Elastic neutralino-matter scattering

Volume 237, number 1 PHYSICS LETTERS B 8 March 1990

ELASTIC NEUTRALINO-MATTER SCATTERING

Ricardo FLORES Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455. USA

Keith A. OLIVE School qfPhysics and Astronomy. University ofMinnesota, Minneapolis, MN 55455, USA

and

Mark SREDNICKI Department ofphysics, University of Califorma, Santa Barbara, CA 94306, USA

Received 12 December 1989

The lightest neutralino is a natural candidate for the dark matter in the Universe. It is very nearly a pure bino or a symmetric

or antisymmetric higgsino over a wide region of parameter space of the supersymmetric standard model. We consider here their

elastic scattering off of matter in the laboratory. We find that while binos are very much like photinos, the symmetric or antisym-

metric higgsinos would be invisible to proposed laboratory detectors.

Most of the matter in the Universe is invisible in the electromagnetic spectrum, and is only indirectly detected by its gravitational pull on the visible matter. While it appears unlikely that all the dark matter could be ordinary, baryonic matter, it could very nat- urally be relic supersymmetric fermions. Their relic density is determined by their annihilation cross section, which, because they are Majorana fermions, is typically small enough to imply cosmologically significant relic densities.

Under the simplest assumption, all the dark matter in the Universe is the same. Thus, the particle dark matter hypothesis is, hopefully, directly testable in the laboratory through dark matter particle interactions in proposed cryogenic detectors [ 11. In the minimal supcrsymmetric standard model, the lightest neutralino, x, is generally a mixture of two gauginos and two higgsinos [2],

x=a!,W3+p,B+y,fi, +s,I&, (1)

where the wino W3 and the bino B are the supersymmetric partners of the corresponding standard model gauge bosons, and fi, (a,) is the partner of the Higgs

72

scalar H, (H,) that gives mass to up (down)-type quarks. Here we study the laboratory interaction of binos, x= B, symmetric higgsinos, x= ( 1 /fi) (Hi +

fi+&2, and antisymmetric higgsinos x=

(l/J% (I%-&)=H,,,,. Typically x has been assumed to be a photino [ 3 1,

X=Y=sin 8,W3+cos &B. In the minimal model the composition of x depends on three parameters #I: MZ,

the supersymmetry breaking SU (2)-gaugino mass; E, the Higgs fermion mixing mass: and tan p= v, /v2 = (H,)/(H,). Indeed, in the limit M2+0, x=7 whereas in the limit E+O, x=3’, the higgsino defined

by a, = /3, = 0, y, = cos p and 6, = sin p. However, when m,k20 GeV, x is never a Jo nor so [4]. As it turns out, however, over a very wide region of the parameter space defined by M2, q and vi /v2, x is still a state of very nearly constant composition: x= B, PI, i2), or H [, z I [ 4 1. This region of the parameter space has not been considered in this context before.

Because the photino-nucleon elastic cross section

!+’ .4s usual, we assume M,= (5g:/3g:)M, upon unification,

where M, is the U( I )-gaugino mass.

Volume 237, number 1 PHYSICS LETTERS B 8 March 1990

is well known, it will be useful to compare the bino- nucleon elastic cross section to it. Recall the effective lagrangian for the interaction of a photino with an

ordinary fermion (quark or lepton) f.

&= (2KaQf/m:)~y~Y5~~y,ySf, (2)

where Qr is the electric charge off and my is the mass of the scalar partner off. The elastic cross section for

y+p-ty+p can be written as

where the factorf, is

.f-,=47ra C Q;&, 4

(4)

where g, =e/cos 19, is the hypercharge coupling con-

stant, Y,,,, is the hypercharge of the left (right)- handed fermion f. In eq. (6), we have only included

terms which contribute to the elastic cross section in the non-relativistic limit. From eq. (6), it is straight- forward to obtain the bino-proton elastic cross

section and Aq is the fraction of the spin ofthe proton carried

by the quark q

(p>sIS:Ip>s)=S;Aq, (5) where now

here S; = tqYWy5q and 5’; is the proton spin. Assum- ing SU(2) symmetry, the combination Au- Ad=

g, v 1.26 [ 5 ] is determined from neutron decay. As- suming SU (3) symmetry, hyperon P-decay data tell

us that Au+Ad-2As=0.60+0.12 [6]; thus, in the naive quark model (NQM ) As= 0 yields Au 2: 0.93, Adz -0.33 and .f; L 1.2~ 10W3. However, recent data by the European Muon Collaboration (EMC) [7] challenge this simple picture; when combined with P-decay data, their results imply #’ [ 8 ] Au r 0.73, Ad-_ -0.53 and As= -0.20; thus f; -5.0x 10p4,

.&=s:C ;(Y:,+Y$.)Aq, 4

(8)

and is given in table 1. As one can see, for equal LSP and sfermion masses,

we expect that the bino cross sections and hence scattering rates in detectors will no differ greatly from the photino rates.

w These numbers differ from ref. [ 81 because of the different

hyperon data fit [ 6 ] used here, but are consistent at the 1-u

level.

The elastic cross section of the So higgsino is very accurately determined through Z” exchange (squark exchange processes involve the Yukawa couplings to up and down quarks and are far less important than Z” exchange processes). The so-p elastic cross section can be written as (NQM)

Table 1

Values off2 in elastic neutralino (X) nucleon (N) scattering, u(XN+XN)af’, for X=7, B and fi,,,, or fitlzl and N=proton, neutron.

The values are given for a ratio FIDzO.58 [ 61, but they are fairly insensitive to it, except for those marked by an asterisk: these are small

but sensitive to F/D, we give the largest value for &F/D< 3, where the lowest (highest) value is favored by the Skyrme model (NQM).

lowering op by -0.42 relative to the NQM prediction. More significant, on the other hand, is the fact that neutrons are predicted to be much better photino scatterers, as can be seen in table 1.

For binos we can write an analogous effective lagrangian,

X &IV2 Proton

EMC NQM

Neutron

EMC NQM

5.0x 1o-4 1.2x 1o-3 2.6x 1O-4 2.9x lO-5 *

2.1 x lo-4 5.4x 1OW l.lxlo-4 7.1 x 10-6*

2 7.9x IO-i2 2.9x10-“* 1.0x lo-‘* 1.2x lo-”

4 1.3x lo-I0 1.4x lo-” 2.3x lo-” 1.5x lo-lo

8 2.1xlo-9 2.9x 1O-1o 3.8x lo-“’ 2.3x 1O-9

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Volume 237, number I PHYSICS LETTERS B 8 March 1990

For cos 2p= 1, this is exactly the cross section for

Majorana mass neutrinos. For other LSP states, the Z”-exchange cross section is obtained by replacing cos 2/3 with ( y: - 8: ). As one can readily see, for the

R - (r2) or H,rzl pure LSP states one has y: =S: and this contribution to the total cross section vanishes. Of course in reality states are never 100% pure and there will always be some admixture of other states. However, for a 99% pure HCr2) or Ht12,, as was dis- cussed in ref. [ 41, the Z”-exchange elastic cross section is suppressed at least by a factor of 0.08 relative to the Majorana neutrino cross section. Typical sup- pressions are much larger as the purity increases.

For pure R, r2)’ s the remaining scattering channel is given by squark exchange and now

(10)

where

(~,sICa,tn~S::Ip,s)=~,,,,m~S~ 4

and

(11)

aq= (t+T~,)-L- (

1

sm2 /3 +(f-T,,) - > cos2p .

(12)

The evaluation of the hadronic matrix element

.fa ,,2) requires some care since heavy quarks do not decouple from .fn,,,, due to the tn2 factor, but effec- tively the rnz in (11) can be replaced by the non- relativistic, constituent quark masses [ 91. Taking tn,=md=mp/3and tn,ltn,=(m,-m,)l(m,.-m,) as an estimate [ 91

fa,,,, = 5 GFmi&(cos2 /3sin’f3)-’

(13)

where Au-Ad=g,=F+D, Au+Ad-2As=3F-D and gi = Au + Ad+ As. This is given in table 1. We

see that&,, of-, due to the suppression factor ( mp/ tnw)4 in ( 13), and will result in extremely small interaction rates.

In order to calculate laboratory interaction rates we

must now consider the nuclear matrix elements

<NI&,“, IN) =J.J (14)

where J is the nuclear spin. Estimates of I [ 10,111 have used the single-particle shell model, which is fairly accurate near closed shells, but further away it tends to overestimate the spin contribution to magnetic moments as can be seen by comparing the predicted and measured magnetic moments [ 111. A better way to proceed has been pointed out by Engel and Vogel [ 121. Assuming that the even system in odd- even or even-odd nuclei makes a vanishingly small contribution to the total and spin angular momen- tum, as suggested by theoretical arguments and nuclear transfer data, they use the measured value of the nuclear magnetic moment p to determine 1 [ 12 ]

(15)

where (J’) =J( J+ 1) and the g’s are the free-nucleon g-factors. Where a comparison is possible, the prediction ( 15 ) agrees quite well with full-scale shell model calculations [ 13 1, and we shall use it here. For odd-odd systems we follow ref. [ 111.

We can now write down the rate of neutralino elastic scattering off nuclei (mass m,) in the laboratory as [ 111 (assuming spin dependent interactions dom- inate; see below )

X&I Px >

VE

0.3 GeV cmp3 320 km s-’ ’ (16)

where the figure of merit FM = (isotopic abundance) [4m,mN/(m,+mN)2]~~ZJ(~+ 1 ), vE is the particle’s mean speed in the Earth’s rest frame, m is a com- mon sfermion mass scale. When evaluating ( 16 ) for x=7, B we shall assume equal sfermion masses rn7, such that C$h2=t. For x=H(,~, or HtlzI, we have taken the minimally allowed squark mass compatible with experiment m,> 74 GeV [ 141 and varied other

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Volume 237, number I PHYSICS LETTERS B 8 March 1990

Table 2

Spin-dependent, elastic interaction rate R expected in the laboratory detector for different nuclei, assuming pure states x=1, B (R/kg

day) and H,,,, ( Htlzl ) (R,,,/ 106 kg day). The upper (lower) value assumes the EMC (NQM) value off:. The last three columns are the maximum expected rate for V,/V~Q 10.

Isotope X

I (m [GeVl) B (m[GeV]) H(U) andHti,r (m[GeVl)

10 20 I5 20 40 15 20 40

‘H 0.15 0.023 0.028 0.013 0.002 I 1.4 1.1 0.57

0.36 0.055 0.07 1 0.033 0.0053 0.20 0.15 0.08

2H 0.01 I 0.0018 0.002 1 0.0010 0.000 17 0.49 0.39 0.21

0.28 0.046 0.064 0.031 0.0052 0.70 0.56 0.30

‘He 0.22 0.039 0.044 0.022 0.0038 0.79 0.65 0.37

<0.025 * < 0.007 * < 0.003 * < 0.002 * i 0.003 * 4.8 3.9 2.2

‘Li 0.15 0.032 0.034 0.018 0.0036 1.7 I.5 0.99

0.35 0.076 0.087 0.046 0.0093 0.24 0.21 0.14

9Be 0.054 0.013 0.013 0.007 1 0.0015 0.23 0.21 0.15

(0.006 * < 0.002 * i 0.0008 * < 0.0005 * <0.0001 * 1.4 1.3 0.88

‘OB 0.015 0.0038 0.0038 0.002 I 0.00046 0.87 0.80 0.57

0.39 0.095 0.12 0.064 0.014 1.3 1.2 0.82

19F 0.37 0.11 0.10 0.064 0.017 5.4 5.4 4.6

0.89 0.27 0.27 0.16 0.043 0.75 0.75 0.64

S’V 0.061 0.025 0.020 0.014 0.005 I 1.0 1.2 1.4

0.014 0.059 0.052 0.036 0.013 0.14 0.17 0.20

parameters (such as Mz ) to obtain QRh2 = f We present the expected rates in table 2 for a few nuclei, cho- sen not by their technical merit, but by their high figure of merit (FM z 0.2 in at least some mass range for /nx= 1 O-40 GeV). In the first five columns we give R

for x=7, B; we see that the rates are comparable and relatively small, RSO( 10-l) kg-’ day-‘, compared to the lowest background attained in double beta decay experiments [ 151. By contrast, in the last three columns we give the rnaxirnurn rate expected for

x=I%i,, or 13tizl and v,/vz < 10. This is extremely

small, R,,, ,<O( 10P6) kg-’ day-‘; for a more likely ratio U, /v2 - 2 the rates are at least two orders of mag- nitude smaller. One should note that for m~~,~, = 15 GeV avoiding a light chargino requires v, /vz I 2 and for the same reason rne,,z, = 15 GeV is not possible for vi /v2> 2. Thus in the 15 GeV higgsino column, the values are all overestimates of the possible rates. One should also note that because these rates are all so small, any impurity (y: -8: f0) in PI,,,, or Rt,21 could result in a larger rate than quoted in the table. In fig. 1 we show the regions where R/F,< lo-’ in the MZ-e plane for the Z-exchange process (9) (with the substitution cos 2j3-1~: -8: ). In fig. 2, the shaded

le(/GeV

Fig. 1. Regions of the parameter space (Mz, t) of the minimal

supersymmetric standard model in which R/F,+,> 1O-3 for Z-ex-

change processes the upper (lower, left, right) panels are for v,/

vz=2 (v,/v2=8,e<0,t>O).

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Volume 237. number 1 PHYSICS LETTERS B 8 March 1990

10 ‘1”“1 1”““’ I”“‘)> “““” j “““” “““’ 104 103 102 IO 102 103 104

jel/GeV

Fig. 2. Regions of the parameter space (M,, t) of the minimal

super-symmetric standard model excluded by double B-decay ex-

periments [ 151 if one assumes the existence of a light Higgs sca-

lar. The light shaded (dark shaded) region is excluded if mu2 = 10

(20) GeV and the upper (lower, left, right) panels are for t.t/

u*=2 (v,/v,=S, tie, t>O).

region corresponds to the parameter space in which elastic scattering mediated by a light Higgs boson [ 161 ( mH = IO GeV, light shading and lrlH = 20 GeV, dark shading) would have been detected in the double beta decay experiments [ 151. To produce fig. 2, we have simply extended the parameter space de- scribed in ref. [ 161. What is clear from both of these figures is that these processes always avoid the “pure” regions we have been describing. In view of this we have also looked at the rates for radiative processes, xN-xNy, which would have the added signature of the photon and for which coherence in the rate. RX Z2 (Z= nuclear charge), could help overcome the small- ness due to the extra powers of (Y. Unfortunately. we

find that the rates are suppressed by powers of (p,/

mx)2- lop6 and, therefore, are also small. Thus, un- less there is significant squark mixing present [ 171, it appears that the symmetric (antisymmetric) higgsino would be invisible to proposed cryogenic

detectors.

The work of K.A.O. was supported in part by DOE- AC02-83ER-40105 and by a Presidential Young In- vestigator Award. The work of M.S. was supported in part by NSF grant PHY-86- 14 185 and by a Alfred P. Sloan Research Fellowship.

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Elastic neutralino-matter scattering