In this work we present the first steps towards benchmarking isospin symmetry breaking in ab initio nuclear theory for calculations of superallowed Fermi $\beta$ 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 analog triplets of relevance to superallowed $\beta$ decay for masses $A=10$ to $A=74$ and find an overall agreement within approximately 250 keV of experimental data for both $b$ and $c$ coefficients. A greater level of accuracy, however, is obtained by a phenomenological Skyrme interaction or a classical charged-sphere estimate. Finally, we show that evolution of the valence-space operator does not meaningfully improve the quality of the coefficients with respect to experimental data, which indicates that higher-order many-body effects are likely not responsible for the observed discrepancies.

We present a detailed account of neutron capture experiments of astrophysical relevance on ${}^{78,80,84,86}\mathrm{Kr}(n,\gamma )$ reactions at the border between weak and main $s$ process. The experiments were performed with quasi-Maxwellian neutrons from the Liquid-Lithium Target (LiLiT) and the mA-proton beam at 1.93 MeV (2–3 kW) of the Soreq Applied Research Accelerator Facility (SARAF). The setup yields high-intensity $\approx 40$ keV quasi-Maxwellian neutrons (3–5 $\times {10}^{10} n$/s) closely reproducing the conditions of $s$-process stellar nucleosynthesis. A sample of 100 mg of atmospheric, pre-nuclear-age Kr gas contained in a Ti spherical shell was activated in the LiLiT neutron field. The abundances of long-lived Kr isotopes (${}^{81,85g}\mathrm{Kr}$) were measured by atom counting via atom trap trace analysis (ATTA) at Argonne National Laboratory and low-level counting (LLC) at University of Bern. This work is the first measurement of a nuclear cross section using atom counting via ATTA. The activities of short-lived Kr isotopes (${}^{79,85m,87}\mathrm{Kr}$) were measured by $\gamma$-decay counting with a high-purity germanium detector. Maxwellian-averaged cross sections for $s$-process thermal energies are extracted. By comparison to reference values, our nucleosynthesis network calculations show that the experimental cross sections have a strong impact on calculated abundances of krypton and neighboring nuclides, in some cases improving agreement between theory and observations.

We discover new, hitherto unknown, shadow poles in Brune's alternative parametrization of $R$-matrix theory [Brune, Phys. Rev. C 66, 044611 (2002)]. Where these poles are, and how many there are, depends on how one continues $R$-matrix operators to complex wave numbers (especially the shift $S$ and penetration $P$ functions). This has little consequence for the exact $R$-matrix formalism (past the last energy threshold), as we show one can still always fully reconstruct the scattering matrix with only the previously known alternative parameters (poles and corresponding resonance widths), for which there were as many poles as the number of levels $N_{\lambda }$. However, we generalize the alternative parametrization to the Reich-Moore formalism and show that the choice of continuation is now critical as it changes the alternative parameters values (poles and residue widths are now complex). In order to establish nuclear libraries with alternative parameters, the nuclear community will thus have to decide what convention to adopt. We argue in favor of analytical continuation (against the legacy of Lane and Thomas approach) in the second article [P. Ducru, B. Forget, V. Sobes, G. Hale, and M. Paris, Phys. Rev. C 103, 064609 (2021)] of this trilogy on pole parametrizations of $R$-matrix theory, on which we build to establish the windowed multipole representation or $R$-matrix cross sections in the third and last article [P. Ducru, A. Alhajri, I. Meyer, B. Forget, V. Sobes, C. Josey, and J. Liang, Phys. Rev. C 103, 064610 (2021)]. We observe the first evidence of shadow poles in the alternative parametrization of $R$-matrix theory in the isotope ${}^{134}\mathrm{Xe}$ spin-parity group $J^{\pi }=1/2^{(-)}$ and show how they indeed depend on the choice of continuation to complex wave numbers.

First generation, or Population III, stars have a different evolution than those of later generations owing to their initial primordial abundance composition. Most notably, the lack of carbon, oxygen, and nitrogen means that primordial massive stars must rely on the less efficient $p\text{−}p$ chains, thereby requiring the star to contract to reach temperatures high enough to eventually trigger $3\alpha$ reactions. Even small amounts of the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reactions begin feeding the CNO mass range and enable the CNO cycle to generate energy, but this occurs at higher temperature compared to later stellar generations. It is currently controversial if the observed enhanced abundances of Ca in the most metal-poor stars could be a result of the high temperature H-burning conditions in the first massive stars. The level of this enrichment depends on the hot breakout path from the CNO cycles via the ${}^{19}\mathrm{F}(p,\gamma ){}^{20}\mathrm{Ne}$ reaction. In this work, the rates of both the ${}^{19}\mathrm{F}(p,\gamma ){}^{20}\mathrm{Ne}$ and competing ${}^{19}\mathrm{F}(p,\alpha ){}^{16}\mathrm{O}$ reactions are re-evaluated using the phenomenological $R$-matrix approach, simultaneously considering several ${}^{19}\mathrm{F}(p,\gamma ){}^{20}\mathrm{Ne}$, ${}^{19}\mathrm{F}(p,\alpha ){}^{16}\mathrm{O}$, and ${}^{19}\mathrm{F}(p,p){}^{19}\mathrm{F}$ data sets, to better characterize the rate uncertainties. It is found that the rate uncertainty for ${}^{19}\mathrm{F}(p,\gamma ){}^{20}\mathrm{Ne}$ reaction is considerably larger than previously reported. This is the result of undetermined interferences between observed resonances, a possible threshold state, possible subthreshold states, direct capture, and background levels. Additional experimental measurements are therefore needed to determine if ${}^{19}\mathrm{F}(p,\gamma ){}^{20}\mathrm{Ne}$ CNO breakout is responsible for Ca enrichment in metal-poor stars. Astrophysically, the breakout reaction revision makes it less likely that Ca observed in the most Fe-poor stars can originate in hot CNO breakout H-burning nucleosynthtesis, thereby casting doubt on the prevailing faint supernova scenario to explain the abundances observed in these stars.

We present a systematic investigation of few-nucleon systems and light nuclei using the current Low Energy Nuclear Physics International Collaboration interactions comprising semilocal momentum-space regularized two- and three-nucleon forces up to third chiral order (${\mathrm{N}}^2\mathrm{L}\mathrm{O}$). Following our earlier study utilizing the coordinate-space regularized interactions, the two low-energy constants entering the three-body force are determined from the triton binding energy and the differential cross-section minimum in elastic nucleon-deuteron scattering. Predictions are made for selected observables in elastic nucleon-deuteron scattering and in the deuteron breakup reactions, for properties of the $A=3$ and $A=4$ nuclei, and for spectra of $p$-shell nuclei up to $A=16$. A comprehensive error analysis is performed including an estimation of correlated truncation uncertainties for nuclear spectra. The obtained predictions are generally found to agree with experimental data within errors. Similarly to the coordinate-space regularized chiral interactions at the same order, a systematic overbinding of heavier nuclei is observed, which sets in for $A\sim 10$ and increases with $A$.

We present a precise measurement of the cross section, proton and ${}^3\mathrm{He}$ analyzing powers, and spin correlation coefficient $C_{y,y}$ for $p\text{−}{}^3\mathrm{He}$ elastic scattering near 65 MeV, and a comparison with rigorous four-nucleon scattering calculations based on realistic nuclear potentials and a model with $\mathrm{\Delta }$-isobar excitation. Clear discrepancies are seen in some of the measured observables in the regime around the cross section minimum. Theoretical predictions using scaling relations between the calculated cross section and the ${}^3\mathrm{He}$ binding energy are not successful in reproducing the data. Large sensitivity to the $NN$ potentials and rather small $\mathrm{\Delta }$-isobar effects in the calculated cross section are noticed as different features from those in the deuteron-proton elastic scattering. The results obtained above indicate that $p\text{−}{}^3\mathrm{He}$ scattering at intermediate energies is an excellent tool to explore nuclear interactions not accessible by three-nucleon scattering.

While Josephson-like junctions, transiently established in heavy-ion collisions (${\tau }_{\mathrm{coll}}\approx {10}^{-21}\mathrm{s}$) between superfluid nuclei—and through which Cooper-pair tunneling ($Q$-value $Q_{2n}$) proceeds mainly in terms of successive transfer of entangled nucleons—are deprived of the macroscopic aspects of a supercurrent, they display many of the special effects associated with spontaneous symmetry breaking in gauge space (BCS condensation), which can be studied in terms of individual quantum states and of tunneling of single Cooper pairs. From the results of studies of one- and two-neutron transfer reactions carried out at energies below the Coulomb barrier we estimate the value of the mean-square radius (correlation length) of the nuclear Cooper pair. A quantity related to the largest distance of closest approach for which the absolute two-nucleon tunneling cross section is of the order of the single-particle one. Furthermore, emission of $\gamma$ rays of (Josephson) frequency ${\nu }_J=Q_{2n}/h$ distributed over an energy range $\hslash /{\tau }_{\mathrm{coll}}$ is predicted.

Neutron star (NS) merger ejecta offer viable sites for the production of heavy $r$-process elements with nuclear mass numbers $A\gtrsim 140$. The crucial role of fission recycling is responsible for the robustness of this site against many astrophysical uncertainties. Here, we introduce improvements to our scission-point model, called SPY, to derive the fission fragment distribution for all neutron-rich fissioning nuclei of relevance in $r$-process calculations. These improvements include a phenomenological modification of the scission distance and a smoothing procedure of the distribution. Such corrections lead to much better agreement with experimental fission yields. Those yields are also used to estimate the number of neutrons emitted by the excited fragments on the basis of different neutron evaporation models. Our fission yields are extensively compared to those predicted by the GEF (general description of fission observables) model. The impact of fission on the $r$-process nucleosynthesis in binary neutron mergers is also reanalyzed. Two scenarios are considered, the first one with low initial electron fraction subject to intense fission recycling, in contrast to the second one, which includes weak interactions on nucleons. The various regions of the nuclear chart responsible for fission recycling during the neutron irradiation and after freeze-out are discussed. The contribution fission processes may have to the final abundance distribution is also studied in detail in the light of newly defined quantitative indicators describing the fission recycling, the fission seeds, and the fission progenitors. In particular, those allow us to estimate the contribution of fission to the final abundance distribution stemming from specific heavy nuclei. Calculations obtained with SPY and GEF fission fragment distributions are compared for both $r$-process scenarios.

Background: Many quantal many-body methods that aim at the description of self-bound nuclear or mesoscopic electronic systems make use of auxiliary wave functions that break one or several of the symmetries of the Hamiltonian in order to include correlations associated with the geometrical arrangement of the system's constituents. Such reference states have been used already for a long time within self-consistent methods that are either based on effective valence-space Hamiltonians or energy density functionals, and they are presently also gaining popularity in the design of novel ab initio methods. A fully quantal treatment of a self-bound many-body system, however, requires the restoration of the broken symmetries through the projection of the many-body wave functions of interest onto good quantum numbers.

Purpose: The goal of this work is threefold. First, we want to give a general presentation of the formalism of the projection method starting from the underlying principles of group representation theory. Second, we want to investigate formal and practical aspects of the numerical implementation of particle-number and angular-momentum projection of Bogoliubov quasiparticle vacua, in particular with regard of obtaining accurate results at minimal computational cost. Third, we want to analyze the numerical, computational, and physical consequences of intrinsic symmetries of the symmetry-breaking states when projecting them.

Methods: Using the algebra of group representation theory, we introduce the projection method for the general symmetry group of a given Hamiltonian. For realistic examples built with either a pseudopotential-based energy density functional or a valence-space shell-model interaction, we then study the convergence and accuracy of the quadrature rules for the multidimensional integrals that have to be evaluated numerically and analyze the consequences of conserved subgroups of the broken symmetry groups.

Results: The main results of this work are also threefold. First, we give a concise, but general, presentation of the projection method that applies to the most important potentially broken symmetries whose restoration is relevant for nuclear spectroscopy. Second, we demonstrate how to achieve high accuracy of the discretizations used to evaluate the multidimensional integrals appearing in the calculation of particle-number and angular-momentum projected matrix elements while limiting the order of the employed quadrature rules. Third, for the example of a point-group symmetry that is often imposed on calculations that describe collective phenomena emerging in triaxially deformed nuclei, we provide the group-theoretical derivation of relations between the intermediate matrix elements that are integrated, which permits a further significant reduction of the computational cost of the method. These simplifications are valid regardless of the number parity of the quasiparticle states and therefore can be used in the description of even-even, odd-mass, and odd-odd nuclei.

Conclusions: The quantum-number projection technique is a versatile and efficient method that permits to restore the symmetry of any arbitrary many-body wave function. Its numerical implementation is relatively simple and accurate. In addition, it is possible to use the conserved symmetries of the reference states to reduce the computational burden of the method. More generally, the ever-growing computational resources and the development of nuclear ab-initio methods opens new possibilities of applications of the method.

We calculate neutron multiplicities from fission fragments with specified mass numbers for events having a specified total fragment kinetic energy. The shape evolution from the initial compound nucleus to the scission configurations is obtained with the metropolis walk method on the five-dimensional potential-energy landscape, calculated with the macroscopic-microscopic method for the three-quadratic-surface shape family. Shape-dependent microscopic level densities are used to guide the random walk, to partition the intrinsic excitation energy between the two proto-fragments at scission, and to determine the number of neutrons evaporated from the fragments. The contribution to the total excitation energy of the resulting fragments from statistical excitation and shape distortion at scission is studied. Good agreement is obtained with available experimental data on neutron multiplicities in correlation with fission fragments from ${}^{235}\mathrm{U}(n_{\mathrm{th}},f$). With increasing neutron energy a superlong fission mode grows increasingly prominent, which affects the dependence of the observables on the total fragment kinetic energy.

Background: Microscopic description of spontaneous fission is one of the most challenging subjects in nuclear physics. It is necessary to evaluate the collective potential and the collective inertia along a fission path for a description of quantum tunneling in spontaneous or low-energy fission. In past studies of the fission dynamics based on nuclear energy density functional (EDF) theory, the collective inertia has been evaluated with the cranking approximation, which neglects dynamical residual effects.

Purpose: The purpose is to provide a reliable and efficient method to include dynamical residual effects in the collective inertia for fission dynamics.

Methods: We use the local quasiparticle random-phase approximation (LQRPA) to evaluate the collective inertia along a fission path obtained by the constrained Hartree-Fock-Bogoliubov method with the Skyrme EDF. The finite-amplitude method (FAM) with a contour integration technique enables us to efficiently compute the collective inertia in a large model space.

Results: We evaluate the FAM-QRPA collective inertia along a symmetric fission path in ${}^{240}\mathrm{Pu}$ and ${}^{256}\mathrm{Fm}$. The FAM-QRPA inertia is significantly larger than the one of the cranking approximation and shows pronounced peaks around the ground state and the fission isomer. This is due to dynamical residual effects.

Conclusions: To describe the spontaneous or low-energy fission, we provide a reliable and efficient method to construct the collective inertia with dynamical residual effects that have been neglected in most of EDF-based works in the past. We show the importance of dynamical residual effects to the collective inertia. This work will be a starting point for a systematic study of fission dynamics in heavy and superheavy nuclei to microscopically describe the nuclear large-amplitude collective motions.

We study two different methods to prepare excited states on a quantum computer, a key initial step to study nuclear dynamics within linear-response theory. The first method uses unitary evolution for a short time $T=\mathcal{O}\left(\sqrt{1-F}\right)$ to approximate the action of an excitation operator $\widehat{O}$ with fidelity $F$ and success probability $P\approx 1-F$. The second method probabilistically applies the excitation operator using the linear combination of unitaries (LCU) algorithm. We benchmark these techniques on emulated and real quantum devices, using a toy model for thermal neutron-proton capture. Despite its larger-memory footprint, the LCU-based method is efficient even on current generation noisy devices and can be implemented at a lower gate cost than a naive analysis would suggest. These findings show that quantum techniques designed to achieve good asymptotic scaling on fault-tolerant quantum devices might also provide practical benefits on devices with limited connectivity and gate fidelity.

Background: Well-established coupled-channel (CC) and coupled reaction channel (CRC) processes make contributions to elastic scattering that are absent from local density folding models. Little is known concerning the contribution of these processes to ${}^3\mathrm{H}$ optical model potentials. For studying such processes, spin-saturated closed-shell nuclei such as ${}^{16}\mathrm{O}$ and ${}^{40}\mathrm{Ca}$ are particularly suitable target nuclei and the (${}^3\mathrm{H},{}^4\mathrm{He}$) reaction is easily handled within conventional reaction theory since it avoids complications such as breakup.

Purpose: To establish and characterize the contribution to the ${}^3\mathrm{H}$-nucleus interaction generated by coupling to proton pickup (outgoing ${}^4\mathrm{He}$) channels; we also study the contribution of collective states and identify effects of dynamical nonlocality due to these couplings.

Methods: CRC calculations, with CC coupling to collective states, provide the elastic channel $S$ matrix $S_{lj}$ resulting from the included processes. Inversion of $S_{lj}$ produces a local potential that yields, in a single channel calculation, the elastic-scattering observables from the CC/CRC calculation. Subtracting the bare potential yields a local and $l$-independent representation of the dynamical polarization potential (DPP). From the DPPs due to a range of channel couplings the influence of dynamically generated nonlocality can be identified.

Results: Coupling to ${}^4\mathrm{He}$ channels systematically induces repulsion and absorption in the ${}^3\mathrm{H}$ optical model potential (OMP) and generally a reduction in the rms radius of the real part. The qualitative effects, including the general undularity of the DPPs, are similar for all cases. Such coupling cannot be represented by renormalizing folding model potentials. Evidence is presented for substantial dynamical nonlocality of the induced DPPs. Local equivalent DPPs for individual couplings cannot be added to give the DPP for multiple couplings.

Conclusions: The DPPs presented here further challenge the notion that local density folding models provide a satisfactory description of elastic scattering from nuclei. Coupling to proton pickup channels induces dynamical nonlocality in the ${}^3\mathrm{H}$ OMP with implications for direct reactions involving ${}^3\mathrm{H}$. Departures from a smooth radial form for the ${}^3\mathrm{H}$ OMP should be found in good fits to suitable elastic-scattering data.

We present high-precision theoretical predictions for the electron energy spectra for the ground-state to ground-state $\beta$ decays of ${}^{214}\mathrm{Pb}$, ${}^{212}\mathrm{Pb}$, and ${}^{85}\mathrm{Kr}$ most relevant to the background of liquid xenon dark matter detectors. The effects of nuclear structure on the spectral shapes are taken into account using large-scale shell-model calculations. Final spectra also include atomic screening and exchange effects. The impact of nuclear structure effects on the ${}^{214}\mathrm{Pb}$ and ${}^{212}\mathrm{Pb}$ spectra below $\approx 100$ keV, pertinent for several searches for new physics, are found to be comparatively larger than those from the atomic effects alone. We find that the full calculation for ${}^{214}\mathrm{Pb}$ (${}^{212}\mathrm{Pb}$) predicts 15.0%–23.2% (12.1%–19.0%) less event rate in a 1–15 keV energy region of interest compared to the spectrum calculated as an allowed transition when using values of the weak axial vector coupling in the range $g_{\mathrm{A}}=0.7$–1.0. The discrepancy highlights the importance of both a proper theoretical treatment and the need for direct measurements of these spectra for a thorough understanding of $\beta$ decay backgrounds in future experiments.

Recent observations of neutron stars with gravitational waves and x-ray timing provide unprecedented access to the equation of state (EoS) of cold dense matter at densities difficult to realize in terrestrial experiments. At the same time, predictions for the EoS equipped with reliable uncertainty estimates from chiral effective field theory ($\chi \text{EFT}$) allow us to bound our theoretical ignorance. In this work, we analyze astrophysical data by using a nonparametric representation of the neutron-star EoS conditioned on $\chi \text{EFT}$ to directly constrain the underlying physical properties of the compact objects without introducing modeling systematics. We discuss how the data alone constrain the EoS at high densities when we condition on $\chi \text{EFT}$ at low densities. We also demonstrate how to exploit astrophysical data to directly test the predictions of $\chi \text{EFT}$ for the EoS up to twice nuclear saturation density, in order to estimate the density at which these predictions might break down. We find that the existence of massive pulsars, gravitational waves from $\mathrm{GW}170817$, and NICER observations of $\text{PSR}\text{J}0030+0451$ favor $\chi \text{EFT}$ predictions for the EoS up to nuclear saturation density over a more agnostic analysis by as much as a factor of seven for the quantum Monte Carlo (QMC) calculations used in this work. While $\chi \text{EFT}$ predictions using QMC are fully consistent with gravitational-wave data up to twice nuclear saturation density, NICER observations suggest that the EoS stiffens relative to these predictions at or slightly above nuclear saturation density. Additionally, for these QMC calculations, we marginalize over the uncertainty in the density at which $\chi \text{EFT}$ begins to break down, constraining the radius of a $1.4M_{\odot }$ neutron star to $R_{1.4}=11.{40}_{-1.04}^{+1.38}$ ($12.{54}_{-0.63}^{+0.71}$) km and the pressure at twice nuclear saturation density to $p\left(2n_{\mathrm{sat}}\right)=14.2_{-8.4}^{+18.1}$ ($28.7_{-15.0}^{+15.3}$) $\mathrm{MeV}/{\mathrm{fm}}^3$ with massive pulsar and gravitational-wave (and NICER) data.

The energy density is calculated in coordinate space for ${}^{12}\mathrm{C}$, ${}^{40}\mathrm{Ca}$, ${}^{48}\mathrm{Ca}$, and ${}^{208}\mathrm{Pb}$ using a dispersive optical model constrained by all relevant data including the corresponding energy of the ground state. The energy density of ${}^8\mathrm{Be}$ is also calculated using the Green's-function Monte Carlo method employing the Argonne-Urbana two- and three-body interactions. The nuclear interior minimally contributes to the total binding energy due to the $4\pi r^2$ phase-space factor. Thus, the volume contribution to the energy in the interior is not well constrained. The dispersive-optical-model energy densities are in good agreement with ab initio self-consistent Green's-function calculations of infinite nuclear matter restricted to treat only short-range and tensor correlations. These results call into question the degree to which the equation of state for nuclear matter is constrained by the empirical mass formula. In particular, the results in this paper indicate that saturated nuclear matter does not require the canonical value of 16-MeV binding per particle but only about 13–14 MeV when the interior of ${}^{208}\mathrm{Pb}$ is considered.

We develop an effective field theory (EFT) for deformed odd-mass nuclei. These are described as an axially symmetric core to which a nucleon is coupled. In the coordinate system fixed to the core the nucleon is subject to an axially symmetric potential. Power counting is based on the separation of scales between low-lying rotations and higher-lying states of the core. In leading order, core and nucleon are coupled by universal derivative terms. These comprise a covariant derivative and gauge potentials which account for Coriolis forces and relate to Berry-phase phenomena. At leading order, the EFT combines the particle-rotor and Nilsson models. We work out the EFT up to next-to-leading order and illustrate the results in ${}^{239}\mathrm{Pu}$ and ${}^{187}\mathrm{Os}$. At leading order, odd-mass nuclei with rotational bandheads that are close in energy and differ by one unit of angular momentum are triaxially deformed. For bandheads that are well separated in energy, triaxiality becomes a subleading effect. The EFT developed in this paper presents a model-independent approach to the particle-rotor system that is capable of systematic improvement.

The direct Urca process of rapid neutrino emission can occur in nonuniform nuclear pasta phases that are expected in the inner crusts of neutron stars. Here, the periodic potential for a nucleon in nuclear pasta allows momentum conservation to be satisfied for direct Urca reactions. We improve on earlier work by modeling a rich variety of pasta phases (gnocchi, waffle, lasagna, and anti-spaghetti) with large-scale molecular dynamics simulations. We find that the neutrino luminosity due to direct Urca reactions in nuclear pasta can be 3 to 4 orders of magnitude larger than that from the modified Urca process in the NS core. Thus neutrino radiation from pasta could dominate radiation from the core and this could significantly impact the cooling of neutron stars.

A new critical survey of all half-life, decay-energy, and branching-ratio measurements related to 23 superallowed $0^{+}\to 0^{+}\beta$ decays is presented. Included are 222 individual measurements of comparable precision obtained from 174 published references. Compared with our last survey in 2015, we have added results from 28 new publications and eliminated an approximately equal number whose results have been superseded by much more precise modern data. We obtain world-average $ft$ values for each of the 21 transitions that have a complete set of data and then apply radiative and isospin-symmetry-breaking corrections to extract “corrected” $\mathcal{F}t$ values. Fifteen of these $\mathcal{F}t$ values now have a precision of $0.3\%$ or better and all take the same value within statistics, as expected from conservation of the vector current. Their average, $\overline{\mathcal{F}t}$, when combined with the muon lifetime, yields the up-down quark-mixing element of the Cabibbo-Kobayashi-Maskawa matrix, $V_{\mathit{ud}}=0.97373\pm 0.00031$. This is lower than our 2015 result by one standard deviation and its uncertainty is increased by $50\%$. This is a consequence not of any shifts in the experimental data but of new calculations for the radiative corrections. The lower $V_{\mathit{ud}}$ value now leads to greater tension in the top-row test of unitarity in the CKM matrix. Updates in experimental data have independently led to a factor-of-two tighter limit being set on the possible existence of a scalar interaction. The new limit on Fierz interference is $b_F\le$ 0.0033 at the $90\%$ confidence level.

Background: Spectroscopic factors, overlaps, and isospin symmetry are often used in conjunction with single-particle wave functions for the phenomenological analysis of nuclear structure and reactions. Many differing prescriptions for connecting these quantities to physically relevant asymptotic normalization constants or widths are available in the literature, but their relationship and degree of validity are not always clear.

Purpose: This paper derives relationships among the above quantities of interest using well-defined methodology and starting assumptions.

Method: $R$-matrix theory is used as the primary tool to interoperate among the quantities of interest to this work. Particular attention is paid to effects arising from beyond the nuclear surface, where isospin symmetry is strongly violated.

Results: Relationships between the quantities of interest are derived. Example applications of these methods to mirror levels in nucleon $+ {}^{12}\mathrm{C}$, nucleon $+ {}^{16}\mathrm{O}$, and nucleon $+ {}^{26}\mathrm{Al}$ are presented. A new approach to multilevel mirror symmetry is derived and applied to the first three $2^{+}$ states of ${}^{18}\mathrm{O}$ and ${}^{18}\mathrm{Ne}$.

Conclusions: The relationship between the quantities of interest is clarified and certain procedures are recommended. It is found that the asymptotic normalization constant of the second $2^{+}$ state in ${}^{18}\mathrm{Ne}$ deduced from the mirror state in ${}^{18}\mathrm{O}$ is significantly larger than found in previous work. This finding has the effect of increasing the ${}^{17}\mathrm{F}(p,\gamma ){}^{18}\mathrm{Ne}$ reaction rate in novae.