Bayesian probability updates using sampling/importance resampling: Applications in nuclear theory

We review an established Bayesian sampling method called sampling/importance resampling and highlight situations in nuclear theory when it can be particularly useful. To this end we both analyse a toy problem and demonstrate realistic applications of importance resampling to infer the posterior distribution for parameters of ΔNNLO interaction model based on chiral effective field theory and to estimate the posterior probability distribution of target observables. The limitation of the method is also showcased in extreme situations where importance resampling breaks

Bayesian parameter estimation in chiral effective field theory using the Hamiltonian Monte Carlo method

The number of low-energy constants (LECs) in chiral effective field theory (chi EFT) grows rapidly with increasing chiral order, necessitating the use of Markov chain Monte Carlo techniques for sampling their posterior probability density function. For this we introduce a Hamiltonian Monte Carlo (HMC) algorithm and sample the LEC posterior up to next-to-next-to-leading order (NNLO) in the two-nucleon sector of chi EFT. We find that the sampling efficiency of HMC is three to six times higher compared to an affine-invariant sampling algorithm. We analyze the empirical coverage probability and validate that the NNLO model yields predictions for two-nucleon scattering data with largely reliable credible intervals, provided that one ignores the leading-order EFT expansion parameter when inferring the variance of the truncation error. We also find that the NNLO truncation error dominates the error budget

Posterior predictive distributions of neutron-deuteron cross sections

We quantify the posterior predictive distributions (PPDs) of elastic neutron-deuteron (nd) scattering cross sections using nucleon-nucleon (NN) interactions from chiral effective field theory (χEFT) up to and including next-to-next-to-next-to-leading order (N3LO). These PPDs quantify the spread in nd predictions due to the variability of the low-energy constants (LECs) inferred from NN scattering data. We use the wave-packet continuum discretization method to solve the Alt-Grassberger-Sandhas form of the Faddeev equations for elastic scattering. We draw 100 samples from the PPDs of nd cross sections up to 67 MeV in scattering energy, i.e., in the energy region where the effects of three-nucleon forces are expected to be small. We find that the uncertainty about NN LECs inferred from NN scattering data, when assuming uncorrelated errors, does not translate to significant uncertainty in the low-energy nd continuum. Based on our estimates, the uncertainty of nd predictions are dominated by the χEFT truncation error, at least below N3LO. At this order, the 90% credible interval of the PPD and the truncation error are comparable, although both are very small on an absolute scale

Nuclear physics uncertainties in light hypernuclei

The energy levels of light hypernuclei are experimentally accessible observables that contain valuable information about the interaction between hyperons and nucleons. In this work we study strangeness S=-1 systems HΛ3,4 and HeΛ4,5 using the ab initio no-core shell model (NCSM) with realistic interactions obtained from chiral effective field theory (χEFT). In particular, we quantify the finite precision of theoretical predictions that can be attributed to nuclear physics uncertainties. We study both the convergence of the solution of the many-body problem (method uncertainty) and the regulator and calibration-data dependence of the nuclear χEFT Hamiltonian (model uncertainty). For the former, we implement infrared correction formulas and extrapolate finite-space NCSM results to infinite model space. We then use Bayesian parameter estimation to quantify the resulting method uncertainties. For the latter, we employ a family of 42 realistic Hamiltonians and measure the standard deviation of predictions while keeping the leading-order hyperon-nucleon interaction fixed. Following this procedure we find that model uncertainties of ground-state Λ separation energies amount to ≈20(100)keV in HΛ3(HΛ4,He) and ≈400keV in HeΛ5. Method uncertainties are comparable in magnitude for the HΛ4,He 1+ excited states and HeΛ5, which are computed in limited model spaces, but otherwise are much smaller. This knowledge of expected theoretical precision is crucial for the use of binding energies of light hypernuclei to infer the elusive hyperon-nucleon interaction

Bayesian estimation of the low-energy constants up to fourth order in the nucleon-nucleon sector of chiral effective field theory

We use Bayesian methods and Hamiltonian Monte Carlo (HMC) sampling to infer the posterior probability density function (PDF) for the low-energy constants (LECs) up to next-to-next-to-next-to-leading order (N3LO) in a chiral effective field theory (χEFT) description of the nucleon-nucleon interaction. In a first step, we condition the inference on neutron-proton and proton-proton scattering data and account for uncorrelated χEFT truncation errors. We demonstrate how to successfully sample the 31-dimensional space of LECs at N3LO using a revised HMC inference protocol. In a second step we extend the analysis by means of importance sampling and an empirical determination of the neutron-neutron scattering length to infer the posterior PDF for the leading charge-dependent contact LEC in the S01 neutron-neutron interaction channel. While doing so we account for the χEFT truncation error via a conjugate prior. We use the resulting posterior PDF to sample the posterior predictive distributions for the effective range parameters in the S01 wave as well as the strengths of charge-symmetry breaking and charge-independence breaking. We conclude that empirical point-estimate results of isospin breaking in the S01 channel are consistent with the PDFs obtained in our Bayesian analysis and that, when accounting for χEFT truncation errors, one must go to next-to-next-to-leading order to confidently detect isospin breaking effects

Power counting in chiral effective field theory and nuclear binding

Chiral effective field theory (chi EFT), as originally proposed by Weinberg, promises a theoretical connection between low-energy nuclear interactions and quantum chromodynamics (QCD). However, the important property of renormalization-group (RG) invariance is not fulfilled in current implementations and its consequences for predicting atomic nuclei beyond two- and three-nucleon systems has remained unknown. In this work we present a systematic study of recent RG-invariant formulations of chi EFT and their predictions for the binding energies and other observables of selected nuclear systems with mass numbers up to A = 16. Specifically, we have carried out ab initio no-core shell-model and coupled cluster calculations of the ground-state energy of H-3, He-3,He-4, Li-6, and O-16 using several recent power-counting (PC) schemes at leading order (LO) and next-to-leading order, where the subleading interactions are treated in perturbation theory. Our calculations indicate that RG-invariant and realistic predictions can be obtained for nuclei with mass number A <= 4. We find, however, that O-16 is either unbound with respect to the four alpha-particle threshold, or deformed, or both. Similarly, we find that the Li-6 ground-state resides above the alpha-deuteron separation threshold. These results are in stark contrast with experimental data and point to either necessary fine-tuning of all relevant counterterms, or that current state-of-the-art RG-invariant PC schemes at LO in chi EFT lack necessary diagrams-such as three-nucleon forces-to realistically describe nuclei with mass number A > 4

Normal-ordering approximations and translational (non)invariance

Normal ordering provides an approach to approximate three-body forces as effective two-body operators and it is therefore an important tool in many-body calculations with realistic nuclear interactions. The corresponding neglect of certain three-body terms in the normal-ordered Hamiltonian is known to influence translational invariance, although the magnitude of this effect has not yet been systematically quantified. In this paper we study in particular the normal-ordering two-body approximation applied to a single harmonic-oscillator reference state. We explicate the breaking of translational invariance and demonstrate the magnitude of the approximation error as a function of model space parameters for He-4 and O-16 by performing full no-core shell-model calculations with and without three-nucleon forces. We combine two different diagnostics to better monitor the breaking of translational invariance. While the center-of-mass effect is shown to become potentially very large for He-4, it is also shown to be much smaller for O-16 although full convergence is not reached. These tools can be easily implemented in studies using other many-body frameworks and bases

Bayesian predictions for A=6 nuclei using eigenvector continuation emulators

We make ab initio predictions for the A=6 nuclear level scheme based on two- and three-nucleon interactions up to next-to-next-to-leading order in chiral effective field theory (χEFT). We utilize eigenvector continuation and Bayesian methods to quantify uncertainties stemming from the many-body method, the χEFT truncation, and the low-energy constants of the nuclear interaction. The construction and validation of emulators is made possible via the development of jupiterncsm - a new M-scheme no-core shell model code that uses on-the-fly Hamiltonian matrix construction for efficient, single-node computations up to Nmax=10 for Li6. We find a slight underbinding of He6 and Li6, although consistent with experimental data given our theoretical error bars. As a result of incorporating correlated χEFT-truncation errors we find more precise predictions (smaller error bars) for separation energies: Sd(Li6)=0.89\ub10.44MeV, S2n(He6)=0.20\ub10.60MeV, and for the beta decay Q value: Qβ-(He6)=3.71\ub10.65MeV. We conclude that our error bars can potentially be reduced further by extending the model space used by jupiterncsm

Nuclear ab initio calculations of He-6 beta-decay for beyond the Standard Model studies

Precision measurements of beta-decay observables offer the possibility to search for deviations from the Standard Model. A possible discovery of such deviations requires accompanying first-principles calculations. Here we compute the nuclear structure corrections for the beta-decay of He-6 which is of central interest in several experimental efforts. We employ the impulse approximation together with wave functions calculated using the ab initio no-core shell model with potentials based on chiral effective field theory. We use these state-of-the-art calculations to give a novel and comprehensive analysis of theoretical uncertainties. We find that nuclear corrections, which we compute within the sensitivity of future experiments, create significant deviation from the naive Gamow-Teller predictions, making their accurate assessment essential in searches for physics beyond the Standard Model. (C) 2022 The Author(s). Published by Elsevier B.V

What is ab initio in nuclear theory?

Ab initio has been used as a label in nuclear theory for over two decades. Its meaning has evolved and broadened over the years. We present our interpretation, briefly review its historical use, and discuss its present-day relation to theoretical uncertainty quantification