Phase space factors for $β^+β^+$ decay and competing modes of double$β$ decay
Abstract
A complete and improved calculation of phase space factors (PSF) for and decay, as well as for the competing modes , , and , is presented. The calculation makes use of exact Dirac wave functions with finite nuclear size and electron screening and includes lifetimes, single and summed positron spectra, and angular positron correlations.
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) 
Double decay can be classified in various modes according to the various types of particles emitted in the decay. For processes allowed by the standard model, i.e. the two neutrino modes: , , , the halflife can be, to a good approximation, factorized in the form
(2) 
where is a phase space factor and the nuclear matrix element. For processes not allowed by the standard model, i.e. the neutrinoless modes: , , , the halflife can be factorized as
(3) 
where is a phase space factor, the nuclear matrix element and contains physics beyond the standard model through the masses and mixing matrix elements of neutrino species. For both processes, two crucial ingredients are the phase space factors (PSF) and the nuclear matrix elements (NME). Recently, we have initiated a program for the evaluation of both quantities and presented results for decay barea09 (); kotila12 (); barea12 (); barea12b (). This is the most promising mode for the possible detection of neutrinoless double decay and thus of a measurement of the absolute neutrino mass scale. However, in very recent years, interest in the double positron decay, , positron emitting electron capture, , and double electron capture, , has been renewed. This is due to the fact that positron emitting processes have interesting signatures that could be detected experimentally zuber (). With this article we initiate a systematic study of , , and processes. In particular we present here a calculation of phase space factors (PSF). A calculation of nuclear matrix elements (NME), which are common to all three modes, will be presented in a forthcoming publication barprep12 ().
Estimates of the transitions rates for , , and processes were given by Primakoff and Rosen already in the 50’s and 60’s primakoff1 (); primakoff2 (). Haxton and Stephenson haxton () calculated halflives for including relativistic corrections approximately and some nonrelativistic calculations were done in the 1980’s jotain (); jotain2 (). In the 90’s, this subject was revisited by Doi and Kotani doitwoneutrino (); doineutrinoless (); doi () who also presented a detailed theoretical formulation and tabulated results for selected cases. At the same time, Boehm and Vogel boehm () gave more comprehensive results, but without a detailed theoretical description. In these papers, results for the PSFs were obtained by approximating the positron wave functions at the nucleus and without inclusion of electron screening. In this article, we take advantage of some recent developments in the numerical evaluation of Dirac wave functions and in the solution of the ThomasFermi equation to calculate more accurate phase space factors for double decay, decay, and double in all nuclei of interest. While in the case of our results (and corrections) were of particular interest in heavy nuclei, large, where relativistic and screening corrections play a major role, in the case of our results are of interest in all nuclei, since in this case there is a balance between Coulomb repulsion in the final state which favors light nuclei, small, relativistic corrections, which are large for heavy nuclei, large, and screening corrections, which are large in light nuclei due to the opposite sign of relative to . Studies similar to ours were done for single decay and EC in the 1970’s single (); bambynek ().
In this article we specifically consider the following five processes:
i) Two neutrino doublepositron decay, :
(4) 
ii) Positron emitting two neutrino electron capture, :
(5) 
iii) Two neutrino double electron capture, :
(6) 
iv) Neutrinoless doublepositron decay, :
(7) 
v) Positron emitting neutrinoless electron capture, :
(8) 
The neutrinoless double electron capture process cannot occur to the order of approximation we are considering, since it must be accompanied by the emission of one or two particles in order to conserve energy, momentum and angular momentum. It will not be considered here.
Ii Wave functions
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 this article, we need the positron scattering wave functions, and the electron bound state wave functions.
ii.1 Positron scattering wave functions
We use, for decay, negative energy Dirac central field scattering state wave functions,
(9) 
where are spherical spinors and and are radial functions, with energy , depending on the relativistic quantum number defined by . Given an atomic potential the functions and satisfy the radial Dirac equations:
(10) 
The potential appropriate for this case is obtained from that for electrons by changing the sign of ( into ). These scattering positron wave functions are normalized as the corresponding scattering electron wave functions, Eq. (12) of kotila12 (), except for the change in sign in the Sommerfeld parameter .
ii.2 Electron bound wave functions
For electron capture (EC) we use positive energy Dirac central field bound state wave functions,
(11) 
where denotes the radial quantum number and the quantum number is related to the total angular momentum, . For shell electrons , , , while for shell electrons , , . We do not consider here and shells because these are suppressed by the nonzero orbital angular momentum, , . The bound state wave functions are normalized in the usual way
(12) 
ii.3 Potential
The radial positron scattering and electron bound wave functions are evaluated by means of the subroutine package RADIAL sal95 (), which implements a robust solution method that avoids the accumulation of truncation errors. This is done by solving the radial equations by using a piecewise exact power series expansion of the radial functions, which then are summed up to the prescribed accuracy so that truncation errors can be completely avoided. The input in the package is the potential . This potential is primarily the Coulomb potential of the daughter nucleus with charge , in case of decay and the Coulomb potential of the mother nucleus with charge , in case of electron capture. As in the case of single decay and electron capture we include nuclear size corrections and screening.
The nuclear size corrections are taken into account by an uniform charge distribution in a sphere of radius with fm, i.e.
(13) 
. The introduction of finite nuclear size has also the advantage that the singularity at the origin in the solution of the Dirac equation is removed.
The contribution of screening to the phase space factors was extensively investigated in single decay wil70 (); buhring (). The screening potential is of order and thus gives a contribution of order relative to the pure Coulomb potential . We take the screening contribution into account by using the ThomasFermi approximation. The ThomasFermi function , solution of the ThomasFermi equation
(14) 
with and
(15) 
where and is the Bohr radius, is obtained by solving Eq. (14) for a point charge with boundary conditions
(16) 
for decay,
(17) 
for decay (, , respectively), and
(18) 
for decay. This takes into account the fact that the final atom is a negative ion with charge , or a neutral ion depending on the mode (, , , respectively). With the introduction of this function, the potential including screening becomes
(19) 
. This can be rewritten in terms of an effective charge where now depends on . In order to solve Eq. (14), we use the Majorana method described in esp02 () which is valid both for a neutral atom and negative/positive ion. The Majorana method requires only one quadrature and is amenable to a simple solution, the accuracy of which depends on the number of terms kept in the series expansion of the auxiliary function of Ref. esp02 (). The solution is smooth for all three boundary conditions. It is particularly useful here, since we want to evaluate screening corrections in several nuclei. As an example for the resulting functions with the boundary conditions presented in Eqs. (16) (18) we show in Fig. 1 results for Kr decay.
ii.4 Solutions
In order to illustrate the effect of finite size and screening we show in Fig. 2 the positron scattering wave function for MeV, and in Fig. 3 the electron bound wave function for the and states. Comparing Fig. 3 with Fig. 2 of Ref. kotila12 () Fig. 2, one can see that the effect of screening is larger than in and of opposite sign, since the electron cloud decreases the magnitude of the repulsive potential seen by the outgoing positrons.
Iii Phase space factors in double decay
In order to calculate PSFs for , , and , we use the formulation of Doi and Kotani doitwoneutrino (); doineutrinoless ().
iii.1 Decays where two neutrinos are emitted
The decay is a second order process in the effective weak interaction. It can be calculated in a way analogous to single decay. Neglecting the neutrino mass, considering only Swave states and noting that with four leptons in the final state we can have angular momentum , and, , we see that both and decays can occur. We denote by , where , the values of the decay. These can be obtained from the mass difference between neutral mother and daughter atoms, as
(20) 
For the total available kinetic energy one also needs to take into account the binding energy of the captured electron and thus the total available kinetic energies for , and modes are
(21) 
The values of are shown in Table 1. Another quantity of interest in the evaluation of the PSFs is the excitation energy of the intermediate nucleus with respect to the average of the initial and final ground states,
(22) 
illustrated in Fig. 4. As discussed in Ref. kotila12 (), the results for PSF depend weakly on the values of the energies in the intermediate oddodd nucleus, as remarked years ago by Tomoda tom91 () and as shown explicitly in our Ref. kotila12 (), Fig. 4. We therefore perform all calculations in this paper by replacing with an average value and MeV as suggested by Haxton and Stephenson haxton (). The error introduced by this approximation is discussed in the following Sect. IV. We emphasize, however, that our calculation has been set up in such a way as to allow a state by state evaluation, if needed.
Nucleus 
(MeV) 

, and allowed  
Kr  2.8463(7) 
Ru  2.71451(13) 
Cd  2.77539(10) 
Xe  2.8654(22) 
Ba  2.619(3) 
Ce  2.37853(27) 
and allowed  
Cr  1.1688(9) 
Ni  1.9263(3) 
Zn  1.0948(7) 
Se  1.209169(49) 
Sr  1.7900(13) 
Mo  1.651(4) 
Pd  1.1727(36) 
Sn  1.91982(16) 
Te  1.71481(125) 
Sm  1.78259(87) 
Dy  2.012(6) 
Er  1.8440(30) 
Yb  1.40927(25) 
Hf  1.0988(23) 
Os  1.453(58) 
Pt  1.384(6) 
allowed  
Ar  0.43259(19) 
Ca  0.193510(20) 
Fe  0.6798(4) 
Cd  0.27204(55) 
Xe  0.920(4) 
Ba  0.8440(10) 
Ce  0.698(10) 
Gd  0.05570(18) 
Dy  0.284(3) 
Er  0.02507(12) 
W  0.14320(27) 
Hg  0.820(3) 
decay
The formulas for decay are exactly the same as for decay described in kotila12 () where now is the energy of the first positron, , and is the energy of the second positron, . We use here the same approximations as in kotila12 (), that is to evaluate the positron wave functions at the nuclear radius
(23) 
and to replace the excitation energy in the intermediate oddodd nucleus by a suitably chosen energy , giving
(24) 
The phase space factors are then given in terms of quantities kotila12 (); tom91 ()
(25) 
These approximations allow a separation of the PSF from the nuclear matrix elements and the condition under which they are good have been discussed in kotila12 (). Apart from a narrow region around threshold, where the error is , the approximations are good throughout. For decay we have two integrated phase space factors and whose explicit expression are given in Eqs. (21)(28) and (34)(36) of kotila12 (). Since the calculated single decay matrix elements of the GT operator in a particular nuclear model appear to be systematically larger than those derived from measured values of the allowed GT transitions, and this effect is usually taken into account by quenching the axial vector coupling constant , it is convenient to separate it from the phase space factors . Also, it is convenient to scale the matrix elements with the electron mass, . The phase space factors are then in units of yr. From these we obtain
(i) The halflife
(26) 
(ii) The differential decay rate
(27) 
where .
(iii) The summed energy spectrum of the two positrons
(28) 
These three quantities depend only on .
(iv) The angular correlation between the two positrons
(29) 
which depend on both and . Here and in the following subsections 2 and 3,
(30) 
where and . The closure energies and could in principle be different, but in this article we take .
The phase space factors for decay are listed in Table 2 column 2, where they are also compared with values found from literature doitwoneutrino (); boehm () (columns 3 and 4), and in Fig. 5. The values in the literature have been converted to our notation by removing factors of and . The value for Ce should be taken with caution because of the very low value. We also have available upon request for all nuclei in Table 2 the single positron spectra, the summed energy spectra and angular correlations between the two outgoing positrons. As examples, we show the cases of Kr Se decay, Fig. 6, and of Cd Pd, Fig. 7. The use of a screened potential makes a considerable difference compared to the results obtained when taking into account only the finite nuclear size, as shown in Fig. 6. Note the difference between the single positron spectra in Figs. 6 and 7 for and the single electron spectra in Figs. 6 and 7 of kotila12 () for decay. Note also the difference in the scale of Fig. 5, yr, for , as compared with the scale of Fig. 5 of kotila12 (), yr, for .



Nucleus  This work  DK  BV  This work  DK  BV  This work  DK  BV 
, and allowed  
Kr  9770  13600  16000  385  464  390  660  774  136 
Ru  1040  1080  1230  407  454  350  2400  2740  433 
Cd  2000  1970  2420  702  779  652  5410  6220  1120 
Xe  4850  4770  5410  1530  1720  1408  17200  20200  3500 
Ba  110  47.9  59.2  580  549  420  15000  16300  2590 
Ce  0.267  0.559  0.795  190  253  192  12500  15800  2420 
and allowed  
Cr 
1.16  1.05  0.422  0.0887  
Ni  1.11  1.16  1.00  15.3  17.0  3.01  
Zn  3.81  3.83  1.41  0.281  
Se  1.09  8.39  5.656  1.08  
Sr  0.729  0.616  93.6  17.9  
Mo  0.206  0.164  208  24.0  
Pd  1.62  7.16  46.0  9.14  
Sn  4.95  4.33  1150  235  
Te  0.730  0.524  888  173  
Sm  2.49  1.98  5150  982  
Dy  25.3  20.2  17600  3100  
Er  6.40  6.69  5.29  15000  18100  2770  
Yb  0.00979  0.00763  4710  890  
Hf  1.00  2.77  1580  310  
Os  0.0299  0.0156  12900  2240  
Pt  0.00588  0.00235  12900  2290  

allowed  
Ca 
1.25  
Fe  0.0469  
Cd  0.0207  
Xe  46.1  
Ba  39.1  
Ce  18.4  
Dy  0.183  
W  0.00156  
Hg  821 
decay
For the calculation of electron capture processes the crucial quantity is the probability that an electron is found at the nucleus. This can be expressed in terms of the dimensionless quantity doitwoneutrino ()
(31) 
where is the Bohr radius cm and we use for the nuclear radius fm. For capture from the shell , , while for capture from the shell , , .
Denoting by the energy of the emitted positron and by the binding energy of the captured electron, the phase space factor can be written as doitwoneutrino ()
(32) 
where and are the neutrino energies.
Now in the definition of and in Eq. (25), is the energy of the captured electron and is the energy of emitted positron.
Again separating and the electron mass , the PSF are in units of yr. From those, we obtain:
(i) The halflife
(33) 
(ii) The differential decay rate
(34) 
where .
decay
In the case of double electron capture with two neutrinos the energies of the electrons are fixed and the two neutrinos carry all the excess energy. The equation for PSF then reads doitwoneutrino ():
(35) 
In this case, in the definition of and in Eq. (25), is the energy of the first captured electron and is the energy of the second captured electron. The values obtained are listed in Table 2 column 8 where they are compared with previous calculations (columns 9 and 10), and in Fig. 10. From we can calculate:
(i) The halflife
(36) 
While in the case of and decay all three calculations agree within a factor of , in the case of decay, the calculation reported in the book of Boehm and Vogel boehm (), disagrees with other two by a factor of approximately 4. The origin of this discrepancy is not clear. The values in Table 2 have been converted to units yr using the same procedure in all three cases, , , and . Since apart from the factor of 4, the behavior with mass number of in boehm () is the same as in the other two calculations, it may be simply due to a different definition of . Note that the scale in Fig. 10, yr, for is very different from that for , yr, in Fig. 5 due to a much larger value
iii.2 Neutrinoless modes
As discussed in Ref. barea12b (), 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 . In this article we present phasespace factors for the mass mode. Phasespace factors associated with the other modes, called and in Ref. tom91 (), will form the subject of a subsequent publication.
decay
The equations for decay are exactly the same as for decay described in kotila12 () where now is the energy of the first positron, , and is the energy of the second positron, . There are also here two quantities and in units of yr from which one can obtain:
(i) The halflife
(37) 
(ii) The differential decay rate
(38) 
where .
Both the halflife and the differential decay rate, are given in terms of .
(iii) The angular correlation between the two positrons
(39) 
Nucleus  This work  DK  BV  This work  DK  BV 

, and allowed  
Kr 
250  293  59.4  6.37  7.11  
Ru  84.5  90.7  12.2  9.62  10.8  
Cd  96.2  102  14.5  13.0  14.7  
Xe  114  123  18.1  19.7  22.9  
Ba  25.7  21.0  1.67  17.6  19.8  
Ce  2.42  3.55  0.175  15.3  18.7  
and allowed  
Cr 
0.0887  
Ni  1.21  1.30  
Zn  0.0507  
Se  0.230  
Sr  1.94  
Mo  1.92  
Pd  0.287  
Sn  5.20  
Te  3.92  
Sm  8.11  
Dy  15.2  
Er  12.9  15.8  
Yb  4.23  
Hf  0.0272  
Os  7.04  
Pt  5.57  

The values of are shown in Table 3 column 2 where they are compared with previous calculations (column 3 and 4), and in Fig. 11. In this case, our calculation and that of doineutrinoless () disagree with the calculation reported in boehm () by a larger factor.
decay
In case of neutrinoless positron emitting electron capture, the energy of the emitted positron is fixed and the equation for PSF reads doineutrinoless ():
(40) 
The PSF are in units yr, and from those we can calculate:
(i) The halflife