvaM~W)2+12(g′vaM~B)2+1, (35) c~B =−g′va√2M~B√12(gvaM~W)2+12(g′vaM~B)2+1, (36) c~W =gva√2M~W√12(gvaM~W)2+12(g′vaM~B)2+1, (37)

## 3 Constraints from electroweak precision measurement

In the present section we will discuss constraints on our models from electroweak precision measurements (EWPM) and we will show that the MMRSSM parameter space compatible with the EWPM is large. First, we will present bounds on the sneutrino vev coming from lepton mixing and subsequentely we will discuss the EWPM limits on the down type Yukawa couplings that then translate in upper bounds on the sneutrino vev.

As we showed in the previous section, the MMRSSM the charged lepton and the neutrino mix with the adjoint fermions as they both carry charge The mixing changes the coupling of the lepton of flavour to the vector bosons and this will lead to deviations in predictions for EWPM. It is therefore essential to check under which conditions they are compatible with observations.

The mixing of the charged lepton of flavour to the triplet leads the following modifications to its coupling to the boson:

 LNC=g2cosθW¯ψaγμ(gVaSM+δgaV−(gAaSM+δgaA)γ5)ψaZμ (38)

where is the Dirac 4-component spinors for the charged lepton of flavour , while the corrections to the Standard Model coupling can beexpressed in terms of the mixing angles of equation (31):

 δgaV=δgaA=−sin2ϕ2. (39)

We can compare these corrections to the measured values of and [16] shown in table 2. If we impose that and be within the experimental error, we obtain that a mixing smaller than is tolerated at level by EWPM when For and , the limit is Inserting eq.(31) in eq.(39) we obtain bounds on the sneutrino VEV which are shown in fig. 1. For winos at the electroweak scale the region allowed by the experimental data is a fairly high region at level. However, it is possible to enlarge the parameter space by considering heavier gauginos, for example TeV Therefore, the MMRSSM tends to favor a scenario with fairly heavy gauginos .

Since only one of the flavour mixes with the triplet, lepton universality is broken in our model. Charged current universality is verified experimentally to the level for both , and [17, 18], but we find that we do not obtain stronger bounds from this fact than those derived from the coupling. This is shown in fig 2 where we plotted, taking where:

 gτgμ=cosϕ cν+√2sinϕ cψ~W. (40)

In the MMRSSM the down-type Yukawa couplings give extra tree level contributions to electroweak observables which put constraints one those couplings, and therefore put a lower bound on the sneutrino vev. As we have already noticed in the previous section, the MMRSSM down-type Yukawa couplings have the same form as standard violating trilinear couplings. Indeed, the lepton Yukawa couplings correspond in the standard notation to the couplings, while the down type quark Yukawa couplings correspond to Therefore, these extra tree level contribution to the electroweak observables are the same as in standard violating models and we can use result from the literature on those models (see [10] for a review) to put bounds on the Yukawa couplings of our model.

The strongest bound when or comes from the tau Yukawa coupling ( or ) These operators lead to an additional contribution to the leptonic tau decays via exchange. This affects the ratio defined as:

 Rτμ=Γ(τ→μνν)Γ(τ→eνν), (41)

and leads to the following bound:

 yτ<0.07(100GeVm~τc)2. (42)

for GeV. This bound implies a lower limit for the sneutrino vev GeV both for and for We see that this would exclude the region of the parameter space with gauginos with a mass around the electroweak scale. Therefore, the MMRSSM spectrum is characterised by fairly heavy gauginos or in another words the very high region in the MMRSS is excluded by the experimental constraints on the Yukawa coupling.

When the strongest bound on the sneutrino vev comes from the bottom Yukawa coupling. The trilinear coupling leads to an additional contribution to at loop level to the partial width of the to . The comparison with experiment gives the following bound:

 |yb|<0.58(m~bR100GeV)2. (43)

Therefore, the MMRSSM parameter space for is less constrained, and in particular it contains also a very region.

In the standard violating scenario, the EWPM bounds are subleading compared to the bounds that come from the generation of Majorana mass for neutrinos. If we consider for example that is the bottom Yukawa coupling in our model, we see that the constraints on the neutrino mass require: while in our case the same coupling can be several orders of magnitude bigger: We will investigate the phenomenological consequences of this in s ection 5.

Standard violating trilinear couplings are also constrained by cosmological bounds and these constraints can be quite stringent. For example, the requirement that an existing baryon asymmetry is not erased before the electroweak transition typically implies [19] These constraints do not apply to our case, as the model preserves the baryonic number as well as lepton number. However, as we will see in the following section, the MMRSSM requires a very low re-heating temperature and would require a different baryogenesis mechanism.

## 4 R-symmetry breaking

-symmetry is not an exact symmetry because it is broken (at least) by the gravitino mass term that is necessary to cancel the cosmological constant. This breaking is then communicated to the visible sector, through anomaly mediation if nothing else. Therefore, we need to take into account the following additional anomaly-mediated, -symmetry violating soft terms [15]:

 LAM =Au~ur~qLHu−Ad~dR~qL~la−Al~la~l~eR+ (44) Mλ~Bλ~Bλ~B+Mλ~Wλ~Wλ~W+Mλ~gλ~gλ~g,

where:

 Mλi=βiαi4πm32, (45)
 Aijk=−βyijkm32, (46)

where is the gravitino mass, and indicates the SUSY breaking scale. Therefore, the gauginos are not pure Dirac fermions, but pseudo Dirac. For relatively low SUSY breaking scale these contributions will be subdominant compared to the -symmetric SUSY breaking terms in equation (2) and will not have important phenomenological consequences. One important exception is that they will generate neutrino masses that can be above the present bound. Also, the presence of a massive gravitino which in our case is unstable leads to important bound on the reheating temperature.

### 4.1 Neutrino masses

The SUSY breaking term of equation (44) also break the symmetry and will inevitably generate a Majorana mass term for the neutrino of flavour , and this will translate to a limit on the SUSY breaking scale.

At tree level the neutrino remains massless. Indeed, even after introducing the Majorana masses for the gauginos in the neutralino mass matrix of equation (33) the smallest eigenvalue is still zero. At one loop a Majorana mass term for is induced by the diagrams in fig.3. The contribution coming from the insertion of an term is given parametrically by (see [10] for the full expression):

 Mνa∼3(116π2)2(mbm~b)2y2bm32 (47)

where is an averaged sbottom mass parameter. The mass contributions in equation (47) is suppressed by the Yukawa couplings that assume their maximum values at large . For example, when 5 GeV and GeV, requiring eV leads to: MeV which implies GeV. The contribution from the diagram with a Majorana gaugino mass insertion is given parametrically in the large limit by:

 Mνa∼(116π2)m2Zm2χ0Mλtan2β (48)

where is the neutralino mass and is the Majorana gaugino mass insertion. The corresponding bound is then stronger for lower . For GeV and TeV, asking for eV leads to MeV which implies GeV.Therefore, the MMRSSM is compatible with the bounds on the neutrino masses, as long as we consider a fairly low SUSY breaking scale like a scenario of gauge mediated SUSY breaking.

Neutrino masses for the other flavour can be introduced through higher dimensional operators of the form:

 ∫d2θ(HuLb,c)(HuLb,c)Mf, (49)

where the scale is a flavor scale where the overall lepton number is broken.

### 4.2 Cosmological bounds: gravitino LSP

In the MMRSSM the gravitino is the lightest supersymmetric particle. Indeed the bounds on the neutrino mass constrain it to be lighter than MeV. In our model the gravitino is unstable and decays to a neutrino of flavor and a monochromatic photon. Therefore, it is necessary to evaluate its cosmological impact. This requires first computing the gravitino life time. The tree level contribution for the decay [20] is given by :

 Γtree(~G→γνa)∼|U~Bνa|2m33/232πM2P (50)

where is the mixing between the neutrino and the bino and is proportional to the neutrino mass. The leading contribution comes instead from a one loop diagram and is given by [21]:

 Γ(~G→γνa)=α y2b128π4m2bm32M2Pln2m~bmb=α v2a128π4m32m4bM2Plog2m~bmb. (51)

The gravitino lifetime increases with the sneutrino vev and decreases with the gravitino mass .The lifetime of a MeV is approximately s for a sneutrino vev of GeV (see figure 4) . So, the gravitino lifetime is larger than the lifetime of the universe s, and this means that it could be a dark matter candidate. However, because it is unstable, its abundance is constrained by the observed and x rays background and ray lines from the milky way.

Searches for gamma-ray lines from the galactic center [22] put a model independent bound on the mass times the lifetime of an unstable dark matter particle decaying to a monochromatic photon. For photon energy between MeV and MeV, the bound is approximately GeV s. So, for a gravitino mass consistent with the neutrino mass bound: MeV, the bound is , well above what the estimate that equation (51) indicates. This means that the gravitino cannot on its own be the dark matter. We can obtain a bound on the gravitino abundance by rescaling the bound of [22] and making use of (51) for the lifetime 2:

 Ω3/2h2<7×10−11(va1GeV)21log2(m~b/mb) (52)

which is independent of the mass and which for