Nuclear and Nucleon Matter Constraints on Three-Nucleon Forces

Transcription

1 Nuclear and Nucleon Matter Constraints on Three-Nucleon Forces Robert B. Wiringa, Physics Division, Argonne National Laboratory Joe Carlson, Los Alamos Stefano Gandolfi, Los Alamos Alessandro Lovato, Argonne & INFN Trento Saori Pastore, Los Alamos Maria Piarulli, Argonne Steven C. Pieper, Argonne Rocco Schiavilla, JLab & ODU WORK NOT POSSIBLE WITHOUT EXTENSIVE COMPUTER RESOURCES Argonne Laboratory Computing Resource Center (Bebop) Argonne Leadership Computing Facility (Theta) Physics Division Work supported by U.S. Department of Energy, Office of Nuclear Physics

2 Night Sky Mandala Oil on linen 36 x 36 inches Leslie Morgan 1994

3 GOALS Ab Initio CALCULATIONS OF NUCLEI AND NUCLEON MATTER Understand nuclei & matter at level of elementary interactions between individual nucleons: Binding energies, excitation spectra, relative stability, matter saturation Densities, electroweak properties, transitions, neutron star mass & radii Low-energy NA &AA scattering, asymptotic normalizations, astrophysical reactions REQUIREMENTS Two-nucleon potentials that accurately describe elastic NN scattering data Consistent three-nucleon potentials and electroweak current operators Accurate methods for solving the many-nucleon Schrödinger equation RESULTS Quantum Monte Carlo methods evaluate realistic Hamiltonians accurate to 1 2% About 100 states calculated for A 12 nuclei in good agreement with experiment Electromagnetic moments, M1, E2, F, GT transitions, electroweak response Nucleon matter evaluated with Variational Chain Summation methods and/or AFDMC

4 NUCLEAR HAMILTONIAN H = X i K i + X i<j v ij + X V ijk i<j<k K i : Non-relativistic kinetic energy, m n -m p effects included Argonne v 18 : v ij = v γ ij + vπ ij + v I ij + v S ij = P v p (r ij )O p ij 18 spin, tensor, spin-orbit, isospin, etc., operators full EM and strong CD and CSB terms included predominantly local operator structure fits Nijmegen PWA93 data with χ 2 /d.o.f.=1.1 δ (deg) S0 Argonne v 18 np Argonne v 18 pp Argonne v 18 nn SAID 7/06 np Wiringa, Stoks, & Schiavilla, PRC 51, 38 (1995) Urbana & Illinois: V ijk = V 2π ijk + V 3π ijk + V R ijk Urbana has standard 2π P -wave + one central short-short range repulsive π π term for nuclear matter saturation Illinois adds 2π S-wave + 3π rings to provide extra T =3/2 interaction Illinois-7 has four parameters fit to 23 levels in A 10 nuclei Pieper, Pandharipande, Wiringa, & Carlson, PRC 64, (2001) Pieper, AIP CP 1011, 143 (2008) π E lab (MeV) π π π π π π π

5 Norfolk NV2: v ij = v γ ij + vπ ij + vij 2π + vij CT = P v p (r ij )O p ij derived in chiral effective field theory with -intermediate states 16 spin, tensor, spin-orbit, isospin, etc., operators full EM and strong CD and CSB terms included predominantly local operator structure suitable for quantum Monte Carlo multiple models with different regularization fit to Granada PWA2013 data Ia,b fit to E lab = 125 MeV with χ 2 /d.o.f. 1.1 IIa,b fit to E lab = 200 MeV with χ 2 /d.o.f. 1.4 Piarulli, Girlanda, Schiavilla, Kievsky, Lovato, Marcucci, Pieper, Viviani, & Wiringa PRC 94, (2016) Norfolk NV3: V ijk = Vijk 2π + Vijk CT standard 2π S-wave and 2π P -wave terms consistent with chiral NN potential contact terms of c D (π-short range) and c E (short-short range τ i τ k ) type two parameters fit to 3 H binding and nd scattering length Piarulli, Baroni, Girlanda, Kievsky, Lovato, Marcucci, Pieper, Schiavilla, Viviani, & Wiringa PRL 120, (2018)

6 Minimize expectation value of H VARIATIONAL MONTE CARLO E V = Ψ V H Ψ V Ψ V Ψ V E 0 using Metropolis Monte Carlo and trial function 2 3 Ψ V = 4S Y ( U ij + X " # Y U ijk ) 5 f c (r ij ) Φ A (JMTT 3 ) i<j i<j k i,j single-particle Φ A (JMTT 3 ) is fully antisymmetric and translationally invariant central pair correlations f c (r) keep nucleons at favorable pair separation pair correlation operators U ij = P p u p(r ij )O p ij reflect influence of v ij triple correlation operator U ijk added when V ijk is present multiple J π states constructed and diagonalized for p-shell nuclei ability to construct clusterized or asymptotically correct trial functions optimization code COBYLA used to search parameters Ψ V are spin-isospin vectors in 3A dimensions with 2 A `A Z components Lomnitz-Adler, Pandharipande, & Smith, NP A361, 399 (1981) Wiringa, PRC 43, 1585 (1991)

7 GREEN S FUNCTION MONTE CARLO Projects out lowest energy state from variational trial function Ψ(τ) = exp[ (H E 0 )τ]ψ V = X n exp[ (E n E 0 )τ]a n ψ n Ψ(τ ) = a 0 ψ 0 Evaluation of Ψ(τ) done stochastically in small time steps τ Z Ψ(R n, τ) = G(R n,r n 1 ) G(R 1,R 0 )Ψ V (R 0 )dr n 1 dr 0 Mixed estimates used for expectation values; Ψ(τ) = Ψ V + δψ(τ) and neglect O(δψ(τ) 2 ) O(τ) = Ψ(τ) O Ψ(τ) Ψ(τ) Ψ(τ) O(τ) Mixed = Ψ V O Ψ(τ) Ψ V Ψ(τ) O(τ) Mixed + [ O(τ) Mixed O V ] ; H(τ) Mixed = Ψ(τ/2) H Ψ(τ/2) Ψ(τ/2) Ψ(τ/2) Cannot propagate p 2, L 2, or (L S) 2 operators use H = AV8 + Ṽijk Fermion sign problem would limit maximum τ, but... Constrained-path propagation removes steps that have Ψ (τ,r)ψ V (R) = 0 Multiple excited states of same J π stay orthogonal Carlson, PRC 38, 1879 (1988) Pudliner, Pandharipande, Carlson, Pieper, & Wiringa, PRC 56, 1720 (1997) Wiringa, Pieper, Carlson, & Pandharipande, PRC 62, (2000) Pieper, Wiringa, & Carlson, PRC 70, (2004) E 0

8 EXAMPLES OF GFMC PROPAGATION E (MeV) He E(τ) (MeV) Li = g.s α+d τ (MeV 1 ) τ (MeV -1 ) Curve has P i a iexp( E i τ) with E i = 1480, 340 & 20.2 MeV (20.2 MeV is first 4 He excitation) Ψ V has small amounts of 1.5 GeV contamination g.s. ( ) & stable after τ = 0.2 MeV 1 (a broad resonance) never stable decaying to separated α & d E(τ=0.2) is best GFMC estimate of resonance energy

9 H 3 H 1/ Argonne v 18 without & with Urbana IX GFMC Calculations 1 November 2010 Energy (MeV) He 5 He 1/2 α+n 6 He α+2n 6 Li α+d 7 He 5/2 6 He+n 7 Li 7/2 5/2 5/2 7/2 α+t 1/2 8 He 6 He+2n 8 Li 7 Li+n AV18 UIX Exp 8 Be 2α

10 Energy (MeV) He 6 He 6 Li 7 Li 7/2 5/2 5/2 7/2 1/2 8 He 8 Li Argonne v 18 with Illinois-7 GFMC Calculations 10 January Be IL7: 4 parameters fit to 23 states 600 kev rms error, 51 states ~60 isobaric analogs also computed 9 Li 5/2 1/2 AV18 9 Be 7/ 5/ 7/2 7/2 3/ 5/ 1/2 5/2 1/ AV18 +IL7 Expt. 10 Be 3, 10 B 12 C

11 Excitation energy (MeV) Argonne v 18 with Illinois-7 GFMC Calculations Including IL7 gives correct s.-o. splitting & 10 B g.s. 6 He 6 Li 7 Li 7/2 5/2 5/2 7/2 1/2 8 Li 8 Be 9 Be 7/ 5/ 7/2 7/2 3/ 5/ 1/2 5/2 1/ AV18 10 Be AV18 +IL7 Expt. 3, 10 B

12 RMS E for 36 states: AV18+IL7 = 0.80 MeV ; NV2+3-Ia = 0.72 MeV with signed average deviation: MeV and MeV

13 VMC ENERGY EXPECTATION VALUES 4 He T i + V ij V 2π ijk V cd ijk V ce ijk NV2+3-Ia NV2+3-Ib NV2+3-IIa NV2+3-IIb AV18+UX AV18+UXI Li T i + V ij V 2π ijk V cd ijk V ce ijk NV2+3-Ia NV2+3-Ib NV2+3-IIa NV2+3-IIb AV18+UX AV18+UXI

14 OBSERVATIONS FROM LIGHT NUCLEI RESULTS The T i + v ij for all models underbind the light nuclei so need net attraction from V ijk The Vijk 2π is attractive in all cases The net short-range V ijk is usually repulsive The sign of NV3 c D term is not well determined by binding energy alone The τ i τ k in NV3 c E term is negative in light nuclei; will change sign in neutron matter The corresponding central term in Urbana models is repulsive in light nuclei & matter This short-short range term in Urbana V ijk gets most of its contribution by connecting S = 1 2 to S = 1 2 and S = 3 2 to S = 3 2 triples The π-short range term in UXI gets most of its contribution by connecting S = 1 2 to S = 3 2 triples so is sensitive to tensor correlations

15 VARIATIONAL CHAIN SUMMATION Variational energy expectation value of infinite many-body system can be written as: R Q A( i E V = Φ i ) S( Q i<j F ij ) H S(Q i<j F ij) ( Q i Φ i) dτ R Q A( i Φ i ) S(Q i<j F ij ) S(Q i<j F ij) ( Q i Φ i) dτ where F ij = P p fp ij Op ij are correlation operators and Φ i = exp[ik i r i ] are plane-wave states and for convenience only the l.h.s. Ψ is antisymmetrized. This integral can be approximated by expading the dynamical correlations in powers of short-ranged functions Fij c = Fij 1 = (fij) c 2 1 and F p>1 ij = 2fijf c p>1 ij and f p>1 ij f q>1 ij, and in powers of the statistical correlation (Slater function) l(k F r) = 3j 1 (k F r)/(k F r). This expansion is conveniently represented by generalized Mayer diagrams and a very general diagrammatic expansion valid for noncommuting operators has been developed, commonly referred to as the Fermi hypernetted chain + single-operator chain (FHNC+SOC) method. Present calculations of central correlations are now beyond the FHNC/4 level. Pandharipande & Wiringa, RMP 51, 821 (1979) Wiringa, Fiks & Fabrocini, PRC 38, 1010 (1988) Akmal, Pandharipande & Ravenhall, PRC 58, 1804 (1998)

16 ENERGY IN FHNC CALCULATIONS The energy can be computed using distribution functions g and g 3 (in Pandharipande-Bethe form): E P B = T F + W + W F + U + U F W = ρ Z v ij 2 2 f ij g ij d 3 r ij 2 m f ij U = 2 2m The two-body distribution function can be written as: ρ 2 4 Z i f ij i f ik g 3 (r ij,r ik )d 3 r ij d 3 r ik f ij f ik g ij = f 2h ( G de + E de ) G ee + E ee ν(g cc + E cc l/ν) 2i exp(g dd + E dd ). where the chain functions G xy are sums of nodal diagrams, with direct (d), exchange (e) or circular exchange (c) end points and E xy are elementary diagrams. A more complicated expression is available for g 3 : g 3 (r ij, r ik, r jk ) = X n A n ijb n ikc n jkd n ijk Alternatively one can perform an integration by parts to get the Jackson-Feenberg form: E JF = ρ Z h v ij 2 2 f ij ( if ij ) 2 i 2 2m f ij fij 2 g ij d 3 r ij

17 0 Symmetric Nuclear Matter - VCS -5 AV18+UIX E(ρ) MeV NV2-Ib AV18-25 AV18+V2pi NV2-IIb ρ (fm -3 )

18 Pure Neutron Matter - VCS FG AV18+UIX AV18+V2pi NV2-Ib NV2-Ia E(ρ) MeV NV2-IIa AV18 NV2-IIb ρ (fm -3 )

19 OBSERVATIONS FROM NUCLEAR AND NEUTRON MATTER RESULTS Local two-nucleon interactions fit to NN data saturate symmetric nuclear matter (SNM) at 2ρ 0 Vijk 2π by itself is attractive in SNM and pushes saturation to even higher density Shorter-range V ijk must provide net repulsion to saturate at empirical density For UIX this is all c E -like; for UXI it is split between c E - and c D -like terms in same ratio as in light nuclei The NV2-II models fit to higher energy are closer to AV18 in both SNM and pure neutron matter (PNM) In PNM the Vijk 2π is weakly repulsive For UIX the dominant repulsion in PNM is central c E -like term For UXI the c D -like term is much reduced relative to c E -like because of weak tesnor correlations A c E term with τ i τ k dependence is likely to be problemattic

20 SUMMARY AND FUTURE WORK NV2+3-Ia reproduces nuclear binding and excitation energies for A 12 extremely well, comparable to AV18+IL7 However, neither model looks able to support massive neutron stars Energy spectra for other NV2+3 models are being evaluated, but initial c D and c E choices do not give as promising results Determining c D and c E from 3 H binding and nd scattering fits is not easy because these data are highly correlated Alternate strategy is to include 3 H β decay information and energies of larger nuclei like 8 He, 8 Be, 10 B(, ) states; another possibility is nα scattering data Many other electroweak transitions are being evaluated and may help select best models Nuclear and neutron matter provide additional constraints, even if the calculational methods are less precise than for light nuclei Meeting all these demands may well require including sub-leading terms in V 1jk with more spin-isospin operator dependence