Chiral three-body forces: From neutron matter to neutron stars Kai Hebeler (OSU) In collaboration with: E. Anderson(OSU), S. Bogner (MSU), R. Furnstahl (OSU), J. Lattimer (Stony Brook), A. Nogga (Juelich), C. Pethick (Nordita), A. Schwenk (Darmstadt) EMMI program The Extreme Matter Physics of Nuclei: From Universal Properties to Neutron-Rich Extremes Darmstadt, April 20, 2012
Chiral EFT for nuclear forces, leading order 3N forces NN 3N 4N long (2 π ) intermediate ( π ) short-range c E term c D term c 1 , c 3 , c 4 terms large uncertainties in coupling constants at present: 1 . 5 lead to theoretical uncertainties in many-body observables
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Low-momentum interactions: The (Similarity) Renormalization Group • goal: generate unitary transformation of “hard” Hamiltonian H λ = U λ HU † with the resolution parameter λ λ dH λ • basic idea: change resolution in small steps: d λ = [ η λ , H λ ]
Changing the resolution: The (Similarity) Renormalization Group • elimination of coupling between low- and high momentum components, calculations much easier • observables unaffected by resolution change (for exact calculations) • residual resolution dependences can be used as tool to test calculations Not the full story: RG transformation also changes three-body (and higher-body) interactions.
RG evolution of 3N interactions • So far: intermediate (c D ) and short-range (c E ) 3NF couplings fitted to few-body c E term c D term c 1 , c 3 , c 4 terms systems at different resolution scales: and r 4 He = 1 . 95 − 1 . 96 fm E 3 H = − 8 . 482 MeV coupling constants of natural size • in neutron matter contributions from , and terms vanish c D c E c 4 • long-range contributions assumed to be invariant under RG evolution 2 π • Ideal case: evolve 3NF consistently with NN to lower resolution using the RG • has been achieved in oscillator basis (Jurgenson, Roth) • promising results in very light nuclei • problems in heavier nuclei • not suitable for infinite systems
Equation of state: Many-body perturbation theory central quantity of interest: energy per particle E/N H ( λ ) = T + V NN ( λ ) + V 3N ( λ ) + ... kinetic energy E = Hartree-Fock + + V NN V 3N V NN V NN V 3N V 3N V 3N 2nd-order + + + + + V 3N V NN V 3N V NN V 3N 3rd-order and beyond + . . . • “hard” interactions require non-perturbative summation of diagrams • with low-resolution interactions much more perturbative • inclusion of 3N interaction contributions crucial • use chiral interactions as initial input for RG evolution Hartree-Fock
Equation of state of pure neutron matter 20 Hartree-Fock 2nd-order (1) E NN+3N,eff +c 3 +c 1 uncertainties E NN+3N,eff E NN+3N,eff 20 E NN+3N,eff +c 3 uncertainty Energy/nucleon [MeV] Energy/nucleon [MeV] 3N < 2.5 fm -1 2.0 < (1) + E NN (2) 15 E NN 15 3N 10 10 = 1.8 fm -1 = 2.0 fm -1 5 5 = 2.4 fm -1 = 2.8 fm -1 0 0 0 0.05 0.10 0.15 0 0.05 0.10 0.15 0 0.05 0.10 0.15 [fm -3 ] [fm -3 ] [fm -3 ] KH and Schwenk PRC 82, 014314 (2010) • significantly reduced cutoff dependence at 2nd order perturbation theory • small resolution dependence indicates converged calculation • energy sensitive to uncertainties in 3N interaction • variation due to 3N input uncertainty much larger than resolution dependence
Equation of state of pure neutron matter 20 E NN+3N,eff +c 3 +c 1 uncertainties Hartree-Fock 2nd-order (1) E NN+3N,eff E NN+3N,eff 20 Schwenk+Pethick (2005) Energy/nucleon [MeV] Energy/nucleon [MeV] Akmal et al. (1998) 3N < 2.5 fm -1 2.0 < 15 QMC s-wave 15 GFMC v6 GFMC v8’ 10 10 = 1.8 fm -1 = 2.0 fm -1 5 5 = 2.4 fm -1 = 2.8 fm -1 0 0 0 0.05 0.10 0.15 0 0.05 0.10 0.15 0 0.05 0.10 0.15 [fm -3 ] [fm -3 ] [fm -3 ] KH and Schwenk PRC 82, 014314 (2010) • significantly reduced cutoff dependence at 2nd order perturbation theory • small resolution dependence indicates converged calculation • energy sensitive to uncertainties in 3N interaction • variation due to 3N input uncertainty much larger than resolution dependence • good agreement with other approaches (different NN interactions)
Symmetry energy constraints extend EOS to finite proton fractions x and extract symmetry energy parameters S v = ∂ 2 E/N � � � ∂ 2 x � ρ = ρ 0 ,x =1 / 2 ∂ 3 E/N � L = 3 � � ∂ρ∂ 2 x 8 � ρ = ρ 0 ,x =1 / 2 KH, Lattimer, Pethick and Schwenk, in preparation symmetry energy parameters consistent with other constraints
Constraints on the nuclear equation of state (EOS) A two-solar-mass neutron star measured using Shapiro delay P. B. Demorest 1 , T. Pennucci 2 , S. M. Ransom 1 , M. S. E. Roberts 3 & J. W. T. Hessels 4,5 30 a 20 Timing residual ( μ s) 10 0 –10 –20 –30 Credit: NASA/Dana Berry –40 –40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 b Orbital phase (turns) Demorest et al., Nature 467, 1081 (2010) M max = 1 . 65 M ⊙ → 1 . 97 ± 0 . 04 M ⊙ Calculation of neutron star properties requires EOS up to high densities. Strategy: Use observations to constrain the high-density part of the nuclear EOS.
Neutron star radius constraints incorporation of beta-equilibrium: neutron matter neutron star matter parametrize piecewise high-density extensions of EOS: • use polytropic ansatz p ∼ ρ Γ • range of parameters Γ 1 , ρ 12 , Γ 2 , ρ 23 , Γ 3 limited by physics! 37 crust EOS (BPS) 3 neutron star matter 36 with c i uncertainties 2 log 10 P [dyne / cm 2 ] 35 crust 1 34 33 32 31 13.0 13.5 14.0 1 12 23 [g / cm 3 ] log 10 see also KH, Lattimer, Pethick, Schwenk, PRL 105, 161102 (2010)
Constraints on the nuclear equation of state 36 use the constraints: recent NS observation log 10 P [dyne / cm 2 ] 35 M max > 1 . 97 M ⊙ add full band causality 34 � v s ( ρ ) = dP/d ε < c 33 14.2 14.4 14.6 14.8 15.0 15.2 15.4 [g / cm 3 ] log 10 significant reduction of possible equations of state
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