Uncertainty Analysis and Order-by-Order Optimization of Chiral Nuclear Interactions

Chiral effective field theory (chi EFT) provides a systematic approach to describe low-energy nuclear forces. Moreover, chi EFT is able to provide well-founded estimates of statistical and systematic uncertainties-although this unique advantage has not yet been fully exploited. We fill this gap by performing an optimization and statistical analysis of all the low-energy constants (LECs) up to next-to-next-to-leading order. Our optimization protocol corresponds to a simultaneous fit to scattering and bound-state observables in the pion-nucleon, nucleon-nucleon, and few-nucleon sectors, thereby utilizing the full model capabilities of chi EFT. Finally, we study the effect on other observables by demonstrating forward-error-propagation methods that can easily be adopted by future works. We employ mathematical optimization and implement automatic differentiation to attain efficient and machine-precise first-and second-order derivatives of the objective function with respect to the LECs. This is also vital for the regression analysis. We use power-counting arguments to estimate the systematic uncertainty that is inherent to chi EFT, and we construct chiral interactions at different orders with quantified uncertainties. Statistical error propagation is compared with Monte Carlo sampling, showing that statistical errors are, in general, small compared to systematic ones. In conclusion, we find that a simultaneous fit to different sets of data is critical to (i) identify the optimal set of LECs, (ii) capture all relevant correlations, (iii) reduce the statistical uncertainty, and (iv) attain order-by-order convergence in chi EFT. Furthermore, certain systematic uncertainties in the few-nucleon sector are shown to get substantially magnified in the many-body sector, in particular when varying the cutoff in the chiral potentials. The methodology and results presented in this paper open a new frontier for uncertainty quantification in ab initio nuclear theory.

[1]  Stefan M. Wild,et al.  Nuclear Energy Density Optimization , 2010, 1005.5145.

[2]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[3]  D. Bugg.  Coulomb corrections to πN elastic scattering , 1973 .

[4]  N. Metropolis,et al.  Phase shift analysis of 310-MeV proton proton scattering experiments , 1957 .

[5]  B. Karlsson.  Making predictions using χEFT , 2015 .

[6]  Isoscalar Hamiltonians for light atomic nuclei , 1999, nucl-th/9907047.

[7]  D. R. Entem,et al.  Accurate charge dependent nucleon nucleon potential at fourth order of chiral perturbation theory , 2003 .

[8]  D. Higdon,et al.  Nonparametric dark energy reconstruction from supernova data. , 2010, Physical Review Letters.

[9]  Petr Navrátil,et al.  Ab initio no core shell model , 2013 .

[10]  Marja Putus,et al.  A Licentiate thesis , 2008 .

[11]  R. Workman,et al.  Parameterization dependence of T matrix poles and eigenphases from a fit to $\pi$N elastic scattering data , 2012, 1204.2277.

[12]  R. Hackenburg.  Neutron-proton effective range parameters and zero- energy shape dependence , 2006 .

[13]  Giuseppina Orlandini,et al.  Modern ab initio approaches and applications in few-nucleon physics with A≥4 , 2012, 1204.4617.

[14]  Aldemar Torres Valderrama,et al.  Uncertainty Propagation in Nerve Impulses Through the Action Potential Mechanism , 2015, Journal of mathematical neuroscience.

[15]  T. A. Green,et al.  Variable Phase Approach to Potential Scattering , 1968 .

[16]  J. Negele,et al.  Theoretical and Experimental Determination of Nuclear Charge Distributions , 1975 .

[17]  U. Meissner.  Modern theory of nuclear forces A.D. 2006 , 2007 .

[18]  M. Webb,et al.  Quantification of modelling uncertainties in a large ensemble of climate change simulations , 2004, Nature.

[19]  E. Epelbaum,et al.  Chiral three-nucleon force at N^4LO I: Longest-range contributions , 2012, 1203.0067.

[20]  G. Austen,et al.  Improved Coulomb Potential , 1983 .

[21]  I. Angeli,et al.  Table of experimental nuclear ground state charge radii: An update , 2013 .

[22]  A Nogga,et al.  Structure of A=10-13 nuclei with two- plus three-nucleon interactions from chiral effective field theory. , 2007, Physical review letters.

[23]  B. Tromborg,et al.  Electromagnetic Corrections to pi n Scattering , 1977 .

[24]  S. Bogner,et al.  In-medium similarity renormalization group with chiral two- plus three-nucleon interactions , 2012, 1212.1190.

[25]  B. Taylor,et al.  CODATA recommended values of the fundamental physical constants: 2006 | NIST , 2007, 0801.0028.

[26]  Steven Weinberg,et al.  Nuclear forces from chiral Lagrangians , 1990 .

[27]  D. R. Entem,et al.  Chiral effective field theory and nuclear forces , 2011, 1105.2919.

[28]  A. Schwenk,et al.  Three-body forces: From cold atoms to nuclei , 2012, 1210.4273.

[29]  H. Witała,et al.  Three-nucleon forces from chiral effective field theory , 2002, nucl-th/0208023.

[30]  D. Phillips,et al.  Power counting of contact-range currents in effective field theory. , 2014, Physical review letters.

[31]  B. Taylor,et al.  CODATA Recommended Values of the Fundamental Physical Constants: 2010 | NIST , 2007, 0801.0028.

[32]  J. Martorell,et al.  Nuclear sizes and the isotope shift , 1997, nucl-th/9707016.

[33]  W. Nazarewicz,et al.  Accurate nuclear radii and binding energies from a chiral interaction , 2015, 1502.04682.

[34]  E. Epelbaum,et al.  The Two-nucleon system at next-to-next-to-next-to-leading order , 2004, nucl-th/0405048.

[35]  J. E. Amaro,et al.  Statistical error analysis for phenomenological nucleon-nucleon potentials , 2014 .

[36]  Theodor W. Hänsch,et al.  HYDROGEN-DEUTERIUM 1S-2S ISOTOPE SHIFT AND THE STRUCTURE OF THE DEUTERON , 1998 .

[37]  D. R. Entem,et al.  Peripheral nucleon-nucleon scattering at fifth order of chiral perturbation theory , 2014, 1411.5335.

[38]  P. Navrátil,et al.  Few-nucleon systems in a translationally invariant harmonic oscillator basis , 1999, nucl-th/9907054.

[39]  W. Marsden.  I and J , 2012 .

[40]  Evgeny Epelbaum.  Few-nucleon forces and systems in chiral effective field theory , 2006 .

[41]  S. Weinberg.  Effective chiral lagrangians for nucleonpion interactions and nuclear forces , 1991 .

[42]  E. Epelbaum,et al.  Precision Nucleon-Nucleon Potential at Fifth Order in the Chiral Expansion. , 2014, Physical review letters.

[43]  Construction of high-quality NN potential models. , 1994, Physical review. C, Nuclear physics.

[44]  Dean Lee.  Lattice simulations for few- and many-body systems , 2008, 0804.3501.

[45]  L. Durand.  VACUUM POLARIZATION EFFECTS IN PROTON-PROTON SCATTERING , 1957 .

[46]  M. Hjorth-Jensen,et al.  Coupled-cluster computations of atomic nuclei , 2013, Reports on progress in physics. Physical Society.

[47]  U. Meißner,et al.  High-Precision Determination of the Pion-Nucleon σ Term from Roy-Steiner Equations. , 2015, Physical review letters.

[48]  J. E. Amaro,et al.  Coarse grained NN potential with Chiral Two Pion Exchange , 2013, 1310.6972.

[49]  J. Golak,et al.  Efficient calculation of chiral three-nucleon forces up to N3LO for ab initio studies , 2015, 1502.02977.

[50]  S. C. Pieper,et al.  Realistic models of pion-exchange three-nucleon interactions , 2001 .

[51]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[52]  J. Huston,et al.  Uncertainties of predictions from parton distribution functions , 2001 .

[53]  Dean Lee,et al.  Lattice effective field theory for medium-mass nuclei , 2013, 1311.0477.

[54]  Stefan M. Wild.  Chapter 1 Solving Derivative-Free Nonlinear Least Squares with POUNDERS , 2015 .

[55]  Charles Meunier.  [High precision]. , 2008, Perspective infirmiere : revue officielle de l'Ordre des infirmieres et infirmiers du Quebec.

[56]  J. Hamilton.  Electromagnetic Corrections to Hadron Scattering , 1975 .

[57]  J. E. Amaro,et al.  Coarse-grained potential analysis of neutron-proton and proton-proton scattering below the pion production threshold , 2013, Physical Review C.

[58]  J. E. Amaro,et al.  Error analysis of nuclear forces and effective interactions , 2014, 1406.0625.

[59]  J. Dobaczewski,et al.  Error Estimates of Theoretical Models: a Guide , 2014, 1402.4657.

[60]  U. Jentschura,et al.  Proton radius, Darwin-Foldy term and radiative corrections , 2010, 1012.4029.

[61]  T. Duguet,et al.  Ab initio Gorkov-Green's function calculations of open-shell nuclei , 2012, 1208.2472.

[62]  S. C. Pieper,et al.  Quantum Monte Carlo calculations of light nuclei. , 1998, nucl-th/0103005.

[63]  Dimensional Power Counting in Nuclei , 1996, nucl-th/9607020.

[64]  K. Holinde,et al.  Momentum space calculations and helicity formalism in nuclear physics , 1971 .

[65]  Stoks,et al.  Magnetic moment interaction in nucleon-nucleon phase-shift analyses. , 1990, Physical review. C, Nuclear physics.

[66]  E. Epelbaum,et al.  Improved chiral nucleon-nucleon potential up to next-to-next-to-next-to-leading order , 2014, The European Physical Journal A.

[67]  J. E. Amaro,et al.  Phenomenological high precision neutron-proton delta-shell potential , 2012, 1202.2689.

[68]  Stefan M. Wild,et al.  Statistical uncertainties of a chiral interaction at next-to-next-to leading order , 2014, Journal of Physics G: Nuclear and Particle Physics.

[69]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[70]  S. C. Phatak,et al.  Accurate momentum-space method for scattering by nuclear and Coulomb potentials , 1974 .

[71]  Stefan M. Wild,et al.  Optimized chiral nucleon-nucleon interaction at next-to-next-to-leading order. , 2013, Physical review letters.

[72]  M. Kortelainen,et al.  The limits of the nuclear landscape , 2012, Nature.

[73]  R. Vogelaar,et al.  Determination of the axial-vector weak coupling constant with ultracold neutrons. , 2010, Physical review letters.

[74]  G. Morgan,et al.  Search for Resonance Structure in the np Total Cross Section below 800 MeV , 1982 .

[75]  David Higdon,et al.  Cosmic calibration: Constraints from the matter power spectrum and the cosmic microwave background , 2007 .

[76]  J. Pumplin,et al.  Uncertainties of predictions from parton distribution functions. I. The Lagrange multiplier method , 2001 .

[77]  S. Quaglioni,et al.  Three-nucleon low-energy constants from the consistency of interactions and currents in chiral effective field theory. , 2008, Physical review letters.

[78]  H. Bethe.  Theory of the effective range in nuclear scattering , 1949 .

[79]  M. Haftel,et al.  Nuclear saturation and the smoothness of nucleon-nucleon potentials , 1970 .

[80]  B. Tromborg,et al.  ELECTROMAGNETIC CORRECTIONS TO HADRON - HADRON SCATTERING , 1974 .

[81]  I. Šlaus,et al.  Charge symmetry, quarks and mesons , 1990 .

[82]  Stoks,et al.  Partial-wave analysis of all nucleon-nucleon scattering data below 350 MeV. , 1993, Physical review. C, Nuclear physics.

[83]  R. Wiringa,et al.  Accurate nucleon-nucleon potential with charge-independence breaking. , 1995, Physical review. C, Nuclear physics.

[84]  D. Phillips,et al.  Bayesian methods for parameter estimation in effective field theories , 2008, 0808.3643.

[85]  P. Winternitz.  Scattering formalism for nonidentical spinor particles , 1980 .

[86]  Meaningful characterisation of perturbative theoretical uncertainties , 2012 .

[87]  Costas Papadimitriou,et al.  Bayesian uncertainty quantification and propagation in molecular dynamics simulations: a high performance computing framework. , 2012, The Journal of chemical physics.

[88]  S. Quaglioni,et al.  He 4 4He + n + n continuum within an ab initio framework. , 2014, Physical review letters.