Rare weak decays and nuclear structure

Weak interactions cause the atomic nuclei to decay via beta and double beta decays. Double beta decays are extremely rare since they are weak-interaction processes of the second order. Also (single) beta decays can be extremely rare. This can be caused by either a large di fference between the spins of the initial and final state (the so-called “forbidden” beta decays) or an extremely small Q value (decay energy) of the decay. All these cases are discussed in this article, and particular emphasis is given to the neutrinoless double electron capture on the double beta side of decays.


Introduction
Weak interactions, as the name indicates, are indeed weak if we measure weakness in terms of time scales of processes they generate, say in atomic nuclei where they prompt disintegration phenomena in the time scale of seconds. However, notable exceptions to this state of affairs are caused by (a) extremely small decay energies (Q-values), (b) initial and final nuclear states with large difference in angular momentum and (c) weak-interaction processes of higher order (for reviews see e.g. [1][2][3]). These extreme conditions of decay lead to processes that involve time scales far beyond the seconds scale, to scales much longer than the age of the Universe. Typically such processes have half-lives of the order of 10 20 years (practically the age of the Universe squared!) and thus can be called 'ultra slow'. The related transitions need special experimental facilities and dedicated experimental techniques in order to be detected. The detection sites of such rare processes need to be protected against cosmic rays, i.e. the flux of particles from outer space. This is why the dedicated experiments go underground, in deep mine shafts or under huge amounts of massive mountain rock. Hence the related scientific effort is appropriately called "underground physics", or to contrast it with the research done in particle accelerator facilities, "non-accelerator physics".
Examples of the above-mentioned classes of rare decays are the "ultra-low" Q-value decays examined lately both in Penning-trap measurements and some some underground facilities (see Sect. 2), decays that involve large differences in angular momentum, like the β − decay of 96 Zr that competes with the double beta decay of the same nucleus (see Sect. 3) and neutrinoless double electron-capture decays presented in Sect. 4. currently being investigated in experiments that focus on the nuclear double beta decay [1] of various nuclear systems [4] or to the β − decays of tritium (the KATRIN experiment [5]) and 187 Re (the MARE experiment [6]). The latter two experiments are direct searches of the electron-neutrino mass via the slight distortion of the electron end-point spectrum. To detect this distortion, as small as possible Q value of the decay is desirable.  There are some other very interesting cases of potential neutrino-mass studies. One of them is the β − decay of the 9/2 + ground state of 115 In to the first excited state of 115 Sn with spin-parity 3/2 + (see figure 1). This decay transition is second-forbidden unique so that the line shape of the decay is simple (for reviews of the formalism of β − , β + and electron-capture (EC) decays see [7][8][9][10]). Interesting about this decay transition is that it has a world-record small Q value of 0.155(24) keV [11] so that it can be called "ultra-low" (i.e. well below 1 keV). Measurement of such a small Q value is based on the Penning trap techniques [11,12]. The corresponding decay branch was measured by the HADES underground facility in Belgium to have a partial half-life of 4.1(6) × 10 20 yr [12] and it has been speculated that the decay branch could be used as a neutrino-mass detector [13]. Even more intriguing is that such an ultra-low Q value seems to enhance the interference of atomic effects in the nuclear decay, as discussed in [14,15].
An other potential case for such an ultra-low Q-value decay is shown in figure 2. There the 7/2 + ground state of 135 Cs β − decays to the first and second excited state of 135 Ba with spin-parities 1/2 + and 11/2 − which are second-forbidden and first-forbidden unique decays, respectively [16]. The halflife corresponding to the second-forbidden non-unique decay transition to the ground state of 135 Ba can be computed by allowing the decay energy Q and the axial-vector coupling constant g A vary in value. This produces the shaded area of computed half-life values in figure 3. There are also two half-life and Q-value measurements [17,18] that are in strong tension with each other, as shown in figure 3. Depending on which one of the measurements is correct, either the decay to the first or to the second excited state can produce a transition with an ultra-low Q value [16]. So, accurate Penning trap measurement of the mass difference between 135 Cs and 135 Ba is called for.    [17,18] are also given.
Yet another potential ultra-low Q-value decay is presented in figure 4. This is the fourth-forbidden non-unique β − decay of the 1/2 + ground state of 115 Cd to the second excited 9/2 + state in 115 In. Again, the energy difference of the ground states of the mother and daughter nuclei is not known accurately enough to tell whether this transition is possible or not, but if the Q value is positive, it is most likely ultra-low. A thorough treatment of this transition, and also the other transitions of figure 4 has been performed in [19].
In table 1 are listed some other potential ultra-low Q-value transitions in nuclei with mass numbers A ≤ 161 [20]. All the initial states of the first column of the table are ground states of the respective nuclei. The second column indicates the final nucleus and its final state with its excitation energy given in the third column with the experimental error included in the parenthesis. The fourth column of the table indicates the beta-decay type and the last column lists the experimental decay Q value derived from the experimental mass difference between the mother and daughter nuclei and the measured excitation energy of the final state. In the table the decay type is either β − or electron capture (EC).  In table 1 there is only one initial nucleus that is odd-odd ( 146 Pm), all the others are odd-mass nuclei. Particularly interesting cases are the ones with allowed Gamow-Teller β − ( 131 I) or allowed Gamow-Teller ( 159 Dy) and Gamow-Teller/Fermi ( 161 Ho) electron-capture decays. Also the firstforbidden unique β − decay ( 155 Eu) and the second-forbidden unique electron-capture decays ( 111 In) are of high interest. The rest are non-unique decays and depend on several nuclear matrix elements without a unique line shape. The allowed and first-forbidden unique β − transitions ( 131 I and 155 Eu) are optimal for the experimental identification of the neutrino-mass distortions at the end point of the beta spectrum. These can help in determining the absolute neutrino mass in beta-decay experiments. For the investigation of the electron-cloud-nucleus interference effects the allowed, first-forbidden unique and second-forbidden unique β − and electron-capture decays are very useful. In particular, for the EC decays with ultra-low Q values the electron-screening effects become of paramount importance.
From the nuclear-structure point of view the first three nuclei of table 1 are (nearly) spherical or weakly deformed whereas the heavier nuclei are well deformed and require a deformed mean field as the starting point in nuclear-structure calculations.
Finally, It should be stated that verification of the spin-parities of some of the final states in table 1 has to be done by nuclear-spectroscopy methods. At the same time it is imperative to perform highprecision Penning-trap measurements to improve the accuracy of the mass differences of the involved nuclei.  Let us now discuss an example pertaining to the points (b) and (c) of the introduction. In figure 5 the mother nucleus 96 Zr decays to states in 96 Nb via ultra-slow beta transitions, retarded by the large differences in angular momentum between the initial state (spin 0) and the final states (spins 4-6), belonging to the category (b) of the classification of ultra-slow decay transitions. In addition to the ultra-slow beta transitions there is a very interesting ultra-slow direct transition from 96 Zr to the ground state of 96 Mo. In this case the decay jumps past the nucleus 96 Nb and goes directly to the ground state of 96 Mo and thus it falls into the category (c) in our classification of ultra-slow processes. These higher-order transitions form a class of transitions called generically the nuclear double beta decay [1].

Competition of beta and double beta decays in 96 Zr
The half-lives of figure 5 have been calculated [21] by using the experimental Q values listed in the figure by the use of the proton-neutron QRPA (quasiparticle random-phase approximation). A similar situation as here occurs also for the beta and double beta decays of 48 Ca. These decays have been discussed in [22] by the use of the nuclear shell model. In the present case the total beta-decay halflife T 1/2 (β − ) = 2.6 × 10 20 y is determined by the fourth-forbidden unique β − transition to the 5 + state in 96 Nb. The other transitions, the fourth-forbidden non-unique transition to the 4 + state and the sixthforbidden non-unique transition to the 6 + state, do not play a role in the total half-life due to their very long partial half-lives. In fact, it is interesting to note that the computed β − -decay half-life is an order of magnitude longer than the experimental double beta half-life T 1/2 (β − β − ) = (2.3 ± 0.2) × 10 19 y [23]. This bears relevance to the speculated time variation of the weak-interaction constant [24] noticed by comparing the modern laboratory-measured half-lives of double-beta nuclei with the geochemically obtained half-lives of the same nuclei, obtained by applying chemisty on very old terrestial ores [24]. The present result suggests that no severe interference of the β − decay is expected in the β − β − -decay half-life of the old 96 Zr ores.

Resonant neutrinoless double electron capture
Neutrinoless double beta decay involves Majorana neutrinos as propagator between the two decay vertices and thus there are no (anti)neutrinos in the final state [1]. On the double β − side the final state involves two emitted electrons and on the double β + side there are two positrons in the final state [1]. Also the neutrinoless β + EC decay occurs via the one-positron phase space. In the case of the neutrinoless double electron capture, 0νECEC, there are no leptons available in the final state to carry away the decay energy. In this case one has to engage some additional mechanism to rid the initial atom of the excess energy of decay. There are two proposed mechanisms to cope with this situation: the radiative 0νECEC decay [25] and the resonant decay, R0νECEC [26]. The resonance conditionclose degeneracy of the initial and final (excited) atomic states -can enhance the decay rate by a factor as large as 10 6 . The R0νECEC process is of the form where the capture of two atomic electrons leaves the final atom in an excited state, in most cases having the final nucleus in an excited state. The excited state of the nucleus decays by one or more gamma-rays and the atomic vacancies are filled by outer electrons with emission of X-rays. Fulfillment of the resonance condition depends on the so-called degeneracy parameter Q − E, where E is the excitation energy of the final atomic state and Q is the difference between the initial and final atomic masses. Also the nuclear structure is heavily involved through the appropriate nuclear matrix elements [27]. Possible candidates for such resonant decays are many and a representative list is displayed in Table 2 In the table we also list the estimated half-lives for the cases for which such exist. The references of the last column indicates the origin of the Q-value measurement and the possible calculations of the related NME. The Q-value measurements have been performed by the use of modern Penning-trap techniques. In the table the quantity C ECEC is listed and it relates to the R0νECEC half-life as given where the effective neutrino mass should be given in units of eV. In all the listed cases where C ECEC has been computed the decay rates are suppressed by the rather sizable magnitude of the degeneracy parameter. Decays to 0 + states are favored over the decays to 2 + or 1 − , 2 − , 3 − etc. states due to the involved nuclear wave functions and/or higher-order transitions. Also captures from atomic orbitals with orbital angular momentum l > 0 are suppressed [28]. There are some favorable values of degeneracy parameters listed in Table 2, like 106 Cd → 106 Pd(2, 3) − and 156 Dy → 156 Gd(0 + , 1 − , 2 + ) but the associated nuclear matrix elements are not yet evaluated. At the moment the most favorable case with a half-life estimate is the case 152 Gd → 152 Sm(0 + gs ) which describes a decay transition to the ground state.