R-parity violating U(1)'-extended supersymmetric standard model

0.27 MB·English

by  Hye-Sung Lee

ModernPhysicsLetters A
(cid:13)c WorldScientificPublishingCompany
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0 R-parity violating U(1)′-extended supersymmetric standard model
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2

n
a Hye-SungLee
J Department of Physics and Astronomy, Universityof California

8 Riverside, CA 92512, USA
[email protected]

]
h
p Supersymmetryisone of the best motivated new physics scenarios. Tobuildarealistic
- supersymmetric standard model, however, a companion symmetry is necessary to ad-
p
dressvariousissues.WhileR-parityisapopularcandidate thatcanaddresstheproton
e
h and dark matter issues simultaneously, it is not the only option for such a property.
[ We review how a TeV scale U(1)′ gauge symmetry can replace the R-parity. Discrete
symmetries of the U(1)′ can make the model still viable and attractive with distin-
2 guishablephenomenology. Forinstance,witharesidualdiscretesymmetryoftheU(1)′,
v Z6 = B3×U2, the proton can be protected by the baryon triality (B3) and a hidden
9 sectordarkmattercandidatecanbeprotected bytheU-parity(U2).
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5 Keywords:supersymmetry;R-parityviolation;darkmatter.
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PACSNos.:11.30.Er,12.60.Jv,95.35.+d.
.
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8 1. Introduction
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: Though standard model (SM) has been very successful in describing the nature,
v
i there are some issues in the model, which leads to a consensus that it is not an
X
ultimatetheorytodescribetheparticlephysics.Amongthemisthegaugehierarchy
r
a problem,whichis afine-tuningissueinradiativecorrectionofthe Higgsmass.This
can be most naturally addressed by supersymmetry (SUSY).
The following is a superpotential of the general supersymmetric SM before any
extra symmetry is introduced.
W =µH H +y H LEc+y H QDc+y H QUc
u d E d D d U u
+λLLEc+λ′LQDc+µ′LHu+λ′′UcDcDc (1)
η η
+ 1QQQL+ 2UcUcDcEc+··· .
M M
We can see that lepton number (L) violation and/or baryon number (B) viola-
tion at renormalizable and nonrenormalizable levels (in second and third lines) are
one of the most general predictions of SUSY. This superpotential has some prob-
1
lems. First, there is the so-called µ-problem. The µ parameter (in first line) of
the superpotential should be of electroweak (EW) scale to avoid any fine-tuning
in the electroweak symmetry breaking. Supersymmetry itself does not explain why
1
2 Hye-Sung Lee
d e+ q q
(cid:0)
qe lk
λ′′ de λ′ Mη e Wf (cid:31)01 elj(cid:0) (cid:21) (cid:23)i
l
u u¯ q l (cid:1)
+
lj
(a) (b) (c)
Fig. 1. (a) and (b) are proton decay diagrams through renormalizable and nonrenormalizable
operators,and(c)isaneutralinoLSPdecaydiagram.
the SUSY conserving sector parameter µ should have the same scale as the SUSY
breaking sector, not a higher scale such as Planck scale (M ).
Pl
The L violating, B violating terms also cause problems with proton decay. Fig-
ures 1 (a) and (b) are some examples of proton decay diagrams with dimension
4 operators and dimension 5 operators, respectively. Proton decay requires both
L violation and B violation. To construct a proton decay diagram with dimension
4 operators, we need both L violating term and B violating term (e.g. λ′LQDc
and λ′′UcDcDc). With dimension 5 operators, we can violate both of them with
a single operator (e.g. η QQQL). To satisfy a very long proton lifetime constraint
M
(τ > 1029 years), the coefficients of the L, B violating terms should satisfy the
p ∼
following constraints.
Dimension 4: |λ ·λ |<10−27, (2)
LV BV ∼
Dimension 5: |η|<10−7 (for M =M ). (3)
∼ Pl
For dimension 4 operators, if either L violating or B violating operator is absent,
the other could have a sizable coefficient. Dimension 5 L and B violating operators
shouldhaveunnaturallysmallcoefficients,evenifwetakethecutoffscaleM =M ,
Pl
2
in order to ensure long enough life time for proton.
Figure 1 (c) is a decay diagram of the lightest superparticle (LSP) with a neu-
tralino example. The LSP is the most popular dark matter candidate in supersym-
metric models. To be a viable dark matter candidate, however, the LSP lifetime
should be similar to or larger than the Universe age (t ∼14×109 years). It gives
0
severe constraints of
|λ|,|λ′|,|λ′′|<10−20 (4)
∼
for TeV scale superparticle masses.
Therefore, it is clear that SUSY needs a companion symmetry to address these
3
issues. R-parity is a popular companion symmetry, but there are some aspects of
the R-parity that are not completely satisfactory. In this review, we will consider
an alternative companion symmetry, a TeV scale Abelian gauge symmetry and
see how it can play the roles that were played by the R-parity. It will provide a
R-parity violating U(1)′-extended supersymmetric standard model 3
solid theoretical framework to study the R-parity violating phenomenology with
distinguishable predictions.
The rest of the paper is as follows: In Section 2, we consider a popular SUSY
companionsymmetry,R-parity.InSection3,weconsideraTeVscalegaugesymme-
try,U(1)′.InSection4,wereviewthegeneralresidualdiscretesymmetryofanextra
Abelian gauge symmetry. In Section 5, we discuss the residual discrete symmetry
of the R-parity violating scenario and how it can help with the proton stability. In
Section 6, we discuss the extension of the discrete symmetry to the hidden sector
anda dark matter candidate in theR-parityviolating scenario.InAppendix A, we
review the mechanism for the neutrino to acquire mass in the R-parity violating
U(1)′ model, after the summary and outlook in Section 7.
2. R-parity
Under R-parity, the SM fields have even parity and their superpartners have odd
parity.
R [SM]=even, R [superpartner]=odd. (5)
p p
The R-parity is equivalent to matter parity in its effect. While the R-parity is
defined on the component fields, the matter parity is defined on the superfields.
Under matter parity, only matter superfields (fermions, sfermions) have odd-parity
and the others (Higgses, gauge bosons, and their superpartners) have even parity.
R-parity: R =(−1)3(B−L)+2s, (6)
p
Matter parity: M =(−1)3(B−L). (7)
p
With an assumption of R-party, the LSP is absolutely stable, and it would be a
good dark matter candidate if other conditions for a viable dark matter candidate
are satisfied.
The general superpotential with R-parity is given as follows:
W =µH H +y H LEc+y H QDc+y H QUc
Rp u d E d D d U u
η η (8)
+ 1QQQL+ 2UcUcDcEc+··· .
M M
Since R-parity does not address the µ-problem, it still needs a separate solution or
symmetry. R-parity removes all renormalizable L violating terms and B violating
terms,whichisnotreallynecessaryforprotonstability,forbiddingpotentiallyinter-
4
esting phenomenology. Furthermore,the R-paritydoes not preventthe dimension
5 L and B violating operatorswhich still mediate too fast proton decay.Therefore,
the R-parity by itself is incomplete in addressing the proton stability. So we need
to look for an additional or alternative explanation or symmetry.
3. TeV scale U(1)′ gauge symmetry
We will consider a TeV scale U(1)′ gaugesymmetry in this section. When an extra
Abelian gauge symmetry is introduced in the supersymmetric model, its natural
4 Hye-Sung Lee
scaleis TeVscale.SfermionsgetextraD-termcontributionsfromtheU(1)′,andin
order to make sure the soft terms are of TeV scale without fine-tuning, the U(1)′
′
shouldnotbebrokenatlargerthanTeVscale.ForaU(1) breakingmechanismand
5
other details of this model, we refer to a general review and references therein.
The TeV scale U(1)′ can provide a natural solution to the µ-problem with the
chargeassignmentthat canforbid the originalµ term(µH H ) and allow aneffec-
u d
tive µ term (hSH H ).
u u
z[H ]+z[H ]6=0, z[S]+z[H ]+z[H ]=0. (9)
u d u d
S is a Higgs singlet that breaks the U(1)′ spontaneously.a After S gets a vacuum
expectation value (vev)at TeV scale,the effective µ parameter is dynamically gen-
erated at EW or TeV scale.
µ =hhSi∼O(EW/TeV). (10)
eff
This solution to the µ-problem greatly motivates the TeV scale U(1)′ symmetry.
The mass of the new gauge boson Z′ is given by
MZ2′ =2gZ2′ z[Hu]2hHui2+z[Hd]2hHdi2+z[S]2hSi2 , (11)
(cid:16) (cid:17)
wherez[H ],z[H ],z[S](hH i,hH i,hSi)aretheU(1)′ charges(vevs)oftheHiggs
u d u d
fields H , H , and S, respectively. The lower bound on Z′ mass by the Tevatron
u d
7
dilepton search is MZ′ ∼>700∼1000 GeV depending on couplings. Currently, the
most stringent bound on MZ′ is given indirectly by the primordial nucleosynthesis
8
data, and the LHC will be the first collider experiment that can probe this range
(multi TeV) directly.
Consider the MSSM Yukawa terms,b (effective) µ-term, and [SU(2) ]2−U(1)′
L
anomaly condition.
H QUc : z[H ]+z[Q]+z[Uc]=0,
u u
H QDc : z[H ]+z[Q]+z[Dc]=0,
d d
H LEc : z[H ]+z[L]+z[Ec]=0,
d d
(12)
H LNc : z[H ]+z[L]+z[Nc]=0,
u u
(S/M)pSH H : (1+p)z[S]+z[H ]+z[H ]=0,
u d u d
A221′ : Nf(3z[Q]+z[L])+NH(z[Hu]+z[Hd])+δ =0.
We used S pSH H to consider both the originalµ term case (p=−1), and the
M u d
effective µ(cid:0) te(cid:1)rm case (p=0). N , N , δ mean the number of fermion families, the
f H
numberofHiggsdoubletpairs,andtheexoticSU(2) chargedparticlecontribution
L
tothe[SU(2) ]2−U(1)′anomaly.Inthisreview,weconsideronlytheminimalfields
L
assumptionc of
N =3, N =1, δ =0. (13)
f H
aSeeRef.6forageneraldiscussionontheHiggsbosonspectrumwithanadditionalHiggssinglet.
bWealsoincludetheright-handedneutrino.SeeAppendixAfordetails.
cSeeRef.9formoregeneralcases.
R-parity violating U(1)′-extended supersymmetric standard model 5
In order to have an anomaly free gauge theory, we should make sure the other
anomaly conditions are also satisfied for a given particle spectrum. In our rather
general treatment independent of specific spectrum, we do not consider these full
gauge anomaly conditions in this review. However, we refer to Refs. 10, 11, 12, 13,
14, 15 for some examples of anomaly free U(1)′ charge assignments.
WeconsideronlythefamilyuniversalU(1)′ chargesfortheMSSMsector.There
are 9 unknown U(1)′ charges (Q,Uc,Dc,L,Ec,Nc,H ,H ,S) and 6 conditions,
u d
which results in 3 free parameters in determining U(1)′ charges for the MSSM
sector. The general solution for the MSSM sector is then given by
z[Q] 1 1 (1+p)
z[Uc] −1 −4  8(1+p)
z[Dc] −1 2 −(1+p)
       
z[L]  −3 −3  0
  α  β   γ  
z[Ec]=  3+  6+  0 (14)
       
z[Nc] 3  3 6  0 9  9(1+p)
       
z[Hu]  0  3 −9(1+p)
       
z[Hd]  0 −3  0
       
z[S]   0  0  9
       
where the first vector (with α) is the B − L, the second vector (with β) is the
hypercharge (y). Three free parameters can be written in terms of some U(1)′
charges as
α=z[H ]−z[L], β =−2z[H ], γ =z[S]. (15)
d d
Unlike the B violating term (UcDcDc), there are more than one L violating
termsatrenormalizablelevel.Itisusefultonotetheconditionstohavetheseterms
are identical. Since we already have
y H LEc, y H QDc, hSH H , (16)
E d D d u d
allowing the L violating terms
λLLEc, λ′LQDc, h′SH L (17)
u
requires a common condition
z[H ]=z[L]. (18)
d
In other words,if we wantto controlthe L violating terms with a TeV scale U(1)′,
we can only allow either all of them or none of them.
From Eqs. (12), we can get the following relation:
2
z[UcDcDc]−z[LLEc]+ z[H H ]=0. (19)
u d
3
The first term is a total U(1)′ charge of the B violating term (UcDcDc), and the
secondtermisatotalU(1)′ chargeoftheLviolatingterm(LLEc).Thelasttermis
proportionalto a totalU(1)′ chargeofthe originalµ term(H H ) or−(1+p)z[S],
u d
6 Hye-Sung Lee
which would be zero for p = −1 case, but nonzero for p = 0 case (i.e. when the
′
U(1) solves the µ-problem).
When the U(1)′ solves the µ-problem, according to Eq. (19), if the B violating
term exists (i.e. if the first term is zero), the L violating term cannot exist (i.e.
the second term cannot be zero) since the last term is nonzero. Similarly, if the
L violating term exists, the B violating term cannot exist. This means that the L
violating terms and the B violating terms cannot coexist, which is called LV-BV
14
separation. Therefore,the protoncannotdecaythroughthe renormalizablelevel
MSSM fields operators when the µ-problem is solved by the U(1)′.
Buttherearemorefieldsinvolvedinthismodel.The[SU(3) ]2−U(1)′ anomaly
C
condition is
3(2z[Q]+z[Uc]+z[Dc])+A331′[exotic colors]=0. (20)
The first term is equivalent to −3z[H H ] due to the MSSM Yukawa relations,
u d
which would be required to be nonzero unless p=−1.This means we need colored
exotics that can contribute to this [SU(3) ]2−U(1)′ anomaly condition when the
C
µ-problem is solved by the U(1)′.d We should address whether the proton can still
be stable with the exotic fields and also at dimension 5 level. We find that the
residual discrete symmetry of the U(1)′ is a great tool for this argument, and we
will utilize it in following sections.
′
ItwouldbeinstructivetomakeacommentabouttherelationbetweentheU(1)
and R-parity. In principle, it is possible to have the matter parity (equivalent to
R-parity) as a residual discrete symmetry of the U(1)′ so that we can have both
R-parityandtheµ-problemsolutionsimultaneouslyfromasingleU(1)′ symmetry.9
In the R-parity conserving U(1)′ model, the LSP dark matter candidates can be
new neutralino components17,18,19 or evena sneutrino.20 In this review,however,
we will consider only the R-parity violating case.
4. Residual discrete symmetry of an Abelian gauge symmetry
′
Z naturally emerges after U(1) is spontaneously broken by the vev of S. After
N
U(1)′ chargesofallfields z[F ] arenormalizedto integers,the value N and discrete
i
charges q[F ] are determined by the following relations.
i
N = z[S], (21)
q[F ]= z[F ] mod N. (22)
i i
TheHiggssingletS hasq[S]=0,whichkeepsthediscretesymmetryunbrokenafter
the U(1)′ symmetry is broken by the hSi. As long as N 6= 1, there is generically a
residual discrete symmetry of the U(1)′.
dSeeRef.16foradiscussionaboutthegaugecouplingunificationinthepresenceoftheU(1)′.
R-parity violating U(1)′-extended supersymmetric standard model 7
Table 1. Discrete charges of BN and LN, and their relation with
baryonnumber,leptonnumber,andhypercharge.
Q Uc Dc L Ec Nc Hu Hd meaningofq
BN 0 −1 1 −1 2 0 1 −1 −B+y/3
LN 0 0 0 −1 1 1 0 0 −L
y 1 −4 2 −3 6 0 3 −3
21
ThemostgeneralZ oftheMSSMsectorwasfirststudiedbyIbanezandRoss.
N
The family universal Z can be written as
N
Z : g =Bb Lℓ (23)
N N N N
with two generators
BN =e2πiqNB, LN =e2πiqNL. (24)
There are 8 unknown discrete charges for Q,Uc,Dc,L,Ec,Nc,H ,H . We
u d
have 5 conditions from superpotential (H QUc, H QDc, H LEc, H LNc, H H ),
u d d u u d
and another condition from the hypercharge shift invariance (q[F ] → q[F ] +
i i
αy[F ] mod N). Therefore, we have only 2 free parameters (b,ℓ) to determine the
i
MSSM sector discrete charges.
The family universal discrete charges of B and L as well as normalized hy-
N N
perchargesaregiveninTable 1.As the table shows,the discretechargesareclosely
related to the B, L, and y, and the total discrete charge of Z is given by
N
q =bq +ℓq mod N =−(bB+ℓL)+b(y/3) mod N (25)
B L
with a conserved quantity bB+ℓL mod N.
We consideronlythe family universaldiscretecharges.Family nonuniversaldis-
crete charges are not likely at least in the quark sector since the mixing of quarks
would not be allowed in contradiction to the expectation from the CKM matrix.
Family nonuniversal U(1)′ is still possible with the family universal Z as long as
N
the condition z[F ]=q[F ]+n N is kept (z[F ] is family-dependent if n is).e
i i i i i
5. Discrete symmetry of the U(1)′ and the proton stability
AswediscussedinSection3,thereareingeneralexoticfieldsintheU(1)′-extended
supersymmetric standard model, which might change the discrete symmetry. How-
ever,theMSSMdiscretesymmetry(ZMSSM =Bb Lℓ )stillholdsamongtheMSSM
N N N
fields. It is important to note that for a physics process which has only the MSSM
fields in its effective operators (such as proton decay), we can still discuss it with
the ZMSSM.
N
eOne should be careful since the family nonuniversal U(1)′ charges can cause a flavor changing
neutral current by the Z′ in the physical eigenstate.22 This flavor changing Z′, however, may
explain the discrepancies inthe rare B decays.23,24 See alsoRefs. 25, 26, 27for its contribution
totheB-B¯ mixing.
8 Hye-Sung Lee
Now,we wantto consider the L violationcase.We haveanotherconditionfrom
the term LLEc. After the normalization to integers, we have the general U(1)′
charges for the L violating case as
z[Q] 1 0 0
z[Uc] −4  (1+p)  1
z[Dc] 2 0 −1
       
z[L]  −3  0  1
       
z[Ec]=I  6+3 0+(1+p)−2 (26)
  Q     
z[Nc]  0  (1+p)  0
       
z[Hu]  3 −(1+p) −1
       
z[Hd] −3  0  1
       
z[S]   0  1  0
       
withanintegerI =z[Q].TheresidualdiscretesymmetryshouldbeaZ symmetry
Q 3
since N =z[S]=3 (see Eq. (21)). Since the first column is just a hypercharge and
the second column is an integer multiple of N = 3, it is the last column that
determines which Z symmetry it is. Its coefficient should be (1+p) = 3·Z±1
3
to have a discrete symmetry. In the p = −1 case (where we have the original µ
term, H H ), it is not possible to satisfy this unless we change the minimal fields
u d
assumption of Eq. (13). In the p = 0 case (where we have an effective µ term,
SH H ),theconditionisautomaticallysatisfied.Comparingthethirdcolumnwith
u d
general Z = BbLℓ, we can see it is the B symmetry, which is called baryon
3 3 3 3
28
triality.
From the discrete charge of B , q =−B+y/3 mod 3, we have a selection rule
3
B : ∆B =3×integer (27)
3
29
which prevents the proton decay (∆B = 1) completely. In other words, any ob-
servation of the proton decay would invalidate the B scenario.
3
For the B violatingcase,wehavea conditionfromthe UcDcDc term.Following
a similar argument, we will have the L symmetry, which is called lepton triality.
3
The L has a selection rule of
3
L : ∆L=3×integer (28)
3
whichdoes notpreventthe protondecayif the decay productshas3,6,··· leptons.
However,itwasshownthat,withaparticlespectrumspecified,itcanensureproton
stabilityuptodimension5levelwithalittlehelpfromtheU(1)′ gaugesymmetry.28
Any observation of the violation of the selection rule, such as neutrinoless double
beta decay (∆L=2), would rule out the L scenario.
3
6. Discrete symmetry extended to hidden sector and the dark
matter candidate
The LSP is not a good dark matter candidate in general without the R-parity.
Though it is always possible to include an additional symmetry for a new dark
R-parity violating U(1)′-extended supersymmetric standard model 9
matter candidate, e.g. U(1) for axion dark mater, we will see if we can come up
PQ
with a new dark matter candidate without introducing an independent symmetry.
When we try to satisfy the anomaly conditions with a new gauge symmetry, it
is often necessary to include additional SM singlet fields. They contribute to the
[gravity]2−U(1)′ and [U(1)′]3 anomalies.
For simplicity, we consider only Majorana type SM singlets (X), which get a
mass of U(1)′ breaking scale with
ξ
W = SXX. (29)
hidden
2
This hidden sectorfieldcanbe a darkmatter candidate ifit isstable.The question
is how to ensure the stability of this hidden sector field. We consider a Z parity,
2
which we name U-parity, under which the MSSM fields have even parity, while the
30
hidden sector fields have odd parity.
U [MSSM]=even, U [hidden]=odd. (30)
p p
Then the lightest U-parity particle (LUP) would be either a fermionic or a scalar
component of the X field, which is stable due to the U-parity. Now, the important
part is that we do not want to introduce this hidden sector parity as an ad hoc
addition, but we rather want it as a residual discrete symmetry of the U(1)′.
We introduce a new generator U for the hidden sector discrete symmetry.
N
Zhid : U . (31)
N N
This can only be U for the Majorana type case,f under which q[MSSM] = 0 and
2
q[X]=−1.
We take
Ztot =Zobs×Zhid (32)
N N1 N2
′
as a generalizedresidual discrete symmetry of the U(1) gauge symmetry extended
31
to the hidden sector. Z is isomorphic to Z ×Z , if N and N are coprime
N N1 N2 1 2
and N =N N .
1 2
Ztot : gtot = Bb Lℓ ×Uu (33)
N N N1 N1 N2
= BbN2LℓN2UuN1. (34)
N N N
We consider the L violating case, as an example, where the MSSM sector has
theB symmetryaswediscussedintheprevioussection.WiththeU inthehidden
3 2
sector, the total residual discrete symmetry of the U(1)′ is Ztot =B ×U .
6 3 2
AsillustratedinFigure2,aunifiedpictureaboutthestabilityoftheprotonand
thehiddensectordarkmatterarises.AU(1)′ gaugesymmetry,whichinteractswith
boththe MSSMandhiddensectors,providesdiscretesymmetriestoeachsector.In
contrastto the usual R-parity scenario where a single discrete symmetry addresses
fFortheDiractypecaseandmoregeneral discussions,seeRef.9.
10 Hye-Sung Lee
U(1)′ → Ztot =Zobs×Zhid
N N1 N2
MSSM sector Hidden sector
Zobs : B Zhid : U
N1 3 N2 2
stable proton stable dark matter
Fig.2. AsingleU(1)′gaugesymmetryprovidesthestabilityfortheprotonandthehiddensector
darkmattercandidate.
thestabilityofthe protonandthe LSPdarkmattercandidateintheMSSMsector,
inourcaseasinglegaugesymmetry,whichwealreadyhavetosolvethe µ-problem,
addressesthe stability ofthe protonin the MSSM sectorand the LUP darkmatter
candidate in the hidden sector.
The total discrete charge of the Ztot is given by q =2q +3q mod 6.
6 B U
q[Q]=0, q[Uc]=−2, q[Dc]=2,
q[L]=−2, q[Ec]=−2, q[Nc]=0, (35)
q[H ]=2, q[H ]=−2, q[X]=−3.
u d
OtherpossibleexoticfieldsareassumedtobeheavierthantheprotonandtheLUP
so that they are not stable due to the discrete symmetry.
It is useful to know that this B ×U in the U(1)′ gauge symmetry naturally
3 2
ariseswithoutdemandingit.Alloneneedstodoinordertohavethisintheminimal
particle spectrum of Eq. (13) is to require 3 terms: (i) SH H (i.e. solve the µ-
u d
problemwiththeU(1)′),(ii)LLEc(i.e.demandarenormalizableLviolatingterm),
(iii) SXX (i.e. demandaneffective mass termforthe Majoranatype hiddensector
field). Then B ×U is automatically invoked as a residual discrete symmetry of
3 2
the U(1)′ that ensures the stability of the proton and the LUP dark matter in this
R-parity violating U(1)′ model (see Ref. 9 for details).
To be a viable dark matter candidate, however, the LUP should satisfy the
constraints in the relic density and the direct detection. The annihilation channels
include, for the fermionic LUP (ψ ) case,
X
ψ ψ →ff¯, SS, Z′Z′, SZ′, (36)
X X
∗ ′ ′ ′
ψ ψ →ff , SS, Z Z , SZ . (37)
X X
The annihilation channel to the supeeerparetiecleepaeir, ee.ge. ψ ψ → ff∗ (S and Z′
X X
mediated s-channel), is not possible in the usual LSP dark matter scenario unless
ee
the LSP and the next-to-LSP has tiny mass splitting.
Figure 3 shows typical predictions of the relic density and the direct detection
cross section of the LUP dark matter candidate. For parameter values, see Ref. 30.
They show the LUP dark matter can satisfy the experimental constraints from the
32 33 34
WMAP+SDSS and the CDMS /XENON simultaneously. There can also be