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Grand unification, nuclear structure and the double beta-decay
NUCLEAR PHYSICS A
Nuclear Physics A570 (1994) 39c-60~ North-Holland, Amsterdam
GRAND
UNIFICATION,
NUCLEAR AND
THE
University
STRUCTURE
DOUBLE
BETA-DECAY
Amand Faessler of Tiibingen, Institute of Theoretical 72076 Tiibingen, Germany
Physics
ABSTRACT
Unification of the electroweak and the strong interaction prefers that the neutrino is a Majorana particle and therefore essentially identical with its own antiparticle. In such grand unified models the neutrino has also a finite mass and a slight right-handed weak interaction, since the modeJ is left-~ght symmetric. These models have also left handed and right-h~ded vector bosons to mediate the weaJc interactions. If these models are correct the neutrinoless double beta-decay is feasable. Thus if one finds the neutrinoless double beta-decay one knows that the standard model can not be correct in which the neutrino is a Dirac particle and therefore different from its antiparticle. Although the neutrinoless double beta-decay has not been seen it is possible to extract from the lower Jimits of the lifetime against the doubte neutr~oJess beta-decay upper Jim&s for the effective eiectron-neutr~o mass and for the effective mixing a.ngJe of the right-h~ded and the left-handed vector bosons mediating the weak interaction. One also can obtain a.n effective upper limit for the mass ratio of the light and the heavy vector bosons. The extraction of this physical quantities from the data is made difficult due to the fact that the weak interaction must not be diagonal in the representation of the mass matrix of the six neutrinos requested by such left-right symmetric models. A condition for obtaining reliable Jimits for these fundamental quantities from the measured lower limits of the half lifes of the Ov/3p decay are correct calculations of the nuclear matrix elements involved. These nuclear structure calculations can be tested by calculating the two neutrino double beta decay (2~/3/3) for which we have experimental data and not only lower limits as for the OV@/? decay. The Zv/3@ decay is dominated by the C&now TeJler (GT) transitions. The inte~ediate l+ states in the odd-odd mass nucleus axe usua.JJy calculated within the Quasi-particle Random Phase Approcimation (QJZPA). Since the proton-proton and neutron- neutron pairing correlations are treated in the BCS approach particle numbers are only conserved in average for the initial 10:) and the final IOf) states. Particle number projection is used to improve on this point. Since QRPA with the physical p~ticJe-~ticJe interaction from the Bonn potential is close to breakdown, where the excitation energy goes to zero, higher up to third order RPA is included, by allowing anharmonicities up to three bosons in the physicaf wave functions. Third order RPA which is the lowest order to allow modifications of the ground state correlations &ect the results appreciably. Since the intermediate states 0375-9474~4/$07.~ 0 1994 - Elsevier Science B.V. All rights reserved. SSDI 0375-~74(~~~77-2
4oc
A. FaesslerI Grand~az~ca~o~
are supperpositions ofproton- neutron two quasi-particle states one expects that protonneutron pairing affects these states. From the masses of the nuclei one can extract a proton-neutron (pn) pairing gap. Due to the finite basis and the inclusion of only T = 1 (and not also T = 0, J = 1) pn-pairing the bare Brckner matrix elements of the Bonn potential yield no pn-pairing gap in most nuclei. If one fits the experimental pn-pairing gap by renormalizing the pn J” = Ot particfe-particle Brckner matrix elements by a factor dpn R 1.3 to 1.4 one obtains strong pn-pairmg correlations, which reduce the GT strength. Since now part of the pn particle-particle correlations are akeady included in the BCS wave functions the ‘Zv/!Q?strength is reduced and the dependence on the l+ ~tic~~p~tjc~e stren~h parameter gW gets weaker. In heavier nuclei Iike 76Ge to 76Se this dependence on gPPof the second order GT matrix elements C, GT(Of --f 1: -+ 0:) host disappears. The upper limits derived from the neutrinoless double beta decay of 76Ge are for the effective neutrino mass < mue >I 0.88 eV, for the left-right mixing angle < tg< >I 1.55 x lo-* and for the mass ratio of the light over the heavy vector boson squared < MT/M; >< 1.08 x 10B6.
1
Introduction
The most important particle of this talk is the neutrino. It has always been very interesting since it was predicted by Wolfgang Pauli on December 4th, 1930 in a letter to a conference in T~bingen. Still today the neutrino is the only fermion from which we do not know if it is different from its antiparticle (U # F) and therefore a Dirae particle or if it is identical with its antiparticle (u = V) and therefore a Majorana particle. In 1955 the physicists did believe that this question was solved in favour of a Dirac neutrino by an experiment of Davis [I]. He was using antineutrinos from the beta-decay in a reactor to induce the inverse beta-decay in 37Cl for which one needs in the standard model a neutrino: n V+37ca
-+ -+
p+e-+P
(1)
37Ar + e-
Since he did not observe the formation of 37Ar by the inverse beta decay he assumed that the antineutrino v is different from the neutrino u and thus the neutrino is a Dirac particle. But this conclusion was outdated already two years later. In 1956 Lee and Yang[Z] proposed to test if in the weak interaction parity is conserved, and in 1957 Wu et al.[S] did find that indeed parity is violated in the beta-decay and soon it turned out that it is violated maximally and the weak interaction is purely left-handed. Therefore, the neutrino created in the beta-decay of the neutron in eq. (1) has a positive helicity (is right-handed), while the neutrino needed for the inverse beta-decay must have a negative helicity and therefore needs to be left-handed. Thus, even if the neutrino is a Majorana particle, reaction (1) would be forbidden due to helicity mismatch of the two reactions. Thus, we know already since about 30 years that the problem, if the neutrino is a Dirac or if it is a Majorana particle, is not solved. But why are we just today discussing this question so intensively? The reason is that the grand unified theories from which we
A. Faessler I Grand unification
41c
think that they can be successful predict that the neutrino is a Majorana particle[4]. Measurements of the lower limit of the proton lifetime seem to exclude SU5 and thus one concentrates on SOlO, which is also a subgroup of dynamical groups discussed in superstring theories. Left-right symmetric theories inaugurated by Mohapatra, Pati and Senjanovic [5] and especially theories based on SO10 which have first been proposed by Fritzsch and Minkowski[S] predict in improved versions[4] not only, that the neutrino is a Majorana particle, but automatically predict dso that the neutrino has a mass and a weak right-handed interaction. The basic idea behind grand unified models is an extension of the local gauge invariance from quantum chromodynamics (SU3) involving only the coloured quarks also to electrons and neutrinos. The presently favoured models are left-right symmetric models. They contain left- and right-handed vector bosons W: and W;.
W*
=
cos<.WLf+sin6.W;
w:2
=
-sin<.
W:
+cosC.
(2) Wi
The left- and right-handed vector bosons are mixed if the mass eigenstates are not identical with the weak eigenstates, which have a definite handedness. The left-right symmetry is broken since the vector bosons W: and W$ obtain different masses by the Higgs mechanism. Since we have not seen a right-handed weak interaction the mass of the heavy, mainly right-handed vector boson must be much larger than the mass of the light (81 GeV) vector boson, which is mainly left-handed. The weak interaction Hamiltonian must now be generalized. Hw
x
GF [(L.l)+tgC(R.I)+tg((L.r)+ Jz
-
LIR = &(svr, =FgA?p?'5)$'n gv
=
l;gA = 1.25
l/r
=
2(-Y, T
(g)‘(R-4] (3)
Ya^l5k
The capital L and R indicate the hadronic right- and left-handed currents changing a neutron into a proton. The lower case 1 and r are the left and right handed leptonic currents which annihilate a neutrino (or create an antineutrino) and create an electron (or annihilate a positron). C is the mixing angle of the vector bosons (2) and MI and Ms are the light and heavy vector boson masses. The weak interaction Hamiltonian is given for C << 1 and Mz >> Ml. Most of the left-right symmetric models predict that the neutrino is a Majorana particle that means that they are loosely speaking identical with their antiparticles (v = v). A Majorsna neutrino must have automatically a mass and due to construction it has also a slight right-handed interaction (3). As we will see below these facts allow the double beta-decay without neutrinos. Or inversely: The existence of the double neutrinoless beta-decay would establish that the neutrino is a Majorana particle. Figure 1 shows the diagram for the neutrinoless double beta-decay. This is naturally only possible if the neutrino is identical with its antiparticle
42c
A. Faessler I Grand un@cation
since at the first vertex, one must emit an antineutrino and at the second vertex one needs to absorb a neutrino in the standard model. Since in the final state we have only two particles in the continuum in the process in Figure 1, the phase space is bigger by a factor lo6 compared to the double beta-decay with two neutrinos. Thus, even if the matrix element squared is reduced by a factor 10m6,one has the same transition probability for the neutrinoless double beta decay as for the double beta-decay with two neutrinos.
P
P
ei
v, ’ ’ .’
n
r
@2
.’
1
:
-
n
Fig.1. Diagrams for the neutrinoless double beta-decay with a Majorana neutrino. By having only two particles in the final states in the continuum, the phase space is increased by a factor of about lo6 compared to the 2upp decay. Even with a Majorana neutrino this process is only possible if the neutrino has a finite mass and/or also a right-handed weak interaction. But a right handed weak current must be accompagnied by a finite neutrino mass to yield a finite o&3 decay.
But even if the neutrino is a Majorana particle, the process in Figure 1 can not happen since for a pure left-handed weak interaction theory, the emitted neutrino must be righthanded (positive helicity), while the absorbed neutrino must be left-handed (negative helicity). But grand unified theories predict also that the neutrino has a mass and a slight right-handed weak interaction. With a finite mass the neutrino has not any more a good helicity and the interference term between the leading helicity and the small admixtures allows a neutrinoless double beta-decay. The Ovpp decay propability is further increased with a massive neutrino and also right-handed vector bosons which allow a right-handed weak interaction. If the first vertex is a left handed V-A interaction and the second vertex is a right-handed V+A interaction, the process in Figure 1 is also allowed. (But a detailed analysis shows that also in this case one has to request a finite neutrino mass to allow the neutrinoless double beta-decay.) Thus, the neutrinoless double beta-decay can only happen due to the finite mass of the neutrino and can be additionally increased due to a small admixture of right-handed leptonic currents. The matrix element for the double neutrinoless beta-decay consists of three parts: One proportional to the neutrino mass mev and the other to the small right- handedness tg[ or (Mi/M2)2 of the weak interaction.
43c
A. Faessler I Grand un@ication
2
Description
of the two-neutrino
double
beta-decay
Since there are measurements available for the two neutrino double beta-decay with the geochemical method [6,7], and for three nuclei even laboratory measurements[S - 111, one could try to calculate for a test of the theory the double beta-decay with two neutrinos and compare them with the data. If one performs this calculation, one finds that the calculated transition probability for the double beta-decay with two neutrinos (2vp/3) is by a factor 50 to 100 too large[l9]. This discrepancy which is typical for the Zv/?/3decay naturally casts doubt on the reliability of the calculations of the OY@/~ transition probabilities. The nuclear many-body methods for calculating the double beta-decay are either shell model calculations[l2] or extended shell model calculations based on the Hartree-FockBogoliubov approach called MONSTER [13] or they are based on the Random Phase Approach (RPA). Let’s try to asialyse the 2u@ decay in the RPA approximation. 6
E’ig. 2. The upper part shows the way how in the Random Phase App~xjmation (EPA] the S/?/3 decay is calculated. For the Fermi transitions the @-(a + p) amplitude moves just a neutron into the same proton level and the p+(p + n) amplitude moves a proton into the same neutron level. For the Gamow-Teller transitions it can also invoke a spin flip, but the orbital part remains the same. One immediately realizes that the occupation and non-occupation amplitudes favour the p- amplitude, but disfavour the p+ amplitude. There one has a transition from an unoccupied to an occupied single particle state, which is two-fold small (s’) first by the fact that the occupation amplitude far the proton v* and secondly that the unoccupation amp~jtude for the neutron state 21, are both small. Therefore the 2vflfl is drastically reduced. Figure 2 explains why the 2vpp decay amplitude is so drastically reduced. Therefore the small effects which normally do not play a major role can affect the 2~~~ transition
44c
A. Faessler I Grand u~t~cation
probability. If one looks to the second leg of the double beta-decay which is calculated backwards as a p+(p -+ n) decay from the final nucleus to the intermediate nucleus one finds that the matrix elements involved in these diagrams are Pauli suppressed by a factor (u,~r)~ = (s~a~~)4. The neut~n-p~ticle proton-hole force in the isovector channel, which is usually included is repulsive while the particle-particle force usually neglected is attractive. Therefore both excitations tend to cancel each other and therefore the amplitude pf is drastically reduced. In this way one can obtain a better agreement with the experimental data. We calculated the 2v/3@decay half-lives for the following 0: -_) 0: transitions: 4sCa +48 Td t*s Cc -+46 Ti >76 Ge -+ 76Se 9 g2Se -+ 82Kr 2 ‘?l’e --f lBXe ad 130Te -+ lmXe. Although one can obtain agreement in this way with the measured 2v/3@data multiplying the particle-particle matrix elements with a factor g,,, in a range of 0.8 2 g,, 2 1.2, one observes a tendency for the unrenormalized Bfickner reaction matrix elements 9r,, = 1 of the different realistic forces, that the particle-hole and the particle-particle contributions to the 2v/3/3transition cancel each other to zero. The physical reason for this cancellation can be understood in Wigner’s supermultiplet model SU4. The Gamow-Teller (GT) operator ,& ait+; are generators of the Wigner supermultiplet symmetry SU4. If this would be a dynamical symmetry of the nuclear many body Hamiltonian the 2v@ GT transition would lead only to the double GT resonance and not to the ground state. But it is well-known that the spin-orbit splitting destroys the SU4 symmetry in the wave functions of medium heavy and heavy nuclei and this is only an approximate symmetry. The more technical reason for this cancellation is that for the second leg the backgoing amplitudes and thus groundstate correlations cancel the leading term.
Eq. (4) defines the QBPA boson for an l+ state la) in the intermediate nucleus based on the ground state of the &ml nucleus IOf). Eq (2) shows the expression for the Gamow Teller (GT) strength from the final state IOF> into the 1+ state IQ) in the intermediate nucleus. The operators a$, a& create neutron and proton quasi-particles. a+ Cc S Cijm
=
U;C+
=:
(-)j-"Cij_m
+
ViC; (6)
The cancellation of the GT strength in the second leg for Briickner matrix elements of the Bonn potential (gm, = 1) can be seen in eq. (5) by looking to the expression in the round brackets ( ). For nuclei with a neutron excess (see fig. 2) the amplitudes t&?&p are small and uipZ)knare large for the second leg. Although X,$,+ is large and x&,, is small, the products in the round brackets cancel each other for gpp = 1 (bare Brckner matrix elements).
A. Faessler I Grand un@ication
4%
Figure 3 shows the second order GT matrix element as function of the particle-particle strength parameter gPP for 76Ge to ‘6Se.
(7)
Here Ia) are the l+ state in the intermediate nucleus built on the initial nucleus and 1~) are the l+ states built on the final nucleus.
0.8
_-
L-
is
+
SP
+ -%
tY
+7
\\
0.4
7 + .. 0 t&J
g
0.0
--_-t-w,___-_-_
I_--_-__
I
__.
._
._
\
W-0.2
r0.0
0.4 g
PP
0.8 ___ --_._*
1.2
Fig. 3: Second order Gamow-Teller (GT) matrix elements for 76Ge to 76Se (7) as a function of the particle-particle strength parameter gPP with which all particleparticle matrix elements of the Bonn potential in the QRPA equation are multiplied.
46c
A. Faessler / Grand unification
Fig. 4.: Hdf life of 2upp “‘Se to “‘h’r as a function of the particle-particle strength gP,,. The data are from geochemicd measurements in refk [6,7] and from the first Isboratory measurements of Moe et al f8].
Figure 4 shows the half life of “Se to s21’r as a function of the particle-particle strength g,,. At gpP = 1 the particle-hole and the particle-particle contributions cancel (see eq. (2)) and the half life goes to infmity. Thus one sees that one is in principle able to understand the data by slightly changing ga, around unity (0.8 2 gpp 5 1.2), but the predictive power is very small [26]. The cancellation of the particle-particle and the particle-hole matrix dements for the 2upp at around gpp = 1 means that the QRPA in its present form is not accurate enough. We have therefore improved it in three ways:
47c
2t4
2
zt2
3
E &Z
z-2 N-4
N-2
Iv
Nt2
Nclutrons
The second leg 4”Sc to “Ti is calculated in the inverse direction. The leading ampijtude for the second leg is proportional to u~~~~~zL,,~~~~ (plus a smaller additional term proportional to v,,~~~~u~~~~~). This amplitude is small since in average one has not more than two protons in the j712 JeveI in 48Ti and one has for the neutrons the f712 almost full (94,/7/z = small). To iaTiN one admixes the four nuclei .Z z?c2, N f 2 by amplitudes around 0.6. For Z = 24, N = 26, which has a wrong particie number for the final nucleus one has twice as many protons in the fTi2 sheik One could therefore expect a iarge error on the 2u@ decay probability from particle number non-conservation.
Fig. 5.: The Bv&3 decay 4sCa to 4sTi in the isotopic chart.
First we included paxticle number projection on the correct proton and neutron number before the BCS and the RPA variation 1251. The B~den-Cooper-Schriefer (BCS) model used for treating the pairing correlations does not conserve the particle number.
IBCS)
=
IIi>*(Ui t WiQC;t)lO)
(8)
48c
A. Faessler I Grand unification
Although one conserves the particle numbers for protons and neutrons in average,
c vf= 2
(BCSlilBCS) =
(9)
i=p>o
c v:= N
(BCSlAIBCS) =
k=n>o
one admixes nuclei with two protons or two neutrons more or less by about 0.6 if the nucleus with 2 and N has the amplitude 1. One can expect in nuclei with neutron excess a large effect for 2vp,f3. Let us consider the decay $$ass -+g Sczr +g T&e in figure 5. We therefore used particle number projection to circumvent this difficulty. The ui,ui coefficients of the Bogoliubov transformation (6) are now determined by minimizing the projected and normalized expectation value of the Hamiltonian in the BCS states.
(BCSlI?&&lBCS)
= Min
(10)
(BCSl&&IBCS)
The states in the intermediate nucleus are given by:
II:, QRPA;
Z + 1, N - 1)
=
&+&-I
=
e+,kd
z
{
x~k,da~ka$
-
~,nlhd%k}l+
b)
(11)
The Euler-Lagrange eqs. for the coefficients Xk, and Yk, are determined by varying the expectation value of the commutator:
(BCSll?,
p,‘;]
IBCS)
=
E,(BCS@‘,fp+IBCS)
(12)
Here we approximated as usual the ground state by the BCS wave function of the initial or the final IO+) state. The variation of (12) yields the projected QRPA (PQRBA) equations [27]. The particle number projection stabilizes the results around gpp x 1. Figure 6 shows the IvP&Gamow-Teller matrix element as a function of gpp for QTDA (no ground state correlations) and for QRPA with (PQTDA and PQRPA) and without (QTDA and QRPA) particle number projection for 13’Te to 13’Xe.
49c
Fig. 6. Dependence of the ZV,&?Gamow-Teller matrix element (271in QTDA (Quasi particle Tamm-Dancov~ and &EPA with (PQTDA~ and without (Q~PA~ particle number projection before variation Table 1 shows a comparison of PQRPA half lives with experimental data. The agreement between theory and experiment normally is satisfactory. Due to the fast variation of the GT matrix elements with g,,(a 1) for ‘OOMo the predictive power of the theory is very low for this and also in other nuclei.
I’QWA
Table I: 2~~~ theoretical 127,331 and experimental f7,14,29,30,3i,32~ half~iv~, The theoretical results are obtained with the particle number Projected Random Phase Approximation (PQRPA) [27,33].
Second we allowed for the higher order random phase approach (H&FLPA) to describe the states in the intermediate nucleus [28]. We mixed to the one phonon 1: states also two [[lf
> I2f >11* and three phonon
111;’ > 12: > 12: >11+ states.
We included
excitations of the first 2f state in the initial and the final nucleus in the description of the intermediate l+ states. Figure 7 shows the GT matrix element[MeV”j of the 2v/3@ transition in 82Se to 82.Kp.. Again HQRPA stabilizes the values around gpp M 1 to be different from zero. The third order contribution is large since ~~~~~~2~~~2~~~ can be coupled to angular momentum zero and thus modifies the ground state correlations. v
r-
-,
i
8’
I’
r-
3
RP.4
-
RPA.t2b t
-
-
-
RPA+Zb+3D
_ 0.0
0.2 o.* 0.8 0.8 LO Particle-Partick Strength go.
12
Fig. 7. Gamow-Teller matrix element [MeV-*] of the 2vp,O transition in szSe to The solid fine is the usual QRPA with the Bonn potential. The dotted curve includes two bosons 111: > 12: >I%+and the ~~hed”dotted curve also three boson states [ll;f > (2: > 12: >]i+ for the jnter~edjate nucleus in a higher order QRPA (HQRPA) [ZSl. The results are plotted as a function of the pa~tjc~~partic~e strength
d2X~.
%P
A third point we improved is the inclusion of the proton-neutron pairing f37,38,39]. The intermediate states in the add-odd mass nucleus are proton-particle and neutron-hole or neutron-particle and proton-hole states relative to the initial or the final nucleus, respectively. For proton-neutron excitated states one expects that pn-pairing plays an important role. If one uses the base pn Brckner matrix elements of the Bonn potential 137,381 one obtains the lower energy so&on either for pn-paired or unpaired solutions. But in spherical nuclei J” = O+ pairing can only include T = 1, ‘S pairing. T = 0, 35'( J" = l+) pairing must be considered by renormalizing the pn, J = O+ particle-particle matrix elements by a factor dpn > 1. This parameter is adjusted to the pn-pairing gap defined experimentally by the equations [40,41]:
SIC
A. Faessler I Grand uni~ca~i~n
+ 1, N) - $i(Z,
4
x
M(Z
A,,
=
M(Z,N-l)-
6pn M +
$(Z,
~[~(2,~)+~(2,~-2)]
N) + M(Z
AP+A,
N) + M(Z + 2, N)] (131
+ 2, N - 2)]
-M(Z+l,N-1)
These formulas are for the double beta decay (2, N) + (2 -i- 1, N - 1) -+ (2 + 2, N - 2). In redity we use higher order inte~olation form&s than linear ones. We a&socompare the first two equation with the lowest quasi-particle energies Es = [(Z&i - X,)’ + A;]““. pn-pairing is described with the Bogoliubov transformation:
A$ =
?&piC~
+
Uynic$ + VrpiCpZ+ t'~n
(14)
With an isospin qu~i-p~ti~Ie quantum number 7 = 1,2. The intermediate states axe now defined by applying the boson operator to the initial or the fkmI nuclear ground state [O&). (15) The theoretical gap is now determined by
46 22Tj24 /
d pn
___-
---_)
Fig. 8: Theoretical pn gaps S,, (13) as a function of the renormaiisation factor (a,, for the pn Ji = O+ Bruckner particle-partjc~e matrix elements of the Bonn poteat~~ (421. The experimental gaps 6;7(13) determine the renormalization factors dPn = 1.41 and dP,, = 1.36 for *“Ti and 46Ti, respectively.
52c
A. Faessler I &arid ~n~~co~ion
bpn =
{(~lfilpn) + JY + E,p”- [(fi) + El
-
l~nf is the BCS state with pn-pairing and E: and Er two lowest quasi-particle energies with pn-pairing.
1.6
MBCS
/ _
-----_.___
.:’ .
0.4.-
o.o-
,:
/’
-___
k.”
.’
_...' '. _,.,..,.__....~~~'~~~ 1 t \ \\
-I '1 -.4 I, 5, (1 * f 8, 1 0.0 0.2 0.4 0.6 0.8 1.0 1.2
9
06)
are for the odd-odd nucleus the
0.8 L
+ &I}
PP
Fig. 9: The second order Gamow-Teller (GT) matrix element for the 2u&3 decay 4sCa d4s SC d4’ Ti. The results for BCS and BCS,, with pn-pairing are calculated with the completeness relation using an energy denominator determined by the systematics of the GT resonance 1431. The figure shows further the QTDA and the &WA results without and with pR-pairing.
A. Faessler I Grand unification
----
QRPA without
53c
np pair.
Ge76+Se76
1.6 -I
‘< :
-A/ 0.0
0.2
0.4
0.6
9
0.8
1.0
1. 2
PP
Fig.10: Gamow-Teller matrix element of the 2@/3 decay for ‘6Ge to 76Se as a function of the particle- particle strength parameter gPP. The solid curve geves the result of QRpa with pn-pairing, while the same result without pn- pairing is given as the dashed line. Figure 8 shows the theoretical proton-neutron (pn) “gap” S,,, as defined in eq. (16) as a function of the,@ctor of the pn O+ particle-particle matrix element dpn. dpn = 1 characterizes the bare Brckner reaction matrix elements of the Bonn potential. The proton-proton and the neutron-neutron matrix elements are multiplied by dpp and A, to reproduce the odd-even mass differences A, and AR defined in eq. (13). While dp,, and dnn are close to unity (bare Brckner reaction matrix elements of the Bonn potential [42]), one needs renormalisation factors dpn of 1.3 to 1.5 for the pn monopole matrix elements (see figure 8), to reproduce S,, defined experimentally by the masses in eq. (13). Fig. 9 gives the dependence on the factor of the pn particle- particle matrix elements gpp in the QRPA equation of the second order Glow-Teller (GT) matrix element from the ground state of &Ca to the ground state of 48Ti. One sees the well known collapse of the QRPA result with and without pn-pairing around gpp = 1. Since already at gpp = 0 QR%n includes the J” = O+ pn particle-particle matrix element with pn-pairing is smaller over the whole range of gpp. This reduction is due to the cancellations of the
54c
A. Faessler I Grand un@cation
forward and backward going diagrams
in the second leg of the 2upp decay. (17)
The RPA coefficients XF= are naturally always larger than Yg. But the coefficient u,v,, in front of Xpun is small, while urv,, in front of Yr: is large for nuclei with a neutron excess. The ground state correlations and thus Y,*n are enlarged with g,, and around = 1 the two terms in (17) cancel each other. For QTDA (quasi-particle Tamm QPP Dancov approximation) the second term in (17) is missing. The GT matrix element is in this case increasing with gpp (see fig. 6 and 9) since the GT collectivity is increasing with gpp. This qualitative features are not changed with pn-pairing (see fig. 9). pn-pairing is generally reducing the absolute value of the GT matrix element since the monopole pn matrix element is already included in its full size in BCS,, (for all gpp. Fig. 10 shows that in some nuclei inclusion of pn-pairing can lead to about a constant or even increasing GT (2vpp) matrix element over the interesting range of gpp. In these nuclei the absolute values of the first term of (14) increases stronger than the second in QRPA, if pn-pairing
is included.
1oz4
1oz3
10
r
22
cd cJ
10
I9 0.0
0.8
0.4
1.2
g PP Fig. 11: Half life of 2upp decay of 76Ge to 76Se as a function of the particle-particle strength gpp. The solid line is the experiment of Klapdor and coworkers [ll]. The dashed line represents QRPA,, (with pn- pairing) and the dotted line is QRPA without pn-pairing. Figure 11 shows how the half life ri,z(2vp/3 is stabilized by pn-pairing against the in variation of gpp. As discussed in connection with Fig. 10. the two contributions
A.
Faessler I Grand
K~l~Cati~n
5%
eq. (17) to the GT(BvPP) matrix element have both an encreasing absolute value as a function of gPP. Together with the opposite sign both increases cancel each other and one obtains a good agreement with the new data of Klapdor and coworkers [ll].
3
The
neutrinoless
beta-decay
double
For the neutrinoless double beta-decay we have only lower limits for the lifetimes [10,11,14, 15,161. The theoretical transition probability for the zero neutrino double beta-decay can be written as P 04P
=
const
@iM,
< tgr’ >
+
MC+ <
M;/M;
> A&l2
(18)
with generally positive nuclear matrix elements M,,M( and MM, which include also the phase space factors. The experiment yields an upper limit for Pa”pp. Thus in the parameter space of the effective electron-neutrino mass < WC, > and the right-left (lepton and baryon vertices) handedness < tg(<) > and ~ght-~ght (lepton and baryon vertices) handedness < M:/M,2 > of the weak interaction relative to the left-handed one, one finds an ellipsoid, in which all allowed values for the neutrino mass < mve > and the right-handedness < tg[ > and < Mf/M,2 > must lie. Here Mr and M2 are the left and right-handed vector boson masses, respectively. C is the mixing angle WI = cos f ’ WL + sin < . W, of the left and right-handed vector bosons. Stringent limits for < a,, > and < tg{ > are obtained from the lower limit of the lz8Te lifetime ~11s> 5.10” years. The results are given in Figure 12 and in the conclusions including an analysis of the new data of Beck, Klapdor et al. [ll].
-2
-1
2
0
R ighthandedness
<$ x IO6
Fig. 12.: The allowed regions (inside the eilipsis~ deduced from the experjment~ bounds ~C~dwe~~,Berkeley Conference [15] and refs. 64 and 17) for the neutrinoless double P-decay 12sTe + “‘Xe with qla > 5. 10z4years. The upper limit for the effective neutrino mass is < m, >I 1.9eV and for the effective right-handedness < q >=< tg< >I L&10-” and < Mf/M.$ >I 6~10~~. Thelowerlimit ofKlapdoret al. rI/2 > 1.9x 1O24years [III for 76Ge yields: (m,,) 5 0.88eV; < tgC >I 1.55~ 10-s and (MfIM,Z) 5 1.08 x 10W6.
56c
4
A. Faessler I Grand un$cation
Conclusions
The neutrinoless double beta-decay (OY@/?) can distinguish if the neutrino is a Dirac particle, that means if the neutrino is different from the antineutrino, or if it is a Majorana particle and therefore identical with its antiparticle. Only in the case of a Majorana particle, the (OV@~) is p ossible. Grand unified theories predict that the neutrino is a Majorana particle, especially the ones which are built on the SO(10) group and thus are left-right symmetric. But being a Majorana particle the neutrino has to have a finite mass and for a left-right symmetric model also a slight right-handed weak interaction. Since the Oupp decay needs always a finite mass of the neutrino and is further increased by a right-handedness of the weak interaction. If one tries to test the calculation by the known 2v/3p decays one finds that the theory is by a factor 10 to 100 larger than the experimental data. But if one includes the particleparticle correlations of protons with neutrons, which are attractive, they cancel by a large part the neutron-particle and proton-hole correlations. Including these additional ground state correlations one finds [20 - 231 a strong quenching of the 2v/?p transitions in agreement with the experimental data. The quenching of the ,L?+branch in the 2vpp decay can also be tested in the single /3+ decay of the neutrino deficient nuclei. Including the particle-particle correlations one finds also agreement for these transitions which could not be understood before. For the Ovp/3 transition amplitude this quenching is for the leading recoil term only about a factor 0.7 or 30 %. This difference between the 2~p/3 and Ovp/? decay stems from the fact that in the Ovpp one has for the transition operator a dependence on the distance between the two vertices and thus, by a multipole expansion one gets a transition operator which excites also higher multipoles in the intermediate nucleus[22]. These higher multipoles are not strongly quenched and thus the Ovp,6 transitions are not so drastically affected by the particle-particle correlations. The lower experimental limit of the OV@/?half life in “Ge [ll] of ~112> 1.9 x 10z4 years allows to extract with our results [44] upper limits for the effective neutrino mass < meu >, the effective righthandedness < tgC > and < @/IV; >. < mve >
X
mue 5 0.88 eV
< tgc >
I
1.55.1o-s
5
1.08. 1O-6
< M;/M;
>
(19)
To extract an upper limit for the bare neutrino mass mev the bare mixing angle 6 of the left- and right-handed vector bosons and a lower limit for the mass of the heavy mainly right-handed vector boson M2, one has to know a specific model of Grand Unification. The upper limit of the effective neutrino mass is probably close to the bare one, while the bare values C and Mf/Mi of the two other quantities can be appreciably larger [4,34,35].
A. Faessler I Grand unification
57c
References [l] R. Davis , Phys. Rev. 97 (1955) 766 [2] T. D. Lee and C. N. Yang, Phys. Rev. 104 (1956) 254 [3] C. S. Wu , E. Ambler, Rev. 105 (1957) 1413
R. W. H ay ward , D. D. Hopper and R. P. Hudson, Phys.
[4] P. Langacker, Phys. Rep. 72 (1981) 185 [5] H. Fritzsch and R. Minkowski, Phys. Rep. 73 (1981) 67 R.N. Mohapatra, J.C. Pati, Phys. Rev. D 11 (1975) 2558 R.N. Mohapatra, G. Senjanovic, Phys. Rev. D 12 (1975) 1502 [6] Kirsten 1986 P rot. Int. Symp. Nuclear Beta-Decays and Neutrino,(Osaka, ed. Kotani, Ejiri, Takasugi (Singapore, World Scientific, p. 81)
1986)
[7] 0. K. Manuel, Proc. Int. Symp. on Nuclear Beta- Decays and Neutrino (Osaka June 1986, eds. Kotani, Ejiri, Takasugi (Singapore, World Scientific, p. 71) [8] S. R. Elliott, A. A. Hahn and M. K. Moe, Phys. Rev. Lett. 59 (1987) 2020 [9] S. R. Elliott, M. K. Moe, M. A. Nelson, M. A. Vient, J. Phys. G. 17 (1991) 145. H. Ejiri et al, J. Phys. G. 17 (1991) 155-164 [lo] F.T. Avignone, R.L. Brodzinski, C.G. Guerard, I.V. Kirpichnikov, H.S. Miley, V.S. Pogosov, J.H. Reeves, A.S. Starostin, A.G. Tamanyan, Phys. Lett. B 256 (1991) 559 [ll]
M. Beck et al. (Heidelberg-Moscow Collaboration, Jan.-Feb. 1993 and these Proceedings (1994)
[12] W. C. Haxton and G. J. Stephenson,Progr. [13] T. Tomoda, 591
Proc. XIII. Moriond Workshop,
Part. Nucl. Phys. 12 (1984) 409
A. Faessler, K. W. Schmid and F. Griimmer, Nucl. Phys. A452(1986)
[14] F. T. Avignone III, R. L. Brodzinski, J. C. Evans, Ir., K. Hensley, H. S. Miley and J. H. Reeves, Phys. Rev. C34 ( 1986) 666 [15] D. 0. Caldwell, R. M. Eisberg, D. M. Grumm, D. L. Hale, M. S. Witherell, F. S. Goulding, D. A. Laudis, N. W. Madden, D. F. Malone, R. H. Pehl and A. R. Smith, Phys. Rev. D33 (1986) 2737 [16] D. 0. Caldwell, Osaka Conference, ley Conference,
June 1986 (r$z(‘6Ge)
July 1986 (r1”/“2(76Ge)> 4.7. 1O23years)
[17] E. Nolte et al., Z. Physik A306
(1985) 223
[18] P. Kleinheina et al., Phys. Rev. Lett. 55 (1985) 2664
> 3.9. 1O23years); Berke-
58~
A. Faessler I Grand unification
[19] 0. Civitarese, A. Faessler and T. Tomoda,
Phys. Lett. B194 (1987) 11
[20] A. Bohr and B. R. Mottelson,
Nucl. Structure, Vol. 1 (Benjamin,
[21] G. Bertsch, The Practitioners
Shell Model (A merican Elsevier, New York, 1972)
[22] T. Tomoda
New York, 1969)
and A. Faessler, Phys. Lett. B199 (1987) 2383
[23] P. Vogel and M. R. Zirnbauer, Phys. Rev. Lett. 57 (1986) 3148 [24] K. Muto and H. V. Klapdor.,
Phys. Lett.201B
(1988) 420
[25] J. Engel, P. Vogel, 0. Civitarese and M. R. Zimbauer, [26] 0. Civitarese, A. Faessler, T. Tomoda, [27] 0. Civitarese, 424
Phys.Lett.
B208
Phys. Lett. B194(1987)
(1988) 187
11
A. Faessler, J. Suhonen, X. R. Wu, Nucl. Phys. A524
(1991) 404 -
[28] A. A. Raduta, A. F aessler, S. Stoica, W. A. Kaminski, Phys. Lett. B254 [29] L. V. Kirpich ni kov: Proc. Conf. on Lepton-Photon
Interaction,
(1991) 7
Stanford (1989)
[30] P. Fisher, F. Boehm, E. Bovet, J. P. Egger, H. Henrikson, K. Gabathuler, L. W. Mitchell, D. Renser, M. Treichel, J. L. Vuillenmier, Phys. Lett. I3218 (1989) 257 [31] T. Kirsten et al. Proc. Int. Symp. on Nuclear Beta- Decays and Neutrino; 1986); (Singapore, World Scientific) p. 251
(Osaka,
[32] S. R. Elliott, A. A. Hahn, M. K. Moe, Phys. Rev. Lett. 56 (1986) 2582 [33] 0. Civitarese, A. Faessler, J. Suhonen, X. R. Wu, Nuclear structure calculation of the two neutrino pp- decay transition ‘mMo _tlOo Ru; J. Phys. G. 17 (1991) 943 953 [34] Faessler A., Progr. Part. Nucl. Phs. 21 (1988) 183 [35] T. Kotani, 14th EPS Nucl. Phys. Conf. Bratislawa, Oct. 22 - 26, 1990; J. Phys. G [36] A. A. Klimenko et al., Institute of Nuclear Research Moscow and B&an, Comm. [37] A. Bobyk, M.K. Ch eoun, G. Teneva, A. Faessler, to be published
Personal
1993
[38] M. Cheoun, A. Bobyk, A. Faessler, F. Simkovic, G. Teneva, to be published [39] M.K. Cheoun, A. Faessler, to be published
1993
[40] P. Moeller, J.R. Nix, Nucl. Phys. A 536 (1992) 20 [41] D.G. Madland, J.R. Nix, Nucl. Phys. A 476 (1988) 1
1993
A. Faessler I Grand uni$cation
59c
1421 K. Holinde, Phys. Rep. 68 (1981) 121 [43] D.J. Horen, C.D. Goodman, D.E. Bainum, C.C. Forster, C. Gsarde, C.A. Goulding, M.B. Greenfield, J. Rapaport, T.N. Taddeuci, E. Sugarbaker, T. Masterson, SM. Austin, A. Galonsky and W. Sterrenberg, Phys. Lett. B 99 (1981) 383 1441 J. Subonen, 0. Civitarese,
A. Faessler, Nucl. Phys. A 543 (1992) 645
Nuclear Physics A570 (1994) 39c-60~ North-Holland, Amsterdam
GRAND
UNIFICATION,
NUCLEAR AND
THE
University
STRUCTURE
DOUBLE
BETA-DECAY
Amand Faessler of Tiibingen, Institute of Theoretical 72076 Tiibingen, Germany
Physics
ABSTRACT
Unification of the electroweak and the strong interaction prefers that the neutrino is a Majorana particle and therefore essentially identical with its own antiparticle. In such grand unified models the neutrino has also a finite mass and a slight right-handed weak interaction, since the modeJ is left-~ght symmetric. These models have also left handed and right-h~ded vector bosons to mediate the weaJc interactions. If these models are correct the neutrinoless double beta-decay is feasable. Thus if one finds the neutrinoless double beta-decay one knows that the standard model can not be correct in which the neutrino is a Dirac particle and therefore different from its antiparticle. Although the neutrinoless double beta-decay has not been seen it is possible to extract from the lower Jimits of the lifetime against the doubte neutr~oJess beta-decay upper Jim&s for the effective eiectron-neutr~o mass and for the effective mixing a.ngJe of the right-h~ded and the left-handed vector bosons mediating the weak interaction. One also can obtain a.n effective upper limit for the mass ratio of the light and the heavy vector bosons. The extraction of this physical quantities from the data is made difficult due to the fact that the weak interaction must not be diagonal in the representation of the mass matrix of the six neutrinos requested by such left-right symmetric models. A condition for obtaining reliable Jimits for these fundamental quantities from the measured lower limits of the half lifes of the Ov/3p decay are correct calculations of the nuclear matrix elements involved. These nuclear structure calculations can be tested by calculating the two neutrino double beta decay (2~/3/3) for which we have experimental data and not only lower limits as for the OV@/? decay. The Zv/3@ decay is dominated by the C&now TeJler (GT) transitions. The inte~ediate l+ states in the odd-odd mass nucleus axe usua.JJy calculated within the Quasi-particle Random Phase Approcimation (QJZPA). Since the proton-proton and neutron- neutron pairing correlations are treated in the BCS approach particle numbers are only conserved in average for the initial 10:) and the final IOf) states. Particle number projection is used to improve on this point. Since QRPA with the physical p~ticJe-~ticJe interaction from the Bonn potential is close to breakdown, where the excitation energy goes to zero, higher up to third order RPA is included, by allowing anharmonicities up to three bosons in the physicaf wave functions. Third order RPA which is the lowest order to allow modifications of the ground state correlations &ect the results appreciably. Since the intermediate states 0375-9474~4/$07.~ 0 1994 - Elsevier Science B.V. All rights reserved. SSDI 0375-~74(~~~77-2
4oc
A. FaesslerI Grand~az~ca~o~
are supperpositions ofproton- neutron two quasi-particle states one expects that protonneutron pairing affects these states. From the masses of the nuclei one can extract a proton-neutron (pn) pairing gap. Due to the finite basis and the inclusion of only T = 1 (and not also T = 0, J = 1) pn-pairing the bare Brckner matrix elements of the Bonn potential yield no pn-pairing gap in most nuclei. If one fits the experimental pn-pairing gap by renormalizing the pn J” = Ot particfe-particle Brckner matrix elements by a factor dpn R 1.3 to 1.4 one obtains strong pn-pairmg correlations, which reduce the GT strength. Since now part of the pn particle-particle correlations are akeady included in the BCS wave functions the ‘Zv/!Q?strength is reduced and the dependence on the l+ ~tic~~p~tjc~e stren~h parameter gW gets weaker. In heavier nuclei Iike 76Ge to 76Se this dependence on gPPof the second order GT matrix elements C, GT(Of --f 1: -+ 0:) host disappears. The upper limits derived from the neutrinoless double beta decay of 76Ge are for the effective neutrino mass < mue >I 0.88 eV, for the left-right mixing angle < tg< >I 1.55 x lo-* and for the mass ratio of the light over the heavy vector boson squared < MT/M; >< 1.08 x 10B6.
1
Introduction
The most important particle of this talk is the neutrino. It has always been very interesting since it was predicted by Wolfgang Pauli on December 4th, 1930 in a letter to a conference in T~bingen. Still today the neutrino is the only fermion from which we do not know if it is different from its antiparticle (U # F) and therefore a Dirae particle or if it is identical with its antiparticle (u = V) and therefore a Majorana particle. In 1955 the physicists did believe that this question was solved in favour of a Dirac neutrino by an experiment of Davis [I]. He was using antineutrinos from the beta-decay in a reactor to induce the inverse beta-decay in 37Cl for which one needs in the standard model a neutrino: n V+37ca
-+ -+
p+e-+P
(1)
37Ar + e-
Since he did not observe the formation of 37Ar by the inverse beta decay he assumed that the antineutrino v is different from the neutrino u and thus the neutrino is a Dirac particle. But this conclusion was outdated already two years later. In 1956 Lee and Yang[Z] proposed to test if in the weak interaction parity is conserved, and in 1957 Wu et al.[S] did find that indeed parity is violated in the beta-decay and soon it turned out that it is violated maximally and the weak interaction is purely left-handed. Therefore, the neutrino created in the beta-decay of the neutron in eq. (1) has a positive helicity (is right-handed), while the neutrino needed for the inverse beta-decay must have a negative helicity and therefore needs to be left-handed. Thus, even if the neutrino is a Majorana particle, reaction (1) would be forbidden due to helicity mismatch of the two reactions. Thus, we know already since about 30 years that the problem, if the neutrino is a Dirac or if it is a Majorana particle, is not solved. But why are we just today discussing this question so intensively? The reason is that the grand unified theories from which we
A. Faessler I Grand unification
41c
think that they can be successful predict that the neutrino is a Majorana particle[4]. Measurements of the lower limit of the proton lifetime seem to exclude SU5 and thus one concentrates on SOlO, which is also a subgroup of dynamical groups discussed in superstring theories. Left-right symmetric theories inaugurated by Mohapatra, Pati and Senjanovic [5] and especially theories based on SO10 which have first been proposed by Fritzsch and Minkowski[S] predict in improved versions[4] not only, that the neutrino is a Majorana particle, but automatically predict dso that the neutrino has a mass and a weak right-handed interaction. The basic idea behind grand unified models is an extension of the local gauge invariance from quantum chromodynamics (SU3) involving only the coloured quarks also to electrons and neutrinos. The presently favoured models are left-right symmetric models. They contain left- and right-handed vector bosons W: and W;.
W*
=
cos<.WLf+sin6.W;
w:2
=
-sin<.
W:
+cosC.
(2) Wi
The left- and right-handed vector bosons are mixed if the mass eigenstates are not identical with the weak eigenstates, which have a definite handedness. The left-right symmetry is broken since the vector bosons W: and W$ obtain different masses by the Higgs mechanism. Since we have not seen a right-handed weak interaction the mass of the heavy, mainly right-handed vector boson must be much larger than the mass of the light (81 GeV) vector boson, which is mainly left-handed. The weak interaction Hamiltonian must now be generalized. Hw
x
GF [(L.l)+tgC(R.I)+tg((L.r)+ Jz
-
LIR = &(svr, =FgA?p?'5)$'n gv
=
l;gA = 1.25
l/r
=
2(-Y, T
(g)‘(R-4] (3)
Ya^l5k
The capital L and R indicate the hadronic right- and left-handed currents changing a neutron into a proton. The lower case 1 and r are the left and right handed leptonic currents which annihilate a neutrino (or create an antineutrino) and create an electron (or annihilate a positron). C is the mixing angle of the vector bosons (2) and MI and Ms are the light and heavy vector boson masses. The weak interaction Hamiltonian is given for C << 1 and Mz >> Ml. Most of the left-right symmetric models predict that the neutrino is a Majorana particle that means that they are loosely speaking identical with their antiparticles (v = v). A Majorsna neutrino must have automatically a mass and due to construction it has also a slight right-handed interaction (3). As we will see below these facts allow the double beta-decay without neutrinos. Or inversely: The existence of the double neutrinoless beta-decay would establish that the neutrino is a Majorana particle. Figure 1 shows the diagram for the neutrinoless double beta-decay. This is naturally only possible if the neutrino is identical with its antiparticle
42c
A. Faessler I Grand un@cation
since at the first vertex, one must emit an antineutrino and at the second vertex one needs to absorb a neutrino in the standard model. Since in the final state we have only two particles in the continuum in the process in Figure 1, the phase space is bigger by a factor lo6 compared to the double beta-decay with two neutrinos. Thus, even if the matrix element squared is reduced by a factor 10m6,one has the same transition probability for the neutrinoless double beta decay as for the double beta-decay with two neutrinos.
P
P
ei
v, ’ ’ .’
n
r
@2
.’
1
:
-
n
Fig.1. Diagrams for the neutrinoless double beta-decay with a Majorana neutrino. By having only two particles in the final states in the continuum, the phase space is increased by a factor of about lo6 compared to the 2upp decay. Even with a Majorana neutrino this process is only possible if the neutrino has a finite mass and/or also a right-handed weak interaction. But a right handed weak current must be accompagnied by a finite neutrino mass to yield a finite o&3 decay.
But even if the neutrino is a Majorana particle, the process in Figure 1 can not happen since for a pure left-handed weak interaction theory, the emitted neutrino must be righthanded (positive helicity), while the absorbed neutrino must be left-handed (negative helicity). But grand unified theories predict also that the neutrino has a mass and a slight right-handed weak interaction. With a finite mass the neutrino has not any more a good helicity and the interference term between the leading helicity and the small admixtures allows a neutrinoless double beta-decay. The Ovpp decay propability is further increased with a massive neutrino and also right-handed vector bosons which allow a right-handed weak interaction. If the first vertex is a left handed V-A interaction and the second vertex is a right-handed V+A interaction, the process in Figure 1 is also allowed. (But a detailed analysis shows that also in this case one has to request a finite neutrino mass to allow the neutrinoless double beta-decay.) Thus, the neutrinoless double beta-decay can only happen due to the finite mass of the neutrino and can be additionally increased due to a small admixture of right-handed leptonic currents. The matrix element for the double neutrinoless beta-decay consists of three parts: One proportional to the neutrino mass mev and the other to the small right- handedness tg[ or (Mi/M2)2 of the weak interaction.
43c
A. Faessler I Grand un@ication
2
Description
of the two-neutrino
double
beta-decay
Since there are measurements available for the two neutrino double beta-decay with the geochemical method [6,7], and for three nuclei even laboratory measurements[S - 111, one could try to calculate for a test of the theory the double beta-decay with two neutrinos and compare them with the data. If one performs this calculation, one finds that the calculated transition probability for the double beta-decay with two neutrinos (2vp/3) is by a factor 50 to 100 too large[l9]. This discrepancy which is typical for the Zv/?/3decay naturally casts doubt on the reliability of the calculations of the OY@/~ transition probabilities. The nuclear many-body methods for calculating the double beta-decay are either shell model calculations[l2] or extended shell model calculations based on the Hartree-FockBogoliubov approach called MONSTER [13] or they are based on the Random Phase Approach (RPA). Let’s try to asialyse the 2u@ decay in the RPA approximation. 6
E’ig. 2. The upper part shows the way how in the Random Phase App~xjmation (EPA] the S/?/3 decay is calculated. For the Fermi transitions the @-(a + p) amplitude moves just a neutron into the same proton level and the p+(p + n) amplitude moves a proton into the same neutron level. For the Gamow-Teller transitions it can also invoke a spin flip, but the orbital part remains the same. One immediately realizes that the occupation and non-occupation amplitudes favour the p- amplitude, but disfavour the p+ amplitude. There one has a transition from an unoccupied to an occupied single particle state, which is two-fold small (s’) first by the fact that the occupation amplitude far the proton v* and secondly that the unoccupation amp~jtude for the neutron state 21, are both small. Therefore the 2vflfl is drastically reduced. Figure 2 explains why the 2vpp decay amplitude is so drastically reduced. Therefore the small effects which normally do not play a major role can affect the 2~~~ transition
44c
A. Faessler I Grand u~t~cation
probability. If one looks to the second leg of the double beta-decay which is calculated backwards as a p+(p -+ n) decay from the final nucleus to the intermediate nucleus one finds that the matrix elements involved in these diagrams are Pauli suppressed by a factor (u,~r)~ = (s~a~~)4. The neut~n-p~ticle proton-hole force in the isovector channel, which is usually included is repulsive while the particle-particle force usually neglected is attractive. Therefore both excitations tend to cancel each other and therefore the amplitude pf is drastically reduced. In this way one can obtain a better agreement with the experimental data. We calculated the 2v/3@decay half-lives for the following 0: -_) 0: transitions: 4sCa +48 Td t*s Cc -+46 Ti >76 Ge -+ 76Se 9 g2Se -+ 82Kr 2 ‘?l’e --f lBXe ad 130Te -+ lmXe. Although one can obtain agreement in this way with the measured 2v/3@data multiplying the particle-particle matrix elements with a factor g,,, in a range of 0.8 2 g,, 2 1.2, one observes a tendency for the unrenormalized Bfickner reaction matrix elements 9r,, = 1 of the different realistic forces, that the particle-hole and the particle-particle contributions to the 2v/3/3transition cancel each other to zero. The physical reason for this cancellation can be understood in Wigner’s supermultiplet model SU4. The Gamow-Teller (GT) operator ,& ait+; are generators of the Wigner supermultiplet symmetry SU4. If this would be a dynamical symmetry of the nuclear many body Hamiltonian the 2v@ GT transition would lead only to the double GT resonance and not to the ground state. But it is well-known that the spin-orbit splitting destroys the SU4 symmetry in the wave functions of medium heavy and heavy nuclei and this is only an approximate symmetry. The more technical reason for this cancellation is that for the second leg the backgoing amplitudes and thus groundstate correlations cancel the leading term.
Eq. (4) defines the QBPA boson for an l+ state la) in the intermediate nucleus based on the ground state of the &ml nucleus IOf). Eq (2) shows the expression for the Gamow Teller (GT) strength from the final state IOF> into the 1+ state IQ) in the intermediate nucleus. The operators a$, a& create neutron and proton quasi-particles. a+ Cc S Cijm
=
U;C+
=:
(-)j-"Cij_m
+
ViC; (6)
The cancellation of the GT strength in the second leg for Briickner matrix elements of the Bonn potential (gm, = 1) can be seen in eq. (5) by looking to the expression in the round brackets ( ). For nuclei with a neutron excess (see fig. 2) the amplitudes t&?&p are small and uipZ)knare large for the second leg. Although X,$,+ is large and x&,, is small, the products in the round brackets cancel each other for gpp = 1 (bare Brckner matrix elements).
A. Faessler I Grand un@ication
4%
Figure 3 shows the second order GT matrix element as function of the particle-particle strength parameter gPP for 76Ge to ‘6Se.
(7)
Here Ia) are the l+ state in the intermediate nucleus built on the initial nucleus and 1~) are the l+ states built on the final nucleus.
0.8
_-
L-
is
+
SP
+ -%
tY
+7
\\
0.4
7 + .. 0 t&J
g
0.0
--_-t-w,___-_-_
I_--_-__
I
__.
._
._
\
W-0.2
r0.0
0.4 g
PP
0.8 ___ --_._*
1.2
Fig. 3: Second order Gamow-Teller (GT) matrix elements for 76Ge to 76Se (7) as a function of the particle-particle strength parameter gPP with which all particleparticle matrix elements of the Bonn potential in the QRPA equation are multiplied.
46c
A. Faessler / Grand unification
Fig. 4.: Hdf life of 2upp “‘Se to “‘h’r as a function of the particle-particle strength gP,,. The data are from geochemicd measurements in refk [6,7] and from the first Isboratory measurements of Moe et al f8].
Figure 4 shows the half life of “Se to s21’r as a function of the particle-particle strength g,,. At gpP = 1 the particle-hole and the particle-particle contributions cancel (see eq. (2)) and the half life goes to infmity. Thus one sees that one is in principle able to understand the data by slightly changing ga, around unity (0.8 2 gpp 5 1.2), but the predictive power is very small [26]. The cancellation of the particle-particle and the particle-hole matrix dements for the 2upp at around gpp = 1 means that the QRPA in its present form is not accurate enough. We have therefore improved it in three ways:
47c
2t4
2
zt2
3
E &Z
z-2 N-4
N-2
Iv
Nt2
Nclutrons
The second leg 4”Sc to “Ti is calculated in the inverse direction. The leading ampijtude for the second leg is proportional to u~~~~~zL,,~~~~ (plus a smaller additional term proportional to v,,~~~~u~~~~~). This amplitude is small since in average one has not more than two protons in the j712 JeveI in 48Ti and one has for the neutrons the f712 almost full (94,/7/z = small). To iaTiN one admixes the four nuclei .Z z?c2, N f 2 by amplitudes around 0.6. For Z = 24, N = 26, which has a wrong particie number for the final nucleus one has twice as many protons in the fTi2 sheik One could therefore expect a iarge error on the 2u@ decay probability from particle number non-conservation.
Fig. 5.: The Bv&3 decay 4sCa to 4sTi in the isotopic chart.
First we included paxticle number projection on the correct proton and neutron number before the BCS and the RPA variation 1251. The B~den-Cooper-Schriefer (BCS) model used for treating the pairing correlations does not conserve the particle number.
IBCS)
=
IIi>*(Ui t WiQC;t)lO)
(8)
48c
A. Faessler I Grand unification
Although one conserves the particle numbers for protons and neutrons in average,
c vf= 2
(BCSlilBCS) =
(9)
i=p>o
c v:= N
(BCSlAIBCS) =
k=n>o
one admixes nuclei with two protons or two neutrons more or less by about 0.6 if the nucleus with 2 and N has the amplitude 1. One can expect in nuclei with neutron excess a large effect for 2vp,f3. Let us consider the decay $$ass -+g Sczr +g T&e in figure 5. We therefore used particle number projection to circumvent this difficulty. The ui,ui coefficients of the Bogoliubov transformation (6) are now determined by minimizing the projected and normalized expectation value of the Hamiltonian in the BCS states.
(BCSlI?&&lBCS)
= Min
(10)
(BCSl&&IBCS)
The states in the intermediate nucleus are given by:
II:, QRPA;
Z + 1, N - 1)
=
&+&-I
=
e+,kd
z
{
x~k,da~ka$
-
~,nlhd%k}l+
b)
(11)
The Euler-Lagrange eqs. for the coefficients Xk, and Yk, are determined by varying the expectation value of the commutator:
(BCSll?,
p,‘;]
IBCS)
=
E,(BCS@‘,fp+IBCS)
(12)
Here we approximated as usual the ground state by the BCS wave function of the initial or the final IO+) state. The variation of (12) yields the projected QRPA (PQRBA) equations [27]. The particle number projection stabilizes the results around gpp x 1. Figure 6 shows the IvP&Gamow-Teller matrix element as a function of gpp for QTDA (no ground state correlations) and for QRPA with (PQTDA and PQRPA) and without (QTDA and QRPA) particle number projection for 13’Te to 13’Xe.
49c
Fig. 6. Dependence of the ZV,&?Gamow-Teller matrix element (271in QTDA (Quasi particle Tamm-Dancov~ and &EPA with (PQTDA~ and without (Q~PA~ particle number projection before variation Table 1 shows a comparison of PQRPA half lives with experimental data. The agreement between theory and experiment normally is satisfactory. Due to the fast variation of the GT matrix elements with g,,(a 1) for ‘OOMo the predictive power of the theory is very low for this and also in other nuclei.
I’QWA
Table I: 2~~~ theoretical 127,331 and experimental f7,14,29,30,3i,32~ half~iv~, The theoretical results are obtained with the particle number Projected Random Phase Approximation (PQRPA) [27,33].
Second we allowed for the higher order random phase approach (H&FLPA) to describe the states in the intermediate nucleus [28]. We mixed to the one phonon 1: states also two [[lf
> I2f >11* and three phonon
111;’ > 12: > 12: >11+ states.
We included
excitations of the first 2f state in the initial and the final nucleus in the description of the intermediate l+ states. Figure 7 shows the GT matrix element[MeV”j of the 2v/3@ transition in 82Se to 82.Kp.. Again HQRPA stabilizes the values around gpp M 1 to be different from zero. The third order contribution is large since ~~~~~~2~~~2~~~ can be coupled to angular momentum zero and thus modifies the ground state correlations. v
r-
-,
i
8’
I’
r-
3
RP.4
-
RPA.t2b t
-
-
-
RPA+Zb+3D
_ 0.0
0.2 o.* 0.8 0.8 LO Particle-Partick Strength go.
12
Fig. 7. Gamow-Teller matrix element [MeV-*] of the 2vp,O transition in szSe to The solid fine is the usual QRPA with the Bonn potential. The dotted curve includes two bosons 111: > 12: >I%+and the ~~hed”dotted curve also three boson states [ll;f > (2: > 12: >]i+ for the jnter~edjate nucleus in a higher order QRPA (HQRPA) [ZSl. The results are plotted as a function of the pa~tjc~~partic~e strength
d2X~.
%P
A third point we improved is the inclusion of the proton-neutron pairing f37,38,39]. The intermediate states in the add-odd mass nucleus are proton-particle and neutron-hole or neutron-particle and proton-hole states relative to the initial or the final nucleus, respectively. For proton-neutron excitated states one expects that pn-pairing plays an important role. If one uses the base pn Brckner matrix elements of the Bonn potential 137,381 one obtains the lower energy so&on either for pn-paired or unpaired solutions. But in spherical nuclei J” = O+ pairing can only include T = 1, ‘S pairing. T = 0, 35'( J" = l+) pairing must be considered by renormalizing the pn, J = O+ particle-particle matrix elements by a factor dpn > 1. This parameter is adjusted to the pn-pairing gap defined experimentally by the equations [40,41]:
SIC
A. Faessler I Grand uni~ca~i~n
+ 1, N) - $i(Z,
4
x
M(Z
A,,
=
M(Z,N-l)-
6pn M +
$(Z,
~[~(2,~)+~(2,~-2)]
N) + M(Z
AP+A,
N) + M(Z + 2, N)] (131
+ 2, N - 2)]
-M(Z+l,N-1)
These formulas are for the double beta decay (2, N) + (2 -i- 1, N - 1) -+ (2 + 2, N - 2). In redity we use higher order inte~olation form&s than linear ones. We a&socompare the first two equation with the lowest quasi-particle energies Es = [(Z&i - X,)’ + A;]““. pn-pairing is described with the Bogoliubov transformation:
A$ =
?&piC~
+
Uynic$ + VrpiCpZ+ t'~n
(14)
With an isospin qu~i-p~ti~Ie quantum number 7 = 1,2. The intermediate states axe now defined by applying the boson operator to the initial or the fkmI nuclear ground state [O&). (15) The theoretical gap is now determined by
46 22Tj24 /
d pn
___-
---_)
Fig. 8: Theoretical pn gaps S,, (13) as a function of the renormaiisation factor (a,, for the pn Ji = O+ Bruckner particle-partjc~e matrix elements of the Bonn poteat~~ (421. The experimental gaps 6;7(13) determine the renormalization factors dPn = 1.41 and dP,, = 1.36 for *“Ti and 46Ti, respectively.
52c
A. Faessler I &arid ~n~~co~ion
bpn =
{(~lfilpn) + JY + E,p”- [(fi) + El
-
l~nf is the BCS state with pn-pairing and E: and Er two lowest quasi-particle energies with pn-pairing.
1.6
MBCS
/ _
-----_.___
.:’ .
0.4.-
o.o-
,:
/’
-___
k.”
.’
_...' '. _,.,..,.__....~~~'~~~ 1 t \ \\
-I '1 -.4 I, 5, (1 * f 8, 1 0.0 0.2 0.4 0.6 0.8 1.0 1.2
9
06)
are for the odd-odd nucleus the
0.8 L
+ &I}
PP
Fig. 9: The second order Gamow-Teller (GT) matrix element for the 2u&3 decay 4sCa d4s SC d4’ Ti. The results for BCS and BCS,, with pn-pairing are calculated with the completeness relation using an energy denominator determined by the systematics of the GT resonance 1431. The figure shows further the QTDA and the &WA results without and with pR-pairing.
A. Faessler I Grand unification
----
QRPA without
53c
np pair.
Ge76+Se76
1.6 -I
‘< :
-A/ 0.0
0.2
0.4
0.6
9
0.8
1.0
1. 2
PP
Fig.10: Gamow-Teller matrix element of the 2@/3 decay for ‘6Ge to 76Se as a function of the particle- particle strength parameter gPP. The solid curve geves the result of QRpa with pn-pairing, while the same result without pn- pairing is given as the dashed line. Figure 8 shows the theoretical proton-neutron (pn) “gap” S,,, as defined in eq. (16) as a function of the,@ctor of the pn O+ particle-particle matrix element dpn. dpn = 1 characterizes the bare Brckner reaction matrix elements of the Bonn potential. The proton-proton and the neutron-neutron matrix elements are multiplied by dpp and A, to reproduce the odd-even mass differences A, and AR defined in eq. (13). While dp,, and dnn are close to unity (bare Brckner reaction matrix elements of the Bonn potential [42]), one needs renormalisation factors dpn of 1.3 to 1.5 for the pn monopole matrix elements (see figure 8), to reproduce S,, defined experimentally by the masses in eq. (13). Fig. 9 gives the dependence on the factor of the pn particle- particle matrix elements gpp in the QRPA equation of the second order Glow-Teller (GT) matrix element from the ground state of &Ca to the ground state of 48Ti. One sees the well known collapse of the QRPA result with and without pn-pairing around gpp = 1. Since already at gpp = 0 QR%n includes the J” = O+ pn particle-particle matrix element with pn-pairing is smaller over the whole range of gpp. This reduction is due to the cancellations of the
54c
A. Faessler I Grand un@cation
forward and backward going diagrams
in the second leg of the 2upp decay. (17)
The RPA coefficients XF= are naturally always larger than Yg. But the coefficient u,v,, in front of Xpun is small, while urv,, in front of Yr: is large for nuclei with a neutron excess. The ground state correlations and thus Y,*n are enlarged with g,, and around = 1 the two terms in (17) cancel each other. For QTDA (quasi-particle Tamm QPP Dancov approximation) the second term in (17) is missing. The GT matrix element is in this case increasing with gpp (see fig. 6 and 9) since the GT collectivity is increasing with gpp. This qualitative features are not changed with pn-pairing (see fig. 9). pn-pairing is generally reducing the absolute value of the GT matrix element since the monopole pn matrix element is already included in its full size in BCS,, (for all gpp. Fig. 10 shows that in some nuclei inclusion of pn-pairing can lead to about a constant or even increasing GT (2vpp) matrix element over the interesting range of gpp. In these nuclei the absolute values of the first term of (14) increases stronger than the second in QRPA, if pn-pairing
is included.
1oz4
1oz3
10
r
22
cd cJ
10
I9 0.0
0.8
0.4
1.2
g PP Fig. 11: Half life of 2upp decay of 76Ge to 76Se as a function of the particle-particle strength gpp. The solid line is the experiment of Klapdor and coworkers [ll]. The dashed line represents QRPA,, (with pn- pairing) and the dotted line is QRPA without pn-pairing. Figure 11 shows how the half life ri,z(2vp/3 is stabilized by pn-pairing against the in variation of gpp. As discussed in connection with Fig. 10. the two contributions
A.
Faessler I Grand
K~l~Cati~n
5%
eq. (17) to the GT(BvPP) matrix element have both an encreasing absolute value as a function of gPP. Together with the opposite sign both increases cancel each other and one obtains a good agreement with the new data of Klapdor and coworkers [ll].
3
The
neutrinoless
beta-decay
double
For the neutrinoless double beta-decay we have only lower limits for the lifetimes [10,11,14, 15,161. The theoretical transition probability for the zero neutrino double beta-decay can be written as P 04P
=
const
@i
< tgr’ >
+
MC+ <
M;/M;
> A&l2
(18)
with generally positive nuclear matrix elements M,,M( and MM, which include also the phase space factors. The experiment yields an upper limit for Pa”pp. Thus in the parameter space of the effective electron-neutrino mass < WC, > and the right-left (lepton and baryon vertices) handedness < tg(<) > and ~ght-~ght (lepton and baryon vertices) handedness < M:/M,2 > of the weak interaction relative to the left-handed one, one finds an ellipsoid, in which all allowed values for the neutrino mass < mve > and the right-handedness < tg[ > and < Mf/M,2 > must lie. Here Mr and M2 are the left and right-handed vector boson masses, respectively. C is the mixing angle WI = cos f ’ WL + sin < . W, of the left and right-handed vector bosons. Stringent limits for < a,, > and < tg{ > are obtained from the lower limit of the lz8Te lifetime ~11s> 5.10” years. The results are given in Figure 12 and in the conclusions including an analysis of the new data of Beck, Klapdor et al. [ll].
-2
-1
2
0
R ighthandedness
<$ x IO6
Fig. 12.: The allowed regions (inside the eilipsis~ deduced from the experjment~ bounds ~C~dwe~~,Berkeley Conference [15] and refs. 64 and 17) for the neutrinoless double P-decay 12sTe + “‘Xe with qla > 5. 10z4years. The upper limit for the effective neutrino mass is < m, >I 1.9eV and for the effective right-handedness < q >=< tg< >I L&10-” and < Mf/M.$ >I 6~10~~. Thelowerlimit ofKlapdoret al. rI/2 > 1.9x 1O24years [III for 76Ge yields: (m,,) 5 0.88eV; < tgC >I 1.55~ 10-s and (MfIM,Z) 5 1.08 x 10W6.
56c
4
A. Faessler I Grand un$cation
Conclusions
The neutrinoless double beta-decay (OY@/?) can distinguish if the neutrino is a Dirac particle, that means if the neutrino is different from the antineutrino, or if it is a Majorana particle and therefore identical with its antiparticle. Only in the case of a Majorana particle, the (OV@~) is p ossible. Grand unified theories predict that the neutrino is a Majorana particle, especially the ones which are built on the SO(10) group and thus are left-right symmetric. But being a Majorana particle the neutrino has to have a finite mass and for a left-right symmetric model also a slight right-handed weak interaction. Since the Oupp decay needs always a finite mass of the neutrino and is further increased by a right-handedness of the weak interaction. If one tries to test the calculation by the known 2v/3p decays one finds that the theory is by a factor 10 to 100 larger than the experimental data. But if one includes the particleparticle correlations of protons with neutrons, which are attractive, they cancel by a large part the neutron-particle and proton-hole correlations. Including these additional ground state correlations one finds [20 - 231 a strong quenching of the 2v/?p transitions in agreement with the experimental data. The quenching of the ,L?+branch in the 2vpp decay can also be tested in the single /3+ decay of the neutrino deficient nuclei. Including the particle-particle correlations one finds also agreement for these transitions which could not be understood before. For the Ovp/3 transition amplitude this quenching is for the leading recoil term only about a factor 0.7 or 30 %. This difference between the 2~p/3 and Ovp/? decay stems from the fact that in the Ovpp one has for the transition operator a dependence on the distance between the two vertices and thus, by a multipole expansion one gets a transition operator which excites also higher multipoles in the intermediate nucleus[22]. These higher multipoles are not strongly quenched and thus the Ovp,6 transitions are not so drastically affected by the particle-particle correlations. The lower experimental limit of the OV@/?half life in “Ge [ll] of ~112> 1.9 x 10z4 years allows to extract with our results [44] upper limits for the effective neutrino mass < meu >, the effective righthandedness < tgC > and < @/IV; >. < mve >
X
mue 5 0.88 eV
< tgc >
I
1.55.1o-s
5
1.08. 1O-6
< M;/M;
>
(19)
To extract an upper limit for the bare neutrino mass mev the bare mixing angle 6 of the left- and right-handed vector bosons and a lower limit for the mass of the heavy mainly right-handed vector boson M2, one has to know a specific model of Grand Unification. The upper limit of the effective neutrino mass is probably close to the bare one, while the bare values C and Mf/Mi of the two other quantities can be appreciably larger [4,34,35].
A. Faessler I Grand unification
57c
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(1985) 223
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> 3.9. 1O23years); Berke-
58~
A. Faessler I Grand unification
[19] 0. Civitarese, A. Faessler and T. Tomoda,
Phys. Lett. B194 (1987) 11
[20] A. Bohr and B. R. Mottelson,
Nucl. Structure, Vol. 1 (Benjamin,
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Shell Model (A merican Elsevier, New York, 1972)
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and A. Faessler, Phys. Lett. B199 (1987) 2383
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(1988) 187
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A. Faessler, J. Suhonen, X. R. Wu, Nucl. Phys. A524
(1991) 404 -
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Personal
1993
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1993
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59c
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