Abstract
The dominant decay mode of atomic nuclei is beta decay (βdecay), a process that changes a neutron into a proton (and vice versa). This decay offers a window to physics beyond the standard model, and is at the heart of microphysical processes in stellar explosions and element synthesis in the Universe^{1,2,3}. However, observed βdecay rates in nuclei have been found to be systematically smaller than for free neutrons: this 50yearold puzzle about the apparent quenching of the fundamental coupling constant by a factor of about 0.75 (ref. ^{4}) is without a firstprinciples theoretical explanation. Here, we demonstrate that this quenching arises to a large extent from the coupling of the weak force to two nucleons as well as from strong correlations in the nucleus. We present stateoftheart computations of βdecays from light and mediummass nuclei to ^{100}Sn by combining effective field theories of the strong and weak forces^{5} with powerful quantum manybody techniques^{6,7,8}. Our results are consistent with experimental data and have implications for heavy element synthesis in neutron star mergers^{9,10,11} and predictions for the neutrinoless doubleβdecay^{3}, where an analogous quenching puzzle is a source of uncertainty in extracting the neutrino mass scale^{12}.
Main
Gamow–Teller transitions are a form of βdecay in which the spins of the βneutrino pair emitted during the nuclear decay are aligned. Remarkably, calculated Gamow–Teller strengths appear to reproduce most of the experimental data if the fundamental constant g_{A} ≈ 1.27 characterizing the coupling of the weak interaction to a nucleon is quenched by a factor of q ≈ 0.75 (refs. ^{13,14,15,16}). Missing nuclear correlations (that is, a lack of complexity in nuclear wavefunctions due to the limitations of nuclear models) as well as neglected contributions from mesonexchange currents (that is, coupling of the weak force to two nucleons) have been proposed as possible causes of the quenching phenomenon^{4}. However, a solution has so far remained elusive. To address the quenching puzzle, we carry out a comprehensive study of Gamow–Teller decays through manybody computations of nuclei based on effective field theories (EFTs) of quantum chromodynamics^{5,17}, including an unprecedented amount of correlations in the nuclear wavefunctions. The EFT approach offers the prospect of accuracy, by encoding the excluded highenergy physics through coefficients adjusted to the data, and precision, from the systematically improvable EFT expansion. Moreover, EFT enables a consistent description of the coupling of weak interactions to two nucleons via twobody currents (2BCs). In the EFT approach, 2BCs enter as subleading corrections to the onebody standard Gamow–Teller operator στ^{+} (with Pauli spin and isospin matrices σ and τ, respectively); they are smaller but significant corrections to weak transitions as threenucleon forces are smaller but significant corrections to the nuclear interaction^{5,17}.
In this work we focus on strong Gamow–Teller transitions, where the effects of quenching should dominate over cancellations due to fine details (as occur in the famous case of the ^{14}C decay used for radiocarbon dating^{18,19}). An excellent example is the superallowed βdecay of the doubly magic ^{100}Sn nucleus (Fig. 1), which exhibits the strongest Gamow–Teller strength so far measured in all atomic nuclei^{20}. A firstprinciples description of this exotic decay, in such a heavy nucleus, presents a significant computational challenge. However, its equal ‘magic’ numbers (Z = N = 50) of protons and neutrons arranged into complete shells makes ^{100}Sn an ideal candidate for largescale coupledcluster calculations^{21}, while the daughter nucleus ^{100}In can be reached via novel extensions of the highorder chargeexchange coupledcluster methods developed in this work (see Methods and Supplementary Figs. 4, 12 and 15 for details). This method includes correlations via a vast number of particle–hole excitations of a reference state and also employs 2BCs in the transition operator.
Figure 1 shows our results for the strength (that is, the absolute square of the transition matrix element, M_{GT}) of the Gamow–Teller transition to the dominant J^{π} = 1^{+} state in the ^{100}In daughter nucleus (see Supplementary Table 1 and Supplementary Fig. 12 for more details). To investigate systematic trends and sensitivities to the nuclear Hamiltonian, we employed a family of established EFT interactions and corresponding currents^{22,23,24}. For increased precision, we also developed a new interaction labelled NNN^{4}LO + 3N_{lnl} which is constrained to reproduce the triton halflife (see Methods for details on the Hamiltonians considered). The open symbols in Fig. 1 depict the decay with the standard, leadingorder coupling of the weak force to a single nucleon in the nonrelativistic limit (that is, via the standard Gamow–Teller operator στ^{+}). The differences with respect to the extreme singleparticle model (ESPM), which approximates both ^{100}Sn and its ^{100}In daughter as a single shellmodel configuration, reveals the influence of correlations among the nucleons. The full symbols include 2BCs, using consistent couplings as in the employed EFT interactions. Finally, the partially filled symbols in Fig. 1 represent results from other models from ref. ^{20}, where the standard Gamow–Teller operator has been multiplied by a quenching factor of q ≈ 0.75.
Based on the results shown in Fig. 1, we predict the range \(5.2(5) \lesssim M_{{\mathrm{GT}}}^2 \lesssim 7.0(7)\) for the Gamow–Teller strength. This range overlaps with the evaluation in ref. ^{25}, based on systematic experimental trends in tin isotopes, and the lower end of the measurement in ref. ^{20}. The quenching factor we obtain from 2BCs depends somewhat on the employed Hamiltonian and is in the range q_{2BC} = 0.73–0.85. This range is consistent with the value q = 0.75(2) from ref. ^{25}. In the present work we used the spread of results obtained with the selected set of EFT interactions and 2BCs as an estimate of the systematic uncertainty. A more thorough quantification of the uncertainties associated with the manybody methods and EFT truncations is beyond the scope of this work, and will be addressed in future studies. We note that neglected higherorder correlations in our coupledcluster approach will further reduce the Gamow–Teller strength (see Supplementary Information for details).
Moreover, we observe that the spread for the ^{100}Sn Gamow–Teller strength obtained for the family of EFT interactions used here is significantly reduced (by a factor two) when 2BCs are included. This is consistent with ideas from EFT that the residual cutoff dependence is due to neglected higherorder terms in the Hamiltonian and 2BCs. In addition, we find that the relative contributions to the quenching of the Gamow–Teller strength coming from correlations and 2BCs vary as a function of the resolution scale of the underlying EFT interactions.
Starting from the extreme singleparticle model, and adding first correlations and then the effects of 2BC, we find that the quenching from correlations typically increases with increasing resolution scale of the interaction, and that most of the quenching stems from correlations. However, adding first the effects of the 2BCs and then the correlations shows that the quenching from 2BCs increases with decreasing resolution scale and that most of the quenching stems from 2BCs for all but the ‘hardest’ potentials considered in this work (see Supplementary Fig. 6 for details).
For a comprehensive study, we now turn to βdecays of light and mediummass nuclei. Using a selection of the EFT interactions and 2BCs adopted for ^{100}Sn, we achieved an overall good description of βdecays in light nuclei. Figure 2 shows theorytoexperiment ratios for large Gamow–Teller transitions in light nuclei. Here, we highlight the results obtained for the highprecision NNN^{4}LO + 3N_{lnl} interaction and corresponding 2BCs developed in this work. As detailed in the Methods, the 2BCs and threenucleon forces 3N_{lnl} are parametrized consistently and are constrained to reproduce the empirical value of the triton βdecay halflife. Our calculations were carried out with the nocore shell model (NCSM)^{6}, a virtually exact treatment of correlations in the nuclear wavefunctions (see Methods for details). The role of 2BCs is relatively small in light nuclei with mass numbers A ≤ 7. Full nuclear wavefunctions already provide a rather satisfactory description of the transitions with the standard Gamow–Teller operator. Furthermore, the inclusion of 2BCs may enhance (for example, ^{8}He → ^{8}Li), quench (for example, \(\,{}^7{\mathrm{Be}}_{{\textstyle{3 \over 2}}} \to \,{}^7{\mathrm{Li}}_{\frac{1}{2}}\)), or have virtually no impact on the computed transition (for example, \(\,{}^7{\mathrm{Be}}_{{\textstyle{3 \over 2}}} \to \,{}^7{\mathrm{Li}}_{\frac{3}{2}}\); see also Supplementary Fig. 13). The small role of 2BCs in A ≤ 7 nuclei is similar to what was found in the Green’s function Monte Carlo calculations of ref. ^{26}. We find a rather substantial enhancement of the ^{8}He Gamow–Teller matrix element due to the 2BC. Let us mention, though, that this transition matrix element is the smallest of those presented in Fig. 2. We note that, for the other Hamiltonians employed in this work, the 2BCs and 3N were not fit to reproduce the triton halflife; nevertheless, the inclusion of 2BCs for most of these cases also improves the agreement with data for the light nuclei considered in Fig. 2 (see Supplementary Fig. 9 for results obtained with NNLO_{sat} and NNN^{3}LO + 3N_{lnl}). The case of ^{10}C is special because the computed Gamow–Teller transition is very sensitive to the structure of the J^{π} = 1^{+} state in the ^{10}B daughter nucleus. Depending on the employed interaction, this state can mix with a higherlying 1^{+} state, greatly impacting the precise value of this transition. We finally note that benchmark calculations between the manybody methods used in this work agree to within 5% for the large transition in ^{14}O. For smaller transitions discrepancies can be larger (see Supplementary Information for details).
Historically, the most extensive evidence for the quenching of Gamow–Teller βdecay strength comes from mediummass nuclei^{14,16,27}, and we now show that our calculations with these consistent Hamiltonians and currents largely solve the puzzle here as well. We use the valencespace inmedium similarity renormalization group (VSIMSRG) method^{8} (see Methods for details) and compute Gamow–Teller decays for nuclei in the mass range between oxygen and calcium (referred to as sdshell nuclei) and between calcium and vanadium (lower pfshell nuclei), focusing on strong transitions. Here, we highlight the NNN^{4}LO + 3N_{lnl} interaction and corresponding 2BCs.
Figure 3 shows the empirical values of the Gamow–Teller transition matrix elements versus the corresponding unquenched theoretical matrix elements obtained from the phenomenological shell model with the standard Gamow–Teller στ operator and the firstprinciples VSIMSRG calculations. Perfect agreement between theory and experiment is denoted by the diagonal dashed line. The results from the phenomenological shell model clearly exemplify the state of theoretical calculations for decades^{13,14,15,16,27}; as an example, in the sdshell shell, a quenching factor of q ≈ 0.8 is needed to bring the theory into agreement with experiment^{14}. The VSIMSRG calculations without 2BCs (not shown) exhibit a modest improvement, with a corresponding quenching factor of 0.89(4) for sdshell nuclei and 0.85(3) for pfshell nuclei, pointing to the importance of consistent valencespace wavefunctions and operators (Supplementary Fig. 10). As in ^{100}Sn, the inclusion of 2BCs yields an additional quenching of the theoretical matrix elements, and the linear fit of our results lies close to the dashed line, meaning our theoretical predictions agree, on average, with experimental values across a large number of mediummass nuclei.
Another approach often used in the investigation of Gamow–Teller quenching is the Ikeda sumrule: the difference between the total integrated β^{−} and β^{+} strengths obtained with the \(\sigma \tau ^ \mp\) operator yields the modelindependent sumrule 3(N – Z). We have computed the Ikeda sumrule for ^{14}O, ^{48}Ca and ^{90}Zr using the coupledcluster method (see Methods for details). For the family of EFT Hamiltonians used for ^{100}Sn we obtain a quenching factor arising from 2BCs that is consistent with our results shown in Fig. 3 and the shellmodel analyses from refs. ^{14,15,16,27}. (Supplementary Fig. 7). We note that the comparison with experimental sumrule tests using chargeexchange reactions^{28,29} is complicated by the use of a hadronic probe, which only corresponds to the leading weak onebody operator, and by the challenge of extracting all strength to high energies. Here, our developments enable future direct comparisons.
It is the combined proper treatment of strong nuclear correlations with powerful quantum manybody solvers and the consistency between 2BCs and threenucleon forces that largely explains the quenching puzzle. Smaller corrections are still expected to arise from neglected higherorder contributions to currents and Hamiltonians in the EFT approach we pursued, and from neglected correlations in the nuclear wavefunctions. For beyondstandardmodel searches of new physics such as neutrinoless doubleβdecay, our work suggests that a complete and consistent calculation without a phenomenological quenching of the axialvector coupling g_{A} is called for. This Letter opens the door to ab initio calculations of weak interactions across the nuclear chart and in stars.
Methods
Hamiltonians and model space
In this work we employ the intrinsic Hamiltonian
Here p_{i} is the nucleon momentum, m the average nucleon mass, A the mass number of the nucleus of interest, V_{NN} the nucleon–nucleon (NN) interaction and V_{3N} the threenucleon (3N) interaction.
We use a set of interactions from ref. ^{22} labelled 1.8/2.0 (EM), 2.0/2.0 (EM), 2.2/2.0 (EM), 2.8/2.0 (EM) and 2.0/2.0 (PWA). These consist of a chiral NN interaction at order N^{3}LO from ref. ^{32} evolved to the resolution scales λ_{SRG} = 1.8, 2.0, 2.2 and 2.8 fm^{−1} by means of the similarity renormalization group (SRG)^{33} plus a chiral 3N interaction (unevolved) at order N^{2}LO, using a nonlocal regulator with momentum cutoff Λ_{3N} = 2.0 fm^{−1}. Note that the 2.0/2.0 (PWA) interaction employs different longrange pion couplings in the NN and 3N sectors. The lowenergy couplings entering these interactions were adjusted to reproduce NN scattering data as well as the ^{3}H binding energy and ^{4}He charge radius. With the exception of 2.8/2.0 (EM), this set of interactions was recently used to describe binding energies and spectra of neutronrich nuclei up to ^{78}Ni (refs. ^{34,35}) and of neutrondeficient nuclei around ^{100}Sn (ref. ^{21}). The results with the 1.8/2.0 (EM) interaction, in particular, reproduce groundstate energies very well.
In addition, we also employ the NNLO_{sat} interaction, which was constrained to reproduce nuclear binding energies and charge radii of selected p and sdshell nuclei^{23}. Ab initio calculations based on NNLO_{sat} accurately describe both the radii and binding energies of light and mediummass nuclei^{7}.
Finally, we employ two consistently SRGevolved NN and 3N interactions, namely NNN^{3}LO + 3N_{lnl} (ref. ^{24}) and the NNN^{4}LO + 3N_{lnl} introduced in this work. The NN interactions at N^{3}LO and N^{4}LO are from refs. ^{32,36}, respectively. The 3N interactions 3N_{lnl} use a mixture of local^{37} and nonlocal regulators. The local cutoff is 650 MeV and the nonlocal cutoff of 500 MeV is the same as in the NN interactions. In the case of the NNN^{4}LO + 3N_{lnl}, the parameters of the twopionexchange 3N forces (c_{1}, c_{3} and c_{4}) are shifted with respect to their values in the NN potential following the recommendation of ref. ^{36}. The couplings of the shorterrange 3N forces (c_{D} and c_{E}) are constrained to the binding energies and radii of the triton and ^{4}He in the NNN^{3}LO + 3N_{lnl} model, and to the triton halflife and binding energy in the NNN^{4}LO + 3N_{lnl} model. We note, however, that the NNN^{3}LO + 3N_{lnl} interaction also reproduces the triton halflife as shown in Supplementary Fig. 9. The NN and 3N interactions are consistently SRG evolved to the lower cutoff λ_{SRG} = 2.0 fm^{−1} (or λ_{SRG} = 1.8 fm^{−1} in the case of some of our lightnuclei calculations).
In our NCSM calculations of light nuclei we employ the harmonicoscillator basis varied in the range N_{max} = 4–14 and with frequency ℏω = 20 MeV. In our coupledcluster and VSIMSRG calculations we start from a Hartree–Fock basis built from the harmonicoscillator basis with modelspace parameters in the range N_{max} = 6–14 and ℏω = 12–16 MeV, respectively. Finally, the 3N interaction is truncated to threeparticle energies with \(E_{{\mathrm{3max}}} \le 16\hbar \omega\).
Gamow–Teller transition operator
The rate at which a Gamow–Teller transition will occur is proportional to the square of the reduced transition matrix element
Here, i and f label the initial and final states of the mother and daughter nuclei, respectively. (Note that, throughout this work, we quote the reduced matrix element \(\left\langle {f\left\ {O_{{\mathrm{GT}}}} \right\i} \right\rangle\), using the Edmonds convention^{38}). The transition operator O_{GT} is defined in terms of the J = 1 transverse electric multipole of the chargechanging axialvector current^{39} \({\bf {J}}^A(\vec {\bf {K}})\)
Here, K denotes the momentum transferred to the resulting electron and antineutrino pair (or positron and neutrino in \(\beta ^ +\)decay). Because the change in energy between mother and daughter states is typically very small (a few MeV) compared to other relevant scales, setting \({\bf {K}} = 0\) is a very good approximation that significantly simplifies the calculation of M_{GT}.
The standard (onebody) chargechanging axialvector current is given by
Here \({\bf{r}}_j,\sigma_j,\tau _j^ \pm\) are the position, Pauli spin and chargeraising (lowering) operators for the jth particle. In this work we use axialvector currents derived within the same chiral EFT framework used for the strong interactions, including the leading 2BC^{40}. For a diagrammatic picture of the relevant contributions (at \({\bf{K}} = 0\)) see Supplementary Fig. 5. The leftmost diagram corresponds to the leadingorder onebody current. In addition, we include two classes of 2BC: a shortrange term that shares a parameter (c_{D}) with the onepionexchange 3N force, as well as two longrange terms that share parameters (c_{3} and c_{4}) with the twopionexchange NN and 3N forces. Within this framework, the Gamow–Teller operator naturally decomposes into two major terms, the standard onebody current \(\left( {O_{\sigma \tau }^{1b} = \sigma \tau ^ \pm } \right)\) and a 2BC:
For each of the chiral EFT Hamiltonians employed in this work, the parameters c_{D}, c_{3} and c_{4} are taken consistently in the 3N force and 2BC, and the momentum cutoff for the regularization of the currents, Λ_{2BC}, is set to the value used in the nonlocal regulator of the 3N interaction (Supplementary Table 2). We found that the choice of a local^{41} versus nonlocal regulator in the 2BC has a negligible effect on the Gamow–Teller transition strength. The majority of our results were obtained using a local regulator. When appropriate, the currents were consistently evolved with the nuclear forces to a lower resolution using the SRG (for NNN^{3}LO + 3N_{lnl} and NNN^{4}LO + 3N_{lnl}), keeping only up to twobody contributions. In light nuclei, three and higherbody SRGinduced terms are very small.
Quantum manybody methods
In what follows we describe the manybody methods used in this work: coupledcluster theory, the NCSM and the IMSRG. The coupledcluster calculations required new methodological developments, which are described in detail below.
Coupledcluster method
Our coupledcluster calculations start from a Hamiltonian H_{N} that is normalordered with respect to a singlereference HartreeFock state Φ_{0}〉. We approximate the full 3N interaction by truncating it at the normalordered twobody level. This approximation has been shown to work well for light and mediummass nuclei^{42,43,44}. The central quantity in the coupledcluster method is the similarity transformed Hamiltonian \(\overline H _N = e^{  T}H_Ne^T\), with T = T_{1} + T_{2} + … being a linear expansion of particle–hole excitations with respect to the reference state Φ_{0}〉. The truncation of this expansion at some loworder particle–hole excitation rank is the only approximation that occurs in the coupledcluster method^{45,46}. The nonHermitian Hamiltonian \(\overline H _N\) is correlated and the reference state Φ_{0}〉 becomes the exact ground state.
We compute ground and excited states using the CCSDT1 and EOMCCSDT1 approximations^{47,48}, respectively. These approximations include iterative singles and doubles and leadingorder triples excitations, and capture ~99% of the correlation energy in closed (sub) shell systems^{4}. For the Gamow–Teller transitions and expectation values, we solve for the left ground state of \(\overline H _N\):
Here, Λ is a linear expansion in particle–hole deexcitation operators. We truncate Λ at the EOMCCSDT1 level consistent with the right CCSDT1 groundstate solution^{48}.
The Gamow–Teller transition of a J^{π} = 0^{+} ground state occupies lowlying 1^{+} states in the daughter nucleus. These states in the daughter nucleus are calculated by employing the chargeexchange equationofmotion coupledcluster method^{49}, and we also include the leadingorder threeparticle–threehole excitations as defined by the EOMCCSDT1 approximation^{48}. The absolute squared Gamow–Teller transition matrix element is then
Here, \(R_\mu ^{1^ + }\) is the right and \(L_\mu ^{1^ + }\) the corresponding left excited 1^{+} state in the daughter nucleus, and \(\overline {O_N} = e^{  {\mathrm{T}}}O_Ne^{\mathrm{T}}\) is the similarity transform of the normalordered Gamow–Teller operator O_{GT} (equation (5)). In O_{N} we approximate the twobody part of the operator O_{GT} at the normalordered onebody level, neglecting the residual twobody normalordered part^{49,50}. Note that the construction of \(\overline O _N\) induces higherbody terms, and we truncate \(\overline O _N\) at the twobody level. This approximation is precise for the case of electromagnetic sumrules in coupledcluster theory^{51}.
We evaluate the total integrated Gamow–Teller strengths as a groundstate expectation value
For \(O = O_{\sigma \tau }^{1b}\) the Ikeda sumrule is S^{−} − S^{+} = 3(N − Z) (refs. ^{52,53}). As a check of our code, we have verified that this sumrule is fulfilled.
The inclusion of triples excitations of the right and left eigenstates R_{μ} and L_{μ}, respectively, is challenging in terms of CPU time and memory. To limit CPU time, we restrict the employed threeparticle–threehole configurations in the EOMCCSDT1 calculations to the vicinity of the Fermi surface. This is done by introducing a singleparticle index \(\tilde e_{\mathrm{p}} = N_{\mathrm{p}}  N_{\mathrm{F}}\) that measures the difference between the numbers of oscillator shells N_{p} of the singleparticle state with respect to the Fermi surface N_{F}. We only allow threeparticle and threehole configurations with \(\tilde E_{pqr} = \tilde e_p + \tilde e_q + \tilde e_r < \tilde E_{{\mathrm{3max}}}\). This approach yields a rapid convergence in EOMCCSDT1 calculations, as seen in Supplementary Fig. 12.
The storage of the included threeparticle–threehole amplitudes exceeds currently available resources and had to be avoided. We follow ref. ^{54} and define an effective Hamiltonian in the P space of singles and doubles excitations, so that no explicit triples amplitudes need be stored. Denoting the Qspace as that of all triples excitations below \(\tilde E_{{\mathrm{3max}}}\), the right eigenvalue equation can be rewritten as
This yields
and
Here, we have suppressed the label μ denoting different excited states. Solving equation (11) for the triples component of R, and then substituting into equation (10), we arrive at
In the EOMCCSDT1 approximation \(\overline H _{QQ} = \langle TFT\rangle\), where F is the Fock matrix. In the Hartree–Fock basis \(\overline H _{QQ}\) is diagonal, and its inversion is trivial. We solve this energydependent, effective Hamiltonian selfconsistently to arrive at exact eigenstates of the EOMCCSDT1 Hamiltonian. This allows for only one state to be constructed at a time. For the computation of higher spin excited states in the daughter nucleus ^{100}In (ref. ^{55}), we combine the iterative EOMCCSDT1 approach, with a perturbative approach that accounts for all excluded threeparticle–threeexcitations outside the energy cut \(\tilde E_{3{\mathrm{max}}}\). This approach is analogous to the active space coupled cluster methods of refs. ^{56,57}. By denoting the Q′space as that of all threeparticle–threehole excitations above \(\tilde E_{3{\mathrm{max}}}\), we arrive at the following perturbative noniterative energy correction:
Here, R_{μ} and L_{μ} are the right and corresponding left EOMCCSDT1 eigenstates obtained from diagonalization of the energydependent similarity transformed Hamiltonian given in equation (12). We label this approach EOMCCSDt1, and it drastically improves convergence to the fullspace EOMCCSDT1 energies (see Supplementary Fig. 15 for details).
NCSM
The NCSM^{6,58} treats nuclei as systems of A nonrelativistic pointlike nucleons interacting through realistic internucleon interactions. All nucleons are active degrees of freedom. The manybody wavefunction is cast into an expansion over a complete set of antisymmetric Anucleon harmonicoscillator basis states containing up to N_{max} harmonicoscillator excitations above the lowest Pauliprincipleallowed configuration:
Here, N denotes the total number of harmonicoscillator excitations of all nucleons above the minimum configuration, J^{π}T are the total angular momentum, parity and isospin, and i denotes additional quantum numbers. The sum over N is restricted by parity to either an even or odd sequence. The basis is further characterized by the frequency ω of the harmonic oscillator. Squareintegrable energy eigenstates are obtained by diagonalizing the intrinsic Hamiltonian, typically by applying the Lanczos algorithm. In the present work we used the importancetruncation NCSM^{59} to reduce the basis size in the highest N_{max} spaces of the A = 10 and A = 14 nucleus calculations.
VSIMSRG
The IMSRG^{60,61} transforms the manybody Hamiltonian H to a diagonal or blockdiagonal form via a unitary transformation U; that is, it generates \(\tilde H = UHU^\dagger\). To achieve this, one expresses the transformation as the exponential of an antiHermitian generator, U = e^{Ω}. Here, Ω encodes information on the offdiagonal physics to be decoupled^{62}. Beginning from some singlereference groundstate configuration Φ_{0}〉 (for example, the Hartree–Fock state based on initial interactions), we map the reference to the fully correlated ground state Ψ_{0}〉 via a continuous sequence of such unitary transformations U(s). With no approximations, this gives the exact groundstate energy, but in the IMSRG(2) approximation used here, all operators are truncated at the twobody level.
In the valencespace formulation, VSIMSRG^{8,63}, the unitary transformation is constructed (based on a redefinition of Ω) to also decouple a valencespace Hamiltonian H_{vs} from the remainder of the Hilbert space. We use an ensemble reference^{8} state for normal ordering to capture the main effects of threebody operators within the valence space. The eigenstates are obtained by a subsequent diagonalization of H_{vs} within the valence space. Furthermore, any general operator \({\cal O}\) can then be transformed by \(\tilde {\cal O} = e^\Omega {\cal O}e^{  \Omega }\), to produce an effective valencespace operator consistent with the valencespace Hamiltonian^{64}. The expectation value of \({\cal O}\) between initial and final states is obtained as usual by combining the matrix elements of \({\cal O}\) with the one and twobody shellmodel transition densities. Note that there is some ambiguity about which reference we should take when normal ordering: the parent or the daughter. If we were able to perform the unitary transformation without approximation, either choice should give exactly the same answer, as long as we use the same transformation on the wavefunctions and the operators. However, because we truncate at the twobody level, the transformation is not unitary and the error made is referencedependent. Comparing results obtained by normal ordering with respect to the parent or the daughter nucleus then provides a (lower bound) estimate of the error due to the truncation. In this work, we find that the different choices give transition matrix elements that differ on the order of ~5%. The results presented are those obtained with the parent as the reference. As an example, if we use ^{14}N as the reference, the numbers in the third line of Supplementary Table 4 become 1.77, 1.81, 1.88 and 1.87 (refs. ^{65,66}).
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Additional information
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References
 1.
Janka, H.T., Langanke, K., Marek, A., MartínezPinedo, G. & Müller, B. Theory of corecollapse supernovae. Phys. Rep. 442, 38–74 (2007).
 2.
Schatz, H. et al. Strong neutrino cooling by cycles of electron capture and βdecay in neutron star crusts. Nature 505, 62–65 (2013).
 3.
Engel, J. & Menéndez, J. Status and future of nuclear matrix elements for neutrinoless doublebeta decay: a review. Rep. Prog. Phys. 80, 046301 (2017).
 4.
Towner, I. S. Quenching of spin matrix elements in nuclei. Phys. Rep. 155, 263–377 (1987).
 5.
Epelbaum, E., Hammer, H.W. & Meißner, U.G. Modern theory of nuclear forces. Rev. Mod. Phys. 81, 1773–1825 (2009).
 6.
Barrett, B. R., Navrátil, P. & Vary, J. P. Ab initio no core shell model. Prog. Part. Nucl. Phys. 69, 131–181 (2013).
 7.
Hagen, G. et al. Neutron and weakcharge distributions of the ^{48}Ca nucleus. Nat. Phys. 12, 186–190 (2016).
 8.
Stroberg, S. R. et al. Nucleusdependent valencespace approach to nuclear structure. Phys. Rev. Lett. 118, 032502 (2017).
 9.
Korobkin, O., Rosswog, S., Arcones, A. & Winteler, C. On the astrophysical robustness of the neutron star merger rprocess. Mon. Not. R. Astron. Soc. 426, 1940–1949 (2012).
 10.
Mumpower, M. R., Surman, R., McLaughlin, G. C. & Aprahamian, A. The impact of individual nuclear properties on rprocess nucleosynthesis. Prog. Part. Nucl. Phys. 86, 86–126 (2016).
 11.
Pian, E. et al. Spectroscopic identification of rprocess nucleosynthesis in a double neutronstar merger. Nature 551, 67–70 (2017).
 12.
Barea, J., Kotila, J. & Iachello, F. Limits on neutrino masses from neutrinoless doubleβ decay. Phys. Rev. Lett. 109, 042501 (2012).
 13.
Wilkinson, D. H. Renormalization of the axialvector coupling constant in nuclear βdecay (II). Nucl. Phys. A 209, 470–484 (1973).
 14.
Brown, B. A. & Wildenthal, B. H. Experimental and theoretical Gamow–Teller betadecay observables for the sdshell nuclei. At. Data Nucl. Data Tables 33, 347–404 (1985).
 15.
Chou, W.T., Warburton, E. K. & Brown, B. A. Gamow–Teller betadecay rates for A ≤ 18 nuclei. Phys. Rev. C 47, 163–177 (1993).
 16.
MartínezPinedo, G., Poves, A., Caurier, E. & Zuker, A. P. Effective G _{A} in the pf shell. Phys. Rev. C 53, R2602–R2605 (1996).
 17.
Machleidt, R. & Entem, D. R. Chiral effective field theory and nuclear forces. Phys. Rep. 503, 1–75 (2011).
 18.
Holt, J. W., Kaiser, N. & Weise, W. Chiral threenucleon interaction and the ^{14}Cdating β decay. Phys. Rev. C 79, 054331 (2009).
 19.
Maris, P. et al. Origin of the anomalous long lifetime of ^{14}C. Phys. Rev. Lett. 106, 202502 (2011).
 20.
Hinke, C. B. et al. Superallowed Gamow–Teller decay of the doubly magic nucleus ^{100}Sn. Nature 486, 341–345 (2012).
 21.
Morris, T. D. et al. Structure of the lightest tin isotopes. Phys. Rev. Lett. 120, 152503 (2018).
 22.
Hebeler, K., Bogner, S. K., Furnstahl, R. J., Nogga, A. & Schwenk, A. Improved nuclear matter calculations from chiral lowmomentum interactions. Phys. Rev. C 83, 031301 (2011).
 23.
Ekström, A. et al. Accurate nuclear radii and binding energies from a chiral interaction. Phys. Rev. C 91, 051301 (2015).
 24.
Leistenschneider, E. et al. Dawning of the N = 32 shell closure seen through precision mass measurements of neutronrich titanium isotopes. Phys. Rev. Lett. 120, 062503 (2018).
 25.
Batist, L. et al. Systematics of Gamow–Teller beta decay ‘southeast’ of ^{100}Sn. Eur. Phys. J. A 46, 45–53 (2010).
 26.
Pastore, S. et al. Quantum Monte Carlo calculations of weak transitions in A = 6–10 nuclei. Phys. Rev. C 97, 022501 (2018).
 27.
Langanke, K., Dean, D. J., Radha, P. B., Alhassid, Y. & Koonin, S. E. Shellmodel Monte Carlo studies of fpshell nuclei. Phys. Rev. C 52, 718–725 (1995).
 28.
Gaarde, C. et al. Excitation of giant spin–isospin multipole vibrations. Nucl. Phys. A 369, 258–280 (1981).
 29.
Wakasa, T. et al. Gamow–Teller strength of ^{90}Nb in the continuum studied via multipole decomposition analysis of the ^{90}Zr(p,n) reaction at 295 MeV. Phys. Rev. C 55, 2909–2922 (1997).
 30.
Bhat, M. R. in Qaim, S. M. (ed.) Nuclear Data for Science and Technology, 817 (Springer, Berlin, 1992).
 31.
Brown, B. A. & Richter, W. A. New ‘USD’ Hamiltonians for the sd shell. Phys. Rev. C 74, 034315 (2006).
 32.
Entem, D. R. & Machleidt, R. Accurate chargedependent nucleon–nucleon potential at fourth order of chiral perturbation theory. Phys. Rev. C 68, 041001 (2003).
 33.
Bogner, S. K., Furnstahl, R. J. & Perry, R. J. Similarity renormalization group for nucleon–nucleon interactions. Phys. Rev. C 75, 061001 (2007).
 34.
Hagen, G., Jansen, G. R. & Papenbrock, T. Structure of ^{78}Ni from firstprinciples computations. Phys. Rev. Lett. 117, 172501 (2016).
 35.
Simonis, J., Stroberg, S. R., Hebeler, K., Holt, J. D. & Schwenk, A. Saturation with chiral interactions and consequences for finite nuclei. Phys. Rev. C 96, 014303 (2017).
 36.
Entem, D. R., Machleidt, R. & Nosyk, Y. Highquality twonucleon potentials up to fifth order of the chiral expansion. Phys. Rev. C 96, 024004 (2017).
 37.
Navrátil, P. Local threenucleon interaction from chiral effective field theory. FewBody Systems 41, 117–140 (2007).
 38.
Edmonds, A. R. Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, NJ, 1957).
 39.
Krebs, H., Epelbaum, E. & Meißner, U.G. Nuclear axial current operators to fourth order in chiral effective field theory. Ann. Phys. 378, 317–395 (2017).
 40.
Park, T.S. et al. Parameterfree effective field theory calculation for the solar protonfusion and hep processes. Phys. Rev. C 67, 055206 (2003).
 41.
Gazit, D., Quaglioni, S. & Navrátil, P. Threenucleon lowenergy constants from the consistency of interactions and currents in chiral effective field theory. Phys. Rev. Lett. 103, 102502 (2009).
 42.
Hagen, G. et al. Coupledcluster theory for threebody Hamiltonians. Phys. Rev. C 76, 034302 (2007).
 43.
Roth, R. et al. Mediummass nuclei with normalordered chiral NN + 3N interactions. Phys. Rev. Lett. 109, 052501 (2012).
 44.
Hergert, H. et al. Inmedium similarity renormalization group with chiral two plus threenucleon interactions. Phys. Rev. C 87, 034307 (2013).
 45.
Bartlett, R. J. & Musiał, M. Coupledcluster theory in quantum chemistry. Rev. Mod. Phys. 79, 291–352 (2007).
 46.
Hagen, G., Papenbrock, T., HjorthJensen, M. & Dean, D. J. Coupledcluster computations of atomic nuclei. Rep. Prog. Phys. 77, 096302 (2014).
 47.
Lee, Y. S., Kucharski, S. A. & Bartlett, R. J. A coupled cluster approach with triple excitations. J. Chem. Phys. 81, 5906–5912 (1984).
 48.
Watts, J. D. & Bartlett, R. J. Economical triple excitation equationofmotion coupledcluster methods for excitation energies. Chem. Phys. Lett. 233, 81–87 (1995).
 49.
Ekström, A. et al. Effects of threenucleon forces and twobody currents on Gamow–Teller strengths. Phys. Rev. Lett. 113, 262504 (2014).
 50.
Menéndez, J., Gazit, D. & Schwenk, A. Chiral twobody currents in nuclei: Gamow–Teller transitions and neutrinoless doublebeta decay. Phys. Rev. Lett. 107, 062501 (2011).
 51.
Miorelli, M., Bacca, S., Hagen, G. & Papenbrock, T. Computing the dipole polarizability of ^{48}Ca with increased precision. Phys. Rev. C 98, 014324 (2018).
 52.
Ikeda, K., Fujii, S. & Fujita, J. The (p,n) reactions and beta decays. Phys. Lett. 3, 271–272 (1963).
 53.
Yako, K. et al. Gamow–Teller strength distributions in ^{48}Sc by the ^{48}Ca(p,n) and ^{48}Ti(n,p) reactions and twoneutrino doubleβ decay nuclear matrix elements. Phys. Rev. Lett. 103, 012503 (2009).
 54.
Smith, C. E., King, R. A. & Crawford, T. D. Coupled cluster methods including triple excitations for excited states of radicals. J. Chem. Phys. 122, 054110 (2005).
 55.
Faestermann, T., Górska, M. & Grawe, H. The structure of ^{100}Sn and neighbouring nuclei. Prog. Part. Nucl. Phys. 69, 85–130 (2013).
 56.
Shen, J. & Piecuch, P. Biorthogonal moment expansions in coupledcluster theory: review of key concepts and merging the renormalized and activespace coupledcluster methods. Chem. Phys. 401, 180–202 (2012).
 57.
Shen, J. & Piecuch, P. Combining activespace coupledcluster methods with moment energy corrections via the CC(P;Q) methodology, with benchmark calculations for biradical transition states. J. Chem. Phys. 136, 144104 (2012).
 58.
Navrátil, P., Vary, J. P. & Barrett, B. R. Largebasis ab initio nocore shell model and its application to ^{12}C. Phys. Rev. C 62, 054311 (2000).
 59.
Roth, R. & Navrátil, P. Ab Initio study of ^{40}Ca with an importancetruncated nocore shell model. Phys. Rev. Lett. 99, 092501 (2007).
 60.
Tsukiyama, K., Bogner, S. K. & Schwenk, A. Inmedium similarity renormalization group for nuclei. Phys. Rev. Lett. 106, 222502 (2011).
 61.
Hergert, H., Bogner, S. K., Morris, T. D., Schwenk, A. & Tsukiyama, K. The inmedium similarity renormalization group: a novel ab initio method for nuclei. Phys. Rep. 621, 165–222 (2016).
 62.
Morris, T. D., Parzuchowski, N. M. & Bogner, S. K. Magnus expansion and inmedium similarity renormalization group. Phys. Rev. C 92, 034331 (2015).
 63.
Bogner, S. K. et al. Nonperturbative shellmodel interactions from the inmedium similarity renormalization group. Phys. Rev. Lett. 113, 142501 (2014).
 64.
Parzuchowski, N. M., Stroberg, S. R., Navrátil, P., Hergert, H. & Bogner, S. K. Ab initio electromagnetic observables with the inmedium similarity renormalization group. Phys. Rev. C 96, 034324 (2017).
 65.
Brown, B. A. & Wildenthal, B. H. Status of the nuclear shell model. Annu. Rev. Nucl. Part. Sci. 38, 29–66 (1988).
 66.
Wildenthal, B. H., Curtin, M. S. & Brown, B. A. Predicted features of the beta decay of neutronrich sdshell nuclei. Phys. Rev. C 28, 1343–1366 (1983).
Acknowledgements
The authors thank H. Grawe and T. Faestermann for useful correspondence, J. Engel, E. Epelbaum, D. Gazit, H. Krebs, D. Lubos, S. Pastore and R. Schiavilla for useful discussions and K. Hebeler for providing us with matrix elements in Jacobi coordinates for the threenucleon interaction at nexttonexttoleading order^{22}. This work was prepared in part by Lawrence Livermore National Laboratory (LLNL) under contract DEAC5207NA27344 and was supported by the Office of Nuclear Physics, US Department of Energy, under grants DEFG0296ER40963, DEFG0297ER41014, DESC0008499, DESC0018223 and DESC0015376, the Field Work Proposals ERKBP57 and ERKBP72 at Oak Ridge National Laboratory (ORNL), the FWP SCW1579, LDRD projects 18ERD008 and 18ERD058 and the Lawrence Fellowship Program at LLNL, and by NSERC grant no. SAPIN201600033, ERC grant no. 307986 STRONGINT and the DFG under grant SFB 1245. TRIUMF receives federal funding through a contribution agreement with the National Research Council of Canada. Computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) programme. This research used resources of the Oak Ridge Leadership Computing Facility located at ORNL, which is supported by the Office of Science of the Department of Energy under contract no. DEAC0500OR22725. Computations were also performed at LLNL Livermore Computing under the institutional Computing Grand Challenge Program, at Calcul Quebec, Westgrid and Compute Canada, and at the Jülich Supercomputing Center (JURECA).
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Affiliations
TRIUMF, Vancouver, British Columbia, Canada
 P. Gysbers
 , J. D. Holt
 , P. Navrátil
 & S. R. Stroberg
Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada
 P. Gysbers
Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA
 G. Hagen
 , G. R. Jansen
 , T. D. Morris
 & T. Papenbrock
Department of Physics and Astronomy, University of Tennessee, Knoxville, TN, USA
 G. Hagen
 , T. D. Morris
 & T. Papenbrock
National Center for Computational Sciences, Oak Ridge National Laboratory, Oak Ridge, TN, USA
 G. R. Jansen
Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA
 T. D. Morris
Nuclear and Chemical Science Division, Lawrence Livermore National Laboratory, Livermore, CA, USA
 S. Quaglioni
 & K. A. Wendt
Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt, Germany
 A. Schwenk
ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany
 A. Schwenk
MaxPlanckInstitut für Kernphysik, Heidelberg, Germany
 A. Schwenk
Physics Department, Reed College, Portland, OR, USA
 S. R. Stroberg
Department of Physics, University of Washington, Seattle, WA, USA
 S. R. Stroberg
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Contributions
G.H., T.D.M. and T.P. performed the coupledcluster calculations. G.R.J. computed threenucleon forces for the coupledcluster calculations. P.G., S.Q., P.N. and K.A.W. performed calculations for the twobody currents. P.N. developed the higherprecision chiral threenucleon interactions used in this work and performed nocore shell model calculations. G.H. and T.D.M. derived and implemented the new formalism to incorporate higherorder excitations in coupledcluster theory. S.R.S. and J.D.H. performed VSIMSRG calculations. All authors discussed the results and contributed to the manuscript at all stages.
Competing interests
The authors declare no competing interests.
Corresponding author
Correspondence to G. Hagen.
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