1207
Progress of Theoretical Physics, Vol. 87, No.5, May 1992
Neutrino Emitting Modes of Double Beta Decay
Masaru DOl and Tsuneyuki KOTANI*
Osaka University of Pharmaceutical Sciences, Matsubara 580
* Osaka Gakuin University, Suita 564
(Received December 19, 1991)
Four types of nuclear double beta decays accompanied with emission of two neutrinos are
compared: (1) capture of double atomic electrons, (2) capture of one atomic electron and emission of
one positron, (3) emission of two positrons and (4) emission of two electrons. Under the assumption
that a charge distribution is uniform inside nucleus, bound state solutions of the Dirac equation are
given analytically by using transcendental functions. Practically, both captures of the K and L/
electrons are taken into account.
Half·lives for the 0+ ... 0+ transitions are calculated by using relativistic wave functions and are
shorter than those by solutions of the Schrodinger equation. It is also shown that the approximated
non· relativistic Coulomb correction factor used previously is not appropriate to discuss positron
energy spectra and total transition rates. Concerning the capture process of the atomic electron, the
124Xe double beta decay is the shortest half· life, if there is no anomalous cancellation of the nuclear
matrix elements.
§ 1.
Introduction
Much interest has been drawn to the double beta decay of the nucleus, because
this decay without emission of a neutrino (the 0)) rpode) is a sensitive probe to the
theory beyond the standard electroweak model. The 0)) mode takes place only when
the neutrino is a Majorana type with finite mass. We can also extract some information on the structure of the weak interaction such as the strength of the right-handed
current and the existence of right-handed heavy neutrinos. On the other hand, the
double beta decay emitting two neutrinos (the 2)) mode) is allowed even within the
standard model. Importance of the 2)) mode is to get some information on nuclear
matrix elements relevant to the 0)) mode, in addition to the interest in the nuclear
physics itself, that is, how to treat the intermediate nuclear levels theoretically.
Laborious efforts by experimentalists have confirmed the 2)) mode, but no decisive
indication of the 0)) mode has yet been reported.!)
So far, most attention has been paid to the 2)) mode of (3-(3- decay,
the «(3- (3-b mode;
(1'1)
where A and Z are the mass and atomic numbers of the daughter nucleus, respectively. The (3-(3- decay has advantages that the available kinetic energy is relatively
large and that the Coulomb attraction makes the overlapping of the electron wave
function with the nucleus increase to some extent. But experimental sensitivity in
searching for this event is limited mainly by the presence of electron background.
Detailed theoretical studies have been done, 'for example, in Refs. 2) and 3), each of
which will be referred to throughout this paper as I and II, respectively.
Another possible process is the (3+(3+ decay,
1208
M. Doi and T. Kotani
the (/3/3)2V mode;
(A, Z +2)--t(A, Z)+2/3++2ve.
(1·2)
This decay is relatively easy to be separated from the background contaminations.
However, this advantage is considerably diminished by the smaller available kinetic
energy in comparison with the /3-/3- decay, as shown in Tables 1.1 and 11.1 of I, and
also by the Coulomb repulsion on the positron.
If the /3+ /3+ decay takes place, there should accompany two following decays: The
capture of one bound electron is
the (e/3)2V mode;
(1·3)
and the capture of double bound electrons is
the (ee )2V mode;
(1·4)
It has been pointed oue) that these capture processes are more favorable than the
/3+ /3+ decay, because their available energies become larger and the suppression by the
nuclear Coulomb repulsion on the positron is avoided. From these viewpoints,
experimental investigations of these modes have revived recently.5),6)
Some theoretical studies have been done for the (/3/3)2V, (e/3)2V and (ee )2V
modes. 4),7)-9) In previous calculations, the plane wave for the positron and the
Schrodinger wave function for the captured electron have been used. The Coulomb
correction tQ the positron was taken into account by using the energy independent
multiplicative factor which was first introduced for the electron case by Primakoff
and Rosen. 10) The use of this approximation is not adequate for the positron, because
the Coulomb repulsion suppresses the energy spectrum in the lower momentum region.
Furthermore, it has been known for the (/3- /3-)2V mode that the relativistic wave
function gives the larger decay rate than the use of the Schrodinger wave function.
The reason for this difference is as follows: It is well· known that the solution of the
Schrodinger equation in the Coulomb field of a point charge Ze has no singularity at
the origin r=O, while the Dirac equation for the j=1/2 state gives singularity like
(pr )71-1 for the continuum state ll ) or (r/ao)71-1 for the bound state/2 ) where P is a
momentum, ao is the Bohrradius and Y1 =j1-(aZ)2 , a being a fine structure constant.
One of our purposes is to evaluate contributions due to this difference. Haxton and
Stephenson have made some rough estimates for it. 13 )
In order to avoid this· singularity at the origin, some model of the extended
nucleus should be introduced. It is generally expected that the Dirac wave function
for a bound electron does not seriously depend on the form of the nuclear charge
distribution, because the nuclear radius (R) is much smaller than the Bohr radius and
also the wave function is restricted by the normalization condition. Therefore, the
electrostatic potential based on the uniform charge distribution will be assumed inside
nucleus (r~R), because the solution of the Dirac equation can be expressedanalytically by using transcendental functions. Since this solution has been expressed only
by the integral form previously/4) their derivation will be shown in Appendix B of this·
paper. The screening correction due to the atomic charge is not taken into account,
because· it does not change the characters of these solutions drastically.14)
In the previous investigations, the K shell electron has been considered. It will
Neutrino Emitting Modes of Double Beta Decay
1209
be confirmed in Appendix B that contributions from the higher shells are small even
for the Dirac solution with the extended nucleus. The captures from both the K and
LJ shells will be taken into account.
In this paper, we shall make the systematic study of the (/3(3)2)), (e/3)2)) and (ee)2))
modes in conjunction with the (/3- /3-)2)) mode. For electrons and positrons, we shall
use the relativistic wave functions. The decay formulas for the 0+ -> 0+ transition are
derived by using the second order perturbation. It will be shown in Appendix A that
the 0+ -> 2+ transition rate is small. The OJ) mode will be discussed in the forthcoming
paper.
The necessary properties of the Dirac wave functions are summarized and
discussed in § 2.. The .detailed characters of the bound state for the extended nucleus
will be given in Appendix B. The decay rate formulas will be compared in § 3, and
their brief derivations are mentioned in Appendix A. In § 4, the energy spectra and
the relative contributions among four different modes will be discussed numerically.
Finally, the numerical comparisons with the previous estimations are discussed in § 5.
§ 2.
Wave functions of charged lepton
Let us briefly summarize the necessary characters of the Dirac wave functions for
both the bound and emitted charged leptons. It is enough to consider only the inner
wave functions for r < R, because their overlapping with the nucleus is relevant.
Let us first discuss the bound electron whose energy Ce is less than an electron
mass me. The electron capture from the s·states is most probable, because the inner
s-wave does not vanish at the origin. The normalization constant of its radial part
is expressed as Bn"tc with K= -1, which is defined in Eq. (B·30) of Appendix B. Here
a radial quantum number is denoted by n' and the quantum number K is related to the
total angular momentum jtc=IKI-(1/2) with K=±l, ±2, ... , -n, where n is the
principal quantum number, n=k+ n' with k=IKI. Then let us introduce a dimensionless quantity Jl n',tc defined as
(2'1)
This Jl n',-l means a probability that an electron in the s-state is found in the volume
1/me3 around the origin.
The wave function for the K shell (n'=O, K=-l; ls1!2) gives the largest Jln"tc, as
it is well known. For the LJ shell (n'=l, K= -1; 2S1!2), this probability is suppressed
by about 1/8 even in the relativistic case, as shown in Appendix B. The L II (n'=l,
K=l; 2P1!2) and Lm (n'=O, K=-2; 2P3/2) shells are further suppressed because their
waves are pushed outside due to the centrifugal energy for the non·zero orbital
angular momentum or the s·wave inner solution of the LII shell is connected with the
so-called small component of the Dirac outer wave function. Thus both captures
from the K and LJ shells will be taken into account for the (e/3)z)) and (ee)2)) modes.
Let us next turn to the subject of the continuum energy state where the electron
(positron) energy c is larger than me. The wave functions for the emitted positron
are obtained from the case of the electron merely by changing a sign of Z. Normali·
1210
M. Doi and T. Kotani
zation constants of an inner radial wave function are denoted as A-k(c) and A+k(c),
see Eqs. (3 -1- 21) ~ (3 -1- 23) of I. It is convenient to express them by products of two
factors,
(2-2)
The factor R±k(C) reflects the charge distribution inside the nucleus and is expressed
by
R ±k()
C
me D2 ( )
2c
±k C ,
C =+=
(2-3)
where D±k(C) take values around unity within a few percent errors, so that R-k+ R+k
::::::1, except in the low momentum region, see Fig. 2 of II. The details of D±k(C) for
positrons are obtained from those in Appendix A of II by changing the sign of Z.
The other factor Fk-1(Z, c) in Eq. (2-2) is the relativistic Coulomb factor:
Fk-1(Z, £)=CF,k_dk_l(C)(me/p)2k-l(c/m e)2 rd ,
(2-4)
where Yk=j e-(aZ)2 and the constant factor CF,k-l is defined as
CF,k-l =2 7fa IZI(2 a IZI meR)2<r.-ll[ 2k(2k
r(2Yk-1)!!
+ 1)
J2
(2-5)
The important difference between an electron and a positron appears through the
Coulomb factor dk-l(C) in Eq. (2-4) which is defined as
dk-1(C)= 217[ lyll-2r· e 1r IYISplr( Yk + iy)1 2 ,
(2-6)
Energy dependence of do(ep)
for positron
0.3
--- -----
where y=aZ(c/p) and Sp is a discriminator between an electron (Sp
= + 1) and a positron (Sp= -1). This
dk-1(C) is so defined as
-----78
Kr
0.2
...--
1.0
--.-
for an electron,
for a positron,
----------
2.0
«,- m,)/m,
(2-7)
124.Xe
3.0
4.0
Fig. 1. Energy dependence of the Coulomb factor
do(E) in Eq. (2·6) for the positron, ct. Fig. 1 of
Ref. 3) for the electron. Two cases are drawn
for parent nuclei 78Kr and 124Xe. The solid
lines are for the Dirac solution, the dot-dashed
lines for the Schrodinger solution doNR in
Eq. (2'8) and the dashed lines for the approximated one dlR in Eq. (2'9).
and the probability of the positron emission in the low momentum region
becomes smaller. The e1rIYISp factor in
dk-l(C) suppresses one positron emission
by e-21rIYI ~ e-21raIZI ~ 1/10 in the large
momentum region in comparison with
one electron emission.
Let us introduce the non-relativistic
(NR) approximation where the solution
of the Schrodinger equation with the
Neutrino Emitting Modes of Double Beta Decay
1211
point nuclear charge is used. Practically, it is realized from the above formulas by
taking the limits rk~ k, R- k(E)=l and R+k(E)=O. For the s-wave case, we have
doNR(E)=SfJ/(1-e-21CIYISn) , CP'~=2;'WIZI
and
J1~~I=(aIZlnJr.
(2'8)
In order to perform the phase space integral analytically, Primakoff and Rosen lO )
introduced a further approximation such that y in dONR(E) is replaced by an energy
independent constant, i.e.,
(2·9)
This will be called the PR approximation hereafter.
Energy dependence of do(E), dONR(E) and d OPR for a positron is shown in Fig. 1 for
two parent nuclei, 78Kr and 124Xe. We can say that dONR(E) has a similar energy
dependence to does) over the whole region, although dONR(E) is somewhat larger (at
most about 10 %). On the other hand, the deviation of d/R from do(E) is serious in the
small momentum region of the positron, because do PR remains non-zero in the p~ 0
limit. While, the use of do PR is not so wrong in the P-P- decay, because do(E)~ 1 in
the p~ 0 limit for an electron, d. Fig. 1 of II. Thus the positron energy spectra
derived by using doPR are much different from those by do(E), and the decay rates are
overestimated in the PR approximation.
§ 3.
Decay formulas for the 211 modes
The decay formulas for the O/~O/ transition in various 21/ modes will be
presented in this section by using the second order perturbation. First let us summarize main assumptions and approximations:
1. The non-relativistic impulse approximation is used for the hadronic current.
2. The captured bound electrons are in either the K or LJ shells, as discussed in the
preceding section.
3. Only non-vanishing leading contributions to the decay amplitudes are retained.
That is, the s-wave is used for all leptons, and only first terms of the power series
expansion of radial wave functions are kept. The recoil effect of nucleons is
neglected.
4. Masses of neutrinos are neglected in calculating the phase space integral in the
final states.
5. The recoil energy of the daughter nucleus is neglected because it is of the order of
0.1 ke V at most.
The total kinetic energy released in each of the decay modes is defined as follows:
the (P- P- )2V mode,
the (PPb mode,
the (ep)2V mode,
the (ee hv mode,
(3·1)
1212
M. Doi and T. Kotani
where Mi and Mf are masses of parent and daughter nuclei, respectively. One of
necessary conditions for the double beta decay to take place is that the total kinetic
energy release is positive. We see that the (/3/3)21) mode always accompanies the
(e/3)2V and (ee)2V modes.
Half-life formulas for the 0/-->0/ transition are expressed in a unified form for
three (/3/3)2V, (e/3)2V and (ee)2V modes, namely, a product of common nuclear matrix
elements and different integrated kinematical factors,
(3-2)
where a stands for either one of /3/3, e/3 and ee corresponding to three modes. They
are derived from Eqs. (A -17) and (A - 20) of Appendix A. The formula for the
(/3- /3-)2V mode will be discussed at the end of this section.
The common nuclear matrix elements are defined as follows,
Mh2f)
( fl-o
)=2:( Mh2fd) ,
a
fJ.a
(M}.2V)
fl-oF
)=2:( M}'~V)) ,
a
(3-3)
fJ.a
where fJ.a stands for an excitation energy of theintermediate nuclear state
energy E a , i.e.,
la> with an
(3-4)
The Gamow-Teller and Fermi types of the nuclear matrix elements are defined
respectively by
(3-5)
(3-6)
where nand m refer to nucleons in the nucleus as a result of the non-relativistic
impulse approximation for the hadronic current. It should be noted that the closure
approximation has not been made, as it will be explained below Eq. (A-20) in
Appendix A. Concerning two constants gv and gA, the conserved vector current
hypothesis requires gv to be so normalized that gv=cosBe=0.9744 where Be is a
Cabbibo-Kobayashi-Maskawa mixing angle between d and s quarks. The axial
vector form factor gA is taken to be gA/gv=1.261.
Let us next discuss the integrated kinematical factors Gha; which include the
phase space integral over emitted particles. They are expressed as a sum over the
product of three factors,
(a)-K ~
7\T(a') T(a')(T. )
GGT211~a'lV2J.1 1211
a',
(3-7)
where a' of the sum over mode dependent constants NJ~') and integral factors IJ~')
means no capture or to capture the atomic electon from the K or/and L1shell in the
a mode, and K 2V is a common mode-independent constant,
1213
Neutrino Emitting Modes of Double Beta Decay
K2V=C~2 )C40 )a2v=9.563
0
10- 24 [yr- 1] .
(3 8)
0
Here a2v is defined in Eq. (3 2 2a) of 1.
The mode dependent factors NiP are defined as
0
0
7I.T(PP)-(C
lV2v
F,O )2 ,
(3 9)
0
7IT(ep,K)=2CF,O (2.".2)'Jl
_
'"
0, 1,
lV2v
71 T(ee,KK) = (2 .".2)2('Jl
)2
lV2v
"
0,-1,
7I.T(ep,L)=2CF,O (2.".2)'Jl
_
n,
1, 1,
(3°10)
71 T(ee,KL) = 2(2 .".2)2'Jl
'Jl 1,-1,
lV2v
"0,-1
(3°11)
lV2v
where 'Jl n',K and CF,O are given in Eqs. (2 °1) and (2 °5), respectively. There are
systematic rules among N~~'). They contain 27[ 2'Jl n ',K or CF,O corresponding to one
captured electron or one emitted positron, respectively. This factor 27[2 is introduced
if one positron is replaced by one bound electron, because it originates from the state
density for one particle in the final state, i.e., dp/(27[)3~ p2dP/(27[2). The factor 2 in
N~~P,K) or NW,L) comes from the existence of two possible spin states in the K or L/
shell. The other factor 2 in N~~e,KL) is due to the fact that only anti-parallel spin states
of the K and LI electrons are allowed if all leptons are in the s-wave state. We have
neglected the case where both of captured electrons are in the L/ shell.
The remaining factors I~~') in Eq. (3 °7) are defined as follows,
p
ne )( To) = me -2(1+271)
ff
([)5
h2Vdo(€P1)do(€P2)R1,1(€P1)RI,I(€P2)(€PI€P2)2r'd€Pld€P2 ,
(3 °12)
n~p,KOrL)( T 1 )= me -(1+2rll
f
([)5
h2Vdo(€p)RI,I(€p)€p2r'd€p ,
(ee,KKorKL)( rr'
) h,5h 211,
I 2v
12
(3°13)
(3°14)
-\,.V
where ([) and h2V are related to neutrinos and will be explained later. Note that the
kinetic energy releases TI and T2 in Eqs. (3°1) depend on the different binding energies
of the K and L/ shells. The Coulomb effect on one positron gives a factor
doRI,I€p2n-1p-r, where
(3°15)
by using R±1(€P) given in Eq. (2°3).
Concerning neutrinos, let us define the total kinetic energy ([) which is carried off
by neutrinos,
-1
([)-
TO-(€PI + €P2-2me)/me ,
TI-(€P- me)/me,
T2 ,
for the (/3/3hv mode,
for the (e/3)2V mode,
for the (ee )2V mode.
(3°16)
The function h2v is the integral over both the phase space· and the square of energy
denominators in the second order perturbation,
(3 °17)
1214
M. Doi and T. Kotani
where
.
( Kn)=l[~±(O(2X-1)J
Ln
2 me
(3·18)
Here the charged leptonic parts are eI2=epI- ep2, ee+ ep and ee2- eel for the (/3/3)2V,
(e/3)2V and (ee)2V modes, respectively. This h2v is so defined that it tends to unity in
the limit fla~OO. Strictly speaking, the function h 2V depends on intermediate nuclear
states la> through the excitation energies fla in Eq. (3·4). However, h2V changes so
weakly under the variation of fla, as shown in Fig. 2. In fact, fla is generally larger
than the lepton energy differences, Kn and Ln. Therefore, fla in h2V can be replaced
by some average value <fla> independent of the intermediate nuclear state la>.
Furthermore, the ep dependence of h2v is
4.0
shown in Fig. 3 by taking <fla>=20.0 in
1'. dependence of h,.
the (e/3)2V mode. It varies scarcely for
the range of ep from me to (TI+ 1) me.
Therefore, we shall assume h2v = 1 in the
3.0
NR and PR approximations.
The decay formulas of the (/3/3)2V
and (/3- /3-)2V modes are similar formally.
2.0
Different points are as follows: (1) The
sign of Z in the Coulomb potential for
electrons should be changed into (- Z)
1.0
for positrons. The effect of this change
is minor for RI,1 in Eq. (3 ·15) of s-wave
20
o
40
60
80
case. However, the factor do(ep)
definied in Eq. (2·6) gives the repulsive
Fig. 2. Dependence of h,v on nuclear excitation
character for the positron, while attracenergy f.1.a, see Eqs. (3'17) and (3'4). The
tive for the electron, as discussed in § 2.
(e/3b mode is shown where the parent nuclei
(2) Different isospin operators appear in
7BKr captures the K·electron and the positron
energy C:p is either me or 4me.
the nuclear matrix elements in Eqs. (3·5)
5
.----r----.----r----.---~----._--~--__.
Energy dependence of
for positron
o
-
h,.
3
I
~
2
o
1.0
2.0
3.0
4.0
(e,-m,)/m,
Fig. 3. Positron energy dependence of the function h,v in the (ef3),v mode where the parent nuclei 7BKr
captures the K·electron. The energy level of the intermediate nucleus is fixed at <f.1.a>=20.0.
Neutrino Emitting Modes of Double Beta Decay
1215
and (3·6), namely, r- and r+ for the (/3/3)21) and (/3-/3-)21) modes, respectively.
§ 4.
Numerical results
The nuclear matrix'elements in the half-life formula Eq. (3·2) are energy independent and common to three (/3/3)21), (e/3b and (ee)21) modes. Therefore, positron
energy spectra and relative contributions from these modes can be discussed numerically. Concerning the (/3-/3-)21) and (/3/3)z1) modes, their relative characters can be
compared, although their nuclear matrix elements are different.
4.1. Energy spectra
Let us first discuss the positron energy spectrum derived from Eq. (3 ·13) for the
(e/3)21) mode. It is shown in Fig. 4 for parent nuclei 78Kr. It has a peak around 1/4
Positron energy spectrum in the (e,B) mode
78 Kr
1.0
(0+ ..... 0+)
0.8
Ov mode
2v mode
,
0.6
,,
/',
0.4
PR-approx. '"
0.2
,,
"
1.0
2.0
3.0
(cp-m,)/m,
Fig. 4. Positron energy spectrum in the (ef3)2" mode for the 0+ -> 0+ transition of the parent nuclei 78Kr.
The solid line is drawn for the exact Dirac case in Eq. (3·13), while the dashed line for the PR
approximation. The spectra are normalized to unity.
Single positron energy spectrum
in the (f3f3hv mode
3.0
\
\
\ PR-approx.
2.0
1.0
o
0.5
1.0
1.5 To
2.0
(ep-m,)/m,
Fig. 5. Single positron energy spectrum in the
(Sf3)2" mode for the 0+ -> 0+ transition of the
parent nuclei 78Kr, see Eq. (3'12). Other notation is the same as in Fig. 4.
of the maximum kinetic energy release
T!. The probability to find the zero
momentum positron becomes zero,
because of the Coulomb repulsion character. . The spectrum also becomes
gradually zero in the maximum energy
limit, because of the presence of a/ in the
decay formula. For comparison, the
spectrum of the PR approximation is
shown in Fig. 4. It does not give a
correct distribution especially in the low
energy region, while the NR approximation gives almost the same spectrum as
the exact one.
Concerning the (/3/3)21) mode, the single positron energy spectrum from
1216
M. Doi' and T. Kotani
Sum energy spectrum in the (f3f3) mode
78 Kr
2.0
( 0+ -+ 0+ )
Oll mode
".-- ....
/
/
I
1.0
I
I
I
I
I
I
I
0.5
1.0
1.5
To
2.0
(,,1 + ',2 - 2m,)/m,
Fig. 6. Positron sum energy spectrum in the (flflb mode for the 0+ -> 0+ transition of the parent nuclei
78Kr, see Eq. (3 '12). Other notation is the same as in Fig. 4.
Eq. (3·12) is shown in Fig. 5. This is quite different from the electron spectrum in the
(/3- /3-)22/ mode where the probability of finding zero momentum electrons remains
finite, as seen from Fig. 6.1 of 1. It comes from the different behavior ofdo(c) in
Eq. (2' 7) for an electron and a positron. On the other hand, the sum energy spectrum
of two positrons shown in Fig. 6 does not have such a remarkable difference in
comparison with the (/3- /3-)22/ mode, although the maximum is shifted to the higher
energy region, d. Fig. 6.4 of I.*) This is because the characteristic feature of the sum
energy spectrum requires the zero probability in the low energy limit, and in addition
the ul term in the decay formula of Eq. (3 ·12) forces the spectrum gradually down to
zero in the high energy region. The NR approximation reproduces these spectra
almost correctly again. On the other hand, the PR approximation is not good for the
positron case as shown in Figs. 5 and 6, although it was allowable for the electron
case.
Emitted positrons in the (/3/3)22/ mode have the same angular correlation as
electrons in the (/3-/3-)22/ mode; namely [1 - (PIP2!clCl)cOS e] where e is an opening
angle between two electrons (positrons). The reason why two positrons (electrons)
are emitted in favor of the backward direction is that they are both in the s-wave
states and likely to be emitted in opposite direction in order to satisfy the angular
momentum conservation.
4.2. Integrated kinematical factors
The integrated kinematical factors Gift in Eq. (3·7) are computed fOF eight
typical isotopes which have relatively large total kinetic energy releases. They are
listed in Table I for the (ee)z2/ and (e/3)22/ modes. The (/3/3)22/ mode is given in Table
II where the (/3- /3-)22/ mode for 82Se and 136Xe is shown as examples, d. Table I of Ref.
3).**)
First, let us examine three (ee )22/, (e/3)22/ and (/3/3)22/ modes.· We see that the (ee >zv
*) Note that inl, Fig. 6.4 is for the 0+->0+ transition, while Fig. 6·2 for the 0+->2+ transition.
**) The numerical values of GeT of the (fl- fl-)2V mode in Table II are a little larger than those in Table
I of Ref. 3), because gv and gA given below Eq. (3'6) in this paper are a little larger than'previous values in
Ref. 3), ·gv=0.9729 iJ-nd gA/gv=1.254.
Neutrino Emitting Modes
0/ Double Beta Decay
1217
Table I. Integrated kinematical factors G&'!' and G&1' in Eq. (3'7) for the O+~O+ transition in the (e{J)2V and (ee)2V modes. Kinetic
energy releases T, and T2 in Eqs. (4·6) and (4' 7) are listed only for the K·electron capture in units of m, and the atomic mass
difference - Q in Eq. (4'2) is taken from Wapstra and AudL 15' The abbreviations NR and PR mean the use of the NR and
PR approximations, respectively. The averaged constant <I'a> is taken to be 20.0.
nucleus
~Ni
~~Kr
~7Ru
I~Cd
l~Xe
l~gBa
I~Ce
l~r
-Q[m,]
3.771
5.630
5.325
5.436
5.608
5.053
4.716
3.608
1. 750
3.595
3.272
3.373
3.527
2.966
2.622
1.476
T,[m,]
G&1'[yr-']
Gif!'(NR)[yr-']
G&'!'(PR)[yr-']
T,[m,]
G&'i'[yr-']
G&1'(NR)[yr-']
2.940' 10-24 1.174' 10- 21 1.148 • 10- 21 1.970 • 10-21 4.353' 10- 21 1. 387 • 10-21 6.399' 10- 22 1.692. 10- 23
1. 762 • 10-24 .5257, 10- 21 .3541 • 10-21 .4940 • 10-21 .7798' 10- 21 .2174' 10- 21
4.646' 10-
24
3.729
.9164 • 10-
2
'
5.560
.7035 • 10-
21
5.219
.9945 • 10-
21
5.309
1.597' 10-
21
5.446
2.413' 10-22 .6965' 10- 23
2o
3.988' 10- 20 4.575 • 10- 20
.5229, 104.878
4.289' 10- 23 1. 957 • 10-21 6.936 ~ 10- 21 1.573' 10- 20 5.101 • 10- 20 4.134'
.8757' 10c22 .1087 • 10-23
21
4.528
io-
3.344
2.486' 10- 23 .8280' 10- 21 2.018' 10-21 .3710 • 10-20 .8562 • 10-20 .6156 • 10-20 .5246 • 10-20 .3040 • 10-20
Table II. Integrated kinematical factor GIt!' in Eq. (3'7) for the O+~O+ transition in the ({J{J)2V and ({J- {J-)2V modes. Other notation
is the same as in Table I.
({J- {J-)2V mode
({JfJ)2V mode
nucleus
~~Kr
~~Ru
l~Cd
Ig:Xe
l~gBa
l~Ce
To[m,]
1.630
1.325
1.436
1.608
1.053
<I'a>
20.0
20.0
20.0
20.0
20.0
~Se
l~Xe
.7162
5.861
4.851
20.0
19.73
20.0
3.445' 10-
25
GIt!'(NR)[yr-']
1.561 • 10-
25
G!!'!'(PR)[yr-']
25.95' 10- 25 33.23' 10-26 48.21' 10-26 7.938' 10-25 28.09' 10- 27 141.9' 10-29 2.388' 10- 18 .8447, 10- 18
GIt!'[yr-']
26
2.737 • 10.8271 • 10-
26
4.991' 10-
26
1.213 • 10-
26
1. 205 • 10-
25
.2056' 10-
25
1. 211 • 10-
27
.1737, 10-
27
1. 414 • 10-
29
.1710' 10-
29
4.519' 10-
18
5.010' 10- 18
2.078' 10-
18
.7822, 10- 18
Table III. Comparison of integrated kinematical factors G&'9 in Eq. (3'7) for the O+~O+ transition in the (ee)2V, (e{JJzv, ({J{J),v and
({J- {J-)2V modes. Only the case of the K·electron capture is listed. The kinetic energy release Ta , the mode dependent constant
factor Ni~' and the integral factor Ii~' are defined in Eqs. (4·4-7), (3'9-11) and (3'12-14), respectively.
g~Se
~gKr
l~:Xe
l~Xe
mode
(ee)2V
(e{J)2V
({J{J)2V
({J- {J-)2V
(ee)2V
(e{J)2V
(m;v
({J- {J-)"
Ta[m,]
5.560
3.595
1.630
5.861
5.446
3.527
1.608
4.851
.03012
.8301
5.719
7.049
.8494
11.16
36.64
55.75
N,(a)
2v
KNi~'[yr-']
pal
2v
G!!''![yr-']
2.880' 10-25 7.938' 10-24 5.469. 10- 23 6.741' 10- 23 8.122' 10-24 1.067. 10-22 3.504' 10-22 5.331' 10-22
5.342' 103
1.539' 10-
21
1. 298 • 10 2
1.030' 10-
21
6.298' iO- 3
3.445' 10-
25
6.705' 10'
4.519' 10-
18
4.817' 103
3.913' 10-
20
3.508' 10'
3.743' 10-
21
3.440' 10-'
1.205 • 10-
25
9.398' 103
5.010' 10- 18
mode has the largest decay probability and the (/3/3)211 mode is the smallest for the
definite parent nuclei. In order to understand these tendencies qualitatively, numerical values of NJ~), IJ~) and Ghat for parent nuclei 78Kr and l24Xe are shown in Table III
together with the cases of the (/3- /3-)~11 mode for their neighboring nuclei 82Se and
136Xe. Only the K-electron capture case is listed. The variation is NJ~P,K) /2NJ~e,KK)
p
2
~ 2NJe ) / NW,K) ~ CF,O/ (27r 'Jl 0,-1) ~ 13 and 6 for 78Kr and 124Xe, respectively, from Eqs.
(3·9)~(3·11). Note that this ratio is l/(aZ)2 in the NR approximation, as seen from
Eq. (2·8). Physically, this mode d~pendence can be understood from the fact that the
overlapping of the created charged lepton waves with nucleus is greater than the one
of the bound electron outside nucleus.
On the other hand, the greater decrease of integral factors IJ~) is seen from Table
1218
M. Doi and T. Kotani
III as the number of emitted positrons increases. This overcomes the increase of
Ni~) mentioned above. This greater variation of li~) comes from the following three
reasons: (1) The available total kinetic energy decreases by 2me when one captured
electron is replaced by one positron, as seen from the definition of Ta in Eq. (3 1). (2)
Let us denote the number of charged leptons in the final states as n and the available
kinetic energy release as T, respectively. Then the integral factors li~) are roughly
given by relations,
0
for relatively low T, say T
for higher T, say T >7.
<2 ,
(4°1)
These are derived from the plane wave approximation for positrons. (3) The suppression due to the Coulomb repulsion factor do(cp) becomes important for the smaller
T, see Fig. 1. In summary, it can be said that as the number of positrons increases
in the final states, the total kinetic energy releases become smaller rapidly and
accordingly the integral factors Ii~) decrease from the second and the third reasons.
Next let us compare the (/3/3)21' and (/3- /3-)21' modes. At first glance, the Coulomb
repulsion for a positron contrasts with the attraction for an electron. This difference
makes the decay rate in the (/3/3)21' mode smaller by a factor 10-2 in comparison with
the (/3-/3-b mode, see below Eq. (2 ° 7). However, as it can be seen from Table III,
the integral factor Iiep) is smaller by 10-7 than lie-po). This large suppression can be
understood roughly by the smaller kinetic energy release, i.e. (1.63/5.86)1;:::;1.29 10- 4
from Eq. (4 °1) for the ratio of 78Kr and 82Se cases. Thus the detection of the (/3/3)21'
mode becomes difficult, although nuclear matrix elements are different between these
two modes.
0
4.3.
Total kinetic energy release
There is some problem in determining the total kinetic energies Ta. The double
beta decays are processes inside the nucleus essentially, and the energy balance should
be satisfied among the parent and daughter nuclei and leptons. The Ta in Eq. (3 °1)
have been derived from this point of view. However, it is difficult to measure nuclear
masses Mi and Mf themselves, and usually we have to use the atomic masses
measured by-the technique of the mass spectroscopy. In the table by Wapstra and
Audi/5) the Q-value for the double beta decay is given as a mass difference between
two neutral atoms,
Q=J}t(A, Z)-J}t(A, Z +2),
(4°2)
where J}t(A, Z) and J}t(A, Z +2) stand for the atomic masses. Note that this Q value
is defined such that the /3-/3- decay is allowed for the positive Q and the (ee)21' mode
for the negative Q.
The absolute Q-value is of order of a few MeV for the double beta decaying
nuclei. In order to obtain T2 from the Q-value for the (ee)21' mode, corrections due
to other bound electrons are necessary: A relation between the nuclear mass Mf and
the atomic mass J}t(A, Z) is expressed as
J}t(A, Z) = Mf + Zme - B(Z). ,
(4°3)
Neutrino Emitting Modes of Double Beta Decay
1219
where B(Z) stands for the total binding energy of the bound Z electrons. The kinetic
energy release Ta can be given in terms of the atomic masses by comparing Eqs. (3-1),
(4-2) and (4-3),
Tme=Q'+[B(Z-2)-B(Z)] ,
(4-4)
Tome= - Q+[B(Z +2)- B(Z)]-4me,
(4 -5)
Tlme= - Q+[B(Z +2).,- B(Z)]-2me-cb,
(4 -6)
T2me= - Q+[B(Z +2)- B(Z)]-Cbl-Cb2,
(4-7)
where Cbj=me-cej(j=I, 2) are binding energies of captured electrons.
It is a complicated problem to evaluate B(Z) and Cb for high Z atoms. By
assuming the hydrogen like model, the binding energy Cb is calculated and listed in
Table IV of Appendix B. It amounts to the order of magnitude of 50 keV for heavier
atoms. Since the decay rate is proportional to the higher power of Ta as seen from
Eq. (4 -I), we have taken into account those values in Table IV of Appendix B for Cb.
If the inore accurate rate is required in future, we have to consider the correction due
to the screening effect. Also we have assumed [B(Z +2)- B(Z)]=O in our numerical
calculation.
§ 5_
Discussion and conclusions
We have studied the (/3/3)2)), (e/3)~)), (ee)2)) and (/3-/3-)2)) modes of the double beta
decay. It was assumed that the nuclear charge distribution is uniform inside a radius
R. The analytic -solutions for the Dirac wave functions in this Coulomb potential
were derived in Appendix B for a bound state and Appendix D of I for a continuum
state.
The overlapping of the bound electron wave with the nucleus, 'Jl n',K=-l for K and
LI shells, is larger by about factors 1.3 ~ 3.8 in comparison with the Schrodinger
solution for a point charge (the NR approximation), d. Table V. Therefore, the
decay rates of the (ee)2)) mode are enhanced by about 1.8 and 15 for the lower and
higher Z nuclei, respectively, see Table I. This is mainly due to the multiplicative
factor (2 aZmeR/N) 71-1 ~ 1 which originates from the singularity of the s-wave solution
of the Dirac equation for the point charge and comes into the normalization of the
inner wave function through the continuity condition at the nuclear surface r = R, see
Eq. (B-31). A similar factor (2aZm eR)"1-1 in Eq. (2-5) appears for the continuum
state by the same reason. All integrated kinematical factors Gsa; for the Dirac
solution are larger by these multiplicative factors than those for the NR approximation, as seen from Table I and II. Therefore, the NR approximation is not appropriate, when the total half-lives are discussed.
On the other hand, all Gsa; for the (/3/3)2)), (e/3)2)) and (ee)2)) modes are smaller than
that for the (/3- /3-)2)) mode, as seen from Tables I and II. The main reason is that the
available kinetic energy releases are small for the (/3/3)2)) and (e/3)2)) modes, as
discussed in § 4.2. Even in the (ee )2)) mode whose energy release is comparable with
that of the (/3- /3-)2)) mode, G?fP is smaller by an order of 10-2 than Gl!i P->, as seen from
1220
M. Doi and T. Kotani
Table III. This is because the overlapping of the bound electron wave with the
nucleus, i.e. B';",IC=-I, is smaller in comparison with that of the electron in the continuum state, i.e. IA±112, d. Nif} in Table III.
Next, let us discuss how the energy spectrum is affected by different Coulomb
effects on the positron and electron in the continuum state. The Coulomb repulsion
makes the positron spectrum small in the low energy region, while the probability of
emitting electron with zero momentum in the /3-/3- decay remains non-zero, d. Fig. 5
of this paper and Fig. 6.1 of I. These features of spectra <;:an be well reproduced by
the NR approximation do NR in Eq. (2·8). Therefore, this NR approximation can be
adopted, when the Monte Carlo calculation, for example, is performed.
However, the PR approximation in which doPR in Eq. (2·9) is used can be adopted
for electrons, but not for positrons, see Figs. 4~6. The PR approximation gives the
larger decay probabilities for positrons with smaller momenta. In fact, GI!P(PR) is
overestimated by about 8 and 100 for 82Kr and 136Ce, respectively, in comparison with
the correct Dirac case GI!P, as seen from Table II. For the (e/3)2V mode, GifP(PR) is
about 0.4 ~ 1.5 of Ghep, because the overestimation due to the emitting positron is
compensated by the underestimation due to the bound electron. Thus, results in this
paper are different from the previous ones which are derived by using the PR
approximation. 4),7)-9)
The screening effect induced by atomic electron cloud has not been taken into
consideration in this paper. This effect will give a tendency to weaken the long range
character of the Coulomb potentiFll. For charged leptons in the continuum state, the
corrections due to this effect are expected to be relatively larger in the low energy
region, especially for positrons. However, since the probability to find such low
energy positrons is small as shown iri Figs. 4 ~ 6, the maximum of energy spectrum for
positrons is shifted slightly to the lower energy side and the transition rate increases
by an order of 10 %.14) For electrons, these tendencies are opposite: Namely, the
maximum of electron energy spectrum is shifted slightly to the higher energy side and
the transition rate decreases by an order of 2 %. On the other hand, in the case of the
discrete state, if the atomic electron cloud is taken into consideration, the overlapping
between wave functions for the bound state electron and nucleus decreases by an
order of 10 % and also the binding energy becomes smaller by an order of 30 % than
the one in Table IV.14) Therefore, it is necessary to check this correction, when the
half-life is compared with the theoretical estimate in detail. In any case, the characteristic features mentioned in this paper will not be altered drastically.
In summary, when half-lives of the 2v modes are discussed, the integrated
kinematical factors Ghat from the exact Dirac solutions should be used. The halflives for the (/3/3)2V (e/3)2V and (ee)2V modes are longer in comparison with the ordinary
(/3- /3-hv mode. Thus the observation of such modes becomes difficult in the present
accuracy of experiments. However, it is known experimentally that the nuclear
matrix elements for the (/3- /3-)2V mode are smaller than the naive theoretical estimates. 16 ) Therefore, it is desirable to check whether such a situation happens also in
the double electron capture. If it is not, there may be some possibility to observe the
(ee )2V and (e/3)2V modes for the heavier nuclei like 124Xe.
Neutrino Emitting Modes of Double Beta Decay
1221
Appendix A
- - Derivation of the Decay Formula for the 2v Mode--
In this Appendix, we shall show mainly how to derive the decay formula for the
(e/3)2V mode. It will be shown also that the decay rates for the other (/3/3)2V, (ee)2V and
(/3- /3-)2V modes are easily obtained from the above mode by replacing some quantities
appropriately.
The 2 v mode of the double beta decay is well described by the standard ( V - A)
theory of the weak interaction. However, in order to see the effect due to the finite
neutrino mass, we shall assume that a left-handed electron neutrino VeL is expressed
as superposition of mass eigenstate neutrinos N j by using a neutrino mixing matrix
element Uej,
a
(A·l)
The sum. over j runs from j = 1 to j = n for the Dirac type of neutrino, while from j = 1
to j=2n for the Majorana type of neutrino, where n is a number of the generation, d.
Eqs. (3 ·1· 5) of I for the detailed definition. Note that mass eigenstate neutrinos N jL
with mass mj are emitted in the 2v modes. Of course, their masses should be less than
the total available energy Ta defined in Eq. (3 ·1).
The R matrix element R£'{fl for the Oi + ~ lf + transition of the nucleus is expressed
as follows by applying the second order perturbation theory,
R£'{fl=(
Jzy U6U6'Ejj'~
!dxdyjPp(x, y; a)[I- P(Nj , Nj')]
(A·2)
where Oh =/ q12+ m/ , W2=/ q22+ m7', Ee and Ep are energies of emitted neutrinos N j
and Nj', a captured electron and an emitted positron, respectively. The sum of a runs
over all possible intermediate nuclear states la> whose energy is Ea. A statistical
factor for emitted neutrinos N j and Nj' is denoted by Ejj' which is equal to 1 and 1/!2
for j '*' j' and j = j', respectively, and peN;, N j ,) means to take exchange of quantum
numbers (energy and spin) of these two neutrinos.
The leptonic part in Eq. (A·2) is
Lpp(WIX, EpX; OhY, Eey)=[¢,Jwl,.x)rp(l- rs)ep/(Ep, x)]
X
[¢VAW2, Y )rp(l- rs)<Pe(Ee, y)] ,
(A·3)
where <Pe, epee and ¢V} are wave functions for the captured electron, the emitted
positron and the neutrino Nj, respectively. The leptonic wave functions will be
rearranged by using the Fierz transformation such that Lpp in Eq. (A ·3) is expressed
as a product of two parts, one for the neutrinos and the other for charged leptons, see
Eq. (B·2·14) of I.
On the other hand, the hadronic part is defined by
(A·4)
1222
M. Doi and T. Kotani
where hP(x) is a hadronic (V - A) current. The non-relativistic impulse approximation for the hadronic currents is introduced, d. Eq. (3 -1-16a) of I, and nucleon recoil
effect is neglected.
The derivation 'of the decay formula for the (ej3)2V mode proceeds almost parallel
to the case of the (j3-j3-)2V mode, because Eq. (A-2) is quite similar to Eq. (B-2-2) of
1. It is convenient to separate the R-matrix into symmetric and anti-symmetric parts
with respect to quantities related with two decaying protons. This is because the
anti-symmetric part does not contribute to the decay rate when the so-called closure
approxi~ation is adopted. The closure approximation means to use the completeness character of the intermediate nuclear states ~ala><al=l in Eq. (A-2) by replace
ing other quantities depending on the intermediate states with some appropriate
constants. It will be shown that the contribution from this anti-symmetric part is
small independently of the closure approximation.
Then let us rewrite the R-matrix as follows,
(A-5)
where {M2V (a)}n and {M2V(a)}C stand for the non-vanishing and vanishing terms under
the closure approximation, and (lime) is introduced to make them dimensionless.
They are expressed as follows,
(A-6)
and
(A-7)
where the summation operator Ta is defined by
(A-S)
and it is understood that the integrals of x and y extend over the position variables
x and y of wave functions in the leptonic E±f.I. and M±f.I. parts defined below, and that
the sums of nand m also include all nucleonic suffices nand m in the hadronic (X±,
y± land Xkl) and leptonic parts.
Hadronic operators are defined as
(A-g)
(A -10)
Neutrino Emitting Modes of Double Beta Decay
1223
(A·n)
°
The first tensor operator X± is of a rank and consists of the double Fermi and double
Gamow-Teller types of the nuclear operator. The remaining y±l and X kl operators
are of ranks 1 and 2, respectively_ Note that X denotes symmetric under the
exchange of nucleon suffices nand m, while Y is anti-symmetric_
On the other hand, the neutrino parts in Eqs. (A· 6) and (A· 7) are
(A·12)
where
(A ·13)
The charged leptonic parts are
(A· 14)
where
(A·15)
Energy denominators in Eq. (A ·2) are included as the following combinations Ka±
and La± in {MzlI(a)}n and {MzlI(a)}c: Ka± is defined as
Ka±
me
+
me
f.1.ame-(clz+ahz)/2 f.1.ame+(clz+wd/2'
(A ·16)
and La± is obtained from Ka± by changing the sign of WIZ. Here, the excitation
energy f.1.a is defined in Eq_ (3·4) and shorthand notations clz=ce+cp and WIZ=WI-WZ
are used. Note that Ka+ and La+ appear in {MzlI(a)}n, while Ka- and La- in {MzuCa)}c.
This characteristic feature can be understood easily for the case of the s-wave leptons
by dividing the hadronic currents rp(x, y; a) in Eq. (A·4) into symmetric and
anti-symmetric parts with respect to x and y.
Thus far, we have not made any assumptions on lepton wave functions. Now let
us assume that all leptons are in the s-wave state, and only the leading terms of the
power series expansion for their radial wave functions are retained. Then we have
M-P=O and E-P=O, because all lepton wave functions are independent of space
coordinates. Under this assumption, it folows from Eqs. (A·6) ad (A·7) that
{Mzv(a)}n contains three terms, (Ka+ + La+)X+, CKa+ - La+)X- and CKa+ - La+)X kl ,
while {MzlI(a)lc consists of two terms (Ka- + La-) y+ I and (Ka- - La-) y_l. These are
the same as in the «(3-(3-) case and summarized in Table B·l of I.
Let us show that contributions from three combinations, (Ka+ - L a+), (Ka- + La-)
and (Ka- - La-) in Eqs. (A ·6) and CA· 7), can be neglected in comparison with
(Ka+ + L a+), if other coefficients are of the same order. The average excitation
energy f.1.a is expected to be about 20, while wlz/(2m e) and clz!(2m e) are at most 1.5 and
2.5, respectively, for the typical nuclei which take the (e(3)zv mode. Then we have
approximately (Ka+ + L a+) ~4/f.1.a, (Ka+ - L a+) ~2Wlzclz/(f.1.a3meZ), and therefore
1224
M. Doi and T. Kotani
(Ka+-La+)/(Ka++La+)~cIzWlz/(2f1.a2me2)s;.1/50. While, we have (Ka-+La-)~2cI2/
(f1.a zme) and (Ka-- La_)~2wIz/(f1.aZme2), so that they are of the same order of magnitude but smaller than (Ka+ + L a+) by a factor of 10. If these small contributions are
neglected, only one remaining term is (Ka++La+)X+ in {M2U(a)}n of Eq. (A·6). It is
clear that the contribution from {M2U(a)}C is already small independently of the
closure approximation.
Now let us consider the selection rules for the 0+--' J+ nuclear transition.
Nuclear tensor indices are carried only by the X±, y± I and X kl operators defined in
Eqs. (A ·9)~(A ·11), as a result of the s-wave character of lepton wave functions.
The nucleus which induces the 0+ --. 1+ transition has not yet been reported, .so the rank
one operator y± I in {Mzu(a)}c need not be considered. The contributions from
operators X- and X kl .is negligible, because of their small coefficients (Ka+ - La+).
Therefore, the 0+--.2+ transition rate is small, because it is derived from Xkl. Thus,
the 2)) mode takes place mainly through the 0+--.0+ transition induced by the X+
operator in {M2U (a)}n.
The half-life for the 0+ --. 0+ transition of the (e(3)2U mode is expressed as follows
from Eqs. (A·5), (A·6) and (A·9),
[T2~Pl(0+--'0+)]-I=C~2 )a2ufd.Qi~P)IM2uI2Ni~P)do(cp)RI.I(cP)'
(A ·17)
where the constant factor a2u is defined in Eq. (3·2·2a) of I, and the modified phase
space factor is
d.Qi~P)=me-(27I+6)qIq2wIW2C/718(Mz+ ce -
Mf
-
cp -
WI -
(2)dwIdw2dcp. (A ·18)
The wave function for the emitted charged lepton CPe(cp, 0) in the leptonic part E+f.I. of
Eq. (A ·14) has been separated into a product of three factors; (1) the directly energy
dependent part (me/P) (cp/m e)271-I from Eq. (2·4), (2) the energy independent constant
CF •O which is included in NW) of Eq. (3·10) in combination with the normalization of
the bound electron wave function and (3) the Coulomb correction term doRI.1 in
Eqs. (2·2), (2·6) and (3·15).
Since the s-wave lepton part of {M2U(a)}n in Eq. (A·6) is independent of the
nucleon suffices, its hadronic part can be separated out and is expressed as
(A·19)
where ACT and AF are written as follows in terms of the hadronic parts X+ in
Eq. (A·9) and the energy denominator Ka+ and La+ in Eq. (A·16),
_ ( Mb2td ) f1.a
_ ( M}~u) ) f1.a
ACT-~ ---;;:;- T(Ka++La+) , AF-~ ---;;:;- T(Ka++L a+).
(A·20)
Here the Gamow-Teller and Fermi types of the nuclear matrix elements Mgtd and
are defined in Eqs. (3·5) and (3·6), respectively. The excitation energy f1.a is
introduced to make the factor (f1.a/4)(Ka+ + La+) almost unity, as it is clear from
Eq. (A·16).
Now, let us introduce an approximation, under which (f1.a/2)(Ka++ La+) can be
treated independently of the intermediate state la>: That is, f1.a is replaced by some
M}~U)
Neutrino Emitting Modes of Double Beta Decay
1225
appropriate average value <f1.a>. This approximation is good enough, because generally f1.a210 and f1.ame::?>(€lZ+Ul!Z), see Fig. 2. Thus, this factor can be separated from
the sum over a in Eq. (A· 20) and included in the integrand of the phase space
integration for the emitted neutrinos as in hZII defined by Eq. (3·17). The simpler
nuclear matrix elements in Eq. (3·3) are derived from Eq. (A·20). Finally, the
half-life formula in Eq. (A·17) becomes Eq. (3·2).
The derivation of R -matrix for the (ee )ZII mode proceeds similarly to the (e/3)zII
mode. This time, two wave functions for captured electrons r/Je(€el, x) and r/Je(€ez, Y)
are used in tp.( €pX; €eY) of Eq. (A ·15) like
(A·21)
Accordingly, another replacement €lZ=€eZ-€el is necessary in the energy denominators Ka± and La±. Note that €lZ is zero or a very small constant for the (ee)zII mode.
Then the half-life formula is easily obtained from that of the (e/3)zII mode through the
following procedures: (1) The constant normalization of the positron wave function is
replaced by that of the captured electron, namely, CF,O-'> (2J[Z)'Jl n ',_1. (2) Correspondingly, the phase space integral over positron is omitted like Eq. (3·14). (3) The
re-interpretation of (J) and €lZ is required as shown in Eq. (3·16) and below Eq. (3·18).
(4) In the case of the double K-electron capture, Eq. (A·17) is divided by 2.
The decay formula of the (/3/3)ZII mode is obtained from the (e/3)zII mode by takihg
the backward procedures explained above. It is also derived from the (/3- /3-)ZII mode
by two following changes; the sign of Coulomb potential for charged leptons (Z -'> - Z)
and the isospin operator for nucleons in the nuclear matrix elements (Tn + -'> Tn -).
Appendix B.
- - Wave Function of the Bound Lepton--
The relativistic wave functions for the charged lepton bounded by nucleus are
summarized to explain notation and to compare contributions from various lower
bound states.
In order to solve the Dirac equation, the wave function for the bound lepton is
expressed as
Gn"tc(r)xtc,p.(e, r/J) )
r/Jn',tc,p. ( r, e, r/J ) = ( ·F ()
(e A..) ,
Z n',tc r X-tc,p.
,'f'
(B· 1)
and normalized such that there is one lepton in the space. H ) Here n' is the radial
quantum number whic.h takes non-negative integers, K takes either positive or negative integers*) and f1. is the magnetic quantum number related to the total angular
momentum jtc.
- Since the angular part is defined in Eq. (D·4) of I as
(B·2)
*)
The orbital angular momenta are written in terms of K as IK=k, i-K=k-l for K=k>O and IK=k-l,
i-K=k for K=-k<O.
1226
M. Doi and T. Kotani
the radial part is restricted by the normalization condition,
1 dr{[ rGn',K( r))2 + [rFn',K( r))2} =
00
1.
(B·3)
In this appendix, the radial quantum number n' in the suffix will not be written
explicitly, except some special cases.
Let us briefly summarize how to determine the discrete energy and the normalization constant under the general assumption that the nuclear charge is confined inside
the finite nuclear radius R.
The regular inner solutions for the radial part are expressed as follows in the
region of r~R,
(B·4)
where BK is a normalization constant related with the outer solutions, k=IKI and
.,l=hne2-e;"K' where en',K is the total energy of a bound lepton and expressed as ee
in the text. The expansion coefficients aK,1 and bK,1 depend on the assumed charge
distribution inside nucleus.
On the other hand, the outer solutions for r;p R are written as
(B·5)
Here the regular (GK (out), FK (out)) and irregular (GK (out), FK (out)) outer solutions mean to
become zero and singular for r->oo, respectively.
The normalization condition in Eq. (B·3) requires the following boundary condition: rGir) and rFK(r) should be integrable, that is,
.
(B·6)
93=0.
The discrete energy levels for the bound lepton are determined from Eq. (B· 6).
are expressed as
They
(B·7)
where epo!nt is the total energy obtained from the Dirac equation for the point nuclear
charge/ 2 ),ll) () denotes the deviation due to the finite extension of nucleus and
N=jn,2+2n'Yk+k 2 • If the principal quantum number (n=n' +k) is given primarily,
then the allowed values are K=±l, ±2, "', -n and n'=(n-1), (n-2), "', O. Note
that the case with K= n is not allowed for the case of the point nuclear charge, because
G~o~N(r) does not satisfy th~ boundary condition in Eq. (B·3).
According to Kramers,17),14) after determining en',K, the overall normalization
constant is determined as
. Ji (-:\a93 )
(-B1)2 = .11m
"
e-el}',1C
uS
Wron,
~
(B·8)
1227
Neutrino Emitting Modes of Double Beta Decay
where the r-independent Wronskian is
Wron =
r2[ G" (out) ( r )F" (out)( r) - F" (out) (r ) G" (out) ( r )] .
(B·9)
In general, these sets of outer solutions are not convenient to treat Ji ~nd 93,
because they are characterized in the region of r -> ex). Then, it is useful to introduce
other sets of outer solutions which are classified by the characters near the origin.
They are the regular (G" (reg), F" (reg») and irregular (G" (Irr), F" (Irr») outer solutions for
r -> O. The former is related with the previous outer solutions as
G,,(reg )( r
)=DI G,,(out)( r) + D 2G,,(out)( r) ,
(B·10)
where coefficients DI and D2 depend on the assumed form of the outer potential for
r > R. Similarly, the irregular outer solutions are related with the previous outer
solutions by using other constants D3 and D4 in place of DI and D2 in Eq. (B ·10).
Let us require a continuity condition at the nuclear surface, r = R;
G,,(i)(R)=KIG,,(reg)(R) + K 2G,,(lrr)(R) ,
F,,(i)(R) =KIF,,(reg)(R)+ K 2F,,(lrr)(R) .
(B·ll)
Here two constants, KI and K 2, are solutions of Eq. (B ·11), namely,
(B ·12)
where
KIN = [G,,(i)(R)F,,(lrr)(R)- F,,(i)(R)G/lrr)(R)] ,
K2N=[ G,/il(R)F/reg)(R)- F,,(il(R)G,/reg)(R)] ,
KD=[G,,(reg)(R)F,,(lrr)(R)- F,,(reg)(R)G/lrr)(R)] .
Thus, by substituting Eq. (B·10) and similar relations for
Eq. (B·ll), we find from Eq. (B·S),
(B ·13)
G,,(lrr)
and
F,,(lrr)
into
Ji = KIDI + K2D3 ,
(B·14)
93 = KID2 + K2D4 .
(B· IS)
So far, any assumptions on the nuclear charge distribution have not been used,
except its confinement inside the nuclear radius. Hereafter, let us assume the potential of the uniform charge distribution, that is,
-1· -(~~)[3-(~rJ
a;)
V(r)-
_(
for
r~R,
for
r~R.
(B·16)
Concerning the inner functions, the expansion coefficients a",l and b",l in Eq. (B·4)
are derived from the same definition as in Eq. (D·ll) of I by replacing the charge of
1228
M. Doi and T. Kotani
daughter nucleus (Z) with the parent one (Z +2).
The well-known outer solutions which are regular near r=O are I2 ),1l)
(B·17)
where p=2ilr, y=aZ(c"jil) and two confluent hypergeometric functions are FI
=F(Yk-Y, I+2Yk; p) and F2=F(I+Yk-Y, 1+2Yk; p). The corresponding irregular
outer solutions, G,,(irr)(r) and F,,(irr)(r), are obtained from the above G,,(reg)(r) and
F,,(reg)(r), respectively, by replacing Yk with -Yk. Thus KD in Eq. (B·I3) becomes
(B·I8)
On the other hand, the outer solutions wl:tich are regular for r . . HX) are
(B ·19)
where two new confluent hypergeometric functions defined by Bateman l8 ) are
lJfl= lJf(Yk-Y, 1 +2Yk; p)and lJf2= lJf(I + Yk-Y, 1 +2Yk; p). The corresponding irregular outer solutions are
(B·20)
where lJf3 =lJf(I+Yk+Y,I+2Yk; -p) and lJf4 =lJf(Yk+Y,I+2Yk; -p). Note that the
previous confluent hypergeometric function F(a, c; x) is the same as a>(a, c; x) defined
by Bateman. 18 )
Thus, the Wronskian in Eq. (B·9) becomes
(B·2!)
where the phase is defined as (- p)«=exp( - iJr({!pa)p«, ({!p= + 1 if 1m p>Orand ({!p=-I
if 1m p < O. This phase will cancel with the phase of the following D j in the final
result. The coefficiens D j in Eq. (B ·10) are
D - ( me _ )
1-
Y
CIC
K
r(I + 2 Yk) i7r<pP(7k ....Y)
r(I + Yk + y) e
,
me _ ) r(I +2Yk) i7r<pp( .... I .... 7k-CY)
D 2-(
- Y
,
c" K r( Yk-Y ) e
where
(B·22)
rea) is a gamma function, and similarly
D - -( me _ ) r(I -2Yk) i7r<pp( .... 1 .... 7k .... Y)
3Y c" K r(I-Yk+y)e
,
me _ ) r(I-2Yk) i7r<pp( .... 1 .... 7k ....Y)
D 4-(
- Y
K r(
)e
.
CIC
- Yk-Y
(B·23)
1229
Neutrino Emitting Modes of Double Beta Decay
Note that these outer solutions are nothing but the solutions of the Dirac equation for
the point nuclear charge.*)
Now, since all necessary quantities are given analytically, each discrete energy
level can be determined as a root of the following equation derived from Eqs. (B-6)
and (B-15),
(B-26)
It is worthwhile to point out that there is no root of Eq. (B-26) for the K=n case, as
it is so for the point charge. Practically, in order to reduce numerical errors in the
computer calculation, the following deviation (0') from the binding energy for the
point charge is calculated,
(B-27)
where Eb,P=me-EpOint. In Table IV, Eb,~ in units of me and 0' are listed for four low
lying shells of eight nuclei which take the (e!3)ZJJ mode. As it is expected, the
deviation due to the uniform charge distribution is negligibly small. In the text, these
calculated values of En"~( = Ee) have been used.
Finally, the normalization constant is determined from Eq. (B -8). Let us list
some simplified expressions: By using !B = KJDz + KzD4 =0, Jl becomes
Jl=K( me
J
Y E~
-K)( r(I+Yk+Y)
r(I+2Yk») sin(27rYk)
sin[7r(Yk+Y)] '
(B-28)
and o!B IOE can be written as
l!!L= KJ[D4~ (Dz) - Dz( Dz)~ (KJ)] .
OE
OE D4
D4 OE Kz
(B-29)
When o!B IOE is multiplied by Wron as in Eq. (B-8), all phase factors cancel each other
in the combinations like D4 Wron and Dz Wron. Thus, the normalization constant B~ in
Table IV. Binding energies in units of m, for the low atomic shells of double beta decaying nuclei with the uniform charge
distribution. Lower values are 8' in Eq. (B' 27) which stands for the deviation from the binding energy for the Dirac equation
with a point nuclear charge. Values of quantum numbers n' and K are shown for each shelL
shell
58Ni
78Kr
96Ru
IO'Cd
lUXe
130Ba
136Ce
162Er
K(ls1J2)
2.11 • 10 '
6.34' 10-6
5.29 • 10 '
3.28' 10-6
5.29'10'
2.63'10- 8
-5.23 '10'
3.07' 10- 13
3.51' 10 2
1.49' 10-5
8,82' 10 '
7.89' 10-6
8.82' 10 '
1.06. 10-7
8.66'10'
1.37' 10- 12
5.29' 10 '
3.09' 10-5
1.33' 10 '
1.68. 10-5
1.33' 10 '
3.46' 10-7
1.30' 10 '
4.51' 10- 12
6.33'10'
4.36' 10-5
1. 60 • 10 '
2.41. 10-5
1.60'10'
5.98' 10-7
1.55· 10 '
7.76 • 10- 12
8.09'10'
7.26 • 10-5
2.04 • 10 '
4.13' 10-5
2.04' 10 '
1. 32 • 10-'
1. 96 • 10 '
1. 70 • 10- II
8.73'10'
8.57' 10-5
2.21'10'
4.92' 10- 5
2.21 • 10 '
1. 71 • 10-6
2.11 • 10 '
2.17' lO- II
9.40 • 10 '
1.01 . 10- 4
2.38' 10 '
5.85' 10-5
2.38' 10 '
2.20' 10-6
2.26' 10 '
2.75' lO- II
1.32· 10 '
2.20' 10- 4
3.35' 10 '
1.35. 10- 4
3.35 '10 '
7.30' 10-6
3.13 • 10 '
8.07' 10- II
(n'=O,K=~l)
L/(2Sl/')
(n'=I, K=~I)
L ll (2p,,,)
(n'=l, K=l)
L m (2/Ja,,)
(n'=O, K=~2)
*)
The solutions of the Dirac equation for the point nuclear charge are expressed as
G.(r)=KK,pGK(reg)(r),
FK(r)=KK,pFK(reg)(r).
(B'24)
The discrete energy levels, cpolnt, are determined from the condition D 2 =0 in Eq. (B '10) and the normalization
constant corresponding to Eq. (B'8) is determined as
(l/KK,p)2= lim D.(aD.joc) Wron.
e--->epOlnt
(B'25)
1230
M. Doi and T. Kotani
Eq. (B·S) becomes real,
(B·30)
where
(B·31)
This Btc=-I,P is essentially the same as the value of Ktc=-I,pG1r::~i(R), the normalized
regular solution for the point charge evaluated at the nuclear surface, d. Eq. (B·24).
Each term in Eq. (B·30) has been derived as follows: The first correction Cl is defined
as Kl =2Yk(2aZm eR/N)(7k-1) Cl, where C1 depends on the quantum numbers and the
assumed inner potential; namely, Cl is a little less than unity for both the K and LJ
shells, of order of 10 for the LIJ shell and of order of 10-3 for the Lm shell.
Corrections C2 and C3 come from the first and second terms of JfB /Je: in Eq. (B'29),
where C2 is always of order of unity and C3 is negligibly small, because the latter
includes (D2)2 which is required to be zero for the point charge case.
In Table V, the dimensionless normalization quantity 'Jln',tc defined in Eq. (2'1) is
listed numerically. As you see, those values for the LIJ and Lm shells are smaller by
one order. Strictly speaking, this 'Jl n',tc for the p-state should be multiplied by another
additional small factor, for example, (2.-tR)2 for the P3/2 case, as seen from Eq. (B·4).
The corresponding nuclear matrix elements become smaller by including a factor
(r/R) in comparison with the s-state. Therefore, we do not consider the p-state.
Concerning the s-state, there are three allowed states as seen from Eq. (B '1); namely,
Gg:~l of the K shell, Gl:~l of the LJ shell and Fl,i21 of the LIJ shell. This last Fl,i21 has
larger contribution than Gl:Ll of the Pl/2 wave. However, its 'Jl1,+l was smaller, as
shown in Table V, because it connected with the so-called small outer component.
Thus, we have considered only the K and LJ shells iIi the text.
As a reference, the corresponding normalization constants for the non-relativistic
case are listed in Table V:
'Jl n ,I=-4
1 3 (B n ,l)2
Jrme
(B·32)
,
where
Table V. N onnalizations of wave functions for the bound electron in the low lying shells of double beta decaying nuclei, ']I n·.<
in Eq. (2 ·1). Upper values are those calculated by the Dirac solution for the uniform nuclear charge distribution, while lower
values by the Schrodinger solution for a point nucleus.
shell
58Ni
78Kr
96Ru
''''Cd
124Xe
130Ba
K(ls1/2)
(n'=O, F - l )
L 1 (2SI/ 2 )
(n'=I, K=-l)
L II (2PI/2)
(n'=I, Fl)
L III (2/h,,)
(n'=O, K=-2)
3.55. 10 3
2.72 . 10-3
8.79.10 3
5.77.10- 3
1.17'10 3
7.21 • 10-'
1.58. 10 5
6.01 . 10- 5
7.83'10 5
6.01' 10-5
1.93.10 2
1.05. 10- 2
2.64' 10 3
1. 32 • 10-3
5.45. 10 5
1.10 • 10-'
1. 61 • 10 '
1.10 • 10-'
2.78.10 2
1.37· 10- 2
3.87'10 3
1. 71 • 10-3
9.61 • 10 5
1.42 • 10-'
4.67. 10 2
1. 95' 10- 2
6.71'10 3
2.43' W 3
2.15 • 10 '
2.03' 10-'
3.53' 10 '
2.03' 10-'
5.52. 10 2
2.17' 10-2 .
8.02' 10 3
2.72' 10-3
2.79' 10 '
2.26' 10-'
4.09'10'
2.26' 10-'
4.60' 10 '
3.39' 10-'
3.70' 10 6
2.83' 10-5
3.34'10 5
2.83' 10-5
2.23' 10 '
1.42' 10-'
136Ce
162Er
2
6.52'10
2.41' 10- 2
9.57 • 10 3
3.02' 10-3
3.60'10'
2.51' 10-'
4.73 • 10 '
2.51' 10-'
1.47' 10 '
3.89' 10- 2
2.29' 10 2
4.86' 10-3
1. 24 • 10 3
4.05' 10-'
9.42 • 10 '
4.05' 10-'
Neutrino Emitting Modes of Double Beta Decay
(aZme)3 [
2
n4
(2/+1)1
J2[ (n-I-l)l'
(n+l)l ]
1231
(B·33)
For the case K= - k, (Bn,IY is reduced from (B",pY in Eq. (B· 31) by taking /k
.... k= 1+1 (N .... n= n' + k). The factor (2aZm eR/n)21(r/R)21 obtained from Eq. (B·31)
is included in the solution of the Schrodinger equation.
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
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Atoms (Springer, Berlin, 1957).
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H. Behrens and]. Janecke, Numerical Tables for Beta Decay and Electron Capture (Springer,
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H. Behrens and W. Biihring, Electron Radial Wave Functions and Nuclear Beta Decay (Clarendon
Press, Oxford, 1982).
A. H. Wapstra and G. Audi, Ntic!. Phys. A432 (1985), 55.
K. Muto and H. V. Klapdor, Neutrinos, ed. H. V. Klapdor (Springer·Verlag, 1988), p. 183.
T. Tomoda, Rep. Prog. Phys. 54 (1991), 53.
H. A. Kramers, Quantum Mechanics (North·Holland Publishing Co., Amsterdam, 1957), § 66.
E. L. Hill and R. Landshoff, Rev. Mod. Phys. 10 (1938), 87, see Appendix D.
H. B. Bateman, Higher Transcendental Functions (McGraw·Hill Book Co., New York, 1953), see
pp. 252, 258.
Note added in proof: After this paper was submitted, we received the internal report, "Phase·Space Factors
for the Double Beta Decays Z .... Z - 2 with Emission of Two Neutrinos", written by Prof. P. Vogel at Caltech.
We would like to express our thanks for his sending it before publication.