Nuclear Physics from Lattice Effective Field Theory Dean Lee (NCSU/Bonn) work done in collaboration with Evgeny Epelbaum (Bochum) Hermann Krebs (Bochum) Ulf-G. Meißner (Bonn/Jülich) Buḡra Borasoy (now at EnBW) DGH Fest, UMass Amherst Oct. 22-23, 2010 1
Outline Effective field theory for nucleons Lattice effective field theory Lattice interactions and scattering data Euclidean time projection and auxiliary fields Neutron matter Three-nucleon forces Isospin breaking and Coulomb effects Results for A = 3, 4, 6, 12 Summary and future directions 2
Chiral EFT for low-energy nucleons Weinberg, PLB 251 (1990) 288; NPB 363 (1991) 3 Construct the effective potential order by order N N N N N N N N p p p N N N N N N N N Contact interactions Leading order (LO) Next-to-leading order (NLO) 3
Nuclear Scattering Data Effective Field Theory Ordonez et al. 94; Friar & Coon 94; Kaiser et al. 97; Epelbaum et al. 98, 03; Kaiser 99-01; Higa et al. 03; 4
Lattice EFT for nucleons n n p n p 5
T [MeV] Accessible by Lattice QCD 100 early universe quark-gluon plasma 10 1 gas of light nuclei neutron star crust nuclear liquid superfluid heavy-ion collisions Accessible by Lattice EFT excited nuclei neutron star core 10-3 10-2 10-1 1 r [fm -3 ] r N 6
Lattice interactions Leading order on the lattice p n p 7
Next-to-leading order on the lattice p p p p p p... 8
Computational strategy LO NLO NNLO NNNLO 9
Non-perturbative Monte Carlo Perturbative corrections LO LO NLO NLO Improved LO NNLO NNLO NNNLO NNNLO 10
LO 1 : Pure contact interactions LO 2 : Gaussian smearing LO 3 : Gaussian smearing only in even partial waves 11
Physical scattering data Unknown operator coefficients Lüscher s finite-volume formula Lüscher, Comm. Math. Phys. 105 (1986) 153; NPB 354 (1991) 531 Two-particle energy levels near threshold in a periodic cube related to phase shifts L L L Not so useful for higher partial waves and partial wave mixing 12
Physical scattering data Unknown operator coefficients Spherical wall method Borasoy, Epelbaum, Krebs, D.L., Meißner, EPJA 34 (2007) 185 Spherical wall imposed in the center of mass frame R wall 13
Energy levels with hard spherical wall Energy shift from free-particle values gives the phase shift 14
LO 3 : S waves 15
LO 3 : P waves 16
Euclidean time projection p p 17
Auxiliary fields We can write exponentials of the interaction using a Gaussian integral identity We remove the interaction between nucleons and replace it with the interactions of each nucleon with a background field. 18
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Take any initial state with the desired quantum numbers and which is an antisymmetric product of A single nucleon states (i.e., a Slater determinant) For any configuration of the auxiliary and pion fields, For A nucleons, the matrix is A by A. We use Monte Carlo to integrate over all possible configurations of the auxiliary and pion fields. 20
Schematic of calculations Hybrid Monte Carlo sampling 21
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Layers of a neutron star inner crust outer crust inner core outer core 23
Dilute neutron matter at NLO 24
N = 8, 12, 16 neutrons at L 3 = 4 3, 5 3, 6 3, 7 3 Epelbaum, Krebs, D.L, Meißner, EPJA 40 (2009) 199 25
Dilute neutrons and the unitarity limit Neutron-neutron scattering amplitude: Unitarity limit: Free Fermi gas ground state Unitarity limit ground state Neutron matter close to unitarity limit for ξ is a dimensionless number 26
P-wave pairing? 27
Three-nucleon forces Fit c D and c E to spin-1/2 nucleon-deuteron scattering and 3 H binding energy E D Spin-1/2 nucleon-deuteron 3 H binding energy Fitting point 28
Isospin breaking and Coulomb effects Isospin-breaking and power counting [Friar, van Kolck, PRC 60 (1999) 034006; Walzl, Meißner, Epelbaum NPA 693 (2001) 663; Friar, van Kolck, Payne, Coon, PRC 68 (2003) 024003; Epelbaum, Meißner, PRC72 (2005) 044001 ] Pion mass difference 29
Coulomb potential p p Charge symmetry breaking Charge independence breaking p p n p n n 30
Neutron-proton Proton-proton Epelbaum, Krebs, D.L, Meißner, PRL 104:142501, 2010 Epelbaum, Krebs, D.L, Meißner, EPJA 45 (2010) 335 31
Triton and Helium-3 32
Relative contribution of omitted operators Approximate universality 33
Helium-4 Epelbaum, Krebs, D.L, Meißner, PRL 104:142501, 2010 Epelbaum, Krebs, D.L, Meißner, EPJA 45 (2010) 335 34
Helium-4 LO NLO NLO + IB + EM NNLO + IB + EM NNLO + IB + EM + 4N contact Physical (infinite volume) -30.5(4) MeV -30.6(4) MeV -29.2(4) MeV -30.1(5) MeV -28.3(5) MeV -28.3 MeV 35
Lithium-6 Epelbaum, Krebs, D.L, Meißner, PRL 104:142501, 2010 Epelbaum, Krebs, D.L, Meißner, EPJA 45 (2010) 335 36
Lithium-6 LO NLO NLO + IB + EM NNLO + IB + EM NNLO + IB + EM + 4N contact Physical (infinite volume) -32.6(9) MeV -34.6(9) MeV -32.4(9) MeV -34.5(9) MeV -32.9(9) MeV -32.0 MeV 37
Carbon-12 Epelbaum, Krebs, D.L, Meißner, PRL 104:142501, 2010 Epelbaum, Krebs, D.L, Meißner, EPJA 45 (2010) 335 38
Carbon-12 LO NLO NLO + IB + EM NNLO + IB + EM NNLO + IB + EM + 4N contact Physical (infinite volume) -109(2) MeV -115(2) MeV -108(2) MeV -106(2) MeV -99(2) MeV -92.2 MeV 39
Summary and future directions Lattice effective field theory is a promising tool that combines the framework of effective field theory and computational lattice methods Applications to zero and nonzero temperature simulations of neutron matter, nuclear matter, etc. nuclei, In progress: Energy splitting between different spin channels and excited resonances; nuclear wavefunctions order by order; multinucleon correlations; storing lattice configurations for future calculations Near future: Larger nuclei, higher-density neutron matter, nuclear matter, smaller lattice spacing, higher orders 40