The modulation effect for supersymmetric dark matter detection with asymmetric velocity dispersion

35 Pages·0.38 MB·English

by  J. D. Vergados

THE MODULATION EFFECT FOR SUPERSYMMETRIC
DARK MATTER DETECTION WITH ASYMMETRIC
VELOCITY DISPERSION.
0
0
J. D.VERGADOS
0
2 Theoretical Physics Section, University of Ioannina, GR-45110, Greece
E-mail:[email protected]
n
a
J Thedetectionofthetheoreticallyexpecteddarkmatteriscentraltoparticlephysics
cosmology. Current fashionable supersymmetric models provide a natural dark
1
mattercandidate whichisthelightestsupersymmetricparticle(LSP).Suchmod-
1 els conmbined with fairly well understood physics like the quark substructure of
the nucleon and the nuclear form factor and/or the spinresponse function of the
1 nucleus,permittheevaluationoftheeventrateforLSP-nucleuselasticscattering.
v Thethusobtainedeventratesare,however,veryloworevenundetectable. Soitis
0 imperativetoexploitthemodulationeffect, i.e. thedependence oftheeventirate
9 ontheearth’sannualmotion. Inthispaperwestudysuchamodulationeffectboth
1 innondirectional anddirectional experiments. Wecalculate boththe differential
1 and the total rates using symmetric as well as asymmetric velocity distributions.
0 We find that in the symmetric case the modulation amplitude is small, less than
0 0.07. Thereexist,however,regionsofthephasespaceandexperimentalconditions
0 suchthattheeffectcanbecomelarger. Theinclusionofasymmetry,witharealistic
/ enhancedvelocitydispersioninthegalactocentricdirection,yieldsthebonusofan
h enhanced modulation effect, with an amplitude which for certain parameters can
p becomeaslargeas0.46
-
o
r
t 1 Introduction
s
a
Itisknownthatthatdarkmatterisneededtoclosethe Universe1,2. Itisalso
:
v known that one needs two kinds of dark matter. One composed of particles
i
X which were relativistic at the time of structure formation. These constitute
the hot dark matter component (HDM). The other is made up of particles
r
a which were non-relativistic at the time of freeze out. This is the cold dark
matter component (CDM). The COBE data 3 suggest that CDM is at least
60% 4. OntheotherhandrecentdatafromtheSupernovaCosmologyProject
suggest 5 ,6 thatthereisnoneedforHDMandthesituationcanbeadequately
describedbyΩ<1,e.g. Ω =0.3andΩ =0.6. Inamorerecentanalysis
CDM Λ
Turner7 gives Ω =0.4.
m
Sincethenonexoticcomponentcannotexceed40%oftheCDM 2,8,there
is room for the exotic WIMP’s (Interacting Massive Particles). Recently the
DAMAexperiment 9 hasclaimedtheobservationofonesignalindirectdetec-
tionofaWIMP,whichwithbetterstatisticshassubsequentlybeeninterpreted
as a modulation signal 10.
1
Inthe currentlyfavoredsupersymmetricextensionsofthe standardmodel
themostnaturalWIMPcandidateistheLSP,i.e. thelightestsupersymmetric
particle,whosenaturecanbedescribedinmostsupersymmetric(SUSY)mod-
els to be a Majorana fermion, a linear combinationof the neutral components
of the gauginos and Higgsinos 11 26.
−
Since this particle is expected to be very massive, m 30GeV, and
χ
≥
extremely non relativistic with averagekinetic energy T 100KeV, it can be
≤
directly detected 11 12 only via the recoiling of a nucleus (A,Z) in the elastic
−
scattering process:
χ + (A,Z) χ + (A,Z) (1)
∗
→
(χ denotes the LSP). In order to compute the event rate one proceeds with
the following steps:
1)Write downthe effective Lagrangianatthe elementary particle (quark)
level obtained in the framework of supersymmetry as described in Refs. 2,
Bottino et al.23 and26.
2)Gofromthequarktothenucleonlevelusinganappropriatequarkmodel
for the nucleon. Special attention in this step is paid to the scalar couplings,
which dominate the coherent part of the cross section and the isoscalar axial
current, which, as we will see, strongly depend on the assumed quark model
13,27,28
3)Compute the relevantnuclearmatrix elements14 17 using asreliable as
−
possiblemanybodynuclearwavefunctionshopingthat,byputtingasaccurate
nuclear physics input as possible, one will be able to constrain the SUSY
parameters as much as possible.
4) Calculate the modulation of the cross sections due to the earth’s revo-
lution around the sun by a folding procedure assuming some distribution2,18
of velocities for LSP.
The purpose of our present review is to focus on the last point of our
above list along the lines suggested by our recent letter 22, expanding our
previous results and giving some of the missing calculational details. For the
reader’s convenience, however, we will give a brief description on the basic
ingredients on how to calculate LSP-nucleus scattering cross section, without
elaborating on how one gets the needed parameters from supersymmetry. For
the calculationof these parametersfrom representativeinput in the restricted
SUSYparameterspace,wereferthereadertotheliterature,e.g. Bottinoet al.
23, Kaneet al. , Castanoet al. andArnowitt et al.24. Thenwe will specialize
our study in the case of the nucleus 127I which is one of the most popular
targets19 9. To this end we will include the effect of the nuclear form factors.
−
WewillconsiderbothasymmetricMaxwell-Boltzmanndistribution2aswellas
asymmetricdistributionsliketheonesuggestedbyDrukier18. Wewillexamine
2
theeffectmodulationinthedirectionalaswellasthenondirectionaldetection,
both in the differential as well as the total event rates. We will present our
resultsafunctionoftheLSPmass,m ,forvariousdetectorenergythresholds,
χ
in a way which can be easily understood by the experimentalists.
2 The Basic Ingredients for LSP Nucleus Scattering
Because of lack of space we are not going to elaborate here further on the
construction of the effective Lagrangian derived from supersymmetry, but re-
fer the reader to the literature11,12,21,23,29. The effective Lagrangian can be
obtained in first order via Higgs exchange, s-quark exchange and Z-exchange.
Wewilluseaformalismwhichisfamiliarfromthetheoryofweakinteractions,
i.e.
G
L = F (χ¯ γλγ χ )J +(χ¯ χ )J (2)
eff 1 5 1 λ 1 1
−√2{ }
where
J =N¯γ (f0 +f1τ +f0γ +f1γ τ )N (3)
λ λ V V 3 A 5 A 5 3
and
J =N¯(f0+f1τ )N (4)
s s 3
We have neglected the uninteresting pseudoscalar and tensor currents.
Note that, due to the Majorana nature of the LSP, χ¯ γλχ = 0 (identically).
1 1
The parameters f0,f1,f0,f1,f0,f1 depend on the SUSY model employed.
V V A A S S
In SUSY models derived from minimal SUGRA the allowed parameter space
is characterized at the GUT scale by five parameters, two universal mass pa-
rameters, one for the scalars, m , and one for the fermions, m , as well as
0 1/2
the parameters tanβ, one of A and mpole and the sign of µ24. Deviations
0 t
fromuniversalityattheGUTscalehavealsobeenconsideredandfounduseful
25. We will not elaborate further on this point since the above parameters in-
volvinguniversalmasseshavealreadybeencomputedinsomemodels11,29 and
effects resulting from deviations from universality will be published elsewhere
31 (see also Arnowitt et al in Ref. 25 and Bottino et al in Ref. 23). For some
choicesinthe allowedparameterspace the obtainedcouplingscanbe found in
a previous paper29.
The invariantamplitude in the case ofnon-relativistic LSP canbe cast 11
in the form
E E m2 +p p
2 = f i− x i· f J 2+ J2+ J 2
|M| m2 | 0| | | | |
x
β2 J 2+ J2+ J 2 (5)
0
≃ | | | | | |
3
where m is the LSP mass, J and J indicate the matrix elements of the
x 0
| | | |
time and space components of the current J of Eq. (3), respectively, and J
λ
represents the matrix element of the scalar current J of Eq. (4). Notice that
J 2 is multiplied by β2 (the suppression due to the Majorana nature of LSP
0
| |
mentioned above). It is straightforwardto show that
2
A 2Z
J 2 =A2 F(q2)2 f0 f1 − (6)
| 0| | | (cid:18) V − V A (cid:19)
2
A 2Z
J2 =A2 F(q2)2 f0 f1 − (7)
| | (cid:18) S − S A (cid:19)
1
J2 = J [f0Ω (q) + f1Ω (q)] J 2 (8)
| | 2J +1|h i|| A 0 A 1 || ii|
i
with F(q2) the nuclear form factor and
A A
Ω0(q)= σ(j)e−iq·xj, Ω1(q)= σ(j)τ3(j)e−iq·xj (9)
Xj=1 Xj=1
where σ(j), τ (j), x are the spin, third component of isospin (τ p = p )
3 j 3
| i | i
and coordinate of the j-th nucleon and q is the momentum transferred to the
nucleus.
The differential cross section in the laboratory frame takes the form11
dσ σ µ 2η+1
= 0( r )2ξ β2 J 2[1 ξ2]+ J2+ J 2 (10)
dΩ π m { | 0| − (1+η)2 | | | | }
N
where m is the proton mass, η = m /m A, ξ = pˆ qˆ 0 (forward
N x N i
· ≥
scattering) and
1
σ0 = (GFmN)2 0.77 10−38cm2 (11)
2π ≃ ×
The reduced mass µ is given by
r
m
χ
µ = (12)
r
1+η
For the evaluation of the differential rate, which is the main subject of the
present work, it will be more convenient to use the variables (υ,u) instead of
the variables (υ,ξ). Thus integrating the differential cross section, Eq. (10),
with respect to the azimuthal angle we obtain
du υ2
dσ(u,υ)= [(Σ¯ +Σ¯ F2(u))+Σ¯ F ] (13)
2(µ bυ)2 S V c2 spin 11
r
4
with
µ A 2Z
Σ¯ =σ ( r )2 A2[(f0 f1 − )2] (14)
S 0 m { S − S A
N
µ F (u) F (u)
Σ¯ =σ ( r )2[f0Ω (0))2 00 +2f0f1Ω (0)Ω (0) 01 +(f1Ω (0))2]
spin 0 m A 0 F (u) A A 0 1 F (u) A 1
N 11 11
(15)
µ A 2Z 1 2η+1 2u
Σ¯ =σ ( r )2A2(f0 f1 − )2[1 h i] (16)
V 0 m V − V A − (2µ b)2(1+η)2 υ2
N r
h i
We shouldremarkthateventhoughthe quantityΣ¯ canbe afunctionofu,
spin
in actual practice it is indepenent of u. The same is true of te less important
term Σ¯ In the above expressions F(u) is the nuclear form factor and
V
Ω(ρλ,κ)(u) Ω(ρλ′,κ)(u)
Fρρ′(u)= , ρ,ρ′ =0,1 (17)
Xλ,κ Ωρ(0) Ωρ′(0)
are the spin form factors with
u=q2b2/2 (18)
bbeingthe harmonicoscillatorsizeparameterandqthe momentumtrasferto
the nucleus. The quantity u is also related to the experimentally measurable
energy transfer Q via the relations
1
Q=Q u, Q = (19)
0 0 Am b2
N
The detection rate for a particle with velocity υ and a target with mass m
detecting in the direction e will be denoted by R( e). Then one defines the
→
undirectiona rate R via the equations via the equations
undir
dN ρ(0) m
R = = σ(u,υ)[ υ.eˆ + υ.eˆ + υ.eˆ ] (20)
undir x y z
dt m Am | | | | | |
χ N
ρ(0)=0.3GeV/cm3 is the LSP density in our vicinity. This density has to be
consistent with the LSP velocity distribution (see next section).
The differential undirectional rate can be written as
ρ(0) m
dR = dσ(u,υ)[ υ.eˆ + υ.eˆ + υ.eˆ ] (21)
undir x y z
m Am | | | | | |
χ N
where dσ(u,υ) is given by Eq. ( 13)
5
The directional rate in the direction eˆtakes the form:
ρ(0) m
R =R( e) R( e)= υ.e σ(u,υ) (22)
dir
→ − → − m Am
χ N
and the corresponding differential rate is given by
ρ(0) m
dR = υ.e dσ(u,υ) (23)
dir
m Am
χ N
3 Convolution of the Event Rate
We have seen that the event rate for LSP-nucleus scattering depends on the
relativeLSP-targetvelocity. Inthissectionwewillexaminetheconsequencesof
theearth’srevolutionaroundthesun(theeffectofitsrotationarounditsaxisis
expectedtobenegligible)i.e. themodulationeffect. Thiscanbeaccomplished
by convoluting the rate with the LSP velocity distribution.Hitherto such a
consistent choice can be a Maxwell distribution 2
f(υ )=(√πυ ) 3e (υ′/υ0)2 (24)
′ 0 − −
v = (2/3) v2 =220Km/s (25)
0
h i
p
i.e. v isthevelocityofthesunaroundthecenterofthegalaxy. Inthepresent
0
paper following the work of Drukier, see Ref. 18, we will assume that the
velocity distribution is only axially symmetric, i.e. of the form
f(υ ,λ)=N(y ,λ)(√πυ ) 3)[f (υ ,λ) f (υ ,υ ,λ)] (26)
′ esc 0 − 1 ′ 2 ′ esc
−
with
(υ )2+(1+λ)((υ )2+(υ )2)
x′ y′ z′
f1(υ′,λ)=exp[(− υ2 ] (27)
0
υ2 +λ((υ )2+(υ )2)
f (υ ,υ ,λ)=exp[ esc y′ z′ ] (28)
2 ′ esc − υ2
0
whereυ is the escapevelocityinthe gravitationalfieldofthe galaxy,υ =
esc esc
625Km/s18. In the above expressions λ is a parameter, which describes the
asymmetry and takes values between 0 and 1 and N is a proper normalization
constant given by
1 = 1 [erf(yesc) e−(λ+1)ye2sc erf(i √λ yesc)
N(λ,yesc) λ+1 − i √λ
e−ye2sc[ 2 yesc e−λ ye2sc erf(i √λ yesc)] (29)
− λ √π − i √λ
6
with y = υesc and erf(x) the error function given by
esc υ0
erf(x)= 2 xdte t2 (30)
−
√π Z
0
For y we get the simple expression N 1 =λ+1
esc −
→∞
The z-axis is chosen in the direction of the disc’s rotation, i.e. in the
directionofthe motionofthe the sun, the y-axisis perpendicular to the plane
of the galaxy and the x-axis is in the radial direction. In the above frame we
find that the position of the axis of the ecliptic is determined by the angle
γ 29.80(galactic latitude) andthe azimuthalangleω =186.30 measuredon
≈
the galactic plane from the ˆz axis12. Thus, the axis of the ecliptic lies very
close to the y,z plane and the velocity of the earth around the sun is
υ = υ + υ = υ +υ (sinαxˆ cosαcosγyˆ+cosαsinγˆz) (31)
E 0 1 0 1
−
where α is the phase of the earth’s orbital motion, α = 2π(t t )/T , where
1 E
−
t is around second of June and T =1year.
1 E
Wearenowinapositiontoexpresstheabovedistributioninthelaboratory
frame, i.e.
f(υ, λ, υ )=f (υ,υ ,λ) f (υ,υ ,λ) (32)
E 3 E 4 esc
−
with
(υ +υ sinα)2
x 1
f (υ,υ ,λ) =exp[ ]
3 E − υ2
0
(1+λ)((υ +υ cosγsinα)2+(υ +υ +υ sinγcosα)2)
y 1 z 0 1
exp[ ]
× − υ2
0
(33)
υ2 +λ((υ +υ cosγsinα)2+(υ +υ +υ sinγcosα)2)
f (υ,υ ,λ)=exp[ esc y 1 z 0 1 ]
4 esc − υ2
0
(34)
4 Expressions for the Differential Event Rate in the Presence of
Velocity Dispersion
We will begin with the undirectional rate.
7
4.1 Expressions for the Undirectional Differential Event Rate
The mean value of the undirectional event rate of Eq. (22), is given by
dR ρ(0) m dσ(u,υ)
undir = f(υ,υ )[ υ.eˆ + υ.eˆ + υ.eˆ ] d3υ
E x y z
D du E mχ AmN Z | | | | | | du
(35)
Fromnow onwe will omit the subscript undir in the case ofthe undirectional
rate. The above expression can be more conveniently written as
dR ρ(0) m dΣ
= υ2 (36)
DduE mχ AmNph ihdui
where
dΣ [ υ.eˆ + υ.eˆ + υ.eˆ ] dσ(u,υ)
= k xk k yk k zk f(υ,υ ) d3υ (37)
E
hdui Z υ2 du
h i
p
It is convenient to work in spherical coordinates. But even then the angular
sun is small. Thus introducing the parameter
2υ
1
δ = = 0.27, (38)
υ
0
expanding in powers of δ and keeping terms up to linear in it we can manage
to perform the φ integration using standard contour integral techniques and
expresstheresultintermsofthetwomodifiedBesselfunctionsI (λυ2(1 t2))
m 2υ02 −
with t=cosθ and m=0,1. Thus the angular integration of Eq. 32 yields
M˜ (λ,y) =2π
i
exp[ (y2+1)(1+λ)]Λ˜ (λ,y) exp[ (y2 +λy2)]Λ˜′(λ,y),i=1,2
× − i − − esc i
(39)
where Λ˜ ,Λ˜′ come from f ,f respectively. We find
i i 3 4
Λ˜ (λ,y)=Λ˜ (λ,y)+Λ˜ (λ,y)+Λ˜ (λ,y) (40)
1 1,1 1,2 1,3
with
1
Λ˜ (λ,y)= dtexp[(ζ2/2 2(λ+1)yt)] t I (ζ2/2) (41)
1,3 0
Z − | |
1
−
1 1 erf(i ζ)
Λ˜ (λ,y)= dtexp[( 2(λ+1)yt)](1 t2)1/2 (42)
1,2
√π Z − − i ζ
1
−
8
1 1 exp(ζ2) erf(ζ)
Λ˜ (λ,y)= dtexp[( 2(λ+1)yt)](1 t2)1/2 (43)
1,1
√π Z − − ζ
1
−
where ζ = y (λ (1 t2))1/2 and the function erf(iζ) given by Eq. 30.
−
Furthermore
Λ˜ (λ,y)= (λ+1)Λ˜′′(λ,y) (44)
2 − 1
whereΛ˜′′(λ,y)isobtainedfromΛ˜ (λ,y)byaddinginthe integrandstheextra
1 1
factorty+1. Weshouldmentionthatinthelastintegralwehaveommittedthe
numericalfactor δcosαsinγ. Note that inthe caseΛ˜ (λ,y), whichis asociated
2
with the modulation amplitude, only the z-component of the velocity in the
exponential contributes. Hence the dependence on the earth’s phase is cosα.
The formulas for the second term in Eq. 39 for Λ˜′(λ,y) are obtained by a
i
mere replacement of the expression λ+1 by λ. In all the above expressions
y =(υ/υ ) (not to be confused with the y-coordinate).
0
Itisconvenienttoseparateouttheasymmetriccontributionfromtheusual
one by writing
2y2Λ˜ (λ,y)=F˜ (λ,(λ+1)2y)+G˜ (λ,y) (45)
1 0 0
2y2Λ˜ (λ,y)=F˜ (λ,(λ+1)2y)+G˜ (λ,y) (46)
2 1 1
The functions F˜ havebeen obtainedby consideringthe leadingnon vanishing
i
terminthezetaexpansionoftheintegrandsoftheexpressions(41)-(43). Thus
G˜ (0,y)=0 , G˜ (0,y)=0 (47)
0 1
F˜ (λ,x)=(λ+1) 2[x sinh(x) cos(x)+1+x I (x)] (48)
0 − 1
−
F˜1(λ,x) =(1+λ)−2 [(2+λ)((x2/(2(2+λ))+1) cosh(x) x sinh(x) 1)
− −
+x2 I [x] (λ+1) x I (x)] (49)
2 1
−
note that here x = (λ+1)2y. I (x) is the modified bessel function of order
m
m. The funcions G˜ cannot bo obtained analytically, but they can easily be
expressed as a rapidly convergent series in y = υ , which will not be given
υ0
here.
Similarly
G˜′(λ,y)=2y2Λ˜′(λ,y) , i=1,2 (50)
i i
Thus the folded non-directional event rate takes the form
dΣ υ2
=Σ¯ F¯ (u)+ h iΣ¯ F¯ (u)+Σ¯ F¯ (u) (51)
hdui S 0 c2 V 1 spin spin
9
where the Σ¯ ,i=S,V,spin are given by Eqs. (14)- (16).
i
ThequantitiesF¯ ,F¯ ,F¯ areobtainedfromthecorrespondingformfac-
0 1 spin
tors via the equations
(1+k)a2
F¯ (u)=F2(u)Ψ (u) , k =0,1 (52)
k k
2k+1
F¯ (u)=F (u)Ψ (u)a2 (53)
spin 11 0
(54)
Ψ˜ (u)=[ψ˜ (a√u)+0.135cosαψ˜ (a√u)] (55)
k (0),k (1),k
with
1
a= (56)
√2µ bυ
r 0
and
ψ˜ (x)=N(y ,λ)e λ(e 1Φ˜ (x) exp[ y2 ]Φ˜′ (x)) (57)
(l),k esc − − (l),k − − esc (l),k
2 yesc
Φ˜ (x)= dyy2k 1exp( (1+λ)y2))(F˜(λ,(λ+1)2y)+G˜ (λ,y)))
(l),k √6π Z − − l l
x
(58)
Φ˜′ (x)= 2 yescdyy2k 1exp( λy2))G˜′(λ,y)) (59)
(l),k √6π Z − − l
x
The undirectional differential rate takes the form
dR
=R¯tT(u)[(1+cosαH(u))] (60)
hdui
In the above expressions R¯ is the rate obtained in the conventional approach
11 byneglectingthe foldingwiththeLSPvelocityandthemomentumtransfer
dependence of the differential cross section, i.e. by
ρ(0) m υ2
R¯ = v2 [Σ¯ +Σ¯ + h iΣ¯ ] (61)
m Am h i S spin c2 V
χ Np
where Σ¯ ,i=S,V,spin have been defined above, see Eqs (14) - (16).
i
The factor T(u) takes care of the u-dependence of the unmodulated dif-
ferential rate. It is defined so that
umax
duT(u)=1. (62)
Z
umin
10