14 June 2001

Physics Letters B 509 (2001) 197–203www.elsevier.nl/locate/npe

Investigating the DAMA annual modulation datain a mixed coupling framework

R. Bernabeia, M. Amatob, P. Bellia, R. Cerullia, C.J. Daic, H.L. Hec, A. Incicchitti b,H.H. Kuangc, J.M. Mac, F. Montecchiaa, D. Prosperib

a Dipartimento di Fisica, Università di Roma “Tor Vergata” and INFN, sezione di Roma II, I-00133 Rome, Italyb Dipartimento di Fisica, Università di Roma “La Sapienza” and INFN, sezione di Roma, I-00185 Rome, Italy

c IHEP, Chinese Academy, P.O. Box 918/3, Beijing 100039, PR China

Received 8 January 2001; accepted 17 April 2001Editor: K. Winter

Abstract

In this Letter the data, collected by the� 100 kg NaI(Tl) DAMA set-up deep underground in the Gran Sasso NationalLaboratory of I.N.F.N. during four annual cycles (57986 kg day statistics), are analyzed in terms of WIMP annual modulationsignature considering a candidate with mixed coupling to ordinary matter. 2001 Elsevier Science B.V. All rights reserved.

1. Introduction

WIMPs (Weakly Interacting Massive Particles) areexpected to be a primary component of the dark matterhalo in the Milky Way. The DAMA experiment issearching for WIMPs by investigating the so-calledannual modulation signature [1–7] by means of the� 100 kg NaI(Tl) set-up [8] running deep undergroundin the Gran Sasso National Laboratory of the I.N.F.N.The data collected during four annual cycles show, ina model-independent way, an annual modulation ofthe low energy rate with peculiar features [6,7]. Asregards possible model-dependent analysis, a purelyspin-independent coupled candidate, which equallycouples to proton and neutron, has been consideredso far [2–6]. In this Letter, instead, a candidate havingnot only a spin-independent, but also a spin-dependentcoupling different from zero is investigated. This is

E-mail address: rita.bernabei@roma2.infn.it (R. Bernabei).

possible for the neutralino in supersymmetric theoriessince both the squark and the Higgs bosons exchangesgive contribution to the coherent (SI) part of the crosssection, while the squark and the Z0 exchanges givecontribution to the spin-dependent (SD) one. However,the results of the analyzes presented here and inRef. [6] are not restricted to the neutralino case.

2. Theoretical framework

The studied process is the WIMP–nucleus elasticscattering and the measured quantity in undergroundset-ups is the recoil energy.

The differential energy distribution of the recoilnuclei can be calculated [9,10] by means of thedifferential cross section of the WIMP–nucleus elasticprocesses

dσ

dER(v,ER)=

(dσ

dER

)SI

+(dσ

dER

)SD

0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)00493-2

198 R. Bernabei et al. / Physics Letters B 509 (2001) 197–203

(1)

= 2G2FmN

πv2

{[Zgp + (A−Z)gn]2F 2SI(ER)

+ 8Λ2J (J + 1)F 2SD(ER)

},

where:GF is the Fermi coupling constant;mN is thenucleus mass;v is the WIMP velocity in the labora-tory frame;ER =m2

WNv2(1− cosθ∗)/mN (with mWN

WIMP–nucleus reduced mass andθ∗ scattering anglein the WIMP–nucleus c.m. frame) is the recoil energy;Z is the nuclear charge andA is the atomic number;gp,n are the effective WIMP–nucleon couplings forSI interactions;Λ2J (J + 1) is a spin factor. More-

over,F 2SI(ER)= 3j1(q·r0)

q·r0 e− 12s

2q2is the SI form factor

according to Ref. [11];q2 = 2mNER is the squaredthree-momentum transfer,j1(q · r0) is the sphericalBessel function of index 1,s � 1 fm is the thick-ness parameter of the nuclear surface,r0 = √

r2 − 5s2

and r = 1.2A1/3 fm. As regards the SD form factorF 2

SD(ER) an universal formulation is not possible; infact, in this case the internal degrees of the WIMPparticle model (e.g., supersymmetry in case of neu-tralino) cannot be completely separated from the nu-clear ones. It is worth to notice that this adds sig-nificant uncertainty in the final results. In the calcu-lations presented here we have adopted the SD formfactors of Ref. [12] estimated by considering the Nij-mengen nucleon–nucleon potential. It can be demon-strated thatΛ= (ap〈Sp〉 + an〈Sn〉)/J with J nuclearspin,ap,n effective WIMP–nucleon couplings for SDinteraction and〈Sp,n〉 mean value of the nucleon spinin the nucleus. Therefore, the differential cross sec-tion and, consequently, the expected energy distribu-tion depends on the WIMP mass and on four unknownparameters of the theory:gp,n andap,n.

The total cross section for WIMP–nucleus elasticscattering can be obtained by integrating Eq. (1) overER up toER,max= 2m2

WNv2/mN:

σ(v)=ER,max∫

0

dσ

dER(v,ER) dER

= 4

πG2

Fm2WN

{[Zgp + (A−Z)gn

]2GSI(v)

(2)+ 8J + 1

J

[ap〈Sp〉 + an〈Sn〉

]2GSD(v)

}.

Here GSI(v) = 1ER,max

∫ ER,max0 F 2

SI(ER) dER; GSD(v)

can be derived straightforward.

The standard point-like cross section can be evalu-ated in the limitv→ 0 (that is in the limitGSI(v) andGSD(v)→ 1). Knowing that〈Sp,n〉 = J = 1/2 for sin-gle nucleon, the SI and SD point-like cross sections onproton and on neutron can be written as:

σSIp,n = 4

πG2

Fm2W(p,n)g

2p,n,

(3)σSDp,n = 32

π

3

4G2

Fm2W(p,n)a

2p,n,

wheremWp � mWn are the WIMP–nucleon reducedmasses.

As far as regards the SI case, the first term withinsquared brackets in Eq. (2) can be rewritten in the form[Zgp + (A−Z)gn

]2

=(gp + gn

2

)2[1− gp − gn

gp + gn(

1− 2Z

A

)]2

A2

(4)= g2A2.

SinceZ/A is nearly constant for the nuclei typicallyused in direct searches for Dark Matter particles, thecouplingg can be assumed — in a first approxima-tion — as independent on the used target nucleus.Therefore, it is convenient to introduce a generalizedSI WIMP–nucleon cross sectionσSI = 4

πG2

Fm2Wpg

2.Let us now introduce the useful notations

a =√a2p + a2

n, tgθ = an

ap,

(5)σSD = 32

π

3

4G2

Fm2Wpa

2,

where σSD is a suitable SD WIMP–nucleon crosssection. The SD cross sections on proton and neutroncan be, then, written as:

(6)σSDp = σSDcos2 θ, σSD

n = σSDsin2 θ.

In conclusion, Eq. (1) can be rewritten in terms ofσSI, σSD andθ as:

(7)dσ

dER(v,ER)= mN

2m2Wpv

2Σ(ER)

with

Σ(ER)={A2σSIF

2SI(ER)+ 4

3

(J + 1)

JσSD

(8)

× [〈Sp〉cosθ + 〈Sn〉sinθ]2F 2

SD(ER)

}.

R. Bernabei et al. / Physics Letters B 509 (2001) 197–203 199

The mixing angleθ is defined in the[0,π) interval; inparticular,θ values in the second sector account forapandan with different signs. As it can be noted from itsdefinition [10],F 2

SD(ER) depends onap andan onlythrough their ratio and, consequently, depends onθ ,but it does not depend ona.

Finally, setting the local WIMP density,ρW, andthe WIMP mass,mW, one can write the energydistribution of the recoil rate(R) in the form

dR

dER=NT

ρW

mW

vmax∫vmin(ER)

dσ

dER(v,ER)vf (v) dv

(9)=NTρWmN

2mWm2Wp

Σ(ER)I (ER)

where:NT is the number of target nuclei andI (ER)=∫ vmaxvmin(ER)

dvf (v)v

with f (v) WIMP velocity distribu-

tion in the Earth frame [10];vmin =√mNER/(2m2

WN)

is the minimal WIMP velocity providingER recoil en-ergy;vmax is the maximal WIMP velocity in the haloevaluated in the Earth frame. The extension of for-mula (9), e.g., to multiple nuclei detectors can be eas-ily derived.

3. Data analysis and results

According to Section 2, we extend here the dataanalysis of Refs. [6,7] to the case of a WIMP hav-ing also a SD component different from zero. In ad-dition, we account for the uncertainties on some of theused astrophysical, nuclear and particle physics pa-rameters. Following the procedure already describedin Refs. [2–6], we build they log-likelihood func-tion which depends on the experimental data and onthe theoretical expectations in the given model frame-work.1 Then,y is minimized searching for parame-ters’ regions allowed at given confidence level. Obvi-ously, different model frameworks vary the expecta-tions and, therefore, the values of the four free para-meters:ξσSI, ξσSD, θ andmW (whereξ is the WIMP

1 A model framework is identified not only by the generalastrophysical, nuclear and particle physics assumptions, but alsoby the set of the used parameters values (such as WIMP localvelocity, v0, form factors parameters, etc.).

local density in 0.3 GeV cm–3 units), corresponding tothey minimum as discussed, e.g., in Ref. [5] forv0 inthe purely spin-independent case.2 For simplicity theeffect of the inclusion of known uncertainties on theparameters is generally expressed by making a super-imposition of all the allowed regions [5,6].

As regards the WIMP velocity distribution the sameassumptions as in Ref. [6] have been adopted here;in particular,v0 can range between 170 km/s and270 km/s [5,6]. Moreover, to clearly point out thepossible increase of sensitivity associated with the un-certainties on some other parameters we have varied:(i) the measured23Na and127I quenching factors [9]from their mean values up to+2 times the errors;3

(ii) the nuclear radius,r, and the nuclear surface thick-ness parameter,s, in the SI form factor [11] from thevalues quoted in Section 2 down to−20%; (iii) theb parameter in the considered SD form factor fromthe given value [12] down to−20%. In addition, asin Ref. [6] masses above 30 GeV and the physicalconstraint of the limit on the recoil rate measured inRef. [9] have been taken into account.

When the SD component is different from zero,a very large number of possible configurations areavailable; here for simplicity we show the results ob-tained only for 4 particular couplings, which corre-spond to the following values of the mixing angleθ :(i) θ = 0 (an = 0 andap �= 0 or |ap| � |an|); (ii) θ =π/4 (ap = an); (iii) θ = π/2 (an �= 0 andap = 0 or|an| � |ap|); (iv) θ = 2.435 rad (an/ap = −0.85, pureZ0 coupling). The caseap = −an is nearly similar tothe case (iv).

In Fig. 1 the regions, allowed by the data of Refs.[6,7], at 3σ C.L. (colored areas) in the (ξσSI, ξσSD)plane are shown for givenθ values and for someWIMP masses. Note that the calculation has beenperformed by minimizing they function with therespect to theξσSI, ξσSD andmW parameters for eachgiven θ value; Fig. 1 shows slices for given masses.In the same Fig. 1 the dashed lines marked when

2 For example, in Ref. [6]mW = 72+18−15 GeV and ξσSI =

(5.7 ± 1.1) × 10−6 pb correspond to the position ofy minimumwhenv0 = 170 km/s, whilemW = 43+12

−9 GeV andξσSI = (5.4 ±1.0)× 10−6 pb are found whenv0 = 220 km/s.

3 Note that the usual notation to omit the uncertainty when equalto one unity on the less significant digit was used in Ref. [9].

200 R. Bernabei et al. / Physics Letters B 509 (2001) 197–203

Fig. 1. Regions allowed at 3σ C.L. for WIMP configurations corresponding to: (i)θ = 0; (ii) θ = π/4; (iii) θ = π/2; (iv) θ = 2.435 rad. Theseregions have been calculated taking into account the uncertainties onv0 on the quenching factors and on the SI and SD nuclear form factorsparameters as mentioned in the text. The dashed lines given for the casemW = 50 GeV represent the limit curves calculated forv0 = 220 km/sfrom the data of the DAMA liquid xenon experiment [13]. See text.

R. Bernabei et al. / Physics Letters B 509 (2001) 197–203 201

Fig. 2. 3σ C.L. allowed region for a mixed coupled candidate: (a)ξσSI = 3× 10−6 pb; (b)ξσSI = 1× 10−6 pb; (c)ξσSI = 5× 10−8 pb. Forsimplicity, the calculations have been performed here by fixingv0 = 220 km/s and the quenching factors and the parameters of the SI and SDnuclear form factors at their mean values. See text.

mW = 50 GeV represent the limit curves calculatedfor v0 = 220 km/s from the data of the DAMA liquidxenon experiment [13]; regions above these dashedlines could be considered excluded at 90% C.L. Sincethe 129Xe nucleus — on the contrary of the23Naand of the127I — has the neutron as odd nucleon,a reduction of the allowed region is obtained onlyfor the caseθ � π/2; similar results are obtainedfor all the other WIMP masses. However, we furtherremark that the comparison of results achieved bydifferent experiments — even more when differenttarget nuclei and/or different techniques have beenused — is affected by intrinsic uncertainties.

We take this occasion to comment that no quantita-tive comparison can be directly performed between theresults obtained in direct and indirect searches becauseit strongly depends on assumptions and on the consid-

ered model framework. In addition, very large uncer-tainties are present in the evaluation of the results ofthe indirect searches themselves. In particular, a com-parison would always require the calculation and theconsideration of all the possible WIMP configurationsin the given particle model (e.g., for neutralino: in theallowed parameters space), since it does not exist a bi-univocal correspondence between the observables inthe two kinds of experiments: WIMP–nucleus elasticscattering cross section (direct detection case) and fluxof muons from neutrinos (indirect detection case).4

4 In fact, the counting rate in direct search is proportional tothe ξσSD and ξσSI, while the muon flux is connected not only toξσSD andξσSI but also to the WIMP annihilation cross section. Inprinciple, the three cross sections can be correlated, but only when aspecific model is adopted and by non directly proportional relations.

202 R. Bernabei et al. / Physics Letters B 509 (2001) 197–203

Fig. 3. Minimum of they function in the plane (ξσSI, ξσSD) when fixingmW andθ at the quoted values. Two cases are represented: (a) whenfixing v0 = 220 km/s and the quenching factors and the SI and SD nuclear form factors parameters at their mean values; (b) when consideringv0 = 170 km/s and taking into account the uncertainties on the quenching factors and on the SI and SD nuclear form factors parameters asmentioned in the text. We do not report here the cases withθ = π/2 andv0 = 270 km/s because they do not provide any minimum having bothξσSI andξσSD different from zero. See text.

As already pointed out in Ref. [6], when theSD contribution goes to zero (y axis), an intervalnot compatible with zero is obtained forξσSI (seeFig. 1). Similarly, when the SI contribution goes tozero (x axis), finite values for the SD cross sectionare obtained. Large regions are allowed for mixedconfigurations also forξσSI � 10−5 pb andξσSD �1 pb; only in the particular case ofθ = π/2 (that isap = 0 andan �= 0) ξσSD can increase up to� 10 pb,since the23Na and127I nuclei have the proton as oddnucleon.

At this point, we give in Fig. 2 a simple verificationof the following items: (i) finite values can be allowedfor ξσSD even whenξσSI � 3 × 10−6 pb as in theregion allowed in the pure SI scenario of Ref. [6](contour a); (ii) regions not compatible with zero intheξσSD versusmW plane are allowed even whenξσSIvalues much lower than those allowed in the pure SI

scenario of Ref. [6] are considered (contours b and c).For simplicity, in this case the calculations have beenperformed by usingv0 = 220 km/s and the quenchingfactors and the parameters of the SI and SD nuclearform factors at their mean values. The general trendfor other ξσSI and θ values can be easily inferred.Fig. 2 clearly shows the existence of allowed mixedconfigurations up to masses of� 85 GeV in this modelframework; moreover, it has been calculated that theinclusion of known uncertainties in the parametersas well as of possible dark halo rotation will furtherextend the allowed regions up to� 140 GeV mass (1σC.L.).

Finally, to further point out the existence of min-imum with both SI and SD couplings we have min-imized — at variance of the previous approach —the y function with the respect toξσSI andξσSD forfixed mW andθ values. The obtained minima in the

R. Bernabei et al. / Physics Letters B 509 (2001) 197–203 203

(ξσSI, ξσSD) plane for somemW andθ pairs are shownin Fig. 3, where two particular different model frame-works (see figure caption) have been considered as ex-amples. The associated C.L. ranges roughly between� 3 and� 4 σ .

4. Conclusion

In conclusion the analysis discussed in this paperhas shown that the DAMA data of the four annual cy-cles, analyzed in terms of WIMP annual modulationsignature, can also be compatible with a mixed sce-nario where bothξσSI and ξσSD are different fromzero. The pure SD and pure SI cases have been im-plicitly given.

Further investigations are in progress. Moreover, thedata of the 5th annual cycle are already at hand, while— after a full upgrading of the electronics and ofthe data acquisition system — the set-up is runningto collect the data of a 6th annual cycle. Finally, the

exposed mass will be increased in near future up to� 250 kg to achieve higher experimental sensitivity.

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