Gravitino dark matter with constraints from Higgs boson mass and sneutrino decays

We investigate gravitino dark matter produced thermally at high temperatures and in decays of a long-lived sneutrino. We consider the Non-Universal Higgs Model and a generalized gauge mediation model, and in each case identify sneutrino LOSP regions of the parameter space consistent with the mass of the Higgs-like boson observed at the LHC. We apply relevant collider and cosmological bounds, including constraints from Big Bang Nucleosynthesis and from warm dark matter on large scale structures. Generally, we find allowed values of the reheating temperature TR below 109 GeV, i.e. somewhat smaller than the values needed for thermal leptogenesis, even with a conservative lower bound of 122 GeV on the Higgs mass. Requiring mass values closer to 126 GeV implies TR below 107 GeV and the gravitino mass less than 10 GeV.


Contents 1 Introduction
Of all extensions of the Standard Model of elementary particles, its supersymmetric version (Minimal Supersymmetric Standard Model, MSSM) still remains particularly well motivated (for a review, see, e.g., [1]).Several mechanisms have been proposed to describe the origin and mediation of the necessary supersymmetry breaking in the MSSM, yielding distinctive mass spectra for the supersymmetric partners of the known particles.Among these, models equipped with R-parity predict that the lightest supersymmetric particle (LSP) is stable which allows for a possibility that it constitutes dark matter (DM) in the Universe.
In the MSSM, the most popular DM particles are: the lightest neutralino, the gravitino (present in the MSSM embedded in supergravity) or the axino (in the MSSM extended with a U (1) PQ symmetry) [2].Different properties of these particles require different variants of the history of the early Universe, including the observationally determined abundance of DM.The lightest neutralino LSP is considered as perhaps the most natural choice for DM as its relic abundance from thermal freeze-out can agree with observations, Ω CDM h 2 = 0.113 ± 0.004 [3], for a O(100) GeV, although scenarios in which the lightest neutralinos are produced in decays of heavier particles previously dominating the energy density of the Universe, have also been proposed, see, e.g., [4].
The abundance of extremely weakly interacting particles (EWIMPs), like the gravitino or the axino LSP, is determined in a way markedly different from that of the lightest neutralino.They are produced in scatterings of other particles in the primordial plasma; the abundance of such thermally produced EWIMP is proportional to the reheating temperature T R .In addition to thermal production (TP), EWIMPs can also be produced in the decays of the next-to-lightest supersymmetric particles (NLSP), which are usually the lightest ordinary supersymmetric particles (LOSP).While in general nearly any MSSM particle can be the LOSP, the most natural choices, other than the lightest neutralino, are: the lighter stau (or stop), or the sneutrino.It is this last case that we will consider in this paper.
In non-thermal production (NTP) from LOSP decays, the LOSP lifetime may be long enough for energetic decay products to affect the abundances of the light elements (particularly in the case of gravitinos, while much less so for axinos).A good agreement of predictions of the standard Big Bang Nucleosynthesis (BBN) with observations sets stringent limits on such additional contributions (for reviews, see e.g.[5][6][7]).The resulting picture can be roughly summarized as follows: the constrains are weakest for light (small energy release in decay), short-lived (τ < O(100 sec)) and not too-abundant decaying particles, but above all for decays with a small hadronic branching ratio.Therefore, of all possible LOSPs, sneutrino decays are the least constrained by BBN since hadronic showers can only be produced in (strongly suppressed) 3-and 4-body sneutrino decays.It should also be noted that the present LHC data, albeit quite restrictive for the gluinos and the colored scalar partners of quarks of the first and second generation, still allow sleptons and sneutrinos with much lower masses.All this makes supersymmetric models with gravitino LSP and sneutrino LOSP quite attractive phenomenologically from the bottom-up perspective. 1n the simplest scenarios of supersymmetry breaking the sneutrino is not the LOSP and there have been just a handful of studies devoted to analyzing BBN constraints on sneutrino LOSP [9][10][11][12] in theoretically motivated scenarios, such as the Non-Universal Higgs Model (NUHM) [13][14][15] or the Generalized Gauge Mediation (GGM) model [16][17][18][19].
In the present letter we re-visit the possibility of gravitino dark matter from both TP and from NTP of sneutrino decays in light of recent discovery at the LHC of a Higgs-like boson with a mass of 126 GeV [20,21].We will show that taking this new result at face value implies a stringent upper bound on the reheating temperature T R below 10 7 GeV, and also favors a low gravitino mass region, below 10 GeV.Assuming a conservative lower bound on the Higgs boson mass of 122 GeV leads to weaker constraints on T R and allows a larger gravitino mass.
So far T R has been allowed to take the largest values for the sneutrino LOSP of all MSSM choices owing to the lowest yield at freeze-out.This had a double effect of having the weakest effect on BBN and also requiring largest T R for the TP contribution to make up for the reduced gravitino relic abundance from NTP.However, the relatively large Higgs mass implies larger SUSY breaking mass scale, thus also larger masses of gauginos whose scatterings dominate TP.This in turn boost the TP contribution (which is proportional to their square) and, as a consequence, requires lower T R .A larger SUSY breaking scale also implies larger sneutrino mass, and, as a consequence, larger yield at freeze-out and more energetic hadronic showers, which translates to more stringent BBN bounds.
In addition to analyzing the effect of the Higgs mass on the scenario, we extend and update previous analyses in several directions.First, for small and moderate tan β we identify the patterns of soft supersymmetry breaking masses at the high scale that can lead to the sneutrino NLSP.In realistic models, sneutrinos are often degenerate in mass with right-handed sleptons and/or neutralinos (in addition to degeneracy with corresponding left-handed charged sleptons).This leads to many coannihilation channels, which may affect the relic density Ω NLSP h 2 and, as a consequence, also BBN bounds, as they are sensitive to Ω NLSP h 2 .Second, we implement the BBN bounds using a state-of-the-art numerical code for solving the relevant Boltzmann equations [22].As an input parameter to that code, we perform a full computation of the hadronic branching fraction of sneutrino decays, including quark-antiquark production from on-shell and off-shell electroweak gauge bosons and gauginos.
The paper is organized as follows.In Section 2, we review the constraints in ν LOSP scenarios, presenting the assumptions leading to sufficiently light sneutrinos, together with the constraints from the Higgs boson mass measurement and BBN.In Section 3, we perform a numerical analysis of ν LOSP scenarios, discussing the impact of various constraints on the NUHM and the GGM model.We present our conclusions in Section 4.
2 Review of constraints in sneutrino LOSP scenarios

Soft supersymmetry breaking masses at the low scale
The τ -sneutrino, ντ , is the lightest of the sneutrinos due to the τ -Yukawa coupling driving its mass slightly below the sneutrinos of the other two generations, and from now on we will refer to it as simply the sneutrino.The sneutrino can become lighter than its charged slepton partner thanks to the electroweak D-term contributions to the slepton and the sneutrino masses.For moderate tan β the sneutrino mass after electroweak symmetry breaking (EWSB) reads (see e.g.[23]) whereas the mass matrix of the charged sleptons of the third generation is given by In (2.1) and (2.2) m2 L and m 2 E denote soft supersymmetry breaking mass parameters of the superpartners of the left-handed and the right-handed leptons, respectively, and A τ stands for the τ trilinear parameter, with all the parameters evaluated at the EWSB scale.The D-term contribution to the sneutrino mass, D 2 ν = − 1 2 M 2 Z , is negative, while analogous contributions to the masses of the charged sleptons, W , are positive.The sneutrino mass (2.1) is smaller than the smaller of the eigenvalues of the slepton mass matrix (2.2) if the condition is satisfied.For typical values of the parameters, it follows from the inequality (2.3) that the sneutrino is the lightest slepton if m 2 L < m 2 E and the left-right mixing in the slepton sector is not too large.For example, with µ = 1000 GeV and tan β = 10, the condition (2.3) is satisfied if the splitting between m 2 E and m 2 L is of at least about 100 GeV.It is also clear that increasing tan β while keeping the other parameters in (2.1) and (2.2) fixed decreases the mass of the lighter charged slepton of each generation, eventually closing the region of the parameters where the sneutrino is the lightest slepton.

Conditions for m 2
L < m 2 E from renormalization group running from a high scale Since sneutrino LOSP disfavors large tan β, we can, following the method outlined in [24], obtain solutions of renormalization group equations (RGEs) for the soft supersymmetry breaking parameters m 2 E and m 2 L : where by m 2 S,0 (with an additional index 0) for S = E, L, Q, U, D we denote sfermion masses at the high scale, while M 1,2 are low-scale U (1) and SU (2) gaugino soft mass parameters.The coefficients c E1 and c Li can be found by solving the 1-loop RGEs, whereas cEUi , cLQ by solving the 2-loop RGEs and identifying the leading effects; they are given in Table 1 for some representative choices for the high scale Q and the scale m S = √ m t1 m t2 at which electroweak symmetry breaking is evaluated. 2D 2 (denoted in literature also as S 0 ) is defined as ) where m 2 S,0 are the 3 × 3 sfermion mass matrices at the high scale, m 2 Hu and m 2 Hu are the soft supersymmetry breaking masses of the Higgs doublets at the high scale, and g 1 (g 1,0 ) is the U (1) Y gauge coupling at the low (high) scale.Leading corrections arising due to the 16 GeV 500 GeV 0.47 0.12 0.52 −0.0027 −0.0049 0.62 0.15 0.64 −0.0038 −0.0060 1000 GeV 0.45 0.11 0.51 −0.0026 −0.0048 0.59 0.15 0.62 −0.0037 −0.0059Table 1.Numerical values of the coefficients c E1 , c L1 , c L2 , cEU , cLQ in (2.4) for two representative choices of the high scale Q and of the EWSB mass scale m S = √ m t1 m t2 .
τ Yukawa couplings are denoted by δ 2 E,yτ and δ 2 L,yτ ; for small and moderate values of tan β they are small and their only role is to make the third generation of sleptons slightly lighter than the first two, but they can become important if the mass parameters m 2 H d at the high scale or the coefficient A τ in the trilinear coupling of staus are much larger than m 2 L and m 2 E .In the case when the colored particles are much heavier than the sleptons, as is usually the case, one should in principle include the leading two-loop contributions to the RGEs in order to obtain O(10 GeV) accuracy in mass determination.
Substituting (2.4) and (2.5) into (2.3),we see that the sneutrino can be the LOSP in two (mutually not exclusive) cases.Note first that in models with D 2 = 0 and universal gaugino masses, such as the CMSSM or the NUHM1 model (i.e. the NUHM with m Hu = m H d ), for which M 2 ≈ 2M 1 , and a high scale > 10 14 GeV, the sneutrino cannot be the LOSP, since it is always heavier than the bino.We can then firstly demand D 2 < 0, which gives m 2 L < M 2 1 , and then the sign difference in the coefficients multiplying D 2 in (2.4) and (2.5) can lead to m 2 L < m 2 E .This possibility is realized in the general NUHM and later we shall also discuss the corresponding mass spectra in some detail.The second option is to relax the gaugino mass universality.This possibility is naturally realized in GGM models, leading to the sneutrino LOSP and we shall later present some representative examples of mass spectra arising in such models, as well. 3

Higgs boson mass of 126 GeV
Recent data from the LHC [20,21] strongly suggest that the lightest Higgs boson has a mass of approximately 126 GeV.As we mentioned in the Introduction, this implies a larger supersymmetry breaking scale and implies non-trivial consequences for the possibility of having sneutrino NLSP with gravitino LSP.
At one loop, the lightest Higgs boson mass can be approximated as [26] Approximate values of the coefficients c α β in (2.8)-(2.11)for m S varying from 1 to 5 TeV and two patterns of gaugino masses at the high scale Q = 2 × 10 16 GeV: universal gaugino case and mass measurement at ∼ 126 GeV points toward large values of m S > ∼ O(1) TeV and values of X t maximizing the second term in the square bracket in (2.7), with largest values achieved for X t ∼ ± √ 6m S .(The other option of increasing m S so that the logarithmic correction in (2.7) gives the whole necessary contribution is less natural as it requires very heavy stops.)A solution of the one-loop MSSM RGEs gives [24]: At one loop the values of the numerical coefficients c α β can be expressed as functions of the gauge and top Yukawa couplings.In Table 2 we indicate typical values of these coefficients for different choices of M S and gaugino mass patterns.
For brevity, in (2.9)-(2.11)only the terms depending on the high-scale parameters m 1/2 and A t,0 are shown, as they suffice for the following argument.From (2.8)-(2.9) it is obvious that the easiest way of obtaining a large negative X t is to make the gluino rather heavy; increasing A t,0 by an equal amount is about five times less effective and may threaten to make the stops tachyonic.However, it should be kept in mind that a large m 1/2 tends to make |µ| large; it is of no particular consequence for X t , as µ enters this quantity multiplied by 1/ tan β, but a large |µ| additionally increases left-right sfermion mixing, which, as we discussed in Section 2.1 tends to make charged sleptons lighter than sneutrinos (for fixed sfermion masses at the high scale).
If the soft supersymmetry breaking mass parameters A t , m 2 Q and m 2 U are dominated by the RGE contributions from the gluino mass, one obtains which is not very close to the maximal stop mixing scenario [32], optimal for a large Higgs boson mass (the second term in the square bracket in (2.7) is ∼ 2 times smaller than its maximal value).A Higgs boson mass of 126 GeV can be then obtained either by assuming a rather large m 1/2 , or by taking a large negative A t,0 , preferably A t,0 ∼ −(1 − 3)m 1/2 [33], or else by admitting tachyonic stops at high scales [30], which we shall not pursue further here.
We are therefore led to the conclusion that a Higgs boson mass of 126 GeV puts an important constraint on the possibility of sneutrino LOSP by implying a higher scale of supersymmetry breaking.Lower bounds from direct SUSY searches are consistent with this trend but currently not yet as strong.A large Higgs boson mass favors large negative A t which is usually correlated with large A τ via RG running; this increases left-right mixing in the charged slepton sector and makes a stau lighter than the sneutrino.Such a large A t most easily originates from a large m 1/2 (or a combination of slightly smaller m 1/2 and a large negative A t,0 ), which increases µ, thereby also increasing left-right mixing in the charged slepton sector.In the following we will illustrate within two SUSY models employing different supersymmetry breaking mechanisms, and both allowing sneutrino LOSP, how a large value of m 1/2 implied by a heavy Higgs boson leads to strong constraints on the reheating temperature resulting from BBN bounds.

Bound on the reheating temperature from BBN
For gravitinos with a mass significantly smaller than the Fermi scale, their present abundance resulting from scatterings in thermal plasma [34][35][36] can be approximated by [37]: where m G is the gravitino mass, M r denote gaugino mass parameters at the low scale and the coefficients γ r can be calculated from 1-loop RGEs for the gaugino masses and gauge couplings: they can be evaluated for T R = 10 9 (10 7 ) GeV as γ 3 = 0.50 (0.67), γ 2 = 0.51 (0.49), γ 1 = 0.20 (0.15), for the gluino masses of 900 GeV.It is easy to read from this estimate that with m 1/2 ∼ 1 TeV and m G = 100 GeV the observed dark matter abundance implies a reheating temperature of T R ∼ 5 × 10 8 GeV, which is close to minimal values ∼ 2× 10 9 GeV (∼ 2× 10 8 GeV) required by simple models of thermal leptogenesis with zero (thermal) initial abundance of the lightest right-handed neutrinos and sneutrinos [38,39].
The lifetime of sneutrino LOSP can be approximated as which can easily be of the order of 10 5 − 10 7 sec.For such long lifetimes it is then possible that hadro-dissociation processes induced by a subdominant decay process of sneutrino LOSP where a quark-antiquark pair is produced can alter the BBN predictions beyond the current observational uncertainties.We shall study this issue in the following Section, by numerically analyzing representative examples in two models of supersymmetry breaking which allow for sneutrino LOSP.

Numerical analysis
As we have argued in Section 2.2, models of supersymmetry breaking at the high scale allow a sneutrino LOSP only if at least one of the two conditions: D 2 = 0 or M 1 : M 2 : M 3 = α 1 : α 2 : α 3 is violated at the high scale.(At 1 loop these conditions are renormalization group invariants, hence they can be evaluated at any scale.)A violation of the former is manifest in the NUHM while the latter condition can be satisfied in many ways.A recently considered scenario is the GGM where it is assumed that the hidden and the messenger All scans: m Hu = 500 GeV, m H d = 4000 GeV, µ > 0 Case varied parameters fixed parameters Table 3. Description of scans over the parameters in the NUHM presented in Figures 1 and 2.
sectors in models of gauge mediation can be more complicated than what is required in a minimal theory.(The soft supersymmetry breaking parameters in both models are given in the Appendix.)

The NUHM
Armed with the above analytical considerations, we will now identify regions of the NUHM parameter space where the sneutrino is the LOSP.We will determine if these solutions are consistent with the Higgs mass, low-energy observables and early Universe.The details of the scans are given in Table 3.In our numerical work we used suspect [40] to solve the renormalization group equations and calculate mass spectra, micrOMEGAs [41] for the LOSP relic abundance and SuperIso [42] for flavor observables.
In Figures 1 and 2 we present the LOSP identity and its mass, as well as the mass of the Higgs boson.In both panels of Figure 1 and in the right panel of Figure 2 the sneutrino LOSP region is bounded from above at large enough values of m 1/2 .This can be easily understood since m 2 L is a much faster growing function of m 1/2 than M 2 1 which is the bino mass squared.At fixed m 1/2 and increasing m 0 , sfermion masses grow and they eventually become larger than the bino mass, which explains the bending of the boundary between the sneutrino and bino LOSP regions in the left panel of Figure 1.A negative contribution to m 2 L , which is proportional to D 2 , has to be overcome by some other positive contributions proportional to m 0 or m 1/2 ; otherwise we find unphysical regions (marked white) with tachyonic sleptons.It should also be mentioned that the negative contribution to m 2 U (proportional to D 2 ) is larger by a factor of 3/2 than that to m 2 L ; the former parameter also receives a much larger renormalization group correction proportional to m 2 1/2 than the latter.As a consequence, for sufficiently small values of m 1/2 the lighter stop becomes lighter than the sleptons; the corresponding region of stop LOSP is visible in the left panel of Figure 1.All these effects leads to a lower bound on m 1/2 ; in our scans we find no sneutrino LOSP models for m 1/2 < 800 GeV, which, as we shall discuss later, has the important consequences for the maximum reheating temperature.As it can be seen in the right panels of Figures 1 and 2, for µ > 0 large negative values of A 0 result in large off-diagonal entries in the stop mass matrix and lead to very light and even tachyonic stops.The appearance of the bino LOSP region in Figure 2 results from the τ -Yukawa effect in the renormalization group equations: in the leading logarithm approximation, the quantities δ 2 E,ytau in (2.4) and δ 2 L,yτ in (2.5) can be approximated by δ   [22].The observables are ratios of the element abundances, with the obvious exception of Y p which is 4 He mass fraction.The lower limit on Y p is irrelevant for constraining the abundance of decaying particles.The upper limit on D/H represents a compromise between the commonly used average of the best determinations of this quantity and the large spread of the individual results.The use of two bounds for 6 Li/ 7 Li, a stringent and a conservative one, reflects the uncertainty in estimating the efficiency of production/destruction of this element in stellar environment.
increases with growing tan β, we see that moderate values of tan β actually help one of the sleptons to become the LOSP.
Unsurprisingly, for fixed A 0 = −3000 GeV, we find a lower bound m 1/2 > ∼ 1 TeV resulting from the requirement that the Higgs boson mass exceeds 122 GeV, the value that we adopt as a conservative lower bound on the observable.For fixed m 0 , the shape of the constant Higgs boson mass contours in the (A 0 , m 1/2 ) plane agrees with the requirements for maximal stop mixing [33].For all the points shown in Figures 1 and 2 the low-energy observables lie within a conservative 95% CL range quoted in Table 4.As in many unified models, the supersymmetric contribution to δa µ is too low to explain the observed anomaly [43].We also checked that for all the points of interest squark masses of the first and second generations are well above 1400 GeV, required by the LHC data [44].In the stop LOSP regions, stop masses are often much smaller that ∼ 450 GeV which is the lower limit from the LHC, but these regions are disfavored anyway, because the Higgs boson mass drops below 120 GeV there.
For the region with sneutrino LOSP shown in the left panel of Figure 1 we calculate the abundances of light elements following the method outlined in [22] and apply the observational limits shown in Table 5.A representative sample of our results is shown in Figures 3 and 4. We find no constraints for the gravitino masses smaller than 7.5 GeV.At m G = 10 GeV a part of the parameter space corresponding to m ν > ∼ 500 GeV or, equivalently, to τ ν > ∼ 10 3 s, is excluded because of too large D/H abundance.For all values of m G the bounds from 6 Li/ 7 Li are always more stringent than the D/H bounds.A further increase of m G does not change this picture much, until the gravitino becomes degenerate with the sneutrino, which introduces a strong phase-space enhancement of the sneutrino lifetime.This is illustrated in the case with m G = 250 GeV, for which the BBN bounds, while still dominated by 6 Li/ 7 Li, become weaker.
A closer look at the actual predictions for D/H in the considered parameter range of Τ Ν s the NUHM reveals that even at points consistent with the allowed bounds, the abundance of D is altered with respect to the standard BBN value.It is also quite sensitive to the hadronic energy release: if we approximated it as (m ν − m G)/3, as is often done in the literature, instead of calculating the energy of the q q pair produced in the sneutrino decay, then with the conservative 6 Li/ 7 Li limit the lower boundary of the respective excluded region in the right panel of Figure 3 would shift downwards by as much as ∼ 100 GeV (see the right panel of Figure 5).In other words, one would significantly underestimate the sneutrino LOSP region allowed by the constraint.
In order to understand better the origin of the BBN constraints, we first project all the analyzed points onto the τ ν vs m ν Y ν plane; this is shown in the left panel of Figure 5.We also show there the bounds from the abundances of those light elements that constrain parameter space regions with sneutrino LOSP.Since Ω ν h 2 is roughly proportional to m 2 ν (neglecting the opening of additional annihilation channels for increased ν masses) and since for m G ≪ m ν the sneutrino lifetime scales as τ ν ∝ m 2 Gm −5 ν , it is easy to understand why, with increasing m G, the constraints from D/H and 6 Li/ 7 Li first appear, next tighten up and then eventually become weaker.As can be also easily seen from (2.13), a partial degeneracy between m G and m ν causes a much larger increase of τ ν than the simple power law above implies, hence the BBN bounds become correspondingly weaker.Those features can easily be seen for the results of our scan with a three values of fixed gravitino mass of m G = 2.5, 20, 250 GeV, as a band of dark red dots.
One may worry that for long sneutrino lifetimes, τ ν > 10 7 s, the electromagnetic show- ers produced in scatterings of energetic neutrinos from sneutrino decays off neutrinos of cosmic background can affect the BBN by altering the 3 He/H abundance [9,49].In order to verify this we determined that the exclusion plots from [9] do not provide additional constraints on our parameter space; we also interpolated the exclusion plots from [49] in (m ν , τ ν , B h ) plane and found no significant impact from 3 He/D.
As we have seen, increasing m G for a given m ν tends to alleviate the BBN constraints.However, for large gravitino masses there is another factor that we have to take into account.Non-thermal gravitinos produced in sneutrino LOSP decays will have velocities much larger than those characteristic for thermal distribution.Such fast moving dark matter particles tend to erase small scales of Large Scale Structures (LSS), especially when they constitute a sizable fraction of the dark matter density.Following [50], we account for these LSS constraints by requiring that the root mean square velocity of the non-thermally produced dark matter gravitinos does not exceed 1 km/s and that the non-thermal component makes less than 20% of the total dark matter abundance.
The impact of this bound on the NUHM parameter space is shown in Figure 6, where we show the superposition of the BBN bound discussed previously and the above LSS bounds for m G = 175 and 250 GeV.At such large m G, the LSS bounds become more stringent than the BBN ones (at m ν ≥ 300 GeV the LOSP relic abundance, Ω LOSP h 2 , exceeds 20% of the total dark matter abundance, hence Ω NTP G h 2 is also of this order), leaving just a small allowed strip in the parameter space.For m G > 270 GeV, we find that the LSS bounds exclude the entire section of the parameter space that we analyze here.This has important consequences for the maximum reheating temperature, since limits on the maximum reheating temperature become weaker with increasing gravitino mass. .For 6 Li/ 7 Li the stringent limit was used; the boundary of the excluded region with the more conservative constraint for 6 Li/ 7 Li is represented by a red dash-dotted line.
A summary of our results is presented in Figure 7 which shows regions in the (m G, m ν ) plane excluded by our constraints.It is clear that the BBN bounds alone allow two distinct regions in the parameter space.For small m G < 10 GeV, there are no constraints on m ν but the allowed maximum reheating temperature is relatively low, T max R ∼ 10 7 GeV.For larger m G, the BBN bounds start constraining the sneutrino mass and the maximum reheating temperature increases to ∼ 10 9 GeV when m ν ∼ m G. Imposing the LSS bounds closes this second region, thus slightly reducing the reheating temperature down to ∼ 9 × 10 8 GeV.However, now the points for which T R is maximal correspond to Higgs boson masses much smaller than the LHC measurement.The requirement that the Higgs boson mass is at least 122 GeV, brings T max R down to 7 × 10 8 GeV.These bounds on maximum T R as a function of m G are shown in the left panel of Figure 8 for the same sets of constraints.In the panel we impose the BBN bounds and we show the results with and without the LSS bounds and with and without the requirement that the Higgs boson mass is at least 122 GeV.We see that in each case the maximum T R lies close to 10 9 GeV, depending on the set of bounds imposed.Without the LSS or the Higgs boson mass bounds, this constraint mainly results from the lower bound on m 1/2 , as the maximum T R scales roughly as m −2 1/2 .This can be seen in the right panel of Figure 8 where we show the maximum T R versus the Higgs boson mass with and without BBN and These maximum values of T R is close to the quoted above lower bound required by simple thermal leptogenesis.It should be noted that the quoted leptogenesis bounds should be treated as indicative rather than absolute, since a rather mild mass degeneracy in the right-handed neutrino sector may lower the minimum reheating temperature for successful leptogenesis by a factor of a few [51].

GGM models
Another class of theoretically motivated scenarios in which it is possible to obtain sneutrino LOSP are models of Generalized Gauge Mediation.Unlike in the NUHM, where the condition tr(Y M 2 scalars ) = 0 is violated, the feature of GGM models that allows for a sneutrino LOSP is a non-universality of the gaugino masses.In particular, it follows from (2.4) that sneutrino LOSP is viable for M 2 /M 1 < ∼ 2 at the electroweak scale.We shall therefore utilize the freedom of gaugino mass assignment offered by GGM models to reduce M 2,0 at the messenger scale without breaking the universality of the two remaining gauginos, i.e. we shall adopt M 1,0 = M 3,0 .More specifically, we shall assume M 1,0 : M 2,0 : M 3,0 = 5 : 2 : 5, which predicts that the lightest gaugino-like neutralino is a wino.Eqs.(2.4) also show that for sneutrino LOSP m 2 L cannot be too large, which for fixed gaugino mass scale places upper bounds on the parameters Λ1 and Λ2 , whose relation to scalar masses at the messenger scale is shown in the Appendix.These bounds can be seen in the left panel of Figure 9, which also shows that increasing Λ2 with fixed Λ1 increases m 2 L,0 with respect to m 2 E,0 , which may lead to a right-handed stau LOSP.
The interplay between the gaugino and scalar mass scales is shown in the right panel of Figure 9, where we keep Λ1 = 2 Λ2 .Increasing gaugino masses while keeping the mass ratios fixed enlarges the range of Λ2 (and Λ1 ) for which one of the sleptons is the LOSP.With small values of Λ1 slepton masses are governed by 1-loop corrections proportional to gaugino masses and, as follows from (2.4), right-handed staus are the lightest.By increasing Λ one can obtain sneutrino LOSP, because the slepton masses at the messenger scale, m 2 L,0 and m 2 E,0 , contain contributions proportional to the product of Λ2 1 and the square of the respective hypercharge, which is larger for right-handed sleptons.
GGM models do not allow A-terms at the messenger scale.(This can, however, be circumvented by adding direct messenger-matter couplings in the superpotential [52], consistently with a Higgs mass of 126 GeV [53].)Therefore, in order to have a sufficiently large Higgs boson mass, we have to consider much larger gluino mass M 3,0 at the messenger scale, which generates radiatively large stop masses and a large negatively A t at the electroweak scale.The left panel of Figure 9 corresponds to fixed gaugino mass parameters and, therefore, to an almost constant Higgs boson mass of 126 GeV; in the right panel, the Higgs boson mass becomes close to 126 GeV for large values of M 1,0 = M 3,0 .At first sight it might seem that one could use the non-universality of the gaugino masses to make M 1,0 and M 2,0 much smaller than M 3,0 , but a large M 3,0 also results in a large µ parameter at the electroweak scale, which prevents the sneutrino from being the LOSP.This is also the reason for adopting a relatively small value of Λ3 : too large squark masses at the messenger scale also increase µ.All this results in sneutrino LOSP masses of about 1 TeV, which is of the parameter space between this calculation and one using a simplified formula for the hadronic energy release.
We showed that in models of gravitino dark matter with sneutrino LOSP, the four classes of constraints that can be applied: (i) the BBN constrains, (ii) the constraints on the large structure formation due to a presence of free-streaming decay products of ν, (iii) the Higgs boson mass bounds derived from the LHC data and (iv) the bounds on reheating temperature required by simple models of thermal leptogenesis are inconsistent, albeit the maximum reheating temperature is only 2 − 3 times smaller than the value suggested by the leptogenesis bound.(This was clearly visible in the NUHM; with gauge mediation, such large reheating temperatures were unattainable due to theoretical constraints on the gravitino mass.)Therefore, our results challenge the notion that models of gravitino dark matter with sneutrino LOSP are compatible with simple thermal leptogenesis.

Figure 1 .Figure 2 .
Figure 1.Slices of the NUHM parameter space: m 0 vs m 1/2 (left panel) and A 0 vs m 1/2 (right panel) with the values of m Hu = 500 GeV, m H d = 4000 GeV fixed at the unification scale and tan β = 10, µ > 0. Contours of constant LOSP (Higgs boson) masses are shown as dashed (solid) lines.Unphysical regions are marked in white.

Figure 3 .Figure 4 .
Figure 3. BBN bounds for the sneutrino LOSP region in the NUHM shown in the left panel of Figure1for the values of gravitino mass of m G = 10 and 20 GeV.For 6 Li/ 7 Li the stringent limit was used; the conservative limit does not constrain the parameter space.

Figure 5 .
Figure 5. Left panel: BBN constraints shown in the τ ν vs m ν Y ν plane for the sneutrino LOSP region shown in the left panel of Figure 1.Dots show the results of our scan with fixed m G = 2.5, 20 and 250 GeV.Right panel: The impact of different estimates of hadronic energy release on the D/H bounds for m G = 20 GeV.For the excluded region marked 'simple Ehad' an approximation E had = (m ν − m G)/3 was used, while the excluded region marked 'full Ehad' corresponds to a computation of E had involving integration over the full 4-body phase space.

Figure 7 .
Figure 7.A summary of the bounds in the NUHM in the (m G, m ν ) plane.The thick dashed line bounds the region excluded by BBN, the solid red line marks the boundary of the region excluded by LSS.Thinner dashed lines show the maximum reheating temperature, T max R , and thinner dotted lines show the Higgs boson mass corresponding to T max R

Figure 8 .
Figure 8. Left panel: the maximum reheating temperature as a function of m G with the BBN, the LSS and the Higgs boson mass constraints (m h > 122 GeV) applied, as well as without one or both of the LSS and the Higgs boson mass constraints.Right panel: maximum reheating temperature versus the Higgs boson mass without the LSS constraint (upper red dashed line) and with the LSS constraint (lower line).The solid (dotted) segments correspond to the cases where the BBN bound is (is not) applied.
)where M GUT is the unification scale at which m H d and y τ are evaluated here.As y τ observable BR(b → sγ)