Session GK: Nuclear Theory III

Bayesian uncertainty quantification in effective field theories

Effective field theories (EFTs) offer a rigorous connection of quantum chromodynamics to the low energy regime of nuclear structure and interactions. High-quality EFT interactions provide precise, accurate results when paired with modern many-body methods, such as quantum Monte Carlo. However, despite the advancements made with techniques and interactions, there is still a lack of rigorous uncertainty quantification in many theoretical calculations. One way to remedy this is by introducing Bayesian methods in EFT parameter estimation, which can be accomplished by implementing Markov Chain Monte Carlo (MCMC) to sample the EFT parameter posterior. This route, however, generates its difficulties by requiring significant computational resources. To counteract the computational cost, we explore the use of emulation to speed up objective function evaluations in the MCMC algorithm.   

Effective Field Theory Convergence Pattern of Modern Nucleon-Nucleon Potentials

We apply our model of correlated effective field theory (EFT) truncation errors to several nucleon-nucleon (NN) potentials developed using chiral EFT (χEFT). We use the associated graphical diagnostics to assess which potentials have a χEFT convergence pattern consistent with the assumed statistical model based on Gaussian processes. We find that regular EFT convergence over a wide range of angles and energies requires careful identification, using the diagnostics, of the EFT expansion parameter Q, reference scale, and independent variables for the observables, and we demonstrate that concordance with our model can be improved for a given choice of potential by varying these parameters. For potentials that do yield observables that converge in accordance with the Q expansion, we infer a breakdown scale and correlation lengths for the EFT description of NN scattering observables from the convergence of the series. We illustrate how the diagnostics reveal deficiencies of a potential with non-standard order-by-order convergence. All our results can be reproduced through a publicly available Jupyter notebook, which can be straightforwardly modified to analyze other χEFT NN potentials.   

Uncertainty quantification of optical models in fission fragment angular momentum removal

Understanding the post-scission state of fission fragments, and their subsequent energy and angular momentum removal by emitted prompt neutrons and gammas, is an important open problem. Recent measurements have assumed that prompt neutrons are predominantly emitted as s-waves, which is contradictory to phenomenological optical model predictions. Despite this contradiction, Monte Carlo Hauser fesbach (MCHF) fission event generators like CGMF are still generally predictive of fission observables, such as prompt neutron and gamma multiplicities and energy distributions. Are the optical models, level densities, and models for post-scission fragment energy and angular momentum valid individually, or do canceling errors lead to the predictive powers of MCHF? Workhorse phenomenological optical models, Koning Delaroche and Chapel Hill 89, have been re-fit to their corpora of experimental data using Markov Chain Monte Carlo. We propagate the resulting posterior distributions, representing model parameter uncertainty, through CGMF, an MCHF fission event generator, to determine the uncertainty in angular momentum and energy removed from the fragments by neutrons. These models are trained on experimental scattering data on β-stable targets, but using them for fission requires extrapolation to neutron-rich regions, where their reliability is unknown. The Whitehead-Lim-Holt optical model is derived ab-initio from chiral effective field theory, and should not suffer from extrapolation error. We add this model to CGMF, and propagate its theoretical uncertainties as well, comparing its predictions to the phenomenological models.   

Theoretical Uncertainty Quantification for Heavy-ion Fusion

Despite recent advances and focus on rigorous uncertainty quantification for microscopic models of quantum many-body systems, the uncertainty on the dynamics of those systems has been under-explored. To address this, we have used time-dependent Hartree-Fock to examine the model uncertainty for a collection of low-energy, heavy-ion fusion reactions. Fusion reactions at near-barrier energies represent a rich test-bed for the dynamics of quantum many-body systems owing to the complex interplay of collective excitation, transfer, and static effects that determine the fusion probability of a given system. While the model uncertainty is sizable for many of the systems studied, the primary contribution comes from ill-constrained static properties, such as the neutron radius of neutron-rich nuclei. These large uncertainties motivate the use of information from reactions to better constrain existing models and to infer static properties from reaction data.   

Quantifying uncertainties due to irreducible three-body forces in deuteron-nucleus reactions

Deuteron-induced nuclear reactions are typically described within a Faddeev three-body model consisting of a neutron, proton, and the nucleus interacting through pairwise forces. While Faddeev techniques enable the exact description of the three-body dynamics, their predictive power is limited in part by the omission of irreducible three-body nucleon-nucleon-nucleus forces. An alternative approach for describing deuteron-nucleus reactions is ab initio theory, where the system is described from first principles, starting from individual nucleons and the interactions amongst them. We adopt the ab initio no-core shell model (NCSM) coupled with the resonating group method (RGM) to compute microscopic nucleon-nucleus interactions and use them to describe deuteron-induced reactions by means of momentum space Faddeev calculations, beginning with ²H+⁴He scattering. Simultaneously, we also carry out ab initio calculations of the same deuteron-induced scattering process within the NCSM/RGM approach. I will show that the effects of the irreducible three-body force arising from antisymmetrization of  ²H+⁴He system have significant impact on bound state energies as well as cross sections. This finding lays the groundwork for improved Faddeev calculations of deuteron-nucleus reactions in which the effective three-body force is explicitly included.

    

Uncertainties here, there, and everywhere: interpolating between small- and large-g expansions using Bayesian Model Mixing

Bayesian Model Mixing (BMM) is a statistical technique that can be used to combine models that are predictive in different input domains into a composite distribution that has improved predictive power over the entire input space. We explore the application of BMM to the mixing of two expansions of a function of a coupling constant g that are valid at small and large values of g respectively. This type of problem is quite common in nuclear physics, where physical properties are straightforwardly calculable in strong and weak interaction limits or at low and high densities or momentum transfers, but difficult to calculate in between. Interpolation between these limits is often accomplished by a suitable interpolating function, e.g., Padé approximants, but it is then unclear how to quantify the uncertainty of the interpolant. We address this problem in the simple context of the partition function of zero-dimensional Φ⁴ theory, for which the (asymptotic) expansion at small g and the (convergent) expansion at large g are both known. We consider three mixing methods: linear mixture BMM, localized bivariate BMM, and localized multivariate BMM with Gaussian processes. We find that employing a Gaussian process in the intermediate region between the two predictive models leads to the best results of the three methods. The methods and validation strategies we present here should be generalizable to other nuclear physics settings.   

Snapshot-based Emulators for Chiral EFT Observables

Bayesian nucleon-nucleon (NN) uncertainty quantification (UQ) has proven useful in helping understand to what extent our models work, but the process is known to be computationally expensive when using direct calculation methods. Recently, emulators have shown to provide fast & accurate™ predictions of bound state and scattering observables for applications that require repeated calculations with different parameters. The emulation process is performed by creating a basis of eigensolutions for several sets of known parameters to accurately interpolate and extrapolate solutions for the same Hamiltonian with different parameters. Here we provide a comparison between a Lippmann-Schwinger (LS) equation emulator and the KVP momentum-space emulator for a representative set of NN observables, with an alternate quasi-spline-based approach to speed up the KVP-based emulator. In addition, we apply a method to deal with the anomalies that appear in the emulation result.   

(Towards) Emulators for the In-Medium Similarity Renormalization Group

The in-medium similarity renormalization group (IMSRG) is an ab-initio method for computing the properties of medium mass and heavy nuclei. The IMSRG flow is a continuous unitary transformation which decouples a target reference state from all excitations in the many-body basis, thereby constructing a map to the true ground state (up to truncation errors).

We show that the solution to the IMSRG flow equation can be emulated via the data-driven discrete Koopman operator, effectively bypassing effort of solving the flow equation with standard ODE methods—at the cost of a few initial iterations of polynomial complexity. The Koopman operator characterizes a linear flow map which propagates a dynamical system. Evaluating the Koopman eigenfunctions, tuned by measurements of the flowing system, allows us to reproduce the system dynamics over a given range. Using this method, we construct a linear IMSRG transformation which emulates the conventional solution to a high degree of accuracy. We also apply a similar method to the parametric version of the IMSRG to allow sensitivity studies and uncertainty propagation for the parameters of the nuclear interactions.

    

Training and Projecting: A Reduced Basis Method Emulator for Many-Body Physics

We present the reduced basis method (RBM) as a tool for developing emulators for equations with tunable parameters within the context of the nuclear many-body problem. The RBM uses a basis expansion informed by a set of solutions for a few values of the model parameters and then projects the equations over a well-chosen low-dimensional subspace. We connected some of the results in the eigenvector continuation literature to the formalism of RBMs and show how RBMs can be applied to a broader set of problems. We applied the RBM to the one-dimensional Gross-Pitaevskii equation with a harmonic trapping potential and to nuclear density functional theory for 48Ca. The outstanding performance of the approach, together with its straightforward implementation, show promise for its application to the emulation of computationally demanding calculations, including uncertainty quantification.