TRIUMF. Three-body forces in nucleonic matter. Weakly-Bound Systems in Atomic and Nuclear Physics. Kai Hebeler (TRIUMF) INT, Seattle, March 11, 2010

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1 Three-body forces in nucleonic matter Kai Hebeler (TRIUMF) INT, Seattle, March 11, 21 TRIUMF A. Schwenk, T. Duguet, T. Lesinski, S. Bogner, R. Furnstahl Weakly-Bound Systems in Atomic and Nuclear Physics

2 The nuclear landscape Nuclear systems are complex manybody systems with rich properties No one size fits all method All theoretical approaches need to be linked Nucleonic matter: Infinite system of interacting neutrons and protons in the thermodynamic limit.

3 Significance of nuclear and neutron matter results for the extremes of astrophysics: neutron stars, supernovae, X-ray burst Crab pulsar SN1987a neutrino interactions with nuclear matter Nova microscopic constraints of nuclear energy-density functionals, next-generation Skyrme functionals UNEDF universal properties at low densities ultracold Fermi gases my focus: development of efficient methods to include 3N forces in microscopic many-body calculations of neutron and nuclear matter applications to finite systems

4 Reminder: Chiral EFT for nuclear forces NN 3N 4N long (2) intermediate () short-range c 1,c 3,c 4 terms c D term c E term large uncertainties in coupling constants at present: 1.5 Meissner et al., Machleidt

5 Chiral 3N interaction as density-dependent two-body interaction antisymmetrized 3N interaction (at N 2 LO) in neutron matter: = 1 2 ( ga 2f ) 2 i j k A ijk (σ i q i )(σ j q j ) (q 2 i + m2 )(q 2 j + m2 ) [ 4c 1m 2 f 2 + 2c 3 f 2 q i q j ] = c 4, c D and c E terms vanish in neutron matter in nuclear matter all terms contribute

6 Chiral 3N interaction as density-dependent two-body interaction antisymmetrized 3N interaction (at N 2 LO) in neutron matter: = 1 2 ( ga 2f ) 2 i j k A ijk (σ i q i )(σ j q j ) (q 2 i + m2 )(q 2 j + m2 ) [ 4c 1m 2 f 2 + 2c 3 f 2 q i q j ] = c 4, c D and c E terms vanish in neutron matter in nuclear matter all terms contribute Basic idea: Sum one particle over occupied states in the Fermi sea = q,σ n(k F q) = q, σ

7 Chiral 3N interaction as density-dependent two-body interaction antisymmetrized 3N interaction (at N 2 LO) in neutron matter: = 1 2 ( ga 2f ) 2 i j k A ijk (σ i q i )(σ j q j ) (q 2 i + m2 )(q 2 j + m2 ) [ 4c 1m 2 f 2 + 2c 3 f 2 q i q j ] = c 4, c D and c E terms vanish in neutron matter in nuclear matter all terms contribute Provides 3N corrections to free space NN interaction: V = V NN + 1/c

8 Operator form of general momentum dependence : P-dependence only weak, evaluate for P =: P = = 1 2 ( ga 2f ) 2 [ 4c 1m 2 f 2 B(k, k )= 1 { ρ(k, k )(k + k ) 4 } 3 ((k + k ) 2 + m 2 ) 2 +2Bs 1(k, k ) B1(k, s k ) (B2(k, s k )+B2(k s, k)) + 1 { 2 3 (σ ρ(k, k )(k k ) 4 σ ) 3 ((k k ) 2 + m 2 ) ρ(k, k )(k + k ) 4 3 ((k + k ) 2 + m 2 ) 2 +B1(k, s k ) 1 3 [Bs 2(k, k )+B2(k s, k)] 2 } 3 [Bs 2(k, k )+B2(k s, k)] + 2 [ ρ(k, k )(k + k ) 2 S 12 (k + k ) 3 ((k + k ) 2 + m 2 ) 2 ρ(k, k )(k k ) 2 S 12 (k k ] ) ((k k ) 2 + m 2 ) σa σ b [ Bab(k, t k ) Bab(k, t k )+Bab(k t, k) Bab(k t, k) ] i (σa + σ a )[B v a(k, k ) B v a(k, k )] in neutron matter = (k, k, P) A(k, k )+ 2c 3 f 2 ] B(k, k )

9 Operator form of general momentum dependence : P-dependence only weak, evaluate for P =: P = = 1 2 ( ga 2f ) 2 [ 4c 1m 2 f 2 in neutron matter = (k, k, P) A(k, k )+ 2c 3 f 2 ] B(k, k ) B s 1(k 1, k 2 ) = B s 1(k, k ) d 3 q (2) 3 n(q)f R(Λ 3N, q, k 1 )f R (Λ 3N, q, k 2 ) ((k 1 + q) (k 2 + q)) 2 ((k 1 + q) 2 + m 2 )((k 2 + q) 2 + m 2 ) neglect P-dependence in the following, set P= in fixed-p approximation matrix elements have the same form like genuine free-space NN matrix elements straightforward to incorporate in existing many-body schemes

10 Partial wave matrix elements (Λ 3N =2. fm 1 ) 3N (, ) [fm] S F F F F = 1. fm-1 = 1.2 fm-1 = 1.4 fm-1 = 1.6 fm-1 3N (, ) [fm].2 3 P P [fm -1 ] 3N (, ) [fm] non-trivial density dependence (k, k ; 1 S ) k 4 F ρ 4/3 dominant central contributions non-central tensor and spin-orbit components 3N (, ) [fm] P F F F F = 1. fm-1 = 1.2 fm-1 = 1.4 fm-1 = 1.6 fm [fm -1 ] KH and A.Schwenk arxiv:

11 Low-momentum interactions from the RG Argonne v 18 AV18 k k fundamental ingredient of microscopic nuclear many-body calculations interaction constructed from experimental phase shifts and scattering lengths, interaction no physical observable! well constrained long-distance part, ill-defined short distance part hard core leads to technical problems in many-body calculations, nonperturbative

12 Low-momentum interactions from the RG Argonne v 18 AV18 k k RG evolution decouples low and high momenta universal interaction for low momenta V low k (Λ)

13 Equation of state (EOS): Many-body perturbation theory central quantity of interest: energy per particle E/N H(Λ) =T + V NN (Λ)+ (Λ)+.... E = + V NN + V NN V NN V NN V NN for low momentum interactions no resummation of diagrams necessary self-consistent single-particle propagators thermodynamic consistency

14 Neutron matter: EOS (first order), Test of fixed-p approximation E (1) full = + V NN + V E (1) eff = + V relative difference of 3N contributions only ~3% [fm -3 ] P-independent effective NN interaction is a very good approximation Energy/nucleon [MeV] (1) E NN (1) E NN+3N,full (1) E NN+3N,eff KH and A.Schwenk arxiv:

15 Neutron matter: EOS (second order) 2 (1) E NN+3N,eff E NN+3N,eff Energy/nucleon [MeV] [fm -3 ] 2. < 3N < 2.5 fm [fm -3 ] reduced cutoff dependence at 2nd order self-energy effects small system is perturbative for low-momentum interactions = 1.8 fm -1 = 2. fm -1 = 2.4 fm -1 = 2.8 fm -1 KH and A.Schwenk arxiv:

16 Neutron matter: EOS (second order) Energy/nucleon [MeV] E NN+3N,eff +c 3 +c 1 uncertainties E NN+3N,eff +c 3 uncertainty E (1) (2) NN + E NN 3N Energy/nucleon [MeV] E NN+3N,eff +c 3 +c 1 uncertainties Schwenk+Pethick (25) Akmal et al. (1998) Gezerlis+Carlson, QMC Carlson et al. (v6) Carlson et al. (v8) [fm -3 ] [fm -3 ] KH and A.Schwenk arxiv: energy sensitive to c 3 variations uncertainty due to coupling constants much larger than cutoff variation

17 Basics about neutron stars Structure of a neutron star is determined by Tolman-Oppenheimer-Volkov equation: dp dr = GMɛ r 2 [ 1+ P ][ ɛc r3 P Mc 2 ][ 1 2GM c 2 r ] 1 crucial ingredient: energy density ɛ = ɛ(p )

18 Neutron star radius Problem: Solution of TOV equation requires EOS up to very high densities. Radius of a typical NS (M~1.6 M ) theoretically not well constrained. But: Radius of NS is only relatively insensitive to high density region! parametrize piecewise EOS: 2 p 1 ρ 1 ρ 12 ρ Schwenk, Lattimer, Pethick, KH in prepararion

19 Neutron star radius Problem: Solution of TOV equation requires EOS up to very high densities. Radius of a typical NS (M~1.6 M ) theoretically not well constrained. But: Radius of NS is only relatively insensitive to high density region! p parametrize piecewise EOS: 1 2 M [M solar ] =1.5 1 =2.5 1 =3.5 1 =4.5 2 =1.5 2 =2.5 2 =3.5 2 =4.5 = 12 = 1 = 12 =1.5 s = 12 =2.5 s = 12 =3.5 s = 12 =4.5 s 1 =1.1 s ρ 1 ρ ρ.5 radius constrained to about 11 km ± 1.5 km R [km] Schwenk, Lattimer, Pethick, KH in prepararion

20 Symmetric nuclear matter Energy/nucleon [MeV] (1) E NN+3N,eff = 1.8 fm -1 = 2. fm -1 = 2.4 fm -1 = 2.8 fm [fm -3 ] (2) E NN+3N,eff [fm -3 ] Bogner, Furnstahl, Schwenk, Nogga, KH in preparation 3N forces crucial for saturation cutoff dependence at 2nd order significantly reduced self-energy effects significant, self-consistency of approximation crucial saturation density and binding energy consistent with experiment for coupling constants fitted to E3 H = MeV and 3rd order pp and hh contributions small r4 He = fm

21 Pairing gap in semi-magic nuclei Three-body mass difference: (3) (N) = ( 1)N 2 [E(N + 1) 2E(N)+E(N 1)] repulsive 3N contributions lead to suppression of the pairing gap Lesinski, Duguet, Schwenk, KH in preparation

22 Conclusions and Outlook derivation of density dependent effective NN interactions from 3N interactions in the zero P-approximation effective NN interaction efficient to use and accounts for 3N effects in nucleonic matter to good approximation neutron matter from low-momentum interactions more perturbative than nuclear matter due to large 3N and tensor contributions saturation properties of SNM consistent with experiment many-body constraints to c i couplings? generalization and application of to finite nuclei microscopic constraints of non-empirical EDFs