values of the mirror transitions
A complete survey is presented of all half-life and branching-ratio measurements related to the isospin mirror transitions ranging from He to Mo. No measurements are ignored, although some are rejected for cause. Using the decay energies obtained in the 2003 Mass Evaluation experimental values are then determined for the transitions up to V. For the first time also all associated theoretical corrections needed to convert these results into ”corrected” values, similar to the superallowed pure Fermi transitions, were calculated. Precisions of the resulting values are in most cases between 0.1% and 0.4%. These values can now be used to extract precise weak interaction information from past and ongoing correlation measurements in the beta decay of the mirror transitions.
pacs:21.10.Tg, 23.40.Bw, 24.80.+y, 27.20.+n, 27.30.+t, 27.40.+z, 27.50.+e
In the past, several experiments in nuclear -decay searching for non-Standard Model contributions to the weak interaction were performed with mirror nuclei calaprice75 ; schreiber83 ; garnett88 ; severijns89 ; masson90 ; converse93 ; melconian07 ; scielzo03 ; scielzo04 ; iacob06 ; vetter08 . Whereas originally the accuracy of these measurements was still rather limited (at best 2 %), first precision results were recently obtained with Na scielzo03 ; scielzo04 ; iacob06 ; vetter08 while several other experiments are ongoing (with Ar beck03 and K behr05 ) or in preparation (Ne berg03 ; broussard05 and Na sohani06 ). In order to extract reliable information from such measurements, precise knowledge of the value of the mirror transition under investigation is required. We have therefore performed a thorough survey of all data in the literature related to the values of the mirror transitions and calculated the values for the cases up to V, thereby updating the previous work of Raman et al. raman78 .
On the experimental side, half-lives, , branching ratios, , and values are required for the determination of values. As for the first two, the literature was searched and data were evaluated, leading to adopted values for each isotope. The values were taken from the 2003 Mass Evaluation audi03 . Since for most nuclei up to 40 the experimental data turned out to be sufficiently precise to yield values with a precision at the few 10 level we decided to perform, for the first time for these mirror transitions, a full analysis of all radiative and nuclear structure corrections leading to the corrected values. Up to now such complete evaluation of the value was only carried out for the superallowed pure Fermi transitions hardy05 . For all mirror nuclei up to V values with a precision ranging from 0.10 % to about 2.3 % were obtained. For the heavier nuclei experimental data are either not available or not sufficiently precise. Nevertheless, all experimental data reported in the literature are listed here.
In a first section the equation for the value of an allowed transition, including all corrections, is derived. From this the equation for the value for the mirror transitions is then deduced. The next section explains the selection and treatment of the experimental data, while the last section deals with the values themselves. At the end of this paper tables are given that list all experimental data and adopted values leading to the values of the mirror transitions, the values for the different correction factors applied for the nuclei up to V and, finally, the derived results for the values.
The decay rate for an allowed -decay from an unpolarized nucleus is written jackson57
where is the total electron energy in electron rest-mass units, its maximum value, its momentum and the electron rest mass. Further, , with the fine structure constant and the charge of the daughter nucleus (taken positive for electron emission, negative for positron emission), is the fundamental weak interaction coupling constant taken from muon decay, GeV, is the up-down quark-mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, the Fermi-function, and is the shape-correction function the value of which is unity in the allowed approximation, but whose value differs weakly from one when this approximation is relaxed. In addition, we define
where and are the Fermi and Gamow-Teller matrix elements respectively, and and are the strength of the weak vector and axial-vector interactions (in units of ) as defined in the Hamiltonian of Jackson, Treiman and Wyld jackson57 . We have assumed maximal parity violation for V- and A-currents. Finally, is the Fierz interference term jackson57 . The mean lifetime of the decaying state is , which after integrating over neutrino and electron directions, yields
We isolate the partial half-life by correcting for electron capture competition, , and selecting the branching ratio, , for the particular transition under study, to obtain
The statistical rate function, , and the Fierz correction factor, , are defined as
Inserting these definitions into Eq. (5), we come to our principal result
We now introduce two classes of small corrections: those due to radiative processes that go undetected in the experiment, and those due to isospin not being an exact symmetry in nuclei. Details on the nature of these corrections can e.g. be found in ref. hardy90 . We discuss the radiative corrections first. These are divided into terms that depend on the nucleus in question (’outer radiative correction’), , and those that do not (’inner radiative correction’), :
The nuclear-dependent term can be further divided into those pieces that depend trivially on the nucleus, (depending only on and ), and those that require a detailed nuclear-structure calculation, :
The term is mainly obtained from a standard QED calculation that has been completed to orders and and estimated to order sirlin86 ; jaus87 ; sirlin87 . These three contributions we will call , and respectively
while the -term is a leading log extrapolation of a low-energy term in the evaluation of the inner radiative correction marciano06 that turned out to be weakly nucleus-dependent and was therefore shifted from the inner radiative correction to the outer one towner08 . All four contributions in Eq. 14 are the same for both Fermi and Gamow-Teller transitions. By contrast, the contributions and differ between Fermi and Gamow-Teller transitions and so their notation will include a superscript of or as required. Details of the calculation of can be found in refs. jaus90 ; barker92 ; towner92 ; towner94 ; towner08 . The nucleus-independent radiative correction was originally evaluated by Marciano and Sirlin marciano86 and Sirlin sirlin95 , yielding = 2.40(8) % and has recently been addressed again by Marciano and Sirlin marciano06 leading to the new value = (2.361 0.038)%, in agreement with the previous value, but a factor of about two more precise. The reduction of the central value by approximately 0.04% is due to the fact that the aforementioned term was shifted from the inner radiative correction to the outer one.
The Fermi matrix element in the isospin-symmetry limit is precisely known – it is given in terms of an isospin Clebsch-Gordan coefficient. In practice, however, nuclei are impacted by Coulomb and other charge-dependent forces that weakly break the isospin symmetry. So we write
where is the isospin-symmetry breaking correction in Fermi transitions ormand95 ; towner02 and is the isospin symmetry limit value of the matrix element squared given by for transitions, and for transitions. By contrast, the Gamow-Teller matrix element is not known in the isospin symmetry limit. Nevertheless, to maintain a consistency in the equations, we write
although separate values of the symmetry-limit matrix element, , and the symmetry-breaking correction, , are not required for the development here. The isospin-symmetry breaking correction in Fermi transitions, , is typically separated into two components towner08
where the first term quantifies the impact of charge-dependent configuration mixing leading to differing wave functions for the parent and daughter nuclei, while the second term accounts for the 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 less than unity.
Including now all corrections, and noting the shape-correction function in the statistical rate function differs between Fermi and Gamow-Teller transitions, we have (setting )
For the superallowed pure Fermi transitions, with = 2 and = 0, one then has
For a mixed Fermi and Gamow-Teller transition, we can recast Eq. (18) into the form
where a mixing ratio is defined as
Lastly, restricting our attention to the mirror -transitions, for which = 1, Eq. (21) reduces to
This is our master equation. Our goal now is to extract values of the mixing ratio squared using data on the partial half-lives, , for mirror transitions in odd-mass nuclei. To this end we need apart from experimental data also calculations of the statistical rate function, and the ratio , the nucleus-dependent radiative corrections, and , and the isospin-symmetry breaking correction, . Further, we take the current best value of from the most recent work of Towner and Hardy towner08 .
Iii Experimental data
To determine the value for a transition three measured quantities are required: the half-life, , of the parent state, the branching ratio, , of the particular transition of interest, and the total transition energy, . The half-life and the branching ratio combine to yield the partial half-life, , (Eq. (6)), whereas the value is required to determine the statistical rate function, , (Eq. (8)). In our treatment of the data all half-life and branching ratio measurements published before January 2008 are considered. Since the evaluation of the values from different types of measurements would be too vast a project in itself it was decided to rely for these on the very extended 2003 Mass Evaluation audi03 . Half-life and branching ratio data are available for mirror nuclei up to Mo. All original experimental data were checked in detail. In Tables I and 2 we present all measured values for the half-life and the branching ratio that were used in our analysis. References to these data are listed in Tables 7 and 9. Each datum appearing in these tables is attributed to its original journal reference via an alphanumeric code comprising the initial two letters of the first author’s name and the last two digits of the publication date. If data were obviously wrong they were rejected. All rejected data are listed in Tables 8 and 10, with the reason for this rejection.
Similar evaluation principles and statistical procedures as those that are adopted for the analysis of the superallowed pure Fermi transitions hardy05 were used. Thus, of the surviving results, only those with uncertainties that are within a factor of 10 of the most precise measurement for each quantity were retained for averaging in the tables.
The statistical procedures followed in analyzing the tabulated data are based on those used by the Particle Data Group in their periodic reviews of particle properties (e.g. Ref. yao06 ). In the tables and throughout this work, ”error bars” and ”uncertainties” always refer to plus/minus one standard deviation (68% confidence level).
For a set of independent measurements, , of a particular quantity, a Gaussian distribution is assumed, the weighted average being calculated according to the equation
and the sums extend over all measurements. For each average the is also calculated and a scale factor, , determined from
This factor is then used to establish the quoted uncertainty. If 1, the value of from Eq. (24) is left unchanged. If 1 and the input are all about the same size, then is increased by the factor , which is equivalent to assuming that all the experimental errors were underestimated by the same factor. Finally, if 1 but the are of widely varying magnitudes, is recalculated with only those results for which being retained; the recalculated scale factor is then applied in the usual way. In all three cases, no change is made to the original average calculated with Eq. (24).
Adopted values for the half-life and the branching ratio are listed in Table 3, together with the calculated electron-capture fraction, , the deduced partial half-life, , (cf. Eq. (6)) and the value from ref. audi03 . The values were obtained from the tables of Bambynek et al. bambynek77 and Firestone firestone96 . No errors were assigned to these values as they are expected to be accurate to a few parts in 100 hardy05 ; bambynek77 such that they do not contribute perceptibly to the overall uncertainties.
Iv The values
Having surveyed the experimental data we can now turn to the determination of the values. The statistical rate function, , for each transition was calculated using the procedure and the code described in hardy05 . Results appear in column 2 of Table 4. To obtain values according to Eq. (23) we must still deal with the small correction terms. The values for the nucleus dependent radiative correction are listed in columns 5 to 9 of Table 4. Similar to the superallowed Fermi decays we have assigned an uncertainty equal to the term as an estimate of the error made in stopping the calculations at the order . Finally, one still has to deal with the nuclear-structure dependent corrections and . Two of these corrections, and , are very sensitive to the details of the shell-model calculation used in their evaluation. Fortunately, these two terms are also the smallest of the corrections we need in Eq. (23). We have mounted shell-model calculations using standard effective interactions and modest-size model spaces to evaluate them following exactly the same procedures as discussed in ref.towner02 . Further we assigned a generous error to account for their inherent model dependence. Less dependent on nuclear structure is the larger radial overlap correction, . Here we are guided by the recent work of Towner and Hardy towner08 , who pointed out the importance of including ’core’ orbitals in the shell-model evaluation of spectroscopic amplitudes. A decision has to be made as to which core orbitals should be included in the active model space. Towner and Hardy’s criterion is that experimental neutron pick-up reactions should observe strong spectroscopic factors for the orbitals in question. We have followed this criterion in obtaining our values for . All these corrections are listed in columns 10 to 12 in Table 4 with their sum in column 13. In total, these nuclear-structure dependent corrections are of order one percent or less.
One other quantity that depends weakly on a shell-model calculation is the ratio . Here a modest shell-model calculation is sufficient. We can also use these shell-model calculations to determine the relative sign of the Fermi and Gamow-Teller matrix elements, which can then be taken as the sign of in Eq. (22). Finally, the resulting values and corresponding values for (using = (3071.4 8) s towner08 ) are recorded in Table 5. As can be seen, for most of the nineteen transitions the precision on the value is better than 1 %, except for Ti and V, while it is even better than 0.3 % in nine cases. The highest precision is reached for H, N and Ar.
In figure 1 the fractional uncertainties attributed to each experimental and theoretical input factor that contributes to the final value are shown in the form of a histogram for all nineteen transitions. Clearly, to bring all contributions at the level of 1 part in 1000 or better, new and more precise measurements of the half-lives, , are required for almost all transitions. Better values are needed for almost half of the transitions, i.e. C, O, Na, Mg, S, Ca, Ti and V, while more precise measurements of the branching ratio, , are needed for Mg, Cl, K, Ti and V. The theoretical corrections, and contribute less than 1 part in 1000 to the final values in all cases except Ti and V.
V Standard model values for the decay correlation coeficients
With these values for we can now calculate the standard model values for correlation coefficients in decay jackson57 that are of interest to search for physics beyond the standard electroweak model (e.g. herczeg01 ; erler05 ; severijns06 ; abele08 ). The standard model assumes only vector and axial-vector interactions with maximal parity violation. In addition it is expected that the effects due to CP (or T) violation are negligible in the light quark sector at the present level of precision. These assumptions result in the conditions and for . Neglecting Coulomb as well as induced recoil effects one then obtains (the upper sign is for decay, the lower sign for decay), for the -neutrino angular correlation coefficient
for the asymmetry parameter
for the neutrino asymmetry parameter
and for the particle longitudinal polarization
where is the Kronecker delta and
for the mirror transitions.
Note that the coefficients in the standard model. When including also the effect of the Coulomb interaction of the charged nucleus and emitted particle (i.e. final state interaction, FSI) it turns out that, to first order in , this depends for the , , , , and correlation coefficients on interferences between the standard model coupling constants and the non-standard model coupling constants jackson57 , and therefore vanishes in the standard model. For the and correlation coefficients, however, the final state effects contain terms that depend on the time reversal invariant parts of the vector and/or axial-vector coupling constants and are thus non-zero in the standard model. To first order in one has jackson57
with the total electron energy. Numerical calculations vogel83 have shown that the values obtained for and within the used approximation are accurate at the 10% level.
The standard model values for the coefficients , and as well as the values for and at the spectrum endpoint, all calculated with the values for obtained from our value analysis, are listed in Table 6. A full analysis of the sensitivity of the different correlation coefficients to several types of physics beyond the standard model as well as the effect of recoil order corrections (i.e. weak magnetism) on the correlation coefficients is in preparation and will be published elsewhere tandecki08 .