The extent of the stop coannihilation strip

Many supersymmetric models such as the constrained minimal supersymmetric extension of the Standard Model (CMSSM) feature a strip in parameter space where the lightest neutralino \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi $$\end{document}χ is identified as the lightest supersymmetric particle, the lighter stop squark \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{t}_1}$$\end{document}t~1 is the next-to-lightest supersymmetric particle (NLSP), and the relic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi $$\end{document}χ cold dark matter density is brought into the range allowed by astrophysics and cosmology by coannihilation with the lighter stop squark \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{t}_1}$$\end{document}t~1 NLSP. We calculate the stop coannihilation strip in the CMSSM, incorporating Sommerfeld enhancement effects, and we explore the relevant phenomenological constraints and phenomenological signatures. In particular, we show that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{t}_1}$$\end{document}t~1 may weigh several TeV, and its lifetime may be in the nanosecond range, features that are more general than the specific CMSSM scenarios that we study in this paper.

Generally speaking, the allowed parameter space of the CMSSM for any fixed values of tan β and A 0 /m 0 may be viewed as a wedge in the (m 1/2 , m 0 ) plane. Low values of m 0 /m 1/2 are excluded because there the LSP is the lighter stau slepton, which is charged and hence not a suitable dark matter candidate. The stau coannihilation strip runs along the boundary of this forbidden region [42][43][44][45][46][47][48]. High values of m 0 /m 1/2 are also generically excluded, though for varying reasons. At low A 0 /m 0 , the reason is that no consistent electroweak vacuum can be found at large m 0 /m 1/2 , and close to the boundary of this forbidden region the Higgs superpotential mixing parameter μ becomes small, the Higgsino component of the χ gets enhanced, and one encounters the focus-point strip [52][53][54][55][56]. However, when A 0 /m 0 is larger, the issue at large m 0 /m 1/2 is that the LSP becomes the lighter stop squark t 1 , which is also not a suitable dark matter candidate. Close to this boundary of the CMSSM wedge, thet 1 is the nextto-lightest supersymmetric particle, and the relic χ den-sity may be brought into the cosmological range byt 1 χ coannihilation [57][58][59][60][61][62]. The length of thet 1 χ coannihilation strip is increased by Sommerfeld enhancements in somẽ t 1t 1 annihilation channels [64][65][66][67][68][69], which we include in our analysis.
In this paper we study the extent to which portions of thist 1 χ strip may be compatible with experimental and phenomenological constraints as well as the cosmological dark matter density, paying particular attention to the constraint imposed by the LHC measurement of the mass of the Higgs boson. Other things being equal, the measurement m H = 125.9 ± 0.4 GeV tends to favour larger values of A 0 such as those featuring at 1 χ coannihilation strip, reinforcing our interest in this region of the CMSSM parameter space [37][38][39][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84]. We use FeynHiggs 2.10.0 to calculate the lightest supersymmetric Higgs mass and to estimate uncertainties in this calculation [85]. We find that the stop coannihilation strip may extend up to m 1/2 13000 GeV, corresponding to m χ = mt 1 6500 GeV, that the endpoint of the stop coannihilation strip may be compatible with the LHC measurement of m H for tan β = 40 or large A 0 /m 0 = 5.0 within the FeynHiggs 2.10.0 uncertainty, and that the stop lifetime may extend into the nanosecond range.
The layout of this paper is as follows. In Sect. 2 we review relevant general features of the CMSSM, setting thet 1 χ coannihilation strip in context and describing our treatment of Sommerfeld enhancement effects. In Sect. 3 we study the possible extent of this strip and the allowed range of thet 1 mass. Although our specific numerical studies are the framework of the CMSSM, we emphasise that our general conclusions have broader validity. In Sect. 4 we discusst 1 decay signatures, which are also not specific to the CMSSM, and in Sect. 5 we summarise our conclusions.

Anatomy of the stop coannihilation strip
We work in the framework of the CP-conserving CMSSM, in which the soft supersymmetry-breaking parameters m 1/2 , m 0 and A 0 are assumed to be real and universal at the GUT scale. We treat tan β as another free parameter and use the renormalisation-group equations (RGEs) and the electroweak vacuum conditions to determine the Higgs superpotential mixing parameter μ and the corresponding soft supersymmetry-breaking parameter B (or, equivalently, the pseudoscalar Higgs mass M A ). We concentrate in the following on the choices μ > 0 and A 0 > 0.

Sommerfeld effect
We evaluate the dark matter density in the regions of the stop coannihilation strips including the Sommerfeld effect, which may enhance the annihilation rates at low velocities, and which is particularly relevant for strongly interacting particles such as the stop squark. As we discuss in more detail below, the general effect of including the Sommerfeld factors is to increase substantially the length of the stop coannihilation strip.
In general, the Sommerfeld effect modifies s-wave cross sections by factors [64] where β is the annihilating particle velocity and α is the coefficient of a Coulomb-like potential whose sign is chosen so that α < 0 corresponds to attraction. In the case of annihilating particles with strong interactions, the Coulomb-like potential may be written as [86][87][88] where α 3 is the strong coupling strength at the appropriate scale, C i and C i are the quadratic Casimir coefficients of the annihilating coloured particles, and C f is the quadratic Casimir coefficient of a specific final-state colour representation. 1 In our case, we always have C i = C i = C 3 = 4/3. Int 1 −t 1 annihilations the possible s-channel states are singlets with C 1 = 0 and octets with C 8 = 3, whereas int 1 −t 1 annihilations Bose symmetry implies that the only possible final colour state is a sextet with C 6 = 10/3. The factors in the square parentheses [...] for the singlet, octet and sextet final states are therefore −8/3, +1/3 and +2/3, respectively, corresponding to α = −4α 3 /3, α 3 /6 and α 3 /3, respectively. Only the singlet final state exhibits a Sommerfeld enhancement: s-wave annihilations in the other two colour states actually exhibit suppressions. We implement the Sommerfeld effects in the SSARD code [89] for calculating the relic dark matter density, which is based on a non-relativistic expansion for annihilation cross sections: where ... denotes an average over the thermal distributions of the annihilating particles, the coefficient a represents the contribution of the s-wave cross section, x ≡ T /m, and the dots represent terms of higher order in x. When α < 0 in (1), 1 It is well established that perturbative QCD can be used to describe cross sections for processes involving heavy coloured particles just above their thresholds, cf, the use of perturbative QCD to describe the e + e − →tt threshold. Since mt 1 > m t in the region of interest, perturbative QCD will be even more reliable for calculatingt 1t1 annihilation close to threshold. Specifically, this is done in [69]. as in the singlet final state discussed above, the leading term in (3) acquires a singularity where the dots again represent terms of higher order in x. The Sommerfeld correction to the annihilation cross section that we include is parametrically enhanced by a factor 1/x close to threshold, cf. our Eq. (4). Going beyond this term to include non-enhanced corrections would require a complete calculation of O(α s ) corrections, which lies far beyond the scope of this paper. Along the stop coannihilation strip, the dominantt 1 −t 1 s-wave annihilation cross sections are typically those into colour-singlet pairs of Higgs bosons (∼60-70 % in the CMSSM before incorporating the Sommerfeld effect) and into gluon pairs (∼20-30 %), which are a mixture of 2/7 colour-singlet and 5/7 colour-octet final states, followed by the colour-octet Z + gluon final state (∼5 % in the CMSSM). We have implemented the Sommerfeld effects for theset 1 −t 1 final states, and also fort 1 −t 1 → t + t annihilations, whose s-wave annihilation cross section ∼5 % of the totalt 1 −t 1 s-wave annihilation cross section before including the Sommerfeld effect.
We emphasise that the Sommerfeld factors in different channels depend only on the final states and are independent of the specific CMSSM scenario that we study. We also emphasise that many other supersymmetric models feature the same suite of final states in stop-neutralino coannihilation. Moreover, some of the couplings to these final states are universal, e.g.,t 1 −t 1 annihilations to gluon pairs mediated by crossed-channelt 1 exchange and directchannel gluon exchange. The similarities imply that results resembling ours would hold in many related supersymmetric models. 2

The end-point of the stop coannihilation strip
As we shall also see, there are differences in the lengths of the stop coannihilation strips for different values of the model parameters. Looking at the dominantt 1 −t 1 annihilation mechanisms, it is clear that the matrix elements for annihilations to some final states are universal, e.g., to gluon pairs. However, the dominantt 1 −t 1 annihilations to pairs of Higgs bosons are model dependent. The dominant contributions tot 1 −t 1 → h + h annihilation, in the notation of the appendix in [61], are I × I, II × II, I × II, I × III and II × III with i = 2, corresponding to t− and u-channel exchanges of the heavier stopt 2 , the exchange of the lighter stop exchange being suppressed by sin θ t , where θ t is thet 1 −t 2 mixing angle. Thet 1 −t 2 − h coupling takes the form which depends on A t , sin β, the Higgs mixing angle α and μ, as well as θ t , and the annihilation cross section also depends on mt 2 . Thet 1 −t 1 → h + h annihilation rate is therefore model dependent, depending primarily on the combination Ct 1 −t 2 −h /mt 2 , which causes m χ at the tip of the stop coannihilation strip to vary as we see later.
3 Representative parameter planes in the CMSSM We display in Fig. 1 some representative CMSSM (m 1/2 , m 0 ) planes for fixed tan β = 20, μ > 0 and different values of A 0 /m 0 that illustrate the interplay of the various theoretical, phenomenological, experimental and cosmological constraints. 3 In each panel, any region that does not have a neutral, weakly interacting LSP is shaded brown. Typically there are two such regions which appear as triangular wedges. The wedge in the upper left of the (m 1/2 , m 0 ) plane contains a stop LSP or tachyonic stop, and the wedge in the lower right of the plane contains a stau LSP or tachyonic stau. The dark blue strips running near the boundaries of these regions have a relic LSP density within the range of the cold dark matter density indicated by astrophysics and cosmology [91] 4 : that near the boundary of the upper left wedge is due to stop coannihilation, and that near the boundary of the lower right wedge is due to stau coannhilation. As we discuss later, the stop coannihilation strips typically extend to much larger values of m 1/2 than the stau coannhilation strips, indeed to much larger values of m 1/2 than those displayed in Fig. 1, reaching as far as 7000-13000 GeV in the models studied. The green shaded regions are incompatible with the experimental measurement of b → sγ decay [92], and the green solid lines are 95 % CL constraints from the measured rate of B s → μ + μ − decay [93][94][95]. The solid purple lines show the constraint from the 3 We have not taken into account the possible role of charge and colour breaking minima here. For a recent study of these effects see [90]. 4 The widths of these dark matter strips have been enhanced for visibility. Barely visible in the lower parts of the unshaded wedges between the strips in some panels of Figs. 1 and 2 are low densities of points where annihilations of other sparticles coannihilating with the neutralino are enhanced by direct-channel Higgs poles, reducing χ h 2 into the allowed range.  [85,[98][99][100][101]. We note that the multiple RGE solutions found in [102,103] appear in regions of parameter space with either 5 The sensitivity of the CMSSM limits to tan β and A 0 has been studied in both previous experimental papers and in [96], where it was shown that the limits form jets + MET searches in the region of interest are insensitive to these parameters. μ < 0 and/or small A 0 < m 0 (mostly A 0 = 0)-whereas the stop coannihilation strips we study appear for μ > 0 and A 0 /m 0 ≥ 2.
In general, we identify stop coannihilation strips in CMSSM (m 1/2 , m 0 ) planes for 2.1 m 0 A 0 5.5 m 0 , and the panels in  Here we see that the opening angle of the stop LSP wedge is rather insensitive to tan β, that of the stau coannihilation strip being more sensitive. Also, we recall that studies indicate that the LHC / E T constraint is essentially independent of tan β. On the other hand, the impacts of the b → sγ and B s → μ + μ − constraints increase with tan β. They only ever exclude a fraction of the stop coannihilation strip, but the B s → μ + μ − constraint does exclude the entire stau coannihilation strip for tan β = 40. The m H contours calculated using FeynHiggs 2.10.0 are quite similar for tan β = 10, 20 and 30. However, we find smaller values of m H for tan β = 40, a feature whose implications we discuss in more detail later.  Fig. 3 we see that along all of the displayed portion of the dark matter contour extending from (tan β, A 0 ) = (6, 7500) to (25,7800 GeV) corresponding to A 0 /m 0 ∼ 2.2 we have 127 < m H < 128 GeV, which is also compatible with the experimental measurement within the estimated theoretical uncertainties. 7 7 We note that the ATLAS search for jets + / E T events, the measurement by CMS and LHCb of B s → μ + μ − decay and the experimental constraint on b → sγ do not constrain any of the strip regions shown in Figs. 3, 4, 5 and 6.    We display in Table 1 Figure 9 shows the behaviours of δm and m H along coannihilation strips for fixed m 0 = m 1/2 for the choices tan β = 10, 20, 30 and 40. In the upper left panel for tan β = 10 we see that δm is maximised at ∼83 GeV for the nominal value χ h 2 = 0.120, when m 1/2 ∼ 4000 GeV. This value of δm is just below the threshold fort 1 → χ + b + W decay. The end-point of this strip is at m 1/2 ∼ 12000 GeV corresponding to m χ = mt 1 ∼ 5900 GeV, and the portion of the strip with m 1/2 ∈ (4000, 10000) GeV has a value of m H compatible with the LHC measurement within the FeynHiggs 2.10.0 uncertainties. The upper right panel for tan β = 20 is quite similar, with δm rising slightly higher, but still below m χ + m W + m b for χ h 2 = 0.120. The lower panels for tan β = 30 and 40 are very different. Indeed, in these cases the appropriate relic density is found along the stau coannihilation strip, and the ends of the blue lines in these panels mark the tips of the corresponding stau coannihilation strips. In the tan β = 30 case, the whole strip with m 1/2 600 GeV is compati-

Stop decay signatures along the coannihilation strip
We now consider the stop decay signatures along the coannihilation strips discussed in the previous section. Generally speaking, one expects the two-body decayst 1 → χ + c to dominate as long as δm > m D ∼1.87 GeV [104][105][106]. Below this threshold, the dominant two-body decay processes aret 1 → χ + u, which would lead to decays of a mesinõ t 1q → χ + non-strange mesons and of a sbaryont 1 qq → χ + baryon, etc. Four-body decayst 1 → χ + b + + ν andt 1 → χ + b + u +d are also important as long as δm > m B ∼5.3 GeV, together witht 1 → χ +b +c +s when δm > m B s +m D ∼ m B +m D s ∼ m B c +m K ∼7 GeV. Above this threshold, the total four-body decay rate ∼9 (t 1 → χ + b + + ν). Our results are, in general, qualitatively consistent with those of previous authors [104][105][106]. In general, we see that the lifetime τt 1 increases as m 1/2 increases monotonically towards the end of the coannihilation strip, reaching τt 1 ∼ 1 ns near the end of the strip for A 0 = 2.3 m 0 and tan β = 10. 8 The lifetime would be further enhanced when δm < m D , by a   (20) as well as by phase-space suppression, but we do not discuss this possibility in detail. In the lower left panel of Fig. 11 we display the corresponding calculations of the totalt 1  is formally accessible. In our treatment of this case we calculatet 1 → χ + b + (W * → f +f ), where W * denotes an (in general) off-shell W boson represented by a Breit-Wigner line shape. This yields a larger (and more accurate) decay rate than calculating naively the three-body decay to b and an onshell W boson, and we find that BR(t 1 → χ + b + f +f ) may exceed BR(t 1 → χ + c) by over an order of magntitude.

Summary and conclusions
We have shown in this paper that the existence of a long stop coannihilation strip where the relic neutralino density χ h 2 falls within the cosmological range is generic in the CMSSM for 2.2 m 0 A 0 5.5 m 0 . It is essential for calculating the length of this strip and the mass difference δm = mt 1 − m χ along the strip to include Sommerfeld effects. The two annihilation processes that are most important for determining the length of this strip aret 1t * 1 → 2 gluons via t-channelt 1 exchange and s-channel gluon exchange, which are completely model-independent, andt 1t * 1 → 2 Higgs bosons, which is more model dependent. Specifically, the cross section for the latter process is mediated byt 2 in the cross channel, and hence it depends on mt 2 and on thet 1 −t 2 − h coupling Ct 1 −t 2 −h (5) in the combination Ct 1 −t 2 −h /mt 2 . We therefore expect that the location of the end-point of the stop coannihilation strip should depend primarily on this ratio.
In Tables 1 and 2 we have listed the parameters of the endpoints in the various cases we have studied, including those appearing in the expression for Ct 1 −t 2 −h (5). In Fig. 13 we display a scatter plot of the end-point values of m χ = mt 1 vs. the quantity Ct 1 −t 2 −h /mt 2 . We see that, to a good approximation, the end-point of the stop coannihilation strip is indeed a simple, monotonically increasing function of Ct 1 −t 2 −h /mt 2 . As seen in Fig. 13, in the models we have studied the maximum value of m χ = mt 1 compatible with the cosmological Fig. 12 The branching ratios fort 1 → χ + c decay in the same models as in Fig. 11 and using the same colours for the lines  Tables  1 and 2 dark matter constraint is ∼6500 GeV. As seen in the tables, these scenarios yield large values of m H as calculated using FeynHiggs 2.10.0, but when tan β = 40 the end-points are compatible with the measured value of m H within the calculational uncertainty of ∼3 GeV. It seems possible that larger values of m χ = mt 1 would be possible in models with larger values of Ct 1 −t 2 −h /mt 2 . We infer that a high-mass end-point for a stop coannihilation strip is likely to be a general feature of a broad class of models. Its appearance is not restricted to the CMSSM and closely related models such as the NUHM [107][108][109][110][111][112], and its location depends primarily on the combination Ct 1 −t 2 −h /mt 2 . However, the extent of the stop coannihilation strip might be increased further in models in which other sparticles are (almost) degenerate with thet 1 and χ . This might occur, for instance, in circumstances under which the lighter sbottomb 1 or one or more squarks of the first two generations happened to be nearly degenerate with thet 1 and χ , but this is unlikely to be a generic model feature.
We note also that the dominantt 1 decay mode along the stop coannihilation strip is likely to bet 1 → χ + c, since the mass difference δm = mt 1 − m χ < m B + m W in general and four-body decayst 1 → χ + b + f +f are strongly suppressed by phase space. This is likely to be a generic feature of stop coannihilation strips. We also note that thẽ t 1 lifetime may approach a nanosecond near the tip of the stop coannihilation strip, which is also likely to be a generic feature.
We conclude that the stop coannihilation strip may be distinctive as well as generic.