Extra Dimensions and Neutrinoless Double Beta Decay Experiments

0.17 MB·English

by  Marek Góźdź

Extra Dimensions and Neutrinoless Double Beta Decay Experiments
Marek G´o´zd´z,1,∗ Wiesl aw A. Kamin´ski,1,† and Amand Faessler2,‡
1Theoretical Physics Department, Maria Curie–Sk lodowska University, Lublin, Poland
2Institute fu¨r Theoretische Physik, Universita¨t Tu¨bingen,
Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany
The neutrinoless double beta decay is one of the few phenomena, belonging to the non-standard
physics,whichisextensivelybeingsoughtforinexperiments. Inthepresentpaperthelinkbetween
the half-life of the neutrinoless double beta decay and theories with large extra dimensions is ex-
plored. The use of the sensitivities of currently planned 0ν2β experiments: DAMA, CANDLES,
COBRA, DCBA, CAMEO, GENIUS, GEM, MAJORANA, MOON, CUORE, EXO, and XMASS,
gives the possibility for a non-direct ‘experimental’ verification of various extra dimensional sce-
narios. We discuss also the results of the Heidelberg–Moscow Collaboration. The calculations are
2 based on the Majorana neutrino mass generation mechanism in the Arkani-Hamed–Dimopoulos–
1 Dvalimodel.
0
2 PACSnumbers: 11.10.Kk,12.60.-i,14.60.St,14.60.Pq
Keywords: neutrinomass,extradimensions,ADDmodel,neutrinolessdoublebetadecay
n
a
J
I. INTRODUCTION shapes with tiny radii (the so-calledCalabi–Yaushapes)
5
whichhasverysmallinfluenceonourlife. Itis,however,

possible that the extra dimensions are open, large, and
] The standardmodel (SM), despite being very success-
h the difficulty in observing them has a different basis.
p ful, has many drawbacks and cannot be regarded as the
There are three main models used in building theo-
- ultimate theory of elementary particles and their inter-
p ries with large extra dimensions. Firstly, we have the
actions. One of the most severe are the big number of
e descriptionbasedonthe originalKaluza–Kleinapproach
free parameters and the hierarchy problem. The idea of
h [6,7],whichpredictsaloadofnewparticles,theso-called
[ going beyond the SM is old, but only recently has been
towersofKKexcitations. Secondly,therearemodelsfor-
fully backedup by experimentalevidence of neutrino os-
1 mulated by Randall and Sundrum (RS models) [8, 9], in
cillations [1–5]. This observation clearly calls for a more
v whichtheextradimensionsarecompactifiedonorbifolds
fundamental theory, which will extend the SM. One of
8 withaZ symmetry. Lastbutnotleast,therearepropo-
the best candidates, whichcures mostof the SM’s weak- 2
2 sitionsofArkani-Hammed,DimopoulosandDvali(ADD
2 nesses, is the theory of superstrings. Within this theory
models) [10–14], in which the standard model particles
1 onegetsridoftheproblematicpoint-likenatureofparti-
are trapped on a 3D brane, which in turn floats in a
. cles, therefore removing ultravioletdivergences. What is
1 higher dimensional bulk. In what follows we will use the
0 more,thereisnoneedtointroduceandfittoexperiments
ADD approach.
2 so many free parameters (more than 20 in the SM). The
Asmentionedabove,themainevidenceforphysicsbe-
1 stringtheoryisaconsistent,divergent-freequantumfield
: theory. Itallowsalsotogenerate,inanaturalway,aspin- yondthe SM arethe oscillationsofneutrinos. According
v
to most theories, neutrino oscillations imply a non-zero
i 2 massless field, which represents the graviton,therefore
X neutrino mass and a difference between the flavor and
unifying gravity and electroweak interactions.
mass eigenstates of neutrinos. The physical mechanism
r
a However, the string theory requires two new features generating this difference, as well as masses, remain still
for consistent formulation: the supersymmetry and ad-
an open issue.
ditional spatial dimensions. Supersymmetry introduces
Assuming the existence of supersymmetry, or non-
newkindofoperators,whichchangethe spinofparticles
standard physics in general, one may expect the phe-
by 1/2. Therefore a boson may be turned into a fermion
nomena, normally forbidden by the symmetries of SM.
and vice versa, which results in unification of forces and
One of these is the neutrinoless channel of double beta
matter. The idea of additional spatial dimensions sound
decay (0ν2β) [15]. This decay, still not confirmed, is the
evenmorestrange. Becausethesetwophenomenaarenot
onlyexperimentalpossibilityofdeterminingthenatureof
observed in the low energy domain, the supersymmetric
neutrino (Majorana or Dirac particle). It may also help
particles must be heavier than our current experimental
in obtaining the absolute mass of these particles, since
capabilities,andextradimensionsmuchsmallerthanour
the oscillation experiments are sensitive to differences of
ability to observe them. They may form closed, curled
masses squared only.
In our earlier work [16, 17] we have shown, that it is
possibletorelatetheparametersdescribingextradimen-
sionstothehalf-lifeofthe0ν2β decay. Inthepresentpa-
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] per we continue this topic with the analysis of the data
‡Electronicaddress: [email protected] provided by the Heidelberg–Moscow Collaboration (H–
2
M) as well as sensitivities of planned 0ν2β experiments, “shines” it everywhere, in particular also on our brane.
from the point of view of large extra dimensions. The strength of the shined χ is in a natural way sup-
Inthe nextsectionwerecallthe necessaryinformation pressed by the distance r between branes, and therefore
about the ADD model of large extra dimensions. In the one can write for the messenger
following section we shortly discuss the formula describ-
ing the half-life of neutrinoless double beta decay. After χ = φ ∆n(r), (2)
h i h i
that, we arrive at the final formula and present and dis-
where ∆ (R) is the n-dimensional propagator given by
cuss the results. A summary follows at the end. n
log(Rm ), (Rm 1)
χ χ
II. EXTRA DIMENSIONS ∆2(R) ∼ (−√e−RRmmχχ, (Rmχ ≪≫1) (3)
1 , (Rm 1)
twIonmthaeinADpaDrtsa.pprOoanceh itshethsepascoe-ccaalnledbebduilvki,dead 3int+o ∆n>2(R) ∼ (eRR−nnR−−m22χ, (Rmχχ ≪≫1) (4)
n dimensional space in which the gravity propagates.
−
Besides gravitons, it may be also populated with other We introduce a lepton field l(x) and a Higgs scalar field
particles and fields, which are not contained in the stan- h(x) localized on our brane. They can interact with the
dard model. Floating in the bulk, there is at least one messenger and the interaction is given by the following
3-dimensionalbrane,anobjectwhichispredictedby the Lagrangian[14]:
string theory as a higher-dimensional generalization of
thestring. Thestandardmodelisassumedtobeconfined Mn−1 int d4x′ φ χ(x′,y )+
on suchbrane. It means that all the SM interactionsare ∗ L ∼ h i ∗
Z
propagating only within the brane. The same goes to
d4x α[l(x)h∗(x)]2χ(x,0), (5)
fermions, which must also be restricted to live on the
Z
brane. The just presented setting immediately explains
the weakness of gravity in our universe, thus solving the where the first part represents the lepton number viola-
hierarchyproblem. Namely,usingforexamplethegener- tion, occurring on the other brane, and the second part
alized Gauss Law, one arrives at the so-called reduction is responsible for the interaction between SM fields and
formula [13] the messenger on our brane. Let us assume for simplic-
ity, that the second brane is as far away from ours as
M2 RnM2+n, (1) possible, i.e. the distance between the branes is approx-
Pl ∼ ∗
imately equal to R, the compactification radius. After
where M 1028 eVis the Planckmass, M is the true
Pl ∗ spontaneous symmetry breaking we substitute (2) into
∼
scaleofgravity,andRis the assumedcommoncompact-
(5), write the Higgs field in terms of its vev v, and iden-
ification radius of extra dimensions. One sees that, by
tifyl withν . WearriveatamasstermoftheMajorana
L
properly adjusting the values of R and n, it is possible formm νT ν withthe massgivenapproximatelyby
Maj L L
tolowerthe true scaleofgravitytothe electroweakscale
[14]:
1TeV,thusgettingridofthe hierarchyproblem. This
∼
is the main motivation for the ADD approach. v2∆(r)
m . (6)
Introducing a second brane, parallel to ours, one may Maj ∼ Mn−1
∗
generate, under special conditions, a Majorana neutrino
mass term. Let us denote the coordinates by xµ,ym , which, using Eq. (1), may be rewritten as [16]:
{ }
where µ = 0...3 labels the ordinary space-time coordi-
indaetnetsifaynindgmy =y1+..2.πnRlawbeelcsotmhpeaecxtitfryatdhiemeexntsriaondsi.meBny- mMaj ∼v2Rn(nn+−21)MP2(ln1+−2n)∆n(R), (7)
∼
sions on circles. (From now on we will drop the indices withv2 =(174 GeV)2 1022 eV2 beingthe Higgsboson
µ and m for simplicity.) ∼
vev.
Letusassume[13,14]thatleptonnumberisconserved
onourbrane,locatedaty =0,butmaximallybrokenon
the other one, placed on y = y∗. The breaking occurs III. NEUTRINOLESS DOUBLE BETA DECAY
in a reaction where a particle χ, with lepton number
L = 2 and mass m , escapes the other brane into the
χ Neutrinoless double beta decay (0ν2β) is a process in
bulk. This particle, called the messenger, may interact
whichanucleusundergoestwosimultaneousbetadecays
withourbraneandtransmittoustheinformationabout
without emission of neutrinos
lepton number breaking.
To be more specific, let us introduce a field φL=2 lo- A(Z,N) A(Z+2,N 2)+2e−. (8)
cated on the other brane, whose vacuum expectation → −
value (vev) breaks the lepton number. What is more, It requires neutrino to be a Majorana particle, so that
it acts as a source for the bulk messenger field χ and twoneutrinosemittedinbetadecaysannihilatewitheach
3
other. It is readily seen that this process violates the
TABLEI:Sensitivitiesofsome0ν2β experimentsplannedfor
lepton number by two units, thus it is forbidden in the
the future. The half-life Texpt. is expressed in years, while
frameworkofSM.The0ν2βdecayhasbeenclaimedtobe 1/2
effective neutrino mass in eV; ξ = Texpt.hm i2 . Values
observed[18], butthis informationhasnotyetbeencon- 1/2 ν expt.
firmed and has met some strong criticism from the com- taken from Refs. [21–25].
munity (see e.g. [19, 20]). In any case, even the limit for Texpt. hm i ξ
non-observability of this decay sets valuable constraints 1/2 ν expt.
expt. [y] [eV] [y eV2]
on the shape of physics beyond the SM. It is the main
tool to verify the nature of neutrino (Dirac or Majorana DAMA (136Xe) 1.2×1024 2.3 6.3×1024
particle). It may also help in determining the absolute MAJORANA(76Ge) 3×1027 0.044 5.8×1024
value of the neutrino mass. EXO 10t (136Xe) 4×1028 0.012 5.7×1024
Ignoring the contributions from right-handed weak GEM (76Ge) 7×1027 0.028 5.5×1024
currents, the half-life of 0ν2β can be written in the form GENIUS (76Ge) 1×1028 0.023 5.3×1024
[15]: CANDLES (48Ca) 1×1026 0.2 4.0×1024
m2 MOON (100Mo) 1×1027 0.058 3.4×1024
T1/2 =M−sp1ec me2. (9) XMASS (136Xe) 3×1026 0.10 3.0×1024
h νi CUORE (130Te) 2×1026 0.10 2.0×1024
One can write the same for any performed or planned COBRA (116Cd) 1×1024 1 1.0×1024
0ν2β experiment, replacing the relevant values with the DCBA (100Mo) 2×1026 0.07 9.8×1023
sensitivities of the experiments DCBA (82Se) 3×1026 0.04 4.8×1023
CAMEO (116Cd) 1×1027 0.02 4.0×1023
m2
Texpt. = −1 e , (10) DCBA (150Nd) 1×1026 0.02 4.0×1022
1/2 Mspec m 2
h νiexpt.
Inrelations(9)and(10) isthe nuclearmatrixele-
spec
mentforaspecificnucleuMswhichcanbecalculatedwithin can be treated as nearly degenerate, with m1 m2 [28].
≈
certainnuclearmodels,andme istheelectronmass. The Within these approximations one may express mee as a
so-called effective neutrino mass mν is defined by the function of m1 and two mixing angles in the following
relation h i form [28]:
m2 =[1 sin2(2θ )sin2(φ /2)]m2 κ−1m2, (13)
m = m U2 ee − Solar 12 1 ≡ 1
h νi (cid:12)(cid:12)(cid:12)(cid:12)Xmi Ui e2i(cid:12)(cid:12)(cid:12)(cid:12) m U 2 m U 2 , (11) wthheeKreasminL2a(n2θdSroelasur)lt≈s[10–.832,;2t9h].isBvyalCuePtsaykmesminettroyaccocnosuenrt-
⇒ (cid:12) 1| e1| (cid:12)± 2| e2| ± 3| e3| vation, the relative phase factor φ takes values either
12
(cid:12) (cid:12)
where U is the(cid:12)neutrino mixing matrix and m a(cid:12)re neu- 0 or π/2. For φ12 = 0, mee = m1. The more interesting
(cid:12) i (cid:12) case involves the mixing angle θ therefore we chose
trinomasseigenvalues. ThelastformulationassumesCP Solar
the value φ =π/2 having
invariance. One sees from this equation that it is possi- 12
bmleattroixidinentthiefyflhamvoνribwaistihs the ee entry of neutrino mass m2ee =1.69 m21, (14)
Infactthe exactvalue ofκis irrelevantinourdiscussion
m =m , (12)
h νi ee and the whole coefficient can in principle be set to one.
What we are interested in is the order of magnitude of
whichisgivenexactlybythesuperpositionofmasseigen-
the half-life of 0ν2β decay.
values from Eq. (11).
By rewriting T in terms of Texpt. we get rid of the
In the next section the link between T and the pa- 1/2 1/2
1/2
unwanted nuclear matrix elements
rametersdescribingextradimensionswillbeestablished.
m 2
T =Texpt.h νiexpt. =ξ m −2, (15)
1/2 1/2 m 2 h νi
IV. RESULTS h νi
where we have gathered all experimental values in the
In the calculations we have neglected the contribution parameter ξ =Texpt. m 2 . Eqs. (12) and (14) allow
comingfromthethirdneutrinointhemassbasis,setting 1/2 h νiexpt.
to rewrite this expression as
U 2m =0. This is justified by the results of CHOOZ
e3 3
| |
experiment [26, 27], which showed that there is a negli- T =ξm−2 =ξκm−2. (16)
1/2 ee 1
gible admixture of ν in the third mass eigenstate (less
e
than 3%; most of the analyses place the value of U 2 Finally,weassumethatneutrinosareMajoranaparticles
e3
| |
between0and0.05). Theremainingtwomasseigenstates in order to discuss the neutrinoless double beta decay.
4
Sinceweareleftwithonlyoneindependentmass,wemay specificvaluesofT . Ifapositivesignalwillberecorded
1/2
replace m with the expression for Majorana neutrino by that collaboration, it may be compared with our val-
1
mass, Eq. (7), finishing with ues to estimate the values of n, R and Rm .
χ
Letusstartwiththediscussionoftheclaimofevidence
T1/2 = κ ξ v−4 R2n(1−n)/n+2 of 0ν2β decay. Klapdor et. al. in Ref. [18] reported a
M4(n−1)/n+2 [∆ (R)]−2. (17) positive signal for the 0ν2β transition in 76Ge and, after
× Pl n reanalyzing the data, announced the following values at
more than 99.9% confidence level [37]:
One sees that Eq. (17) consists of two parts. The first
one is connected with 0ν2β experiments, for which the TH−M =1.19 1025 y, m =0.44 eV, (21)
relevantvaluesarepresentedinTab.I.Wehaveincluded 1/2 × h νiH−M
tha values as given by various collaboration assuming, which corresponds to ξ = 2.3 1024 y eV2. This claim,
that these represent the best sensitivity of the projects. ×
although not confirmed, has not been withdrawn, so it
Therefore the effective neutrino mass, which depends on
deserves a careful analysis. The results are presented in
the nuclear matrix elements, may be different for the
Fig. 1.
sameisotope. Howeverthecalculationsofthematrixele-
Oneseesthatforalightmessengerparticle,theresults
mentsusingvariousapproachesgiveonaverageachange
do not depend on Rm , except for n=2, where the de-
χ
in m 200–300%which is well below the level of accu-
ν pendenceisveryweak. Infacttheyareevenindependent
h i
racy of our discussion.
of R and m separately. Obtaining constant values for
χ
The second part of Eq. (17) contains parameters of
thatcase(forn 3)isageneralfeatureofthe model,as
the extra dimensions. At this point we need to include ≥
will be seen in further discussion. Assuming that the H–
constraints coming from other sources, like supernova
M values correctly describe 0ν2β decay, and taking into
and neutron star data [30–34], and cosmological mod-
account possible errors, the closest match will be n= 3.
els [35, 36]. Altogether, one of the most complete limits
The n = 2 case is practically ruled out, since the sensi-
have been derived in Ref. [32] and read:
tivities of the experiments exceeded the obtained value
by more than 15 orders of magnitude, and all of them
R<1.5 10−7 mm for n=2, (18)
× reported negative results. Another possible scenario is
R<2.6 10−9 mm for n=3, (19) the n=4, but its verification will have to wait for more
×
R<3.4 10−10 mm for n=4. (20) powerful experiments than those planned today.
×
In the case of heavy messenger, the first solution is
Taking into account bounds Eqs. (18)–(20) the formula n=2andRm 17. IftheH–Mresultswereconfirmed,
χ
forT1/2 becomesaninequality. ExplicitlyEq. (17)takes this possibility i≈s promising. Another choice could be
the following forms of lower bounds on the half-life of n = 3 and Rm being close to 1. This, however, puts
χ
0ν2β: a question mark on our prediction, since we have used
Case n=2 theasymptoticalformsofthe propagator,whicharenot
validforRm 1. Moreextradimensionsblowthehalf-
T >1.11 10−15 ξ′ (log(Rm ))−2 , (Rm 1); χ ≈
1/2 × χ χ ≪ life to extremely huge values, making the discussion at
T >1.11 10−15 ξ′ e2Rmχ Rm , (Rm 1). this stage meaningless.
1/2 χ χ
× ≫
The H–M data’s ξ value places itself in the middle of
Case n=3 considered projects. Let us now say a few words about
the currently planned 0ν2β experiments. The results for
T >0.039 ξ′ , (Rm 1);
1/2 χ ≪ the DCBA experiment, which has the smallest ξ value,
T >0.039 ξ′ e2Rmχ , (Rm 1). and for DAMA experiments with the biggest ξ are de-
1/2 χ
≫
picted on Figs. 2 and 3, respectively. For the remaining
Case n=4 projects the results are summarized in Tabs. II and III.
Allthevaluespresentedinthetablesareboundsfrombe-
T >1.69 1011 ξ′ , (Rm 1);
1/2 × χ ≪ low on the 0ν2β half-life. In any case one should always
T >1.69 1011 ξ′ e2Rmχ . (Rm 1). bear in mind the upper limit on R and from the value
1/2 χ
× ≫
of Rm deduce the appropriate mass of the messenger
χ
We have denoted ξ′ = ξ/(1 eV2). In the above calcu- particle.
lations the value κ = 1.69 has been used, which is con- The results for n = 2 are summarized in Tab. II. The
sistent with the KamLand results [29]. The inequalities caseoflightmessengerisexcludedbythenegativeresults
represent lower bounds on the half-life. The true value of 0ν2β experiments, which sensitivities have exceeded
of T may be much bigger than the bounds itself. We thegiventhresholdbymorethan15ordersofmagnitude.
1/2
wouldliketo stress,thatthe followingdiscussionis valid In the case of heavy messenger,the quantity Rm turns
χ
onlyunderourassumptions,i.e. weliveinabraneworld out to be equal to at least 15. This value matches well
andgenerateneutrinomassesaccordingtotheADDsug- the Rm 17 obtained for H–M experiment.
χ
≈
gestion. If not, our bounds onT are not valid. Insert- Case n = 3 is presented in Tab. III for heavy messen-
1/2
ing ξ corresponding to various experiments one obtains ger. One sees that this case is not limited by the ex-
5
1e+40 1e+65
1e+60
1e+35
1e+55
y] 1e+30 y] 1e+50
[1/2 [1/2 1e+45
T T
on 1e+25 on 1e+40
d d
n n
u u 1e+35
o 1e+20 o
b b
er H-M result er 1e+30
w n=2 w
lo 1e+15 n=3 lo 1e+25
n=4
H-M result
1e+20 n=2
1e+10 n=3
1e+15 n=4
100000 1e+10
1e-05 0.0001 0.001 0.01 0.1 5 10 15 20 25 30
R mχ R mχ
FIG.1: Lowerboundsonthehalf-lifeofneutrinolessdoublebetadecayinthecaseoflight(leftpanel)andheavy(rightpanel)
messenger particle for theHeidelberg–Moscow experiment.
1e+35 1e+60
1e+55
1e+30
1e+50
y] y] 1e+45
[2 1e+25 [2
T1/ T1/ 1e+40
n n
o o
d 1e+20 d 1e+35
n n
ou n=2 ou
wer b 1e+15 nn==34 wer b 1e+30
o o 1e+25
l l
n=2
1e+20 n=3
1e+10 n=4
1e+15
100000 1e+10
1e-05 0.0001 0.001 0.01 0.1 5 10 15 20 25 30
R mχ R mχ
FIG.2: Lowerboundsonthehalf-lifeofneutrinolessdoublebetadecayinthecaseoflight(leftpanel)andheavy(rightpanel)
messenger particle for theplanned DCBA (150Nd) experiment.
periments. In fact the values Rm around 3 correspond periment. It may, however, serve as clue if all of them
χ
to the sensitivities of currently performed experiments, report negative results. For a light messenger, we face
like the Germanium neutrinoless double beta decay in a similar situation as previously, i.e. the m depen-
χ
Gran Sasso. We repeat here, that for such small value, dencedropsout. Forfourextradimensionswefinishwith
the validity of the results may be questioned. For light T >1.69 1011 ξ′, therefore obtaining T >1035 y.
1/2 1/2
χ particle the calculations give T > 3.9 10−2 ξ′, so ×
1/2
×
thereisnom dependence. Sincetheexperimentalfactor
χ
is of the order of 1024, we end up with T >1022−23 y.
1/2
Ifthenumberofextradimensionsisfour,asimplecal- Thediscussionmaybeextendedformoreextradimen-
culation gives T > 1038 y for heavy messenger. This sions, if one is interested in the half-life of 0ν2β being
1/2
value isbeyondthe abilities ofanycurrentlyplannedex- longer than 1038 years.
6
1e+40 1e+65
1e+60 n=2
1e+35 n=3
1e+55 n=4
n=2
[y]1/2 1e+30 nn==34 [y]1/2 11ee++4550
T T
on 1e+25 on 1e+40
d d
n n
u u 1e+35
o 1e+20 o
b b
er er 1e+30
w w
lo 1e+15 lo 1e+25
1e+20
1e+10
1e+15
100000 1e+10
1e-05 0.0001 0.001 0.01 0.1 5 10 15 20 25 30
R mχ R mχ
FIG.3: Lowerboundsonthehalf-lifeofneutrinolessdoublebetadecayinthecaseoflight(leftpanel)andheavy(rightpanel)
messenger particle for theplanned DAMA experiment.
TABLE II: Lower boundson T1/2 in years for two additional dimensions (R<1.5×10−7 mm).
Rm = 0.001 0.01 0.1 10 15 20 25
χ
DAMA 1.6×108 3.6×108 1.4×109 3.7×1019 1.2×1024 3.6×1028 1.0×1033
MAJORANA 1.5×108 3.3×108 1.3×109 3.4×1019 1.1×1024 3.3×1028 9.2×1032
EXO (10t) 1.4×108 3.2×108 1.3×109 3.3×1019 1.1×1024 3.2×1028 9.0×1032
GEM 1.3×108 3.1×108 1.2×109 3.2×1019 1.0×1024 3.1×1028 8.6×1032
GENIUS 1.3×108 3.0×108 1.2×109 3.1×1019 1.0×1024 3.0×1028 8.3×1032
CANDLES 9.7×107 2.2×108 8.7×108 2.3×1019 7.5×1023 2.2×1028 6.2×1032
MOON 8.2×107 1.9×108 7.4×108 2.0×1019 6.3×1023 1.9×1028 5.3×1032
XMASS 7.2×107 1.7×108 6.5×108 1.7×1019 5.6×1023 1.7×1028 4.7×1032
CUORE 4.7×107 1.1×108 4.3×108 1.1×1019 3.7×1023 1.1×1028 3.1×1032
COBRA 2.4×107 5.5×107 2.1×108 5.7×1018 1.8×1023 5.5×1027 1.5×1032
DCBA (100Mo) 2.3×107 5.4×107 2.1×108 5.6×1018 1.8×1023 5.4×1027 1.5×1032
DCBA (82Se) 1.1×107 2.6×107 1.0×108 2.7×1018 8.8×1022 2.6×1027 7.4×1031
CAMEO 9.5×106 2.2×107 8.5×107 2.3×1018 7.3×1022 2.2×1027 6.1×1031
DCBA (150Nd) 9.5×105 2.2×106 8.5×106 2.3×1017 7.3×1021 2.2×1026 6.1×1030
V. SUMMARY use the results for exotic nuclear processes also to con-
strainthe theorieswithlargeextradimensions. Needless
to say, we have much better control over these experi-
Theories which deal with extra dimensions face one ments, than those aforementioned.
basic problem, namely the obvious difficulty of verifica- To sum up the results of the present paper, one can
tion. Therearetwomaintypesofexperimentsperformed say that within the model used, the current0ν2β exper-
nowadays. One are the tabletop gravity experiments, iments have reached the sensitivity to explore the possi-
which test the Newton 1/r2 law on very small distances. bility of two, and maybe three additional spatial dimen-
Thebestaccuracyofpresentsetupsis0.1mm[38]which sions. The future experiments shouldbe able to rule out
isnotsufficient. Theotherarebasedonastronomicalob- the possibility of three extra dimensions, provided they
servations,mainlyconcerningsupernovasandblackholes finish with negative results. The sensitivity needed to
[30–36]. These, however, are very difficult and highly reach the n = 4 threshold should be at least 1035 y,
∼
model-dependent. The work presented first in Refs. [16] which is 7 to 9 orders of magnitude better than the cur-
and [17], and continued here, shows that it is possible to rently planned experiments can achieve.
7
TABLE III: Lower bounds on T1/2 in years for three additional dimensions (R < 2.6×10−9 mm). The case of light mχ is
discussed in thetext.
Rm = 3 5 10 15
χ
DAMA 1.1×1026 6.0×1027 1.3×1032 2.9×1036
MAJORANA 1.0×1026 5.5×1027 1.2×1032 2.7×1036
EXO (10t) 1.0×1026 5.4×1027 1.2×1032 2.6×1036
GEM 9.5×1025 5.2×1027 1.1×1032 2.5×1036
GENIUS 9.1×1025 5.0×1027 1.1×1032 2.4×1036
CANDLES 6.8×1025 3.7×1027 8.1×1031 1.8×1036
MOON 5.8×1025 3.2×1027 6.9×1031 1.5×1036
XMASS 5.1×1025 2.8×1027 6.0×1031 1.3×1036
CUORE 3.4×1025 1.8×1027 4.0×1031 9.0×1035
COBRA 1.7×1025 9.2×1026 2.0×1031 4.5×1035
DCBA (100Mo) 1.6×1025 9.0×1026 2.0×1031 4.4×1035
DCBA (82Se) 8.1×1024 4.4×1026 9.6×1030 2.1×1035
CAMEO 6.7×1024 3.6×1026 8.0×1030 1.8×1035
DCBA (150Nd) 6.7×1023 3.6×1025 8.0×1029 1.8×1034
Acknowledgments national Graduiertenkolleg GRK683 by the DFG (Ger-
many). Twoofus(MG,WAK)wouldliketothankProf.
This work has been supported by the Polish A. Faessler for his warm hospitality in Tu¨bingen during
State Committee for Scientific Research under grants the Summer 2004.
no. 2P03B 071 25 and 1P03B 098 27 and the Inter-
[1] Q. R. Ahmad et al. (SNO Collaboration), Phys. Rev. [16] M.G´o´zd´zandW.A.Kamin´ski,Phys.Rev.D68,057901
Lett.89, 011301 (2002). (2003).
[2] Q. R. Ahmad et al. (SNO Collaboration), Phys. Rev. [17] M.G´o´zd´zandW.A.Kamin´ski, Int.J.Mod.Phys.E13,
Lett.89, 011302 (2002). 367 (2004).
[3] S.N.Ahmedetal.(SNOCollaboration),Phys.Rev.Lett. [18] H. V. Klapdor-Kleingrothaus, A. Dietz, H. Harney, and
92, 181301 (2004). I. Krivosheina, Mod. Phys.Lett. A 16, 2409 (2001).
[4] S. H. Ahn et al. (K2K-KEK Experiment), Phys. Rev. [19] C.E.Aalsethetal.,Mod.Phys.Lett.A17,1475(2002).
Lett.90, 041801 (2003). [20] A. M. Bakalyarov et al. (2003), hep-ex/0309016.
[5] Y.Ashieetal.(Super-KamiokandeCollaboration),Phys. [21] S. R. Elliott and P. Vogel, Annu. Rev. Nucl. Part. Sci.
Rev.Lett. 93, 101801 (2004). 52, 115 (2002).
[6] T.Kaluza, Preuss. Akad.Wiss., Phys.-Math. Kl.,Berlin [22] N. Ishihara, talk given at the APPI conference, Japan
Math. Phys.Bd. K1, 966 (1921). (2004).
[7] O.Klein, Z. Phys. 37, 895 (1926). [23] G. Gratta, talk given at the Lepton–Photon 03 confer-
[8] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 ence, Fermilab, USA(2003).
(1999). [24] R.Bernabeietal.(DAMACollaboration), Phys.Lett.B
[9] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 546, 23 (2003).
(1999). [25] M. Danilov et al. (EXO Collaboration), Phys. Lett. B
[10] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. 480, 12 (2000).
Lett.B 516, 70 (1998). [26] M. Apollonio et al.,Phys. Lett.B 420, 397 (1998).
[11] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. [27] M. Apollonio et al.,Eur. Phys.J. C 27, 331 (2003).
Rev.D 59, 086004 (1999). [28] H.Pa¨sandT.J.Weiler,talk givenatthe3rd Workshop
[12] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and onNeutrinoOscillationsandtheirOrigin(NOON2001),
G. Dvali, Phys.Lett. B 436, 257 (1998). Kashiwa, Japan, hep-ph/0205191 (2001).
[13] N. Arkani-Hamedand S. Dimopoulos, Phys. Rev. D 65, [29] S. M. Bilenky, A. Faessler, and F. Sˇimkovic, Phys. Rev.
052003 (2002). D 70, 033003 (2004).
[14] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, and [30] S.HannestadandG.Raffelt,Phys.Rev.Lett.87,051301
J. March-Russell, Phys.Rev.D 65, 024032 (2002). (2001).
[15] M. Doi, T. Kotani, and E. Takasugi, Prog. of Theor. [31] S.HannestadandG.Raffelt,Phys.Rev.Lett.88,071301
Phys.Suppl.83, 1 (1985). (2002).
8
[32] S. Hannestad and G. Raffelt, Phys. Rev. D 67, 125008 [36] S. Hannestad, Phys.Rev.D 64, 023515 (2001).
(2003). [37] H. V. Klapdor-Kleingrothaus, I. Krivosheina, A. Dietz,
[33] C. Hanhart et al., Nucl.Phys. B 595, 335 (2001). and O.Chkvorets, Phys. Lett. B 586, 198 (2004).
[34] C. Hanhart et al., Phys.Lett. B 509, 1 (2001). [38] J. C. Long et al., Nature421, 922 (2003).
[35] L. J. Hall and D. R. Smith, Phys. Rev. D 60, 085008
(1999).