Light magnetic dark matter in direct detection searches

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JCAP08(2012)010

ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

Light magnetic dark matter in directdetection searches

Eugenio Del Nobile, Chris Kouvaris, Paolo Panci,Francesco Sannino and Jussi Virkajarvi

CP3-Origins & DIAS, Department of Physics, Chemistry and Pharmacy,University of Southern Denmark,Campusvej 55, Odense, Denmark

E-mail: delnobile@cp3-origins.net, kouvaris@cp3-origins.net,panci@cp3-origins.net, sannino@cp3.dias.sdu.dk, virkajarvi@cp3-origins.net

Received April 16, 2012Revised July 8, 2012Accepted July 11, 2012Published August 9, 2012

Abstract. We study a fermionic Dark Matter particle carrying magnetic dipole moment andanalyze its impact on direct detection experiments. In particular we show that it can accom-modate the DAMA, CoGeNT and CRESST experimental results. Assuming conservativebounds, this candidate is shown not to be ruled out by the CDMS, XENON and PICASSOexperiments. We offer an analytic understanding of how the long-range interaction modifiesthe experimental allowed regions, in the cross section versus Dark Matter mass parameterspace, with respect to the typically assumed contact interaction. Finally, in the context of asymmetric Dark Matter sector, we determine the associated thermal relic density, and furtherprovide relevant constraints imposed by indirect searches and colliders.

Keywords: dark matter theory, dark matter detectors, dark matter experiments

ArXiv ePrint: 1203.6652

c© 2012 IOP Publishing Ltd and Sissa Medialab srl doi:10.1088/1475-7516/2012/08/010

JCAP08(2012)010

Contents

1 Introduction 1

2 The event rate 32.1 Kinematics 32.2 Model and differential cross section 42.3 Nuclear recoil rate 5

3 Theoretical predictions 83.1 Light Dark Matter 113.2 Heavy Dark Matter 12

4 Data sets and analysis technique 13

5 Fit to the direct detection experiments 15

6 Relic abundance 18

7 Constraints 197.1 Epoch of reionization and CMB 207.2 Present epoch γ-rays 217.3 Collider and other astrophysical constraints 22

8 Conclusions 22

A Relevance of spin-dependent interaction 23

1 Introduction

The presence of a relatively big dark component in the matter content of the Universe hasbeen now established, and part of the scientific community is trying to unravel its stillenigmatic properties. Despite the conspicuous number of experiments that probed the grav-itational effects of this Dark Matter (DM) on the known matter, we still lack a more directevidence of what it actually is and what are its properties.

Under the assumption that the DM is constituted by one or more unknown particles,several experiments have attempted recently to probe its microscopic nature and its interac-tions. Different search strategies are possible. For indirect detection experiments, traces ofDM annihilations to known particles are looked for in cosmic rays or in the spectrum of solarneutrinos. Collider searches are also performed, in case the DM can be produced in sizableamounts to have an impact in the channels featuring missing energy. Direct detection exper-iments rely instead on direct scatterings of DM particles off nuclei of ordinary matter, insideshielded detectors placed underground in order to diminish the cosmic rays background.

DM direct detection experiments are providing exciting results. For example theDAMA/NaI collaboration [1] has claimed to have observed the expected annual modula-tion of the DM induced nuclear recoil rate [2, 3], due to the rotation of the Earth around theSun. The rotation, in fact, causes a different value of the flux of DM particles hitting the

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JCAP08(2012)010

Earth depending on the different periods of the year. The upgraded DAMA/LIBRA detectorhas further confirmed [4] the earlier result adding much more statistics, and it has reached asignificance of 8.9σ C.L. for the cumulative exposure [5].

Interestingly the DAMA annual modulation effect has been shown to be compatiblewith a DM interpretation which, for the case of coherent spin-independent scattering, leadsto a range of DM masses spanning from a few GeV up to a few hundred GeV, and crosssections between 10−42 cm2 to 10−39 cm2 [1, 4, 5], with some noticeable differences due tothe galactic halo modeling [6, 7].

More recently, the CoGeNT experiment first reported an irreducible excess in the count-ing rate [8], which could also be in principle ascribed to a DM signal. In the last months, thesame experiment reported an additional analysis which shows that the time-series of theirrate is actually compatible with an annual modulation effect [9]. The evidence of such amodulation for CoGeNT is at the level of 2.8σ C.L.

Also the CRESST collaboration observed an excess [10]. In particular, 67 counts werefound in the DM acceptance region, where the estimated background from leakage of e/γevents, neutrons, α particles and recoiling nuclei in α decays, is not sufficient to account forall the observed events. The analysis made by the collaboration rejects the background-onlyhypothesis at more than 4σ [10].

The interesting feature is that the DAMA and CoGeNT results appear to be compatiblefor relatively light DM particles, in the few GeV to tens of GeV mass range and coherentscattering cross section around 10−40 cm2. CRESST points somehow to larger DM masses,but it is still compatible with the range determined by the other two experiments. Theactual relevant range of masses and cross sections depends on assumptions of the galacticDM properties, namely the velocity distribution function and the local DM density [7].Other relevant analyses can be found in refs. [11–28].

The CDMS and XENON100 experiments have recently reported a small number ofevents which pass all the selection cuts. Specifically, they have 2 events for CDMS [29] and6 events for XENON100, reduced to 3 events after post-selection analysis [30], which arestill too few to be interpreted as potential DM signal. They can therefore provide upperbounds on the DM scattering cross section. These constraints seem to set severe boundson the DM parameter space, and therefore to rule out much of the allowed regions of theother experiments. Very recently the PICASSO experiment [31] seems to provide even morestringent constraints for the low mass DM interpretation of DAMA, CoGeNT and CRESST.

However, there are at least two caveats when interpreting the results from theexperiments mentioned above. The first is that one has to pay attention to the fine detailsassociated to the results quoted by each experiment, since a number of factors can affectthe outcome. For example the actual response of the XENON and CDMS detectors canbe uncertain and model dependent for a low energy signal [32–34]. Moreover, for crystaldetectors like DAMA, an additional source of uncertainty is provided by the presence of anunknown fraction of nuclear recoils undergoing channeling. This effect is currently beinginvestigated [35–37]. If confirmed, it would lead to a significant shift of the DAMA allowedregions in the DM parameter space. Another source of uncertainty is associated with thedetails of the nuclei form factors for each experiment.

The second caveat is associated to the interpretation of the actual data withina very simple-minded model of the DM interaction with nuclei. For example, for thespin-independent case it is often assumed that DM couples via a contact interaction withequal strength to the proton and neutron. Upon relaxing some of these assumptions, lessstringent constraints can be drawn.

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JCAP08(2012)010

In this work we perform a thorough investigation of the effects of a magnetic dipolemoment of the DM on the direct detection experiments. We refer to [23, 38–50] for a limitedsample of the earlier literature.

A fermionic DM particle with a magnetic dipole moment can arise for example in modelsof composite DM [51–53], or more generally in models featuring a new strong interaction (seee.g. [54]). The application to cosmic rays as well as a complete discussion also of the lightasymmetric DM paradigm, and last, but not the least, the first link to the study of grandunifications using DM appeared in [55]. Recently it was shown, for the first time, via firstprinciple lattice simulations [56] that one has, indeed, for minimal models of Technicolor thecorrect pattern of chiral symmetry breaking leading to natural candidates of light compositeDM. Despite the possibility to link the DM issue to other unresolved fundamental questionslike e.g. the (dynamical) breaking of the Electroweak symmetry or to the grand unificationproblem, we investigate this DM candidate in a model independent way addressing only itsDM phenomenology. The main feature of the present DM candidate is that it couples mainlyto the photon and therefore interacts mostly with protons, and secondly and more impor-tantly it has long-range type interactions modifying substantially the recoil spectrum withrespect to a contact interaction. In fact, as pointed out in [57], low-energy threshold detectorslike CoGeNT and DAMA are especially sensitive to this kind of interaction, and thereforethe hope exists to get a better agreement among the various direct detection experiments.

Here we will show how the DM magnetic moment interaction affects the interpretationof the different direct detection experiments. In section 2 we present the DM interactionLagrangian and provide some considerations concerning the scattering of DM off nuclei.This will allow us to determine the nuclear recoil rate for a given experiment. In section 3we show how the experimental favored regions and constraints are modified with respect tothe naıve contact interaction case. In section 4 we describe the experimental data set we useand our statistical analysis performed to determine the favored regions and constraints forthe DM candidate envisioned here. We summarize these results in section 5. We determinethe associated thermal relic density section 6, and in section 7 we report the constraintsimposed by indirect searches, colliders and by the observations of compact stars. Finally,we conclude in section 8.

2 The event rate

2.1 Kinematics

When a DM particle scatters off a nucleus, depending on the DM properties, one can envi-sion at least two distinct kinematics, the elastic and the inelastic. The elastic scattering isrepresented by

χ+N (A,Z)at rest → χ+N (A,Z)recoil , (2.1)

while the inelastic isχ+N (A,Z)at rest → χ′ +N (A,Z)recoil . (2.2)

In (2.1) and (2.2), χ and χ′ are two DM particle states, and A, Z are respectively the mass andatomic numbers of the nucleus N . In the detector rest frame, a DM particle with velocity vand mass mχ can scatter off a nucleus of mass mN , causing it to recoil. The minimal velocityproviding a recoil energy ER is:

vmin(ER) '√mN ER2µ2

χN

(1 +

µχN δ

mN ER

), (2.3)

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JCAP08(2012)010

where µχN is the DM-nucleus reduced mass and δ = m′χ −mχ is the mass splitting betweenχ and χ′, and the equation above holds for δ m′χ,mχ. Elastic scattering occurs for δ = 0,while δ 6= 0 implies inelastic scattering. In this paper we will only consider the case of elasticscattering.

2.2 Model and differential cross section

We concentrate on the possibility of having a massless mediator of the DM-nuclei interactionable to yield long-range interactions absent in contact interactions usually assumed. Theobvious candidate mediator in the SM is the photon while other exotic possibilities can bealso envisioned such as a dark photon, see e.g. [57, 58].

Assuming that the dark matter is either a neutral boson or fermion one can then usethe language of the effective theories to select the relevant operators. Stability of the specificcandidate can be easily ensured by charging it under an unbroken global symmetry. Themost popular choices are either a new U(1) or a Z2 depending on the reality of the specificcandidate DM field.

In [59] one can find a general classification, up to dimension six, of all the possibleinteractions between a scalar DM and the SM fields. To this order one discovers that thereis only one gauge-invariant operator coupling the DM complex boson to a photon, namely

i(ϕ†←→∂µϕ)∂νF

µν ; (2.4)

DM-nucleus scattering mediated by this operator has been studied for instance inref. [18, 60, 61], where it is shown that it actually leads to a contact interaction and willhenceforth not be used here.

On the other hand, considering a fermionic DM χ one can show that the onlygauge-invariant couplings to the photon, up to dimension five, are the magnetic and electricmoment-mediated interactions:

LintM = −1

2λχ χσµνχF

µν , LintE = −1

2i dχ χσµνγ

5χFµν . (2.5)

For a Majorana fermion both LintM and Lint

E vanish identically, therefore χ has to be a Diracfermion. λχ and dχ are the magnetic and electric dipole moment, respectively, and areusually expressed in units of e×cm. The scales ΛM ≡ e/λχ and ΛE ≡ e/dχ can be interpretedas the energy scales of the underlying interaction responsible for the associated operators toarise. For instance, in models in which the interactions (2.5) are explained by the DM beinga bound state of charged particles, the Λ’s could represent the compositeness scale.

The differential cross sections for elastic scattering are [41]

dσM(v,ER)

dER=dσSI

M(v,ER)

dER+dσSD

M (v,ER)

dER=

=αλ2

χ

ER

Z2

[1−ER

v2

(2mN+mχ

2mNmχ

)]F 2

SI(ER)+

(λnuc

λp

)2ERv2

mN

3m2p

F 2SD(ER)

,

(2.6)

dσE(v,ER)

dER=

αZ2

ERv2d 2χ F

2SI(ER) , (2.7)

where v is the speed of the DM particle in the Earth frame, α = e2/4π ' 1/137 is thefine structure constant, λp = e/2mp is the nuclear magneton and FSI (FSD) denotes the

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JCAP08(2012)010

spin-independent (spin-dependent) nuclear form factor which takes into account the finitedimension of the nucleus. Here

λnuc =

∑isotopes

fiµ2i

Si + 1

Si

1/2

, (2.8)

is the weighted dipole moment of the target [42], where fi, µi and Si are respectively theabundance, nuclear magnetic moment and spin of the i-th isotope; the values for thesequantities are taken from [62, 63] and agree exactly with the values provided in figure 1 of [42].

The differential cross section in (2.6) features both a spin-independent (SI) and aspin-dependent (SD) part. The SI part, neglecting for a moment the form factor, containstwo terms: an energy dependent term with a E−1

R drop-off of the cross section, and anenergy independent one. In contrast the common contact interactions only feature theconstant term, typically with the 1/v2 dependence on the DM velocity; the SD part is ofthis kind. Notice that, for low enough energies (that might be also below threshold for someexperiments, in principle), the interaction is always SI due to the E−1

R divergence. As therecoil energy rises, the interaction becomes mostly SD for target nuclei with large magneticmoment, namely 19F, 23Na and 127I, (λnuc/λp = 4.55, 2.86, 3.33 respectively); the SD termis instead negligible for all the other nuclei.

The differential cross section in (2.7), instead, has only an energy dependent term. Weobserve that the cross section in the electric case is enhanced with respect to the magneticone by a factor 1/v2 ∼ 106 translating in a value for dχ circa 103 times lower than the onefor λχ, when one tries to fit the experiments. This is confirmed for instance by ref. [41],where the authors find that, in order to fit the CoGeNT data alone, a ΛM of the order of theTeV is needed for a magnetic moment interaction, while for electric moment interaction ΛE

is around the PeV. Given that such a high scale is hardly reconcilable with other attemptsto study DM and in general new physics, we will treat from now on only the magnetic dipolemoment interaction Lint

M .

As for FSI and FSD, we use the nuclear form factors provided in ref. [64]. For the SIinteraction we have checked that FSI matches with the standard Helm form factor [65]. Werecall that all the parameters used in the parameterization of the nuclear form factors maybe affected by sizable uncertainties.

2.3 Nuclear recoil rate

The differential recoil rate of a detector can be defined as:

dR

dER= NT

∫dσM(v,ER)

dERv dnχ , (2.9)

where NT = NA/A is the total number of targets in the detector (NA is the Avogadro’snumber) and dnχ is the local number density of DM particles with velocities in the elementalvolume d3v around ~v. This last factor can be expressed as a function of the DM velocitydistribution fE(~v) in the Earth frame, which is related to the DM velocity distribution inthe galactic frame fG(~w) by the galilean velocity transformation fE(~v) = fG(~v+~vE(t)); here~vE(t) is the time-dependent Earth (or detector) velocity with respect to the galactic frame.The prominent time-dependence (on the time-scale of an experiment) comes from the annualrotation of the Earth around the Sun, which is the origin of the annual modulation effect of

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JCAP08(2012)010

the direct detection rate [2, 3]. More specifically:

~vE(t) = ~vG + ~vS + ~v⊕(t) . (2.10)

The galactic rotational velocity of our local system ~vG and the Sun’s proper motion ~vS arebasically aligned and their absolute values are vG ≡ v0 = 220± 50 km/s and vS = 12 km/s,while the Earth rotational velocity ~v⊕(t) has a size v⊕ = 30 km/s, period of 1 year and phasesuch that it is aligned to ~vG around June 2nd and it is inclined of an angle γ ' 60 with respectto the galactic plane. More details can be found, for instance, in ref. [66]. Summarizing:

dnχ = nχfE(~v) d3v , (2.11)

where nχ = ξχρ0/mχ is the local DM number density in the Galaxy and is determined bythe local Dark Matter density ρ0 and, in general, by a scaling factor ξχ which accounts forthe possibility that the specific DM candidate under consideration does not represent thewhole amount of DM. Here we assume ξχ = 1. In eq. (2.11) the velocity distribution functionneeds to be properly normalized: this can be achieved by requiring that in the galactic frame∫

v6vescd3v fG(~v) = 1 , (2.12)

where vesc denotes the escape velocity of DM particles in the Milky Way. For definiteness,we will adopt here vesc = 650 km/s.

When considering the differential cross section given in equation (2.6), the rate of nuclearrecoils reduces to

dR

dER(t)=NA

ξχ ρ0

Amχ

αλ2χ

ER

(Z2GSI(vmin, t)F

2SI(ER)+

(λnuc/λp

)2 GSD(vmin, t)F2SD(ER)

), (2.13)

where

GSI(vmin(ER), t) = I(vmin(ER), t)

[I1(vmin(ER), t)

I(vmin(ER), t)− ER

(2mN +mχ

2mNmχ

)], (2.14)

GSD(vmin(ER), t) = I(vmin(ER), t)mNER

3m2p

, (2.15)

and

I(vmin, t) =

∫v>vmin(ER)

d3vfE(~v)

v, I1(vmin, t) =

∫v>vmin(ER)

d3v v fE(~v) , (2.16)

with vmin(ER) given by eq. (2.3). The detection rate is function of time through the velocityintegrals I(vmin, t) and I1(vmin, t) as a consequence of the annual motion of the Earth aroundthe Sun. Their actual form depends on the velocity distribution function of the DM particlesin the halo. In this paper we will consider an isothermal sphere density profile for the DM,whose velocity distribution function in the galactic frame is a truncated Maxwell-Boltzmann:

fG(~v) =exp(−v2/v2

0)

(v0√π)3 erf(vesc/v0)− 2v3

0π(vesc/v0) exp(−v2esc/v

20). (2.17)

Under this assumption, and defining the normalized velocities η ≡ (vG + vS)/v0 ≡ v/v0,ηE(t) ≡ vE(t)/v0, ηmin(ER) ≡ vmin(ER)/v0 and ηesc ≡ vesc/v0, the velocity integrals can bewritten analytically as [41]

I(ηmin, t) =1

2 v0ηE(t)[erf(η+)− erf(η−)]− 1√

π v0ηE(t)(η+ − η−) e−η

2esc (2.18)

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JCAP08(2012)010

and

I1(ηmin, t) = v0

[(η−

2√π ηE(t)

+1√π

)e−η

2− −

(η+

2√π ηE(t)

− 1√π

)e−η

2+

]+

v0

4 ηE(t)

(1 + 2η2

E(t))

[erf(η+)− erf(η−)] (2.19)

− v0√π

[2 +

1

3ηE(t)

((ηmin + ηesc − η−)3 − (ηmin + ηesc − η+)3

)]e−η

2esc ,

where η±(ER) = min(ηmin(ER)± ηE, ηesc).

Since in eq. (2.10) the rotational velocity of the Earth around the Sun v⊕, is relativelysmall compared to the main contribution represented by vG + vS, we can approximate ~vE(t)with its component directed toward the galactic center. We can then write [66]

ηE(t) ' η + ∆η cos [2π(t− φ)/τ ] , (2.20)

where ∆η = v⊕ cos γ/v0, with ∆η η, and where φ = 152.5 days (June 2nd) is the phaseand τ = 365 days is the period of the Earth motion around the Sun. By means of eq. (2.20)we can then expand the recoil rate, assuming that the velocity distribution is not stronglyanisotropic:

dR

dER(t) ' dR

dER

∣∣∣∣ηE=η

+∂

∂ηE

dR

dER

∣∣∣∣ηE=η

∆η cos [2π(t− φ)/τ ] . (2.21)

To properly reproduce the recoil rate measured by the experiments, we should takeinto account the effect of partial recollection of the released energy (quenching), and theenergy resolution of the detector:

dR

dEdet(Edet) =

∫dE′K(Edet, E

′)∑i

dRidER

(ER =

E′

qi

). (2.22)

Here the index i denotes different nuclear species in the detector, Edet is the detected energyand qi are the quenching factors for each of the nuclear species. The function K(Edet, E

′)reproduces the effect of the energy resolution of the detector; as is generally done, we assumefor it a Gaussian behavior.

Finally, the recoil rate of eq. (2.22) must be averaged over the energy bins of the detector.For each energy bin k of width ∆Ek we therefore define the unmodulated components of therate S0k and the modulation amplitudes Smk as:

S0k =1

∆Ek

∫∆Ek

dEdetdR

dEdet

∣∣∣∣ηE=η

, (2.23)

Smk =1

∆Ek

∫∆Ek

dEdet∂

∂ηE

dR

dEdet

∣∣∣∣ηE=η

∆η . (2.24)

S0k and Smk are the relevant quantities that we use for the analysis of the experimentswhich address the annual modulation effect, namely DAMA and CoGeNT. For the otherexperiments, only the S0k are relevant.

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JCAP08(2012)010

10 102

10-44

10-43

10-42

10-41

10-40

10-39

10-38

DM Mass mΧ @GeVD

Con

tact

SIC

ross

Sect

ion

Σp

@cm2

D

MB Halo Hv0=220 kmsL

DAMA H8Σ,7ΣLCoGeNT H1ΣLCRESST H4Σ,3ΣL

CD

MS H5

ΣLXENON H5ΣL

PICASSO H5ΣL

Figure 1. DM-proton spin-independent interaction cross section σp as a function of the DarkMatter mass mχ, in the “standard” case of coherent contact interaction. The galactic halo has beenassumed in the form of an isothermal sphere with velocity dispersion v0 = 220 km/s and local densityρ0 = 0.3 GeV/cm3. In this figure we show the allowed regions compatible with the annual modulationeffects in DAMA and CoGeNT, as well as the region compatible with the CRESST excess, wheninterpreted as a DM signal. Specifically, the solid green contours denote the regions compatible withthe DAMA annual modulation effect [4, 5], in absence of channeling [35]. The short-dashed bluecontour refers to the region derived from the CoGeNT annual modulation effect [9], when the boundfrom the unmodulated CoGeNT data is included. The dashed brown contours denote the regionscompatible with the CRESST excess [10]. For all the data sets, the contours refer to regions wherethe absence of excess can be excluded with a C.L. of 7σ (outer region), 8σ (inner region) for DAMA,1σ for CoGeNT and 3σ (outer region), 4σ (inner region) for CRESST. For XENON, the constraintsrefer to a threshold of 4 photoelectrons (published value [30], lower line) and 8 photoelectrons (ourconservative estimate, upper line), as discussed in section 4. The two lines for PICASSO enclose theuncertainty in the energy resolution [31].

3 Theoretical predictions

We attempt here an analytical study of the effect of the magnetic moment interaction on theobserved differential rate. More precisely we provide a simple comparison between the resultsarising from the interaction studied here and the standard picture, i.e. the spin-independentcoherent contact interaction, shown in figure 1. To this aim, eq. (2.13) can be rewritten as

dR

dER(t) = NA

ξχ ρ0

Amχ

mN

2µ2χn

A2αλ2χ Θ(ER)I(ηmin(ER), t)F 2

SI(ER) , (3.1)

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JCAP08(2012)010

0 5 10 15 20 2510-5

10-4

10-3

10-2

10-1

Recoil Energy ER @keVD

QHE

RL

O

F

Na

CaGe I

Xe W

PIC

ASS

O

DA

MA

-N

a

CR

ESS

T

CoG

eNT

2ndbi

n

CD

MS

DA

MA

-I

XE

NO

N4P

E

XE

NO

N8P

E

mΧ = 10 GeV

0 5 10 15 20 2510-5

10-4

10-3

10-2

10-1

Recoil Energy ER @keVD

QHE

RL

O

F

Na

Ca

GeI

XeW

PIC

ASS

O

DA

MA

-N

a

CR

ESS

T

CoG

eNT

2ndbi

n

CD

MS

DA

MA

-I

XE

NO

N4P

E

XE

NO

N8P

E

mΧ = 50 GeV

Figure 2. The function Θ(ER) (dashed lines), parametrizing the deviation of the allowed regions andconstraints with respect to the standard case of contact interaction without isospin violation presentedin figure 1. Each line is for one of the target elements used in the experiments taken into accountin the text. The vertical lines indicate the low energy thresholds for the various experiments (apartfrom CoGeNT, for which we show the threshold of the second energy bin, where much of the signal isrecorded). For DAMA two thresholds are shown, corresponding to the corrections due to the quenchingfactors for sodium (qNa = 0.3, in the assumption of scattering mainly off Na) and iodine (qI = 0.09,for the assumption of scattering mainly off I). For XENON the 4 photoelectrons (4PE, publishedvalue [30]) and 8 photoelectrons (8PE, our conservative estimate) thresholds are shown, as discussedin section 4. We have shown also the lowest threshold for the PICASSO experiment, although we referto the main text for a more precise interpretation of its temperature dependent thresholds. The pictureis reported for two values of the DM mass, mχ = 10 GeV (left panel) and mχ = 50 GeV (right panel).

where µχn is the DM-nucleon reduced mass, αλ2χ plays the role of the spin-independent

DM-proton cross section σp and we defined

Θ(ER) = ΘSI(ER)

(1 +

ΘSD(ER)

ΘSI(ER)

F 2SD(ER)

F 2SI(ER)

)= ΘSI(ER) (1 + r(ER)) , (3.2)

where

ΘSI(ER)=

(Z

A

)2(

2µ2χn

mNER

)(GSI(ηmin(ER), t)

I(ηmin(ER), t)

), (3.3)

ΘSD(ER)=

(1

A

)2(λnuc

λp

)2(

2µ2χn

mNER

)(GSD(ηmin(ER), t)

I(ηmin(ER), t)

)=

2

3

(1

A

)2(λnuc

λp

)2(µχnmp

)2

. (3.4)

The function Θ measures the deviation of the allowed regions and constraints with respectto the standard spin-independent picture, and the ratio r parametrizes the relevance of theSD interaction respect to the SI one: if r > 1 the interaction is largely SD, while it is mostlySI for r < 1 (see appendix A for a more detailed analysis).

Since GSI/I depends itself on mχ, we expect an asymmetric shift of the favored regionsand constraint lines in the (mχ, σp) plane due exclusively to the modified dynamics of the

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JCAP08(2012)010

interaction considered here. There are only two free parameters in this model: mχ, and σp(or alternatively λχ). This is not true for instance in models with contact interaction in whichthe DM interacts differently with protons and with neutrons (the so-called “isospin violating”scenario, see e.g. [17, 18, 22, 58, 67–74]). Indeed in such models the role of Θ is played bythe expression ([Z + fn/fp(A− Z)]/A)2, and therefore there are three free parameters thatmust be fitted: σp, mχ, and the ratio of the DM-neutron and DM-proton couplings fn/fp.

In figure 2 we show the behavior of Θ as a function of ER both for small (left panel)and large (right panel) DM masses, considering several targets, v0 = 220 km/s and vE = v(being v⊕ cos γ v). In order to simplify the reading of the figures, we take the limitvesc → ∞; one can figure out the effect of a finite escape velocity as a target-dependentcut-off at high recoil energies, that sets in when the minumum velocity required to have arecoil above threshold exceeds vesc + vE, the maximum escape velocity in the Earth frame.There, the function Θ loses its meaning since the expected rate is zero both in the standardpicture and in the case considered here.

A first striking feature of Θ is its overall magnitude: depending on the DM mass andon the experiment, we have in fact a suppression of the expected rate of roughly 1 to 4orders of magnitude. For nuclei in which the interaction is mostly SD (r > 1), Θ providesa suppression of 1 to 3 orders of magnitude for the well known reason that the rate doesnot carry any A2 factor, contrarily to the usual SI case. Due to this fact, the SD part of thecross section is usually considered negligible respect to the SI one, but this turns out to befalse in our case, for nuclei with large magnetic moment, since also the SI part is stronglysuppressed. This suppression is due to the interplay of the last two factors in eq. (3.3),while the first term plays only a marginal role given that A ∼ 2Z roughly for all the targetnuclei. The second term, 2µ2

χn/mNER, enhances Θ by 2 to 4 orders of magnitude due to thepresence of ER, whose typical scale is few (tens of) keV and therefore very small comparedto all the other mass scales involved. The last term GSI/I, containing the velocity integrals,gives instead a suppression of roughly 6 orders of magnitude. Given the measured rate,this overall suppression provided by the Θ function will reflect in the fit pointing to higherinteraction cross sections, with respect to the standard case of figure 1. This fact will playa role in fitting the relic abundance (section 6).

The steep rise at low energies is always due to the long-range type of the SI partof the interaction, being the SD one of a contact type; in the particular case of magneticdipole moment interaction considered here, the differential cross section diverges as 1/ER.At energies higher than the steep rise, for nuclei with large magnetic moment (r > 1),the interaction rapidly becomes of a contact type and therefore the function Θ exhibits aplateau whose value is ∼ (λnuc/Aλp)

2. Instead, for the other nuclei for which r < 1, tworegimes are possible, depending on the ratio vmin/vE. These reflect the two trends assumedby the function1 ζ(vmin) ≡ I1(vmin)/I(vmin), namely a plateau ζ ∼ 1.8 v2

0 for vmin vE anda rise ζ ∼ v2

min for vmin vE, with an intermediate value of ζ(vE) ∼ 2.8 v20.

Light DM particles require a higher minimum velocity to recoil compared to heavierones. For the range of DM masses we are interested in, and for the range of recoil energiesrelevant for the direct DM search experiments, the value of vmin/vE is controlled by the ratiomN/mχ. We will see that, depending on this ratio, the function ΘSI can assume a constantbehavior or scale as E−1

R , while the function ΘSD is always constant.

1Here we ignore the time dependence, which gives only a negligible contribution.

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JCAP08(2012)010

3.1 Light Dark Matter

For DM particles much lighter than the target nuclei, we have that vmin > vE already at lowenergies for all the targets considered, and therefore

GSI(ηmin(ER), t)

I(ηmin(ER), t)'[v2

min(ER)− ER(

2mN +mχ

2mNmχ

)]=mNER2m2

χ

; (3.5)

accordingly, Θ simplifies to (µχn/(Amp))2[Z2 (mp/mχ)2 + 2/3

(λnuc/λp

)2], displaying the

plateau shown in the left panel of figure 2. As expected, the plateau sets in earlier for heavytargets if the SI interaction dominates (Z2 (mp/mχ)2 2/3

(λnuc/λp

)2), while its starting

point is independent on the target mass for SD interactions. We notice that, for high enoughrecoil energies, Θ always gets to this constant behavior; it is interesting though that it exists aregime in which this happens at the low energies relevant for DM direct detection experiments.An indicative mχ ∼ 10 GeV falls into this case, and as we will show in section 5 can fit the re-gions of the parameter space allowed by CoGeNT, DAMA-Na, CRESST-O and CRESST-Ca.2

For an experiment with a lower threshold above the steep rise, the differential rate doesnot feature any dependence on ER and it is therefore similar to the rate of spin-independentcontact interactions, apart from a different functional dependence on mχ. Notice that dueto this different dependence on the mass of the DM, the rate is more suppressed for heavierDM particles, and therefore we expect the favored regions in the (mχ, σp) plane as well asthe exclusion lines to raise and slightly tilt at higher mχ (compared to the spin-independentcontact interaction).

Considering now that most of the signal in a given detector comes generally from thefirst energy bin,3 i.e. close to the low energy threshold, we can roughly estimate, before doingany statistical analysis of the data, the shift of the allowed regions and constraints comparedto the standard picture. For example, the expected rate of DM scatterings off Na nuclei isreduced by a factor ∼ 10−2 with respect to the standard case (see figure 2), and therefore theDAMA favored region, when assumed that most of the signal comes from DM scattering onNa, is shifted up by ∼ 102 in the (mχ, σp) plane. Taking now the DAMA-Na allowed region asbenchmark, we can see how the other favorite regions and exclusion lines move with respect toit. CoGeNT moves a factor ∼ 5 up, making the agreement with DAMA almost perfect. Con-cerning CRESST, the fit at small DM masses is due equally to O and Ca [10] in the standardscenario, from which we conclude that the cross section for DM magnetic dipole interactionwith Ca is ∼ 1.5 times bigger than that with O; the overall effect is an increase in σp, that liftsthe favored region to better agree with DAMA and CoGeNT. In detail we expect that theCRESST allowed region moves a factor ∼ 4 towards the DAMA-Na ballpark. Finally, CDMSand XENON move up by a factor ∼ 6.5 and ∼ 8.5 respectively, roughly irrespective of thechoice of the threshold. This improves once again the agreement with the other experiments.For PICASSO the situation is different due to the special experimental setup in which thelow energy threshold is a function of the temperature, and therefore one should be careful es-pecially if the differential cross section is energy dependent. However, since the scattering inthe PICASSO experiment is dominated by the dipole-dipole interaction which is of a contacttype, a simple rescaling is again applicable, and we expect that the constraints will becomea factor 2.5 more stringent (see left-panel of figure 2), compared to the standard picture.

2DAMA and CRESST are multi-target detectors and allowed regions at large DM mass correspond toscattering on I for DAMA and W for CRESST, while at small DM mass regions correspond to scattering onNa for DAMA and both O and Ca for CRESST.

3In CoGeNT, the larger part of the signal comes instead from the second energy bin.

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JCAP08(2012)010

3.2 Heavy Dark Matter

For higher DM masses, where for instance one can find the regions of the parameter spaceallowed by DAMA-I and CRESST-W, the effect of the long-range interaction turns to bevery evident. As the minimal velocity for a given nuclear recoil vmin becomes smaller thanvE, the function ζ(vmin) changes behavior. For vmin vE (i.e. mχ mN )4

GSI(ηmin(ER), t)

I(ηmin(ER), t)'[1.8 v2

0 − ER(

2mN +mχ

2mNmχ

)]. (3.6)

In this regime, for the nuclei whose interaction is SI, Θ scales as E−1R for the whole range of

the recoil energies relevant for the direct detection experiments. If ER increases sufficientlyenough, vmin becomes eventually larger than vE and Θ assumes the constant behavior wedescribed in the previous section. The difference in Θ between the low and high DM masseswe depicted in figure (2) is that in the first case the plateau sets in at small recoil energies(smaller than the energy threshold of most experiments), whereas in the second case, theplateau sets in at large recoil energies (more than 25 keV). Instead for the other nuclei withlarge magnetic moment, the interaction is SD and as in the case of light DM the plateaumanifests itself at low recoil energy.

Due to the pronounced energy dependence in this case, the allowed regions and theconstraints will shift in the (mχ, σp) plane considerably more than in the case of low DMmass (still compared to the standard contact spin-independent interaction). By taking againthe DAMA-Na allowed region as a benchmark from the right panel of figure (2), we estimatethat both DAMA-I and CRESST-W shift more than one order of magnitude up in σp. Thesame happens for CDMS and XENON. On the other hand PICASSO, like DAMA-Na, doesnot change its behavior respect to the light DM case: this is due to the fact that 19F and23Na enjoy mostly spin-dependent interactions, that depend only slightly on the DM mass.

To summarize, taking into account the whole DM mass range, we see that, apart froman overall shift upwards in σp, there is a general trend of the various experiments to “gather”,getting closer to each other, with respect to the standard case. This behavior is not homoge-neous in the DM mass and is more pronounced for masses above ∼ 25 GeV. The effect is tofavor a better fit compared to the standard case, with a better agreement of all the experi-ments. Even though this is true in general for any DM mass, we don’t expect anyway to get agood agreement for higher masses: in that region in fact the XENON experiment rules out thesignal featured by other experiments by several orders of magnitude in the standard picture(figure 1), and the shifts that take place in our case are not big enough to change this situation.

Finally we comment on the dependence of the function Θ from the halo modelconsidered, and in particular from the choice of the dispersion velocity v0. As we have seen,the transition between the two regimes discussed above is driven by the ratio vmin/vE, andit is only important for the nuclei which experience SI interactions with a DM particle.Therefore in this case we expect that for larger values of vE ≈ v0, the plateau behavior ofΘ sets in at higher recoil energies (or equivalently at smaller DM masses). This would makethe shifts more pronounced even for light DM. In section 5 and figures 3 and 4 we presentthe exact numerical results.

4This is generally true provided we consider a recoil energy ER within our interest: from a minimumthreshold value of a few keV up to 25− 30 keV.

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JCAP08(2012)010

4 Data sets and analysis technique

In this section we discuss the techniques used to analyze the various data sets. In particularwe adopt the approach of [57], and we summarize below the details of how we perform thefits to data and constraints from null results experiments.

For DAMA, CoGeNT and CRESST, we test the null hypothesis (absence of signalon top of estimated background for CRESST and absence of modulation for DAMA andCoGeNT). From this we infer:

i) the confidence level for the rejection of the null hypothesis (we find 8÷ 9σ for DAMA,1÷ 2σ for CoGeNT, and 4σ for CRESST);

ii) the domains in the relevant DM parameter space (defined by the DM mass mχ and theDM magnetic dipole moment λχ) where the values of the likelihood function departfor more than nσ from the null hypothesis, and thus the corresponding evidence of theDM signal. We use n = 7, 8, n = 1, and n = 3, 4 for DAMA, CoGeNT, and CRESST,respectively [7].

Our statistical estimator is the likelihood function of detecting the observed number ofevents L =

∏i Li, where the index i indicates the i-th energy bin in DAMA and CoGeNT,

and the i-th detector in CRESST. For DAMA and CoGeNT Li are taken to follow aGaussian distribution and for CRESST, since in this case the number of events in eachsub-detector is low, a Poissonian one. Defining Lbg as the likelihood of absence of signal, weassume the function y = −2 lnLbg/L to be distributed as a χ2-variable with one degree offreedom for a given value of the DM mass (notice that for DAMA and CoGeNT y reducesto y = χ2

bg − χ2). From the y function, the interval on λχ where the null hypothesis(i.e. λχ = 0) can be excluded at the chosen level of confidence are extracted: 7σ (outerregion) or 8σ (inner region) for DAMA, 1σ for CoGeNT and 3σ (outer region), 4σ (innerregion) for CRESST. We then plot allowed regions in the (mχ, λχ) plane.

We derive constraints from CDMS and XENON100 with a similar likelihood functionλ = −2 lnL/Lbg; here L is the likelihood of detecting the observed number of events (2 forCDMS and 3 for XENON100), while in Lbg the DM signal is not included. Both likelihoodsare taken as Poissonian variables and λ is assumed to follow a χ2-distribution. For PICASSOwe instead derive constraints by using a ∆χ2 method of the data points shown in figure 5of [31]. Bounds are conservatively shown at 5σ C.L.

Concerning the experimental data sets and statistical methods, we refer to [57]; forcompleteness we summarize below for each experiment the most important ingredients usedin the analysis.

DAMA: we use the entire set of DAMA/NaI [4] and DAMA/LIBRA [5] data, cor-responding to a cumulative exposure of 1.17 ton×yr. We analyze the modulationamplitudes Sexp

mk reported in figure 6 of ref. [5] requiring that the DM contributionto the unmodulated component of the rate, S0, does not exceed the correspondingexperimental value Sexp

0 in the 2÷ 4 keV energy range. We compute

y = −2 lnL ≡ χ2(λχ,mχ) =8∑

k=1

(Smk − Sexp

mk

)2σ2k

+(S0 − Sexp

0 )2

σ2θ(S0 − Sexp

0 ) , (4.1)

where σk and σ are the experimental errors on Sexpmk and Sexp

0 , respectively. The lastterm in eq. (4.1) implements the upper bound on S0 by penalizing the likelihood

– 13 –

JCAP08(2012)010

when S0 exceeds Sexp0 with the Heaviside function θ. The detector energy resolution is

parametrized by a Gaussian function of width σres(E) = E(0.448/√E + 0.0091) [75],

using for the quenching factor the central values quoted by the collaboration, namelyqNa = 0.3 and qI = 0.09 [76]. We don’t take into account the possibility of a nonzerochanneling fraction [35].

CoGeNT: we consider the time-series of the data, treating the measured total rate asa constraint. Similarly to the analysis done on the DAMA data, we define

y = −2 lnL ≡ χ2(λχ,mχ) =

=16∑k=1

(Sm1,k−Sexp

m1,k

)2

σ2k

+16∑k=1

(Sm2,k−Sexp

m2,k

)2

σ2k

+31∑j=1

(S0j−Sexp

0j

)2

σ2j

θ(S0j−Sexp0j ) ;

(4.2)

here Smk = 1/∆tk∫

∆tkSmk cos [2π(t− φ)/τ ] dt, with ∆tk the temporal bin of the data,

and Sexpmk = Rexp

mk − 〈Rexpmk 〉, where Rexp

mk is the total rate (taken from figure 4 of ref. [9])and 〈Rexp

mk 〉 is its annual average. The subscripts 1 and 2 in eq. (4.2) refer to the firstand second energy bins. The total rate in the 0.9 − 3.0 keVee energy-bin is computedby subtracting the rate in the 0.5− 0.9 keVee bin to the rate in the 0.5− 3.0 keVee bin,with a Gaussian propagation of the errors. Sexp

0j and σj denote the experimental countsand the corresponding errors as given in ref. [9] (31 energy bins in the interval 0.4− 2keVee), after removal of the L-shell peaks but without removing any other background.The total fiducial mass is 330 g, the energy resolution is given by a Gaussian withwidth taken from [77], and the quenching factor below 10 keV is described by therelation E = 0.2E1.12

R [78].

CRESST: we compute the expected DM signal in each of the 8 CRESST detectormodules. The acceptance regions and the number of observed events are provided intable 1 of ref. [10], and we derive background events according to estimates in section 4of ref. [10]. A likelihood-ratio test yields a 4.1σ C.L. evidence for the best-fit of a DMsignal over the background-only hypothesis, in good agreement with the result quotedby the collaboration. We use the published value of 730 kg×days for the exposureand assume an even contribution among the different modules5 (each module accountstherefore for an exposure of 730/8 kg×days); we consider moreover a constant efficiency.

CDMS: we use the “standard” 2009 CDMS-II results based on Ge data [29]; these areobtained employing conservative nuclear recoil selection cuts and assuming an energythreshold of 10 keV. The total exposure is 612 kg×days and we take the efficiencyfrom the black curve of figure 5 in ref. [79] with q ' 1 as quenching factor.6 In spiteof an expected background of 0.9 ± 0.2 events, two signal events were found in the10− 100 keV energy interval [29] (we use these numbers to derive the constraints).

XENON100: we use the results presented in ref. [30], with an exposure of 100.9 daysin a fiducial volume of 48 kg. After all the cuts, three events were reported in theDM signal region in spite of an expected background of 1.8 ± 0.6 events. We model

5This is the same analysis performed in [57], although a value of 400/9 kg days for the exposure waserroneously reported in the text of the published version.

6In the case of light DM, one can perform a similar analysis by using combined data from the CDMS andEDELWEISS experiments (see figure 1 in ref. [80]), with basically the same results.

– 14 –

JCAP08(2012)010

the data using a Poissonian distribution of photoelectrons, with a single-photoelectronresolution equal to 0.5. The shape of the Leff function is very crucial for such a lownumber of photoelectrons and for small DM masses. Following [57] we try to enclosea possible (but not exhaustive) uncertainty on the bounds derived for XENON100 byadopting two different approaches:

i) we adopt as threshold the published value of 4 photoelectrons and the nominalcentral value of Leff as shown in figure 1 of ref. [30], which relies heavily on linearextrapolation below 3 keVnr;

ii) more conservatively, we raise the threshold for the photomultipliers to 8 photo-electrons: this value is the lowest one for which the analysis is nearly independenton the shape of Leff below 3 keVnr.

Notice that these two approaches are not exhaustive of all the possible assumptionsone can do to determine the XENON100 response to light DM (for further discussionand considerations, see e.g. refs. [32, 33]). It appears therefore still preliminary, giventhese large uncertainties, to assume the bounds quoted by the collaboration as strictlyfirm. The 8 photoelectrons bound is less dependent on the Leff extrapolation, andtherefore, conservatively, we consider it as more appropriate.

Finally, we follow eqs. (13)–(16) in ref. [81] to compute the expected signal. We deriveupper bounds for both CDMS and XENON as mentioned at the beginning of thissection.

PICASSO: the PICASSO experiment, located at SNOLAB [31], is very different fromthe ones discussed above; it is in fact based on the superheated droplet technique, avariant of the bubble chamber technique, to search for DM recoiling on 19F in a C4F10

target. The experimental procedure consists in measuring the acoustic signal releasedby the nucleation of a bubble as a function of the temperature T . Details of thedetector principle can be found in [82, 83]. Since bubble formation is only triggeredabove a certain energy threshold Eth(T ), the spectrum of the particle-induced energydepositions can be constructed by varying the temperature. We compute the predictedDM rate as a function of Eth(T ) from eq. (3) of [31] and we compare such predictionwith the experimental rate shown in figure 5 by using a ∆χ2 method. In this analysiswe adopt two reference values of the parameter a(T ) which describes the steepness ofthe energy threshold; namely we take a = (2.5, 7.5) in order to encapsulate as muchas possible the experimental uncertainties [31]. We checked our result against the oneof the collaboration given in figure 7 of ref. [31], and we found an excellent agreement.Remaining coherent with our choice of being conservative we show here the result at5σ, as we did also for the other exclusion experiments.

5 Fit to the direct detection experiments

We analyze the direct detection data sets by using a standard isothermal halo model,which basically implies a truncated Maxwell-Boltzmann velocity distribution function (seediscussion in section 2.3). Since the response of direct detection experiments is quite sensitiveto the DM distribution in the galactic halo [6], especially in the scenario we are interestedto study, we take into account uncertainties on the velocity dispersion v0, as discussed inref. [6]. We will use the three values v0 = 170, 220, 270 km/s, which bracket the uncertainty

– 15 –

JCAP08(2012)010

10 102

10-19

10-18

10-17

DM Mass mΧ @GeVD

DM

Mag

netic

Dip

ole

ΛΧ

@ecm

D

MB Halo Hv0=220 kmsL

DAMA H8Σ,7ΣLCoGeNT H1ΣLCRESST H4Σ,3ΣL

CD

MS

H5ΣL

XEN

ON

H5ΣL

PICASSO H5ΣL

WDM

Figure 3. DM magnetic dipole moment λχ as a function of the Dark Matter mass mχ. The galactichalo has been assumed in the form of an isothermal sphere with velocity dispersion v0 = 220 km/s andlocal density ρ0 = 0.3 GeV/cm3. Notations are the same as in figure 1; to match the two figures, onehas to note that the role of σp is played here by αλ2χ. The orange strip shows the values for (mχ, λχ)that fit the relic abundance ΩDM assuming a completely thermal DM production (see section 6).

in the local rotational velocity. Let us notice that the value of the local DM density ρ0 iscorrelated to the adopted value of v0, as discussed e.g. in ref. [6]. This corresponds to themodel denoted as A0 in [6], and we adopt here the case of minimal halo, which implieslower values of the local DM density (since a fraction of the galactic potential is supportedby the disk/bulge). In turn, this implies the adoption of ρ0 = 0.18, 0.30, 0.45 GeV/cm3 forv0 = 170, 220, 270 km/s, respectively [6].

Figure 3 shows the constraints and the favored regions coming from DM direct detec-tion experiments in the (mχ, λχ) plane. The galactic halo has been assumed to be in theform of an isothermal sphere with velocity dispersion v0 = 220 km/s and local density ρ0 =0.3 GeV/cm3. The solid green contours denote the regions compatible with the DAMA annualmodulation effect [4, 5], in absence of channeling [35]. The short-dashed blue contour refersto the region derived from the CoGeNT annual modulation signal [9], when the bound fromthe unmodulated CoGeNT data is taken into account. The dashed brown contours denote theregions compatible with the CRESST excess [10]. For all the data sets, the contours refer toregions where the absence of modulation can be excluded with a C.L. of 7σ (outer region), 8σ(inner region) for DAMA, 1σ for CoGeNT, and the absence of an excess can be excluded at 3σ(outer region), 4σ (inner region) for CRESST. Constraints derived by the null result experi-ments are shown at 5σ as gray, magenta and red dashed lines for CDMS, XENON100 and PI-CASSO respectively. For the XENON detector, as discussed in the previous section, the con-straints refer to thresholds of 4 and 8 photoelectrons, while for PICASSO we take into accountthe uncertainty in the intrinsic energy resolution of the employed detection technique [31].

– 16 –

JCAP08(2012)010

10 102

10-19

10-18

10-17

DM Mass mΧ @GeVD

DM

Mag

netic

Dip

ole

ΛΧ

@ecm

D

MB Halo Hv0=170 kmsL

DAMA H8Σ,7ΣLCoGeNT H1ΣLCRESST H4Σ,3ΣL

CD

MS H5

ΣLX

ENO

NH5ΣL

PICASSO H5ΣL

WDM

10 102

10-19

10-18

10-17

DM Mass mΧ @GeVD

DM

Mag

netic

Dip

ole

ΛΧ

@ecm

D

MB Halo Hv0=270 kmsL

DAMA H8Σ,7ΣLCoGeNT H1ΣLCRESST H4Σ,3ΣL

CD

MS

H5ΣL

XEN

ON

H5ΣL

PICASSO H5ΣL

WDM

Figure 4. DM magnetic dipole moment λχ as a function of the Dark Matter mass mχ. The galactichalo has been assumed in the form of an isothermal sphere with velocity dispersion v0 = 170 km/s andlocal density ρ0 = 0.18 GeV/cm3 (left panel); v0 = 270 km/s and local density ρ0 = 0.45 GeV/cm3

(right panel). Notations are the same as in figure 1; to match the two figures, one has to note thatthe role of σp is played here by αλ2χ. The orange strip shows the values for (mχ, λχ) that fit the relicabundance ΩDM, in the assumption of thermal DM production (see section 6).

We can see that, as expected from the discussion in section 3, for a given value of mχ,all the experiments determine a DM-proton cross section σp = αλ2

χ that is higher than theone for the standard case of non isospin violating contact interaction depicted in figure 1; togive a benchmark value, DAMA points now to σp ∼ 1.5× 10−38 cm2, about 102 times abovethe standard case, as shown in section 3. Moreover we observe the expected gathering ofthe various experiments, that get closer to each other with respect to the standard scenario,for an overall better agreement, and with DAMA, CoGeNT and CRESST featuring a largeroverlap at low masses. Both the DAMA and CoGeNT regions point towards a DM mass inthe 10 GeV ballpark (more specifically, from about 7 up to about 12 GeV) and DM magneticdipole moment around 1.5× 10−18 e cm without exceeding the constraints, corresponding toan inverse mass energy scale of circa ΛM ∼ 10 TeV. CRESST allows for even heavier DMmasses, but still compatible with the range determined by the other two experiments. CDMSand XENON, on the other hand, exclude a smaller part of the parameter space with respectto the standard case. These two experiments do not exclude the overlapping region once oneaccepts our conservative choice of the XENON threshold at 8 photoelectrons. At 5σ it is seenthat PICASSO cannot exclude the common regions of the experiments reporting a signal.Finally, we also notice the occurrence of the expected flattening of the various experiments,meaning that the tilt featured in the standard case (higher mass regions pointing to lowerσp) is very suppressed in the case of magnetic moment interaction, as explained in section 3.

These results depend on the galactic halo model assumed. The effect induced by thevariation in the DM dispersion velocity is shown in the two panels of figure 4. In the case ofv0 = 170 km/s, the regions are not significantly modified as compared to the case v0 = 220km/s, except for the overall normalization due to the different values of the local DM density

– 17 –

JCAP08(2012)010

in the two cases. Larger dispersion velocities, instead, lead to the second regime discussedin section 3 also at small DM masses (see right panel of figure 2). Due to the high energydependence of the differential rate in this regime, experiments that use heavy targets and/orhave high thresholds, get a suppression of the event rate, and point therefore to relativelyhigher values of the interaction cross section compared to the standard case. Since theDM signal is expected at lower recoil energies, smaller DM masses are now favorite. Theright panel of figure 4 shows in fact that the allowed regions are shifted toward smaller DMmasses for v0 = 270 km/s.

6 Relic abundance

In addition to a satisfactory fitting of the direct detection data, the DM magnetic dipoleinteraction can accommodate the annihilation cross section appropriate for thermal produc-tion. Since it couples to the photon, the DM annihilates to any charged Standard Modelparticle-antiparticle pair, with mass lower than mχ. Considering a DM particle with 10 GeVmass, that as we argued fits well the direct detection experiments, the allowed primarychannels are e+e−, µ+µ−, τ+τ−, uu, dd, ss, cc, bb and γγ.

The DM annihilation cross sections to Standard Model fermions and photons at treelevel are

σχχ→ff (s) =αQ2

fNfC

12

βfβχ

1

s2×

×[λ2χ

(s2(3− β2) + 12m2

χs+ 48m2fm

2χ

)+ d2

χ

(s2(3− β2)− 12m2

χs)], (6.1)

σχχ→γγ(s) =λ4χ

4πβχ

(m2χ +

s

8−sβ2

χ

24− 4

m4χ

s

arcth(βχ)

βχ

), (6.2)

where βχ,f = (1−4m2χ,f/s)

1/2, β ≡ βχβf , Qf is the charge of the fermion and finally N `C = 1

for leptons and N qC = 3 for quarks. In the non-relativistic limit, eqs. (6.1) and (6.2) reduce to

〈σχχ→ff vrel〉 ' N αλ2χ · BR(f) , (6.3)

〈σχχ→γγ vrel〉 '1

4πλ4χm

2χ , (6.4)

where N =∑

f Q2fN

fC = 20/3 accounts for the number of degrees of freedom of the Standard

Model fermions into which the DM can annihilate; the branching ratios BR(f) are defined

as BR(f) = Q2fN

fC/N . As one can see, for the low values of λχmχ pointed by the direct

detection experiments, the two photons final state is suppressed with respect to the fermionicone. Using a DM energy density ΩDMh

2 = 0.1126 ± 0.0036 measured by WMAP [84], weshow7 with an orange strip in the (mχ, λχ) plane of figures 3 and 4 the phase space wherea thermal production of the magnetic DM is possible.

It can be seen in figures 3 and 4 that the agreement between the allowed regions andthe relic abundance depends strongly on the dispersion velocity v0. The best agreement

7The figures are produced by solving numerically the Boltzmann equation

ntot + 3Hntot = −1

2〈σannvrel〉

(n2tot − n2

eq

), (6.5)

where ntot is the total number density of particles and antiparticles, H is the Hubble parameter and for〈σannvrel〉 we use the more precise formula given in eq. (3.8) of [85].

– 18 –

JCAP08(2012)010

takes place for lower values of v0. This is due to the fact that, as commented already insection 5, ρ0 is linked to the dispersion velocity v0. By varying v0, ρ0 is modified accordingly,changing therefore the expected rate and the fitting values for the interaction cross sections,or in other words λχ. As a consequence, the favored regions and constraint lines of directdetection experiments move along the vertical axis in the (mχ, λχ) plane. For lower velocitydispersion, both the local DM density and the expected rate decrease. Therefore, the favoredregions will point to a higher DM magnetic moment, getting closer to the relic abundancestrip. The value of the relic abundance is obviously independent on the local DM density.

The fact that the magnetic DM (with only two parameters) provides a good fit to the di-rect detection experiments and simultaneously can accommodate a thermal annihilation crosssection (which is several orders of magnitude stronger than the DM-nucleon cross section inthe context of “standard” contact interaction) is not easily met by other DM candidates. Al-though there are candidates that can have a DM-nucleon cross section much lower than theirthermal annihilation cross section, these candidates have usually spin-dependent interactionsand cannot fit nicely the direct search experiments with a positive DM signal. For a typicalcandidate with spin-independent contact interactions, and for a strength of interaction thatleads to a DM-nucleon cross section pointed by the direct detection experiments featuring asignal, the annihilation cross section is way too small to produce this candidate thermally.In order to match, a suppression mechanism for the direct detection event rate should takeplace. In the case of magnetic moment DM, this mechanism exists and has two differentreasons for the SI and the SD parts of the interaction. As already explained in section 3, thissuppression is encoded into the function Θ in eq. (3.2), that spans roughly from 10−3 to 10−1

for a 10 GeV DM (see figure 2). This suppression is a result between two competing factorsfor the SI case: the enhancement provided by the E−1

R dependence of the differential crosssection, and the suppression provided by the kinetic integral; in the SD case, the suppressionis instead due to the lack of the A2 enhancement usually present in the standard SI case.

Notice that in the case of electric dipole DM one faces a very different situation: here thedifferential cross section has the same dependence on the velocity as in a contact interaction(eq. (2.7)), and therefore there is no kinetic suppression, while on the contrary the rate isenhanced by the E−1

R dependence. Moreover, the annihilation to fermions is a p-wave process,eq. (6.1), and therefore the annihilation cross section is suppressed, making the value of dχneeded to fit the relic abundance even bigger.

Furthermore, it is worth to point out that neither asymmetric/mixed DM [86], noroscillating DM [87–89] can improve the agreement between the relic density and the allowedregions of direct DM searches in the context of magnetic DM interaction.

7 Constraints

Given the qualitative agreement between the direct detection allowed regions and the fitof the relic abundance, it is natural to ask whether constraints coming from other searchescould limit the parameter space of the magnetic moment DM. This is even more pressingsince the bounds coming from indirect DM searches are on the verge of constraining thethermal annihilation rate for light DM particles. We will discuss the constraints imposed byindirect searches, colliders and by the observations of compact stars, and then identify themost stringent ones.

– 19 –

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10 102

10-19

10-18

10-17

DM Mass mΧ @GeVD

DM

Mag

netic

Dip

ole

ΛΧ

@ecm

D

MB Halo Hv0=220 kmsL

DAMA H8Σ,7ΣLCoGeNT H1ΣLCRESST H4Σ,3ΣL

CD

MS

H5ΣL

XENON H5ΣL

PICASSO H5ΣL

WDM

DwarfGalacticCMB

Γ-ray lines

Figure 5. DM magnetic dipole moment λχ as a function of the Dark Matter mass mχ. The galactichalo has been assumed in the form of an isothermal sphere with velocity dispersion v0 = 220 km/sand local density ρ0 = 0.3 GeV/cm3. Notations are the same as in figure 1. The orange stripshows the values for (mχ, λχ) that fit the relic abundance ΩDM, in the assumption of thermal DMproduction (see section 6). The shaded regions refer to the constraints from γ-ray lines, CMB, galacticγ-rays and dwarf galaxies. These constraints are only valid in the assumption of symmetric DM. Thegalactic photons constraint enforces total annihilation of the DM into bb, and therefore a less stringentconstraint is expected for magnetic moment DM.

7.1 Epoch of reionization and CMB

Strong constraints are imposed on DM annihilations from considering the effect on thegeneration of the CMB anisotropies at the epoch of recombination (at redshift ∼ 1100)and their subsequent evolution down to the epoch of reionization. The actual physicaleffect of energy injection around the recombination epoch results in an increased amountof free electrons, which survive to lower redshifts and affect the CMB anisotropies [90–93].Detailed constraints have been recently derived in [94, 95], based on the WMAP (7-year)and Atacama Cosmology Telescope 2008 data. The constraints are somewhat sensitiveto the dominant DM annihilation channel: annihilation modes for which a portion of theenergy is carried away by neutrinos or stored in protons have a smaller impact on the CMB;on the contrary the annihilation mode which produces directly e+e− is the most effectiveone. Usually the approach here is to consider 100% annihilation rate in a single final state;anyway, in our case several annihilation channels are open, and therefore we expect a smallerenergy injection in the interstellar medium with respect to the case of annihilation only intoe+e−. In figure 5 we reproduce the constraints as obtained in refs. [93–95] considering nowall the channels, each with its branching ratio defined in section 6.

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7.2 Present epoch γ-rays

For most of the DM annihilation modes, another relevant constraint is in fact imposed by theindirect DM searches in the present epoch. The DM constraints provided by the FERMI-LATγ-ray data are particularly relevant as they are now cutting into the thermal annihilationcross section for low DM masses (. 30 GeV) and a variety of channels.

In particular, dwarf satellite galaxies of the Milky Way are among the most promisingtargets for Dark Matter searches in γ-rays because of their large dynamical mass to lightratio and small expected background from astrophysical sources. No dwarf galaxy hasbeen detected in γ-rays so far and stringent upper limits are placed on DM annihilation byapplying a joint likelihood analysis to 10 satellite galaxies with 2 years of FERMI-LAT data,taking into account the uncertainty in the Dark Matter distribution in the satellites [96].The limits are particularly strong for hadronic annihilation channels, and somewhat weakerfor leptonic channels as diffusion of leptons out of these systems is poorly constrained. Inour case, having both hadronic as well as leptonic annihilation channels, we expect again aweaker constraint with respect to the pure hadronic annihilation (see e.g. figure 2 in ref. [96]).

Other strong limits on annihilation channels are set by, for example, the γ-ray diffuseemission measurement at intermediate latitudes, which probes DM annihilation in ourMilky Way halo [97–99]. In particular, the most recent limits come from 2 years of theFERMI-LAT data in the 5 6 b 6 15, −80 6 ` 6 80 region [99], where b and ` are thegalactic latitude and longitude. Since bounds on all annihilation channels are not availablewith the latest data, we report only the most stringent one, coming from bb.

The last kind of constraints that we can set on DM annihilation in the case of magneticmoment interaction comes from γ-ray lines. Indeed, given the possibility of annihilation intwo photons, we expect a line in the cosmic photon spectrum at energies equal to the massof the DM. Constraints on the annihilation cross section into photons can be drawn fromthe latest FERMI-LAT data [100]. For instance, the FERMI-LAT collaboration excludes at95% the annihilation 〈σχχ→ff vrel〉 < 0.5 × 10−27 cm3/s for a DM mass mχ = 30 GeV inthe case of an isothermal halo. In our case we use the latest (preliminary) data on the fluxfrom spectral lines (shown in slide 39 of [101]) to constrain the annihilation cross section ofeq. (6.4).8 We consider an all-sky region with the Galactic plane removed (|b| > 10), plusa 20 × 20 square region centered on the Galactic center [100]. Notice that, for small DMmasses and dipole moments, the one-loop contribution becomes comparable with the treelevel one; a rough estimate of the loop cross section in the non-relativistic limit is α3λ2

χ/4π,

and therefore we expect it to become sizeable for λχmχ < α3/2 ' 6 × 10−4. In the part ofthe parameter space probed by γ-ray lines searches the loop correction is negligible.

We superimpose all these constraints in figure 5. We see that, apart from the γ linesbound, they are somewhat stronger than the CMB one considered above. We keep howeverthe latter as it is less model dependent.

As shown in figure 4, a change in the DM local density modifies the direct searchesresults. This does not happen for indirect searches, as they are not very sensitive to ρ0.Therefore the constraints shown in figure 5 also apply, unchanged, to this case. For lowerDM dispersion velocity v0 these constraints start to play an important role in cutting theparameter space for direct detection searches.

8A similar study has been published in ref. [102] for DM masses above 30 GeV.

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7.3 Collider and other astrophysical constraints

The constraints we have addressed above apply to the case of symmetric (thermally produced)DM. These constraints are not relevant in the case where DM is of asymmetric nature. Thereare two extra type of constraints: constraints imposed by collider searches, and constraintsimposed by observations of compact stars such as white dwarfs and neutron stars. Thecollider constraints are applicable wether DM is symmetric or not, whereas the compact starconstraints are valid only for asymmetric DM.

The collider constraints emerge from the fact that for a given λχ and mχ that fit thedirect DM searches experimental data (and even provide the proper DM annihilation crosssection for thermal production), the cross section for production of a pair of DM particlesin colliders is completely fixed. The processes that lead to constraints are mono-photonproduction (from initial or final state) plus missing energy due to the pair of DM particlesin e+e− collisions at LEP, or mono-jet production plus missing energy in proton-antiprotoncollision in Tevatron, or proton-proton collisions at LHC. In the case of magnetic DM theseconstraints have been studied in [48, 49] where it is found that the upper bound on λχ issafely above the range of values of λχ relevant for the direct search experiments (for therange of mχ ∼ 10 GeV).

In the case of asymmetric DM, constraints can be imposed by compact star observationsbased on the fact that a substantial number of captured DM particles might lead to gravita-tional collapse and formation of a black hole that can destroy the host star. The magnetic DMbeing a fermion evades the severe constraints on asymmetric bosonic DM based on neutronstars [103–105]. The constraints on self-DM cross section with Yukawa interactions presentedin [106] are avoided for several reasons: firstly DM-DM interactions scale as m2

χλ4χ (leading to

a typical DM-DM cross section of ∼ 10−43 cm2 which is much smaller than the constraint).Secondly the constraints are not directly applicable because the mediator of the magnetic DMis a massless photon (and not a massive mediator necessary for the constraint) and the DM-DM interaction is repulsive. The latter adds up to the effect of the Fermi pressure of the DMparticles and therefore the amount of particles needed for gravitational collapse cannot beaccumulated within the age of the Universe. A potential attractive photon interaction takesplace in the symmetric case, which will however lead also to annihilation of the DM populationinside the neutron star invalidating thus the constraint derived from black hole formation.

Finally the magnetic DM evades constraints on the spin-dependent part of the crosssection imposed by observations of white dwarfs [107]. Although these constraints aretypically weaker than the ones derived from direct searches at the range mχ ∼ 10 GeV, theycould become stricter if spin-dependent interactions scale as some (positive) power of therecoil energy. Since DM particles acquire high velocities when entering the white dwarf,such constraints could in principle exclude such a candidate. However, as it was pointed outin [41], the spin-dependent cross section does not scale with the recoil energy and thereforethe white dwarf constraints can be safely ignored.

8 Conclusions

We investigated a fermionic Dark Matter particle carrying magnetic dipole moment andanalyzed its impact on direct detection experiments. We provided an analytic understandingof how the photon induced long-range interaction neatly modifies the experimental allowedregions with respect to the typically assumed contact interaction. We showed that thiscandidate can accommodate the DAMA, CoGeNT and CRESST experimental results. By

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0 5 10 15 20 25

10-6

10-5

10-4

10-3

10-2

10-1

1

10

Recoil Energy ER @keVD

rHER

L

O

F

Na

Ca

Ge

I

Xe

W

PIC

ASS

O

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MA

-N

a

CR

ESS

T

CoG

eNT

2ndbi

n

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MS

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MA

-I

XE

NO

N4P

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XE

NO

N8P

E

mΧ = 10 GeV

0 5 10 15 20 25

10-6

10-5

10-4

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Recoil Energy ER @keVD

rHER

L

O

FNa

Ca

Ge

I

Xe

W

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ASS

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DA

MA

-N

a

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ESS

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eNT

2ndbi

n

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MS

DA

MA

-I

XE

NO

N4P

E

XE

NO

N8P

E

mΧ = 50 GeV

Figure 6. The SD to SI rates ratio r defined in eq. (A.1) for the two cases of mχ = 10 GeV (left) andmχ = 50 GeV (right). We consider here a Maxwellian halo with local dispersion velocity 220 km/s.Supposing the major part of the signal to come from the lower energy threshold (or from the secondbin in the case of CoGeNT), r can be determined in the figure by the point where the dashed linemeets the vertical line with the same color, for each nuclear element.

assuming conservative bounds we have demonstrated that this candidate is not ruled outby the CDMS, XENON and PICASSO experiments. Assuming a symmetric Dark Mattersector, we also determined the associated thermal relic density and provided relevant boundsfrom indirect search experiments. We found that all the experimental results are compatiblewith a Dark Matter particle with mass around 10 GeV and magnetic moment 1.5 × 10−18

e cm, which corresponds to a new physics scale ΛM ∼ 10 TeV.

Acknowledgments

We thank Robert Foot, Aldo Morselli and Kimmo Kainulainen for useful discussions. We areparticularly grateful to Marco Regis for sharing parts of the code used for the data analysis.

A Relevance of spin-dependent interaction

In this appendix we estimate the effect of the dipole-dipole interaction term in eq. (2.6) forall the targets considered in our analysis. To this aim, one can define the useful ratio

r(ER) ≡ dRSD

dER·(dRSI

dER

)−1

=

∫dσSD/dER v dnχ∫dσSI/dER v dnχ

, (A.1)

which measures the main contribution of the total recoil rate for a given nuclear speciein a detector. If r > 1 the interaction is largely spin-dependent, while it is mostlyspin-independent for r < 1.

For elastic DM, which features δ = 0, the ratio r can be computed from eqs. (3.3)and (3.4), yielding

r(ER) =2

3ΘSI(ER)−1

(1

A

λnuc

λp

µχnmp

)2

, (A.2)

– 23 –

JCAP08(2012)010

where for the illustrative purposes of this appendix we approximated hereFSI(ER) ' FSD(ER). In figure 6 we show the ratio r as a function of the nuclear re-coil energy considering a DM mass of 10 GeV, which gives a good combined fit to theexisting experiments, and 50 GeV, for a comparison. One can see that the dipole-dipoleterm plays a major role for DAMA (Na, I) and PICASSO (F), while being negligible for allthe other experiments considered, as stated in the main text.

While the dipole-charge interaction already provides a good fit, the effect of thedipole-dipole interaction contributes to a better agreement between the experiments. Infact, the increase in the cross section in DAMA favors a lower value for the DM magneticmoment λχ, and therefore the DAMA region shifts down towards CoGeNT and CRESST.The overlap becomes then almost perfect, for a Maxwellian velocity distribution, especiallywith velocity dispersion 220 km/s. On the other hand also the bound by PICASSO lowers,but even this enhanced constraint excludes only a minor part of the overlap zone.

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