

 3 Nov 1995

On the observability of Ma joron emitting

double beta decays

M. Hirsch, H.V. Klapdor-Kleingrothaus, S.G. Kovalenko, 1

H. P"as

Max-Planck-Institut f"ur Kernphysik, P.O.Box 10 39 80, D-69029 Heidelberg,

Germany

Abstract

Because of the fine-tuning problem in classical Majoron models in recent years several new models were invented. It is pointed out that double beta decays with Majoron emission depend on new matrix elements, which have not been considered in the literature. A calculation of these matrix elements and phase space integrals is presented. We find that for new Majoron models extremly small decay rates are expected. PACS 13.15;23.40;21.60E;14.80

Keywords: Majoron, double beta decay, QRPA, neutrino interactions

In many theories of physics beyond the standard model neutrinoless double beta decays can occur with the emission of new bosons, so-called Majorons [1-4]:

2n ! 2p + 2e\Gamma + OE (1) 2n ! 2p + 2e\Gamma + 2OE: (2) Since classical Majoron models [1,5] require severe fine-tuning in order to preserve existing bounds on neutrino masses and at the same time get an observable rate for Majoron emitting double beta decays in recent years several new Majoron models have been constructed [6-8], where the terminus Majoron means in a more general sense light or massless bosons with couplings to neutrinos. The main novel features of the "New Majorons" are that they can carry units of leptonic charge, that there can be Majorons which are no Goldstone bosons [6] and that decays with the emission of two Majorons [4,7] can occur. The latter can be scalar-mediated or fermion-mediated. In vector Majoron models the Majoron becomes the longitudinal component of

1 on leave from Joint Institute for Nuclear Research, Dubna, Russia

Preprint submitted to Elsevier Science 3 November 1995

case modus Goldstone boson L n Matrix element

IB fifiOE no 0 1 MF \Gamma MGT IC fifiOE yes 0 1 MF \Gamma MGT ID fifiOEOE no 0 3 MF !2 \Gamma MGT !2

IE fifiOEOE yes 0 3 MF !2 \Gamma MGT !2 IIB fifiOE no -2 1 MF \Gamma MGT IIC fifiOE yes -2 3 MCR IID fifiOEOE no -1 3 MF !2 \Gamma MGT !2

IIE fifiOEOE yes -1 7 MF !2 \Gamma MGT !2

IIF fifiOE Gauge boson -2 3 MCR Table 1 Different Majoron models according to Bamert/Burgess/Mohapatra9. The case IIF corresponds to the model of Carone10.

a massive gauge boson [8] emitted in double beta processes. For simplicity we will call it Majoron, too. In tab. 1 the nine Majoron models we considered are summarized. [7,8] It is divided in the sections I for lepton number breaking and II for lepton number conserving models. The table shows also whether the corresponding double beta decay is accompanied by the emission of one or two Majorons. The next three entries list the main features of the models: The third column lists whether the Majoron is a Goldstone boson or not (or a gauge boson in case of vector Majorons IIF). In column four the leptonic charge L is given. In column five the "spectral index" n of the sum energy of the emitted electrons is listed, which is defined from the phase space of the emitted particles, G , (Qfifi \Gamma T )n, where Qfifi is the energy release of the decay and T the sum energy of the two electrons. The different shapes can be used to discriminate the different decay modes from each other and the double beta decay with emission of two neutrinos. In the last column we listed the nuclear matrix elements which will be defined in more detail later. Nuclear matrix elements are necessary to convert half lives (or limits thereof) into values for the effective Majoron-neutrino coupling constant, using the approximate (see below) relations: [4,9]

[T1=2]\Gamma 1 = j ! gff ? jm \Delta jMffj2 \Delta GBBff (3) with m = 2 for fifiOE-decays or m = 4 for fifiOEOE-decays. The index ff in eq. (3) indicates that effective coupling constants gff, nuclear matrix elements Mff and phase spaces GBBff differ for different models. As shown in tab. 1, several Majoron models with different theoretical motivation can lead to signals in double beta decays which are experimentally

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indistinguishable. The interpretation of experimental half life limits in terms of the "effective Majoron-neutrino coupling constant" is therefore model dependent. Subsequently we give a brief summary of the theoretical background on which our conclusions on the different Majoron models are based. Single Majoron emitting double beta decays (0*fifiOE) can be roughly divided into two classes, n = 1 (case IB, IC and IIB) and n = 3 (IIC and IIF) decays. As has been noted in [7] as long as 0*fifi decay has not been observed, the three n = 1 decays are indistinguishable from each other. We will call these Majorons "ordinary", since they contain the subgroup IC, which leads to the classical Majoron models. [1,2,10] For all ordinary Majorons the effective Majoron- neutrino interaction Lagrangian, leading to 0*fifiOE decay is [2,6]

LO:M:OE** = \Gamma 12 *i(aijPL + bijPR)*j OE\Lambda + h:c: (4) Here, PR=L = 1=2(1 \Sigma fl5). Using eq. (4) the amplitude corresponding to the Feynman graph is, in the notation of [6]

AO:M:(0*fifiOE) = 4p2 X

i;j V

eiVej Z d

4q

(2ss)4

mimjaij + q2bij (q2 \Gamma m2i + iffl)(q2 \Gamma m2j + iffl) (wF \Gamma wGT )(5)

Vei, Vej are elements of the neutrino mixing matrix, mi and mj denote neutrino mass eigenvalues and wF=GT are nuclear matrix elements containing double Fermi and Gamow-Teller operators. To arrive at the factorized decay rate eq. (3), the usual assumption mi;j o/ q ss pF ss O(100M eV ), where pF is the typical Fermi momentum of nucleons, is made. By this assumption the term proportional to aij can be dropped and the effective coupling constant is defined as:

hgiO:M: = X

i;j V

eiVej bij: (6)

In this approximation matrix elements for ordinary Majoron decays coincide with the leading terms MGT and MF of the well-known mass mechanism of 0*fifi decay. Burgess and Cline advocated the so-called charged Majoron model IIC. [6] In this model the effective interaction Lagrangian is

LC:M:OE** = \Gamma i2f *fl_(ALPL + ARPR)*@_OE + h:c: (7) Note, that in the charged Majoron model the two additional powers of n in the phase space integrals originate from the derivative coupling of the Majoron in LC:M:OE** . As shown in [6], for charged Majorons the contribution from the leading order matrix elements to the decay rate vanishes identically, so that one has to go to the next higher order in the non-relativistic impulse approximation

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of hadronic currents. The amplitude for 0*fifiOE decay is then given by

AC:M:(0*fifiOE) = 8p2 X

i;j V

eiVej Z d

4q

(2ss)4

q2bij (q2 \Gamma m2i + iffl)(q2 \Gamma m2j + iffl)(w5 + w6)(8)

which leads to an effective coupling constant hgiC:M: as in the ordinary Majoron case, but with bij given by b = if (ALm\Lambda + m\Lambda AR), with the neutrino mass matrix m, generator matrices AL=R and the decay constant f . The hadronic term w6 is similar but not identical to the recoil matrix element of 0*fifi decay induced by right-handed currents. This difference has turned out to be important. In the notation of [6]

w5 = i_~qjqj2(q2

0 \Gamma _2 + iffl)hF je

\Gamma i~q~r[g2A(Cn ~oem \Gamma Cm ~oen) + g2V (Dm \Gamma Dn)]jIi;(9)

w6 = _gAgV ~qjqj2(q2

0 \Gamma _2 + iffl)hF je

\Gamma i~q~r[Dn \Theta ~oem + Dm \Theta ~oen]jIi; (10)

in which the summation over Pnm o/ +n o/ +m is suppressed. Here, Cn and Dn are nuclear recoil terms [9]

Cn = (Pn + P0n) \Delta ~oen=(2Mn) \Gamma (En \Gamma E0n)(Pn \Gamma P0n) \Delta ~oen=(2m2ss) (11) Dn = [(Pn + P0n) + i_fi(Pn \Gamma P0n) \Theta ~oen]=(2Mn) (12) Pn (En) and P0n (E0n) are momenta (energies) of initial and final state nucleons, mss is the pion and Mn the nucleon mass and _fi originates from the weak magnetism. The terms of w5 are neglected compared to w6 due to the estimation (Pn + P0n) ^ (Pn \Gamma P0n); (En \Gamma E0n) ^ O(Qfifi). [9] Following [11] we will also keep only the central part of the recoil term D. Although both are approximations, which needs to be checked numerically, we do not expect it to affect any of our conclusions. Finally for vector Majoron models (case IIF) [8]

LV:M:OE** = \Gamma i2f *fl_(cijPL + dijPR)*X_ + h:c: (13) where X_ is the emitted massive gauge boson. The effective coupling constant can be defined as in the ordinary Majoron model, with the replacement bij =

1 2M (cijmj \Gamma midij ), where M is the gauge boson mass. As discussed in [8], thevector Majoron amplitude approaches the charged Majoron one in the limit of

vanishing gauge boson masses, which we assume in the phase space integration. They depend on the same nuclear matrix elements than the charged Majoron discussed above. We will therefore not repeat the definitions here. Double Majoron emitting decays (0*fifiOEOE), mediated by fermions, can have either spectral index n = 7 or n = 3, depending on whether the Majoron couples derivatively suppressed or not. [7]

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In addition, in principle 0*fifiOEOE decays could also be mediated by exotic scalars. The amplitude of scalar-mediated decays, however, is expected to be very much suppressed, since the scalars must have masses larger than about 50 GeV due to the LEP-measurements. [7] We will therefore concentrate on the fermion-mediated decays. The Yukawa coupling of the Majoron to the neutrinos for the n = 3 decays (cases ID, IE, IID) is given as:

LyukOE** = \Gamma *i(AiaPL + BiaPR)NaOE + h:c: (14) where Aia and Bia represent arbitrary Yukawa-coupling matrices and Na are sterile neutrinos. The corresponding amplitude for 0*fifiOEOE decay is

AD:M:(0*fifiOEOE) = s 23ss2 X

i;j;a V

eiVej Z d

4q

(2ss)4

\Theta Nija(q2 \Gamma m2

i + iffl)(q2 \Gamma m2j + iffl)(q2 \Gamma m2a + iffl)(w

F \Gamma wGT ):(15)

Although for AD:M:(0*fifiOEOE) the same combination of nuclear operators appears (wF \Gamma wGT ), note the additional (q2 \Gamma m2)\Gamma 1 compared to AO:M:(0*fifiOE).N

ija in (15) is given by

Nija = \Gamma q2(AiaBjam*i + AjaBiam*j + BiaBjamNa) + AiaAjam*im*j mNa(16) In order to separate the particle physics parameters from the nuclear structure calculation, it is most convenient to neglect the last term in eq. (16). This can be justified by considering that the mass eigenvalues m*i;j o/ pF so that the last term in eq. (16) for not too large mNa is suppressed compared to the first three by at least m*i;j =pF ' O(10\Gamma 5\Gamma 6). Then, the q2 is absorbed into the neutrino potentials and we redefine Nija to obtain the effective coupling constant as

hgi = ( 1m

e Xij V

eiVej [AiaBjam*i + AjaBiam*j + BiaBjamNa])

1 2 (17)

Note, that we have arbitrarily absorbed a factor of m\Gamma 1e into the definition ofh

gi here to get for the effective coupling a dimensionless quantity. For the n = 7 0*fifiOEOE decays, the effective Lagrangian is (IIE/fermion mediated):

LOE** = \Gamma i*ifl_(XiaPL + YiaPR)Na@_OE + h:c: (18) Again, Na denotes a sterile neutrino and the derivative coupling of OE accounts for the additional powers of n in the phase space integrals. The amplitude for n = 7 decays is the same as for the n = 3 case, discussed above, with the replacement: Nija = XiaYiamNa.

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Note, that Xia and Yia have the dimension of an inverse mass. Therefore, alsoh

gi has a dimension of an inverse mass. To define a dimensionless coupling constant in this case one would have to specify the symmetry breaking scale, which is however undetermined by the model. For the Majoron models considered in this work there are five nuclear matrix elements to be calculated. Within the closure approximation they are defined as:

MF = ( g

2V

g2A ) ! Nf khmass(_; ~r)o/

+n o/ +mkNi ? (19)

MGT =! Nf khmass(_; ~r)o/ +n o/ +m~oen~oemkNi ? (20) MCR = ( gVg

A )

fW

3 ! Nf khR(_; ~r)o/

+n o/ +m~oen~oemkNi ? (21)

MF !2 = ( g

2V

g2A ) ! Nf kh!

2 (_; ~r)o/ +n o/ +m kNi ? (22)

MGT !2 =! Nf kh!2 (_; ~r)o/ +n o/ +m~oen~oemkNi ? (23) where hff denote the neutrino potentials

hmass(_; ~r) = R2ss2 Z d

3~q

!

ei~q~r ! + _ (24)

hR(_; ~r) = 14ss2 ( 1M ) Z d

3~q

! e

i~q~r _ + 2!

(_ + !)2 (25)

h!2 (_; ~r) = m

2eR

16ss2 Z d

3~qq2ei~q~r 3_2 + 9_! + 8!2

!5(_ + !)3 (26)

Here _ = hEN \Gamma EI i denotes the average excitation energy of the intermediate nuclear states. ! = pq2 + m2 is the energy of the neutrino and since we assume all neutrinos to be light, the indices on neutrino masses have been dropped. Note, that in order to define matrix elements dimensionless we follow the convention of [9]. That is hmass(~r) and h!2(~r) are arbitrarily multiplied by the nuclear radius R = r0A

1 3 with r0 = 1:2 fm, while hR(~r) includes the

nucleon mass. Compensating factors appear in the prefactors of the phase space integrals. We have carried out a numerical calculation of these matrix elements within the pn-QRPA model of [12,13]. To estimate the uncertainties of the nuclear structure matrix elements the parameter dependence of the numerical results has been investigated. Since the matrix elements MGT and MF have been studied before, [12] we will concentrate on MCR, MGT !2 and MF !2 . MGT and MF can be calculated with an accuracy of about a factor of 2[12].

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The matrix element MCR shows a very similar behaviour as MGT . This is in agreement with the expectation, since only the central part of the recoil terms is taken into account, so that apart from the different neutrino potential MCR has the same structure as MGT . Neither variations of the strength of the particle-particle force gpp nor a change in the intermediate state energies significantly affects the numerical value of MCR. We therefore conclude that MCR should be accurate up to a factor of 2, as is expected for MGT . Unfortunately, in the case of the matrix elements MGT !2 and MF !2 the situation is very different. Both, variations of gpp or _, can change the numerical results drastically (fig. 1). In fact, it is found that MGT !2 displays a very similar dependence on gpp as has been reported in pn-QRPA studies of 2*fifi decay matrix elements. [12] Especially important is that in the region of the most probable value of gpp MGT !2 crosses zero. Also for variations of the assumed average intermediate state energy a rather strong dependence of the results on the adopted value of _ has been found. As a consequence of this unpleasant strong dependence, for an accurate prediction of MGT !2 and MF !2 it seems necessary to go beyond the closure approximation. The basic reason for the unusual sensitivity of MGT !2 and MF !2 on _ can be traced back to a certain difference in the neutrino potential of these matrix elements compared to MGT=F , hmass(_; ~r) , !\Gamma 2 while h!2 (_; ~r) , !\Gamma 4. Contributions from very low momenta are therefore much preferred in h!2(_; ~r) compared to hmass(_; ~r). (Note, that this leads also to a much smaller value for a typical ! than the naive expectation of ! , pF , O(50 \Gamma 100) MeV!). With typical ! of only O(few) MeV the strong dependence of h!2 (_; ~r) on _ becomes obvious. Results of the calculation for various experimentally interesting isotopes are summarized in table 2. Note that the matrix elements are valid for the limit of small intermediate particle masses, up to the order of 10 MeV. If any of the virtual particles in the Feynman graphs can have masses larger than 10 MeV, the matrix elements are no longer constant and the values in table 2 should only be taken as upper limits for the analysis of data. In comparison to the nuclear matrix elements phase space integrals can be calculated very accurately, so uncertainties of this calculation will not be discussed. We define the phase space integral as

GBBff = aff \Delta Z (Qfifi \Gamma ffl1 \Gamma ffl2)n Y

k

pkfflkf (fflk)dfflk (27)

where the prefactor aff depends on the Majoron mode under consideration. A summary of the definitions is given in table 3. Qfifi is the maximum decay energy, fflk and pk are the energies and momenta of the outgoing electrons and f (fflk) is the Fermi function calculated according to the description of. [9] Note the large difference in the phase space values of the old (n = 1) and new Majoron models.

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nucleus MF \Gamma MGT MCR MF !2 \Gamma MGT !2

Ge 76 4:33 0:16 , 10\Gamma 3\Sigma 1

Se 82 4:03 0:14 , 10\Gamma 3\Sigma 1 Mo 100 4:86 0:16 , 10\Gamma 3\Sigma 1

Cd 116 3:29 0:10 , 10\Gamma 3\Sigma 1

Te 128 4:49 0:14 , 10\Gamma 3\Sigma 1 Te 130 3:90 0:12 , 10\Gamma 3\Sigma 1 Xe 136 1:82 0:05 , 10\Gamma 3\Sigma 1 Nd 150 5:29 0:15 , 10\Gamma 3\Sigma 1 Table 2 Dimensionless nuclear matrix elements of Majoron emitting modes calculated in this work

nucleus fifiOE fifiOE fifiOEOE fifiOEOE

n=1 n=3 n=3 n=7

aff (GF gA)

4\Delta 2\Delta m2e

256ss7ln(2)_h(meR)2

(GF gA)4\Delta 2 64ss7ln(2)_h

(GF gA)4\Delta 2 12288ss9ln(2)_h(meR)2

(GF gA)4\Delta 2 215040ss9m4eln(2)_h(meR)2

Ge 76 1:25 \Delta 10\Gamma 16 2:07 \Delta 10\Gamma 19 6:32 \Delta 10\Gamma 19 1:21 \Delta 10\Gamma 18

Se 82 1:03 \Delta 10\Gamma 15 3:49 \Delta 10\Gamma 18 1:01 \Delta 10\Gamma 17 7:73 \Delta 10\Gamma 17 Mo 100 1:80 \Delta 10\Gamma 15 7:28 \Delta 10\Gamma 18 1:85 \Delta 10\Gamma 17 1:54 \Delta 10\Gamma 16

Cd 116 1:75 \Delta 10\Gamma 15 6:95 \Delta 10\Gamma 18 1:60 \Delta 10\Gamma 17 1:03 \Delta 10\Gamma 16

Te 128 1:02 \Delta 10\Gamma 17 5:96 \Delta 10\Gamma 21 1:28 \Delta 10\Gamma 20 1:20 \Delta 10\Gamma 21 Te 130 1:35 \Delta 10\Gamma 15 4:97 \Delta 10\Gamma 18 1:06 \Delta 10\Gamma 17 4:83 \Delta 10\Gamma 17 Xe 136 1:40 \Delta 10\Gamma 15 5:15 \Delta 10\Gamma 18 1:06 \Delta 10\Gamma 17 4:54 \Delta 10\Gamma 17 Nd 150 1:07 \Delta 10\Gamma 14 7:27 \Delta 10\Gamma 17 1:41 \Delta 10\Gamma 16 1:85 \Delta 10\Gamma 15 Table 3 Values of phase space integrals calculated in this work

Having calculated nuclear matrix elements and phase space integrals, it is straightforward to derive limits on the effective Majoron-neutrino coupling constants for the various Majoron models from experiment. Although experimental half life limits are comparable for all decay modes, as observed recently for 76Ge decay [14,15], restrictive limits on the coupling constants of ordinary Majoron models contrast with limits on any of the new Majoron models, which will be weaker by (3-4) orders of magnitude. The surprisingly weak limits which one obtains for the neutrino-Majoron coupling constant due to small matrix elements and phase spaces for all of the new Majoron models, require further explanation. (Note that the follow8

ing discussion is independent of the isotope under consideration.) Consider, for example, ordinary and charged Majoron 0*fifiOE decays. Limits on the effective coupling constant for single Majoron emitting decays will scale ash

gi , A\Gamma 1(T1=2 \Delta GBB)\Gamma

1

2 . Thus, the relative sensitivity of a double beta decay experiment on ordinary and charged Majoron decays can be expressed as

hgiO:M:

hgiC:M: ,

AC:M: AO:M: i

T C:M:1=2 T O:M:1=2 j

1 2 \Delta (Qfifi \Gamma T ).

Inserting the definitions of the corresponding amplitudes, it is clear that even if the half life limit derived for the charged Majoron decay equals that of the ordinary Majoron mode, limits on the charged Majoron-neutrino coupling constant will be weaker by Mn=(Qfifi \Gamma T ) ' 1000 ! (Note, that this crude estimation is to first approximation independent of nuclear structure properties.) A similar analysis can be easily done for double Majoron emitting decays. Again, very crudely, a reduced sensitivity of (48ss2) \Delta pF =(Qfifi \Gamma T ) ' (few)\Theta

104 for n = 3 double Majoron decay, compared to ordinary Majoron decays, is expected. Here, the factor (48ss2) is due to the phase space integration over the additional emitted particle, while the latter factor comes from the additional propagator. One might think that since our definition of the effective coupling constant for the n = 3 0*fifiOEOE decays includes a factor mNa=me, where mNa is the sterile neutrino mass, one could get hgi easily as large as wanted, since the mass of the sterile neutrino is not bounded experimentally. However, matrix elements will fall off M , m\Gamma 2Na as soon as mNa is larger than the typical momenta. While for the matrix elements MGT=F for ordinary Majoron decays such a reduction occurs starting from masses of exchanged virtual particles in the region of 100 \Gamma 1000 MeV, for MGT !2=F !2 the suppression will be important already for much smaller masses (see the En-dependance fig. 1). Since the sensitivity of double beta decay experiments to the new Majoron models is so weak, it might be interesting to compare expected half lives for the different models for different hgi, hgi ss 10\Gamma 4 as a typical sensitivity in coupling constant for ordinary Majoron models and hgi = 1 as an upper possible limit allowed by perturbation theory, with current experimental limits of O(1022) years (see tab. 4). From this consideration it is very unlikely that any of the new Majoron models can produce an observable rate in planned or ongoing double beta decay experiments. Only the charged and the vector Majoron model [6,8] could produce an observable effect if Pij VeiVej is not smaller than 0:1 and the real coupling constant of order O(1).

Acknowledgements The authors would like to thank C.P. Burgess and E. Takasugi for several discussions on the theoretical aspects of Majoron models. The research described in this publication was made possible in part (M.H.) by the Deutsche

9

model T1=2(! g ?= 10\Gamma 4) T1=2(! g ?= 1) T1=2exp IB,IC,IIB 4 \Delta 1022 4 \Delta 1014 1:67 \Delta 1022 ID,IE,IID 1038\Gamma 42 1022\Gamma 26 1:67 \Delta 1022

IIC,IIF 2 \Delta 1028 2 \Delta 1020 1:67 \Delta 1022

IIE 1038\Gamma 42 1022\Gamma 26 3:37 \Delta 1022 Table 4 Comparison of half lives calculated for different ! g ?-values for the new Majoron models with experimental best fit values16;18

Forschungsgemeinschaft (446 JAP-113/101/0 and Kl 253/8-1) and (S.G.K.) by Grant No. RFM300 from the International Science Foundation.

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[16] M. G"unther et al. (HEIDELBERG-MOSCOW collaboration), in Proc. Int.

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11 0 0.2 0.4 0.6 0.8 1 -0.02

-0.01

0 0.01 0.02 0.03 0.04

Fig. 1. MGT !2 \Gamma MF !2 dependence of gP P for different intermediate state energies En =4 (top on the left),8,12,16,20,24 (bottom on the left) MeV for 76Ge

MGT !2 \Gamma MF !2

gP P 12

