Constraints on UED KK-neutrino dark matter from magnetic dipole moments

0.15 MB·English

by  Thomas Flacke

OUTP-0605P
Constraints on UED KK-neutrino dark matter from magnetic dipole moments
Thomas Flacke∗ and David W. Maybury†
Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
1 Keble Road, Oxford OX1 3NP, United Kingdom
(Dated: January 24, 2006)
Generically, universal extra dimension (UED) extensions of the standard model predict the sta-
bility of the lightest Kaluza-Klein (KK) particle and hence provide a dark matter candidate. For
UEDscenarioswithoneextradimension,wemodel-independentlydeterminethesizeoftheinduced
dimension-five magnetic dipole moment of the KK-neutrino, ν(1). We show that current obser-
vational bounds on the interactions of dipole dark matter place constraints on UED models with
KK-neutrinodark matter.
PACSnumbers: 12.60.-i,14.60.St,95.30.Cq,95.35.+d
6 Keywords: extradimensions,darkmatter,dipolemoments
0
0
2 I. INTRODUCTION As an example, a UED model with one extra-dimension
compactified on an S /Z orbifold of radius R, leads to
n 1 2
the tree level mass relation
a While the standard model proves an excellent frame-
J
work for fundamental interactions at energy scales up m(n) = (n/R)2+(m(0))2 (1)
4
to at least the sub TeV range, it nevertheless leaves a q
2
number of fundamental problems. Theoretically, one of for the n-th KK mode, where m(0) constitutes the zero-
2 the most outstanding puzzles centers on the origin and mode mass (i.e. the standard model particle value).
v mechanism of electroweak symmetry breaking, and the Quantum corrections typically dominate over zero mode
1 quantum mechanicalstability of the hierarchygenerated levelcontributionsandthereforetheresultingmassspec-
6 between the electroweak scale and the Planck scale. In trum depends crucially on radiative effects. In general,
1 addition, recent astrophysical observations [1] concord a moderately split UED mass spectrum [3] develops. In
1 with 0.094 < ΩCDMh2 < 0.129, indicating the pres- the minimal UED model (MUED) [3], one-loop calcula-
0
6 ence of cold non-baryonic dark matter as the principle tions suggest that the LKP is well approximated by the
0 form of matter in the Universe, of which the standard KKhyperchargeboson,B(1),andnumerousstudieshave
/ model provides no explanation. The most popular can- examinedthethermalproductionandprospectsofdirect
h
didate for dark matter assumes a non-standard model, andindirectdetectionofB(1) andγ(1) LKPdarkmatter
p
- stable, electrically neutral, and weakly interacting parti- [4, 5].
p cle – the WIMP hypothesis. Clearly, from both a the- However,thenon-renormalizabilityofUEDmodelsim-
e oretical and phenomenologicalperspective, the standard plytheexistenceofanultravioletcut-off,typicallyofthe
h
model requires extension. In this letter we wish to ex- order of a few tens of TeV, at which point the model
:
v plore the consequences of the universal extra dimension requires UV completion. As such, UED models must be
Xi (UED) [2] extension of the standard model. regardedasaneffectivetheory. Thepresenceofincalcula-
In the UED scenario,all standard model particles can bleboundarytermsarisingfromtheUVcompletetheory
r
a freely propagatein the bulk of one ormore extra dimen- can potentially change the mass spectrum, resulting in
sionsandthuseachstandardmodelparticleisassociated differentLKPcandidates. Recentstudiesoftherelicden-
with a Kaluza-Klein(KK) tower of states. Each state in sity of B(1) LKP dark matter with the full MUED spec-
the KK tower has the same spin as its standard model trum [6, 7] reveal substantial observational tension with
counterpart. An important consequence of UED models constraints from electroweak precision data [8]. Thus,
concernsthe existence ofa conserveddiscrete symmetry, non-minimal models with brane-localized terms appear
KK-parity, which guarantees the stability of the lightest as a likely alternative if UED models are to provide a
KKparticle(LKP)andthusprovidesadarkmattercan- successfulphenomenology. Furthermore,modelindepen-
didate. Suitable thermal relic dark matter candidates dent studies [4] show that B(1), γ(1), and ν(1) can all be
that have been studied extensively [4] include the first thermallyproducedwithabundancessufficienttoprovide
KK-excitations of the hypercharge boson, the photon, the dark matter. Constraints on minimal UED models
and the neutrino i.e. B(1), γ(1), or ν(1). from limits on weak neutral current nucleon-ν(1) elastic
The tree-level mass spectrum of the KK-excitations scattering in direct searches,together with thermal dark
of UED models reveals a nearly degenerate spectrum. matterproductionmechanisms,disfavourν(1) darkmat-
ter [9]. However, given the need for possible new non-
minimal interactions, we consider further consequences
of KK-neutrino dark matter model-independently.
∗Electronicaddress: t.fl[email protected] While there exists compelling evidence for dark mat-
†Electronicaddress: [email protected] ter in the form of WIMPS, there also exists strong con-
2
straints on possible electromagnetic interactions of dark (1 γ )/2,and ˆisassociatedwiththe(4Dleft-handed)
5
matter,eveninthelimitofcompletechargeneutrality. A lep±tondoubletvLiatheidentificationP (xµ) (ν ,e ).
L L L
L ↔
neutralDiracfermioncanpossesbothapermanentmag- We see from eq(3) that ˆ contains a purely left-handed
netic dipole moment, µ, and a permanent electric dipole zero mode while all highLer KK modes contain both chi-
moment, d, arising from the dimension-five operator, ralities. Therefore,thenon-zeromodeKK-levelfermions
i appear as Dirac spinors from the 4-dimensional perspec-
D = f¯σµν(µ+γ5d)fFµν. (2) tive. Specifically, the KK-neutrino charged under the
L 2
weak SU(2) appears with both chiralities.
While the presence of the magnetic dipole moment does
Translational invariance in the fifth direction implies
not violate any discrete symmetries, the electric dipole
(once the extra dimension becomes integrated out) that
moment requires the violation of parity and CP. Severe
a UED model compactified solely on S conserves KK-
1
constraintsexistonthedipole momentsof 1TeVdark
∼ excitation-number in every vertex. The orbifold reduces
matter WIMPS [10]. (We are aware that the authors of
the conserved KK-number to a discrete Z symmetry,
2
[10]arecurrentlyrevisingtheirestimatesand,asaresult,
calledKK-parity. TheconservationofKK-parityinevery
the strength of the constraints in [10] may change sub-
vertex implies the stability of the lightest KK-particle.
stantially[11].) Thus,ifamodelpredictsDiracfermionic
Since the fermions receive mass through Yukawa cou-
darkmatter,itisimportanttodeterminethe strengthof
plings afterelectroweaksymmetry breakingandthrough
the induced dipole moment.
the KK-level expansion itself, the theory, in general, re-
Since the KK-neutrino of UED models is a Dirac
quires a unitary transformation to connect mass and
fermion from the 4-dimensional perspective, UED mod-
gauge eigenstates. For example, in the lepton sector at
elsthatassumeKK-neutrinodarkmatterareconstrained
the jth KK-level we have,
by the strength of the induced dipole operator.
In this letter we derive model independent bounds j γ cosα sinα ′j
on KK-neutrino dark matter by examining the induced (cid:18) Ej (cid:19)=(cid:18)−γ55sinαjj cosαjj (cid:19)(cid:18) E′j (cid:19) (4)
L L
dipole moment and comparing the predictions with the
current observational bound. Section II provides a brief where j and j denote the jth KK mode of the lepton
E L
review on the properties of KK-fermionsin UED models SU(2) singlet and doublet respectively, and where
alongwithadiscussionondipolemomentsrelevanttothe
calculationpresentedinsectionIII.Finally,insectionIV m(0)
tan(2α )= l . (5)
we present our conclusions. j j/R
As the lepton masses are small compared to j/R, we
II. KK-NEUTRINO LKP AND INDUCED ignorethe effects ofthe mixing matrix for the remainder
DIPOLE MOMENTS ofthisletter. Furthermore,weignoretheeffectsoflepton
flavour violation and neutrino mixing.
In UED models, standard model fields become iden- A Dirac fermion can posses a dipole moment, derived
tified with the zero modes of KK towers of states once from the transition amplitude (c.f. e.g. [12]),
the extra dimensions are integrated out of the theory.
T = iǫµqνf¯(p′)σ (F +G γ )f(p) (6)
For concreteness, we will restrict our discussion to five − µν 2 2 5
dimensionalUED models compactifiedonan S1/Z2 orb- where q = p′ p. The magnetic moment is defined
ifold. Five dimensional theories do not posses a chirality by µ = F (0)−while the electric dipole is defined by
2
condition since γ5 becomes part of the five dimensional d = G2(0). At low energies compared to the mass of
Clifford Algebra. Thus, in order to arrive at a chiral the particle, the photon does not distinguish between µ
theoryatthe zeromodelevel,werequireonefivedimen- ordprovidedthatoneignoresothertime-reversalviolat-
sionalDirac spinorforeveryWeyl spinorofthestandard ing observables. We will focus on the limits established
model. By use of the orbifold boundary conditions half on µ throughout. On dimensional grounds, we naively
the number ofstates projectout ofthe spectrum leaving expectthe induced magnetic dipole momentto scale as,
a zero mode level chiral theory. In the case of the lep-
ton doublet, the decompositionofthe correspondingfive µ.eMν(1) e =1.022 10−6µ TeV (7)
dimensional Dirac spinor reads (c.f. e.g. [2]), R−2 ≃ M × B(cid:18)M (cid:19)
ν(1) ν(1)
1 2 ny where M denotes the mass of KK-neutrino, ν(1), and
ˆ(xµ,y)= P (xµ)+ P (xµ)cos( ) ν(1)
L √πR LL rπRnX=1h LL R R indicates the radius of compactification.
ny
+P (xµ)sin( )
R
L R i III. CALCULATION
(3)
where ˆdenotesthe fivedimensionalDiracspinor, de- The KK-neutrino develops a magnetic dipole moment
L L
notesafourdimensional4component spinorwithP = through the diagrams tabulated in figure 1. In general,
L,R
3
x 10−8
a) (cid:13)(0) b) (cid:13)(0) 10
9
W(1;0) W(1;0)
8
(cid:23)(1) (cid:23)(1) (cid:23)(1) (cid:23)(1)
l(0;1) 7
6
c) (cid:1)(cid:13)(0) d) (cid:1)(cid:13)(0) µµ/)B 5
(
4
3
(cid:23)(1) (cid:23)(1) (cid:23)(1) (cid:23)(1)
2
(cid:1) (cid:1) 1
e) f) 500 1000 M(ν1) (GeV) 1500 2000
(cid:23)(1) (cid:23)(1) (cid:23)(1) (cid:23)(1)
(cid:1) (cid:1) FIG. 2: Magnetic moment of the KK-neutrino ν(1) vs. KK
(cid:13)(0) (cid:13)(0) darkmattermassM forvaryingM ,andM . Upper
ν(1) e(1) W(1)
FIG. 1: Magnetic dipole moment inducing one loop Feyn- line (solid): Me(1)-Mν(1) degeneracy with tree-level MW(1)
man graphs for the first KK-excitation of the neutrino con- (c.f. analytical result eq(8)). Middle line (dashed-dotted):
tained in the SU(2)L doublet in UED models. KK-number 5% splitting of Me(1)/Mν(1) with with tree-level MW(1).
assignmentsaredenotedingrapha),tobeunderstoodanalo- Lower line (dashed): Me(1)-Mν(1) degeneracy with 5% split-
gouslyingraphsb)-f). Graphswithdashedlinesdenotethe ting of MW(1)/Mν(1)
Goldstone modes along their KK-excitations, and the KK-
excitation of the Higgs scalar. The Feynman rules are taken
from [13]. In figure 2 we display the result of the dipole moment
asa function ofthe KK-neutrinomass. The upper curve
illustrates the magnetic dipole moment with exact de-
generacy M -M while holding the KK-W mass at
the entire Kaluza-Kleintower of states participate, how- e(1) ν(1)
its tree level value. The middle curve plots the effect of
ever we estimate the leading order effect by considering
a KK-electron 5% heavier than the KK-neutrino. This
only the first level KK excitations. Furthermore, we re-
has an (1) effect on the dipole moment. The predicted
strict our calculation to KK-number conserving graphs O
value of the dipole moment exceeds the current upper
since KK-number violation proceeds with volume sup-
bound for a KK-neutrino mass M 1TeV by more
pression. For simplicity, we ignore flavour violation in ν(1) ∼
than five orders of magnitude [10]. The lower curve dis-
the lepton sector and we assume that the lightest KK-
plays the effect of maintaining M -M degeneracy
(1) ν(1) e(1)
neutrinoisνe . Relaxingtheseassumptionswillnotsig- whilevaryingtheKK-W mass. Allowingthemassdiffer-
nificantly alter our conclusions. ence, M -M , to vary by up to 5% has at most an
W(1) ν(1)
As we make no assumptions on the exact UED spec- (10) effect.
trum, we consider the mass of the KK-W (W(1)) and O
We should note that the calculation presented above
the KK-electron(e(1))asfree parameters. Inournumer-
determines only the radiatively induced part of the KK-
ical calculations we do not consider a KK-electron/KK-
neutrino magnetic dipole moment. The presence of
neutrino mass difference in excess of 5% as any substan-
boundary terms or effects arising from the UV com-
tialsplittingwillleadtounacceptablylargecontributions
plete theory may also contribute a non-renormalizable
to the T parameter.
dimension-fivedipole operatorwhich,apriori,maybe of
TherelevantUEDFeynmanrulesarelistedin[13]and
the same order as the radiative part itself.
wecalculateinthe Feynman-’tHooftgauge. Inthe limit
of exact M -M degeneracy and where the effects
ν(1) e(1)
of Yukawa couplings are ignored, we arrive at the semi- IV. CONCLUSION
analytic result,
eg2 1 UEDmodelshaveattractedattentionasapossibleex-
µ=
(4π)2M × tension to the standard model. A particular appealing
ν(1)
feature of the model class centers on the existence of
3 7 1 5 r
ln(ǫ)+r+ + (r 1)ln (8) plausible dark matter candidates as the result of KK-
(cid:26)2 2 2r − 2 − (cid:18)r 1(cid:19)
parityconservation. While theminimalUEDmodelsug-
−
r gests B(1) dark matter [3], detailed studies of the relic
(r 1)2ln + (√ǫ)
− − (cid:18)r 1(cid:19) O (cid:27) abundance in the minimal UED model [6, 7, 14] in com-
− bination with electroweak precision constraints [8] show
with the approximation ǫ M2 /M2 1 and r strong observational tension, and thus provide motiva-
≡ W(0) ν(1) ≪ ≡
M2 /M2 1. Numerically, we find agreement with tion for new possible UED model building avenues. The
W(1) ν(1) ≃
our semi-analytical result as seen in figure 2. need for non-minimality has been reported [15]. Exten-
4
sions of the MUED scenario by incalculable boundary dimensional models compactified on S /Z , we expect
1 2
terms arising from the UV completion of the model can that the qualitative features carry over to UED mod-
lead to a different LKP and therefore different possible els with multiple extra dimensions. We have also as-
dark matter candidates. We have taken a model inde- sumed the absence of any fine-tuning between the ra-
pendent approach, following [4], and examined the con- diatively induced magnetic dipole moment and possible
sequences of UED KK-neutrino dark matter. non-renormalizable dimension-five dipole operators aris-
As the KK-neutrino is a Dirac fermion, UED models ing from the UV complete theory or boundary terms.
predictaninducedKK-neutrinodipolemoment. Wefind Ourresultsindicatethatobservationallimitsondipole
that the induced dipole moment, typically µ . 10−7µB, darkmattercanplacesignificantconstraintsonUEDsce-
strongly conflicts – by over five orders of magnitude – narios where the KK-neutrino is the LKP dark matter
with the current observational bounds stated in [10] for candidate.
TeVscaledipole darkmatter. We reiteratethatthecon-
straintsprovidedby[10]arecurrentlyunderrevision,and
thestrengthofthestatedboundsareexpectedtochange
V. ACKNOWLEDGMENTS
[11]. The constraints on magnetic dipole moments given
in[10]wouldprovidethestrongestlimitsonKK-neutrino
darkmatter. Evenintheabsenceofthestronglimitspro- WewouldliketothankJ.March-Russell,G.Starkman,
vided by [10], the bounds on dipole moments remain an B.A.Campbell, andK.Sigurdsonforusefuldiscussions.
importantconstraintonfuture modelbuilding. Notonly DM wishes to acknowledge the support of the Natural
will new, non-minimal models that predict KK-neutrino ScienceandEngineeringResearchCouncilofCanadaand
LKP need to circumvent the constaints provided by [9], the Canada-United Kingdom Millennium Research Fel-
but will also have to evade the constraints provided by lowship. The work of TF is supported by “Evangelis-
radiatively induced magnetic dipole moments, which are ches Studienwerk Villigst e.V.” and PPARC Grant No.
generically at least as large as the current experimental PPA/S/S/2002/03540A. This work was also supported
bounds. by the “Quest for Unification” network, MRTN 2004-
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