GamowTeller and doublebeta decays of heavy nuclei within an effective theory
Abstract
We study decays within an effective theory that treats nuclei as a spherical collective core with an even number of neutrons and protons that can couple to an additional neutron and/or proton. First we explore GamowTeller decays of parent oddodd nuclei into lowlying ground, one, and twophonon states of the daughter eveneven system. The lowenergy constants of the effective theory are adjusted to data on decays to ground states or GamowTeller strengths. The corresponding theoretical uncertainty is estimated based on the power counting of the effective theory. For a variety of mediummass and heavy isotopes the theoretical matrix elements are in good agreement with experiment within the theoretical uncertainties. We then study the twoneutrino double decay into ground and excited states. The results are remarkably consistent with experiment within theoretical uncertainties, without the necessity to adjust any lowenergy constants.
I Introduction
Atomic nuclei are sensitive to fundamental interactions beyond the strong force that binds them. Excited states typically decay due to electromagnetic interactions emitting rays, while unstable nuclear ground states decay via weak interactions emitting or capturing electrons, neutrinos, or their antiparticles. Nuclei are also used as laboratories due to their sensitivity to newphysics interactions beyond the Standard Model Severijns and NaviliatCuncic (2011); Baudis (2012); Engel et al. (2013).
The weak interaction is closely connected to groundstate decays. Almost every unstable isotope lighter than Pb decays either via decay or electron capture. The associated halflives can range from milliseconds to billions of years, with the corresponding nuclear transition matrix elements varying by three or more orders of magnitude. This wide range makes theoretical predictions of decay and electron capture particularly challenging tests of nuclearstructure calculations. Reliable predictions are also especially important for astrophysics because decay halflives of experimentally inaccessible very neutronrich nuclei set the scale of the rapidneutron capture, or process, which is responsible for the nucleosynthesis of heavy elements Langanke and MartínezPinedo (2003); Arnould et al. (2007); Arcones and Thielemann (2013).
Ab initio calculations of decays are still limited to fewnucleon systems (see, e.g., Refs. Gazit et al. (2009); Baroni et al. (2016); Klos et al. (2016)) or focus on selected lighter isotopes Maris et al. (2011); Ekström et al. (2014). For mediummass and heavy nuclei theoretical studies typically use the quasiparticle randomphase approximation (QRPA) (see, e.g., Refs. Nikšić et al. (2005); Marketin et al. (2007); Niu et al. (2013a, b); Fang et al. (2013); Mustonen et al. (2014); Shafer et al. (2016)), sometimes combined with more macroscopic calculations Möller et al. (2003), and when possible the nuclear shell model (see, e.g., Refs. Caurier et al. (2005); Suzuki et al. (2012); Zhi et al. (2013); Kumar et al. (2016)). The agreement of these manybody calculations with experimental data demands fitting part of the nuclear interactions and/or the effective operators, such as the isoscalar pairing for the QRPA or a renormalization (“quenching”) factor for the transition operator in shell model calculations Brown and Wildenthal (1988); MartínezPinedo et al. (1996). At present, the predictions made by different methods for nonmeasured decays disagree by factors of a few units. Furthermore, in these phenomenological calculations it is difficult to provide well founded estimates of the associated theoretical uncertainties.
Secondorder processes in the weak interaction, double () decays, have been observed in otherwise stable nuclei. They exhibit the longest halflives measured to date, exceeding years Barabash (2015). This decay mode is via twoneutrino () decay, to distinguish it from an even more rare type of decay without neutrino emission, neutrinoless () decay, which is so far unobserved. The latter process is not allowed by the Standard Model since it violates lepton number conservation, and can only occur if neutrinos are their own antiparticles. decay is the object of very intense experimental searches Agostini et al. (2017); Gando et al. (2016); Alfonso et al. (2015); Albert et al. (2014) with the goal to elucidate the nature of neutrinos. The calculation of matrix elements for decays is specially subtle Caurier et al. (2012); Vergados et al. (2012); Suhonen and Civitarese (2012); Engel and Menéndez (2017). It faces similar challenges to those of decays, with the additional difficulty that decays are suppressed. As with decays, calculations by different manybody approaches of unknown nuclear matrix elements vary by factors of a few units. The prediction of reliable matrix elements with theoretical uncertainties is specially pressing for the interpretation and planning of present and future decay experiments.
The goal of this work is to use an effective theory (ET) framework to study the , electron capture, and decays of mediummass and heavy nuclei. Calculations within an ET can provide transition matrix elements with quantified theoretical uncertainties, and are therefore a good complement to existing manybody decay studies. We focus on GamowTeller (GT) transitions for which experimental data is available, leaving the study of decay for future work. The ET used here is valid for transitions involving spherical systems, because nuclei are treated as a spherical collective core coupled to a few nucleons.
Over the last decades ETs have been applied to describe the lowenergy properties of nuclei. Effective theories exploit the separation of scales between the lowenergy physics governing the processes of interest, which are treated explicitly, and the highenergy physics whose effects are integrated out and encoded into lowenergy constants (LECs) that must be fit to experimental data. The ET is formulated in terms of the lowenergy degrees of freedom (DOF) and their interactions, consistent with the symmetries of the underlying theory. The ET offers a systematic orderbyorder expansion based on a power counting given by the ratio of the lowenergy over the breakdown scale, at which the ET is no longer valid. This also allows for the quantification of the theoretical uncertainties at a given order in the ET Dobaczewski et al. (2014); Furnstahl et al. (2015a, b); Wesolowski et al. (2016).
Effective field theories have been very successful in describing two and threenucleon forces in terms of nucleon and pion fields van Kolck (1999); Bedaque and van Kolck (2002); Epelbaum et al. (2009); Machleidt and Entem (2011); Hammer et al. (2013). In combination with powerful ab initio methods, chiral interactions have been employed to calculate lowenergy properties of light and mediummass isotopes (see, e.g., recent reviews Barrett et al. (2013); Hagen et al. (2014); Hebeler et al. (2015); Navrátil et al. (2016); Hergert et al. (2016)). Complementary, a different set of ETs has been proposed to describe heavy nuclei in terms of collective DOF Papenbrock (2011); Zhang and Papenbrock (2013); Papenbrock and Weidenmüller (2014); Coello Pérez and Papenbrock (2015a); Papenbrock and Weidenmüller (2015); Coello Pérez and Papenbrock (2015b, 2016). In particular, the ET developed in Refs. Coello Pérez and Papenbrock (2015b, 2016) describes the lowenergy properties of spherical eveneven and oddmass nuclei in terms of effective singleparticle DOF coupled to a collective spherical core. Using this approach, the lowenergy spectra and electromagnetic properties of eveneven and oddmass nuclei (the latter with ground states) were described consistently, in good agreement with experiment Coello Pérez and Papenbrock (2016). The electromagnetic strength between lowlying states, including magnetic dipole transitions in oddmass systems, was predicted successfully. Since for decays the relevant physics is expected to be like that of magnetic dipole transitions, the ET framework offers a promising approach to describe , electron capture, and decays in nuclei.
This paper is organized as follows. Section II serves as a short summary of key elements from the theory of , electron capture and decays. Section III begins with a brief introduction to the ET of spherical eveneven nuclei followed by an extension of the theory to account for the lowlying states of oddodd nuclei. Next we discuss the GT transition operator that enters decays as well as methods to fix the associated LECs. In Sec. IV we present our ET results for decay matrix elements of spherical oddodd nuclei with ground states into final states of eveneven nuclei corresponding to different collective ET excitations. We compare the ET predictions with experimental data, including the estimated theoretical uncertainties. In Sec. V we test the ability of the ET to consistently describe and decays, without the necessity to adjust additional LECs. We conclude with a brief summary and outlook in Sec. VI.
Ii Types of decays
ii.1 Single decay and electron capture
The most common weak interaction process is decay. Here either one of the neutrons in a nucleus decays into a proton ( decay), or one of the protons decays into a neutron ( decay). The total number of nucleons remains the same. For the electric charge, lepton number, and angular momentum to be conserved, it is required that an electron () or positron () are emitted along with an electron antineutrino () or neutrino ():
(1)  
(2) 
Alternatively, protonrich nuclei can undergo a third process, an electron capture (EC), which involves the capture of an electron by a proton, yielding a neutron. Again, conservation of energy, angular momentum, and lepton number requires an electron neutrino to be emitted:
(3) 
At lowest order in the weak interaction, it is possible to distinguish two types of dominant, socalled allowed, transitions. Gamow Teller and Fermi (F) decays differ in the spin dependence of the associated onebody operator. The GT and F operators are defined as
(4)  
(5) 
where denotes the spin, () is the isospin raising (lowering) operator, and a sum is performed over all nucleons in the nucleus.
In this work, we focus on allowed GT transitions, whose decay rates are related to the reduced matrix elements of the GT operator in Eq. (4) between the corresponding initial () and final () nuclear states:
(6) 
The decay’s halflife is given by
(7) 
Here the phasespace factor contains all the information on the lepton kinematics, s is the decay constant, is the axialvector coupling, and denotes the total angular momentum of the initial state. The quantity is known as the value and is directly comparable to the transition nuclear matrix element.
ii.2 decays
In the decay, two neutrons of the parent eveneven nucleus decay into two protons. Two electrons and antineutrinos are emitted as well due to charge and leptonnumber conservation:
(8) 
Such decays have been detected for several nuclei, including few cases of transitions into excited states Barabash (2015). At present the similar but kinematically less favored and the doubleelectroncapture (ECEC) decays have not been observed experimentally.
In principle, both GT and F operators enter the description of decays. Nevertheless, in decays to lowlying states of the final daughter nucleus only the GT part is relevant. The F operator does not connect states with different isospin quantum numbers, and its strength is almost completely exhausted by the isobaric analogue state, which lies at an excitation energy of tens of MeVs. The decay rate for decay is then Engel and Menéndez (2017)
(9) 
where is a phasespace factor, and the nuclear matrix element is given by
(10) 
where the electron mass is introduced to make the matrix element dimensionless, and the sum runs over all states of the intermediate oddodd nucleus. The energy denominators are given in terms of the energy of the initial, final and intermediate states by
(11) 
Iii ET for Single decay
In this section, we formulate an ET for the GT decays of parent oddodd into daughter eveneven nuclei. Our approach is valid for systems with lowenergy spectra and electromagnetic transitions well reproduced by an ET written in terms of collective DOF, which at leading order (LO) represents a fivedimensional harmonic oscillator. The theoretical uncertainties due to omitted DOF can be propagated in the ET up to the nuclear matrix elements and decay halflives, allowing for a comparison of the ET predictions with experimental data.
iii.1 ET for eveneven and oddodd nuclei
The ET developed in Refs. Coello Pérez and Papenbrock (2015b, 2016) describes the lowenergy properties of spherical eveneven and oddmass nuclei in terms of collective excitations that can be coupled to an odd neutron, neutronhole, proton, or protonhole. The effective operators are written in terms of creation and annihilation operators, which are the DOF of the ET. These include:

Collective phonon operators and , which create and annihilate quadrupole phonons associated with lowenergy quadrupole excitations of the eveneven core.

Neutron operators and , which create and annihilate a neutron or neutronhole in a singleparticle orbital with total angular momentum and parity .

Proton operators and , which create and annihilate a proton or protonhole in a orbital.
Whether the fermion operators represent a particle or a hole depends on the oddmass nucleus we want to describe. In Ref. Coello Pérez and Papenbrock (2016), silver isotopes with ground states were described both as an odd proton coupled to palladium cores and as an odd protonhole coupled to cadmium cores. Both descriptions turned out to be consistent with each other. The above operators fulfill the following relations
(12) 
While the creation operators are the components of spherical tensors, the annihilation operators are not. To facilitate the construction of sphericaltensor operators with definite ranks, we define annihilation spherical tensors with components , where and .
The Hamiltonian describing a particular eveneven and adjacent oddmass pair of nuclei at nexttoleading order (NLO) is given by Coello Pérez and Papenbrock (2016)
(13) 
where , depending in which oddmass nucleus we want to describe, and and are LECs that must be fitted to data. The LEC accompanying the LO term, , may be thought of as the energy of the collective mode. It scales as the excitation energy of the first excited state of the eveneven system of interest. The energy scale , at which the ET breaks down, lies around the threephonon level. Based on previous work Coello Pérez and Papenbrock (2015b), we will set in what follows, even though the breakdown scale might not be exactly the same for every pair of nuclei studied in this work. The effective operators of the theory are constructed orderbyorder adding all relevant terms that correct the previous one by a positive power of .
The reference state of the ET represents the ground state of the eveneven nucleus of interest. Multiphonon excitations of this state represent excited states in the eveneven system. Of particular relevance for our work are one and twophonon excitations:
(14) 
where in the notation , and are the total angular momentum of the state and its projection, and is the number of phonons. We define the coupling of two spherical tensors as in Ref Varshalovich et al. (1988), and refer to Ref. Rowe and Wood (2010) for a detailed description of the construction of multiphonon excitations. We highlight that the ET introduced above reproduces the lowlying spectra and electromagnetic moments and transitions of vibrational mediummass and heavy nuclei within the estimated theoretical uncertainties Coello Pérez and Papenbrock (2015b).
In a similar fashion, the ground states of adjacent oddmass nuclei can be described as fermion excitations of the reference state . Even though several singleparticle orbitals may be relevant, at LO in the ET it is assumed that only one orbital is required to describe the lowenergy properties of the oddmass system. The relevant singleparticle orbital is inferred from the quantum numbers of the ground state of the oddmass nucleus of interest. This assumption works well for oddmass nuclei near shell closures with ground states Coello Pérez and Papenbrock (2016). In these systems, a reasonable agreement was found between the ET predictions and experimental data, regarding not only lowenergy excitations but also electric and magnetic moments and transitions Coello Pérez and Papenbrock (2016).
In order to describe the allowed GT decays of parent oddodd nuclei, we extend the collective ET of Refs. Coello Pérez and Papenbrock (2015b, 2016) and write the lowlying positiveparity states in the oddodd nucleus as
(15) 
where the fermion operators represent particles or holes depending on the oddodd nucleus of interest. For example, can be described coupling a neutron and a protonhole to a core, or coupling a neutronhole and a proton to a core. The and labels in the oddodd state indicate the coupling of the odd neutron and odd proton on top of the collective spherical ground state. The angular momentum and parity of the singleparticle orbitals to be used are inferred from the quantum numbers of the lowlying states of the adjacent oddmass nuclei. Therefore, the total angular momenta and parities of these orbitals must fulfill the relations , and for positiveparity states.
It is important to note that the LO ET calculation of single decay of the groundstates of oddodd nuclei does not require to explicitly construct their energy spectra, simplifying the calculations considerably. Contributions due to additional DOF relevant for excited states are taken into account in the uncertainty estimates associated with the LO results. We stress that these uncertainties must be tested whenever data is available, because they probe the validity of the power counting and the reliability of the LO calculations.
iii.2 Effective GT operator
Next we construct the operator corresponding to the GT operator in Eq. (4), in terms of the effective DOF. We write the most general positiveparity sphericaltensor operator of rank one capable of coupling the lowlying states of the parent oddodd nucleus introduced in Eq. (15) to the ground, one and twophonon states of the daughter eveneven nucleus represented in Eq. (14). At lowest order in the number of operators this operator is given by
(16) 
where , and are LECs that must be fit to experimental data.
Let us discuss the effective operator of Eq. (III.2) in more detail. For the allowed GT decay of the oddodd nucleus with neutrons and protons into the eveneven nucleus with neutrons and protons, the oddodd system is described within the ET as an eveneven core coupled to a neutron and a protonhole. For the decay to take place, the fermion annihilation operators must annihilate the odd neutron and protonhole. This action represents the decay of a neutron in the oddodd system into a proton, which then fills the protonhole yielding the eveneven system. An additional step in which the odd neutron fills a neutronhole takes place if the annihilated neutron is part of the core; however, the ET cannot differentiate between the two processes. For the description of the or EC decay to the eveneven nucleus with neutrons and protons, it is more convenient to describe the oddodd nucleus in terms of the later eveneven core coupled to a neutronhole and a proton. In this case, the fermion annihilation operators annihilate the odd neutronhole and proton, representing the conversion of a proton into a neutron that fills the neutronhole. Again, the additional filling of a protonhole by the oddproton follows if the annihilated proton is in the core.
The reduced matrix elements of the effective GT operator in Eq. (III.2) between lowlying states of the parent oddodd system in Eq. (15) and the daughter eveneven ground, one and twophonon states in Eq. (14) are
(17)  
(18)  
(19) 
where the subscripts and identify the ground and phonon states of the daughter eveneven nucleus, respectively. We also note that the LECs in Eqs. (17)–(19) implicitly take into account additional corrections to the GT operator in Eq. (4), such as a possible “quenching”.
iii.3 ET GT decay to ground and excited states
The first, second, and third terms of the effective GT operator in Eq. (III.2) couple states with phononnumber differences of zero, one, and two, respectively. Thus, they describe the decays from the ground state of the oddodd nucleus to the ground, one and twophonon states in the eveneven nucleus, respectively. Figure 1 schematically shows the case of the decays of into the , , and states of , identified as the ground, one and twophonon states.
The LECs , and encode the microscopic information of the nuclei involved in the decay. While the value of the LECs is not predicted by the ET, the power counting established in previous works Coello Pérez and Papenbrock (2015b, 2016) suggests scaling factors between them. This power counting is based on the assumption that at the energy scale where the ET breaks down, the matrix elements of every term of any effective operator scale similarly.
From this assumption and the effective Hamiltonian describing the eveneven systems, it can be concluded that at the breakdown scale the matrix elements of an operator containing powers of operators scale as Coello Pérez and Papenbrock (2015b)
(20) 
For more details, we refer the reader to Ref. Coello Pérez and Papenbrock (2015b).
The power counting in Eq. (20) implies that at the breakdown scale the matrix elements of the different terms in the effective GT operator (III.2) scale as
(21) 
and
(22) 
The resulting relative sizes of the matrix elements for the decay are also shown schematically in Figure 1. The theoretical uncertainties for the above ratios have been estimated based on the expectation for the nextorder LECs to be of natural size, so that a number of order one has a range from and an associated relative uncertainty. Finally, we emphasize that these theoretical uncertainty estimates must be tested by comparing the ET predictions to experimental data.
iii.4 ET GT decay to ground states and theoretical uncertainties
The matrix elements of single decays to the ground state are set by the values of the LEC , see Eq. (III.2), which are to be fitted to experimental data. Within the ET the uncertainty of these matrix elements comes from two sources:

Omitted terms in the effective GT operator that involve two operators and couple states of oddodd and eveneven nuclei with the same number of phonons. The matrix elements of these terms are expected to scale as
(23) 
Nexttoleading order corrections to the ground state of oddodd nuclei due to terms in the Hamiltonian that can mix states with phononnumber differences of one. These corrections are expected to scale as and are coupled to the eveneven ground state by the second term of the effective GT operator, which contains an additional operator. Therefore, the corrections to the matrix elements also scale as
(24)
Iv Results for single decay
iv.1 GT decays to excited states
We begin our calculations by studying the decay and electron capture of spherical oddodd parent nuclei with ground states into different ground and excited and states of the eveneven daughter nuclei. This will show to which extent can the ET describe processes involving individual nucleons and whether the transitions scale as expected in the ET.
For each parent nucleus, the LEC can be fitted to the transition to the ground state. Then the GT decays into excited collective states are predicted by the ET according to the scaling factors in Eqs. (21) and (22):
(27)  
(28) 
We note that the ET naturally predicts the observed successive hindering reported in Ref. Sakai (1962) of the matrix elements for GT decays from , , and ground states of oddodd nuclei into , , and states of the eveneven daughter.
Figure 2 compares our ET predictions with experiment for GT and EC decay matrix elements for a broad range of mediummass and heavy oddodd nuclei with mass number from to . All parent nuclei have ground states and decay into excited , , and states of the corresponding eveneven nuclei. Within the ET, the states are treated as onephonon excitations of the eveneven core, while the and states are considered to be twophonon excitations. The ET results with uncertainties are calculated according to Eqs. (27) and (28) after adjusting the LEC to the matrix element of the decay to the ground state of the daughter nucleus. The same figure shows that most of the experimental data, including GT and EC decays, is consistent with the ET results. Inconsistencies are larger for the and twophonon states where the ET matrix elements tend to be overestimated, especially for the decays into excited states around mass number . This is not unexpected because twophonon states lie closer to the ET breakdown scale. In Table 1 we list the numerical values corresponding to all the theoretical and experimental matrix elements shown in Fig. 2.
exp  ET  exp  ET  exp  ET  exp  ET  
iv.2 GT decays to ground states using GT transition strengths
In Sec. IV.1, we have used experimental data on single decays to ground states to fit the value of and then predicted the matrix elements for transitions to excited states of the same nucleus. Next, we study whether it is possible to employ other data to fit the LECs and in turn predict the decay matrix elements to ground states.
Besides weak processes, GT strengths studied in chargeexchange reactions (via the strong interaction) are also sensitive to the GT spinisospin operator, because the zerodegree differential cross section of the reaction is proportional to the GT strength Watson et al. (1985)
(29) 
Therefore, reactions such as or have the same form as decays, while reactions are related to like transitions. The GT strengths and decays of isospinmirror nuclei have been found to be consistent in mediummass systems Molina et al. (2015).
In this spirit, we can use the GT transition strengths measured in chargeexchange reactions to fit the LECs of the effective GT operator in Eq. (III.2). In this way, we can predict the GT matrix elements for the transition to the ground states of spherical eveneven nuclei. Figure 3 shows that the ET results for the GT matrix elements including the theoretical uncertainty, agree very well with experiment in three of the four cases where data is available. The experimental values are calculated from the values of the corresponding EC decays from Refs. Singh (2007, 2008); Blachot (2010); Elekes and Timar (2015). The remaining results in Fig. 3 show ET predictions for single decay of additional states, which are, however, excited states of the oddodd system and decay via electromagnetic transitions.
The details of the ET predictions in Fig. 3 are given in Table 2, which lists the GT strengths measured in chargeexchange reactions with initial eveneven and final oddodd nuclei for –130 Popescu et al. (2009); Thies et al. (2012a); Frekers et al. (2016); Thies et al. (2012b); Akimune et al. (1997); Puppe et al. (2012). For each reaction, Table 2 gives the experimental partial GT strength to the lowest state of the oddodd nucleus (the state expected to be well described by the ET). An exception is the case of the reaction, where all the GT strength below 500 keV was taken into account (as reported in Table IV of Ref. Thies et al. (2012a)), corresponding to three closelying states. The resulting values for fit to the partial GT strength are given in Table 2 including the comparison of the ET results to the experimental values.
exp  ET  

