The Lower Bound on the Neutralino-Nucleon Cross Section

10 Pages·0.35 MB·English

by  V. Mandic

UCB-PTH/0023, LBNL-46431,MPI-PhT/2000-26
The Lower Bound on the Neutralino-Nucleon Cross Section
Vuk Mandic
Department of Physics, University of California, Berkeley, CA 94720, USA
Aaron Pierce and Hitoshi Murayama
Department of Physics, University of California, Berkeley, CA 94720, USA;
Theory Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Paolo Gondolo
Max-Planck-Institut fu¨r Physik, F¨ohringer Ring 6, D-80805 Mu¨nchen, Germany
(Dated: February 1, 2008)
We examine if there is a lower bound on the detection cross section, σχ−p, for the neutralino
2 dark matter in the MSSM. If we impose the minimal supergravity boundary conditions as well as
0 the “naturalness” condition, in particular m1/2 < 300 GeV, we show that there is a lower bound
0 of σχ−p > 10−46 cm2. We also clarify the origin for the lower bound. Relaxing either of the
2 assumptions, however, can lead to much smaller cross sections.

n
a
I. INTRODUCTION metric Standard Model (MSSM), and even for a very
J
restrictive framework such as the minimal supergravity
3
(mSUGRA),thenumberofparametersisstillquitelarge.
2 Supersymmetry is considered to be a compelling ex-
We will show that there nonetheless exists a minimum
tension to the Standard Model for several reasons. For
3 cross section in the mSUGRA framework. However,
example, it stabilizes scalar masses against radiative
v we will also show that this result strongly depends on
corrections, allowing theories with fundamental scalars
2 the assumptions of the framework, such as unification
2 to become natural. For a review see [1]. Supersymmetry
of different parameters at the GUT scale, radiative
0 also weighs in on the dark matter problem: stars and
electroweak symmetry breaking and naturalness.
8 other luminous matter contribute a small fraction of the
0 critical density, Ω = (0.003 ± 0.001)h−1, while the Thisargumentisofgreatimportancewhenconsidering
0 lum the upcoming direct detection experiments. For the
amount of matter known to exist from its gravitational
0 mSUGRA framework, one expects that the future am-
effects (both at galaxy and cluster of galaxies scales) is
/ bitious direct detection experiments can explore most of
h much larger, Ω = 0.35±0.07 (Ref. [2]). Furthermore,
p most of the mMissing matter seems to be non-baryonic theparameterspace. However,wefindthatthedetection
- picture is not quite as rosy for a more general MSSM
p in nature. The experimental motivation behind the
framework.
e dark matter problem and different search strategies are
h discussed in more detail in [3, 4, 5].
:
v In supersymmetric models, R-parity is often imposed
II. DEFINITIONS AND APPROACH
i to avoid weak-scale proton decay or lepton number
X
violation. Imposing this symmetry also yields an ideal
r Weadoptthefollowingnotationforthesuperpotential
a fermionic dark matter candidate. Namely, in supersym-
andsoftsupersymmetrybreakingpotentialintheMSSM:
metricmodelswithR-parity,thelightestsupersymmetric
particle(LSP)isstableanditcouldconceivablymakeup
W = ǫ (−eˆ∗h ˆliHˆj −dˆ∗h qˆiHˆj
asubstantialpartofthedarkmatterinthegalactichalo. ij R E L 1 R D L 1
Here we investigate the direct detection of such a +uˆ∗RhUqˆLiHˆ2j −µHˆ1iHˆ2j), (1)
particle. There have been many such studies in the V = ǫ (e∗A h liHj +d∗A h qiHj
literature [4, 6, 7, 8, 9, 10]. More recently, there have soft ij R E E L 1 R D D L 1
been discussions of numerous variables that can effect −u∗RAUhUqLiH2j −BµHˆ1iHˆ2j +h.c.)
direct detection. These studies include an investigation +Hei∗m2 Hei +Hi∗me2 Hi e
1 H1 1 2 H2 2
of the effect of the rotation of the galactic halo [11], the +qei∗M2qi e+li∗M2li +u∗M2u
effectsoftheuncertaintyofthequarkdensitieswithinthe L Q L L L L R U R
nuclei[12,13,14],possibleCPviolation[15,16],andnon- +d∗M2d +e∗M2e
universalityof gauginomasses [13, 17]. Here we attempt e1R DeR eR EeR e e
to address the following question: Is there a minimum +2e(M1BeB+eM2WaeWa+M3gbgb). (2)
crosssectionfortheelasticscatteringofneutralinosoffof
ordinarymatter? Naively,itwouldseemthatajudicious Here the h’s are Yukaweaecouplinfgs,fthe A’s aereetrilinear
choiceofparametersmightallowacompletecancellation couplings, the M are the squark and slepton
Q,U,D,L,E
between different diagrams. After all, the parameter massparameters,theM1,2,3aregauginomassparameters
space is very large in the general Minimal Supersym- and m , m , µ, and B are Higgs mass parameters.
H1 H2
2
Theiandj areSU(2) indices,andaremadeexplicit,so densityofthe neutralinosforthe giveninput parameters
L
as to make our sign conventionsclear. SU(3) indices are following [20], which includes the relativistic Boltzmann
suppressed. In the R-parity invariant MSSM the LSP is averaging, sub-threshold and resonant annihilation and
usuallyaneutralino-amixtureofbino,neutralwinoand coannihilation processes with charginos and neutralinos.
two neutral higgsinos. In our notation, the neutralino Furthermore, DarkSUSY checks for the current con-
mass matrix reads straintsobtainedbyexperiments,includingtheb→s+γ
constraint [21, 22].
M1 0 −mZsθWcβ +mZsθWsβ
0 M2 +mZcθWcβ −mZcθWsβ
 −m s c +m c c 0 −µ 
−mZsθWsβ −mZcθWsβ −µ 0 III. MSUGRA FRAMEWORK
 Z θW β Z θW β 
 (3)
A. Definition of the Framework
Here s = sinβ, c = cosβ, s = sinθ , and c =
β β θW W θW
cosθ . Thephysicalstatesareobtainedbydiagonalizing
W Inthissection,webrieflyoutlinethemSUGRAframe-
thismatrix. Thelightestneutralinocanbewritteninthe
work. In mSUGRA, one makes severalassumptions:
form:
• There exists a Grand Unified Theory (GUT) at
χ01 =N11B˜+N12W˜3+N13H˜10+N14H˜20. (4) some high energy scale. Consequently, the gauge
couplingsunify attheGUT scale. Thevalue ofthe
We are interested in spin independent scattering of
couplings at the weak scale determines the GUT
neutralinos off of ordinary matter. This contribution
scale to be ≈ 2 × 1016 GeV. The gaugino mass
dominates in the case of detectors with large nuclei,
parameters also unify to m at the GUT scale.
such as Ge [18]. As discussed in the literature, in 1/2
most situations the dominant contribution to the spin
• Other unificationassumptionsare: the scalarmass
independentamplitudeistheexchangeofthetwoneutral
parametersunifytoavaluedenotedbym0 andthe
Higgsbosons,althoughinsome casesthe contributionof
trilinear couplings unify to A0 at the GUT scale.
thesquarkexchangeandloopcorrectionsaresubstantial.
Using the MSSM renormalization group equations
The relevant tree-leveldiagrams are shown in Figure 1.
(RGEs) we evaluate all parameters at the weak
WeusetheDarkSUSYpackage[19]inthisstudy. This
scale. We choose to do this using the one loop
code has the following inputs: M1,2,3, µ, the ratio of RGEs that can be found, for example, in [23] or
the vevs of the two Higgs bosons (tanβ = v2/v1), the [24].
mass of the axial Higgs boson (m ), the soft masses of
A
thesparticles(MQ,U,D,L,E)andthediagonalcomponents • Radiative electroweak symmetry breaking
of the trilinear coupling matrices (AE,D,U). All inputs (REWSB) is imposed: minimization of the one-
are to be supplied at the weak scale. DarkSUSY then loop Higgs effective potential at the appropriate
calculates the particle spectrum, widths and couplings scale fixes µ2 and m (we follow the methods
A
based on the input parameters. It evaluates the cross of Refs. [25] and [26]). For completeness, we
section for scattering of neutralinos off protons and reproduce the equation for µ2 at tree level:
neutrons, following Ref. [10]. It also evaluates the relic
m2 −m2 tan2β 1
µ2 = H1 H2 − M2. (5)
tan2β−1 2 Z
With these assumptions, the mSUGRA framework
χ
q q q allows four free parameters (m0, m1/2, A0 and tanβ).
Also, the sign of µ remains undetermined. Starting
~ with these parameters we determine all of the input
q
h,H parameters for the DarkSUSY code. We allow the free
parameters to vary in the intervals
χ q χ χ 0<m1/2 <300GeV, 95<m0 <1000GeV,
−3000<A0 <3000GeV, 1.8<tanβ <25. (6)
The upper bounds on these parameters come from the
naturalness assumption: one of the reasons for using
supersymmetry is its ability to naturally relate high
FIG.1: Theleadingdiagramsfordirectdetection. Notethat and low energy scales; as a result, no parameter in the
thereisalsoau-channeldiagram forsquarkexchange. There theory should be very large. Moreover,the upper bound
are also diagrams where the neutralino scatters off of gluons on m makes our data set insensitive to the stau-
1/2
in thenucleon through heavysquark loops. neutralinocoannihilations,whicharenotincluded inthe
3
calculationofrelic density performedby DarkSUSY. We 10−40
have explicitly checked that there are no models in our
data set which would be cosmologically allowed only if
the stau coannihilations were included. We will also see
laterthatexpandingtheupperboundonm1/2hasserious eon)10−42
consequences regarding the lower bound on the elastic ucl
n
scattering cross section. Also, the low value of tanβ is o
setbytherequirementthatthetopYukawacouplingdoes d t
not blow up before the GUT scale is reached. alise10−44
m
Beforewepresentthedetailedanalysisofcrosssection, or
afewremarksareinorder. First,theb→s+γconstraint 2, n
eliminates large portions of the µ < 0 parameter space, cm10−46
in agreement with [27], [28]. Second, for both µ > 0 n(
o
and µ<0, we find no higgsino-like LSP models that are cti
e
s
cosmologically important, in agreement with [28]. −
s
We plot the variation of the spin independent cross os10−48
Cr
section versus the neutralino mass in Figure 2. Note
that the complete allowed region is split into two parts
by the annihilation channel into W+W−, which affects
10−50
the relic density of neutralino. We consider two relic 101 102
density constraints. Since the present observations favor WIMP Mass (GeV)
the Hubble constant h = 0.7±0.1 and the total matter
density Ω = 0.3 ± 0.1, of which baryons contribute FIG. 2: The cross section for spin-independent χ-proton
M
Ω h2 ≈ 0.02, we consider the range 0.052 < Ω h2 < scattering is shown. Current accelerator bounds, including
b χ
0.236. However, we also examine effects of relaxing the b → s+γ are imposed through the DarkSUSY code. The
relic density constraint to 0.025<Ω h2 <1. The upper top dark region is the DAMA allowed region at 3σ CL. The
χ
dashedcurveistheDAMA90%CLexclusionlimitfrom1996
bound on σχ−p (of the theoretically allowed regions)
(obtainedusingpulse-shapeanalysis). Thethinnersolidcurve
comes from the lower bound on the relic density. The
isthecurrentCDMS90%CLexclusionlimit,thethickersolid
lower bound on M comes from the existing constraints
χ curveistheprojectedexclusionlimitforCDMSIIexperiment
from the accelerator experiments. The upper bound
and the dotted curve is the projected exclusion limit for
on M is a combination of the upper bound on relic
χ the GENIUS experiment. The light shaded regions denote
density and of the bounds on the free parameters. The mSUGRA models passing the 0.025 < Ωh2 < 1 constraint
lower bound on σχ−p is not yet well understood, and andwiththeupperboundm1/2 <1TeV.Asdiscussedinthe
it is the subject of this paper. Figure 2 also includes text, this region shows that including stau coannihilations
some recent and future direct detection experimental into the relic density calculation has a dramatic effect on
results [29, 30, 31, 32, 33]. Note that the parameter the lower bound on σχ−p for m1/2 > 300 GeV (Mχ > 120
space defined by Eq. (6) corresponds to the region of GeV). Restricting the relic density further to 0.052<Ωh2 <
σχ−p - Mχ plane bounded by the two closed solid 0.236 yieldsthedarkerregions within thelighter regions and
lines. Therefore, σχ−p > 10−46cm2 for these models, restricting also the upper bound on m1/2 < 300 GeV (Eq.
or equivalently, assuming a 73Ge target, the dark matter (6)) gives theregions boundedby theclosed solid curves.
densityρ =0.3GeVc−2cm−3,theWIMPcharacteristic
D
evveleonctityratve0 R= 2>300k.1mtso−n1−a1nddayf−o1ll.owiHnegnRcee,f.t[h3e4],mtohset therefore, concentrate on the Higgs boson exchange and
will postpone the discussion of squark exchange to the
ambitious future direct detection experiments may be
end of this section.
able to explore a large portion, if not all, of the these
ThecontributionofHiggsbosonexchangecanbefound
models.
in the literature [4, 10, 13, 15, 35, 36, 37]. It is of the
following form:
B. Results and Analysis σ ∼|(f +f +f )Au+(f +f +f )Ad|2, (7)
h,H u c t d s b
where f ≈ 0.023,f ≈ 0.034,f ≈ 0.14,f = f = f ≈
As mentioned above, the dominant contribution to u d s c t b
spin independent elastic scattering is usually the Higgs 0.0595 parametrize the quark-nucleon matrix elements
and
boson exchange. Figure 3 illustrates this relationship
within our results. The nearly perfect 45 degree line in g2 F cosα F sinα
the figure indicates good agreement between the total Au = 2 h H + H H , (8)
4M m2 sinβ m2 sinβ
cross section as evaluated by DarkSUSY and the cross W h H
(cid:16) (cid:17)
sectioncalculatedincludingtheexchangeofHiggsbosons Ad = g22 −Fh sinαH + FH cosαH , (9)
only (in the approximation explained below). We will, 4M m2 cosβ m2 cosβ
W h H
(cid:16) (cid:17)
4
10−42
140
Case 4 Case 4
0.025< Ω h2 < 1
120
Accelerator bounds not applied
10−43 0 < m1/2 < 300 GeV 100 ase 2
C
80
2m)10−44 GeV) 60
(c− p M (1 40
χ
σ10−45
20
Case 1
0
10−46 −20 oa c−c eClearsaet o3r mbooudnedlss n o t a p p lied Case 1
0.025 < Ω h2 < 1
−40
−1000 −800 −600 −400 −200 0 200 400 600 800 1000
10−47 µ (GeV)
10−47 10−46 10−45 10−44 10−43 10−42
σ (cm2)
H,h exchange only FIG. 4: This figure illustrates the regions of M1 −µ plane
(as defined by the Eq. (6)) in which the different cases
discussed in thetext can besatisfied. For themodels shown,
FIG. 3: The cross section for spin-independent χ-proton the accelerator constraints were not applied, but a relatively
scattering: the complete (DarkSUSY) calculation is shown conservativerelicdensity cut(0.025<Ωh2 <1) was applied.
on the y axis and the contribution to the cross section from Note that almost all of the models of Case 3 shown in this
the Higgs bosons exchange alone is shown on the x axis. A plot are ruled out by the accelerator constraints. Also, there
relatively conservative relic density cut is applied, 0.025 < are no models satisfying the Case 5 for the parameter space
Ωh2 <1. definedby Eq. (6).
and
crosssectionisrelativelysmall. Figure4illustrateswhich
Fh = (N12−N11tanθW)(N14cosαH +N13sinαH) creagseiosn.sInoftthheefMol1lo-wµipnlga,nweesawtiislflyutsheetchoenadpitpiroonxsimofatthieonfivoef
FH = (N12−N11tanθW)(N14sinαH −N13cosαH(1)0.) Eq. (11).
TheN’sarethe coefficientsappearinginEq.(4)andαH • Case1: N12−N11tanθW =0wouldmakebothFh
is the Higgs boson mixing angle (defined after radiative
and F vanish. Intuitively, this is reasonablesince
H
correctionshavebeenincludedintheHiggsmassmatrix).
this condition implies that the neutralino is a pure
Note that there is an upper bound on the light Higgs
photino, and the tree level Higgs coupling to the
boson mass in the MSSM, given by m <130 GeV [38].
h photino vanishes. Using Eq. (11), we may rewrite
Au representstheamplitudeforscatteringoffanup-type
this condition as
quarkinanucleon,whileAd representstheamplitudefor
smcoatdteelrsinwgeogffenaedraowtenh-tayvpeeaqubainrok-ilnikethneenuutrcalelionno.,Sµin>ceMal1l µ(M2−M1)=−MZ2cos2θW (cid:18)sin2β+ Mµ1(cid:19). (12)
and µ > M . Then, following Ref. [35], we can expand
theN1i’souZtinpowersof MµZ. Wereproducetheirresult sSainticsefieMd2on>lyMif1µ><0, 0t.he lIafstweequuasetiotnhecaGnUbTe
here:
relationship betweeen M1 and M2, we get
NN1121 ≈≈ 1−,12MµZ (1−sinM2θ12W/µ2)M2M−ZM1 (cid:20)sin2β+ Mµ(11(cid:21)1,) µM1(cid:18)35csoins22θθWW −1(cid:19)=−MZ2cos2θW (cid:18)sin2β+ Mµ(11(cid:19)3)
MZ 1 M1
N13 ≈ µ 1−M2/µ2 sinθW sinβ 1+ µ cotβ , or, equivalently, for µ and M1 in units of GeV
1 (cid:20) (cid:21)
N14 ≈ −MµZ 1−M12/µ2 sinθW cosβ 1+ Mµ1 tanβ . M1 = −6µ4+496s4i4n92β. (14)
1 (cid:20) (cid:21) µ
Inthisapproximation,wecanidentifyfivepossiblesit- As sin2β ranges from 0 to 1, Eq. (14) spans the
uations in which the neutralino-protonelastic scattering dashed regions marked ’Case 1’ in Figure 4. For
5
sin2β = 1, Eq. (14) implies Mχ ≈ M1 < 40 GeV. we get a roughly linear relationshipbetween µ and
This canbe readdirectly fromFigure4. Note that M1:
such low values of M are excluded by the relic
χ
density constraint and the current experimental M1 =(0.3|µ|−60)±40. (21)
limits (as shown in Figure 2 and Ref. [13]). The
constraint on M is even stronger for other values The spread ±40 comes from the variation in m0
χ
and tanβ and the linearity breaks downsomewhat
ofsin2β,soweconcludethatthisconditioncannot
be satisfied in the mSUGRA framework. atthelowvaluesofM1. Theempiricalrelationship
that we obtain from running the code (see Figure
• Case 2: N14sinαH −N13cosαH = 0 would make 4) is very similar:
only F vanish. In our approximation, this condi-
H
tion translates into M1 =(0.3|µ|−40)±25. (22)
M1/µ=−cotαH −cotβ. (15) The smaller spread comes from the application of
the relic density cut and the current experimental
Tounderstandthemeaningofthisconditionbetter, limits. In any case, we conclude that M1/µ > 0.3
we will use the tree level relationship between αH is not allowedin the mSUGRA framework. This is
and β: in conflict with Eq. (18), implying that the Case
2 cannot be satisfied. Note that the tree level
cot2α =kcot2β, k = m2A−MZ2, (16) relationship in Eq. (16) is altered at higher orders,
H m2 +M2 but we have checked that this does not affect the
A Z
final conclusion.
which, after some trigonometric manipulations,
yields • Case 3: N14cosαH +N13sinαH = 0 would make
F vanish. Manipulation of this condition using
h
k 1 Eq. (11) yields
cotα = −tanβ
H
2 tanβ
(cid:18) (cid:19)
M1/µ=tanαH −cotβ. (23)
k2 1
− +tan2β−2 +1. (17)
s 4 tan2β Figure5showsthatwhenthisconditionis(approx-
(cid:16) (cid:17)
imately) satisfied, the contribution of the heavy
SincebothtermsontherighthandsideofEq.(17) Higgsbosonexchangetotheelasticscatteringcross
are negative because tanβ > 1, the minimum of section is the largest. Hence, even if this condition
|cotαH| = 1 occurs when k = 0 (or, equivalently, is satisfied, σχ−p cannot be arbitrarily small due
when m = M ). Then, since tanβ > 1.8, the to heavy Higgs boson exchange (along with other
A Z
condition of Eq. (15) becomes channels such as the squark exchange). Of course,
this assumes that there is some upper bound on
M1 theheavyHiggsmass,whichistruesimplybecause
>0.5. (18)
µ the parameter space is bounded. Note, also, that
sincetanα <0,theconditionforthevanishingof
H
At this point, we would like to formulate a re-
the light Higgs boson contribution can be satisfied
lationship between µ and M1 resulting from the onlyforµ<0. Itisinteresting,however,thatmost
RGEs and REWSB assumptions. In Ref. [24],
of the models in our data set that approximately
an approximate solution (based on the expansion
satisfy this condition are excluded by the bounds
around the infrared fixed point) to the RGEs for
on Higgs boson masses.
the Higgs mass parameters, m and m , is
H1 H2
presented. Assuming that the value of the top • Case 4: There is one more way of making both F
h
Yukawa coupling is relatively close to the infrared and F small, and that is by making µ very large
H
fixed point, we can write: (N12, N13, and N14 all contain µ in denominator).
However, this possibility is limited by naturalness
m2H1 ≈m20+0.5m21/2, assumption: µ is kept below ≈ 850 GeV by the
m2 ≈−0.5m2−3.5m2 . (19) upper bound we have chosen on m1/2. Hence, in
H2 0 1/2 mSUGRA framework, the naturalness assumption
also keeps the cross section from vanishing.
Theseequations,coupledwithEq.(5),yieldavalue
for µ2 in terms of m1/2,m0, and tanβ. Using the • Case5: Wenowconsiderthepossibilityofcomplete
GUT relationship: cancellation of terms in the Equation (7). Let
us examine the relative signs of F , F and µ.
h H
M1 =m1/2αα1G(mUTZ), (20) SEiqn.ce(1t1a)ntβha>t |1N.813a|n>d|MN114/|µ. ≈Th0e.n3,,sitinfcoellcoowtsαfHrom<
6
10−43 If the upper bound on m1/2 is relaxed to 1 TeV,
the situation changes significantly. The complete
cancellation discussed above is now allowed to
happen. As shown in Figure 6, the lowest values
10−44 of σχ−p are reached exactly when the Ad and Au
terms canceleachother. AsshowninFigure2,the
stringent relic rensity cut 0.052 < Ωh2 < 0.236
rules out all of these models. However, we do
10−45
2m) not include the stau-neutralino coannihilations in
c the calculation of the relic density, so this result
(−p should be taken with caution (see, for example,
χ
σ
10−46 [39, 40]). In fact, our data set contains models
0.025 < Ω h2 < 1 in which the stau-neutralino coannihilation could
make a difference. For this reason, the Figure 2
Accelerator bounds not applied
also contains the allowed region for a conservative
10−47 0 < m1/2 < 300 GeV relic density cut0.025<Ωh2 <1. Clearly,this cut

o ’ s d e n o t e m o d e l s p a s s i n g allows models with very low σχ−p values.
0.8 < (M / µ ) / (tan α − cot β) < 1.2
1 H
10−48
10−48 10−47 10−46 10−45 10−44 10−43
σ (cm2)
H exchange only 10−9
FIG. 5: The proton-neutralino scattering cross section for
models where thecondition in Case 3is approximately satis-
fied(inparticular, modelssatisfying 0.8< M1/µ <1/2
tanαH−cotβ
aredenotedbyo’s). Thecomplete(DarkSUSY)calculationis
shownonthey-axisandthecontributiontothecross section 10−10
dueonly to theexchangeof theheavy Higgs boson, is shown
dA
aopnptlhieed,x0-a.0x2is5. <AΩrehl2at<ive1ly. cMonossterovfattihvee mreolidcedlsendseintyotceudtbiys + f) b
o’sinthisplotareexcludedbytheacceleratorboundsonthe + f s
Higgs boson masses. (f d
10−11
0.025 < Ω h2 < 1
−1 (and sinαH < 0), the N13 term dominates
Accelerator bounds applied
over the N14 term in FH (Eq. (10)). Hence,
F /µ > 0 always, consistent with the analysis of Red o’s denote models
H
Case 2 above. The situation is somewhat more with σχ−p<10−48 cm2
complicatedin the caseofF . For µ>0,following
h 10−12
aintseirmfeilraerncaenbaelytwsiesenwethgeettwFoht/eµrm>s in0.AuT(hEeqn.,(t8h)e) 10−12 10−11 (fu + fc + ft) Au 10−10 10−9
isdestructive andthe interferencebetweenthe two
termsinAd (Eq.(9))isconstructive. Furthermore,
in Eq. (7) we see that Au gets multiplied by a FIG. 6: The absolute values of the Ad and the Au terms
of Eq. (7) are shown. Only Ad < 0 (for which µ < 0)
much smaller form factor than Ad. As a result, Ad
are shown, satisfying a conservative relic density constraint
strongly dominates over Au in Eq. (7), preventing 0.025 < Ωh2 < 1, passing the accelerator bounds, and with
σχ−p from vanishing. m1/2 < 1 TeV. The o’s denote the lowest σχ−p models (<
On the other hand, if µ<0 and if µ>−M1tanβ, 10−48 cm2). Clearly, the lowest values of σχ−p are achieved
N14 can change sign relative to µ. This change when the Ad and Au terms cancel each otherout.
of sign can propagate through Equations (10), (9),
and(7),sothattheAu andAd termsintheEq. (7) With the above discussionin hand, let us go back and
are of opposite sign. If the parameters are tuned consider the squark exchange. The complete calculation
properly,acompletecancellationofthesetermscan of the squark exchange contribution is fairly complex.
be obtained. However, this cancellation happens However,good insights can be gained by making several
onlyifMχ(≈M1)>120GeV,whichisnotallowed simplifying assumptions. First of all, the contribution of
due to the upper bound on m1/2. We conclude thesquarkexchangecanberoughlyapproximatedbythe
that in the parameter space defined by Eq. (6), Ad contribution of the exchange of the u, d, and s squarks.
alwaysdominatesoverAu,soσχ−pdoesnotvanish. In this case, the contribution of the squark exchange to
7
the cross-sectioncan be written as Here we have only kept the contribution of the strange
quark to the Higgs exchange amplitude (Ad) as well.
σq˜∼|fuBu+fdBd+fsBs|2, (24) Note that in general, the light Higgs boson contribution
will dominate. As expected, this is basically due to the
where f is as defined above, and the B represent the
i j fact that squarksare in generalheavier than the lightest
amplitude for scattering off of a quark of type j in the
Higgsboson. Thesquarkexchangecanbeimportantonly
nucleon. Furthermore, in the following considerations
ifthecontributionfromtheexchangeoftheHiggsbosons
we neglect the left-right mixing in these light squarks.
is fine-tuned to be very small.
This should be true over a large class of models, as
the off-diagonal elements in the squark mass matrix
are proportional to the corresponding quark mass. Let
IV. GENERAL MSSM FRAMEWORK
us also neglect the mass splitting of the two different
squarks. Also, since f ≫ f , f , we can neglect all but
s u d A. Definition of the Framework
the B term. In this approximation, following Ref. [15],
s
we can write:
In this framework we relax our assumptions. We
1 1
Bs =−4msMs˜2−Mχ201[2C1C2−2C1C3], (25) kdereopp tthhee ruenqiufiicreamtioenntsofthtahtethgeausgcianloarmmaasssseess, abnudt twhee
trilinear scalar couplings unify. In addition, we drop the
where we have defined the following: REWSBrequirement(i.e.,wetakem2 asindependent
H1,2
g2msN13 parameters from m0). We assume that all scalar mass
C1 = , parameters at the weak scale are equal: m . This
2M cosβ sq
W
assumption is made in order to simplify the calculation,
C2 = eQy1+ g2 y2[T3−Qsin2θW], and it should not affect the general flavor of our results.
cosθ
W Of all trilinear couplings, we keep only A and A and
C3 = eQy1− g2 y2Qsin2θW. (26) we set all others to zero. Then, the frete parambeters
cosθ
W are µ,M2,tanβ,mA,msq,At,Ab. We also relax the
Note that C1 represents the coupling of the down type naturalness assumption, allowing the free parameters to
quark to the Higgsino portion of the neutralino. C2 have very large values. Besides the relatively uniform
and C3 represent the couplings of bino to the left and scans of the parameter space, we also performed special
right handed quark, respectively. Here, T3 is the SU(2) scans in order to investigate the different conditions
quantum number of the squark in question, Q is the mentioned in the previous section. The free parameter
charge,y1 denotesthephotinofractionoftheneutralino, space is then:
while y2 denotes the zino fraction. They are given by:
−300TeV<µ<300TeV, 0<M2 <300TeV,
y1 = N11cosθW +N12sinθW, 95GeV<mA <10TeV, 200GeV<msq <50TeV,
y2 = −N11sinθW +N12cosθW. (27) At,b
−3< <3, 1.8<tanβ <100. (31)
m
After approximating y2 ≈ −sinθW and using sq
′
tanθ = g /g, a brief and straight-forward calculation
W Again, a few comments are in order. First, in this
yields a simple expression for the amplitude due to the
framework we observe higgsino-like (as well as bino-like)
exchange of the strange squarks:
lightest neutralino. In agreement with the Ref. [42],
′ we find very few light higgsino-like models, which will
−g2g N13
Bs = 8M cosβM2−M2. (28) probably be explored soon by accelerator experiments.
W s˜ χ Most of the higgsino-like models (with gaugino content
z < 0.01) have M > 450 GeV, implying very large
Furthermore, we can write the masses Ms˜ and Mχ in g χ
values of m . In particular, in higgsino like models
terms of the input parameters of mSUGRA. This is 1/2
because the Yukawa couplings can be neglected in the M1 > µ ≈ Mχ; our results give M1 > 700 GeV or,
equivalently,m >1700GeV,which canbe considered
RGEs. FollowingthemethodsdescribedinRef.[41],and 1/2
unnatural. For these reasons, we choose not to analyze
using the Eq. (20) we can write
the higgsinocase. Second,b→s+γ is less constraining,
M2 −M2 ≈m2+5.8m2 . (29) but our results are still consistent with Refs. [27], [28].
s˜R χ 0 1/2
We present the plot of σχ−p versus Mχ in this frame-
Using Eqs. (9) and (28), we can compare the squark work(Figure7). WedonotpretendthatFigure7reflects
exchange to the light Higgs boson exchange: allpointsaccessibleinageneralMSSM.However,itdoes
servetoshowsomegenericdifferencesfromthemSUGRA
Ad 2sinαH(N14cosαH +N13sinαH)(m20+5.8m21/2) case. Namely, we can obtain much larger values for the
= .
Bs m2hN13 neutralino mass because of the size of the parameter
(30) space. Inaddition,thelowerboundonσχ−p isalsomuch
8
−40 boson exchange is small and heavy Higgs boson
10
0.025 < Ω h2 < 1 exchange dominates. Unlike in the mSUGRA
b → s + γ applied case, the accelerator bounds do not rule out these
models.
−45 • Case 4: Since the naturalness constraint has been
10
2) relaxed, µ is allowed to have very large values.
m
(c ThenN12,N13,andN14 canbedrivensmall,which
− p in turn would make the Higgs boson exchange
σχ contribution small. Intuitively, large |µ| implies
10−50 that the neutralino is a very pure bino, for which
the Higgs boson scattering channels vanish. If the
squark masses are kept large as well, the squark
contribution will be small too, making the total
elastic scattering cross section very small. This is
1 2 3 4
10 10 10 10 illustrated in Figure 8 - the lowest values of σχ−p
Mχ (GeV) are obtained for the largest values of |µ|.
• Case 5: As discussed in the mSUGRA case, if
FIG. 7: The cross section for spin-independent χ-proton
scattering in the general MSSM framework is shown. A (Mµ1 tanβ) is negative and sufficiently large, N14
relatively conservative relic density cut is applied, 0.025 < can change sign. Following through Eqs. (10) and
Ωh2 < 1. Note also that the constraint on the gaugino (9), this effect can induce destructive interference
fraction, zg > 10, is applied. Current accelerator bounds, between the Au and Ad terms in Eq. (7) and
includingb→s+γareimposedthroughtheDarkSUSYcode. cause the overallσχ−p to vanish. We observedthis
cancellation in the mSUGRA case for M > 120
χ
GeV, and we also observe it in the general MSSM
lowerthaninthemSUGRAcase. Wediscussthespecifics
case. In Table 1 we present some of the models in
of this below.
whichthiskindofcancellationtakesplace. Besides
the models at large M , we also observe models
χ
with relatively low Mχ(≈M1) and very low values
B. Results and Analysis
ofσχ−p(seerows3and4ofTable1). Thesemodels
werenotallowedinthemSUGRAframeworkdueto
We concentrate only on the bino-like lightest neu- the M1−µ relationship(determined by the RGE’s
tralino. All results in this section are presented with andtheREWSBassumption),whichdoesnotexist
this assumption in mind. In particular, we demand in the general MSSM framework.
z =(N2 +N2 )/(N2 +N2 )>10. Inthis case,we can
g 11 12 13 14
relyonthe sameapproximationsweusedinthe previous In other words, in the general MSSM framework it
section. In particular, the expansion of Eq. (11) is valid. is possible to obtain low values of σχ−p in several
So, we can revisit the 5 different cases explored in the different ways: either by tuning parameters to suppress
previous section. the contribution of the heavy or the light Higgs boson
exchange, or by allowing parameters (such as µ) to be
• Case 1: N12−N11tanθW = 0 cannot be satisfied very large, which suppreses both the heavy and the
because, as in the mSUGRA case, it implies Mχ < light Higgs boson exchange channels, or by fine-tunning
40 GeV, which is ruled out by the relic density parameters to achieve a complete cancellation of terms
cut and the experimental limits (as shown on in the Eq. (7). As a result, the lower bound on σχ−p
Figure 7). The models that get close to satisfying vanishes and it is beyond reach of the present and the
this condition have very low contributions due to proposed direct detection experiments.
theHiggsbosonsexchange,sothisisoneoftherare
situationswherethe squarkexchangeisimportant.
V. CONCLUSION
• Case2: N14sinαH−N13cosαH =0isnowpossible
to satisfy because µ and M1 are not related.
We indeed observe that the heavy Higgs boson We summarize our results as follows. The main
exchangecontributionisverysmallinthiscase,but contributions to the cross section for spin independent
since the light Higgs boson exchange dominates, elastic scattering of neutralinos off nucleons come from
σχ−p is kept relatively high in value. the exchange of the Higgs bosons and squarks. The
contribution of the squark exchange is usually much
• Case3: N14cosαH+N13sinαH =0isalsopossible smaller, being of importance only when Higgs boson
tosatisfy. AsinthemSUGRAcase,thelightHiggs exchange contribution is very small.
9
µ (GeV) M1 (GeV) msq (GeV) mA (GeV) tanβ At/msq Ab/msq σχ−p (cm2)
-1794 502 3792 1004 10.1 1.2 2.5 7.9×10−51
-2109 534 3211 1087 11.8 -1.3 1.0 8.4×10−51
-195 55 2995 1120 10.2 -2.0 2.3 2.1×10−50
-182 61 2891 1099 7.0 -0.6 -0.1 1.6×10−50
-274 163 325 1944 3.8 0.6 2.5 7.8×10−50
TABLEI: Some of the models in thegeneral MSSM framework with very low values of σχ−p.
mSUGRA framework to allow larger neutralino masses,
0.025 < Ω h2 < 1 the lowerbound onσχ−p vanishes for those largemasses
10−42 b → s + γ applied due to the occasional complete cancellation of terms in
the Eq. (7).
In the more general MSSM models (as defined in and
above Eq. (31)), the situation is significantly different.
2m) 10−46 The light and/or heavy Higgs boson exchange channel
c
(p can now be suppressed either by tuning parameters to
χ − satisfy Cases2)or3), orby allowingthe free parameters
σ (such as µ) to be very large. Moreover, the complete
−50
10
cancellation of terms in the Eq. (7) is now possible even
at low values of M because the relationship between
χ
M1 and µ that existed in the mSUGRA framework due
to radiative electroweak symmetry breaking, is relaxed
−54
10
1 2 3 4 5 6 in the general MSSM framework. As a result, the lower
10 10 10 10 10 10
| µ | (GeV) bound on σχ−p is much lower than in the mSUGRA
framework and it is beyond reach of the current or
proposed direct detection experiments.
FIG. 8: The cross section for spin-independent χ-proton
scattering in the general MSSM framework is shown. A
relatively conservative relic density cut is applied, 0.025 <
Ωh2 < 1. Note also that the constraint on the gaugino
VI. ACKNOWLEDGEMENTS
fraction, z > 10, is applied. Current accelerator bounds,
g
includingb→s+γareimposedthroughtheDarkSUSYcode.
VM thanks Bernard Sadoulet and Richard Gaitskell
for discussions and suggestions regarding this work. PG
Weinvestigatedifferentconditionswhichcouldleadto thanks Bernard Sadoulet for hospitality at the CfPA.
small Higgs boson exchange contribution. We find that The work of HM and AP was supported in part by the
in mSUGRA framework,with the free parameter ranges Director, Office of Science, Office of High Energy and
defined in Eq. (6), these conditions are not satisfied Nuclear Physics, Division of High Energy Physics of the
due to the relationship between parameters M1 and µ U.S. Department of Energy under Contract DE-AC03-
(coming from the unification and radiative electroweak 76SF00098and in part by the National Science Founda-
symmetry breaking assumptions), the naturalness as- tion under grant PHY-95-14797. AP is also supported
sumption (which keeps different parameters from be- by a National Science Foundation Graduate Fellowship.
coming very large) and the accelerator constraints. We TheworkofVMwassupportedbytheCenterforParticle
find that the light Higgs boson exchange dominates over Astrophysics, a NSF Science and Technology Center
the other channels and it leads to σχ−p > 10−46cm2. operatedbytheUniversityofCalifornia,Berkeley,under
Equivalently, this yields an event rate >0.1ton−1day−1 Cooperative Agreement No. AST-91-20005 and by the
in73Ge target,whichcouldbe withinreachofthe future National Science Foundation under Grant No. AST-
direct detection experiments. However,if we expand the 9978911.
[1] H.Murayama, hep-ph/0002232. Rept. 267, 195 (1996).
[2] M.S.TurnerandJ.A.Tyson,Rev.Mod.Phys.71,S145 [5] R.KolbandM.Turner,TheEarlyUniverse,(NewYork:
(1999). Addison-Wesley),1994.
[3] B. Sadoulet, Rev.Mod. Phys. 71, S197 (1999). [6] P. Nath and R. Arnowitt, Phys. Rev. Lett. 74, 4592
[4] G. Jungman, M. Kamionkowski and K. Griest, Phys. (1995). M. Drees and M. Nojiri, Phys. Rev. D 47, 4226
10
(1993). (1995).
[7] M. Drees and M. Nojiri, Phys.Rev. D 48, 3483 (1993). Abeetal.(CDFCollab.),Phys.Rev.D56,R1357(1997).
[8] A.Bottino et al., Mod. Phys.Lett A7, 733 (1992). Abe et al.(CDF Collab.), Phys. Rev. Lett. 69, 3439
[9] G. Gelmini, P. Gondolo and E. Roulet, Nucl. Phys. B (1992).
351, 623 (1991). Abe et al.(CDF Collab.), Phys. Rev. Lett. 76, 2006
[10] L. Bergstrom and P. Gondolo, Astropart. Phys. 5, 263 (1996).
(1996). Barate et al.(ALEPH Collab.), Phys. Lett. B 433, 176
[11] M. Kamionkowski and A. Kinkhabwala, Phys. Rev. D (1998).
57, 3256 (1998). C. Caso et al.(Particle Data Group), Eur. Phys. J. C3,
F. Donato, N. Fonengo, and S. Scopel, Astropart. Phys. 1(1998),and1999partialupdateforedition2000(URL:
9, 247 (1998). hep-ph/9803295. http://pdg.lbl.gov).
[12] A. Bottino et al., Astropart. Phys. 13, 215 (2000). [23] V. Barger, M.S. Berger and P. Ohmann, Phys. Rev. D
hep-ph/9909228. 47, 1093 (1993).
[13] A.Corsetti and P.Nath, hep-ph/0003186. [24] V.Bargeretal.,“ReportoftheSUGRAWorkingGroup
[14] J. Ellis, A. Fersl and K. Olive, Phys. Lett. B 481, 304 for RunII of theTevatron.” hep-ph/0003154.
(2000). [25] R.ArnowittandP.Nath,Phys.Rev.D46,3981(1992).
[15] Chattopadhyayet al., Phys. Rev.D 60, 063505 (1999). [26] A. Brignole, J. Ellis, G. Ridolfi and F. Zwirner, Phys.
[16] T. Falk, K. Olive and M. Srednicki, Phys. Lett. B 364, Lett. B 371, 123 (1991).
99 (1995). J.Ellis.G.RidolfiandF.Zwirner,Phys.Lett.B257,83
T.Falk,A.FerstlandK.Olive,Phys.Rev.D59,055009 (1991).
(1999). M. Drees and M. Nojiri, Phys.Rev.D 45, 2482 (1992).
K. Freese and P. Gondolo, Nucl. Phys. B (Proc. Suppl.) [27] P.NathandR.Arnowitt,Phys.Lett.B336,395(1994).
87, 519 (2000). hep-ph/9908390. P. Nath and R. Arnowitt, Phys. Rev. Lett. 74, 4592
[17] A.Corsetti and P.Nath, hep-ph 0005234. (1995).
L. Roszkowski and M. Shifman, Phys. Rev. D 53, 404 [28] F.Borzumati,M.DreesandM.Nojiri,Phys.Rev.D51,
(1996). 341 (1995).
[18] M. Goodman and E. Witten, Phys. Rev. D 31, 3059 J. Ellis, T.Falk,G.GanisandK.A.Olive,Phys.Rev.D
(1985). 62, 075010 (2000). hep-ph/0004169.
[19] P.Gondolo et al., DarkSUSYmanual, Unpublished. [29] R. Bernabei et al., Phys. Lett.B 389, 757 (1996).
P. Gondolo, J. Edsjo, L. Bergstrom, P. Ullio, and E. A. [30] R. Bernabei et al., INFN Preprint: ROM2F/2000/01.
Baltz, astro-ph/0012234. [31] R. Abusiadiet al., Phys. Rev.Lett. 84, 5699 (2000).
[20] J. Edsj¨o and P.Gondolo, Phys. Rev.D 56, 1879 (1997). [32] R. W. Schnee et al., Phys. Rept.307, 283 (1998).
P.Gondolo and J. Edsj¨o, Nucl. Phys. B70, 120 (1999). [33] L. Baudis et al., Phys. Rept.307, 301 (1998).
[21] S.Bertolini et al., Nucl. Phys.B353, 591 (1991). [34] J. D. Lewin and P. F. Smith, Astropart. Phys. 6, 87
[22] Abbiendi et al.(Opal Collab.), Eur. Phys. J. C7, 407 (1996).
(1999). [35] R.ArnowittandP.Nath,Phys.Rev.D54,2374(1996).
GaoandGay(ALEPHCollab.),in“HighEnergyPhysics [36] K. Griest, Phys. Rev.Lett. 61, 666 (1988).
99”, Tampere, Finland, July 1999. [37] R.Barbieri,M.FrigeniandG.F.Giudice,Nucl.Phys.B
J. Carr et al., talk to LEPC, 31 March 1998 313, 725 (1989).
(URL: http://alephwww.cern.ch/ALPUB/seminar/ car- [38] See for example: J.R. Espinosa and R.-J. Zhang, JHEP
rlepc98/index.html). 0003, 2000 (026) and hep-ph/0003246, and the refer-
Acciarrietal.(L3Collab.),Phys.Lett.B377,289(1996). ences therin.
Decamp et al.(ALEPH Collab.), Phys. Rept. 216, 253 [39] J. Ellis, T. Falk, K. A. Olive, and M. Srednicki, As-
(1992). tropart. Phys.13, 181 (2000). hep-ph/9905481.
Hidaka,Phys. Rev.D 44, 927 (1991). [40] R.Arnowitt,B.Dutta,andY.Santoso,hep-ph/0102181.
Acciarrietal.(L3Collab.),Phys.Lett.B350,109(1995). [41] D.I. Kazakov, Talk given at “Renormalization Group
Buskulicetal.(ALEPHCollab.),Zeitschriftfu¨rPhysikC at the Turn of the Millennium”, Taxco, Mexico, 1999.
72, 549 (1996). hep-ph/0001257.
Acciarrietal.(L3Collab.),Eur.Phys.J.C4,207(1998). [42] J.Ellis, T.Falk,G.Ganis,K.A.Olive,andM.Schmitt,
Abbiendi et al.(OPAL Collab.), Eur. Phys. J. C8, 255 Phys. Rev.D 58, 095002 (1998). hep-ph/9801445.
(1999).
Abachi et al.(D0 Collab.), Phys. Rev. Lett. 75, 618