Hirsch, Klapdor, Kovalenko, Paes

8/12/2019 Hirsch, Klapdor, Kovalenko, Paes

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ELSEVIER Physics Letters B 372 (1996) 8-14

4 April 1996

PHYSICS LETTERS B

On the observability of Majoron emitting double beta decaysM. Hirsch, H.V. Klapdor-Kleingrothaus, S.G. Kovalenko , H. Pb

Max-Planck-Institutfiir Kemphysik, PO. Box 10 39 80, D-69029 Heidelberg, GermanyReceived 24 October 1995; revised manuscript received 8 January 1996Editor: C. Mahaux

bstractBecause 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 new Majoron emission depend on new matrix elements, which have not beenconsidered in the literature. A calculation of these matrix elements and phase space integrals is presented. We find that fornew Majoron models extremely small decay rates are expected.PAW 13.15; 23.40; 21.6OJ; 14.80Keywords: Majoron; Double beta decay; QRPA; Neutrino interactions

In many theories of physics beyond the standardmodel neutrinoless double beta decays can occur withthe emission of new bosons, so-called Majorons [ l-41:2n --+ 2p + 2e- + 4, (1)2n-+2p+2ee+24. (2)Since classical Majoron models [ I,.5 require severefine-tuning in order to preserve existing bounds onneutrino masses and at the same time get an observablerate for Majoron emitting double beta decays in recentyears several new Majoron models have been con-structed [ 6-81, where the terminus Majoron means ina more general sense light or massless bosons withcouplings to neutrinos. The main novel features of theNew Majorons are that they can carry units of lep-tonic charge, that there can be Majorons which are

On leave from Joint Institute for Nuclear Research, Dubna,Russia.

no Goldstone bosons [6] and that decays with theemission of two Majorons [ 4,7] can occur. The lattercan be scalar-mediated or fermion-mediated. In vectorMajoron models the Majoron becomes the longitudi-nal component of a massive gauge boson [ 81 emittedin double beta processes. For simplicity we will callit Majoron, too.

In Table 1 the nine Majoron models we consideredare summarized [ 7,8]. It is divided in the sections Ifor lepton number breaking and II for lepton numberconserving models. The table shows also whether thecorresponding double beta decay is accompanied bythe emission of one or two Majorons.The next three entries list the main features of themodels: The third column lists whether the Majoronis a Goldstone boson or not (or a gauge boson in caseof vector Majorons IIF). In column four the leptoniccharge L is given. In column five the spectral indexn of the sum energy of the emitted electrons is listed,which is defined from the phase space of the emittedparticles, G N (Qpa - T), where Qpa is the energy

0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reservedPII SO370-2693 (96)00038-X


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M. Hirsch et al. /Physics Letters B 372 11996)8-14 9Table IDifferent Majoron models according to Bamert/Burgess/Mohapatm 191. The case IIF corresponds to the model ofCarone IlOl.Case Modus Goldstone boson L n Matrix elementIBICIDIEIIBIICIIDIIEIIF

BP+ noPPd yesPBd4 noPBJ yesPBdJ noPAfJ yesPP&P noP/W4 yesPP4 gauge boson

0 1 MF - MGT0 1 MF-MGT0 3 hf,2 - MGTo20 3 MFw2 - h&Tw2

-2 I MF-MCT-2 3 MCR-1 3 ,+fF2 - hf~T~2-1 MFw2 - Tw2-2 3 MCR

release of the decay and T the sum energy of the twoelectrons. The different shapes can be used to distin-guish the different decay modes from each other andthe double beta decay with emission of two neutri-nos. In the last column we listed the nuclear matrixelements which will be defined in more detail later.Nuclear matrix elements are necessary to convert half-lives (or limits thereof) into values for the effectiveMajoron-neutrino coupling constant, using the approx-imate (see below) relations [4,9] :V,p- = I(&)Im+ j2. B,r (3)with m = 2 for P/%$-decays or m = 4 for p/3&5-decays. The index cy n Eq. (3) indicates that effectivecoupling constants g,, nuclear matrix elements M,and phase spaces Gns, differ for different models.

As shown in Table 1, several Majoron models withdifferent theoretical motivation can lead to signals indouble beta decays which are experimentally indis-tinguishable. The interpretation of experimental half-life limits in terms of the effective Majoron-neutrinocoupling constant is therefore model dependent. Sub-sequently we give a brief summary of the theoreticalbackground on which our conclusions on the differentMajoron models are based.Single Majoron emitting double beta decays(Ovpp#) 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 [ 71 as long as Ovpp decay hasnot been observed, the three n = 1 decays are indistin-guishable from each other. We will call these Majoronsordinary, since they contain the subgroup IC, which

leads to the classical Majoron models [ 1,2, lo] . For allordinary Majorons the effective Majoron-neutrino in-teraction Lagrangian, leading to Ovp&b decay is [ 2,6]L$.y. = -iq(aijPL + bijPR)vj * + h.c. (4)Here, PR/L = l/2( 1 f ~5). Using Eq. (4) the ampli-tude corresponding to the Feynman graph is, in thenotation of [ 61

i.jd4qx- J

mimjaij + bij(2rr)4 (q2 - 112;+ k)(q2 - rn + k)

x (WF - WGT). (5)&, V, are elements of the neutrino mixing matrix, miand mj denote neutrino mass eigenvalues and W/T/CTare nuclear matrix elements containing double Fermiand Gamow-Teller operators. To arrive at the factor-ized decay rate Eq. (3), the usual assumption mi,j


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10 M . Hir sch et al. Phy sics Letters B 372 (1996) 8- 14Ac.M .(O~/l /3 ) = S& x & i& j

k.i

s d4qx- qbi j(2n)4(q2-m~+ie)(q2-m~+ie)x (w5 + w 6)7 (8)

which leads to an effective coupling constant (g)C.M.as in the ordinary Majoron case, but with bi j given byb = $ ( A~rn + rn*A ~), with the neutrino mass matrixm, generator matrices AL/~ and the decay constant f.The hadronic term w6 is similar but not identical tothe recoil matrix element of Or@ decay induced byright-handed currents. This difference has turned outto be important. In the notation of [6]w5=q(2(qz yL2e) (~l~-i q[g2A(Gw tl - Gt@,)

+ D,, x anI II), (10)in which the summation over C,,,,, 7,~; is suppressed.Here, C, and D, are nuclear recoil terms [9]

The Yukawa coupling of the Majoron to the neutri-nos for the n = 3 decays (cases ID, IE, IID) is givenasLYk

C, = (P, + P;) . a,/Wfn)- (=% - J%) (P, - p,> . a,/Vmi), (11)

D, = [(P, +Pk) -t-O.qdPn-J$) x ~, l / G Md;

&vu= -q( Ai oPL + BiaPR) No4 + h.c. (14)where Aia and Bi, represent arbitrary Yukawa-coupling matrices and N, are sterile neutrinos. Thecorresponding amplitude for Ovfi@$ decay is

( 12) AD.M. Ov@3+4) =P,, (En) and Ph (EA) are momenta (energies) ofinitial and final state nucleons, m, is the pion and M ,the nucleon mass and ,~uporiginates from the weakmagnetism.

X Jxja(q2-m~+ie)(q2-m~+i~)(q2-m~+i~)The terms of wg are neglected compared to Wgdue

to theestimation (P,,+P,) < (P,-P ,), (E,-I $,) 50( Qpp) [ 91. Following [ 111 we will also keep onlythe central part of the recoil term D. Although bothare approximations, which needs to be checked nu-merically, we do not expect it to affect any of ourconclusions.

x WF - WGT . (15)Although for AD.M.Ovp&5~) the same combina-tion of nuclear operators appears ( wF- War ) , note theadditional (q2 - m2) - compared to A.M. Ov@f~) .

JV&, in ( 15) is given by

Finally for vector Majoron models (case IIF) [ 81Lzby = -4Yyp(Ci j PL + di jPR)VX + h.c.,2.f (13)where Xc is the emitted massive gauge boson. Theeffective coupling constant can be defined as in the

Nija = -4 ( AioBjam,,f AjaBiamvi + BiaBjamN,,i- &Aj$%m@N,,. (16)

In order to separate the particle physics parametersfrom the nuclear structure calculation, it is most con-venient to neglect the last term in Fq. ( 16). This canbe justified by considering that the mass eigenvalues

ordinary Majoron model, with the replacement bi j =& (Ci jm j midij) , where M is the gauge boson mass.As discussed in [ 81, the vector Majoron amplitudeapproaches the charged Majoron one in the limit ofvanishing gauge boson masses, which we assume inthe phase space integration. They depend on the samenuclear matrix elements than the charged Majoron dis-cussed above. We will therefore not repeat the defini-tions here.

Double Majoron emitting decays (Ov~@$q5), me-diated by fermions, can have either spectral index n =7 or n = 3, depending on whether the Majoron couplesderivatively suppressed or not. [ 71In addition, in principle Ov/3&5 decays could alsobe mediated by exotic scalars. The amplitude of scalar-mediated decays, however, is expected to be very muchsuppressed, since the scalars must have masses largerthan about 50 GeV due to the LEP-measurements [ 71.We will therefore concentrate on the fermion-mediateddecays.


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M. Hirsch et al. /Physics Letter s B 372 I 996) 8-14

in ,,, ,where h, denote the neutrino potentials

(23)

h,,,,(wr) = T/sjq &r--w a+@ (24)

IIs3geiqr CL + 20u (tL+w)* (25)m2Rfw(p,r) = & J 3 2 iqr 3p2 + 9pW + 8W2d qq e w~(p++)3 .(26)

Here ,Z = (EN - El) denotes the average excitation en-ergy of the intermediate nuclear states. w = J,mis the energy of the neutrino and since we assume allneutrinos to be light, the indices on neutrino masseshave been dropped. Note that in order to define matrixelements dimensionless we follow the convention of[ 91. That is h,,,(r) and h,z (r) are arbitrarily multi-plied by the nuclear radius R = roAf with ra = 1.2 fm,while hR(r) includes the nucleon mass. Compensat-ing factors appear in the prefactors of the phase spaceintegrals.

We have carried out a numerical calculation ofthese matrix elements within the pn-QRPA model of[ 12,131. To estimate the uncertainties of the nuclearstructure matrix elements the parameter dependenceof the numerical results has been investigated. Sincethe matrix elements Mar and MF have been studiedbefore [ 121, we will concentrate on MCR, MCTZand MFJ. MGT and MF can be calculated with anaccuracy of about a factor of 2 [ 121.

The matrix element MCR shows a very similar be-haviour as MGT. This is in agreement with the expec-tation, since only the central part of the recoil termsis taken into account, so that apart from the differ-ent neutrino potential MCR has the same structure asMGT. Neither variations of the strength of the particle-particle force gpp nor a change in the intermediatestate energies significantly affects the numerical valueof MCR. We therefore conclude that MCR should beaccurate up to a factor of 2, as is expected for MGT.Unfortunately, in the case of the matrix elementsMGTo2 and MFJ the situation is very different. Both,variations of g,,,, or p, can change the numerical resultsdrastically (Fig. 1). In fact, it is found that MGT,,,zdisplays a very similar dependence on g,, as has beenreported in pn-QRPA studies of 21@ decay matrixelements [ 121. Especially important is that in the re-gion of the most probable value of g,, MGT,,,zcrosseszero.Also for variations of the assumed average interme-diate state energy a rather strong dependence of the


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12 M Hirsch et al./Physics Letters B 372 (1996) 8-14

Fig. I. MGTo2 - M,z dependenceo gpp for different intermediatestate energies En = 4 (top on the left), 8, 12, 16, 20, 24 (bottomon the left) MeV for %e.

results on the adopted value of ,X has been found. Asa consequence of this unpleasant strong dependence,for an accurate prediction of MG- -~z and MF@Z it seemsnecessary to go beyond the closure approximation.The basic reason for the unusual sensitivity ofMGTo2and MFu2 on ,u can be traced back to a certaindifference in the neutrino potential of these matrixelements compared to MGT/ F, mass ( p , r ) t u r n 2while hw2 ,u, r) N w-4. Contributions from very lowmomenta are therefore much preferred in ho2( p, r)compared to h,,,,( p, r). (Note that this leads also toa much smaller value for a typical w than the naiveexpectation of 0 rV PF N 0(50-100) MeV ). Withtypical u of only U( few) MeV the strong dependenceof &,,2 p. t) on ,U becomes obvious.Results of the calculation for various experimentallyinteresting isotopes are summarized in Table 2. Notethat the matrix elements are valid for the limit of smallintermediate particle masses, up to the order of 10MeV. If any of the virtual particles in the Feynmangraphs can have masses larger than 10 MeV, the matrixelements are no longer constant and the values in Table2 should only be taken as upper limits for the analysisof data.In comparison to the nuclear matrix elements phasespace integrals can be calculated very accurately, souncertainties of this calculation will not be discussed.We define the phase space integral as

Table 2Dimensionless nuclear matrix elements of Majoron emitting modescalculated in this work.

Nucleus MF - MOT MCR Fo2 - M~~o,2Ge 76 4.33 0.16 N lo-3fSe 82 4.03 0.14 N lo-3*1MO 100 4.86 0.16 N 10-349Cd 116 3.29 0.10 N lo-3fTe 128 4.49 0.14 N IO--WTe 130 3.90 0.12 N lo-3*1Xe 136 1.82 0.05Nd 150 5.29 0.15 1 :;I:::

B, = aa *s Qsp - EI - ~2) npkekf(ek)d~k,k (27)where the prefactor a, depends on the Majoron modeunder consideration. A summary of the definitions isgiven in Table 3. Qpa is the maximum decay energy,and pk are the energies and momenta of the outgoingelectrons and f( ) is the Fermi function calculatedaccording to the description of [9]. Note the largedifference in the phase space values of the old (n = 1)and new Majoron models.

Having calculated nuclear matrix elements andphase space integrals, it is straightforward to derivelimits on the effective Majoron-neutrino couplingconstants for the various Majoron models from ex-periment.

Although experimental half-life limits are compa-rable for all decay modes, as observed recently for76Ge decay [ 14,151, restrictive limits on the couplingconstants of ordinary Majoron models contrast withlimits on any of the new Majoron models, which willbe weaker by 3-4 orders of magnitude.The surprisingly weak limits which one obtains forthe neutrino-Majoron coupling constant due to smallmatrix elements and phase spaces for all of the newMajoron models, require further explanation. (Notethat the following discussion is independent of theisotope under consideration.) Consider, for example,ordinary and charged Majoron OV/~/-ecays. Lim-its on the effective coupling constant for single Ma-joron emitting decays will scale as (g) N A- (Tt/2 .

GBB) - I * .hus , he relative sensitivity of a doublebeta decay experiment on ordinary and charged Ma-joron decays can be expressed as


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Table 3hf. Hi rsch et al./Phy sics Lett ers B 372 (1996) R -14 13

Values of phase space integrals calculated in this work.Nucleus p/34

n=l n=3(GFgAp.2.m; (GF%A)~2 (G=~A)~ 2 (GI%A)~.~

In z 2S67r71n(2)fi(t n,R)* an = 64w7 In(2)ti au = 122881rgln(2)ti(m,R)2 aa= 215040?rgnz~In(2)fL(~n,R)2Ce 76 I 25Se 82 I .03MO 100 1.80Cd 116 I .75Te 128 I .02Te I30 1.35Xe 136 I .40Nd IS0 1.07

IO--6IO-15IO_510-U10-17IO_5IO--5IO_4

2.07 IO-93.49 IO_87.28 lo-*6.95 10-l5.96 10-Z4.97 IO_85.15 10-n7.21 IO_7

6.32. 1O-g1.01 . IO_71.85. lo-l71.60. IO-1.28. IO-1.06 . IO-l71.06. IO-l71.41 IO_6

1.21 lo-*7.73 10-171.54 IO--6I .03 IO_6I .20 10-Z4.83 IO_74.54. lo-I .85 IO-l5

(g)O.M. dC.M. qjy. f(~)c.M. N m(y) . QmT).

Table 4Comparison of half-lives calculated for different (&values for thenew Majoron models with experimental best fit values 116,i 8 J

Inserting the definitions of the corresponding am-plitudes, it is clear that even if the half-life limit de-rived for the charged Majoron decay equals that ofthe ordinary Majoron mode, limits on the chargedMajoron-neutrino coupling constant will be weaker byM, / ( Qpp - T) N 1000 (Note that this crude estima-tion is to first approximation independent of nuclearstructure properties.)

A similar analysis can be easily done for doubleMajoron emitting decays. Again, very crudely, a re-duced sensitivity of (483) .pr/(Qap - T) N (few)x I O4 for II = 3 double Majoron decay, compared toordinary Majoron decays, is expected. Here, the fac-tor ( 487r2) is due to the phase space integration overthe additional emitted particle, while the latter factorcomes from the additional propagator.

One might think that since our definition of theeffective coupling constant for the n = 3 Ovpp@#decays includes a factor mNa/ mer where m& is thesterile neutrino mass, one could get (g) easily as largeas wanted, since the mass of the sterile neutrino is notbounded experimentally. However, matrix elementswill fall off M N r n ; f s soon as m& is larger thanthe typical momenta. While for the matrix elementsMot-/r for ordinary Majoron decays such a reductionoccurs starting from masses of exchanged virtual par-ticles in the region of 100-1000 MeV, for Moroz/rozthe suppression will be important already for much

Model 7.1,2((g) = lo-4) Tl/Z((&?) = 1) ~1/2expIBJCJIB 4 . 102* 4 104 5.38. IO**ID,IE,IID I 08-42 1022-26 1.67. lO22IICJF 2. 1028 2. 1020 1.67 lO22IIE 1038-42 ,022-26 3.31 1022

smaller masses (see the Z&-dependence Fig. 1) . Also,the contribution of the last term in Eq. ( 16) for largerneutrino masses can be at most as big as the terms wetook into account. Since our conclusions are based ona very rough estimation of MGTo2 - M~+,,z, hey wouldnot be affected by the omission of these terms.

Since the sensitivity of double beta decay experi-ments to the new Majoron models is so weak, it mightbe interesting to compare expected half-lives for thedifferent models for different (g), (g) M 10m4 as a typ-ical sensitivity in coupling constant for ordinary Ma-joron models and (g) = 1 as an upper possible limitallowed by perturbation theory, with current experi-mental limits of 0( lo**) years (see Table 4). Fromthis consideration it is very unlikely that any of thenew Majoron models can produce an observable ratein planned or ongoing double beta decay experiments.Only the charged and the vector Majoron model [ 631could produce an observable effect if xii VeiV, is notsmaller than 0.1 and the real coupling constant of or-der 0( 1).


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14 M . Hi rsch et al. Ph ysics Lett ers B 372 (1996) 8-14The authors would like to thank C.P. Burgess andE. Takasugi for several discussions on the theoretical

aspects of Majoron models. The research describedin this publication was made possible in part (M.H.)by the Deutsche Forschungsgemeinschaft (446 JAP-113/101/O and K1253/8-1) and (S.G.K.) by GrantNo. RIM300 from the International Science Founda-tion.

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[7] P Bamert, C.P. Burgess and R.N. Mohapatm, Nucl. Phys. B449 (1995) 25.[8] CD. Carone, Phys. Len. B 308 (1993) 85.[9 J M. Doi, T. Kotani and E. Takasugi, Progr. Theor. Phys.Suppl. 83 ( 1985) 1.[lo] C.S. Aulakh and R.N. Mohapatra, Phys. Len. B I19 (1982)

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[16] M. Gunther et al. (HEIDELBERG-MOSCOWcollaboration), in Proc. Int. Workshop on Double Beta Decayand Related Topics, Trento, 24.4.-5.5.95, World ScientificSingapore, eds. H.V. Klapdor-Kleingrothaus and S. Stoica.

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Hirsch, Klapdor, Kovalenko, Paes