Phase space factors and halflife predictions for Majoron emitting $β^β^$ decay
Abstract
A complete calculation of phase space factors (PSF) for Majoron emitting decay modes is presented. The calculation makes use of exact Dirac wave functions with finite nuclear size and electron screening and includes lifetimes, single electron spectra, summed electron spectra, and angular electron correlations. Combining these results with recent interacting boson nuclear matrix elements (NME) we make halflife predictions for the the ordinary Majoron decay (spectral index =1). Furthermore, comparing theoretical predictions with the obtained experimental lower bounds for this decay mode we are able to set limits on the effective Majoronneutrino coupling constant .
pacs:
23.40.Hc, 23.40.Bw, 14.60.Pq, 14.60.StI Introduction
Double decay is a process in which a nucleus decays to a nucleus by emitting two electrons or positrons and, usually, other light particles
(1) 
The mode where two antineutrinos or neutrinos are emitted is predicted by the standard model and has been observed in several nuclei (for a review, see e.g. bara15 ()). The more exotic mode, neutrinoless double beta decay, is not allowed by the standard model, and once observed would offer new information on many fundamental aspects of elementary particle physics. As discussed in Ref. barea13 (), several scenarios of neutrinoless double beta decay have been considered, most notably, light neutrino exchange, heavy neutrino exchange, and Majoron emission. After the discovery of neutrino oscillations, attention has been focused on the first scenario and the mass mode, where the transition operator is proportional to . Even though most current experimental efforts have been focused to the detection of this mode, interest on the mechanism predicting decays through the emission of additional bosons called Majorons has also renewed lately.
Majorons were introduced years ago x (); xx () as massless NambuGoldstone bosons arising from a global (baryon number minus lepton number) symmetry broken spontaneously in the lowenergy regime. These bosons couple to the Majorana neutrinos and give rise to neutrinoless double beta decay, accompanied by Majoron emission xxx (), as shown in Fig. 1 (a).
Although these older models are disfavored by precise measurements of the width of the boson decay to invisible channels xv (), several other models of decay have been proposed in which one or two Majorons, denoted by , are emitted, (see Fig. 1)
(2) 
or
(3) 
Table 1 lists some of the models proposed to describe these decays. The different models are distinguished by the nature of the emitted Majoron(s), i.e. is it a NambuGoldstone boson or not (NG), the leptonic charge of the emitted Majoron(s) (L), and the spectral index of the model (n), which characterizes the shape of the summed electron spectrum, as described in Sect. II.
Model  Decay Mode  NG boson  L  n 

IB  No  0  1  
IC  Yes  0  1  
ID  No  0  3  
IE  Yes  0  3  
IIB  No  2  1  
IIC  Yes  2  3  
IID  No  1  3  
IIE  Yes  1  7  
IIF  Gauge boson  2  3  
“Bulk”  Bulk field  0  2 
In our previous articles we have studied phase space factors (PSF) and prefactors (PF) kotila12 (); kotila14 (); kotila13 (), and nuclear matrix elements (NME) barea (); barea12 (); barea13 (); barea13b (); kotila14 (); bar15 () needed for the theoretical description of , , , , , , , and decay mediated in the case of neutrinoless decay by light or heavy neutrino exchange. In this article we continue our systematic evaluation by presenting phase space factors for the different Majoron emitting mechanisms and combining the results with recent interacting boson model (IBM2) NMEs bar15 () to make halflife predictions for ordinary Majoron decay (). Furthermore, we compare our theoretical predictions with the obtained experimental lower bounds for this decay mode to set some limits on the effective Majoronneutrino coupling constant .
Ii Phase space factors in Majoron emitting double decay
The key ingredients for the evaluation of phase space factors in single and double decay are the scattering wave functions and for EC the bound state wave functions. The general theory of relativistic electrons and positrons can be found e.g., in the book of Rose rose (). The electron scattering wave functions of interest in were given in Eq. (8) of kotila12 ().
In order to calculate PSFs for Majoron emitting we use the formulation of Doi, Kotani, and Takasugi doi (). The differential rate for the decay is given by doi (); tom91 ()
(4) 
where and are the electron energies, the angle between the two emitted electrons, and takes different values depending on the number of emitted Majorons and the spectral index :
(5) 
Here is the Fermi constant, is the Cabibbo angle, and the Majoron energy is determined as , and with fm, is the nuclear radius.The quantities and in Eq. (4) can be written as tom91 ()
(6) 
where is the effective coupling constant of the Majoron to the neutrino, for the emission of one or two Majorons, respectively, and is the nuclear matrix element. The functions , are defined as
(7) 
with
(8) 
where and are obtained from the electron wave functions as explained in Ref. kotila12 ().
All quantities of interest are then given by integration of Eq. (4). Introducing
(9) 
where the axial vector coupling constant is separated from the phase space factors for conveniency, we can calculate:
(i) The halflife
(10) 
(ii) the single electron spectrum
(11) 
where .
(iii) The summed electron spectrum, which shape makes the different Majoron emitting modes experimentally recognizable
(12) 
(iv) and the angular correlation between the two electrons
(13) 
yr  

Nucleus 
m=1,n=1  m=1,n=3  m=2,n=3  m=2,n=7 
Ca  1540  17.1  73.6  690 
Ge  44.2  0.073  0.22  0.420 
Se  361  1.22  3.54  26.9 
Zr  905  4.21  11.0  128. 
Mo  598  2.42  6.15  50.8 
Pd  94.1  0.205  0.487  0.946 
Cd  569  2.28  5.23  33.9 
Sn  209  0.653  1.45  4.45 
Te  3.06  0.001  0.003  0.0003 
Te  413  1.51  3.21  14.4 
Xe  2.92  0.002  0.003  0.0002 
Xe  409  1.47  3.05  12.5 
Nd  197  0.505  0.986  1.72 
Nd  3100  21.1  40.8  538 
Sm  28.2  0.034  0.064  0.021 
Gd  1590  0.361  0.672  0.899 
Pt  60.7  0.068  0.110  0.021 
Th  82.4  0.073  0.105  0.009 
U  337  0.532  0.756  0.213 
We have done a calculation of and in the list of nuclei shown in Table 2. We also plot our results in Figs. 25, where they are compared with previous calculations doi (); tom91 (); suhonen (); gun96 (); arn00 (). For the comparison the values of doi (); suhonen () have been multiplied by a missing factor of two and divided by , and values of tom91 () have been divided with factor . The factor of two is the correct choice, as was acknowledged in doi88 (). This factor of two is also included in Table II of Ref. alb14 (), where the decay of Xe have been studied. In their calculation for the phase space factors they use Fermi functions F(Z,E) that fully include nuclear finite size and electron screening and are evaluated at the nuclear radius. Their result with different and are within 6% of the ones reported here.
We also have available upon request the single and summed electron electron spectra and angular correlation for all nuclei in Table 2. An example, Xe decay, is shown in Fig. 6. The shape of the spectra is determined by the spectral index , i.e. for the case it is the same for both the emission of one or two Majorons, but the overall scale is different ( result should be multiplied by to obtain result). Also, since the angular correlation is obtained by dividing by the calculation becomes unstable for the higher near the endpoint energy and that region is thus excluded from Fig. 6. The distinction between different values of is most prominent in summed electron spectra, where as increases the peak of the spectrum shifts from near the maximum kinetic energy to near the minimum kinetic energy. The difference between different values of is also shown in single electron spectra but not as strongly as in the case of summed electron spectra.
Iii Expected halflives and limits on coupling constant: Ordinary Majoron emitting decay
In the case of ordinary Majoron emitting double decay, , the nuclear matrix elements have the same form as in the mediated by light neutrino exchange
(14) 
The calculation of phase space factors can now be combined with updated nuclear matrix elements in IBM2 bar15 () to produce predictions for halflives for ordinary Majoron decay () in Table 3 (left) and Fig. 7. Judging by the predicted halflives, the most prominent candidates are Nd and Mo. Furthermore, we can compare our predictions to halflife limits coming from experiments to set some limits on the effective Majoronneutrino coupling constant. The obtained limits are shown on Table 3 (right). The most stringent limits for ordinary Majoron decay () at the moment are for Xe coming from KamLANDZen gan12 () and EXO200 alb14 () experiments reaching the order of magnitude of .
Decay  (yr)  (yr)  (eV) 

CaTi  8.19 


GeSe  39.8 


SeKr  7.68 


ZrMo  5.32 


MoRu  3.62 


PdCd  25.0  
CdSn  7.06 


SnTe  18.1  
TeXe  765 


TeXe  6.82 


XeBa  805  
XeBa  10.1 




NdSm  36.8  
NdSm  1.74 


SmGd  173  
GdDy  14.5  
PtHg  132  
ThU  28.7  
UPu  4.94 
Iv Conclusions
In this article, we have reported a complete calculation of phase space factors for decay proceeding through emission of one or two Majorons. The reported results include halflives, single electron spectra, summed electron spectra, and electron angular correlations, to be used in connection with the calculation of nuclear matrix elements. Furthermore, we have combined our results with recent IBM2 nuclear matrix elements to produce predictions of halflives in the case of ordinary Majoron decay, spectral index . Comparing these predictions with experimental lower bounds, we have set some limits on the effective Majoronneutrino coupling constant . At the moment the best limits are coming from Xe experiments reaching the order of magnitude of . Also, the results in Table III are for . If is renormalized to , should be multiplied by and limits on by .
Acknowledgements.
This work was supported in part by US Department of Energy (Grant No. DEFG0291ER40608), Chilean Ministry of Education (Fondecyt Grant No. 1150564), Academy of Finland (Project 266437), and by the facilities and staff of the Yale University Faculty of Arts and Sciences High Performance Computing Center.Footnotes
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