Supersymmetric Dark Matter in Light of WMAP

We re-examine the parameter space of the constrained minimal supersymmetric extension of the Standard Model (CMSSM), taking account of the restricted range of \Omega_{CDM} h^2 consistent with the WMAP data. This provides a significantly reduced upper limit on the mass of the lightest supersymmetric particle LSP: m_\chi<500 GeV for \tan \beta<45 and \mu>0, or \tan \beta<30 and \mu<0, thereby improving the prospects for measuring supersymmetry at the LHC, and increasing the likelihood that a 1-TeV linear e^+ e^- collider would be able to measure the properties of some supersymmetric particles.


Introduction
The recent data from the WMAP satellite [1] confirm with greater accuracy the standard cosmological model, according to which the current energy density of the Universe is comprised by about 73 % of dark energy and 27 % of matter, most of which is in the form of non-baryonic dark matter. The WMAP data further tell us that very little of this dark matter can be hot neutrino dark matter, and the reported re-ionization of the Universe when the redshift z ∼ 20 is evidence against warm dark matter. WMAP quotes a total matter density Ω m h 2 = 0.135 +0.008 −0.009 and a baryon density Ω b h 2 = 0.0224 ± 0.0009 [1], from which we infer the following 2-σ range for the density of cold dark matter: Ω CDM h 2 = 0.1126 +0.0161 −0.0181 . This range is consistent with that inferred from earlier observations [2,3], but is significantly more precise.
It has been appreciated for some time that the lightest supersymmetric particle (LSP) is a suitable candidate for this non-baryonic cold dark matter [4]. The LSP is stable in supersymmetric models where R parity is conserved, and its relic density falls naturally within the favoured range if it weighs less than ∼ 1 TeV. This statement may be made more precise in the minimal supersymmetric extension of the Standard Model (MSSM), in which the LSP is expected to be the lightest neutralino χ, particularly if the soft supersymmetrybreaking mass terms m 1/2 , m 0 are constrained to be universal at an input GUT scale: the constrained MSSM (CMSSM). Since Ω χ h 2 ∝ m χ n χ , where n χ is the relic LSP number density, and n χ typically increases as the universal soft supersymmetry-breaking mass parameters are increased, one would expect the upper limit on m χ to decrease when the upper limit on Ω CDM h 2 is decreased. Compared with taking this upper limit to be 0.3, as we and others [5,6,7] have done previously, taking Ω CDM h 2 < 0.129 as suggested by the WMAP data -which is line with estimates from previous CMB determinations used in [8] -may therefore be expected to improve significantly the corresponding upper limit on m χ . Such is indeed the case for tan β < ∼ 45 and µ > 0, or for tan β < ∼ 30 and µ < 0, as we show below, where the largest values of m χ are found in the χ −τ coannihilation region. We find for these cases that m 1/2 < ∼ 900 − 1200 GeV for Ω CDM h 2 < 0.129, whereas m 1/2 < ∼ 1400 − 1750 GeV would have been allowed for Ω CDM h 2 < 0.3. Correspondingly, the upper limit on the LSP mass becomes m χ < ∼ 400−500 GeV, rather than m χ < ∼ 600−700 GeV as found previously [9,10]. This stronger upper limit improves the prospects for measuring supersymmetry at the LHC. Also, it would put sleptons within reach of a 1-TeV linear e + e − linear collider, whereas previously a centre-of-mass energy above 1.2 TeV might have appeared necessary [11]. All the above remarks would also apply if other particles also contribute to Ω CDM . For any fixed value of tan β and sign of µ, only a narrow region of CMSSM parameter space would be allowed if 0.094 < Ω χ h 2 < 0.129, as would be implied by WMAP if there are no other significant contributors to the density of cold dark matter. The previous 'bulk' regions of parameter space at small values of m 0 and m 1/2 now become quite emaciated, and the previous coannihilation strips now become much narrower, as do the rapid-annihilation funnels that appear at larger tan β [12,5]. However, unlike the coannihilation strips, the rapid-annihilation funnels still extend to very large values of m 0 and m 1/2 , the absolute upper limit on m χ is much weaker for tan β > ∼ 50 if µ > 0, or for tan β > ∼ 35 if µ < 0 1 . The narrowness of the preferred region implies that tan β could in principle be determined from measurements of m 1/2 and m 0 , as we discuss later. b → sγ decay [14] as discussed in [15]. In panels (a) and (d) for µ > 0, we also display the regions favoured by the recent BNL measurement [16] of g µ − 2 at the 2-σ level, relative to the calculation of the Standard Model based on e + e − data at low energies [17]. Also shown in Fig. 1 are the 'old' regions where 0.1 < Ω χ h 2 < 0.3, and the 'new' regions where 0.094 < Ω χ h 2 < 0.129. We see immediately that (i) the cosmological regions are generally much narrower, and (ii) the 'bulk' regions at small m 1/2 and m 0 have almost disappeared, in particular when the laboratory constraints are imposed. Looking more closely at the coannihilation regions, we see that (iii) they are significantly truncated as well as becoming much narrower, since the reduced upper bound on Ω χ h 2 moves the tip where m χ = mτ to smaller m 1/2 . It is this effect that provides the reduced upper bound on m χ advertized earlier. In panels (c) and (d), we see rapid-annihilation funnels that (iv) are also narrower and extend to lower m 1/2 and m 0 than previously. They weaken significantly the upper bound on m χ for tan β > ∼ 35 for µ < 0 and tan β > ∼ 50 for µ > 0.

WMAP Constraint on the CMSSM Parameter Space
We take this opportunity to comment on some calculational details concerning the rapidannihilation funnels. Comparison with other studies of these regions [7,18]  importance of treating correctly the running mass of the bottom quark, as we have done in previous works [5]. Also important is the correct treatment of annihilation rates across the convolution of two Boltzmann distributions when the cross section varies rapidly, as in the rapid-annihilation funnels, and we have taken the opportunity of this paper to improve our previous treatment. The results on the low-m 1/2 sides of the the rapid-annihilation funnels are indistinguishable from those shown previously, apart from the narrowing due to the smaller allowed range of Ω χ h 2 . However, there are more significant differences on the high-m 1/2 sides of the the rapid-annihilation funnels, where our previous approximation was less adequate. This is most noticeable for the case tan β = 50, µ > 0 shown in panel (d) of Fig. 1, where the two strips with relic density in the allowed range are all but merged. A second side of the rapid-annihilation funnel becomes distinctly visible when tan β ≥ 51, but the gap between the two sides is much narrower than we found previously. This is also true for µ < 0, and is exemplified in Fig. 1c for tan β = 35 where both sides of the funnel region are clearly distinct 2 . Before discussing further our results, we comment on the potential impact of a reevaluation of m t . A recent re-analysis by the D0 collaboration favours a central value some 5 GeV higher than the central value m t = 175 GeV that we use [19]. If confirmed, this would shift the displayed contours of m h = 114 GeV to lower m 1/2 , e.g., from m 1/2 ≃ 300 GeV to m 1/2 ≃ 235 GeV in the case tan β = 10, µ > 0. This would not affect the upper bound on m χ that we quote later, but it would weaken the lower limit on m χ in cases where this is provided by the LEP Higgs limit.
The focus-point region would, however, be affected spectacularly by any such increase in m t , shifting to much larger m 0 . In the analysis of [13], for tan β = 10 and m 1/2 = 300 GeV, the focus-point is pushed up from m 0 ≃ 2200 GeV to m 0 ≃ 4200 GeV when m t is increased from 175 GeV to 180 GeV, and for tan β = 50, it is pushed up from m 0 ≃ 1800 GeV to m 0 ≃ 3000 GeV. In our treatment of the CMSSM, we in fact find no focus-point region when m t = 180 GeV. In view of this instability in the fixed-point region, we do not include it in our subsequent analysis: our limits should be understood as not applying to this region, though we do note that it would also be further narrowed by the more restricted range of Ω χ h 2 .
We display in Fig. 2 the strips of the (m 1/2 , m 0 ) plane allowed by the new cosmological constraint 0.094 < Ω χ h 2 < 0.129 and the laboratory constraints listed above, for µ > 0 and values of tan β from 5 to 55, in steps ∆(tan β) = 5. We notice immediately that the strips 2 Note that the irregularities seen in the cosmological regions are a result of the resolution used to produce the figures.  5,10,15,20,25,30,35,40,45,50,55. Also shown in Fig. 2 in darker shading are the restricted parts of the strips that are compatible with the BNL measurement of g µ − 2 at the 2-σ level, if low-energy e + e − data are used to calculate the Standard Model contribution [17]. If this constraint is imposed, the range of m 1/2 is much reduced for any fixed value of tan β, and in particular the upper bound on m 1/2 is significantly reduced, particularly for tan β > ∼ 50. However, there is in general no change in the lower bound on m 1/2 .

Improved Upper Limit on the LSP Mass
We now draw some conclusions from Fig. 2. Its implications for the allowed range of the LSP mass m χ as a function of tan β are displayed in Fig. 3. As already mentioned, the upper limit is rather weak for tan β > ∼ 50 when µ > 0. However, for tan β < 40, we find the absolute upper bound to be compared with the range up to ≃ 650 GeV that we found with the old cosmological relic density constraint. Also shown in Figs. 2 and 3 is the strengthened upper bound on m 1/2 and m χ that would apply if one used the g µ − 2 constraint. We find for all values of tan β. Fig. 3(a) also shows the lower bound on m χ as a function of tan β, leading to for all values of tan β, with the minimum occurring around tan β = 23, when the b → sγ constraint begins to dominate over the Higgs mass constraint. As such, this lower limit depends on the calculation of m h , for which we use the latest version of FeynHiggs [20]. This calculation has an estimated theoretical uncertainty ∼ 2 GeV, and is very sensitive to m t . The lower bound (3) would become > 86 GeV (with the minimum occurring at tan β ≃ 18) if we used m t ≃ 180 GeV in FeynHiggs, or > 84 GeV (with the minimum occurring at tan β ≃ 17) if we allowed for a 2 GeV reduction in the calculated value for the nominal value of m t .
We do not show the plot corresponding to Fig. 2 for µ < 0, but we do show in Fig. 3(b) the corresponding lower and upper bounds on m χ for tan β < ∼ 40. We note again that the upper bound would rise rapidly for larger tan β, due to the appearance of a rapid-annihilation funnel analogous to that appearing for tan β > ∼ 50 when µ > 0. In the µ < 0 case, there is no possibility of compatibility with g µ − 2 when the e + e − data are used. We find the following range for tan β ≤ 30: 160 GeV < m χ < 430 GeV (4) for µ < 0, with the lower bound being provided by b → sγ for tan β > 8. We note that the upper and lower limits meet when tan β = 3.5(4.3) for µ > (<)0, implying that lower values of tan β are not allowed within our analysis of the CMSSM. This lower bound on tan β is strengthened to 7 if the g µ − 2 constraint is included.

Implications for Supersymmetric Phenomenology
The reduced upper limit on the sparticle mass scale improves the prospects for measuring supersymmetry at the LHC. Also, the fact that only a narrow strip in the (m 1/2 , m 0 ) plane is allowed for each value of tan β offers the possibility of determining tan β once m 1/2 and m 0 are known, e.g, from measurements at the LHC. We have discussed previously the sensitivity of Ω χ h 2 to variations in the CMSSM parameters [21], and that analysis can be adapted to the present situation. Since the typical separation between strips with ∆(tan β) = 5 is ∆m 0 ≃ 25 GeV and the width of a typical strip is ∆m 0 ≃ 5 GeV, it would in principle be possible to fix tan β with an accuracy ∆(tan β) ≃ 1 using measurements of m 1/2 and m 0 alone for a fixed value of A 0 (taken to be 0 here). The required accuracy in m 1/2 is not very demanding, since the strips are nearly horizontal, but m 0 would need to be determined with an accuracy ∆m 0 < ∼ 5 GeV. It is interesting to compare with the accuracies in m 0 , m 1/2 and tan β reported in [22] for the case m 0 = 100 GeV, m 1/2 = 300 GeV and tan β = 2.1. This value of tan β is not compatible with our analysis, but the expected accuracies ∆(m 0 ) ∼ 10 GeV and ∆(m 1/2 ) = 20 GeV would already allow interesting crosschecks of the value of tan β extracted from a fit to the LHC data with that required by cosmology.
The strengthened upper bound (1) on m χ may also have important consequences for linear e + e − collider physics [23]. At the tip of the coannihilation region, which corresponds to the upper bound in Fig. 3, we have m χ = mτ 1 , with theμ R andẽ R not much heavier. Therefore, a linear e + e − collider with centre-of-mass energy 1 TeV would be able to produce these sleptons. This conclusion holds only if one restricts attention to the CMSSM, as studied here, ignores the focus-point region as also done here, and discards large values of tan β. Moreover, we note that the left-handed sleptons are somewhat heavier at the tip of the CMSSM cosmological region: ml L > ∼ 700 GeV, as shown by the pale blue dotted lines in Fig. 3, and the lightest chargino has m χ ± > ∼ 800 GeV (green dotted lines). However, the strengthened upper limit on Ω χ h 2 that has been provided by WMAP does strengthen the physics case for a TeV-scale linear e + e − collider, compared with [11].

Perspective
In general, the WMAP constraint on Ω χ h 2 put supersymmetric phenomenology in a new perspective, essentially by reducing the dimensionality of the parameter space: one can now consider m 0 to be (almost) fixed in terms of the other parameters. The 'Snowmass lines' [24] now intersect the allowed cosmological region in just one (fuzzy) point each. The post-LEP benchmark points [25] have values of Ω χ h 2 that lie above the WMAP range. However, most of them can easily be adapted, in the 'bulk' and coannihilation regions simply by reducing m 0 . An exception is benchmark point H, which was chosen at the tip of a coannihilation tail: WMAP would require this to be brought down to lower m 1/2 , which would make it easier to detect at the LHC or a future linear e + e − linear collider. A more detailed update of the CMSSM benchmarks will be presented elsewhere.