Superallowed Fermi $\beta$ Decay

VudErr

There is an intense ongoing focus on experimental and theoretical studies of superallowed 0$^+\rightarrow0^+$ nuclear $\beta$ decays. This class of $\beta$ decay currently provides the most precise determination of the vector coupling constant for weak interactions, $G_V$, which is vital in the extraction of the up-down element of the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix, $V_{\mathrm{ud}}$. Since precision testing of the standard model is heavily dependent on $V_{\mathrm{ud}}$, the importance of accurate superallowed ${\cal F}t$-values is crucial. In order to extract $V_{\mathrm{ud}}$ from the high-precision experimental data, corrections to the almost nucleus-independent $ft$-values for superallowed $\beta$ decays must be made for radiative effects as well as isospin-symmetry-breaking (ISB) by Coulomb and charge-dependent nuclear forces. Although these corrections are small ($\sim1\%$), experimental measurements have provided such precise $ft$-values ($\pm0.03\%$) that the uncertainty on $G_V$ is dominated by the precision of these theoretical corrections. Due to the significant effect these ISB calculations have on the extraction of ${\cal F}t$, the theoretical models must be extensively tested against high-quality experimental data. Superallowed Fermi $\beta$-decay systems have also been explored as a venue for the possible observation of scalar contributions to the weak interaction. The topics below give a more complete theoretical basis for our studies on superallowed $\beta$ decay, as well as the implications from current measurements.

General $\beta$ Decay Introduction

The conservation of angular momentum in $\beta$ decay requires $J_P=J_D+J_{\beta}$, where $J_{\beta}=L_{\beta}+S_{\beta}$ is the total angular momentum of the emitted $\beta$ particle and neutrino, $L_{\beta}$ their total orbital angular momentum and $S_{\beta}$ their total spin angular momentum. The total matrix element is represented as a sum over all angular momentum values allowed by the selection rule $J_P=J_D+J_{\beta}$. However, since the lowest allowed value of $L_{\beta}$ consistent with the selection rules dominates the sum, the decays are classified according to their lowest possible $L_{\beta}$ values (see Table 1).

Table 1: [Total orbital angular momentum classification for $\beta$ decays.] Total orbital angular momentum classification for $\beta$ decays. [1]
Orbital Angular Momentum ($L_{\beta}$) Name Given Typical $\ell og(ft)$ (s)
0 Allowed $\quad\quad 2.9-6.0$
1 First Forbidden $\quad\quad 6-10$
2 Second Forbidden $\quad\quad 10-13$
3 Third Forbidden $\quad\quad 15-21$
4 Fourth Forbidden $\quad\quad\geq21$

The neutron, proton, electron and neutrino are all spin-1/2 fermions, and therefore $S_{\beta}=S_e+S_{\nu}=0$ or 1, and the nucleon spin can either remain the same or undergo an inversion. If $S_{\beta}=0$ it is referred to as a vector or Fermi $\beta$ decay. If instead $S_{\beta}=1$, it is known as an axial-vector or Gamow-Teller $\beta$ decay. The matrix element can subsequently be separated into the vector and axial-vector components which are characterized by different strength constants $G_V$ and $G_A$: \begin{equation} g^2|\overline{M_{fi}^{\prime}}|^2=G_V^2|\overline{M_{fi}^{\prime}(F)}|^2+G_A^2|\overline{M_{fi}^{\prime}(GT)}|^2. \end{equation} In general, this matrix element depends on the complicated initial and final nuclear wave functions for both Fermi and Gamow-Teller decays. However, the matrix element calculation can be largely simplified if, to first order, the axial-vector term is forbidden by the angular momentum selection rules. This occurs for $0^+\rightarrow0^+$ pure allowed Fermi transitions $(L_{\beta}=0, S_{\beta}=0,\pi_P=\pi_D)$. The charge independence of the strong nuclear force allows for the general treatment of protons and neutrons as the same particle, the nucleon. However, the $\beta$-decay process is an electroweak interaction, and does distinguish between these two particles, leading to a projection treatment of the nucleon. This two-state degeneracy is analogous to spin, and is referred to as isospin. An isospin $(t)$ of $1/2$ is defined for each the proton and neutron, with the $z$ projection for the neutron $+1/2$ and the proton
$-1/2$, effectively treating the neutron and proton as two different states of the same particle. Using this concept, the neutron and proton isospin wave functions are: \begin{eqnarray} |n\rangle&=&|t=\frac{1}{2},t_z=+\frac{1}{2}\rangle,\\ |p\rangle&=&|t=\frac{1}{2},t_z=-\frac{1}{2}\rangle, \label{npwf} \end{eqnarray} respectively. This treatment can be expanded to include systems of several nucleons as well, where the isospin coupling follows the same usual rules as ordinary angular momentum [2]. For example, a two-nucleon system can have a total isospin $T=0$ or $1$, which corresponds to the classical notion of aligned or anti-aligned $t=1/2$ vectors. For an A-nucleon system, the projection of the total isospin is given as \begin{equation} T_z=\frac{1}{2}(N-Z), \end{equation} where $N$ and $Z$ are the number of neutrons and protons in the nucleus, respectively. In nearly all situations, the nuclear ground state of any given nucleus has isospin $T=|T_z|$. The only exceptions are some odd-odd $N=Z$ nuclei where the pairing energy is large enough to overcome the symmetry energy. If the symmetry between protons and neutrons were exact, and the nuclear interaction was absolutely charge independent, a $T=1$ nucleus, for example, would have three energetically degenerate projections. In reality, these three isospin states differ in energy due to charge- dependent effects, and are connected via the $\beta$-decay operator. This operator converts nucleons from one type to the other, $\hat{O} (\nu \leftrightarrow \pi)$, and therefore acts on the isospin projection of the particle. For the case of the Fermi $\beta$-decay transition operator, all other characteristics of the nuclear wave function are left unchanged and the operator is simply the isospin ladder operator1, $T^{\pm}$ (for $\beta^{\pm}$ decay), and acts on the entire nuclear wave function isospin $T$. The members of an isospin multiplet (states with the the same $T$, same wave function, but different $T_z$) are referred to as isobaric analogue states (IAS), and are the states connected by superallowed Fermi $\beta$-decay. The isospin ladder operator can be used to calculate the transition matrix elements for them without detailed knowledge of the nuclear wave functions. If the isospin ladder operator is applied to a general nuclear state, \begin{eqnarray} \hat{T}^{\pm}|T,T_z\rangle=\sqrt{(T\mp T_z)(T\pm T_z+1)}|T,T_z\pm 1\rangle, \label{ladderopgeneral} \end{eqnarray} an expression for the matrix element is obtained, \begin{eqnarray} |\overline{M^{\prime}_{fi}(F)}|^2 &=& |\langle T,T_z\pm1|\hat{T}^{\pm}|T,T_z\rangle|^2,\nonumber\\ &=& (T\mp T_z)(T\pm T_z+1). \label{matelisospin} \end{eqnarray} If the case of $\beta^+$ decay between $T=1$ isobaric analogue states is considered, \begin{eqnarray} T_z=-1\rightarrow T_z=0\quad:\quad |\overline {M^{\prime}_{fi}}|^2=(1+1)(1-1+1)=2,\\ T_z=0\rightarrow T_z=+1\quad:\quad |\overline {M^{\prime}_{fi}}|^2=(1-0)(1+0+1)=2. \label{mvalue} \end{eqnarray} The matrix element is greatly simplified, $|\overline{M^{\prime}_{fi}}|^2=2$, independent of the complex nuclear wave functions.

ft-values
Figure 1: The experimental $ft$ values for the 13 high precision superallowed Fermi $\beta$-decaying nuclei using the evaluated values in Table 9 of Ref. [3]. A ~1% error band is also shown here to demonstrate the consistency of the measured values.

Thus, by using the calculated matrix element for superallowed $0^+\rightarrow 0^+$ transitions between $T=1$ isobaric analogue states, \begin{equation} ft=\frac{2\pi^3\hbar^7\ell n(2)}{m_e^5c^4(2G_V^2)}, \label{superallowedft} \end{equation} where $G_V$ is the vector coupling constant. This expression has two very important features:
  • - Every term on the right side is a constant, independent of which nucleus is being studied, and
  • - The left hand side of the equation contains quantities that are experimentally measurable, thereby allowing a direct determination of $G_V$.

The experimental data for the superallowed $ft$ values are indeed constant to ~1% [3], which is a stunning result considering that $ft$ values in general can span over 20 orders of magnitude. However, the above treatment assumes that isospin is an exact symmetry, and given the precision of the current experimental superallowed $ft$ values (0.03% in the best cases), small corrections must be made in order to obtain truly nucleus-independent $ft$ values.

Theoretical Corrections and ${\cal F}t$ values

To provide a more stringent test of the Standard Model, corrections to the extracted $ft$ value in the equation above must be applied. This ${\cal F}t$ value is defined as [3]: \begin{eqnarray} {\cal F}t\equiv ft(1+\delta_R)(1-\delta_C)=\frac{2\pi^3\hbar^7\ell n(2)}{2G_V^2m_e^5c^4(1+\Delta_R)}, \label{Ftvalue} \end{eqnarray} where $\delta_R$ and $\Delta_R$ represent nucleus-dependent and nucleus-independent radiative corrections, respectively. The correction term $\delta_C$ depends explicitly on nuclear structure and accounts for the fact that Coulomb and charge-dependent nuclear forces break the exact isospin symmetry between the parent and daughter nuclear state. Combining the measured $ft$ values for the 13 most precisely determined superallowed nuclei and the $\delta_C$ calculations of [3], the ${\cal F}t$ values in Fig. 2 are obtained. When these corrections are applied, the small (~1%) deviations in the $ft$ values are reduced significantly, and the ${\cal F}t$ values for the 13 high precision cases are constant to 1.3 parts in $10^4$ [3]. Although these data reflect the current status of $\overline{{\cal F}t}$, the history of both the superallowed $\beta$- decay measurements, and the theoretical corrections, tell a much less stable story.

ft-values
Figure 2: The corrected ${\cal F}t$ values for the 13 high-precision superallowed decays using the 2008 $\delta_C$ values from Towner and Hardy [4]. The solid line represents the average of these values and the dashed lines its uncertainty.

Isospin symmetry is always weakly broken in nuclei by charge-dependent interactions, which shift the exact-symmetry hadronic matrix element to some physical value. The calculation of the isospin-symmetry-breaking (ISB), as it pertains to superallowed Fermi $\beta$ decay, is represented as a correction
($\delta_C$) to the experimentally observed $ft$ value, shown above. Although the calculations of Towner and Hardy have been the standard since the 70's, a great deal of interest has recently been paid to the calculation of these corrections, with various methods and varying levels of success [5]. Since the revised ISB calculations were published in 2008 [4], there has been a relative explosion of work on $\delta_C$ calculations, including revisions to the existing formalism [6,7] and approaches using density functional theory [8] and relativistic RPA [9]. Several of these recent calculations are shown in Fig. 3, and demonstrate the large spread of values that arises from the various calculation techniques. Some of these calculations have resulted in drastically different $\delta_C$ values, which consequently shift the extracted mean value of ${\cal F}t$, $\overline{{\cal F}t}$, significantly. A brief summary and review of many of these new techniques is presented in [10]. Although these new techniques have addressed some important concerns, none have yet achieved a satisfactory level of refinement required for Standard Model tests as those of Towner and Hardy. In fact, the calculations of Towner and Hardy have been thoroughly tested [11,12,13] for nuclei where the corrections are large, and are in excellent agreement with the experimentally extracted $\delta_C$ values. The extraction of the vector coupling constant, $G_V$, for weak interactions through the determination of ${\cal F}t$ has large implications for fundamental tests of the Standard Model. The superallowed $\beta$-decay data currently provide the most precise value of this constant, \begin{eqnarray} G_V^2&=&\frac{2\pi^3\hbar^7\ell n(2)}{2m^5_ec^4(1+\Delta_R)\overline{{\cal F}t}},\label{Gvextract}\\ \frac{G_V}{(\hbar c)^3}&=&1.13630(15)(08)(22)\times10^{-5}~~\mathrm{GeV}^{-2},\label{Gvuncert}\\ \frac{G_V}{(\hbar c)^3}&=&1.13630(28)\times10^{-5}~~\mathrm{GeV}^{-2},\label{Gvnumber} \end{eqnarray} where $\overline{{\cal F}t}$ is the so-called world average ${\cal F}t$ value, from the 13 precision superallowed cases. The three uncertainties shown in brackets arise from i) the experimental uncertainties, as well as the calculated ISB and transition dependant radiative correction uncertainties, ii) A systematic uncertainty associated with the ${\cal F}t$ values adopted in [3] to account for the differences between the ISB correction calculations using the Woods-Saxon and Hartree-Fock approaches, and iii) The adopted uncertainty for the inner radiative correction $\Delta_R$. There is a caveat to the above approach, that $G_V$ is constant, which is currently a source of motivation for the study of the superallowed $\beta$ decay ${\cal F}t$ value. This idea is known as the conserved vector current (CVC) hypothesis, which states that $G_V$ is not renormalized within the nuclear medium and should therefore be the same for all $\beta$ decays. The superallowed ${\cal F}t$ values shown in Fig. 2 currently confirm this hypothesis at the level of 1.3$\times 10^{-4}$ [3]. Under the assumption that the CVC hypothesis holds, the extraction of $G_V$ using $\overline{{\cal F}t}$ is valid and can be used to further test the Standard Model. One of the cornerstones of the Standard model of particle physics is the Cabbibo-Kobayashi-Maskawa (CKM) quark-mixing matrix, which relates the weak interaction and mass eigenstates of the quarks: \begin{eqnarray} \left[\begin{array}{c} d^{\prime}\\ s^{\prime}\\ b^{\prime}\end{array}\right]=\left[\begin{array}{ccc} V_{ud} & V_{us} & V_{ub}\\ V_{cd} & V_{cs} & V_{cb}\\ V_{td} & V_{ts} & V_{tb}\end{array}\right]\left[\begin{array}{c} d\\ s\\ b\end{array}\right], \label{CKMmatrix} \end{eqnarray} where $d,s$, and $b$ are the mass eigenstates of the down, strange and bottom quarks, respectively. The prime on each letter in the left hand side of the equation denotes the corresponding weak-interaction eigenstates. In the standard model, this represents a transformation between complete sets of basis states, and as such, has the requirement that it must be a unitary transformation. This means that the sum of the elements squared for every row and column must be unity. A test of this unitarity is a fundamental test of the Standard Model. Since protons and neutrons are comprised of up and down quarks, it is no surprise that the most precisely measured element in the CKM matrix is $V_{ud}$. In fact, it has been the focus of a great deal of experimental efforts over the last 20 years . The experimental determination of this value is conducted by combining the weak vector coupling constant $G_V$, measured through superallowed Fermi $\beta$ decays, with the Fermi coupling constant $G_F$ from muon decay. Presently, the sum satisfies the unitary condition and represents a confirmation of the Standard Model at the $10^{-3}$ level.
The Standard Model description of the weak interaction is an equal mix of vector and axial-vector interactions that maximizes parity violation, and is colloquially known as $V-A$ theory [18]. Despite continually improving experimental sensitivity over many decades, no statistically significant defect with this description of the weak interaction in the Standard Model has ever been observed [19,20]. Although $V-A$ theory assumes that the scalar and tensor interactions are zero, there is no inherent reason why this must be the case. Therefore, superallowed Fermi $\beta$-decay systems have also been explored as a venue for the possible observation of scalar contributions to the weak interaction. The presence of a fundamental scalar interaction would introduce a nonlinear nuclear-charge ($Z$) dependence to the corrected ${\cal F}t$ values (Fig. 4 (top)). In fact, for maximally parity- violating scalar interactions, the ${\cal F}t$ values would contain an additional term that is inversely proportional to the average decay energy, $\langle \frac{1}{Q}\rangle$. Since the smallest superallowed $\beta$-decay $Q$ values exist for the lightest cases of $^{10}$C and $^{14}$O, the respective ${\cal F}t$ values for these nuclei would therefore experience the largest deviation from CVC linearity.

allowed
Figure 4: (top) The ${\cal F}t$ values for the 14 high-precision superallowed cases. The curved lines represent the average path the ${\cal F}t$ values would take if a fundamental scalar interaction exists [21].

The EI group at CSM is currently involved in direct precision measurements of these nuclear decays, as well as detailed nuclear structure measurements related to testing the ISB corrections that are required to test the Standard Model from $0^+\rightarrow0^+$ nuclear $\beta$ decay.




Footnotes
1. It is actually a sum of all single particle isospin ladder operators $\hat{\tau}^{\pm}$
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