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Neutrinoless Double Beta Decay

James Esterline

Abstract

With evidence for massive neutrinos and neutrino oscillations having recently been

established, neutrino physics beyond the Standard Model is of increasing interest. One proposedtreatment of the neutrino is that of a Majorana particle a particle that is its own antiparticle. As

a result of the violation of lepton number conservation that is necessary for a Majorana neutrinoto exist, phenomena such as neutrinoless double beta decay of atomic nuclei would be allowed.

The necessary theoretical modifications to the Standard Model to incorporate Majorananeutrinos, the outline of the calculational procedure of obtaining the neutrinoless double beta

decay half-life, and an overview of the planned experiment of the Majorana Collaboration toobserve the process will be presented.

Table of Contents

Theoretical Background:

Majorana ParticlesPg. 2

Modification to the Standard Model Due to Majorana NeutrinosPg. 3

Double Beta DecayPg. 4

Calculation of Decay Rates for Double Beta Decay Pg. 5

Candidate Nuclei for Double Beta DecayPg. 7

Experimental Measurement of Neutrinoless Double Beta Decay:Experimental Signatures of Double Beta Decay

Pg. 8Experimental Design the Majorana Experiment

Pg. 9Goals of the Majorana Experiment

Pg. 9

References: Pg. 10Appendix:

Pg. 11


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Lorentz boost

Majorana: !L !R

CPT

Modifications to the Standard Model Due to Majorana Neutrinos

Before anything else need be considered, it is immediately obvious that lepton number

conservation is violated by the existence of Majorana neutrinos. In the current formulation of theStandard Model, the neutrino is assigned a lepton number of +1 and the antineutrino is assigned

a lepton number of -1, with a distinction being made between the three lepton flavors. Evidencein the form of neutrino oscillations has already shown that the flavor distinction in lepton number

is violated; the requirement that a neutrino be its own antiparticle would force neutrinos andantineutrinos to have lepton numbers of the same sign, and one can quickly conclude, by

examining beta decays (which would require charged leptons to carry lepton number opposite tothose of associated neutrinos) and other weak processes (muon decay, with the above

assumptions, would be necessarily neutrinoless), that the concept of conserved lepton number isno longer valid.

There is also the possibility, as will be examined in further detail, that right-handedinteractions need be introduced into the Standard Model electroweak Hamiltonian. To do this,

one merely parametrizes the inclusion of right-handed terms in the electroweak interaction(formulation for energies below the Wmass):

( ) ( )( ) ..2

chMMJMMJG

HRLRRLLW

++++= !!!

!!

! "#$

This differs from the Standard Model by the inclusion of right-handed lepton current JRand right-handed quark current MR(# in the above equation denotes four-vector notation);equivalently, it retrieves the Standard Model in the case where the dimensionless parameters $,%, and &are zero. Due to the observation that, if right-handed lepton currents are present at all inthe weak interaction, they are highly suppressed, the coefficients %and &are generally presumedvery small.

Note that a further modification arising from the inclusion of right-handed currents is the

existence of a right-handed Wboson, which presumably interacts much more weakly than its

left-handed counterpart. No deeper explanation of such a particle was to be found in theliterature, however.


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The decay depicted above maintains the left-handedness of the weak interaction, but has

a helicity mismatch in the exchange of the Majorana neutrino. This is only possible if theneutrino is massive (as has been supported experimentally); in this case, the helicity of the

neutrino emitted at the first vertex contains a positive component proportional to the neutrinosmass divided by its energy, which, considering the energy available to double beta decays (on

the order of MeV), is a remarkably small, but nonzero, quantity.

Calculation of Decay Rates for Double Beta Decay

Fermis Golden Rule gives the second-order weak decay (that which proceeds via the

two-nucleon mechanism) to be2

,

2 !!"""#

#$

%&&'

("=

) *

)

)

+,-m emif

fiEEEE

iHmmHfEEd

where the sum over m represents the sum over the intermediate states of the virtual nucleus.

Despite the neutrinos nonvanishing mass, p!= E! is still a good approximation, and is used

henceforth.Substituting in for the weak HamiltonianH', which is the product of nuclear and leptonic

currents, we obtain

( )

2

,, ,

44cos8 ! !!

"""##$

%&&'

("=

))

*+ ,

*+*+

,

,,

-./0m nn emi

nnnn

f

ficF

e

ee

EpEE

JJiMmmMfEEGd

where the notation of [Boe87] has been used: M is the nuclear current, ne(!) is either 1 or 2

(labeling each emitted pair of leptons), and ne(!)is the complement of ne(!)(that is, 3 - ne(!)).

From this, we may extract the decay rate for the two-neutrino double beta decay, w 2!,

using the derivation in [Boe87]:

( )( ) ( ) ( )!!!

"""

"""#$%&

'(=

e

e

e

e

e

e

mE

m ee

EE

m eeee

mE

m eeee

CF dppEEEXpdEEpEZFdEEpEZFG 0100 2

1210

2

1222211117

44

2 ,,

8

cos))))

*

+,

whereE0is the decay energy of the double beta decay, and F(Z,E)is the Fermi function, which

provides the contribution to the decay of the electromagnetic interaction between the nucleus andemitted electrons.

e-

e-

"!e"p

p

Z

( Z+1)

Z+2

n

n

"!e"


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The factor of X in the above expression contains all the nuclear matrix elements; it isgiven for 0

+!0

+decays (a good assumption, considering that all known double-beta-decaying

nuclei are even-even and hence both parent and daughter nuclei have ground states of 0+) in

[Boe87]; combined with the Primakoff-Rosen approximation:

( )!"

!

!"

Z

e

Z

p

EEZF

20 1

2lim,

#$

#

= ,

this results in an energy dependence of the form:

!!"

#$$%

&++++'19809092

1

4

0

3

0

2

007

02

EEEEE

()

The resulting half-life for two-neutrino beta decay is expressed as

( )[ ] ( )2

2

2

2

2

0

22

2/1 ,00 !!!!

F

A

V

GT Mg

gMZEGT "=# ++

where G2!is a lepton phase space factor, MGTandMFare the Gamow-Teller and Fermi matrix

elements, respectively, whose formulae are provided below:

!!!

+"

#=

++++

m fim

k

ikk

l

llf

GTMME

mmM

2/)(

002

$%$%

&

rr

!!!

+"=

++++

m fim

k

ik

l

lf

FMME

mm

M2/)(

002

##

$

where the summation over land kare over individual nucleons;(+and )are the nucleonicisospin raising operator and the nucleonic spin operator, respectively. [Hor02] notes that the

Fermi contribution can generally be eliminated, due to the restriction *T = 0 (that the nuclearisospin must not change) being invalidated by most double beta decays due to proceeding from

prohibited intermediate states.The calculation for neutrinoless double beta decay can proceed over either of the two

postulated channels presented in the previous section (that is, through the right-handedcomponent arising from the massiveness of the neutrino or through weak coupling to right-

handed particles).For the case where right-handed currents have not been added to the electroweak

Hamiltonian, the decay rate may be expressed as:

( )! "++=spins

eeifee pdpdMEEER 23

1

3

21

2

00 2 #$%

&&

whereR0nis the transition amplitude,Efis the energy of the daughter nucleus, andMiis the mass

of the parent nucleus.

Integration over lepton part of the amplitude is given by [Boe87], and yields a neutrinopotential Hn(r,Em), dependent in general on the energy of the intermediate nucleus Em, thatrepresents the propagator between the nucleons involved in the double beta decay. As argued for

in the mentioned source, the potential can be reasonably approximated as a 1/r central potential,roughly:

( ) jrmmn e

r

RErH !,


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where R is the nuclear radius, and mjis the neutrino mass. It is noted that this expression isindependent of the energy of the intermediate nucleus.

With the addition of the parameters &and %(the effect of $on the decay rate is negligibleaccording to [Hor02]), the general expression for the inverse of the neutrinoless double betadecay half-life can be expressed as:

( )[ ] ( )

22

2

2

0

2

20

0

00

2/1 ,00

!!""!" !!"!""#

!

#

"

#

####

CCCm

mC

m

mC

mMg

gMZEGT

e

m

e

m

F

A

VGT

+++++

$=% ++

where the Cijare factors containing the nuclear matrix elements and phase space integrals, andG0!is the phase space integral for the neutrinoless decay.

In the case where right-handed currents are ignored, and the Primakoff-Rosenapproximation is used, G

0!is approximated by:

!

"

#$

%

&'++(

5

2

330

0

000E

EEG

)

It is noted that, since neutrinoless double beta decay has two fewer decay products, the

phase space factor is reduced by a factor of E06, making the neutrinoless mode a very accurate

probe for the measurement of the neutrino mass for low energy decays.

The nuclear matrix elements differ from those for the two neutrino decay only by theinclusion of the neutrino potentialHn(r,Em).

Candidate Nuclei for Double Beta Decay

The kinematics of double beta decay allow a vast number of prospective nuclides toundergo the process. The reaction

AZ!

A(Z+2) + 2e

-(+ 2"!e)

has as the only constraint that the nucleus with Z protons and (A-Z) neutrons (here denoted byAZ) have greater mass than the nucleus with Z+2 protons and (A+Z-2) neutrons (here denoted by

A(Z+2)) and two electron masses (and two neutrino masses, although this is too small to

discriminate between allowed and disallowed processes). Equivalently, neglecting atomicbinding energy, the atomic mass of the parent need exceed that of the daughter. It is noted that

any beta-decaying nucleus that is also the daughter of a beta-decaying nucleus enables its parentto decay to its daughter directly through double beta decay; however, this is in general not

interesting (only where the half-life for the original parent nucleus is sufficiently large undersingle beta decay does the contribution from double beta decay become non-negligible). Instead

of considering these relatively difficult-to-measure processes, the categories of double-beta-decaying isotopes is typically limited to those for which single beta decay is energetically

prohibited. There are relatively few of these nuclides, numbering about 26. Among the isotopeswith the highest decay energies and, hence, the highest decay rates based on available phase

space are48

Ca (decay energy 4.27 MeV),76

Ge (2.04 MeV),82

Se (3.00 MeV),96

Zr (3.35 MeV),100Mo (3.03 MeV), 110Pd (2.01 MeV), 116Cd (2.80 MeV), 124Sn (2.29 MeV), 130Te (2.53 MeV),136

Xe (2.48 MeV),148

Nd (1.93 MeV), and150

Nd (3.37 MeV). These are all the nuclides with


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more than 1.75 MeV decay energy; of these, 48Ca and 96Zr are capable of decaying throughsingle beta decay [Boe87], but the associated half-lives are extremely low.

Experimental Measurement of Neutrinoless Double Beta Decay

The phenomenon of double beta decay is understandably a difficult one to measure, dueto the extremely long half-lives for the decays. The added constraint that the measured decay be

neutrinoless (two-neutrino decays have been observed and are in the literature) complicatesmatters immensely. As shown in the previous section, neutrinoless double beta decays have

much larger half-lives, adding to the difficulty of obtaining a realistically detectable signal(additionally complicated by background and noise, as will be mentioned later). Furthermore,

the distinguishability of neutrinoless- and two-neutrino-double beta decays lies primarily in thesummed energy spectrum of the emitted electrons, and requires precise energy resolution. One

of the proposed experiments to measure neutrinoless double beta decay is the aptly namedMajorana Experiment, whose details follow below.

Experimental Signatures of Neutrinoless Double Beta Decay

The most obvious indication of neutrinoless double beta decay would be a well-definedpeak in the summed electron energy spectrum corresponding to the exact value of the decay

energy. (Two-neutrino double beta decay would have a summed electron energy spectrumqualitatively similar to that of single beta decay: namely, a continuous energy distribution with

end-point energy extremely close to the decay energy, but without any peak in the electronenergy.) This difference between neutrinoless and two-neutrino energy spectra is due to the

absence of invisible neutrinos carrying away any of the decay energy, as they would whenpresent.

It is also conceivable to examine the excitation of the daughter nucleus to provide

evidence for neutrinoless double beta decay. This is due to the This is remarkably moredifficult in that, due to further limitations of phase space, the decay rate is again greatlydecreased, and also in that it is not direct evidence of neutrinoless double beta decay, as the

absence of neutrinos is not directly observed, as it is in the case of detecting electron energies.On the other hand, it is generally easier to detect the emitted de-excitation gamma rays,

especially if two or more are emitted in coincidence. As mentioned in [Tor04], however, the twotechniques observing the electron energy spectrum and the emitted de-excitation gamma rays

would provide positive identification for a decay event.It is worth noting that there has been a controversial claim to a measurement of a direct

observation of neutrinoless double beta decay in76

Ge by the Heidelberg-Moscow Experiment,with the half-life of the decay allegedly being measured to be 1.6+1025years [Kla01] (with an

estimated neutrino mass of about .39 eV. Since the methods of background subtraction appliedto the experimental data appear to be inadequate in determining a definite observation of

neutrinoless double beta decay, this result has been refuted, and a definitive measurement has yetto be reported.


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Experimental Design the Majorana Experiment

There are numerous possible experimental apparatus and configurations thereof that canbe used to obtain a definitive measurement of neutrinoless double beta decay. I choose to detail

only one of them, that of the proposed Majorana Experiment.

The Majorana experiment consists of multiple phases designed to study and optimizeseveral experimental details, such as expected levels of background and detector configuration,culminating in the final phase of the experiment, which is expected to consist of an arrangement

of active high-purity Germanium detectors of total mass 120 kg (previously expected to be 500kg; the revised design will be capable of being extended).

The first of the two preliminary phases of the experiment is currently in operation atDuke University. This phase, the Segmented Enriched Germanium Array (SEGA) consists of a

segmented 1.2 kg high-purity Germanium detector enriched to 85% in 76Ge (the naturalabundance being 7.8%). The intent of SEGA is to evaluate the method of background reduction

proposed for the final phase of the Majorana experiment, namely, the use of a segmenteddetector to distinguish between single-site events corresponding to double beta decay

candidate signals (the distance the emitted electrons travel is at most a few millimeters) andmulti-site events, that is, the occurrence of signals in multiple segments of the detector atapproximately the same time generally associated with background. Pulse-shape discrimination,

a technique applied to distinguish between the types of particles generating the signal in thedetector, is also being evaluated. A schematic cut-away diagram of SEGA is included as Figure

1 in the Appendix.A further phase is also in development. Entitled the Multiple Enriched Germanium Array

(MEGA), it consists of 16 segmented high-purity Germanium detectors it is designed to furtherinvestigate the optimal detector configurations and background reduction techniques. Due to its

larger size and requirements for shielding, MEGA is a closer configurational approximation ofthe final Majorana Experiment. A schematic of the MEGA apparatus is presented as Figure 2 in

the Appendix.The Majorana experiment intends to make use of the ability of one of the Germanium

isotopes,76

Ge, to undergo double beta decay, enabling the massive detector array to servesimultaneously as the radioactive source. This is enabled by the proposal to enrich the desired

quantity of Germanium in76

Ge using existing facilities in Russia, which is purportedly the onlylocation with the appropriate equipment to enrich such a considerable amount of Germanium.

The Majorana phase of the experiment would occur underground, and be in all regardsother than size much like the MEGA experiment in design. The energy spectra would be

narrowly gated about the 2039 keV decay energy of 76Ge. A rendition of the proposedexperimental setup is shown in the Appendix as Figure 3.

Goals of the Majorana Experiment

The original goal of the Majorana Experiment in terms of data acquisition was to run fora period of 5 years with 500 kg of enriched Germanium (hence, have 2500 kg-years of

Germanium to measure). With the reduction of Germanium proposed for the commencement ofthe experiment to 120 kg (as the design has been altered to enable expansion of the detector after

data acquisition has started, this is not anticipated to be the amount of Germanium in theapparatus for the entire experiment), the original sensitivity to neutrino mass has been reduced


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from 50 meV to at most 200 meV, with the focus now being on refuting the purported result ofHeidelberg-Moscow.

References

[Boe87] F. Boehm and P. Vogel. The Physics of Massive Neutrinos. Cambridge Univ.Press, Cambridge, 1987.

[Hor02] M. Hornish. Double Beta Decay of100

Mo and150

Nd to Excited Final States.

Ph.D. thesis, Duke University, 2002.

[Kla01] H. Klapdor-Kleingrothaus, et al. Modern Phys. Lett. A, 37(2001), 2409.

[Tor04] W. Tornow, representing the Majorana Collaboration. Private communication inform of Majorana Experiment Draft Proposal.


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Appendix

Figure 1: SEGAThis is a cut-away view of the SEGA detector, including lead shielding (at top) and

cryostat (the liquid nitrogen dewar for which is shown at the bottom of the figure). The actual

segmented Germanium crystal is the cylindrical gray object encased in the shielding. Source:http://www.wipp.ws/science/DBDecay/SEGAandMEGAdescription.pdf.

Figure 2: MEGA


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A cut-away diagram of the MEGA detector. Visible are the lead shielding and thecryostat. Source: http://www.wipp.ws/science/DBDecay/SEGAandMEGAdescription.pdf.

Figure 3: MajoranaA cut-away diagram of the proposed Majorana experiment (shielding not shown). The

highly enriched, high purity detectors are located in the center; dewars for the cryostat are shownat the sides. Source:

http://www.int.washington.edu/talks/WorkShops/Neutrino2000/WorkingGroups/Underground/DeBraeckeleer_L/ht/07.html

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