Phase space factors for \beta^{+}\beta^{+} decay and competing modes of double-\beta decay

Phase space factors for decay and competing modes of double- 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 life-times, single and summed positron spectra, and angular positron correlations.

pacs:
23.40.Hc, 23.40.Bw, 14.60.Pq, 14.60.St

I 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 half-life 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 half-life 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 half-lives for including relativistic corrections approximately and some non-relativistic 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 Thomas-Fermi 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 double-positron decay, :

(4)

ii) Positron emitting two neutrino electron capture, :

(5)

iii) Two neutrino double electron capture, :

(6)

iv) Neutrinoless double-positron 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 non-zero 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 Thomas-Fermi approximation. The Thomas-Fermi function , solution of the Thomas-Fermi 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.

Figure 1: (Color online) The Thomas-Fermi functions with the boundary conditions of Eqs. (16) -(18) for Kr decay. The dotdashed (gray) curve corresponds to the solution for decay, the dashed (red) curve corresponds to the solution for EC, the dashed (blue) curve corresponds to the solution for decay, and the solid (black) curve corresponds to the ECEC.

ii.4 Solutions

Figure 2: (Color online) Positron radial wave functions , (panels a and c, respectively) and , (panels b and d, respectively) for , MeV and fm (vertical line). The notations WF1 (dotted lines), WF2 (dashed lines), and WF3 (solid lines) correspond to leading finite size Coulomb, exact finite size Coulomb and exact finite size Coulomb with electron screening, respectively.

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.

Figure 3: (Color online) Electron bound state wave functions , (panels a and c, respectively) and , (panels b and d, respectively) for and fm (vertical line) scaled dimensionless with a factor of . The notations WF1 (dotted lines), WF2 (dashed lines), and WF3 (solid lines) correspond to leading finite size Coulomb, exact finite size Coulomb and exact finite size Coulomb with electron screening, respectively.

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 S-wave 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 odd-odd 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.

Figure 4: Notation used in this article. The example is for Cd decay.
Nucleus (MeV)1
, and allowed
Kr 2.8463(7)
Ru 2.71451(13)2
Cd 2.77539(10)3
Xe 2.8654(22)
Ba 2.619(3)
Ce 2.37853(27)4
and allowed
Cr 1.1688(9)
Ni 1.9263(3)
Zn 1.0948(7)
Se 1.209169(49)5
Sr 1.7900(13)
Mo 1.651(4)
Pd 1.1727(36)6
Sn 1.91982(16)7
Te 1.71481(125)8
Sm 1.78259(87)9
Dy 2.012(6)
Er 1.8440(30)10
Yb 1.40927(25)11
Hf 1.0988(23)
Os 1.453(58)12
Pt 1.384(6)
allowed
Ar 0.43259(19)
Ca 0.193510(20)
Fe 0.6798(4)
Cd 0.27204(55)13
Xe 0.920(4)
Ba 0.8440(10)
Ce 0.698(10)
Gd 0.05570(18)14
Dy 0.284(3)
Er 0.02507(12)15
W 0.14320(27)16
Hg 0.820(3)
Table 1: Mass difference used in the calculation.

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 odd-odd 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 half-life

(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
Table 2: Phase space factors obtained using screened exact finite size Coulomb wave functions. For comparison, values of Doi and Kotani doitwoneutrino () and Boehm and Vogel boehm () are also shown. They have been extracted from doitwoneutrino () and boehm () by removing , and converting to yr units.
Figure 5: (Color online) Phase space factors in units yr. The label ”DK” refers to the results obtained by Doi and Kotani doitwoneutrino () using approximate electron wave functions. The figure is in semilogarithmic scale.
Figure 6: (Color online) Single positron spectra (panel a), summed energy spectra (panel b) and angular correlations between the two outgoing positrons (panel c) for the Kr Se -decay. The scale in the left and middle panels should be multiplied by when comparing with experiment. In panels a and b the upper, solid curve is obtained when taking into account finite nuclear size and electron screening, while the lower, dashed curve presents spectra obtained when taking into account only the finite nuclear size. In panel c these two calculations coincide.
Figure 7: Single positron spectra (panel a), summed energy spectra (panel b) and angular correlations between the two outgoing positrons (panel c) for the Cd Pd -decay. The scale in the left and middle panels should be multiplied by when comparing with experiment.

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 half-life

(33)

(ii) The differential decay rate

(34)

where .

Figure 8: (Color online) Phase space factors in units yr. The label ”DK” refers to the results obtained by Doi and Kotani doitwoneutrino () using approximate electron wave functions. The figure is in semilogarithmic scale.

The obtained PSFs are listed in Table 2 column 5 where they are compared with previous calculations (columns 6 and 7), and in Fig. 8. The very small values in the second part of the table should be taken with caution in view of their very small -value.

An example of single positron spectrum is shown in Fig. 9. This figure is for Cd Pd -decay.

Figure 9: Single positron spectra for the Cd Pd -decay. The scale should be multiplied by when comparing with experiment.

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 half-life

(36)
Figure 10: (Color online) Phase space factors in units yr. The label ”DK” refers to the results obtained by Doi and Kotani doitwoneutrino () using approximate electron wave functions. The figure is in semilogarithmic scale.

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 phase-space factors for the mass mode. Phase-space 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 half-life

(37)

(ii) The differential decay rate

(38)

where .
Both the half-life 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


Table 3: Phase space factors obtained using screened exact finite size Coulomb wave functions. For comparison value of Doi and Kotani doineutrinoless () an Boehm and Vogel boehm () are also shown. These are extracted from doineutrinoless () and boehm () by removing and converting to yr units.

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.

Figure 11: (Color online) Phase space factors in units yr. The label ”DK” refers to the results obtained by Doi and Kotani doitwoneutrino () using approximate electron wave functions. The figure is in semilogarithmic scale.

We also have available upon request the single electron spectra and angular correlation for all nuclei in Table 3. An example, Cd decay, is shown in Fig. 12.

Figure 12: Single positron spectra (panel a), and angular correlations between two outgoing positrons (panel b) for the Cd Pd -decay. The scale in the left should be multiplied by when comparing with experiment.

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 half-life