Current state of the investigation of superallowed Fermiβ decays

Nuclear Structure: Superallowed Fermiβ transitions; derivation of the effective vector (G′v) and induced scalar (fs) coupling constants from experimental data.At the present time, the crucial point in a systematic study of superallowed 0+-0+β transitions is the evaluation of the isospin impurity correctionδc. In the literature,δc is decomposed into two parts,δc1 and δc2. Several estimates ofδc1 have been published, while only one is available forδc2. We analyze the compatibility of the different estimates ofδc1 with the most recent surveys of experimental data. The simplest evaluation ofδc1 reported some years ago by Damgaard is found to yield the most satisfactory ℱt values; these provide reliable values of the effective vector coupling constantGv [e.g.,Gv=(1.41242+0.00023)×10−49 ergcm3]. These values are in excellent agreement with a recent valueGv=(1.41248+0.00044)×10−49 erg cm3 obtained by Wilkinson on the basis of a phenomenologic approach toδc. Conversely, the most recent and detailed parentageexpansion approaches toδc1 lead toℱt values which increase with Z, showing pronounced slopes. This fact might be due to a relative overestimation ofδc1 for the lighter nuclei. Using the ℱt values calculated withδc1 as reported by Damgaard, we evaluate the coupling constant for the induced scalar interaction following a procedure described in a previous paper. The mean of such values isfs/fv=(−0.17±0.80)×10−3. In addition, we develop an alternative way of determining a limit forfs/fv using the phenomenological approach toδc suggested by Wilkinson. This new procedure yieldsfs/fv=(−0.16±0.87)×10−3, a result which is in excellent agreement with that obtained using the former method; both values are consistent with a value of zero, supporting the conserved vector current theory. The better accuracy of the experimental data makes it possible to reduce by a factor of two the limit established in a previous work.


Introduction
The main effort in a standard study of superallowed 0+~ + /3 transitions is focused on properties which can lead to a test of the conserved vector current (CVC) theory and the Cabbibo universality hypothesis. In relation to 0+-0 § Fermi decays, the CVC theory predicts that [1][2][3][4][5][6]: (a) the vector coupling constant of the weak interaction, G v (=fv G~, where * Member of the Scientific Research Career of the Consejo Nacional de Investigaciones Cientificas y T6cnicas of Argentina fv is the vector coupling constant and G B is the weak coupling constant for /~ decay), should not be renormalized, and thus should be constant from nucleus to nucleus; and (b) the coupling constant of the induced scalar (IS) interaction, fs, should be zero. Since the three independent comprehensive papers [24] on 0+-0 + /3 transitions were published in late 1975, new experimental improvements have enabled the experimental errors of the relevant measurable magnitudes to be significantly reduced. In other words, the half-lives t o and the maximum energies of the positron spectra, Wo, can now be measured with increased precision, making it possible to calculate more precisely the quantities involved in the evaluation of thef' t values, namely, Finally, substituting the expression (6) in (3), we obtain K f' t-,2 v 0 2 2' (8) Gv (Jgoo0) [l+(fs/fv)Ais] f'=f C~(W) [ and t=~ 1+ . ( Here, f is the integrated statistical rate function defined as in [41, Cp(W) is the shape factor averaged over the energies W of the positron spectrum, fiR(W) is the "outer" model-independent radiative correction which includes corrections of order c~, Z c~ 2, and Z 2 e3 (e being the fine structure constant, and Z the nuclear charge of the daughter nucleus), BR stands for the branching of the superallowed /3 transition, and e/fi+ indicates the electron-capture-to-positrondecay ratio.
Let us now summarize the relevant formulas for the superallowed Fermi /3 transitions. The f't values obey the equation with G~ = Gb(1 +A~) 1/2, where K is a combination of physical constants (see, e.g., [41) which reduces to K = 1.230618 x 10-94ergacm6s in cgs units, VF~ is the Fermi form-factor coefficient (FFC), and A R is the "inner" model-dependent radiative correction. The FFCs are given in terms of the nuclear matrix elements (NMEs) in Table 6 of [7]. Using that formula for the case of fl+ transitions, we arrive at vo vo where V~~ is the Fermi NME, U(r) is the potential of the nuclear charge distribution, and R is the nuclear radius. It has been shown [5,6] The evaluation of the Fermi NME remains as the most outstanding theoretical problem. The superallowed Fermi process involves component states of the T=I isospin multiplet. Assuming that both initial and final states are pure isospin states, vdg~ is equal to ]/2. However such an assumption is no longer valid when one includes Coulomb or chargedependent short-range forces in the nuclear Hamiltonian. Then a small but important breaking of the SU(2) nuclear symmetry occurs. Therefore in the literature the squared Fermi NME is currently written as where ~c accounts for the mismatch of the nuclear states involved. Introducing this expression in (8), we obtain K f' t=ZG'v2(1 -6c) [1 +(fs/fv) AJ 2" (11) From the beginning of the systematic studies of 0 +-0 + superallowed fl transitions, several authors have evaluated 6c, obtaining results which were not completely consistent with one another. Nevertheless such uncertainties in 3o were negligible compared with the relative errors of the experimental f' t values.
However, nowadays the f't values can be determined with enough precision to make accurate knowledge of 6~ crucial. Several efforts have therefore recently been devoted to evaluating 6~. In Sect. 2 we discuss the available estimates of c5. In Sect. 3 we revise the methods in the light of the most up-to-date surveys of experimental data to obtain (a) a value for G~, and (b) a limit for the IS coupling constant fs.

The Isospin Impurity Correction 6 c
The quantity 6 c is nuclear-model dependent and therefore somewhat uncertain. A summary of calculations performed until 1971 can be found in the review articles by Blin-Stoyle [1] and Behrens [8].
Here we concentrate mainly on subsequent work. In the recent literature one can find two different ways of tackling the problem of the evaluation of the isospin impurity correction. One is a microscopic approach, 0.8 where the charge-dependent effects are taken into account be direct computation. The second procedure is a phenomenological one, based on a general ~ 0.6 theorem on renormalization of coupling constants. ,,~ We outline both these estimates in this section. 0.4

Microscopic Calculations of 6~
For the numerical computation, 6~ is broken into two parts: one, 6cl, is due to small differences in the single-particle neutron and proton radial wave functions which cause the radial overlaP integral of the parent and daughter nucleus to be smaller than unity; the other, 6~ arises from charge-dependent configuration mixing with other 0 + states. Although these two aspects cannot be separated completely, such a division has been made in all the available calculations.
Several evaluations of 6ol have been reported. They can be grouped into two approaches:  Damgaard [9]; the difference increases with Z, and reaches 16~ for 5~Co. However, the value 6~=0.47~o for 54Co reported in [12] is still 24~ larger than the value 6~1=0.38~o obtained by Lane and Mekjian [11].
Approach A: This consists in computing the effect of the one-body Coulomb force in admixing, to the states in question, other states of the same or different isospin. Four estimates of this kind are available, published by Damgaard [9], Fayans [10], Lane and Mekjian [11], and Towner, Hardy, and Harvey [12]. We shall not describe those calculations here; for more details, the reader is derived to the original papers. However we shall point out that in all four calculations, the radial integrals were carried out using harmonic-oscillator wave functions. Instead of tabulating the results of [9][10][11][12], we prefer to plot the values of 6cl as a function of Z: see Fig. la. This figure shows a definite trend of 6ol with Z, which is similar for all the calculations. Starting from few hundredths of a percent for ~O, 6cl increases regularly up to few tenths of percent (0.34 ~-0.56 %) for 54Co. Since 6ol is due mainly to the one-body Coulomb potential, it is reasonable to expect on general grounds that it should increase with Z, as is observed in Fig. la. It can be seen that for all Z, the estimates by Damgaard [9] are larger than the others. It is important to point out that the authors of E12] have followed the procedure of Damgaard [9] introducing some modifications: (a) using the radius R of the uniformly charged sphere and the harmonic-oscillator parameter h co appropriate for each nucleus, instead of adopting the standard values R = 1.2A 1/a fm and hco =41 A-1/3 MeV; and (b) including higher-order perturbation theory. The results obtained in both [9] and [12] agree for 140. For larger Z, the estimates by Towner et al. [12] are smaller Approach B. In this case the overlap is calculated using wave functions generated in a standard Woods-Saxon potential including the Coulomb term for protons. Such single-particle wave functions account directly for the different binding energies of the proton and neutron involved by the # process. Configuration mixing has been taken into account by summing over the full parentage spectra in the A-1 nuclei. There are two calculations along these lines, those of Towner, Hardy, and Harvey [12], and those of Wilkinson [13]. The results are displayed in Fig. 1 b. Contrary to the feature found in Fig. 1 a, no general trend can be traced in this case because of a rather random distribution of 6cl versus Z.
For 6~ on the other hand, at present, the only values available are those obtained by Towner and Hardy [14] from calculations with two-body Coulomb forces and wave functions generated by the Oak Ridge-Rochester shell-model code. The numerical results of 6~ reflect the shell effects, as can be expected from a two-body operator for such light nuclei. Table 1 shows all the estimates of 6o=6cl +6~ arising from the combination of the various evaluations of 6ol mentioned above with the unique set of 6c2 reported in [14].

Phenomenological Approach to 6~
The phenomenological approach to 6~ was suggested by Wilkinson [2,15] and subsequently discussed by Wilkinson and coauthors in [16,17]. It is based on a  (13) where k should be obtained from a fitting procedure.
where (f't)z 0 might be considered appropriate to the free nucleon and therefore used to derive G v.
Equation (13) requires that the experimental f't values follow an almost parabolic law as a function of Z. Such a behavior has been observed only after the precise measurements of W 0 carried out using the time-of-flight system at the Munich tandem [20]. For instance, this behavior can be seen looking at Fig. 1 of [-17], where the data are fitted to expression (13). It is interesting to comment on some aspects of such an adjustment. Since the authors of [17] have not published the results for all the parameters, we repeated the fit with the aid of the package of subroutines MINUITS from CERN [21]. A plus-and-minus-onestandard-deviation fit yielded the results quoted in Table2 with Z2=22.3. The value k=(1.63_+0.23) x 10 -5 leads to 6o listed in Table 1. A glance at this table indicates that the smooth trend of ~o with Z obtained from BSAG follows approximately the values of 6c=6c1+6c2 with ~cl taken from [9], of course without the fluctuations due to shell effects taken account of by 6~2.

Analysis and Discussion
In this section we analyze the most up-to-date surveys of the eight best-measured cases -these are the and Hardy [22], and Wilkinson, Gallmann, and A1burger [17]. It should be pointed out that the authors of [17,20,22] have adopted different criteria for selecting values of t o and W 0 corresponding to each nuclei. The technique used by Wilkinson et al. [17] was to accept the most recent precise measurement for each quantity, averaging-in only those earlier measurements with which it is statistically consistent.
Vonach et al. [20] applied a procedure that consisted in taking all measurements whose quoted errors were not larger than ten times the error of the most accurate measurement, and rejecting the data which deviated from the weighted average by more than three times the quoted error. The first step of the method used by Towner and Hardy [22] is the same as that employed by Vonach et al.; however, instead of disregarding incompatible data, they inflated the uncertainties of the average value in a standard way (described in [3]) to cover incompatibility.  [20] are rather closer to Towner and Hardy's data [22].
For our analysis we took the surveys of f't values reported in [17,22] just as they are in the original papers. On the other hand, we used the survey quoted in the note added in proof of [20] including one correction. Since [20] was published, two new precise measurements of the half-life of 46V have been reported -by Squier et al. [23], and by Alburger and Wilkinson [24]. These new data differ significantly from the half-life adopted in [20]. In particular, A1burger and Wilkinson [24]

3.i. Search for a Reliable G' v
Let us define the quantity ~t, which aside from the corrections contained in f't, also includes those due to isospin impurities: Assuming the validity of the CVC theory, the vector second-class current (SCC) does not exist. Under such an assumption, namely setting fs to be zero, if 6r is evaluated correctly, we should expect ~t constant from nucleus to nucleus. In our previous investigations [5,6] we tested the constancy of the ~t values over the known /3 decays performing a leastsquares analysis. Searching for a possible small dependence of ~t on the charge of the daughter nucleus, we fitted the Yt values to the formula No evidence for any lingering Z dependence in the experimental data was found in [5,6]. It was thus reasonable [2][3][4][5][6] to evaluate a weighted average of ~t values and subsequently derive a reliable value for the codpling constant G v. We should point out that the test of Z independence was not applied in the work of Wilkinson et al. [17] or of Towner and Hardy [22]. We calculated several sets of ~ t values using the f' t data quoted in the surveys mentioned above [17,20,22], and combined them with each of the microscopic estimates of 6r listed in Table 1. We evaluated the weighted average ~t for each set and fitted the ~t values to (15). Both the internal and external errors of t were computed according to the formulas quoted in the general instructions of the Nuclear Data Sheets. The internal error reflects the accuracy of the individual data, while the external error accounts also for the dispersion. The ideal situation occurs when both internal and external errors are equal; if not, the larger should be taken. For reference, we calculated the Z2/v of the straight mean of f't, setting 6r = 0. All the results are listed in Table 3.
A glance at Table 3 indicates that we can trace some general conclusions which are independent of a specific survey of f' t values. First of all we can stress that the Z2/v of the averages ~-t corresponding to o~t values evaluated adopting Approach A for 6r are smaller than those •2/v corresponding to calculations using Approach B [~-t(B)]. This behavior has been already found by Wilkinson et al. [17] in the analysis of their own data. The estimate of [9] has not been mentioned at all by Wilkinson et al. [17]. The results obtained adopting Approach B for 6r present other unpleasant features. The external errors of ~t(B) are much larger than the internal ones, and the values of 7~2/v are in general scarcely improved from those obtained with 6r =0; indeed, in th~ case of the survey [20], they are even worse. We analyzed the survey included in the note added in proof; however, instead of using the value 424.0_+ 0.5 ms for the half-life of *6V, we adopted the weighted average 422 34_+0.21 ms calculated with more recent data (for more details see text) Furthermore, the values of ZZ/v are reduced by almost a factor two when'the ~t(B) values are fitted to (15). Such an adjustment indicates pronounced positive slopes and the results of [~t(B)]z_0 are significantly smaller than the corresponding averages ~-t(B). For instance, consider the set built using the survey by Wi!kinson et al. [17] combined with their own ~cl [13]. In this case we have @t(B)=3,080. is reached when the estimates by Damgaard [9] are used. Before focusing our discussion entirely on these results, let us note that when the survey by Vonach et al. [20] is used in conjunction with the estimates by Towner et al. [12], an excellent value of Z2/v = 1.0 is also obtained. In these cases the goodness of the adjust-ment is not improved when the data are fitted to (15). The averages ~---~(A) from the three surveys [17,20,22] are not merely consistent with one other within the quoted errors, but are practically the same. It can be also mentioned that the surveys [17,22] give Y t(A) with external errors slightly larger than the internal ones, showing that the dispersion is larger than expected from errors of individual ~t(A) values. Since the ~-t(A) computed with ~1 from [9] satisfy the minimal conditions to be suitable to provide a reliable effective vector coupling constant, we evaluated G~, getting the values listed in Table 4. For the sake of completeness we tested whether the surveys by Vonach et al. [20] and by Towner and Hardy [22] could be fitted well to the quasi-parabolic formula (13). The numerical computation using the MINUITS code [21] yielded the results quoted in Table 2. The Z2/v of these fits are much better than those obtained with 6=0, and the goodness of the adjustments indicates that the data can be represented satisfactorily well by (13). It was already suggested by Wilkinson et al. [17] that the goodness of the fits can be further improved if the f't values are corrected by the shell effects due to the two-body charge-dependent potential. This can be done by correcting the individual f't values by the departure  Table 3 and text) b Here 6cl was taken from [9] and 6c2, from [14] ~ These results were obtained fitting the f't values to (19) of the individual 6c2 values from 3~ In this way we can eliminate from the experimental data some effects which cannot be accounted for in the BSAG approach. Following this procedure, instead of using (13), we should use Just as an example, the results of such a fit for the survey of [17] are also included in Table 2, where it can be seen that )~2/v = 3.7 is improved to 2.6.
If all is well, the results obtained for ~ t must agree with (f't)z_o. Actually, this is the case, since the reliable ~t values listed in Table 4 are practically the same as the value (f't)z_o=3,084.1++l.9s recommended by Wilkinson et al. [17], and therefore the corresponding values of G v are also in excellent agreement.

Determination of the Upper Limit for fs
To determine the upper limit for fs/fv, we followed the method described in [6] rather than the previous one reported in [5]. The technique introduced in [6] requires experimental ~-t values to be fitted to The advantages of this fit over the straight average of departures from ~t proposed in [5] have been discussed elsewhere [6]. We calculated fs/fv for the sets which provide reliable averages Yt listed in Table 4.
In the least-squares fit, we used the values of Ais listed in Table 1 of the complete version of [6]. The results are included in Table 4. The quality of the adjustments are similar to those obtained with the fit to (15). The results of fs/fv for the three sets are consistent with one other and agree with the value of zero, thus supporting the CVC theory. The actual limit is provided by the quoted errors of fs/fv. The errors of these three values are practically equal.
Therefore we can set as a limit for the IS coupling constant the straight mean value fs/fv = (-0.17 +_ 0.80) x 10-3. For the sake of comparison we included in Table 4 the results of the previous work [6]. It can be seen that the errors obtained in the present work are reduced by a factor of two with respect to that reported before.
To have an alternative way to determine the limit for fs/Jv we developed a new procedure using the BSAG approach to 6o. Starting from (11) and adopting the phenomenological expression (12) Since Ais is about 3, and fs/fv is expected to be of the To derive the parameters (f't)z_o , k, and fs/fv, we fitted the experimental f't values to the expression (19). The numerical computation was performed with the aid of the MINUITS code [21]. As before a plusand-minus-one-standard-deviation fit was employed.
The X2=22.2 at the minimum indicates that the Z2(=22.3) of the fit to (13) could not be improved introducing the extra parameter fs/Jv. The result fs/fv =(-0.16_+0.87) x 10 -3 fixes a similar limit to those obtained from the former procedure. The fact that the limit for fs/fv is independent of the method of managing the experimental data is an attractive and stimulating result.
The author thanks Mr. J. Pouchou for running the MINUITScode. He is also grateful to Dr. M.A.J. Mariscotti for the kind hospitality afforded to him at the Departamento de Fisica of the Comisidn Nacional de Energia Atdmica, Buenos Aires, Argentina, where part of this work was done.