Gravitino Dark Matter in the CMSSM

We consider the possibility that the gravitino might be the lightest supersymmetric particle (LSP) in the constrained minimal extension of the Standard Model (CMSSM). In this case, the next-to-lightest supersymmetric particle (NSP) would be unstable, with an abundance constrained by the concordance between the observed light-element abundances and those calculated on the basis of the baryon-to-entropy ratio determined using CMB data. We modify and extend previous CMSSM relic neutralino calculations to evaluate the NSP density, also in the case that the NSP is the lighter stau, and show that the constraint from late NSP decays is respected only in a limited region of the CMSSM parameter space. In this region, gravitinos might constitute the dark matter.


Introduction
If R parity is conserved, the lightest supersymmetric particle (LSP) is stable, and a possible candidate for the cold dark matter postulated by astrophysicists and cosmologists [1]. Most analyses of such supersymmetric dark matter have assumed that the LSP is the partner of some combination of Standard Model particles, such as the lightest neutralino χ, with an abundance calculated from the freeze-out of annihilation processes in a thermal initial state. However, another generic possibility is that the LSP is the gravitinoG [2] - [8], whose relic abundance would get contributions from the decays of the next-to-lightest supersymmetric particle (NSP) and possibly other mechanisms.
As we discuss in more detail below, the lifetime of the NSP is typically such that it decays between Big-Bang nucleosynthesis (BBN) and the 're-' combination process when the cosmic microwave background (CMB) was released from matter. Since NSP decays release entropy during this epoch, they are constrained by the concordance of the observed light-element abundances with BBN calculations assuming the baryon-to-entropy ratio inferred from CMB observations. For a typical lifetime τ N SP = 10 8 s, the observed 6 Li abundance implies [9] n N SP n γ < 5 × 10 −14 100 GeV m N SP (1) before NSP decay, with the D/H ( 4 He) abundance providing a constraint which is weaker by a factor of about 10 (20). Assuming a baryon-to-entropy ratio η ≡ n B /n γ = 6.0 × 10 −10 , in agreement with the WMAP result η = 6.1 +0.3 −0.2 × 10 −10 [10], (1) implies the constraint n N SP /n B < 10 −4 (100 GeV/m N SP ) before the onset of NSP decay. To assess the power of this constraint, we re-express it in terms of Ω 0 N SP h 2 , the relic density that the NSP would have today, if it had not decayed: where Ω B h 2 ≃ 2 × 10 −2 is the present-day baryon density. However, the requirement (2) would be relaxed for a shorter-lived NSP [7], as we discuss later.
In contrast, assuming that the lightest neutralino χ is the LSP, there have been many calculations of Ω χ h 2 in the constrained minimal supersymmetric extension of the Standard Model (CMSSM), in which the GUT-scale input gaugino masses m 1/2 and scalar masses m 0 are each assumed to be universal [11][12][13][14]. These calculations find generic strips of CMSSM parameter space in which This is similar to the range of the cold dark matter density Ω CDM h 2 favoured by astrophysicists and cosmologists, which is one reason why neutralino dark matter has been quite popular.
In this paper, we assume no a priori relation between m 3/2 and the soft supersymmetrybreaking masses m 1/2 and m 0 of the spartners of Standard Model particles in the CMSSM. This is possible in, e.g., the framework of N = 1 supergravity with a non-minimal Kähler potential [15]. In such a framework, the LSP might well be the gravitinoG. In this case, the NSP would likely be the lightest supersymmetric partner of some combination of Standard Model particles, such as the lightest neutralino χ or the lighter stauτ 1 . Particularly in the χ NSP case, one might expect Ω 0 N SP h 2 to be near the range (3). Comparing this with the condition (2) necessary for gravitino dark matter, we see that, if τ N SP = 10 8 s, gravitino dark matter could be possible only in rather different regions of the CMSSM parameter space, where the NSP density is very suppressed compared with the usual χ density. Moreover, in this case, NSP decays alone could not provide enough gravitinos, since they could only yield Ω 3/2 h 2 < Ω 0 N SP h 2 , so there would need to be some supplementary mechanism for producing gravitinos, if they were to provide all the cold dark matter. For example, gravitino production during reheating after inflation could produce a sufficient abundance of gravitinos if the reheat temperature is relatively large, ∼ O(10 10 ) GeV [7].
The first step in our exploration of the gravitino dark matter possibility is to calculate Ω 0 N SP h 2 throughout the (m 1/2 , m 0 ) planes for different choices of tan β and the sign of µ in the CMSSM, assuming that the trilinear soft supersymmetry-breaking parameter A 0 = 0. In the regions where m χ < mτ 1 , this is essentially equivalent to the usual neutralino dark matter density calculation. However, as we discuss below, this calculation must be adapted in the region where mτ 1 < m χ . Moreover, one must take into account the possibility of a cosmologicalτ 1 asymmetry, in which case the relicτ 1 density would be larger than that given by the standard freeze-out calculation. We next compute the NSP lifetime and use the detailed constraints from the abundances of the light elements as computed in (1) for fixed η = 6 × 10 −10 . This allows us to delineate the regions of the CMSSM (m 1/2 , m 0 ) planes where gravitino dark matter appears possible. We find limited regions of the (m 1/2 , m 0 ) planes that are allowed. In these regions, the density of relic gravitinos due to NSP decay is typically less than the range favoured by astrophysics and cosmology. As noted above, supplementary mechanisms for gravitino production, such as thermal production in the early Universe, might then enable gravitinos to constitute the cold dark matter.

NSP Density Calculations
In the framework of the CMSSM with a light gravitino discussed here, the candidates for the NSP are the lightest partners of Standard Model particles. In generic regions of CMSSM parameter space, these are the lightest neutralino χ and the lighter stauτ 1 1 . In regions where χ is the NSP, the calculation of the NSP density Ω 0 N SP h 2 is identical with that of Ω LSP h 2 in the CMSSM with a heavier gravitino, and we can recycle standard results.
Extending these calculations of Ω 0 N SP h 2 to regions where theτ 1 is the NSP requires some modifications. Whereas the Majorana χ is its own antiparticle, one must distinguish between theτ 1 and its antiparticleτ * 1 , and calculate the sum of their relic densities. This requires a careful accounting of the statistical factors in all relevant annihilation and coannihilation processes. We have also made a careful treatment of the regions where there is rapidτ 1 −τ * 1 annihilation via Higgs poles, and a non-relativistic expansion in powers of the NSP velocity is inadequate. Here our treatment follows that of the neutralino LSP case in [11,13,17].
It is important to note that one would, in general, expect a netτ 1 asymmetry . This would be the expectation, for example, in leptogenesis scenarios, and would also appear in other baryogenesis scenarios, as a result of electroweak sphalerons. However, in the context of the MSSM, there existτ 1τ1 → τ τ annihilation processes which would bleed away any existing lepton asymmetry stored in theτ sleptons, and the final relic density is given by the calculation described above.

NSP Decays
Using the standard N = 1 supergravity Lagrangian [18,19], one can calculate the rates for the various decay channels of candidate NSPs to gravitinos.
The dominant decay of a χ NSP would be into a gravitino and a photon, for which we calculate the width Note that in this and the following equations A χ NSP may also decay into a gravitino and a Z boson, for which we calculate the rate , and we use the auxiliary functions Note that in the limit M Z → 0 we obtain Γ χ→G Z → Γ χ→Gγ by replacing C χZ with C χγ . Decays of a χ NSP into a gravitino and a Higgs boson are also possible, with a rate Analogously, for the heavy Higgs boson H we get Γ χ→Gh → Γ χ→GH by replacing C χh with C χH ≡ (O 4χ sin α + O 3χ cos α), and m h with m H . The corresponding formula for χ →G + A, where A is the CP-odd Higgs boson in the MSSM, is also given by (8) Finally, the dominant decay of aτ NSP would be into a gravitino and a τ , with the rate: where we have neglected the O(m 2 τ /m 2 τ 1 ) terms.

Effects of Gravitino Decay Products on Light-Element Abundances
The effects of electromagnetic shower development between Big-Bang Nucleosynthesis (BBN) and 're-'combination have been well studied, most recently in [9], where the simplest case of χ →G + γ decays were considered. The late injection of electromagnetic energy can wreak havoc on the abundances of the light elements. Energetic photons may destroy deuterium, destroy 4 He (which may lead to excess production of D/H), destroy 7 Li, and/or overproduce 6 Li. The concordance between BBN calculations and the observed abundances of these elements can be used to derive a limit on the density of any decaying particle. In general, this limit will depend on both the baryon asymmetry η B , which controls the BBN predictions, and on the life-time of the decaying particle τ X . For a fixed value η B = 6 × 10 −10 , as suggested by CMB observations, the bounds derived from Fig. 8(a) in [9] may be parameterized approximately as where y ≡ log(ζ X /GeV) ≡ log(m X n X /n γ /GeV) and x ≡ log(τ X /s), for the electromagnetic decays of particles X with lifetimes 10 12 s > τ N SP > ∼ 10 4 s. In our subsequent analysis we use the actual data corresponding to the limit in [9] in order to delineate the allowed regions of the (m 1/2 , m 0 ) planes, but (10) may help the reader understand qualitatively our results. The other NSP decay modes listed above inject electrons, muons and hadrons into the primordial medium, as well as photons. Electromagnetic showers develop similarly, whether they are initiated by electrons or photons, so we can apply the analysis of [9] directly also to electrons. Bottom, charm and τ particles decay before they interact with the cosmological medium, so new issues are raised only by the interactions of muons, pions and strange particles. In fact, if the NSP lifetime exceeds about 10 4 s, these also decay before interacting, and the problem reduces to the purely electromagnetic case. In the case of a shorter-lived NSP, we would need to consider also hadronic interactions with the cosmological medium [20], which would strengthen the limits on gravitino dark matter that we derive below on the basis of electromagnetic showers alone. In the following, we do not consider regions of the (m 1/2 , m 0 ) planes where τ N SP < 10 4 s. It is sufficient for our purposes to treat the decays of µ, π and K as if their energies were equipartitioned among their decay products. In this approximation, we estimate that the fractions of particle energies appearing in electromagnetic showers are π 0 : 100%, µ : 1/3, π ± : 1/4, K ± : 0.3, K 0 : 0.5. Using the measured decay branching ratios of the τ , we then estimate that ∼ 0.3 of its energy also appears in electromagnetic showers. In the case of generic hadronic showers from Z or Higgs decay, we estimate that ∼ 0.6 of the energy is electromagnetic, due mainly to π 0 and π ± production. Our procedure is then as follows. First, on the basis of a freeze-out calculation, we calculate the NSP relic density Ω 0 X h 2 = 3.9 × 10 7 GeV −1 ζ X . Next, we use the calculated life-time τ X to compute the ratio of the relic density to the limiting value, ζ CEF O X provided by the analysis of [9], taking into account the electromagnetic energy decay fractions estimated above. Finally, we require

Results
As compared to the case of CMSSM dark matter usually discussed, in the case of gravitino dark matter one must treat m 3/2 as an additional free parameter, unrelated a priori to m 0 and m 1/2 . We incorporate the LEP constraint on m h in the same way as in [13] 2 , and it appears as a nearly vertical (red) dot-dashed line in each of the following figures. Regions excluded by measurements of b → sγ are shaded dark (green). For reference, the figures also display the strips of the (m 1/2 , m 0 ) planes where 0.094 < Ω 0 N SP < 0.129. This density is the same as Ω LSP h 2 in a standard CMSSM analysis with a heavy gravitino, extended to include the unphysical case where theτ 1 is the LSP. We note the familiar 'bulk' regions and coannihilation 'tails', as well as rapid-annihilation 'funnels' for large tan β [17,21]. If these figures were extended to larger m 0 , there would also be 'focus-point' regions [22,23].
We now summarize our principal results, describing the interplay of these constraints with those associated specifically with gravitino dark matter, studying the (m 1/2 , m 0 ) planes for three choices of tan β and the sign of µ: (1) tan β = 10, µ > 0, (2) tan β = 35, µ < 0, and (3) tan β = 50, µ > 0. In each case, we consider four possibilities for m 3/2 : two fixed values 10 GeV and 100 GeV, and two fixed ratios relative to m 0 : m 3/2 = 0.2 m 0 and m 0 itself. If m 3/2 ≫ m 0 , theG is typically not the LSP, and this role is played by the lightest neutralino χ, as assumed in most analyses of the CMSSM. In each (m 1/2 , m 0 ) plane, we display as a (purple) dashed line the limit where the density of relic gravitinos from NSP decay becomes equal to the highest cold dark matter density allowed by WMAP and other data at the 2-σ level, namely Ω 3/2 h 2 < 0.129: only regions below and to the right of this contour are allowed in our analysis. In the regions below the (purple) dashed line, the relicG density might be increased so as to provide the required cold dark matter density if there were significant thermal gravitino production, in addition to that yielded by NSP decay.
The light-element constraint on NSP decays is shown as the grey (khaki) solid line corresponding to r = 1, where r is defined in (11). Regions to the right and below this line are allowed by this constraint. Here, and in the remaining figures below, the region which satisfies the abundance constraint is labelled r < 1. There is a solid (black) line with m 1/2 ∼ 800 GeV which indicates where τ N SP = 10 4 s. To the right of this line, τ N SP < 10 4 s, the case we do not consider here because additional constraints due to hadronic decays must be included, so this region is left blank 4 . Here and in subsequent figures, the region that is allowed by all the constraints is shaded in light (yellow) color.
We see that there is an extended strip between the grey (khaki) solid line and the solid (black) line. This strip is truncated above m 0 ≃ 650 GeV, because the relic density of gravitinos from NSP decay becomes too large. This is true up to ∼ 2900 GeV, where the relic density drops as we approach the focus-point region. Here a small allowed region opens up as the r = 1 curve bends towards lower values of m 1/2 . The allowed strip broadens in the low-m 0 region where mτ 1 < m χ , below the dotted (red) line where m χ = mτ 1 . In this region, gravitino dark matter is permitted.
Turning now to panel (b) of Fig. 1, where the choice m 3/2 = 100 GeV is made, we see a near-vertical black line at m 1/2 ∼ 250 GeV: the gravitino is the LSP only to its right.
The τ N SP = 10 4 s line has disappeared to larger m 1/2 , and is not shown. In this case the Ω 3/2 h 2 constraint is much more important than in panel (a), forcing m 0 to be relatively small, simply because m 3/2 is larger. The only region allowed by the light-element constraint on NSP decays is in the bottom right-hand corner, in the region where theτ 1 is the NSP.
In panel (c) of Fig. 1, for m 3/2 = 0.2m 0 , there is also a black line to whose right theG is the LSP, which is now diagonal, and the Ω 3/2 h 2 constraint is similar to that in panel (b).

Most of the region allowed by the light-element constraint on NSP decays is in the region
where theτ 1 is the NSP, though a sliver of parameter space runs above the dotted curve. Finally, in panel (d) of Fig. 1, where now m 3/2 = m 0 , theG constraint is more powerful, as is the Ω 3/2 h 2 constraint, and the region finally allowed by the light-element constraint on NSP decays is again in theτ 1 region. Fig. 2 displays a similar array of (m 1/2 , m 0 ) planes for the case tan β = 35 and µ < 0. In the case where mτ 1 = 10 GeV, shown in panel (a), the most significant change compared with panel (a) of Fig. 1 is that the b → sγ constraint is more important, whilst the Ω 3/2 h 2 , NSP  [24], as a near-vertical (red) dot-dashed line, the region excluded by b → sγ is darkly shaded (green), and the region where the NSP density before decay lies in the range 0.094 < Ω 0 N SP h 2 < 0.129 is medium shaded (greyblue). The (purple) dashed line is the contour where gravitinos produced in NSP decay have Ω 3/2 h 2 = 0.129, and the grey (khaki) solid line (r = 1) is the constraint on NSP decays provided by Big-Bang nucleosynthesis and CMB observations. The light (yellow) shaded region is allowed by all the constraints. The contour where m χ = mτ 1 is shown as a (red) diagonal dotted line. Panels (a) and (c) show as a black solid line the contour beyond which τ N SP < 10 4 s, the case not considered here. Panels (b), (c), and (d) show black lines to whose left the gravitino is no longer the LSP. decay and τ N SP constraints do not change so much. The net result is to leave disconnected parts of both the χ andτ 1 regions that are allowed by all the constraints.
The most obvious new feature in panel (b) of Fig. 2 is the rapid-annihilation funnel, which affects both the Ω 3/2 h 2 and NSP decay constraints. The former acquires a strip extending to large m 1/2 and m 0 , whereas the latter would have allowed a region at large m 1/2 > ∼ 1500 GeV that is excluded by Ω 3/2 h 2 . Combining this and the NSP decay constraint, we again find two disconnected allowed regions, one in the χ NSP region and one that is almost entirely in theτ 1 NSP region.
The rapid-annihilation funnel is also very apparent in panel (c) of Fig. 2, which displays the case mτ 1 = 0.2m 0 , where again a strip allowed by both the Ω 3/2 h 2 and NSP decay constraints extends to large m 1/2 and m 0 . There are again disconnected allowed regions in the χ and (mainly) theτ 1 NSP region. Note that this is constrained at large m 1/2 and small m 0 by the τ N SP constraint. Finally, in panel (d) of Fig. 2, for mτ 1 = m 0 , the region allowed by theG LSP, Ω 3/2 h 2 and NSP decay constraints is restricted to the part of the (m 1/2 , m 0 ) plane where theτ 1 is the NSP. Fig. 3 displays a similar array of (m 1/2 , m 0 ) planes for the case tan β = 50 and µ > 0.
The general features of the planes have some similarities to those for tan β = 35 and µ < 0. There are differences in the interplays between the Ω 3/2 h 2 and NSP decay constraints, but an important difference is the relative weakness of the b → sγ constraint. This has the consequence that allowed χ andτ 1 regions are connected for tan β = 50 and µ > 0. It is interesting to note that this is the only case where the putative constraint imposed by the muon anomalous magnetic moment a µ impinges on the allowed region, as shown in panels (a) and (c).
We have seen in the above examples that many of the allowed parts of the (m 1/2 , m 0 ) planes are confined to regions where the NSP is theτ 1 .

Conclusions
We have analyzed in this paper the possibility of gravitino cold dark matter within the CMSSM framework. Combining accelerator and cosmological constraints, particularly those from b → sγ, Ω 3/2 h 2 and the light-element constraint on NSP decays, we have found allowed regions in the (m 1/2 , m 0 ) planes for representative values of tan β and the sign of µ and different values of m 3/2 . Standard calculations of the NSP density before decay based on freeze-out from equilibrium yield allowed regions where either the lightest neutralino χ or the lighter stauτ 1 may be the NSP.  One limitation of our analysis is that it is restricted to τ N SP > 10 4 s, in order to avoid issues related to the hadronic interactions of NSP decay products before they decay. Also, in this paper we have not discussed at much length what part of parameter space may be allowed in the focus-point region. Finally, we have analyzed here only a few examples of the possible relationship between m 3/2 and the CMSSM parameters m 0 and m 1/2 .
For these and other reasons, there are still many important issues to analyze concerning the possibility of gravitino dark matter. We have shown in this paper that such a possibility certainly exists, and that the allowed domains of parameter space are not very exceptional. We consider that gravitino dark matter deserves more attention than it has often received in the past. In particular, this possibility should be borne in mind when considering the prospects for collider experiments, since the allowed regions of the (m 1/2 , m 0 ) are typically rather different from those normally analyzed in the CMSSM. Vive la différence!