We study the shell-model formalism to include the isospin-symmetry-breaking correction ($δ_{C}$) to nuclear matrix element of superallowed $0^+\to 0^+$ Fermi $β$ decays of $T=1$ nuclei. Based on a perturbation expansion in a small quantity, such as the deviation of the overlap integral between proton and neutron radial wave functions from unity or of the transition density from its isospin-symmetry value, we derive that $δ_C$ can be obtained as a sum of six terms, including two leading order (LO) terms, two next-to-leading order (NLO) terms, one next-to-next-to-leading order (NNLO) term and one next-to-next-to-next-to-leading order (NNNLO) term. The first two terms have been considered in a series of shell-model calculations of Towner and Hardy as well as in the recent calculation of the present authors, while the remaining four terms are usually neglected. A numerical calculation has been carried out for 13 transitions in the $p$, $sd$ and the lower part of $pf$ shells. Our results indicate that the magnitude of the sum of all higher order terms is of the order of $10^{-3}$ %. This number is well below typical theoretical errors quantified within the shell model with Woods-Saxon radial wave functions.
Theoretical approaches that use one-body densities as dynamical variables, such as Hartree-Fock or the density functional theory (DFT), break isospin symmetry both explicitly, by virtue of charge-dependent interactions, and spontaneously. To restore the spontaneously broken isospin symmetry, we implemented the isospin-projection scheme on top of the Skyrme-DFT approach. This development allows for consistent treatment of isospin mixing in both ground and exited nuclear states. In this study, we apply this method to evaluate the isospin impurities in ground states of even-even and odd-odd N~Z n…
Recently, we have applied for the first time the angular momentum and isospin projected nuclear density functional theory to calculate the isospin-symmetry breaking (ISB) corrections to the superallowed beta-decay. With the calculated set of the ISB corrections we found |V_{ud}|=0.97447(23) for the leading element of the Cabibbo-Kobayashi-Maskawa matrix. This is in nice agreement with both the recent result of Towner and Hardy [Phys. Rev. {\bf C77}, 025501 (2008)] and the central value deduced from the neutron decay. In this work we extend our calculations of the ISB corrections covering all s…
The superallowed beta-decay rates that provide stringent constraints on physics beyond the Standard Model of particle physics are affected by nuclear structure effects through isospin-breaking corrections. The self-consistent isospin- and angular-momentum-projected nuclear density functional theory is used for the first time to compute those corrections for a number of Fermi transitions in nuclei from A=10 to A=74. The resulting leading element of the CKM matrix, |V_{ud}|= 0.97447(23), agrees well with the recent result by Towner and Hardy [Phys. Rev. C {\bf 77}, 025501 (2008)].
We investigate isospin-symmetry breaking effects in the sd-shell region with large-scale shell-model calculations, aiming to understand the recent anomalies observed in superallowed Fermi beta-decay. We begin with calculations of Coulomb displacement energies (CDE's) and triplet displacement energies (TDE's) by adding the T=1,J=0 isospin nonconserving (INC) interaction into the usual isospin-invariant Hamiltonian. It is found that CDE's and TDE's can be systematically described with high accuracy. A total number of 122 one- and two-proton separation energies are predicted accordingly, and loca…
Self-consistent random phase approximation (RPA) approaches in the relativistic framework are applied to calculate the isospin symmetry-breaking corrections $δ_c$ for the $0^+\to0^+$ superallowed transitions. It is found that the corrections $δ_c$ are sensitive to the proper treatments of the Coulomb mean field, but not so much to specific effective interactions. With these corrections $δ_c$, the nucleus-independent $\mathcal{F}t$ values are obtained in combination with the experimental $ft$ values in the most recent survey and the improved radiative corrections. It is found that the constancy…
After application of an analytical transformation, a new exact representation for the nuclear isospin-symmetry breaking correction $δ_C$ to superallowed beta decay is obtained. The correction is shown to be essentially the reciprocal of the square of an energy parameter $Ω_M$ which characterizes the charge-exchange monopole strength distribution. The proportionality coefficient in this relation is determined by basic properties of the ground state of the even-even parent nucleus, and should be reliably calculable in any realistic nuclear model. Therefore, the single parameter $Ω_M$ contains al…
We investigate the radial-overlap part of the isospin-symmetry breaking correction to superallowed $0^+\to 0^+$-decay using the shell-model approach similar to that of Refs. [1, 2]. The 8 sd-shell emitters with masses between $A=22$ and $A=38$ have been re-examined. The Fermi matrix element is evaluated with realistic spherical single-particle wave functions, obtained from spherical Woods-Saxon (WS) or Hartree-Fock (HF) potentials, fine-tuned to reproduce the experimental data on charge radii and separation energies for nuclei of interest. The elaborated adjustment procedure removes any sensit…
Influence of configuration-interaction on isospin impurities and isospin symmetry breaking corrections to superallowed $0^+\rightarrow 0^+$ beta decays
University of Warsaw（华沙大学）；The University of Manchester（曼彻斯特大学）；Akal University（阿卡尔大学）；Osaka University（大阪大学）
We study the formalism to include isospin-symmetry-breaking corrections when extracting the up-down Cabibbo-Kobayashi-Maskawa matrix element from superallowed 0+ -> 0+ nuclear beta decay. We show that there are no first order isospin-symmetry-breaking corrections to the relevant nuclear matrix elements. We find corrections to the treatment of Towner and Hardy, and assess these using schematic models of increasing complexity.
Based on an exact formalism, we study the effects of isospin-symmetry-breaking interactions on superallowed 0+ -> 0+ transitions. We calculate the second-order renormalization of the Fermi matrix element due to radial contributions and show that radial excitations neglected in the treatment of Towner and Hardy are significant. These are estimated to decrease the isospin-symmetry-breaking corrections. Our results provide a correction term that can be included in existing approaches.
We present a test with which to evaluate the calculated isospin-symmetry-breaking corrections to superallowed 0+-to-0+ nuclear beta decay. The test is based on the corrected experimental Ft values being required to satisfy conservation of the vector current (CVC). When applied to six sets of published calculations, the test demonstrates quantitatively that only one set -- the one based on the shell model with Saxon-Woods radial wave functions -- provides satisfactory agreement with CVC. This test can easily be applied to any sets of calculated correction terms that are produced in future.
Corrections to the superallowed beta decay matrix elements are evaluated in perturbation theory using the notion of the isovector monopole resonance. The calculation avoids the separation into different contributions and thus presents a consistent, systematic and more transparent approach. Explicit expressions for the Coulomb correction as a function of mass number A, are given.
A precision measurement of the gamma yields following the beta decay of 32Cl has determined its isobaric analogue branch to be (22.47^{+0.21}_{-0.19})%. Since it is an almost pure Fermi decay, we can also determine the amount of isospin-symmetry breaking in this superallowed transition. We find a very large value, delta_C=5.3(9)%, in agreement with a shell-model calculation. This result sets a benchmark for isospin-symmetry-breaking calculations and lends support for similarly-calculated, yet smaller, corrections that are currently applied to 0+ -> 0+ transitions for tests of the Standard Mode…
Background: The superallowed beta-decay rates provide stringent constraints on physics beyond the Standard Model of particle physics. To extract crucial information about the electroweak force, small isospin-breaking corrections to the Fermi matrix element of superallowed transitions must be applied. Purpose: We perform systematic calculations of isospin-breaking corrections to superallowed beta-decays and estimate theoretical uncertainties related to the basis truncation, time-odd polarization effects related to the intrinsic symmetry of the underlying Slater determinants, and to the function…
The most precise value of V_{ud}, which is obtained from superallowed nuclear beta decay, leads to a violation of CKM unitarity by 2.2 sigma. Experiments are underway on two continents to test and improve this result through decay studies of odd-odd N = Z nuclei with A >or= 62. We show, in a series of illustrative shell-model calculations, that numerous weak Gamow-Teller branches are expected to compete with the superallowed branch in each of these nuclei. Though the total Gamow-Teller strength is significant, many of the individual branches will be unobservably weak. Thus, new techniques must…
We report the first branching-ratio measurement of the superallowed 0+-to-0+ beta transition from 38Ca. The result, 0.7728(16), leads to an ft value of 3062.3(68)s with a relative precision of +/-0.2%. This makes possible a high-precision comparison of the ft values for the mirror superallowed transitions, 38Ca-to-38mK and 38mK-to-38Ar, which sensitively tests the isospin symmetry-breaking corrections required to extract Vud, the up-down quark-mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, from superallowed beta decay. The result supports the corrections currently used, and poin…
The self-consistent random phase approximation (RPA) approach with the residual interaction derived from a relativistic point-coupling energy functional is applied to evaluate the isospin symmetry-breaking corrections δc for the 0+\to0+ superallowed Fermi transitions. With these δc values, together with the available experimental ft values and the improved radiative corrections, the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix is examined. Even with the consideration of uncertainty, the sum of squared top-row elements has been shown to deviate from the unitarity condition by 0.1% fo…
We report new shell-model calculations of the isospin-symmetry-breaking correction to superallowed nuclear beta decay. The most important improvement is the inclusion of core orbitals, which are demonstrated to have a significant impact on the mismatch in the radial wave functions of the parent and daughter states. We determine which core orbitals are important to include from an examination of measured spectroscopic factors in single-nucleon pick-up reactions. We also examine the new radiative-correction calculation by Marciano and Sirlin and, by a simple reorganization, show that it is possi…
The results of systematic calculations of isospin-symmetry-breaking corrections to superallowed beta-decays based on the self-consistent isospin- and angular-momentum-projected nuclear density functional theory (DFT) are reviewed with an emphasis on theoretical uncertainties of the model. Extensions of the formalism towards no core shell model approach with basis cutoff scheme dictated by the self-consistent particle-hole DFT solutions will be also discussed.
Superallowed $β$-decay studies provide some of the best constraints on the possibility of additional quark generations, as well as limits on exotic currents in the weak interaction. The three experimental quantities that are required for performing these tests using $0^+\rightarrow0^+$ nuclear decays (branching ratio, half-life, and $Q$-value) can all be measured to high-precision with rare-isotope beams at the TRIUMF-ISAC facility in Vancouver, Canada. This proceeding presents a brief outline of the general experimental techniques used at TRIUMF over the past 15 years, as well as recent theor…
Prompted by recent measurements, we have surveyed world data and calculated radiative and isospin-symmetry breaking corrections for the superallowed 0+ -to- 0+ Fermi transitions from 42Ti, 46Cr, 50Fe and 54Ni. This increases the number of such transition with a complete set of calculated corrections from 20 to 23. The results are compared with their equivalents for the mirror superallowed transitions from 42Sc, 46V, 50Mn and 54Co. The predicted ft-value asymmetries of these mirror pairs are sensitive to the correction terms and provide motivation for improving measurement precision so as to be…
The radial overlap correction, $δ_{RO}$ is re-examined for 30 superallowed $β$ decays using the shell model with Hartree-Fock (HF) radial wave functions. Our calculation is based on the Skyrme interaction including Coulomb, Charge-Symmetry-Breaking (CSB) and Charge-Independence-Breaking (CIB) terms. The corrections due to the gradient of charge density, vacuum polarization, Coulomb spin-orbit and finite size of nucleons are also considered. To avoid the spurious isospin mixing, the local equivalent potential is constructed from the solution of a HF calculation for the $Z=N$ nucleus without cha…
We lay out a novel formalism to connect the isospin-symmetry breaking correction to the rates of superallowed nuclear beta decays, $δ_\text{C}$, to the isospin-breaking sensitive combinations of electroweak nuclear radii that can be accessed experimentally. We individuate transitions in the superallowed decay chart where a measurement of the neutron skin of a stable daughter even at a moderate precision could already help discriminating between models used to compute $δ_\text{C}$. We review the existing experimental situation and make connection to the existing and future experimental programs…
We present the self-consistent, non-perturbative analysis of isospin mixing using the nuclear density functional approach and the rediagonalization of the Coulomb interaction in the good-isospin basis. The largest isospin-breaking effects are predicted for N = Z nuclei and they quickly fall with the neutron excess. The unphysical isospin violation on the mean-field level, caused by the neutron excess, is eliminated by the proposed method. We find a significant dependence of the magnitude of isospin breaking on the parametrization of the nuclear interaction term. A rough correlation has been fo…
Quantum Monte Carlo calculation of $δ_C$ in the superallowed beta decay of $^{10}$C
Washington University in Saint Louis（圣路易斯华盛顿大学）；Argonne National Laboratory（阿贡国家实验室）；INFN-TIFPA Trento Institute for Fundamental Physics and Applications（特伦托基础物理与应用研究所 (INFN-TIFPA)）；Instituto de Física Corpuscular (IFIC), Consejo Superior de Investigaciones Científicas (CSIC) and Universidad de Valencia（瓦伦西亚大学与西班牙高等科学研究理事会联合粒子物理研究所 (IFIC)）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
In this work we present the first steps towards benchmarking isospin symmetry breaking in ab initio nuclear theory for calculations of superallowed Fermi $β$-decay. Using the valence-space in-medium similarity renormalization group, we calculate b and c coefficients of the isobaric multiplet mass equation, starting from two different Hamiltonians constructed from chiral effective field theory. We compare results to experimental measurements for all T=1 isobaric analogue triplets of relevance to superallowed $β$-decay for masses A=10 to A=74 and find an overall agreement within approximately 25…
Background: Models to calculate small isospin-symmetry-breaking effects in superallowed Fermi decays have been placed under scrutiny in recent years. A stringent test of these models is to measure transitions for which the correction is predicted to be large. The decay of 32Cl decay provides such a test case. Purpose: To improve the gamma yields following the beta decay of 32Cl and to determine the ft values of the the beta branches, particularly the one to the isobaric-analogue state in 32S. Method: Reaction-produced and recoil-spectrometer-separated 32Cl is collected in tape and transported …
This paper overviews various phenomena related to the concept of isospin symmetry. The focus is on N~Z nuclei, which are excellent laboratories of isospin physics. The theoretical framework applied is nuclear Density Functional Theory and its isospin- and angular-momentum projected extensions, as well as symmetry-projected multi-reference models. The topics covered include: isospin impurities, superallowed beta decays, beta-transitions in mirror nuclei, isospin breaking hadronic interactions, mirror and triplet binding energy differences, and isoscalar pairing.
We present results of systematic calculations of the isospin-symmetry-breaking corrections to the superallowed I=$0+,T=1 --> I=0+,T=1 beta-decays, based on the self-consistent isospin- and angular-momentum-projected nuclear density functional theory (DFT). We discuss theoretical uncertainties of the formalism related to the basis truncation, parametrization of the underlying energy density functional, and ambiguities related to determination of Slater determinants in odd-odd nuclei. A generalization of the double-projected DFT model towards a no core shell-model-like configuration-mixing appro…
A direct $Q_{EC}$-value measurement of the superallowed $β^+$ emitter $^{22}$Mg was performed using TRIUMF's Ion Trap for Atomic and Nuclear science (TITAN). The direct ground-state to ground-state atomic mass difference between $^{22}$Mg and $^{22}$Na was determined to be $Q_{EC}=4781.40(22)$~keV, representing the most precise single measurement of this quantity to date. In a continued push towards calculating superallowed isospin-symmetry-breaking (ISB) corrections from first principles, ab-initio shell-model calculations of the $A=22$ IMME are also presented for the first time using the val…
Superallowed $0^+ \to 0^+$ nuclear beta decay provides a direct measure of the weak vector coupling constant, $\GV$. We survey current world data on the nine accurately determined transitions of this type, which range from the decay of $^{10}$C to that of $^{54}$Co, and demonstrate that the results confirm conservation of the weak vector current (CVC) but differ at the 98% confidence level from the unitarity condition for the Cabibbo-Kobayashi-Maskawa (CKM) matrix. We examine the reliability of the small calculated corrections that have been applied to the data, and conclude that there are no …
The measured $ft$-values for superallowed $0^{+} \to 0^{+}$ nuclear $β$-decay can be used to obtain the value of the vector coupling constant and thus to test the unitarity of the Cabibbo-Kobayashi-Maskawa matrix. An essential requirement for this test is accurate calculations for the radiative and isospin symmetry-breaking corrections that must be applied to the experimental data. We present a new and consistent set of calculations for the nuclear-structure-dependent components of these corrections. These new results do not alter the current status of the unitarity test -- it still fails by m…
The question of isospin-symmetry breaking in nuclei of the $sd$ shell is addressed. We propose a new global parameterization of the isospin-nonconserving (INC) shell-model Hamiltonian which accurately describes experimentally known isobaric mass splittings. The isospin-symmetry violating part of the Hamiltonian consists of the Coulomb interaction and effective charge-dependent forces of nuclear origin. Particular attention has been paid to the effect of the short-range correlations. The behavior of $b$ and $c$ coefficients of the isobaric-mass-multiplet equation (IMME) is explored in detail. I…
The recently observed $α$-decay chain ${}^{108}\text{Xe}\to{}^{104}\text{Te}\to{}^{100}\text{Sn}$ [K.~Auranen \emph{et al.}, Phys.\ Rev.\ Lett.\ {\bf121}, 182501 (2018)] could provide valuable information on the $α$ clustering in even-even nuclei slightly above the major shells $Z=50$ and $N=50$. In this work, this $α$-decay chain is studied theoretically within the framework of the density-dependent cluster model plus the two-potential approach. We calculate the $α$-decay half-lives of ${}^{104}$Te and ${}^{108}$Xe and study in detail their dependence on the $Q$ value and the density-profile …
\textbf{Background}: Superallowed $0^+ \rightarrow 0^+$ $β$ decays of isospin $T=2$ nuclides can be used to test theoretical isospin symmetry breaking corrections applied to extract the CKM matrix element $V_{ud}$ from $T = 0,1$ decays by measuring precise $ft$ values and also to search for scalar currents using the $β-ν$ angular correlation. Key ingredients include the $Q_{\textrm{EC}}$ value and branching of the superallowed transition and the half life of the parent. \textbf{Purpose}: To determine a precise experimental $Q_{\textrm{EC}}$ value for the superallowed $0^+ \rightarrow 0^+$ $β$ …
We present the isospin- and angular-momentum-projected nuclear density functional theory (DFT) and its applications to the isospin-breaking corrections to the superallowed beta-decay rates in the vicinity of the N=Z line. A preliminary value obtained for the Cabbibo-Kobayashi-Maskawa matrix element, |V_{ud}|=0.97463(24), agrees well with the recent estimate by Towner and Hardy [Phys. Rev. C{\bf 77}, 025501 (2008)]. We also discuss new opportunities to study the symmetry energy by using the isospin-projected DFT.
In this work, the Coulomb mixing and the isospin mixing in the ground states of even-even nuclei are evaluated in perturbation theory. The calculation of the isospin mixing is performed by using the connection to isovector monopole resonance properties. The uncertainty in the results that depends on different choices of the Skyrme interactions is shown. While Coulomb mixing turns out to be large in the ground states of heavy nuclei, isospin mixing is very small.
To understand the role of isospin effects on fragmentation due to the collisions of charge asymmetric nuclei, we have performed a complete systematical study using isospin dependent quantum molecular dynamics model. Here simulations have been carried out for $^{124}X_{n}+ ^{124}X_{n}$, where n varies from 47 to 59 and for $^{40}Y_{m}+ ^{40}Y_{m}$, where m varies from 14 to 23. Our study shows that isospin dependent cross-section shows its influence on fragmentation in the collision of neutron rich nuclei.
Mass and isospin dependence of symmetry energy coefficients $a_{sym}$ of finite nuclei is investigated with the measured nuclear masses incorporating the liquid drop mass formula. The enhanced $a_{sym}$ for nearly symmetric nuclei are observed. To describe the mass and isospin dependence of $a_{sym}$, a modified formula based on the conventional surface-symmetry term is proposed and the corresponding rms deviation of nuclear masses is checked.
The relations between the strengths of spin-isospin transition operators extracted from direct nuclear reactions, magnetic scattering of electrons and processes of semi-leptonic weak interactions are discussed.
Isospin singlet (pn) pairing as well as quartetting in nuclei is expected to arise near the symmetry line $N=Z$. Empirical values can be deduced from the nuclear binding energies applying special filters. Within the local density approximation, theoretical estimates for finite nuclei are obtained from results for the condensation energy of asymmetric nuclear matter. It is shown that the isospin singlet condensation energy drops down abruptly for |N-Z|~4 for medium nuclei in the region A=40. Furthermore, alpha-like quartetting and the influence of excitations are discussed.
We present a new procedure of isospin symmetry breaking of cooper-quartets in nuclear matter using Ginzburg-Landau theory, namely we couple the cooper-quartets to the pion-meson field and $SU(2)$ gluons. The paper also describes phenomena resulting from this approach.
The magic numbers in exotic nuclei are discussed, and their novel origin is shown to be the spin-isospin dependent part of the nucleon-nucleon interaction in nuclei. The importance and robustness of this mechanism is shown in terms of meson exchange, G-matrix and QCD theories. In neutron-rich exotic nuclei, magic numbers such as N = 8, 20, etc. can disappear, while N = 6, 16, etc. arise, affecting the structure of lightest exotic nuclei to nucleosynthesis of heavy elements.
It is generally believed that isospin would diminish in its importance as we go towards heavy mass region due to isospin mixing caused by the growing Coulomb forces. However, it was realized quite early that isospin could become an important and useful quantum number for all nuclei including heavy nuclei due to neutron richness of the systems~\cite{robson}. Lane and Soper~\cite{lane} also showed in a theoretical calculation that isospin indeed remains quite good in heavy mass neutron rich systems. In this paper, we present isospin based calculations~\cite{jain,swati} for the fission fragment d…
For nuclei with N = Z, the isospin degree of freedom is important and, for deformed systems, rotational bands of different isospin may be expected at low excitation energies. We have investigated, in a simple model space, the influence of the isospin-breaking Coulomb interaction on the degree of chaoticity of these rotational bands. The statistical measures used rely on an analysis of level-spacing distributions, which are extremely difficult to measure experimentally. We show, however, that the overlap intergrals between states of similar frequency reflect well the degree of chaoticity. This …
This work presents an approach to nuclear spin-isospin response, which is capable of describing the overall strength distribution up to high excitation energies, including the fine structure of the low-lying strength, and resolving the long-standing quenching problem. The model is a covariant realization of the nuclear field theory and based on the self-consistent extensions of the covariant energy density functional (CEDF) theory. Results of the recent calculations for spin-isospin response of ordinary and exotic medium-mass nuclei are presented and discussed.
By the use of the latest experimental data on the spectra of $^{133}$Sb and $^{131}$Sn and on the analysis of properties of other odd nuclei adjacent to doubly magic closed shells the isospin dependence of a mean spin-orbit potential is defined. Such a dependence received the explanation in the framework of different theoretical approaches.
The one- plus two-body isospin nonconserving nuclear interactions are included in the prediction of ground state energies of fp shell nuclei using spectral distribution theory. This in turn is used to calculate the linear term in the isobaric mass-multiplet equation and the predictions are then compared to experimental values after the addition of the Coulomb contribution. The agreement is found to be reasonable as observed for sd shell nuclei earlier. One also sees that in this method the contribution to the linear term comes almost completely from the one body isovector Hamiltonian and that …
A detailed study fusion of neutron-rich colliding nuclei is performed using various isospin dependent potentials. For present study, Three different series namely, Ne-Ne, Ca-Ca, and Zr-Zr are taken into account and N/Z ratio.A monotonous increase (decrease) in the fusion barrier positions (heights) using a unified second order nonlinear parametrization in the normalized fusion barrier positions and heights with (N/Z-1) is presented. These predications are in good agreement with the available theoretical as well as experimental results.
Stability of nuclei beyond the drip lines in the presence of an enveloping gas of nucleons and electrons, as prevailing in the inner crust of a neutron star, is studied in the temperature-dependent Thomas-Fermi framework. A limiting asymmetry in the isospin space beyond which nuclei cannot exist emerges from the calculations. The ambient conditions like temperature, baryon density and neutrino concentration under which these exotic nuclear systems can be formed are studied in some detail.
Nuclear many-body theory is used to study nuclear matter and finite nuclei at extreme isospin. In-medium interactions in asymmetric nuclear matter are obtained from (Dirac-) Brueckner theory. Neutron skin formation in Ni and Sn isotopes is investigated by relativistic mean-field calculations in DDRH theory with density dependent meson-nucleon vertices. Applications to light nuclei are discussed with special emphasis on pairing and core polarization in weakly bound nuclei. Approaches accounting for continuum coupling in dripline pairing and core polarization are presented. Calculations for the …
