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Dark Matter and Detector Cross Sections

We consider here the spin independent neutralino-proton cross section for a variety of SUGRA and D-brane models with R-parity invariance. The minimum cross section generally is>~ 1\times 10^{-(9-10)} pb (and hence accessible to future detectors) except for special regions of parameter space where it may drop to =~ 10^{-12} pb. In the latter case the gluino and squarks will be heavy (>~ 1 TeV).

We consider here the spin independent neutralino-proton cross section for a variety of SUGRA and D-brane models with R-parity invariance. The minimum cross section generally is > ∼ 1 × 10 −(9−10) pb (and hence accessible to future detectors) except for special regions of parameter space where it may drop to ≃ 10 −12 pb. In the latter case the gluino and squarks will be heavy ( > ∼1 TeV).
Dark matter detectors have now achieved a sensitivity that they have begun to probe interesting parts of SUSY parameter space. It is thus of interest to see what sensitivity will be needed to explore the entire space. To examine this we consider here models based on gravity mediated supergravity (SUGRA), where the LSP is generally the lightest neutralino (χ 0 1 ). Neutralinos in the halo of the Milky Way might be directly detected by their scattering by terrestrial nuclear targets. Such scattering has a spin independent and a spin dependent part. For heavy nuclear targets the former dominates, and it is possible to extract then (to a good approximation) the neutralino -proton cross section, σχ0 1 −p . Current experiments (DAMA, CDMS, UKMDC) have sensitivity to haloχ 0 1 for and future detectors (GENIUS, Cryoarray) plan to achieve a sensitivity of We consider here two questions: (1) what part of the parameter space is being tested by current detectors, and (2) what is the smallest value of σχ0 1 −p the theory is predicting (i.e. how sensitive must detectors be to cover the full SUSY parameter space). The answer to these questions depends in part on the SUGRA model one is considering and also on what range of theoretical and input parameter one assumes. In the following, we examine three models that have been considered in the literature based on grand unification of the gauge coupling constants at M G ∼ = 2 × 10 16 GeV: (1) Minimal super gravity GUT (mSUGRA) with universal soft breaking at M G 1 ; (2) Nonuniversal soft breaking models for Higgs and third generation scalar masses at M G , and D-brane models (based on type IIB orientifolds 2 ) which allow for nonuniversal gaugino masses and nonuniversal scalar masses at M G 3 . While each of the above models contain a number of unknown parameters, theories of this type can still make relevant predictions for two reasons: (i) they allow for radiative breaking of SU (2) × U (1) at the electroweak scale (giving a natural explanation of the Higgs mechanism), and (ii) along with calculating σχ0 1 −p , the theory can calculate the relic density ofχ 0 1 , i.e Ωχ0 1 is the relic mass density ofχ 0 1 and ρ c = 3H 2 0 /8πG N (H 0 is the Hubble constant and G N is the Newton constant). Both of these greatly restrict the parameter space. In general one has Ωχ0 where σ ann is the neutralino annihilation cross section in the early universe, v is the relative velocity, 1 , T f is the freeze out temperature, ... means thermal average and h = H 0 /100 km s −1 Mpc −1 ). The fact that these conditions can be satisfied for reasonable parts of the SUSY parameter space represents a significant success of the SUGRA models.
In the following we will assume H 0 = (70 ± 10)km s −1 Mpc −1 and matter (m) and baryonic (b) relic densities of Ω m = 0.3 ± 0.1 and Ω b = 0.05. Thus Ωχ0 1 h 2 = 0.12 ± 0.05. The calculations given below allow for a 2σ spread, i.e. we take 0.02 ≤ Ωχ0 1 h 2 ≤ 0.25. It is clear that when the MAP and Planck sattelites determine the cosmological parameters accurately, the SUGRA dark matter predictions will be greatly sharpened.

Calculational Details
In order to get reasonably accurate results, it is necessary to include a number of theoretical corrections in the analysis. We list here the main ones used in the calculations below: (i) In relating the theory at M G to phenomena at the electroweak scale, the two loop gauge and one loop Yukawa renormalization group equations (RGE) are used, iterating to get a consistent SUSY spectrum. (ii) QCD RGE corrections are further included below the SUSY breaking scale for contributions involving light quarks. (iii) A careful analysis of the light Higgs mass m h is necessary (including two loop and pole mass corrections) as the current LEP limits impact sensitively on the relic density analysis for tanβ ≤ 5. (iv) L-R mixing terms are included in the sfermion (mass) 2 matrices since they produce important effects for large tanβ in the third generation. (v) One loop corrections are included to m b and m τ which are again important for large tanβ. (vi) The experimental bounds on the b → sγ decay put significant constraints on the SUSY parameter space and theoretical calculations here include the leading order (LO) and approximate NLO corrections. We have not in the following imposed b−τ (or t−b−τ ) Yukawa unification or proton decay constraints as these depend sensitively on unknown post-GUT physics. For example, such constraints do not naturally occur in the string models where SU (5) (or SO (10)) gauge symmetry is broken by Wilson lines at M G (even though grand unification of the gauge coupling constants at M G for such string models is still required).
All the above corrections are under theoretical control except for the b → sγ analysis where a full NLO calculations has not been done. (We expect that while the full analysis might modify the regions of parameter space excluded by the b → sγ experimental constraint, the minimum and maximum values of σχ0 1 −p would probably not be significantly changed.) The analysis of σχ0 1 −p , taking into account the above theoretical corrections has now been carried out by several groups obtaing results in general agreement 4,5,6,7,8,9,10 . These results are presented below.
Accelerator bounds significantly limit the SUSY parameter space. In the following we assume the LEP bounds 8 m h > 104(100) GeV for tanβ=3 (5) and m χ ± 1 > 104(100) GeV. (For tanβ > 5, the m h bounds do not produce a significant constraint 11 .) For b → sγ we assume an allowed range of 2σ from the CLEO data 12 . The Tevatron gives a bound of mg ≥ 270 GeV( for mq ∼ = mg) 13 .

mSUGRA model
We consider first the mSUGRA model where the most complete analysis has been done. mSUGRA depends on four parameters and one sign: m 0 (universal scalar mass at M G ), m 1/2 (universal gaugino mass at M G ), A 0 (universal cubic soft breaking mass) , tanβ = H 2 / H 1 (where H 2,1 gives rise to (up, down) quark masses) and µ/|µ|( where µ is the Higgs mixing parameter in the superpotential, W µ = µH 1 H 2 ). One conventionally restricts the range of these parameters by "naturalness" conditions and in the following we assume m 0 ≤ 1 TeV, m 1/2 ≤ 600 GeV (corresponding to mg ≤ 1.5 TeV, mχ0 1 ≤ 240 GeV), |A 0 /m 0 | ≤ 5, and 2≤ tanβ ≤ 50. Large tanβ is of interest since SO(10) models imply tanβ ≥ 40 and also σχ0 1 −p increases with tanβ. σχ0 1 −p decreases with m 1/2 for large m 1/2 , and thus if one were to increase the bound on m 1/2 to 1 TeV (mg ≤ 2.5 TeV), the cross section would drop by a factor of 2-3.
The maximum σχ0 1 −p arise then for large tanβ and small m 1/2 . This can be seen in Fig.1 where (σχ0 1 −p ) max is plotted vs.  (4) which would be accessible to detectors that are currently being planned (e.g. GENIUS).
For larger mχ0 1 , i.e. m 1/2 > ∼ 150 the phenomena of coannihilation can occur in the relic density analysis since the light stau,τ 1 , (and alsoẽ R ,μ R ) can become degenerate with theχ 0 1 . The relic density constraint can then be satisfied in narrow corridor of m 0 of width ∆m 0 < ∼ 25 GeV, the value of m 0 increasing as m 1/2 increases 6 . Since m 0 and m 1/2 increase as one progresses up the corridor, σχ0 1 −p will generally decrease. We consider first the case µ > 0 19 . One finds in general that σχ0 1 −p also decreases as A 0 increases. Fig.2 shows σχ0 1 −p in the domain of large A 0 and for two values of tanβ. One sees that the smaller tanβ still gives the lower cross section, though the difference is mostly neutralized at larger m 1/2 . (For large tanβ, m 0 also becomes large to satisfy the relic density constraint i.e m 0 ∼ = 700 GeV for tanβ=40, m 1/2 = 600 GeV.) We have in general for this regime This is still within the sensitivity range of proposed detectors. When µ is negative an "accidental" cancellation can occur in part of the parameter space in the coannihilation region which can greatly reduce σχ0 1 −p 7 . This can be seen in Fig.3, where starting with small tanβ the cross section decreases, leading to a minimum at about tanβ=10, and then increases again for larger tanβ. At the minimum one has σχ0 1 −p ∼ = 1 × 10 −12 when tanβ=10 and m 1/2 = 600 GeV. More generally one has for the parameter domain when 4 In this domain, σχ0 1 −p would not be accessible to any of the currently planned detectors. However, mSUGRA also then predicts that this could happen only when the gluino and squarks have masses greater than 1 TeV (and for only a restricted region of tanβ) a result that could be verified at the LHC.

Nonuniversal SUGRA Models
In the discussion of SUGRA models with nonuniversal soft breaking, universality for the first two generations of squark and slepton masses at M G is usually maintained to suppress flavor changing neutral currents. One allows, however, the Higgs and third generation squark and slepton masses to become nonuniversal. We maintain gauge and gaugino mass unification at M G .
While these models contain a large number of new parameters, their effects on σχ0 1 −p can be charcterized approximately by the signs of the deviations from universality 9 . One choice can greatly increase σχ0 1 −p , by a factor of 10-100 compared to the universal case, and the reverse choice can reduce σχ0 1 −p ( though by a much lesser amount). Thus it is possible for detectors to probe regions of smaller tanβ with nonuniversal breaking, and detectors obeying Eq. (1) can probe part of the parameter space for tanβ as low as tanβ ≃ 4.
The minimum cross section occurs (as in mSUGRA) at the lowest tanβ and at the largest m 1/2 i.e. in the coannihilation region. We limit ourselves here to the case where only the Higgs masses are nonuniversal. One finds then results similar to mSUGRA i.e. σχ0 1 −p > ∼ 10 −9 pb for µ > 0, m 1/2 ≤ 600 GeV. For µ < 0, there can again be a cancellation of matrix elements reducing the cross section to 10 −12 pb when m 1/2 = 600 GeV in a restricted part of the parameter space when tanβ ≃10.

Summary
We have examined here the neutralino-proton cross section for a number of SUGRA type models. In all the models considered, there are regions of parameter space withχ 0 1 − p cross sections of the size that could be observed with current detectors. Thus with the sensitivity of Eq. (1) The minimum cross sections these models predict are considerably below current sensitivity. Thus for mSUGRA one finds for µ > 0 that σχ0 1 −p > ∼ 1 × 10 −9 pb for m 1/2 ≤ 600GeV, µ > 0, where m 1/2 = 600 GeV corresponds to mg ∼ = 1.5 TeV, mχ0 1 ∼ = 240 TeV. This is still in the range that would be ac-cessible to detectors being planned (such as GENIUS or Cryoarray). For µ < 0, a cancellation can occur in certain regions of parameter space allowing the cross sections to fall below this. Thus and reaching a minimum of σχ0 1 −p ∼ = 1×10 −12 pb for tanβ=10, m 1/2 =600 GeV, µ < 0. This domain would appear not to be accessible to future planned detectors. Since m 1/2 =450 GeV corresponds to mg ∼ = 1.1 TeV, this region of parameter space would imply a gluino squark spectrum at the LHC above 1 TeV.
The above results holds for the mSUGRA model. While a full analysis of coannihilation has not been carried out for the nonuniversal and D-brane models, results similar to the above hold for these over large regions of parameter space. Thus for nonuniversal Higgs masses and for the D-brane model one finds σχ0 1 −p > ∼ 10 −9 pb for µ > 0, while a cancellelation allows σχ0 1 −p to fall to 10 −12 pb for µ < 0 at tanβ ≃ 10 (with again a gluino/squark mass spectrum in the TeV domain).

Acknowledgement
This work was supported in part by NSF grant no. PHY-9722090.   (σχ0