Isospin corrections to super-allowed beta decays in nuclei

Isospin corrections to the super-allowed beta decay matrix elements are evaluated in perturbation theory using the notion of the giant isovector monopole state. The calculation avoids the separation into different contributions and therefore presents a consistent, systematic and more transparent approach. Explicit expressions for δc as a function of the mass number A are derived. These corrections affect the values of the Vud matrix element in the Cabbibo-Maskawa-Kobayashi matrix. Also, it is pointed out that in some nuclei with a low number of excess neutrons (protons) Coulomb mixing with the anti-analog state can introduce significant isospin impurities in the isobaric analog state.


Introduction
Recently there is much activity to determine the corrections one has to introduce when one studies the beta-decay matrix elements for super-allowed decays in , (or ) nuclei [1,2] .This is important because using the measured ft values one can relate these to the u-quark to d-quark transition matrix element (m.el.) in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. In the Standard Model (SM) this matrix satisfies the unitarity condition: (1) In order to use the experimental ft values to determine one has to introduce corrections [1,2]. (There are radiative corrections which we will not treat here, discussions of these can be found abundantly in the literature [1].) The second type of correction, which is usually termed as the isospin symmetry breaking term, denoted as where is the physical Fermi matrix element: present approach we start from a charge-independent Hamiltonian so that the matrix element in eq. (1) is exactly T 2 and we then treat the Coulomb force in perturbation theory. In the way we approach the problem there is no need to break up the contribution of the Coulomb interaction into various separate components. All the effects of Coulomb mixing (such as isospin mixing, the change in the radial part of the wave functions, etc.) are taken into account in a single term.

Coulomb mixing
We start by introducing a nuclear charge independent Hamiltonian -. The eigenstates of this Hamiltonian with isospin 0 H T and will be denoted as and: The action of the isospin lowering and raising operators, , gives: We now add to the charge independent Hamiltonian a charge dependent part .

CD V
The dominant part in the charge dependent interaction is the charge asymmetric Coulomb force . Because of the long range nature of the Coulomb force, the prevailing part will be in such cases the one-body part. Of interest to us here is the isovector part of the potential. For a uniform charge distribution of radius R any off-diagonal matrix element between two states of the isovector part is: Here is the energy of the analog state. 1 E One derives finally for T=1 isotriplets (the detailed derivation can be found in ref. [2]): is the symmetry potential and ξ is a numerical factor which depends on the model used to describe the isovector monopole. The range of values for this parameter is between 3 and 4 [3].

Results
The isospin impurities in the ground state of a nucleus where computed in the past [3] using collective models for the isovector monopole state. Using these impurities one finds the results shown in the These results are a factor 3-8 smaller than in ref. [1].

Isospin mixing and the role of the anti-analog state
In some nuclei with a low number of excess neutrons (protons) Coulomb mixing with the antianalog state can introduce significant isospin impurities in the isobaric analog state. The corrections are particularly sizeable in those medium heavy nuclei that are off the stability line having a small number of excess neutrons. The enhancement of the isospin admixture in the analog can reach a few percent.
The anti-analog A is then: The letters n and p denote neutrons and protons. We will consider here parent nuclei with simple configurations: for even-even nuclei the 1 n and 2 n are even and in each orbit the excess nucleons are coupled to + = 0 J and in odd-even nuclei 1 n is odd and 2 n is even.
The one-body Coulomb matrix element between the analog and anti-analog is then: If the excess neutrons occupy orbits belonging to different major shells, this matrix element is sizable. But even if this is not the case, because of binding energy effects in a finite potential well and angular momentum, this matrix element is of the order of several hundred keV [3,4].
The energy splitting between the analog and anti-analog is given by the symmetry potential. It is easy to show that in this case: