# An Effective Method to Accurately Calculate the Phase Space Factors forβ-β- Decay

Authors: Andrei Neacsu; Mihai Horoi
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

Accurate calculations of the electron phase space factors are necessary for reliable predictions of double-beta decay rates and for the analysis of the associated electron angular and energy distributions. We present an effective method to calculate these phase space factors that takes into account the distorted Coulomb field of the daughter nucleus, yet it allows one to easily calculate the phase space factors with good accuracy relative to the most exact methods available in the recent literature.


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## Body

## 1. Introduction

Double-beta decay(ββ) processes are of considerable importance for the study of neutrinos. They change the charge Z of a nucleus by two units, releasing two electrons, while the mass A remains unchanged. The ββ decay with two associated electron antineutrinos in the final state conserves the lepton number and is permitted within the standard model (SM). This process, called two-neutrino double-beta decay (2νββ), has been experimentally observed for several isotopes with transitions to both ground states and excited states of the daughter nuclei [1]. Should the lepton number conservation be violated, then theories beyond the standard model (BSM) predict that the ββ decay transition could occur without antineutrinos in the final state, called neutrinoless double-beta (0νββ), and this implies that the neutrino is a Majorana fermion [2]. The 0νββ transitions have not yet been confirmed experimentally, but there are many recent experimental and theoretical efforts dedicated to their discovery. Recent reviews on this matter are in [3–5]. There are several mechanisms that could contribute to the 0νββ decay rate, of which the simplest and most studied one involves the exchange of light Majorana neutrinos in the presence of left-handed weak interaction. Other, more complex, mechanisms include contributions from right-handed currents [6, 7] and mechanisms involving supersymmetry [5, 8].The phase space factors (PSF) that enter theββ lifetimes expressions were considered for a long time as being accurately calculated (see, e.g., [9, 10]). Recent reevaluations of the PSF, using methods that take into account the proton distributions distorting the Coulomb field of the daughter nucleus [11–14], have shown considerable differences in some cases when compared to the previous results [9, 10]. A very recent paper [14] presents four of the different methods commonly used to calculate the PSF and compares their results for the case of 0νββ ground state (g.s.) transitions. Table 1 and Figure 2 of [14] show that the Coulomb distortion of the electron wavefunction by inclusion of the finite nuclear size and electron screening effects can produce differences of up to 100%, compared to the point-charge formalism of [9] (see, e.g., the 0νββ PSF G08 of Nd150 in [14]). However, taking into account the charge distributions in the daughter nuclei and solving numerically the Dirac equation with finite nuclear size are very slow and plagued by convergence issues. This makes these complex methods unattractive for the calculations of electron angular and energy distributions, such as those presented in [15, 16].In this paper, we propose an effective method for treating the distortion of the Coulomb field in the daughter nucleus. This method uses the well known formalism of [9] but provides accurate results that are in good agreement with those of [11–14]. This method could be particularly useful when performing complex investigations involving PSF to test BSM physics due to different possible underlying mechanisms contributing to the 0νββ process. These investigations often involve calculations of electron distributions [15, 16], where components of the PSF enter the equations, and it is not possible to only use the tabulated values of [11–14].The paper is organized as follows. Section2 shows the formalism for 0νββ transitions to ground states and for 2νββ transitions to ground and excited states. In Section 3, we present our effective method for the treatment of the distorted Coulomb field in the daughter nucleus. Section 4 is dedicated to the results, and Section 5 shows our conclusions. Appendices A and B summarize the point-charge formalism from [9, 10] that we adjusted to calculate the 0νββ and 2νββ PSF.


## 2. Brief Formalism of theββ Decay

For the0νββ decay, one usually writes the inverse half-life as products of electron PSF, nuclear matrix element (NME) that depends on the nuclear structure of the parent and that of the daughter nuclei, and lepton number violation (LNV) parameters of the BSM mechanisms taken into account. Considering the existence of right-handed currents, one would find several additional contributions to the decay rate [7, 9]. The most studied mechanism is that of the light left-handed neutrino exchange, but other mechanisms could be of importance [5]. One popular model that includes contributions of right-handed currents is the left-right symmetric model [17, 18]. This model assumes the existence of heavy particles that are not included in the standard model (SM). Within this framework, the 0νββ half-life expression is given by(1)T1/20ν-1=G010νgA4M0νην+M0NηNRL+ηNRR+ηλXλ+ηηXη2,where gA is the axial-vector coupling strength, ην, ηNRL, ηNRR, ηλ, and ηη are neutrino physics parameters defined in [19], M0ν and M0N are the light and heavy neutrino-exchange nuclear matrix elements [5, 20], and Xλ and Xη represent combinations of NME and phase space factors. G010ν is a phase space factor [10] that can be calculated with relatively good precision in most cases [11, 12, 14]. Other possible contributions, such as those of R-parity violating SUSY particle exchange [5, 20], are neglected here. With some simplifying notations, the half-life expression [9] (here, we omit the contribution from the ηNRL and ηNRR terms, which share the same PSF as ην2 term, G012ν, and have the same energy and angular distribution as the ην term) is written as (2)T1/20ν-1=gA4C1ην2+cos⁡ϕ1C2ηνηλ+cos⁡ϕ2C3ηνηη+C4ηλ2+C5ηη2+cos⁡ϕ1-ϕ2C6ηληηMGT0ν2,where ϕ1 and ϕ2 are the relative CP-violating phases [19] and MGT0ν is the Gamow-Teller part of the light left-handed neutrino-exchange NME. Different processes give rise to several contributions: C1 comes from the left-handed leptonic currents, C4 from the right-handed leptonic and right-handed hadronic currents, and C5 from the right-handed leptonic and left-handed hadronic currents. The interference between these terms is represented by the C2, C3, and C6 contributions. Neglecting the very small tensor contributions in the mass mechanism, the C1–6 components are defined as products of PSF and NME [9]: (3a)C1=G010ν1-χF2,(3b)C2=G040νχ1++G030νχ2-1-χF,(3c)C3=G030νχ2+-G040νχ1--G050νχP+G060νχR1-χF,(3d)C4=G020νχ2-2+19G040νχ1+2-29G030νχ1+χ2-,(3e)C5=G020νχ2+2+19G040νχ1-2-29G030νχ1-χ2++G080νχP2-G070νχPχR+G090νχR2,(3f)C6=-2G020νχ2-χ2+-19G030νχ1+χ2++χ2-χ1-+19G040νχ1+χ1-,with(4)χ1±=χGTq±3χFq-6χTq,χ2±=χGTω±χFω-χ1±9.The fractions of NME are defined [9] as χα=Mα/MGT0ν, with α=F,GTω,Fω,GTq,Fq,Tq,R, and P indicating other NME. All these nine NME were calculated by several methods, including the interacting shell model (ISM) [16, 21, 22] and quasiparticle random phase approximation (QRPA) [23]. The light-neutrino mass mechanism MGT0ν and MF0ν NME have been extensively studied with many nuclear structure methods, such as interacting boson model (IBM-2) [24–27], interacting shell model (ISM) [20, 21, 28–37], quasiparticle random phase approximation (QRPA) [38–42], projected Hartree-Fock-Bogoliubov (PHFB) [43], energy density functional (EDF) [44], and the relativistic energy density functional (REDF) [45] method. The NME calculated with different methods and by different groups still present large differences, and that has been a topic of many debates in the literature (see, e.g., [46, 47]). Expressions for the G010ν–G090ν PSF are given in Appendix A.For the2νββ process, the half-life for the transition to a state of angular momentum J (J=0 or 2) of the daughter nucleus is given to a good approximation by [48](5)T1/22ν-1=GJ2νgA4mec2J+1MJ2ν2,where G(J)2ν is a phase space factor [9, 10, 13] described in Appendix B, me is the electron mass, and M(J)2ν is the 2νββ NME, which can be calculated as [10, 48](6)MJ2ν=1J+1∑kJf∥στ-∥1k+1k+∥στ-∥0i+Ek+EJJ+1. Here, the k-sum is taken over the 1k+ states with excitation energies Ek in the intermediate odd-odd nucleus. Hence, EJ=1/2Qββ(J)+ΔM, where Qββ(J) is the Q-value for the transition to the state of angular momentum J in the daughter nucleus, and ΔM is the difference in mass between the intermediate nucleus and the decaying nucleus.


## 3. Description of the Effective “Screening” Method

Our approach is based on the formalism from [9, 10] where the nuclear charge is considered point-like, but we replicate the effects of a finite size proton distribution distorting Coulomb field in the daughter nucleus by modifying the charge of the final nucleus (Zf). We multiply Zf with a parameter, called “screening factor” (Sf) in what follows, to obtain an effective “screened charge” (Zs=Sf/100Zf). For large enough energies, the tail of the Coulomb field plays a less significant role when compared to its part close to the nucleus, and the effect resembles charge screening. The PSF calculated with Zs for each nucleus are compared to those of [11–14] (called “data” below), which were obtained with methods that consider Dirac electron wave functions calculated with finite nuclear size and atomic electron screening. References [11, 14] take into account the finite nuclear size by a uniform charge distribution of radius R, while [12, 13] consider a more realistic Woods-Saxon proton distribution inside the nucleus. It was shown [11] that the atomic electron screening effect is small, of the order of 0.1%. The relative deviations between our results and the data, expressed in percentages (Δ=100PSF-data/data), are denoted by ΔG01–090ν for the 9 PSF of 0νββ transitions to ground states, ΔGg.s.2ν for PSF of 2νββ transitions to ground states, ΔG01+2ν for PSF of 2νββ transitions to the first excited 0+ state, and ΔG21+2ν for PSF of 2νββ transitions to the first excited 2+ state.The PSF can be grouped into two classes: those that have onlys-wave electron contributions G010ν,Gg.s.2ν,G01+2ν,G21+2ν and those to which p-wave electrons contribute G020ν–G090ν. Here, we treat them separately naming them s-PSF and p-PSF, respectively. We consider the largest deviation (Δmax) between the PSF of a certain class and the corresponding data, and we search for the value of Sf that minimizes it. Our goal is to maintain Δmax≤10%. This value of the maximum deviation is considerably lower than the uncertainties of the NME contributing to the decay rate, (2). We find that controlling the maximum deviation provides more stable and predictable results than minimizing a χ2 distribution.In our analysis, data is selected as follows:G01–090ν PSF are chosen from Table III of [14]. Other recent results for G010ν [11–13] are within a few percent of these values. For G020ν–G090ν, there are no other results that take into account the finite size effects of the charge distribution. The Gg.s.2ν data is taken from Table 1 of the very recent research [13], and it is in very good agreement with the results of [11]. For the 2νββ transitions to the first excited 0+ states, we take the data from Table 2 of [13]. There are four cases (Pd110, Sn124, Te130, and Xe136) of PSF in [13] that are in significant disagreement with those of [11]. We do not take them into account in our analysis. The data for 2νββ transitions to the first excited 2+ states is taken from Table 3 of [13]. In this case, there are three PSF values (Cd116, Sn124, and Xe136) that seem to deviate significantly from the model results. These 2νββ PSF were not confirmed by other groups, and they were often readjusted [49]. We do not include them in the analysis, but we compare them with our prediction in Table 3. The 2νββ data of Sn124 attributed to [13] in Tables 1–3 was provided to us as private communications by the authors of [13].Table 1
0
ν
β
β to ground states PSF G010ν expressed in yr−1  ×  1015 and 2νββ to ground states Gg.s.2ν expressed in yr−1  ×  1020. Results are calculated with the optimal “screening factor” for s-PSF, Sf=94.5, and compared to those of  [14] for 0νββ and  [13] for 2νββ. Second and third columns display the effective screened charge, Zs, of the daughter nucleus and the energy of the decay, Qββ.
Z
s
Q
β
β [MeV]
G
01
0
ν
G
01
0
ν [14]
Δ
G
01
0
ν
G
g
.
s
.
2
ν
G
g
.
s
.
2
ν [13]
Δ
G
g
.
s
.
2
ν
48Ca
20.79
4.272
24.55
24.83
1.1
1480.46
1553.6
4.7
76Ge
32.13
2.039
2.28
2.37
3.8
4.51
4.65
2.9
82Se
34.02
2.995
9.96
10.18
2.1
150.31
157.3
4.4
96Zr
39.69
3.35
20.45
20.62
0.8
642.0
674.4
4.8
100Mo
41.58
3.034
15.74
15.95
1.3
310.6
323.1
3.9
110Pd
45.36
2.018
4.66
4.83
3.5
12.78
13.25
3.6
116Cd
47.25
2.814
16.57
16.73
1.0
258.78
268.8
3.7
124Sn
49.14
2.289
8.87
9.06
2.1
51.45
50.4
2.1
130Te
51.03
2.527
14.10
14.25
1.0
142.73
144.2
1.0
136Xe
52.92
2.458
14.49
14.62
0.9
133.73
133.2
0.4
150Nd
58.59
3.371
66.00
63.16
4.5
3467.53
3539.7
2.0Table 2
PSF and their deviations for2νββ to the first excited 0+ states and G01+2ν expressed in yr−1  ×  1022. The last two columns present PSF of  [11] and their deviations. The results marked with “∗” and (∗) symbols (see text for details) correspond to the nuclei not included in the analysis.
Z
s
Q
β
β [MeV]
G
0
1
+
2
ν
G
0
1
+
2
ν [13]
Δ
G
0
1
+
2
ν [13]
G
0
1
+
2
ν [11]
Δ
G
0
1
+
2
ν [11]
48Ca
20.79
1.275
3.43
3.52
2.6
3.63
5.5
76Ge
32.13
0.917
0.64
0.61
5.1
0.70
7.7
82Se
34.02
1.508
41.94
41.7
0.6
—
—
96Zr
39.69
2.202
1633.8
1694
3.6
1754
6.9
100Mo
41.58
1.904
562.08
570.8
1.5
605.5
7.2
110Pd∗
45.36
0.547
0.043
0.033
30.9
0.048
10.8
116Cd
47.25
1.057
8.00
7.59
5.4
8.73
8.3
124Sn(∗)
49.14
1.120
15.09
14.1
7.0
—
—
124Sn∗
49.14
0.630
0.180
—
—
0.199
9.7
130Te∗
51.03
0.734
0.69
0.55
25.9
0.76
9.2
136Xe∗
52.92
0.979
3.31
2.82
17.2
3.62
8.7
150Nd
58.59
2.631
40637.5
41160
1.3
43290
6.1Table 3
PSF and their deviations for2νββ to the first excited 2+ states and G21+2ν expressed in yr−1  ×  1021. Denoted with “∗” symbol are PSF and deviations corresponding to the nuclei not included in the analysis.
Z
s
Q
β
β [MeV]
G
2
1
+
2
ν
G
2
1
+
2
ν [13]
Δ
G
2
1
+
2
ν
48Ca
20.79
4.272
3816
4074
6.3
76Ge
32.13
2.039
0.40
0.38
3.5
82Se
34.02
2.995
71.16
69.6
2.2
96Zr
39.69
3.35
730.8
742.5
2.0
100Mo
41.58
3.034
585
569
2.8
110Pd
45.36
2.018
0.46
0.46
0.8
116Cd∗
47.25
1.520
2.11
1.88
12.4
124Sn∗
49.14
1.686
8.89
7.63
16.5
130Te
51.03
2.527
81.09
79.6
1.9
136Xe∗
52.92
1.640
9.03
7.68
17.6
150Nd
58.59
3.371
31964
30308
5.5


## 4. Results and Discussions

For the analysis of thes-PSF, we consider G010ν of [14] and Gg.s.2ν, G01+2ν, and G21+2ν of [13]. We find that the smallest maximum deviations from the data can be obtained using an optimal “screening factor” Sf=94.5. Figure 1 shows how the maximum deviation reaches a minimum when one gets close to the optimal “screening factor.” Table 1 presents s-PSF and their deviations ΔG010ν and ΔGg.s.2ν for transitions to ground states. The adjusted charge of the daughter nucleus is also presented, together with the Qββ values. We find very good agreement for these s-PSF and the data, with deviations smaller than 5%. Should one consider the point-charge formalism [9], the largest deviation goes up to 40% for the case of G010ν of Nd150 (see, e.g., Table 1, columns A and D, of [14]). Table 2 shows PSF and deviations from the data for 2νββ transitions to the first excited 0+ states. The largest ΔG01+2ν=5.4% was for Cd116. The point-charge formalism deviations exceed 38%. The PSF marked with the “∗” symbol correspond to the four nuclei not included in our analysis. Our results for these cases can be considered as predictions for these cases that are not yet validated by other methods. The last two columns show the results of [11] and the corresponding deviations, for comparison. Reference [11] provides no value for Se82. The case of Sn124 is more complicated because of the different values used in the literature for the energy of the first excited 0+ state (see [50] for details). We include here with ∗ the phase space factor corresponding to Qββ used in [11] and with (∗) the phase space factor corresponding to Qββ considered in [51] (see discussion in [50]). The G21+2ν PSF and their deviations are displayed in Table 3. We find the largest deviation ΔG21+2ν=6.3% for Ca48. Neglecting finite nuclear size effects, one would get a deviation of 47% for Nd150. Similar to the previous table, the three results excluded from the analysis are presented for comparison and marked with the “∗” symbol.Figure 1
Fine-tuning the “screening factor” to minimize the maximums-PSF deviations.When calculating thep-PSF, G020ν–G090ν, we find a different optimal “screening factor”, Sf=92, corresponding to a larger maximum deviation, Δmax=18.1%. Figure 2 presents the evolution of Δmax close to the “optimum screening factor” for p-PSF. We attribute this larger deviation to the different kinematic factors of the nine 0νββ PSF (see (A.5a)–(A.5i)).Figure 2
Same as Figure1 for p-PSF.To further minimize the deviations, we obtain eight optimal “screening factors” corresponding toG020ν–G090ν, as seen in Figure 3. The best results are presented in Table 4, where we show the optimal “screening factor,” the maximum deviations, and the PSF values. The last line presents the optimal “screening factor” that minimizes the deviations of all the p-PSF. Alongside G020ν–G090ν, we display G010ν obtained with the optimal “screening factor” common for all s-PSF.Table 4
The calculated0νββ PSF (G010ν–G090ν) expressed in yr−1 for the decay to ground state of the 11 nuclei listed in Table 1. The last line shows the optimal “screening factor” Sf for all 8 p-PSF (G020ν–G090ν). For G01, the s-PSF optimal “screening factor”, Sf=94.5 of Table 1, was used. Shown in the last column are the maximum deviations between our calculations with the indicated parameters and the results from  [14].
48Ca
76Ge
82Se
96Zr
100Mo
110Pd
116Cd
124Sn
130Te
136Xe
150Nd
S
f [%]
Δ
m
a
x [%]
Q
β
β [MeV]
4.272
2.039
2.995
3.350
3.034
2.018
2.814
2.289
2.527
2.458
3.371
G
01
0
ν
⋅
1
0
14
2.454
0.228
0.997
2.045
1.574
0.466
1.657
0.887
1.410
1.449
6.600
94.5
4.5
G
02
0
ν
⋅
1
0
14
16.11
0.376
3.468
8.928
5.733
0.782
5.309
1.919
3.719
3.639
30.65
95.0
5.0
G
03
0
ν
⋅
1
0
15
18.45
1.233
6.671
14.50
10.72
2.548
10.96
5.254
8.853
8.967
47.87
93.0
6.1
G
04
0
ν
⋅
1
0
15
5.283
0.453
2.099
4.382
3.345
0.937
3.511
1.829
2.960
3.035
14.45
95.5
4.2
G
05
0
ν
⋅
1
0
13
3.134
0.559
2.011
4.139
3.464
1.337
4.003
2.427
3.694
3.895
15.27
90.0
4.5
G
06
0
ν
⋅
1
0
14
3.869
0.496
1.655
2.951
2.388
87.46
2.482
1.472
2.157
2.209
7.813
92.0
6.6
G
07
0
ν
⋅
1
0
14
2.790
0.268
1.161
2.432
1.885
0.566
1.984
1.052
1.663
1.703
7.799
91.0
6.0
G
08
0
ν
⋅
1
0
14
1.212
0.154
0.732
1.776
1.417
0.443
1.654
0.891
1.468
1.548
7.946
89.5
9.3
G
09
0
ν
⋅
1
0
14
15.97
1.172
4.647
8.471
6.399
1.863
6.131
3.211
4.884
4.878
20.09
94.0
4.2
p-PSF
CommonSf parameter for G020ν–G090ν
92.0
18.1Figure 3
The behaviour of maximum deviations from the reference values of the 9 PSF involved in the0νββ transitions to ground states. Subfigures (a) to (i) correspond to G010ν to G090ν. The horizontal axis represents the “screening factor” values, and the vertical axis represents the maximum deviations expressed in percentages.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)


## 5. Conclusions

In this paper, we present an effective method to calculate the phase space factors of theβ-β- transitions, which can provide results close to those of methods that consider the finite size of the proton charge and the atomic electron screening. It modifies the point-charge formalism of [9, 10], by considering a constant multiplicative screening factor for the charge of the daughter nucleus. The main advantage of our method consists in its simplicity given its accuracy and its potential to be extended to calculations of the energy and angular electron distributions needed for the analysis of the contributions of the right-handed currents to the 0νββ decay.Our method works well for PSF of0νββ and 2νββ transitions to ground states and also for 2νββ transitions to the first excited 0+ and 2+ states. For PSF where only s-wave electrons contribute, an effective “screening factor”, Sf=94.5, was obtained. Using this Sf value, one finds a maximum deviation of 6.3% between our results and other results in the recent literature [11–14]. In the case of the PSF where p-wave electrons contribute, we obtained another optimal “screening factor”, Sf=92. This corresponds to a maximum deviation of 18.1% between our results and those of [14]. We attribute this large deviation to the kinematic factors of (A.5a)–(A.5i). The deviations are greatly reduced, to less than 10%, when considering individual “screening factors” for each specific PSF (G020ν–G090ν). It is remarkable that, in the case of G08, the original point-charge formalism [9] PSF deviates by over 100% for Nd150, while it is significantly reduced in our model. Similar spectacular reductions are found for other p-PSF. We also provide predictions for the PSF of some isotopes, which can be also used as guidance in cases of disagreement between the more precise methods.In addition, usingSf=92, one gets the largest maximum deviation of 18.1 for all neutrinoless double-beta decay PSF, G010ν–G090ν. This information is relevant for the calculation of the two-electron energy and angular distributions [16].We conclude that this method is well suited for fast and accurate calculations of theββ decay PSF, with uncertainties much lower than those of the associated NME. One could envision further reducing these PSF uncertainties by considering a mass-dependent screening factor. A Mathematica notebook that can be used to obtain all these phase space factors can be downloaded from [52].



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