Lower bound on the neutralino mass from new data on CMB and implications for relic neutralinos
A. Bottino,*F. Donato,†N. Fornengo,‡and S. Scopel§Dipartimento di Fisica Teorica, Universita` di Torino, Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via P. Giuria 1, I–10125 Torino, Italy
共Received 14 April 2003; published 8 August 2003兲
In the framework of an effective minimal supersymmetric extension of the standard model without gaugino-mass unification at a grand unification scale, we set a lower bound on the neutralino gaugino-mass based on the new WMAP data on⍀CDM(R-parity conservation is assumed兲. Our lower bound mⲏ6 GeV leaves much room for relic neutralinos significantly lighter than those commonly considered (mⲏ50 GeV). We prove that these light neutralinos may produce measurable effects in weakly interacting massive particle direct detection ex-periments of low energy threshold and large exposure.
DOI: 10.1103/PhysRevD.68.043506 PACS number共s兲: 95.35.⫹d, 11.30.Pb, 12.60.Jv, 95.30.Cq
I. INTRODUCTION
In a previous paper 关1兴, we called attention to relic neu-tralinos of light masses, i.e., with a mass mⱗ45 GeV. Ac-tually, most analyses on cosmological neutralinos employ a lower bound mⲏ50 GeV, i.e., a bound which rests on the assumption that gaugino masses are unified at the grand uni-fication共GUT兲 scale MGUT⬃1016 GeV. However, this unifi-cation assumption might not be justified, as already dis-cussed some time ago关2,3兴. Furthermore, recent analyses of string models 共see, for instance, Ref. 关4兴兲 indicate that the initial scale for the running of the supersymmetry 共SUSY兲 parameters by renormalization group equations may be much lower than the standard GUT scale, with the implication that at the electroweak 共EW兲 scale the gaugino masses may be quite different from what is expected in a standard super-gravity 共SUGRA兲 scheme with GUT-unification assump-tions. Thus, for instance, the U共1兲 and SU共2兲 gaugino masses M1 and M2 may not be related by the standard formula M1⯝
1
2M2 at the EW scale.
It is then quite natural to discuss supersymmetric models where M1and M2are considered as independent parameters. Previous papers where schemes of this type are considered include the ones in Refs.关2,3,5–18兴. In Ref. 关1兴 we evaluated the neutralino relic abundance ⍀h2 and the neutralino-nucleon scalar cross sectionscalar(nucleon)in an effective minimal supersymmetric extension of the standard model 共EMSSM兲 where the gaugino-mass unification at GUT scale is not as-sumed. The analysis was performed in a scenario where the ratio R⬅M1/ M2 is smaller than the GUT-unification value RGUT⯝0.5, thus allowing the neutralino mass m to be lighter than the commonly used lower bound of about 50 GeV. We showed that, in the derivation of the lower limit on m, the upper bound on the relic abundance for cold dark matter 共CDM兲, ⍀CDMh2, plays a crucial role. A lower limit
of mⲏ5 GeV was derived in Ref. 关1兴 by employing the upper bound ⍀CDMh2ⱗ0.3. It was also shown how this up-per bound on ⍀CDMh2 is instrumental in providing sizeable values for the neutralino-nucleon scalar cross section at small m.
New data on the cosmic microwave background 共CMB兲
关19–21兴, also used in combination with other cosmological
observations, are progressively narrowing down the ranges of the relic abundances for matter (⍀mh2) and for some of its constituents: neutrinos (⍀h2) and baryons (⍀bh2). As a consequence also the range of⍀CDMh2is reaching a unprec-edented level of accuracy. The importance of an improve-ment in the determination of⍀CDMh2 for any cold relic par-ticle is twofold. The upper bound (⍀CDMh2)max obviously establishes a strict upper limit for any specific cold species. On the other side, the lower bound (⍀CDMh2)min fixes the value of the average abundance below which the halo density of a specific cold constituent has to be rescaled as compared to the total CDM halo density. For the determination of the rescaling factor ⬅/0 共where and 0 are the local neutralino density and the total local dark matter density, respectively兲, we use the standard rescaling recipe
⫽min关1,⍀h2/(⍀CDMh2)min兴. In Ref. 关19兴 ranges for
⍀mh2 and ⍀bh2 are derived by employing CMB data of Refs. 关19–21兴, 2dFGRS measurements 关22兴, and Lyman ␣ forest data 关23兴. From the values quoted in Ref. 关19兴 for
⍀mh2 and⍀bh2and allowing for a 2⫺range in⍀CDMh2, one obtains: (⍀CDMh2)min⫽0.095 and (⍀CDMh2)max
⫽0.131. These are the values we will use here. However,
one should cautiously still be open to some possible changes in these values as new cosmological observational data will accumulate in the future.
In the present paper we analyze the supersymmetric sce-nario of Ref. 关1兴 in light of these new determinations of
⍀CDMh2. We first derive approximate analytic expressions which directly relate the lower bound on m to the upper limit on ⍀CDMh2 (R-parity conservation is assumed兲. This requires the evaluation of ⍀h2, which is in itself a very complicated function of the supersymmetric parameters. In order to extract the leading analytic terms at small values of m, we are guided by a detailed numerical analysis of the supersymmetric parameter space. This will allow us to *Electronic address: bottino@to.infn.it
†Electronic address: donato@to.infn.it ‡Electronic address: fornengo@to.infn.it
§Electronic address: scopel@to.infn.it; http://www.to.infn.it/ astropart
present analytic expressions for the lower bounds on m which display the link of these bounds to the relevant particle-physics and cosmological constraints, such as the lower bounds on sfermions and Higgs-bosons masses and the upper limit on ⍀CDMh2. Furthermore, on the basis of these results, we show that light neutralinos, with masses m
ⱗ45 GeV, may actually be probed by weakly interacting
massive particle 共WIMP兲 direct detection experiments with high sensitivities and low energy thresholds.
II. LOWER BOUND ON M
The supersymmetric scheme adopted here is the same as the one described in Ref. 关1兴: an effective MSSM scheme
共EMSSM兲 at the electroweak scale, with the following
inde-pendent parameters: M2,,tan,mA,m˜q,ml˜,A, and R
⬅M1/ M2. Notations are as follows: is the Higgs mixing mass parameter, tan the ratio of the two Higgs vacuum expectation values共VEV’s兲, mAthe mass of the CP-odd neu-tral Higgs boson, mq˜ is a squark soft-mass common to all squarks, ml˜is a slepton soft-mass common to all sleptons, A is a common dimensionless trilinear parameter for the third family, Ab˜⫽At˜⬅Amq˜, and A˜⬅Aml˜ 共the trilinear param-eters for the other families being set equal to zero兲.
The neutralino is defined as the lowest-mass linear super-position of B-ino B˜ , W-ino W˜ (3), and of the two Higgsino states H˜10, H˜20:
⬅a1B˜⫹a2W˜(3)⫹a3H˜1 0⫹a
4H˜2 0
. 共1兲
Since we are here interested in light neutralinos, we con-sider values of R lower than its GUT value: RGUT⯝0.5; for definiteness, we take R in the range 0.01–0.5.
We first outline the procedure for deriving analytical bounds on m from the cosmological upper limit on
⍀CDMh2. As mentioned in the previous section, the identifi-cation of the leading analytic contributions is guided by numerical analysis. This is based on a scanning of the supersymmetric parameter space, with the following ranges of the MSSM parameters: 1⭐tan⭐50, 100 GeV
⭐兩兩,M2,m˜q,ml˜⭐1000 GeV, sign()⫽⫺1,1, 90 GeV
⭐mA⭐1000 GeV, and ⫺3⭐A⭐3. The following experi-mental constraints are imposed: accelerators data on super-symmetric and Higgs boson searches 共CERN e⫹e⫺ collider LEP2 关24兴 and Collider Detector CDF at Fermilab 关25兴兲; measurements of the b→s⫹␥ decay 关26兴; and measure-ments of the muon anomalous magnetic moment a⬅(g
⫺2)/2 关27兴 共the range ⫺160⭐⌬a•1011⭐680 is used here for the deviation of the current experimental world average from the theoretical evaluation within the standard model; for the derivation see Ref.关1兴兲.
The neutralino relic abundance is given by
⍀h2⫽ xf g쐓共xf兲1/2 3.3⫻10⫺38 cm2
具
anngv典
, 共2兲where
具
anngv典
⬅xf具
annv典
int,具
annv典
int being the integral from the present temperature up to the freeze-outtempera-ture Tf of the thermally averaged product of the annihilation cross section times the relative velocity of a pair of neutrali-nos, xf is defined as xf⬅m/Tf and g쐓(xf) denotes the rela-tivistic degrees of freedom of the thermodynamic bath at xf. For
具
anngv典
we will use the standard expansion in S and P waves:具
anngv典
⯝a˜⫹b˜(2xf).A lower bound on m is now derived from Eq. 共2兲, by requiring that
⍀h2⭐共⍀CDMh2兲max. 共3兲 We first work out approximate analytic expressions for
具
anngv典
in the regime of small m(mⱗ45 GeV). By di-agonalizing the usual neutralino mass matrix in the approxi-mation M1ⰆM2,, it turns out that light neutralinos have a dominant B-ino component; a deviation from a pure B-ino composition is mainly due to a mixture with H˜10, i.e., 兩a1兩Ⰷ兩a3兩Ⰷ兩a1兩,兩a4兩. For the ratio 兩a3兩/兩a1兩 one finds
兩a3兩
兩a1兩 ⯝
sinWsinmZ
ⱗ0.42 sin, 共4兲
where in the last step we have taken into account the experi-mental lower bound ⲏ100 GeV.
The dominant terms in
具
anngv典
intare the contributions due to Higgs-exchange in the s channel and sfermion-exchange in the t,u channels of the annihilation process ⫹→ f¯⫹ f共interference terms are neglected兲. We retain only the leading
terms in each contribution. Thus, for the Higgs-exchange contribution, dominated by the S-wave annihilation into down-type fermions, we have, for any final state f¯⫺ f ,
具
anngv典
f Higgs⯝a˜ f Higgs⯝ 2␣e.m. 2 c f sin2Wcos2W a12a32tan2 共1⫹⑀f兲2 ⫻m¯f 2 mW2 m2关1⫺mf2/m2兴1/2 关共2m兲2⫺mA 2兴2 , 共5兲where cf is a standard color factor (cf⫽3 for quarks, cf
⫽1 for leptons兲, m¯
fis the fermion running mass evaluated at the energy scale 2m, and mf is the fermion pole mass.⑀f is a quantity which enters in the relationship between the down-type fermion running mass and the corresponding Yukawa coupling 共see Ref. 关28兴 and references quoted therein兲; in the following evaluations,⑀fis negligible, except for the bottom quark, where⑀b⯝0.2. One easily verifies that when m⬍mb,
具
anngv典
fHiggs
entails a relic abundance ex-ceeding the cosmological bound.
Notice that
具
anngv典
turns out to be an increasing function of m. Then, to obtain a conservative lower bound on m from the condition of Eq. 共3兲, we have first to evaluate an (⍀h2)min which is obtained from Eq. 共2兲, by replacing具
anngv典
with its maximal value (具
anngv典
)max, at fixed m. In the case of Higgs-exchange contributions, this (具
anngv典
)max is obtained by inserting into Eq.共5兲 the maximal value of the product a12a32tan2. Taking into account Eq.共4兲 and that, formA⯝90 GeV, the upper bound of tanis 45关25兴, we obtain (a12a32tan2)
max⯝2.6⫻102, and in turn
共⍀h2兲min Higgs⬅1.5⫻10 ⫺10 GeV2 xf g쐓共xf兲1/2 mW2 m2关共2m兲 2⫺m A 2兴2 ⫻
冉
兺
f m¯2f共1⫹⑀f兲2cf关1⫺mf 2 /m2兴1/2冊
⫺1 ⯝5⫻10⫺11 GeV2 xf g쐓共xf兲1/2 mW2 m¯b2 ⫻ 1 共1⫹⑀b兲2 关共2m兲2⫺mA 2 兴2 m2关1⫺mb2/m2兴1/2. 共6兲 In the last step of this equation we have retained only the dominant contribution due to the b⫺b¯ final state. As far as the value of g쐓(xf)1/2 is concerned, we notice that for these light neutralinos xf⯝21 to 22, so that neutralinos with masses m⯝6 to 7 GeV have a freeze-out temperature Tf⬃TQCD, where TQCDis the hadron-quark transition tempera-ture of order 300 MeV. For definiteness, we describe here the hadron-quark transition by a step function: if TQCD is set equal to 300 MeV, then for mⱗ6 GeV one has g쐓(xf)1/2
⯝4, while for heavier neutralinos g쐓(xf)1/2⯝8 to 9. The quantity (⍀h2)minHiggsas given by Eq.共6兲 is plotted in Fig. 1 as a function of m, for the value mA⫽90 GeV
共cur-rent experimental lower bound兲. The solid curve refers to the case TQCD⫽300 MeV; dashed and dot-dashed curves denote two different representative values of TQCD:TQCD
⫽100 MeV and TQCD⫽500 MeV, respectively. The two horizontal lines denote two representative values for the up-per bound on ⍀CDMh2:(⍀
CDMh2)max⫽0.3 共short-dashed line兲 and (⍀CDMh2)max⫽0.131 共long-dashed line兲. Figure 1 displays how a lower bound on m is derived from an upper limit on⍀CDMh2. In particular, using TQCD⫽300 MeV, one obtains from (⍀CDMh2)max⫽0.3 the bound mⲏ5.2 GeV, a value which increases to mⲏ6.2 GeV, when the new value (⍀CDMh2)max⫽0.131 is employed. Also evident is the varia-tion of the lower bound on m, when the value of TQCD is changed. Finally, to support the validity of the analytical ap-proximations employed to derive Eq.共6兲, in Fig. 1 we also display the scatter plot of⍀h2, when a numerical scanning of the supersymmetric parameter space is performed.
We recall that the bound on m:mⲏ6.2 GeV was de-rived using mA⫽90 GeV 共which is the present experimental lower bound on mA). From our previous formulas one ob-tains that this bound on m simply scales with mA as fol-lows:
m关1⫺mb2/m2兴1/4ⲏ5.3 GeV
冉
mA 90 GeV冊
2
. 共7兲
For the sfermion-exchange contribution,
具
anngv典
sfermionf⯝a˜f
sfermion⫹ b˜(2x f)f
sfermion
, both S-wave and P-wave contri-butions have to be taken into account. In the regime we consider here, the leading terms due to the down-type fermi-ons are a ˜ f sfermion⯝ ␣e.m. 2 8 cos4W a14cf mf2关1⫺m2f/m2兴1/2 mf˜,14 共2⫺m2f˜,1/m2f˜兲2 ⫻
冋
共Yf ,L 2 ⫹Y f ,R 2 兲⫹2Y f ,LY1,R m mf冉
1⫺ m2f˜,1 mf˜2冊册
2 , 共8兲 b ˜ fsfermion⯝ ␣e.m. 2 4 cos4Wa1 4 cf m2关1⫺mf 2 /m2兴1/2 mf˜,14 共2⫺m2f˜,1/m2f˜兲2⫻
冋
2共Yf ,L4 ⫹Yf ,R4 兲⫹3Y2f ,LY2f ,R冉
1⫺ m2f˜,1mf˜2
冊
2册
.共9兲
Notations are as follows: Yf ,L and Yf ,R are the weak hyper-charges for left and right couplings, respectively. mf˜ denotes either ml˜ or m˜qdepending on the nature of the fermion, mf˜,1 is the smallest mass eigenvalue for the sfermion f˜, once the mass matrix in the weak-interaction basis f˜L, f˜R is diagonal-ized. A maximal mixing between the f˜L, f˜R fields has been used in deriving Eqs. 共8兲 and 共9兲, since these equations are meant to provide a conservative lower bound on m. Notice FIG. 1. Neutralino relic abundance ⍀h2 as a function of the
mass m. The solid curve denotes (⍀h2)min Higgs
given by Eq.共6兲 for
TQCD⫽300 MeV. Dashed and dot-dashed curves refer to the repre-sentative values TQCD⫽100 MeV, TQCD⫽500 MeV, respectively. The two horizontal lines denote two representative values of
⍀CDMh2: ⍀CDMh2⫽0.3 共short-dashed line兲 and ⍀CDMh2⫽0.131
共long-dashed line兲. The scatter plot is obtained by a full scanning of
that for l⫽e,, the f˜L⫺ f˜R mixing is negligible; this case is simply recovered from Eqs. 共8兲 and 共9兲 by setting mf˜,1
⫽mf˜.
With the aid of numerical evaluations, it is found that the leading contributions to
具
anngv典
sfermion⬅兺f具
anngv典
fsfermion are provided by the term due to the lepton:
具
anngv典
sfermion⯝
具
anngv典
sfermion. This is in turn maximized by
(
具
anngv典
)sfermionmax⯝␣e.m. 2 8 cos4W m2关1⫺m2/m2兴1/2 m˜4 ⫻
冋冉
2⫹5 2 m m冊
2 ⫹2x23 f册
. 共10兲We denote by (⍀h2)minsfermion the value of ⍀h2 derived from Eq.共2兲 when
具
anngv典
is replaced by (具
anngv典
)sfermionmax of Eq. 共10兲. The quantity (⍀h2)minsfermion is plotted in Fig. 2 for the value m˜⫽87 GeV 共current experimental lower bound兲; also the scatter plot for the quantity ⍀h2 is dis-played, for mA⬎300 GeV. From Fig. 2 one finds for m a lower bound mⲏ14 GeV for (⍀CDMh2)max⫽0.3, and m
ⲏ22 GeV for (⍀CDMh2)max⫽0.131. The scaling of this last bound with the stau mass is approximately given by
m关1⫺m2/m2兴1/4ⲏ22 GeV
冉
m˜ 90 GeV冊
2
. 共11兲
In general, one has conservatively to retain as a lower bound to mthe smaller of the two lower limits given sepa-rately in Eq. 共7兲 and in Eq. 共11兲. From these equations one finds that the lower bound of Eq.共7兲 is less stringent than the one of Eq. 共11兲 as long as mAⱗ2m˜. Due to the present experimental bounds on mAand m˜the lower absolute bound is the one derived from Eq. 共7兲, i.e., mⲏ6.2 GeV. We par-enthetically note that the lower limits mⲏ(15–18) GeV found in Refs. 关17,18兴 are due to the assumption that mA is very large (mA⬃1 TeV).
A plot which shows the transition between the lower bounds established by considering Higgs-exchange contribu-tions and sfermion-exchange contribucontribu-tions is reported in Fig. 3. As was derived by use of the approximate expressions in Eqs. 共7兲–共11兲, the transition point is at mA⬃200–300 GeV.
III. DETECTABILITY OF LIGHT NEUTRALINOS BY WIMP DIRECT MEASUREMENTS
Now we wish to show that a part of the neutralino popu-lation of small mis indeed explorable with experiments of WIMP direct detection with current sensitivities. Let us start by giving an approximate relation between the scalar neutralino-nucleon cross section and the neutralino relic abundance, valid whenscalar(nucleon)is dominated by h-exchange in the t-channel and annby A-exchange in the annihilation s-channel. From the approximate formulas given in Ref.关1兴 for scalar(nucleon) and⍀h2 we obtain
FIG. 2. Neutralino relic abundance ⍀h2 as a function of the mass m. The solid curve denotes (⍀h2)min
sfermion
derived from Eq.
共2兲 when具anngv典 is given by the maximal value of具anngv典 sfermion
关see Eq. 共10兲兴. TQCDis set equal to 300 MeV. The two horizontal lines denote two representative values of ⍀CDMh
2
: ⍀CDMh 2⫽0.3
共short-dashed line兲 and ⍀CDMh2⫽0.131 共long-dashed line兲. The scatter plot is obtained by a full scanning of the supersymmetric parameter space with mA⬎300 GeV.
FIG. 3. Dashed and solid curves give the variation of the lower bound on m as a function of mA for (⍀CDMh2)max⫽0.3 and for (⍀CDMh
2
)max⫽0.131, respectively. For each value of (⍀CDMh 2
)max the region on the left of the relevant curve is forbidden. The two lines are derived from the analytical expressions obtained in the text for the contributions to 具anngv典 due to Higgs-exchange and sfermion-exchange. Also displayed is the scatter plot of a full nu-merical scanning.
共⍀h2兲scalar (nucleon)⯝1.4⫻10⫺40 cm2 T
冉
ms具
N兩s¯s兩N典
200 MeV冊
2 ⫻ GeV 2 m2关1⫺mb2/m2兴1/2冉
mA mh冊
4 , 共12兲where mh is the mass of the lightest CP-even neutral Higgs boson and T is given by
T⫽共a3sin␣⫹a4cos␣兲 2
共a4cos⫺a3sin兲2
关sin␣⫹⑀scos共␣⫺兲sin兴2 sin2共1⫹⑀
b兲2 .
共13兲
In Eqs.共12兲 and 共13兲
具
N兩s¯s兩N典
is the s-quark density matrix element over the nucleonic state and ␣ is the angle which rotates the Higgs fields H1(0) and H2(0) into the mass eigen-states h and H. The approximations employed in deriving Eqs. 共12兲 and 共13兲 imply ␣⬃/2 , mh⬃mA⬃100 GeV, so that T is of order one.Thus for neutralino configurations with mⱗ20 GeV,
scalar
(nucleon) turns out to be bounded by
scalar (nucleon)ⲏ 10⫺40 cm 2 共⍀CDMh2兲max GeV2 m2关1⫺mb2/m2兴1/2. 共14兲 In deriving Eq. 共14兲, we have set ms
具
N兩s¯s兩N典
⫽200 MeV关29兴.
The results of complete numerical evaluations of the quantity scalar(nucleon), where all relevant diagrams for the neutralino-nucleon scalar cross section and for the relic abundance are taken into account, are displayed in Fig. 4. The peculiar funnel in the scatter plot for mⱗ20 GeV is due to the bound of Eq.共14兲.
As was pointed out in Ref.关1兴, the present upper limits to
scalar
(nucleon) provided by WIMP direct detection experiments
关30–33兴 do not significantly constrain the supersymmetric
configurations for the light neutralinos displayed in Fig. 4. Instead, these configurations may be relevant for experi-ments of direct detection with a low energy threshold and a large exposure. An experiment with these features is the DAMA/NaI experiment, whose results, after 4-years running with a total exposure of ⯝58 000 kg day, show an annual-modulation effect at a 4 C.L. which does not appear to be related to any possible source of systematics关34兴. The analy-sis carried out by the DAMA Collaboration to explain their modulation effect in terms of a WIMP with coherent elastic scattering was targeted to a neutralino in the frame of a usual supersymmetric scheme with gaugino-mass unification at GUT, with a consequent lower bound on the neutralino mass above 30 GeV. This interpretation was proved to be consis-tent with supersymmetric models with gaugino-unification at GUT关35兴.
Here we have considered a different supersymmetric scheme which includes significantly lower neutralino masses; thus in order to establish the possible relevance of our low-mass neutralinos for an annual-modulation effect, we have to proceed to an extension of previous analyses. To
put our arguments into a quantitative basis, we evaluate (scalar(nucleon))min, defined as the minimal value of scalar
(nucleon) which may produce an annual-modulation effect at n stan-dard deviations in a detector with a given exposure S, and for a given velocity distribution function f (vជ) for relic neutrali-nos in our galaxy, i.e.,
共scalar (nucleon)兲 min⫽ n2 S I 共⌬I兲2. 共15兲
In Eq. 共15兲 I is defined as the ratio of the expected direct detection rate, integrated over an energy range (E1,E2), to the neutralino-nucleon scalar cross section scalar(nucleon):
I⫽ 1 scalar (nucleon)
冕
E 1 E2dR共E兲 dE dE. 共16兲For a monoatomic material of nuclear mass number A, one has I⫽NT0 mN 2m3
冉
1⫹ m mp冊
2 A2冕
E1 E2 dEF2共E兲冕
vmin(E) ⬁ dvជf共vជ兲 兩vជ兩 , 共17兲FIG. 4. Scatter plot ofscalar (nucleon)
vs m. Crosses共red兲 and dots
共blue兲 denote neutralino configurations with ⍀h2⬎(⍀CDMh2)min and ⍀h2⬍(⍀CDMh
2
)min, respectively. The curves give the sensi-tivity line, (scalar
(nucleon)
)minvs m, for a NaI detector, whose features are discussed in the text. The intermediate curve refers to an iso-thermal DF withv0⫽220 km s⫺1 and0⫽0.3 GeV cm⫺3. The up-per curve refers to a spherical Evans’ power-law DF共denoted as A3 in Ref. 关38兴兲 with v0⫽170 km s⫺1 and 0⫽0.17 GeV cm⫺3; the lower curve refers to an axially symmetric Evans’ logarithmic DF with maximal flatness 共denoted as C2 in Ref. 关38兴兲 with v0
⫽270 km s⫺1and
where NTis the number of the target nuclei per unit of mass, mN is the nuclear mass, F(E) the nuclear form factor, and
vmin is the minimal value of the neutralino velocity to pro-duce an event above the detection threshold. Generalization of Eq. 共17兲 to nonmonoatomic materials is straightforward. In Eq. 共15兲 ⌬I is defined as ⌬I⬅关I(June)
⫺I(December)兴/2.
We emphasize that, by definition, the condition
scalar
(nucleon)⭓( scalar (nucleon))
min only establishes a minimal re-quirement for a neutralino of a given mass mto be detected in a given experiment. It is obvious that the real detectability of the signal depends on the actual amount of the experimen-tal background. For example, an equal amount of signal and background would double the value of (scalar(nucleon))min as given in Eq. 共15兲. Estimates of backgrounds are omitted here.
To establish whether our population of light neutralinos may be relevant for the effect measured by the DAMA ex-periment, we have evaluated (scalar
(nucleon)
)minas a function of m; the energy range of integration employed here is E1
⫽2 keV, E2⫽3 keV, and n is set equal to 4. The nuclear form factors for Na and I are modelled in the Helm form
关36兴, with the values of parameters given in Ref. 关37兴. In Fig.
4 we give the curves of (scalar(nucleon))min versus m for a sample of different galactic distribution functions 共DF兲, taken among those analyzed in Ref. 关38兴. The intermediate curve refers to an isothermal DF with v0⫽220 km s⫺1 and
0⫽0.3 GeV cm⫺3(v0 is the local rotational velocity兲. The upper curve refers to a spherical Evans’ power-law DF 共de-noted as A3 in Ref. 关38兴兲 with v0⫽170 km s⫺1 and 0
⫽0.17 GeV cm⫺3; the lower curve refers to an axially
sym-metric Evans’ logarithmic DF with maximal flatness 共de-noted as C2 in Ref. 关38兴兲 with v0⫽270 km s⫺1 and 0
⫽1.7 GeV cm⫺3.
From the results displayed in Fig. 4 one sees that indeed a part of our population of relic neutralinos of small m may be relevant for the annual-modulation effect discussed above,
especially in case of DF’s which entail higher velocities and/or higher densities as compared to the standard isother-mal distribution. This is a new interesting option which adds to the other, still valid, possibility which we discussed in the papers of Ref. 关35兴, on relic neutralinos with masses above 50 GeV.
IV. CONCLUSIONS
We have considered phenomenological properties of relic neutralinos in a range of low masses (mⱗ45 GeV), which is allowed in a MSSM model without gaugino-mass unifica-tion at a grand unificaunifica-tion scale. We have numerically evalu-ated the relevant relic abundance and the neutralino-nucleon cross section, and discussed our numerical results in terms of analytic formulas, which display the connections among cos-mological properties and particle-physics parameters.
Using the latest determinations of⍀CDMh2 and assuming R-parity conservation, we have shown that in a MSSM model without gaugino-mass unification the lower bound on the neutralino mass is mⲏ6 GeV. We have shown analyti-cally how this bound is linked to the cosmological upper bound on ⍀CDMh2 and to the lower limits on masses of Higgs bosons and sfermions.
The implications of light relic neutralinos with masses
ⱗ45 GeV for WIMP direct searches have been analyzed. It
is found that these neutralinos are actually relevant for WIMP direct detection experiments of low energy threshold and large exposure. The present results extend to small masses our previous analyses about the effects of relic neu-tralinos with masses above 50 GeV关35兴.
ACKNOWLEDGMENTS
We acknowledge Research Grants funded jointly by Min-istero dell’Istruzione, dell’Universita` e della Ricerca
共MIUR兲, by Universita` di Torino, and by Istituto Nazionale
di Fisica Nucleare within the Astroparticle Physics Project.
关1兴 A. Bottino, N. Fornengo, and S. Scopel, Phys. Rev. D 67,
063519共2003兲.
关2兴 J. Ellis, K. Enqvist, D.V. Nanopoulos, and K. Tamvakis, Phys.
Lett. 155B, 381共1985兲.
关3兴 M. Drees, Phys. Lett. 158B, 409 共1985兲; Phys. Rev. D 33,
1468共1986兲.
关4兴 E. Gabrielli, S. Khalil, C. Muno˜z, and E. Torrente-Lujan, Phys.
Rev. D 63, 025008共2001兲.
关5兴 M. Drees and X. Tata, Phys. Rev. D 43, 2971 共1991兲. 关6兴 K. Griest and L. Roszkowski, Phys. Rev. D 46, 3309 共1992兲. 关7兴 S. Mizuta, D. Ng, and M. Yamaguchi, Phys. Lett. B 300, 96
共1993兲.
关8兴 A. Gabutti, M. Olechowski, S. Cooper, S. Pokorski, and L.
Stodolsky, Astropart. Phys. 6, 1共1996兲.
关9兴 G. Be´langer, F. Boudjema, F. Donato, R. Godbole, and S.
Rosier-Lees, Nucl. Phys. B581, 3共2000兲.
关10兴 A. Corsetti and P. Nath, Phys. Rev. D 64, 125010 共2001兲. 关11兴 R. Arnowitt, B. Dutta, and Y. Santoso, hep-ph/001244.
关12兴 D.G. Cerden˜o, S. Khalil, and C. Mun˜oz, hep-ph/0105180. 关13兴 V.A. Bednyakov and H.V. Klapdor-Kleingrothaus, Phys. Rev.
D 63, 095005共2001兲.
关14兴 H. Baer, C. Balazs, A. Belyaev, R. Dermisek, A. Mafi, and A.
Mustafayev, J. High Energy Phys. 05, 061共2002兲.
关15兴 V. Bertin, E. Nezri, and J. Orloff, J. High Energy Phys. 02, 046 共2003兲.
关16兴 A. Birkedal-Hansen and B.D. Nelson, Phys. Rev. D 67,
095006共2003兲; Phys. Rev. D 64, 015008 共2001兲.
关17兴 G. Be´langer, F. Boudjema, A. Pukhov, and S. Rosier–Lees,
hep-ph/0212227.
关18兴 D. Hooper and T. Plehn, Phys. Lett. B 562, 18 共2003兲. 关19兴 D.N. Spergel et al. 共WMAP兲, astro-ph/0302209. 关20兴 T.J. Pearson et al. 共CBI兲, astro-ph/0205388. 关21兴 C.L. Kuo et al. 共ACBAR兲, astro-ph/0212289.
关22兴 W.J. Percival et al., Mon. Not. R. Astron. Soc. 327, 1297 共2001兲.
and A.J.S. Hamilton, Mon. Not. R. Astron. Soc. 334, 107
共2002兲.
关24兴 ALEPH Collaboration, A. Colaleo, talk at SUSY’01, 2001,
Dubna, Russia; DELPHI Collaboration, J. Abdallah et al., DELPHI 2001-085 CONF 513, 2001.
关25兴 T. Affolder et al., Phys. Rev. Lett. 86, 4472 共2001兲.
关26兴 CLEO Collaboration, S. Ahmed et al., CONF 99/10,
hep-ex/9908022; ALEPH Collaboration, R. Barate et al., Phys. Lett. B 429, 169 共1998兲; Belle Collaboration, K. Abe et al.,
ibid. 511, 151共2001兲.
关27兴 Muon (g⫺2) Collaboration, G.W. Bennett et al., Phys. Rev.
Lett. 89, 101804共2002兲; 89, 129903共E兲 共2002兲.
关28兴 A. Bottino, N. Fornengo, and S. Scopel, Nucl. Phys. B608, 461 共2001兲.
关29兴 A. Bottino, F. Donato, N. Fornengo, and S. Scopel, Astropart.
Phys. 13, 215共2000兲; 18, 205 共2002兲.
关30兴 For a summary on WIMP direct detection experiments, see, for
instance, A. Morales, Nucl. Phys. B 共Proc. Suppl.兲 110, 39
共2002兲.
关31兴 DAMA/NaI Collaboration, R. Bernabei et al., Phys. Lett. B 389, 757共1996兲.
关32兴 CDMS Collaboration, D. Abrams et al., Phys. Rev. D 66,
122003共2002兲.
关33兴 A. Benoit et al., Phys. Lett. B 545, 43 共2002兲.
关34兴 DAMA/NaI Collaboration, R. Bernabei et al., Phys. Lett. B 480, 23共2000兲; Eur. Phys. J. C 18, 283 共2000兲.
关35兴 A. Bottino, F. Donato, N. Fornengo, and S. Scopel, Phys. Lett.
B 423, 109 共1998兲; Phys. Rev. D 62, 056006 共2000兲; 63, 125003共2001兲.
关36兴 R.H. Helm, Phys. Rev. 104, 1466 共1956兲.
关37兴 A. Bottino, F. Donato, G. Mignola, S. Scopel, P. Belli, and A.
Incicchitti, Phys. Lett. B 402, 113共1997兲.
关38兴 P. Belli, R. Cerulli, N. Fornengo, and S. Scopel, Phys. Rev. D 66, 043503共2002兲.
