We sample the posterior probability distributions of the low-energy constants (LECs) in $Δ$-full chiral effective field theory ($χ$EFT) up to third order. We use eigenvector continuation for fast and accurate emulation of the likelihood and Hamiltonian Monte Carlo to draw effectively independent samples from the posteriors. Our Bayesian inference is conditioned on the Granada database of neutron-proton ($np$) cross sections and polarizations. We use priors grounded in $χ$EFT assumptions and a Roy-Steiner analysis of pion-nucleon scattering data. We model correlated EFT truncation errors using a two-feature Gaussian process, and find correlation lengths for $np$ scattering energies and angles in the ranges 45--83 MeV and 24--39 degrees, respectively. These correlations yield a non-diagonal covariance matrix and reduce the number of independent scattering data with a factor of 8 and 4 at the second and third chiral orders, respectively. The relatively small difference between the second and third order predictions in $Δ$-full $χ$EFT suppresses the marginal variance of the truncation error and the effects of its correlation structure. Our results are particularly important for analyzing the predictive capabilities in \textit{ab initio} nuclear theory.
The truncation errors in equations of state (EOSs) of nuclear matter derived from the chiral nucleon-nucleon ($NN$) potentials at different expansion orders are analyzed by a Bayesian model. These EOSs are expanded as functions of a dimensionless parameter, $Q$, which is determined by Fermi momentum, $k_F$ and breakdown scale, $Λ_b$. The degree-of-belief (DoB) intervals predicted by the chiral effective field theory are calculated within the corresponding expansion coefficients and the specific prior probability distribution functions in terms of Bayes theorem. The truncation errors of EOSs, g…
Propagation of uncertainties from two-nucleon potential parameters to three-nucleon observables, that is statistical errors for the neutron-deuteron elastic scattering and the deuteron breakup reaction at neutron laboratory energies up to 200 MeV is investigated. To that end we use the chiral nucleon-nucleon interaction with the semi-local momentum-space regularization at various orders of the chiral expansion, exploiting knowledge of the covariance matrix of its parameters. For both reactions we compare statistical uncertainties for chiral predictions with the uncertainties obtained in the sa…
We revisit the $D_s^-D^{*0}$/$D_s^{*-}D^0$ interaction and the $Z_{cs}(3985)$ state in chiral effective field theory(EFT) up to the next-to-leading order. We examine the relative importance of the leading-order contact, one-eta exchange, next-to-leading-order contact, and two-kaon-exchange contributions. We show that the leading-order and next-to-leading-order contact contributions play the most important role such that the $Z_{cs}(3985)$ state qualifies as a $D_s^-D^{*0}$/$D_s^{*-}D^0$ resonance. On the other hand, the weak one-eta-exchange and weakly energy-dependent two-kaon-exchange contri…
Chiral effective field theory (EFT) predictions are necessarily truncated at some order in the EFT expansion, which induces an error that must be quantified for robust statistical comparisons to experiment. In previous work, a Bayesian model for truncation errors of perturbative expansions was adapted to EFTs. The model yields posterior probability distribution functions (pdfs) for these errors based on expectations of naturalness encoded in Bayesian priors and the observed order-by-order convergence pattern of the EFT. A first application was made to chiral EFT for neutron-proton scattering u…
Effective field theories (EFTs) organize the description of complex systems into an infinite sequence of decreasing importance. Predictions are made with a finite number of terms, which induces a truncation error that is often left unquantified. We formalize the notion of EFT convergence and propose a Bayesian truncation error model for predictions that are correlated across the independent variables, e.g., energy or scattering angle. Central to our approach are Gaussian processes that encode both the naturalness and correlation structure of EFT coefficients. Our use of Gaussian processes perm…
We develop a framework for deriving Dyson-Schwinger Equations (DSEs) and Bethe-Salpeter Equation (BSE) in QCD at large $N_c$ limit. The starting point is a modified form (with auxiliary fields) of QCD generating functional. This framework provides a natural order-by-order truncation scheme for DSEs and BSE, and the kernels of the equations up to any order are explicitly given. Chiral symmetry (at chiral limit) is preserved in any order truncation, so it exemplifies the symmetry preserving truncation scheme. It provides a method to study DSEs and BSE beyond the Rainbow-Ladder truncation, and is…
Unquenching of the Polyakov-loop potential showed to be an important improvement for the description of the phase structure and thermodynamics of strongly-interacting matter at zero quark chemical potentials with Polyakov-loop extended chiral models. This work constitutes the first application of the quark backreaction on the Polyakov-loop potential at nonzero density. The observation is that it links the chiral and deconfinement phase transition also at small temperatures and large quark chemical potentials. The build up of the surface tension in the Polyakov-loop extended Quark-Meson model i…
Temperature effects on the neutron matter equation of state are investigated in the framework of chiral effective field theory. Latest, state-of-the-art chiral two-nucleon forces are applied from third to fifth order in the chiral expansion together with chiral three-nucleon forces, allowing for a determination of the truncation error of the theoretical predictions. The thermodynamic quantities considered include the chemical potential, the internal energy, the entropy, and the free energy. In general, good order-by-order convergence of all predictions is observed. As to be expected, temperatu…
Background: Effective interactions for nucleon-nucleus ($NA$) elastic scattering from first principles require the use of the same nucleon-nucleon ($NN$) interaction in the structure and reaction calculations, and a consistent treatment of the relevant operators at each order. Purpose: Truncation uncertainties of chiral $NN$ forces have been studied for scattering observables in few-body systems and for bound state properties of light nuclei. We extend this to $NA$ elastic scattering. Methods: With the spectator expansion of multiple scattering theory and the no-core shell model, we use a chir…
We calculate the energy-dependent cross section of the $np\leftrightarrow dγ$ process in chiral effective field theory and apply state-of-the-art tools for quantification of theory uncertainty. We focus on the low-energy regime, where the magnetic dipole and the electric dipole transitions cross over, including the range relevant for big-bang nucleosynthesis. Working with the leading one- and two-body electromagnetic currents, we study the order-by-order convergence of this observable in the chiral expansion of the nuclear potential. We find that the Gaussian process error model describes the …
We present and discuss properties of isospin asymmetric matter whose equation of state is derived from recent high-quality chiral nucleon-nucleon potentials and chiral effective three-nucleon forces. After a brief review of the chiral few-nucleon forces which we adopt, we concentrate on the symmetry energy and its density derivatives. We also explore the correlation between the symmetry energy at saturation density and its slope parameter, L. We estimate the truncation error across three orders of the chiral expansion for both the symmetry energy as a function of density and the slope paramete…
The properties of nuclear matter are studied using state-of-the-art nucleon-nucleon forces up to fifth order in chiral effective field theory. The equations of state of symmetric nuclear matter and pure neutron matter are calculated in the framework of the Brueckner-Hartree-Fock theory. We discuss in detail the convergence pattern of the chiral expansion and the regulator dependence of the calculated equations of state and provide an estimation of the truncation uncertainty. For all employed values of the regulator, the fifth-order chiral two-nucleon potential is found to generate nuclear satu…
With the goal of using chiral interactions at various orders to explore properties of the few-body nuclear systems, we write the recently developed local chiral interactions as spherical irreducible tensors and implement them in the hyperspherical harmonics expansion method. We devote particular attention to three-body forces at next-to-next-to leading order, which play an important role in reproducing experimental data. We check our implementation by benchmarking the ground-state properties of $^3$H, $^3$He and $^4$He against the available Monte Carlo calculations. We then confirm their order…
We explore the constraints on the three-nucleon force (3NF) of chiral effective field theory ($χ$EFT) that are provided by bound-state observables in the $A=3$ and $A=4$ sectors. Our statistically rigorous analysis incorporates experimental error, computational method uncertainty, and the uncertainty due to truncation of the $χ$EFT expansion at next-to-next-to-leading order. A consistent solution for the ${}^3$H binding energy, the ${}^4$He binding energy and radius, and the ${}^3$H $β$-decay rate can only be obtained if $χ$EFT truncation errors are included in the analysis. All of these excep…
We recently developed a Bayesian framework for parameter estimation in general effective field theories. Here we present selected results from using that framework to estimate parameters with a nucleon-nucleon (NN) potential derived using chiral effective field theory ($χ$EFT): the semi-local NN potential of Epelbaum, Krebs, and Meißner (EKM). There are many NN scattering data, up to high energies, and with rather small errors, so imposing a penalty for unnatural low-energy constants (LECs) usually has a small effect on the fits. In contrast, we have found that including an estimate of higher …
We renormalize the chiral effective field theory (EFT) potential in harmonic-oscillator (HO) model space. The low energy constants (LECs) are utilized to absorb not just the ultra-violet part of the physics due to the cutoff, but also the infrared part due to the truncation of model space. We use the inverse J-matrix method to reproduce the nucleon-nucleon (NN) scattering phase shifts in the given model space. We demonstrate that by including the NLO correction, the nucleon-nucleon scattering in the continuum could be well reproduced in the truncated HO trap space up to laboratory energy $T_{l…
Balancing theory uncertainties in ab initio nuclear structure calculations: Many-body truncation versus finite basis size
CEA（法国原子能委员会）；Université Paris-Saclay（巴黎萨克雷大学）；Laboratoire Matière en Conditions Extrêmes（极端条件物质实验室）；Technische Universität Darmstadt（达姆施塔特工业大学）；Department of Physics（物理系）；ExtreMe Matter Institute EMMI（极端物质研究所EMMI）；GSI Helmholtzzentrum für Schwerionenforschung GmbH（GSI亥姆霍兹重离子研究中心有限公司）；Université Libre de Bruxelles（布鲁塞尔自由大学）；Institut d’Astronomie et d’Astrophysique（天体物理学与天文学研究所）；Brussels Laboratory of the Universe – BLU-ULB（布鲁塞尔宇宙实验室）；IRFU（核物理研究学院）；CEA, DES, IRESNE, DER, SPRC, LEPh；Max-Planck-Institut für Kernphysik（马克斯·普朗克核物理研究所）
We discuss the current status of chiral effective field theory in the three-nucleon sector and present selected results for nucleon-deuteron scattering observables based on semilocal momentum-space-regularized chiral two-nucleon potentials together with consistently regularized three-nucleon forces up to third chiral order. Using a Bayesian model for estimating truncation errors, the obtained results are found to provide a good description of the experimental data. We confirm our earlier findings that a high-precision description of nucleon-deuteron scattering data below pion production thresh…
We introduce novel relations between the derivatives [$\partial^{n}ρ(λ, m_{l})/\partial m_{l}^n$] of the Dirac eigenvalue spectrum [$ρ(λ, m_{l})$] with respect to the light sea quark mass ($m_{l}$) and the $(n+1)$-point correlations among the eigenvalues ($λ$) of the massless Dirac operator. Using these relations we present lattice QCD results for $\partial^{n}ρ(λ, m_{l})/\partial m_{l}^n$ ($n=1, 2, 3$) for $m_{l}$ corresponding to pion masses $m_π=160-55$ MeV, and at a temperature of about 1.6 times the chiral phase transition temperature. Calculations were carried out using (2+1) flavors of …
We investigate the Dirac eigenvalue spectrum ($ρ(λ,m_l)$) to study the microscopic origin of axial anomaly in high temperature phase of QCD. We propose novel relations between the derivatives ($\partial^n ρ(λ,m_l)/\partial m_l^n$) of the Dirac eigenvalue spectrum with respect to the quark mass ($m_l$) and the $(n+1)$-point correlations among the eigenvalues ($λ$) of the massless Dirac operator. Based on these relations, we present lattice QCD results for $\partial^n ρ(λ,m_l)/\partial m_l^n$ ($n=1, 2, 3$) with $m_l$ corresponding to pion masses $m_π=160-55$ MeV, and at a temperature of about 1.…
A common feature of recent functional renormalization group investigations of effective low-energy QCD is the appearance of a back-bending behavior of the chiral phase transition line at low temperatures together with a negative entropy density in the symmetric regime. The regulator scheme dependence of this phenomenon and the necessary modifications at finite densities are analyzed within a two-flavor quark-meson model. The flows at finite densities for three different regulators of three- or four-dimensional momenta are confronted to each other. It is found that the back-bending behavior and…
