Advances in High Energy Physics

Volume 2016 (2016), Article ID 7486712, 8 pages

http://dx.doi.org/10.1155/2016/7486712

## An Effective Method to Accurately Calculate the Phase Space Factors for Decay

Department of Physics, Central Michigan University, Mount Pleasant, MI 48859, USA

Received 21 December 2015; Accepted 28 March 2016

Academic Editor: Aurelio Juste

Copyright © 2016 Andrei Neacsu and Mihai Horoi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### 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.

#### 1. Introduction

Double-beta decay processes are of considerable importance for the study of neutrinos. They change the charge of a nucleus by two units, releasing two electrons, while the mass remains unchanged. The 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 (), 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 decay transition could occur without antineutrinos in the final state, called neutrinoless double-beta (), and this implies that the neutrino is a Majorana fermion [2]. The 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 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 the 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 ground state (g.s.) transitions. Table and Figure 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 PSF of 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 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. Section 2 shows the formalism for transitions to ground states and for 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 and PSF.

#### 2. Brief Formalism of the Decay

For the 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 half-life expression is given bywhere is the axial-vector coupling strength, , , , , and are neutrino physics parameters defined in [19], and are the light and heavy neutrino-exchange nuclear matrix elements [5, 20], and and represent combinations of NME and phase space factors. 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 -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 and terms, which share the same PSF as term, , and have the same energy and angular distribution as the term) is written as where and are the relative CP-violating phases [19] and is the Gamow-Teller part of the light left-handed neutrino-exchange NME. Different processes give rise to several contributions: comes from the left-handed leptonic currents, from the right-handed leptonic and right-handed hadronic currents, and from the right-handed leptonic and left-handed hadronic currents. The interference between these terms is represented by the , , and contributions. Neglecting the very small tensor contributions in the mass mechanism, the components are defined as products of PSF and NME [9]: with

The fractions of NME are defined [9] as , with , and 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 and 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 PSF are given in Appendix A.

For the process, the half-life for the transition to a state of angular momentum ( or ) of the daughter nucleus is given to a good approximation by [48]where is a phase space factor [9, 10, 13] described in Appendix B, is the electron mass, and is the NME, which can be calculated as [10, 48] Here, the -sum is taken over the states with excitation energies in the intermediate odd-odd nucleus. Hence, , where is the -value for the transition to the state of angular momentum in the daughter nucleus, and 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 (). We multiply with a parameter, called “screening factor” () in what follows, to obtain an effective “screened charge” (). 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 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 , 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 (), are denoted by for the 9 PSF of transitions to ground states, for PSF of transitions to ground states, for PSF of transitions to the first excited state, and for PSF of transitions to the first excited state.

The PSF can be grouped into two classes: those that have only -wave electron contributions and those to which -wave electrons contribute . Here, we treat them separately naming them -PSF and -PSF, respectively. We consider the largest deviation () between the PSF of a certain class and the corresponding data, and we search for the value of that minimizes it. Our goal is to maintain . 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 distribution.

In our analysis, data is selected as follows: PSF are chosen from Table III of [14]. Other recent results for [11–13] are within a few percent of these values. For , there are no other results that take into account the finite size effects of the charge distribution. The data is taken from Table of the very recent research [13], and it is in very good agreement with the results of [11]. For the transitions to the first excited states, we take the data from Table of [13]. There are four cases (, , , and ) 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 transitions to the first excited states is taken from Table of [13]. In this case, there are three PSF values (, , and ) that seem to deviate significantly from the model results. These 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 data of attributed to [13] in Tables 1–3 was provided to us as private communications by the authors of [13].