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Calculations of Neutralino-Stop Coannihilation in the CMSSM

We present detailed calculations of the neutralino-stop coannihilation channels that have the largest impact on the relic neutralino density in the constrained minimal supersymmetric extension of the Standard Model (CMSSM), in which scalar masses m_0, gaugino masses m_1/2 and the trilinear soft supersymmetry-breaking parameters A_0 are each assumed to be universal at some input grand unification scale. The most important stop-stop* and stop-stop annihilation channels are also calculated, as well as stop-slepton coannihilation channels. We illustrate the importance of these new coannihilation calculations when A_0 is relatively large. While they do not increase the range of m_1/2 and hence neutralino mass allowed by cosmology, these coannihilation channels do open up new `tails' of parameter space extending to larger values of m_0.


Introduction
A favoured candidate for cold dark matter is the lightest supersymmetric particle (LSP), which is generally thought to be the lightest neutralino χ [1] in the minimal supersymmetric extension of the Standard Model (MSSM). It is common to focus attention on the constrained MSSM (CMSSM), in which all the soft supersymmetry-breaking scalar masses m 0 are required to be equal at an input superysmmetric GUT scale, as are the gaugino masses m 1/2 and the trilinear soft supersymmetry-breaking parameters A 0 . These assumptions yield well-defined relations between the various sparticle masses, and correspondingly more definite predictions for the relic abundance Ω χ h 2 and observable signatures. This paper is devoted to relic-abundance calculations including coannihilations of the lightest neutralino χ witht 1 , the lighter supersymmetric partner of the top quark [2].
The range 0.1 < Ω χ h 2 < 0.3 is generally thought to be preferred by astrophysics and cosmology [3]. Lower values of Ω χ h 2 might be possible if there is some other source of cold dark matter, but higher values are incompatible with observation. The regions of the m 1/2 , m 0 plane where the relic density falls within the preferred range 0.1 < Ω χ h 2 < 0.3 have generally been divided into four generic parts. There is a 'bulk' region at moderate m 1/2 and m 0 [1]. Then, extending to larger m 1/2 , there is a 'tail' of the parameter space where the LSP χ is almost degenerate with the next-to-lightest supersymmetric particle (NLSP), which is in this region theτ 1 , the lighter supersymmetric partner of the τ lepton. Along this 'tail', efficient coannihilations [4,5,6] keep Ω χ h 2 down in the preferred range, even for larger values of m χ [7,8,9,10]. At larger m 0 , close to the boundary where electroweak symmetry breaking is no longer possible, there is the 'focus-point' region where the LSP has a larger Higgsino component and m χ is small enough for Ω χ h 2 to be acceptable [11]. Finally, extending to larger m 1/2 and m 0 at intermediate values of m 1/2 /m 0 , there may be a 'funnel' of CMSSM parameter space where rapid direct-channel annihilations via the poles of the heavier Higgs bosons A and H keep Ω χ h 2 in the preferred range [12,13].
In this paper, we emphasize the significance of coannihilation of the LSP χ witht 1 , the lighter supersymmetric partner of the t quark [2]. This mechanism opens up another 'tail' of parameter space, this time extending to larger values of m 0 . It is not relevant for the small values of A 0 considered in previous coannihilation calculations [8,9,10], but may be important for large A 0 , as we demonstrate in this paper. Coannihilations of χ witht 1 are important when the latter is the NLSP, just as χτ 1 coannihilations are important when theτ 1 is the NLSP. In the latter case, one must also consider coannihilations with theẽ 1 andμ 1 , which are not much heavier than theτ 1 [7,8,9,10]. There are also regions of CMSSM parameter space where both thet 1 andτ 1 are close in mass to the LSP χ, and t 1τ1 coannihilations must also be considered. We present here detailed calculations of the matrix elements and cross sections for all the leading χt 1 andt 1l coannihilation processes, and illustrate their importance for Ω χ h 2 in some instances in the CMSSM when A 0 = 0.
The structure of the paper is as follows. In Section 2 we recall some important features of LSP relic-density calculations in general, and coannihilations in particular. Then, in Section 3 we compare the relative magnitudes of the χ χ, χt 1 ,t 1t

Formalism for Annihilation and Coannihilation
The density of neutralino relics left over from the early Universe may be determined relatively simply in terms of relevant annihilation cross sections, using the Boltzmann rate equation to determine a freeze-out density. The relic density subsequently scales with the inverse of the comoving volume, and hence with the entropy density. In the MSSM framework discussed here, since neutralinos are Majorana fermions, the S-wave annihilation cross sections into fermion-antifermion pairs are suppressed by the masses of the final-state fermions, and it is therefore necessary to compute P -wave contributions to the cross sections [1]. The rate equation for a stable particle with density n is where n eq is the equilibrium number density and σv rel is the thermally averaged product of the annihilation cross section σ and the relative velocity v rel . In the early Universe, we can writeṘ/R = (8πG N ρ/3) 1/2 , where ρ = π 2 g(T )T 4 /30 is the energy density in radiation and g(T ) is the number of relativistic degrees of freedom. Conservation of the entropy density . Defining x ≡ T /m and q ≡ n/T 3 h, we can write The effect of the h ′ term was discussed in detail in [14], and is most important when the mass m is between 2 and 10 GeV. Since we only consider neutralinos that are significantly more massive, we neglect it below (though it is not neglected in our calculations). In the case of the MSSM, freeze-out occurs when x ∼ 1/20, and the final relic density is determined by integrating (1) down to x = 0, and is given by When coannihilations are important, there are several relevant particle species i, each with different mass, number density n i and equilibrium number density n eq,i . Even in such a situation [4], the rate equation (1) still applies, provided n is interpreted as the total number density, n eq as the total equilibrium number density, and the effective annihilation cross section as In (2), m is now understood as the mass of the lightest particle under consideration. For T < ∼ m i , the equilibrium number density of each species is given by [14,15] where g i is a spin and color degeneracy factor and K 2 (x) is a modified Bessel function.
We make the approximation of Boltzmann statistics for the annihilating particles, which is excellent in practice. We now recall how to compute σ 12 v rel for the process 1+2 → 3+4 in an efficient manner.
Suppose we have determined the squared transition matrix element |T | 2 (summed over final spins and averaged over initial spins) and expressed it as a function of the Mandelstam variables s, t, u. We then express |T | 2 in terms of s and the scattering angle θ CM in the center-of-mass frame, as described in [8]. We now define In terms of w(s), the total annihilation cross section σ 12 (s) is given by σ 12 (s) = w(s)/s 1/2 p 1 (s) 1 . The above analysis is exact. To reproduce the usual partial wave expansion, we expand |T | 2 in powers of p 1 (s)/m 1 . The odd powers vanish upon integration over θ CM , while the zeroth-and second-order terms correspond to the usual S and P waves, respectively. Each factor of p 1 (s) is accompanied by a factor of cos θ CM , so we have We can therefore evaluate the S-and P -wave contributions to w(s) simply by evaluating |T | 2 at two different values of cos θ CM ; no integrations are required. The proper procedure for thermal averaging has been discussed in [14,15] for the case of m 1 = m 2 , and in [16,6] for the case of m 1 = m 2 , so we do not discuss it in detail here. One finds where x ≡ T /m 1 (assuming m 1 < m 2 ). In our case, we extract a 12 and b 12 from the transition amplitudes listed in the Appendix for each final state. We set x = 0 to get a 12 , and then compute b 12 by setting x to a numerical value small enough to render the O(x 2 ) terms negligible. We then compute a eff and b eff by performing the sum over initial states as in (6), and integrate the rate equation (2) numerically to obtain the relic LSP abundance. To a fair approximation, the relic density can simply be written as [1,4] Ωh 2 ≈ 10 9 GeV −1 where the freeze-out temperature T f ∼ m χ /20, and g f is the number of relativistic degrees of freedom at T f . This implies that the ratio of relic densities computed with and without coannihilations is approximately whereσ ≡ a+ bx/2 and sub-and superscripts 0 denote quantities computed ignoring coannihilations.
The ratio

Coannihilation Rates fort 1 in the MSSM
We now use the above formalism to estimate the relative importance of thet 1 χ coannihilation processes,t 1t * 1 andt 1t1 annihilations calculated in the Appendix. We also take into account the χl coannihilations calculated previously [7,8] and, for completeness, include thet 1l and t 1l * coannihilations also calculated in the Appendix.
To compute the effective annihilation cross sections for light sparticles in the MSSM, we allow the index i in (4) to run overt 1 ,t * 1 , τ 1 , τ * 1 , e R , e * R , µ R and µ * R , as well as χ. The following is the change in σ eff compared with [8], where 49 of the σ ij in (6) were already included: ∆σ eff = 2 (σt 1t1 + σt 1t * 1 )r 2 t 1 + 4 σ χt 1 r χ rt 1 + 8 (σt 1ẽR + σt 1ẽ * R )rt 1 rẽ R + 4 (σt 1τ1 + σt 1τ * 1 )r τ 1 rt 1 (13) where r i ≡ n eq,i /n eq . We have taken the e R and µ R (but not theτ ) to be degenerate in mass, thus accounting for the 81 possible initial state combinations. Note that we have summed over color states in the cross sections amplitudes listed in the Appendix, and we have taken the stop degeneracy factor gt = 3. We list in Table 1 the sets of initial and final states for which we compute the annihilation cross sections, using the transition amplitudes given in the Appendix. We use q to denote the four light quarks, which we have taken to be massless.
can, however, be large, particularly for the stops, or for sbottoms at large tan β 3 . When A t is sufficiently large, the lighter stop,t 1 , can become degenerate with (or lighter than) the neutralino. Thus, it is when A 0 is large that we expect χt 1 coannihilations to become important.
It is important to distinguish between the effective low-energy parameters A t , etc., and the high-energy input parameter A 0 , which are related through the running of the renormalization-group equations. For example, for tan β = 10 and m 0 = 300 GeV,t 1 χ coannihilations are important when m 1/2 = 200, 450,and 670 GeV and A 0 = 1000, 2000, and 3000 GeV, but these values correspond to A t ≃ 565, 1200, and 1700 GeV respectively. Furthermore, the values of A for the light squarks are different and typically larger than A t .
We display in Fig. 1 numerical values of the contributions toσ ≡ a + bx/2 (see (11)) in χt 1 coannihilation, for the representative values x = 1/23, tan β = 10, µ > 0 and (a) For comparison, the total cross section for χ χ annihilation to all final states is also shown, as a thick dotted line. We see that the χt 1 → tg and th coannihilation cross sections dominate by large factors over the total χ χ annihilation cross section, suggesting that they may have a greater importance than that suggested by simply comparing Boltzmann suppression factors.
The feature in Fig. 1(a) at m 0 ∼ 400 GeV is due to the threshold for the production of th final states. At smaller values of m 0 , this final state is kinematically forbidden. subdominant cross sections such as γg, Zg, gh and the various quark-antiquark channels are far larger than the total χ χ annihilation cross section. Once again, when A 0 = 1000 GeV we see thresholds, in this case corresponding to hh production at m 0 ∼ 180 GeV and tt production at m 0 ∼ 330 GeV. Thet 1t1 annihilation cross sections shown in Fig. 3 show that the cross section for annihilation into the tt final state, when it is kinematically open, is also far larger than the total χ χ annihilation cross section.
A complete study of coannihilation effects must include not only the χl processes considered previously [7,8], and the χt 1 processes considered above, but alsol 1t1 coannihilations. Accordingly, the final set of coannihilation cross sections we present are those forl 1t1 and ℓ * 1t 1 , shown in Fig. 4. We see that, when (a) A 0 = 1000 GeV, the tτ , tτ and bν e final states are the most important, followed by te, bν τ and tē, whereas (b) the tτ and tē final states are relatively much less important when A 0 = 2000 GeV. In all panels of Fig. 4, there are coannihilation cross sections much greater than the total χ χ annihilation cross section, which is also plotted.
The basic reason for the relatively small magnitude of the χ χ annihilation cross section is that it is dominated by the P -wave suppressed cross sections for χ χ annihilation to fermion pairs. This was also the basic reason why χl coannihilation processes were previously found to be so important [7,8,9,10].   The contributions of the various annihilation channels to σ eff are weighted by the relative abundances of the χ,t 1 andl 1 . For a stop degenerate with the χ,t 1t * 1 annihilation, t 1t1 annihilation and χt 1 coannihilation are clearly the dominant contributions to ∆σ eff , and hence to σ eff in (6), and the final neutralino relic density is greatly reduced. As the stops become heavier than the neutralinos, their number densities are exponentially suppressed and the stop contributions to σ eff become less important. Fig. 5 shows the sizes of the separate contributions toσ eff from χ χ annihilation,t 1 χ coannihilation andt 1t * 1 ,t 1t1 annihilations (combined), as functions of the mass difference between thet 1 and χ. In Fig. 5, we have fixed m 0 = 300 GeV, tan β = 10, µ > 0, A 0 = (a) 1000 and (b) 2000 GeV, and computedσ eff for varying m 1/2 , which amounts to varying the fractional mass difference ∆M ≡ (mt 1 − m χ )/m χ . For these choices, the stau mass, mτ 1 , is much larger than mχ. The thin dotted lined is the value ofσ that one would compute if one ignored all coannihilation contributions 4 . Note that, in the case of close degeneracy between the χ andt 1 , thet 1t1 and t 1t * 1 annihilations dominateσ eff . However, since these contributions are suppressed by two powers of n eq,t 1 , they drop rapidly with ∆M, and neutralino-stop coannihilation takes over.   We then see in panel (b) that thet 1 coannihilation tail extends beyond m 0 ∼ 1400 GeV when tan β = 20 and A 0 = 3000 GeV. A similar portion of thet 1 coannihilation tail that is consistent with b → sγ is also visible in panel (c), for tan β = 20 and A 0 = 2000 GeV, where m 0 ∼ 900 GeV. For fixed A 0 , as tan β is increased the b → sγ constraint becomes more severe.
Thet 1 coannihilation tails do not increase the allowed range of m 1/2 , as was the case for thel coannihilation tail [7,8,9,10], but they do add a significant filament to the CMSSM region allowed by experiment and favoured by cosmology.
In the above illustrative examples, we have chosen the sign of A 0 to be the same as that of µ, with the effect of maximizing the stop off-diagonal mass terms, and hence minimizing mt 1 . If the sign of A 0 is switched while keeping µ > 0, an analogous χt 1 coannihilation region is found only at even larger negative A 0 . However, the m h constraint is more severe for A 0 < 0, and also for µ < 0, excluding the region of parameter space of interest in the χt 1 coannihilation context. We have not discussed in this paper the potential constraint imposed by the absence of colour and charge breaking (CCB) vacua, or at least the suppression of transitions to CCB vacua. This constraint would restrict the value of A 0 relative to m 0 [19]. If the CCB constraint is imposed, some of the parameter space for small m 0 will be excluded, but there will still be regions at high m 0 where χt 1 coannihilation is crucial. The issue of compatibity with the potential g µ − 2 constraint would then arise. Since raising m 0 suppresses the neutralino-proton elastic scatttering cross section, χt 1 coannihilation would tend to lower the neutralino direct detection rate for the same range of m χ .

Conclusions and Open Issues
We have documented in this paper the potential importance of χt 1 coannihilation in delineating the preferred domain of CMSSM parameter space for A 0 = 0. In this paper, we have only sctratched the surface of this subject, whose higher-dimensional parameter space merits  more detailed exploration. The Appendix provides details of the diagrammatic calculations that should be sufficient for our results to be verified and used by other authors. Although applied in the context of the CMSSM, our results may also be used in more general MSSM contexts. However, other coannihilation processes are also important in other regions of the general MSSM parameter space. For example, in the CMSSM the sbottom mass is generally larger than the stop mass even for large tan β. However, if one allow non-degeneracy in the scalar soft breaking mass term, a sbottom NLSP becomes possible [10]. A complete calculation of the LSP relic density in the MSSM requires a careful discussion of all such coannihilation possibilities.
We calculate the tree contributions to the coannihilation and annihilation amplitudes that are leading order in α s , α t ≡ λ 2 t /4π and/or α b ≡ λ 2 b /4π. We keep terms that are of the same order in √ α s α em , √ α t α em , √ α b α em , √ α s α W , √ α t α W and/or √ α b α W , but neglect most terms in α em and/or α W . This approximation is motivated by the relatively large magnitudes of α s and α t , and the fact that α b ∼ α t at large tan β. We have however included s-channel Z, and γ exchange fort 1t * 1 → tt and checked that they make negligible contribution to the total cross section and therefore neglect these channels in the remaining amplitudes.

Couplings
Here we list couplings used in the calculation. Factors not written, such as i, γ's, QCD generators and momentum , are taken into account in the calculation of amplitudes squared, which are listed subsequently.