Weak Magnetism in Two Neutrino Double Beta Decay

We have extended the formalism for the two-neutrino double beta decay by including the weak-magnetism term, as well as other second-forbidden corrections. The weak magnetism diminishes the calculated half-lives in $\sim 10%$, independently of the nuclear structure. Numerical computations were performed within the pn-QRPA, for $^{76}Ge$, $^{82}Se$, $^{100} Mo$, $^{128}Te$ and $^{130}Te$ nuclei. No one of the second-forbidden corrections modifies significantly the spectrum shapes. The total reduction in the calculated half lives varies from 6% up to 32%, and strongly depend on the nuclear interaction in the particle-particle $S=1,T=0$ channel. We conclude that the higher order effects in the weak Hamiltonian would hardly be observed in the two-neutrino double beta experiments.

The sensitivity of the double-beta (ββ) decay experiments is steadily and constantly increasing. To become aware of this it suffices to remember that, between the pioneer laboratory measurement on 82 Se [1] and the most recent on 76 Ge [2], the statistics has been improved by a factor of ∼ 5000.
The quantity that is used to discern experimentally between the ordinary Standard Model (SM) two-neutrino decays (ββ 2ν ) and the neutrinoless ββ events not included in the SM, both without (ββ 0ν ) and with Majoron emissions (ββ M ), is the electron energy spectrum dΓ/dǫ of the decay rate Γ. It is usually given as a function of the sum (ǫ) of the energies (ǫ 1 and ǫ 2 ) of the two emitted electrons. The ββ 2ν and ββ M decays exhibit continuous spectra in the interval 2 ≤ ǫ ≤ Q, while the ββ 0ν spectrum is just a line at the released energy Q = E I − E F .
In the evaluation of the ββ 2ν decay rate, the allowed (A) approximation is usually assumed to be valid, i.e., the Fermi (F) and Gamow-Teller (GT) operators are considered at the same level of approximation as in single-allowed β transitions. Besides, as the F operator is strongly suppressed in the ββ 2ν decay by the isospin symmetry, only the dominant contribution of the axial-vector current is considered in practice. Then, the ββ 2ν matrix element, between the initial state |0 I in the (N, Z) nucleus to the final state |0 F in the (N −2, Z +2) nucleus (with energies E I and E F and spins and parities J π = 0 + ) reads: Here the summation goes over the virtual states with spin and parity J π = 1 + . Except for the work of Williams and Haxton [3], all higher order terms (HOT) in the weak Hamiltonian have been almost totally ignored in the past, simply because they are expected to be small. Yet, the question is not so simple. Firstly, from the comparison of the recent experimental results for the ββ half lives in 76 Ge [2]: T 2ν ∼ = 1.77 · 10 21 y, T M > 1.67 · 10 22 y and T 0ν > 1.2 · 10 25 y, it can be stated that presently are being observed effects of the order of 10 −4 at ǫ ∼ Q and of the order of 10 −1 at ǫ ∼ Q/2. Secondly, there are also several ongoing and planned experiments that are supposed to allow for measurements of still smaller effects. Thus, the HOT could, in principle, become relevant.
In recent years we have examined the effects of the first-forbidden operators in the ββ 2ν decay [4,5], both the non-unique (FFNU) and unique (FFU), which contribute via the virtual states J π = 0 − , 1 − and J π = 2 − , respectively. From these studies we have learned that: a) the FFNU transitions might increase the T 2ν up to ∼ 30%, yet they do not modify the A shape of the two-electron spectrum [4], b) the FFU transitions do alter the ββ 2ν spectrum shape but only at the level of 10 −6 and mainly at low two-electron energy, where most backgrounds tend to dominate [5]. Therefore, the effects of the firstforbidden transitions could hardly screen the detection of exotic ββ 0ν and ββ M decays, which are the candidates for testing the physics beyond the SM.
In the present work we want to complete the question of the competition between the standard and exotic ββ-decays [5], by scrutinizing the effects of the nuclear matrix elements, which have been intensively studied in connection with deviations of the simplebeta spectra from the allowed shape [6,7,8,9]. (In particular, it should be reminded that the detection of the weak-magnetism (WM) term in the spectrum of β ± -decays of 12 N and 12 B to 12 C has provided a rather striking test of the CVC theory.) It is customary to denominate these HOT as second-forbidden corrections (SFC), although the most significant among them comes from the WM term, which obeys the same selection rules as the GT operator. One also should keep in mind that the WM plays a very important role in both the ββ 0ν and the ββ M decays, through the so called recoil term [10,11,12].
Proceeding in the same way as in our previous works [4,5,12], we express the ββ 2ν decay rate as: where the symbol Σ represents both the summation on lepton spins, and the integration on neutrino momenta and electron directions. The transition amplitude reads [4,5,12]: where the operator P exchanges the lepton quantum numbers e i ≡ (ǫ i , p i , s e i ) and ν i ≡ (ω i , q i , s ν i ). All other notations have the usual meaning [5].
In the non-relativistic weak Hamiltonian we only retain the terms that are necessary for the evaluation of the SFC, and obtain Here G = (2.996 ± 0.002)×10 −12 is the Fermi coupling constant (in natural units), ξ = αZ/2R = 1.18ZA −1/3 is the Coulomb factor, and are the nuclear operators, with g V = 1 and f W = 4.7. The leptonic part is: Keeping only the interference terms between σ with X, Y and Z, and the linear contributions in the lepton energies, from (3) and (4), we obtain: 1 where and It might be interesting to note that, while the FFNU transitions interfere with the GT operator at the level of the transition rates, the SFC do it already at the level of the transition amplitude. At the same order of approximation the differential transition rate reads: where Finally we derive the expressions for the spectrum shape where For the half-life we get with the kinematical factors As seen from (12) the spectrum shape mainly depends on the factors F i (ǫ). They are displayed for 76 Ge in the upper panel of figure 1, as a function of the energy ǫ. F 0 (ǫ) and F 3 (ǫ) exhibit the same energy dependence, while F 1 (ǫ) shifts the A spectrum slightly to the right and F 2 (ǫ) slightly to the left.
The spectrum shapes without and with the SFC are compared in the lower panel of figure 1. It can be observed that they are quite similar. Analogous behavior was found for other experimentally interesting nuclei, such as 82 Se, 100 Mo, 128 T e, and 130 T e. From (5), (9) and (10) one sees that the main effect of the WM consists in renormalizing the GT matrix element (1) as: i.e., by a factor of ∼ 1.05 for medium heavy nuclei, independently of the nuclear model employed.
The matrix elements M (i) 2ν were evaluated within the pn-QRPA model, following the procedure adopted in our previous works [4,12,13]. We display them in table 1, together with the corresponding kinematical factors G i . Besides the WM term, the velocity dependent matrix elements r × p and 2i(σ · p)r are also important, and particularly in the case of 100 Mo. 2 The moment M (2) 2ν is always relatively small. 2ν (in natural units), evaluated within the pn-QRPA formalism with an effective axial charge g A = 1. As it is well known, within the QRPA the GT moment M (0) 2ν strongly depends on the particle-particle coupling constant in the S = 1, T = 0 channel, denoted by t in ref. [13]. This dependence is particularly pronounced in the physical region for t, where M 2ν also strongly depend on t, but in a slightly different way than M (0) 2ν . Thus the decay rates rely on two or three rapidly varying functions of t, which makes the calculated half-lives (T A+SF C 2ν ) to be even more sensitive on the value of t than within the A approximation (T A 2ν ). The numerical results are shown in table 2. Contrarily to what happens in the case of the FFNU transitions, the SFC decrease the half-lives. The reduction ranges from 6% in 128 T e up to ∼ 32% in 100 Mo.  This work completes our previous inquiries [4,5] on the significance of the higher order corrections in the two-neutrino double beta decay. It can be concluded that: 1) Both the first-forbidden transitions through the J π = 0 − , 1 − virtual states, and the second order corrections to the Gamow-Teller states, affect the transition rates in a significant way.
2) The theoretical uncertainties within the QRPA, in the evaluation of the half-lives, are of the same order of magnitude (or even larger) than the experimental errors.
3) The effect on the energy spectra of the higher order terms in the weak Hamiltonian is too small to shadow the possible exotic neutrinoless events in contemporary experiments.