Isospin Mixing and the Cubic Isobaric Multiplet Mass Equation in the Lowest $T=2,~{}A=32$ Quintet

If nuclear isospin $T$ were a conserved quantity, the members of an isobaric multiplet would be $(2T+1)$-fold degenerate. However, it is known Benenson and Kashy (1979) that this degeneracy is broken by two-body charge-dependent interactions, which can be described at tree-level as the sum of an isoscalar, isovector and isotensor operator of rank 2. To first order, the energy spacings between the multiplet members can be obtained from the expectation value of the charge-dependent perturbation. On applying the Wigner-Eckart theorem to the perturbing Hamiltonian, the mass splittings are described by the isobaric multiplet mass equation (IMME) Wigner (1958); Weinberg and Treiman (1959)

where each member of the multiplet is characterized by its isospin projection $T_{z}=(N-Z)/2$.

The general success of the IMME over a large mass range made it a reliable tool to address a variety of research problems. For example, it was used to test recent advances in nuclear theory Holt et al. (2013); Ormand et al. (2017); Martin et al. (2021), map the proton dripline Ormand (1997), identify candidates for two-proton radioactivity Dossat et al. (2005); Blank and Borge (2008), search for physics beyond the standard model Adelberger et al. (1999), infer rapid proton capture ($rp$) nuclear reaction rates relevant for studies of novae and x-ray bursts Wrede et al. (2009); Richter and Brown (2013); Ong et al. (2017), assess global nuclear mass model predictions Liu et al. (2011) and constrain calculations relevant for CKM unitarity tests Hardy and Towner (2015).

In this context, the lowest isospin $T=2$ quintet for $A=32$ (with spin and parity $J^{\pi}=0^{+}$) is an interesting case. The $\beta$ decay of ³²Ar, the most proton-rich member of the quintet was previously used for searches of exotic scalar Adelberger et al. (1999) and tensor Araujo-Escalona et al. (2020) weak interactions as well as for benchmarking isospin symmetry breaking (ISB) corrections Bhattacharya et al. (2008) important for obtaining a precise value of $V_{ud}$, the up-down element of the CKM quark-mixing matrix Hardy and Towner (2015). In fact, the $A=32$ quintet is one of the most extensively studied and precisely measured multiplets to date Triambak et al. (2006); Kwiatkowski et al. (2009); Kankainen et al. (2010); Blaum et al. (2003); Signoracci and Brown (2011); Lam et al. (2013). It remains an anomalous case, for which the IMME breaks down significantly MacCormick and Audi (2014). A satisfactory fit to the measured masses is only obtained with an additional cubic $dT_{z}^{3}$ term, with $d=0.89(11)$ keV (c.f. Table 1). This is the smallest and most precisely determined violation of the IMME observed so far. Unlike other multiplets, where apparent violations of the IMME were resolved through subsequent measurements Herfurth et al. (2001); Pyle et al. (2002); Gallant et al. (2014); Glassman et al. (2015); Zhang et al. (2012); Su et al. (2016), the $A=32$ anomaly has persisted over several years, despite high-precision remeasurements of ground state masses Kwiatkowski et al. (2009); Kankainen et al. (2010); Shi et al. (2005) as well as excitation energies Triambak et al. (2006); Pyle et al. (2002). A recent compilation MacCormick and Audi (2014) showed the $A=32$ quintet to be a unique case, in which the $\chi^{2}$ value for a cubic fit yields 95% probability that it is the correct model to describe the data. Since there are no known fundamental reasons that preclude a cubic IMME term, it is interesting that the magnitude of the extracted $d$ coefficient for this case agrees well with theoretical estimates that used a simple nonperturbative model Henley and Lacy (1969) or a three-body second-order Coulomb interaction Bertsch and Kahana (1970), both of which allow a non-vanishing cubic term, with $\lvert d\rvert\approx 1~{}{\rm keV}$. Alternatively, the role of isospin-mixing with non-analog $0^{+}$ states was also theoretically investigated in the recent past Signoracci and Brown (2011); Lam et al. (2013).

We delve into the above aspect here, via an analysis of high-resolution experimental data and a comparison with calculations that use recently developed shell model Hamiltonians Magilligan and Brown (2020). For the former, we mainly rely on data from a previous ³²Ar $\beta$ decay experiment at CERN-ISOLDE Adelberger et al. (1999), that acquired $\beta$-delayed protons from unbound states in the daughter ³²Cl ($S_{p}\approx 1581$ keV) with high resolution (full widths at half maximum of $\sim 6$ keV). The primary goal of the ISOLDE experiment was to search for scalar currents in the weak interaction, by determining the $\beta\nu$ angular correlation ($a_{\beta\nu}$) for the decay, via a precise analysis of the shape of the superallowed $\beta$-delayed proton peak Adelberger et al. (1999). Part of the proton spectrum is shown in Fig. 1.

The high resolution nature of the ISOLDE data allow an identification of potential isospin admixtures to the $T=2$ isobaric analog state (IAS) in ³²Cl. The nature of each $\beta$ transition is encoded in the shapes of the proton groups, which would be different if the transitions were Fermi ($0^{+}$ $\to$ $0^{+}$), with $a_{\beta\nu}=1$ or Gamow-Teller ($0^{+}$ $\to$ $1^{+}$), with $a_{\beta\nu}=-1/3$.

We analyzed these data using the R-matrix formalism described in Refs. Barker (1971); Warburton (1986). In the analysis, the proton peaks were grouped as $p_{0},~{}p_{1},~{}p_{2}$ or $p_{3}$ depending on whether the proton emission left the residual ³¹S nucleus in its ground state or any of its first three excited states at 1249, 2234 and 3077 keV (see Fig. 9 in Ref. Bhattacharya et al. (2008)). Interference was allowed between all levels that had the same quantum numbers, transition type (Fermi or Gamow-Teller), and final states in ³¹S. The R-matrix fits folded in the detector response function and lepton recoil effects (described in Ref. Adelberger et al. (1999)), and were parameterized using various $J^{\pi}$ values for the daughter ³²Cl states and associated $a_{\beta\nu}$ coefficients. The fits yielded relative intensities, ³²Cl excitation energies and intrinsic widths. They were repeated for different values of $a_{\beta\nu}$, spin-parity combinations and $p_{0},~{}p_{1},~{}p_{2}$, $p_{3}$ assignments for the daughter levels to obtain best agreement with experimental data. A few important features of the analysis are described below.

Peaks C, E and H were assumed to be from the $p_{1}$ group. These assignments were based on data reported by independent ³²Ar $\beta$-delayed proton-$\gamma$ coincidence measurements Bhattacharya et al. (2008); Blank et al. (2021). We observe that a reasonably a good R-matrix fit is attained (Fig. 1) with the parameters listed in Table 2. The fit assumes that peak B arises from a Fermi transition, while the others (apart from peak I) are exclusively from Gamow-Teller decays. Based purely on $\chi^{2}$ values from independent fits, peak I could be either from a Fermi or Gamow-Teller decay.

We compared these results with ${}^{32}{\rm S}(^{3}{\rm He},t)$ data that were independently obtained at the MLL tandem accelerator facility in Garching, Germany. The experiment used $\sim$300 enA of 33 MeV ${}^{3}{\rm He}^{++}$ ions, incident on a $120~{}\mu$g/cm²-thick natural ZnS target. The tritons exiting the target were momentum analyzed using the high-resolution Q3D magnetic spectrograph Löffler et al. (1973); Dollinger and Faestermann (2018). A sample triton spectrum in the energy range of interest is shown in Fig. 2. These data provided an important confirmation of the $p_{0}$ assignments for peaks A, D, F and G in our R-matrix analysis. Additionally, since the ${}^{32}{\rm S}(^{3}{\rm He},t)$ charge-exchange reaction predominantly populates $J^{\pi}=1^{+}$, $T=1$ levels at forward angles ¹¹1This assumes no anomalous isospin-mixing mechanisms within ³²S., the states observed at these energies in both ³²Ar $\beta$ decay and the ${}^{32}{\rm S}(^{3}{\rm He},t)$ reaction can be ruled out as sources of $J^{\pi}=0^{+}$ isospin impurity. This comparative analysis leaves only the 4443 and 5302 keV levels (c.f. Table 2) as potential admixed states. We find from the $\beta$ decay data that the $p_{1}$ intensity for the latter is around 1.2 times larger than its $p_{0}$ group. In comparison, the $p_{1}$ intensity for the IAS is roughly 80 times smaller than the $p_{0}$. This is due to the low penetrability of $\l=2$ protons from the $J^{\pi}=0^{+}$ IAS. The above discrepancy makes it highly unlikely for the 5302 keV state to have spin-parity $0^{+}$, which rules it out as a source of isospin mixing.

We next used the measured $\beta$-delayed proton intensities in Table 2, together with shell model calculations of isospin mixing to investigate the matter further. For the latter we used newly developed isospin non-conserving (INC) USDC and USDI interactions, described extensively in Ref. Magilligan and Brown (2020). The INC parameters in the new USD Hamiltonians were obtained from a fit to several mirror displacement energies and stringently tested via a comparison with experimental data Magilligan and Brown (2020). The isospin-mixing matrix elements calculated with these Hamiltonians were robustly validated Magilligan and Brown (2020) with results from independent high-precision ^31,32Cl $\beta$ decay experiments Bennett et al. (2016); Melconian et al. (2011, 2012), where large isospin-mixing in the daughter ^31,32S states were observed. More recently, such calculations were used together with a ³²Ar $\beta$ decay measurement Blank et al. (2021), that acquired valuable proton-gamma coincidence data, albeit with lower proton energy resolution. Ref. Blank et al. (2021) identified two possible sources of $T=1$ isospin mixing at 4799 and 4561 keV. However, their measured proton branches were significantly lower than calculated values. We show below that the higher-resolution ISOLDE data justifies ruling out these proposed levels, while providing a viable alternative for the admixed $T=1,0^{+}$ state, which is consistent with both theory predictions as well as experimental observations.

Our shell model calculations show that the isospin mixing within the $T_{z}=1,~{}0,~{}\mathrm{and}~{}-1$ members of the quintet occurs primarily with a single $T=1$ state, located few hundred keV below the $T=2$ IAS in each isobar. The results are summarized in Table 3, which lists the energy differences ($\Delta E=E_{i}-E_{\rm IAS}$) between the admixed $T=1$ and $T=2$ states for each nucleus, and the calculated isospin-mixing matrix element ($v$) for ³²Cl. The evaluated mixing matrix elements for each of the three nuclei are plotted in Fig. 3. We note that the mixing matrix elements obtained with the older USDB-CD interactions Ormand and Brown (1989) are nearly a factor of two smaller than the ones obtained with the newer interactions, for all three isobars. This is consistent with previous observations for ^31,32Cl $\beta$ decay Magilligan and Brown (2020).

The predicted $J^{\pi};T=0^{+};1$ level in ³²Cl can be identified by obtaining an experimental value of $v$ from the data in Fig. 1 and Table 2. For two-state mixing, $v_{\rm expt}$ is simply

where the ratio in the square bracket is the (Fermi) strength to the admixed $T=1$ state, relative to the superallowed decay. This is easily determined from the measured $I_{p}^{\rm rel}$ values in Table 2, the ratio of calculated phase-space factors, a small ISB correction Bhattacharya et al. (2008) and the $p_{0}$ contribution to the total superallowed intensity. On applying this prescription to the only candidate $0^{+}$ level at 4443 keV, we obtain a $v_{\rm expt}=39.0(24)$ keV, in excellent agreement with the calculations. The results in Table 3, together with our aforementioned observations and the experimental values listed in Table 2 allow a credible identification of the 4443 keV level as the predicted admixed $T=1$ state. The discrepancy between theory and experiment for $\Delta E$ should not be surprising, given the $\sim$150 keV root-mean-square (rms) deviation for energies in USD interactions Magilligan and Brown (2020).

We next investigated additional cubic ($dT_{z}^{3}$) and quartic ($eT_{z}^{4}$) terms to the IMME due to such isospin mixing. One can determine the exact solutions for the $d$ and $e$ coefficients by modifying Eq. (1) to incorporate such terms, such that

where the $M_{T_{Z}}$ are isobar masses in the quintet. The results for $d$ and $e$ using the calculated values of $v$ and $\Delta E$ are shown in Fig. 4, and labeled as “unshifted”. We repeated these evaluations by shifting the $T=2$ states in ³²Cl, ³²S and ³²P by the amount needed to reproduce our experimentally determined 603 keV energy difference in ³²Cl. The same $\Delta E$ was used for the three isobars due to the lack of similar experimental information for ³²S and ³²P. The shifts were accomplished by adding a $T^{2}$ term to the Hamiltonian that shifts the $T=2$ states relative to the others, without changing the isospin-mixing. As evident in Fig. 4, the shifts mildly affect the $e$ coefficient (due to changes in the $T=0$ mixing with the IAS in ³²S), but significantly decrease the calculated $d$ coefficient to $\approx 0.3$–$0.4$ keV for the new interactions.

The single-state contributions from $T=0$ and $T=1$ levels are

where $s=-v^{2}/\Delta E$ is the shift in each IAS due to two-state mixing. Thus, one can remove the $T=1$ mixing contribution for further investigation (labeled as “removed” in Fig. 4). We observe that on doing so, the extracted coefficients are mostly consistent with zero. The negative $e$ coefficient from the USDI calculation is due to mixing with a $T=0$ state in ³²S. However such $T=0$ mixing would not explain the non-zero $d$ coefficient required for the quintet, as evident from Eq. (4).

The above analysis validates the contention that isospin mixing with predicted $T=1$ levels necessitates a small cubic term for the multiplet. Our extracted $d$ coefficients for the “shifted” calculations from different USDC and USDI Hamiltonians agree reasonably well with one another, but are smaller than the experimental value $d=0.89(11)$ keV, from Table 1.

As further tests of our calculations, we also evaluated amplitudes for isospin-forbidden proton emission from the two admixed $J^{\pi}=0^{+}$ levels in ³²Cl and the effect of the $T=1$ isospin mixing on the superallowed Fermi decay of ³²Ar. Unlike the energy shift of the $T=2$ IAS in ³²Cl, which is predominantly from isospin mixing with the predicted 0${}^{+}_{2}$ $T=1$ state below the IAS, isospin-forbidden proton emission from the IAS depends on $T=1$ mixing with a large number of states in ³²Cl and isospin mixing within ³¹S, which is dominated by mixing of the lowest $T=3/2$ state into its ground state.

We calculated proton widths for $p_{0}$ and $p_{1}$ transitions from the $0_{2}^{+}$ admixed $T=1$ state and the $T=2$ IAS in ³²Cl. The widths were evaluated using the simple expression

where the $(32/31)^{2}$ factor is a center-of-mass correction Dieperink and de Forest (1974), the $C^{2}S$ are the shell model spectroscopic factors, and $\Gamma_{\rm sp}$ are single-particle proton widths. Similar to Ref. Signoracci and Brown (2011), the $\Gamma_{\rm sp}$ were calculated from $p$ + ${}^{31}{\rm S}$ scattering on potentials obtained with an energy-density functional calculation with the Skx Skyrme-type interaction Alex Brown (1998). On the other hand, the measured proton widths Bhattacharya et al. (2008) for the $T=2$ IAS are known to be $18.2(5)$ eV and $0.233(7)$ eV for the $p_{0}$ and $p_{1}$ protons respectively. Together with the calculated single-particle proton decay widths, we use these results to obtain experimental values for the decay amplitudes $A=(C^{2}S)^{1/2}$. These are simply determined from the relation $\Gamma_{\rm expt}=A_{\rm expt}^{2}(32/31)^{2}\Gamma_{\rm sp}$. The results for $A_{\rm expt}$ are shown in Table 4 and compared with theory predictions, obtained using the ‘shifted’ calculations.²²2For example, for the shifted USDCm calculation, the calculated amplitude can be decomposed as $A$ = 0.0041 (0${}^{+}_{1}$; $T=1$) $+$ 0.0159 (0${}^{+}_{2}$; $T=1$) $+$ 0.0009 (0${}^{+}_{3}$; $T=1$) $+$ 0.0009 (all other $T=1$) $-0.0166$ (³¹S; $T=3/2$) = 0.0052. The USDC result has a larger amplitude, mainly due to a 50% smaller destructive contribution from the $T=3/2$ state in ³¹S. We observe reasonable agreement between theory and experiment, except for the $T=2$ $p_{1}$ transition, whose calculated amplitudes are found to be much larger.

Finally, we also provide isospin-symmetry-breaking (ISB) corrections for ³²Ar $\to$ ³²Cl superallowed Fermi decay, due to the isospin-mixing in ³²Cl. The $T=2$ $\to$ $T=2$ superallowed strength is reduced by a factor $(1-\delta_{C})$, where $\delta_{C}$ is the total ISB correction Hardy and Towner (2015). Such corrections play a critical role in testing the unitarity of the CKM matrix and placing important constraints on beyond the standard model (BSM) physics Hardy and Towner (2015). The ISB correction is generally expressed as a sum of two separate contributions, $\delta_{C}=\delta_{C}^{\rm cm}+\delta_{C}^{\rm ro}$ Bhattacharya et al. (2008), from configuration mixing and a overlap mismatch between the parent and daughter radial wavefunctions. The former are known to quite model dependent as they are very sensitive to the details of their calculation Hardy and Towner (2015). Our calculated results for $\delta_{C}^{\rm cm}$ (from the $T=1$ mixing in ³²Cl) are listed the final column of Table 4. It may be noted that for the shifted USDCm and USDIm calculations, which show best agreement with the measured $T=2$ $p_{0}$ amplitude, we obtain $\delta_{C}^{\rm cm}$ = 0.15% and 0.70% respectively. From a previous evaluation of $\delta_{C}^{\rm ro}=1.4\%$ Bhattacharya et al. (2008), these yield $\delta_{C}=1.6\%$ and $2.1\%$, in agreement with the experimentally extracted value, $\delta_{C}^{\rm expt}=2.1(8)\%$ Bhattacharya et al. (2008).

In summary, we used high-resolution experimental data to validate newly-developed shell model calculations of isospin mixing in ³²Cl. This analysis is used to investigate the observed IMME violation in the first $T=2,~{}A=32$ quintet. We show that isospin mixing with shell-model-predicted $T=1$ states below the IAS necessarily result in a break down of the IMME, leading to the requirement of a small cubic term. However, this alone cannot explain the magnitude of the experimental $d$ coefficient in Table 1. Experimental investigations of intruder $0^{+}$ levels, isospin-mixing in ³²S and ³²P, continuum coupling of the proton unbound states in ³²Cl, and further mass measurements may be useful in this regard.

Our observations pertaining to ³²Ar $\to$ ³²Cl superallowed Fermi decay may also be useful to benchmark theory calculations Bhattacharya et al. (2008) of model-dependent ISB corrections that are important for top-row CKM unitarity tests Hardy and Towner (2015). This is particularly relevant in light of recent evaluations of radiative corrections Seng et al. (2018) that show an apparent violation of CKM unitarity at the $>3\sigma$ level Seng et al. (2021).