Towards chiral three-nucleon forces in heavy nuclei

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1 Towards chiral three-nucleon forces in heavy nuclei Victoria Durant, Kai Hebeler, Achim Schwenk Nuclear ab initio Theories and Neutrino Physics INT, March 2nd /17

2 Why three-body forces? They are key to explain and predict the nuclear chart Medium-mass nuclei properties in good agreement with experiment Oxygen dripline Otsuka et al.,prl 105, (2010) Wienholtz et al., Nature 498, (2013) 2/17

3 Why three-body forces? They are key to explain and predict the nuclear chart Medium-mass nuclei properties in good agreement with experiment Promising results for heavy nuclei Binder et al., PLB 736, (2014) 2/17

4 Why three-body forces? They are key to explain and predict the nuclear chart Medium-mass nuclei properties in good agreement with experiment Promising results for heavy nuclei Tichai et al., PLB 756, (2016) 2/17

5 Chiral EFT Chiral EFT: expansion in powers of Q/Λ b. Q m π 100 MeV; Λ b 500 MeV Long-range physics: given explicitly (no parameters to fit) by pion exchanges. Short-range physics: contact interactions with low-energy constants (LECs) fit to πn, NN, 3N,... data. Many-body forces and currents enter systematically. 3/17

6 Chiral EFT An effective way to treat 3NF is through normal ordered matrix elements, performed in single-particle basis. 4/17

7 Chiral EFT An effective way to treat 3NF is through normal ordered matrix elements, performed in single-particle basis. Residual three-body forces are small. 4/17

8 Chiral EFT An effective way to treat 3NF is through normal ordered matrix elements, performed in single-particle basis. Very good agreement for different methods in medium-mass nuclei and with experiment. Energy (MeV) obtained in large many-body spaces MR-IM-SRG 8 IT-NCSM MBPT SCGF 6 CC Lattice EFT 4 SCGF CC AME MR-IM-SRG Mass Number A Neutron Number N Hebeler et al., Ann. Rev. Nucl. Part. Sci. 65, 457 (2015) S 2n (MeV) 4/17

9 Chiral EFT An effective way to treat 3NF is through normal ordered matrix elements, performed in single-particle basis. Very good agreement for different methods in medium-mass nuclei and with experiment. Energy (MeV) obtained in large many-body spaces MR-IM-SRG 8 IT-NCSM MBPT SCGF 6 CC Lattice EFT 4 SCGF CC AME MR-IM-SRG Mass Number A Neutron Number N Hebeler et al., Ann. Rev. Nucl. Part. Sci. 65, 457 (2015) S 2n (MeV) Challenges to overcome: improvement of the Hamiltonian. expansion of the reachable many-body space. 4/17

10 Many-body space E 1 + E 2 + E 3 E 3max Currently E 3max 16 ω = E 3max 3E occ The space of interaction between three particles is reduced for heavy nuclei! 5/17

11 Many-body space Difference of relative errors for E 3max = 12 ω and E 3max = 14 ω in ground state energies ( E 3max ) α = 0.02 fm 4, α = 0.04 fm 4, α = 0.08 fm 4 E 3max grows with A. Convergence is challenging for heavy nuclei. 5/17

12 Normal ordered matrix elements To determine the normal-ordered interaction, we need the matrix elements evaluated at single-particle coordinates. ab Veff a b = abc V3N a b c Matrix elements are stored in Jacobi basis. c Regular strategy: V 3N stored = pq V 3N p q 6/17

13 Normal ordered matrix elements Challenges: Transformation to m-scheme basis is slow. Storage of matrix elements is challenging memory-wise. Expansion of the model space following the regular strategy is difficult! Roth et al., Phys. Rev. C 90 (2013) 7/17

14 Normal ordered matrix elements Key points of the derivation: Our goal: perform 2BNO in Jacobi basis. Start with 3rd particle in single particle basis Expression involving kak b k c V 3N k ak b k c Transformation to Jacobi basis Only approximation: P = k a + k b = 0 Partial wave decomposition. kakb V eff k a k b = kakb c V 3N k a k c b c 8/17

15 Normal ordered matrix elements Key points of the derivation: Our goal: perform 2BNO in Jacobi basis. Start with 3rd particle in single particle basis Expression involving kak b k c V 3N k ak b k c Transformation to Jacobi basis Only approximation: P = k a + k b = 0 Partial wave decomposition. kakb V eff k a k b = dk 3 dk 3 c k 3 k 3 c kak b k c V 3N k a k b k c c 8/17

16 Normal ordered matrix elements Key points of the derivation: Our goal: perform 2BNO in Jacobi basis. Start with 3rd particle in single particle basis Expression involving kak b k c V 3N k ak b k c Transformation to Jacobi basis Only approximation: P = k a + k b = 0 Partial wave decomposition. pp Veff p P = dk 3 dk 3 c k 3 k 3 c c pq V3N p q δ 3 (P + k 3 P k 3 ) 8/17

17 Normal ordered matrix elements Our goal: perform 2BNO in Jacobi basis. Key points of the derivation: Start with 3rd particle in single particle basis Expression involving kak b k c V 3N k a k b k c Transformation to Jacobi basis Only approximation: P = k a + k b = 0 Partial wave decomposition. Drischler et al., Phys. Rev. C 93 (2016) 8/17

18 Normal ordered matrix elements Final result p(ls)jt V eff p (L S )JT = ( i) L L (4π) 2 (2π) 3 n c,l c J,j c,t occupied 2lc + 1 ( 4π P l c cos(θ k ) 3 2J + 1 2J + 1 2T + 1 2T + 1 k 2 3 dk 3 k 2 3 dk 3 dcos(θ k 3 ) ) R nclc (k 3 )R nclc (k 3 ) p, 2 3 k 3, α V 3N p, 1 3 (k 3 + k 3 ), α 9/17

19 Normal ordered matrix elements Final result p(ls)jt V eff p (L S )JT = ( i) L L (4π) 2 (2π) 3 n c,l c J,j c,t occupied 2lc + 1 ( 4π P l c cos(θ k ) 3 2J + 1 2J + 1 2T + 1 2T + 1 k 2 3 dk 3 k 2 3 dk 3 dcos(θ k 3 ) ) R nclc (k 3 )R nclc (k 3 ) p, 2 3 k 3, α V 3N p, 1 3 (k 3 + k 3 ), α effective two-body matrix elements (relative quantum numbers). 9/17

20 Normal ordered matrix elements Final result p(ls)jt V eff p (L S )JT = ( i) L L (4π) 2 (2π) 3 n c,l c J,j c,t occupied 2lc + 1 ( 4π P l c cos(θ k ) 3 2J + 1 2J + 1 2T + 1 2T + 1 k 2 3 dk 3 k 2 3 dk 3 dcos(θ k 3 ) ) R nclc (k 3 )R nclc (k 3 ) p, 2 3 k 3, α V 3N p, 1 3 (k 3 + k 3 ), α effective two-body matrix elements (relative quantum numbers). occupation number regulated by harmonic oscillator wave function. Σ occ ((2l 3 +1)/4π) R n3 l 3 (3/2q) 2 [fm 3 ] Occupation distribution according to the number of particles in the core 20 particles 28 particles 50 particles 82 particles 126 particles q [fm -1 ] 9/17

21 Normal ordered matrix elements Final result p(ls)jt V eff p (L S )JT = ( i) L L (4π) 2 (2π) 3 n c,l c J,j c,t occupied 2lc + 1 ( 4π P l c cos(θ k ) 3 2J + 1 2J + 1 2T + 1 2T + 1 k 2 3 dk 3 k 2 3 dk 3 dcos(θ k 3 ) ) R nclc (k 3 )R nclc (k 3 ) p, 2 3 k 3, α V 3N p, 1 3 (k 3 + k 3 ), α effective two-body matrix elements (relative quantum numbers). occupation number regulated by harmonic oscillator wave function. three-body matrix elements in Jacobi basis as a function of k c and k c. Σ occ ((2l 3 +1)/4π) R n3 l 3 (3/2q) 2 [fm 3 ] Occupation distribution according to the number of particles in the core particles 28 particles 50 particles particles 126 particles q [fm -1 ] 9/17

22 Normal ordered matrix elements Extra steps towards single-particle basis: Transformation to harmonic oscillator basis n(ls)jt Veff n (L S )JT = dpp 2 R nl (p)dp p 2 R n L (p ) p V p Talmi-Moshinsky transformation. n(ls)jt Veff n (L S )JT [ ] [ 1 n 1 n 2 (l 1 2 )j 1 1(l 2 2 )j 2 J V eff n 1 n 2 (l )j 1 (l 2 2 )j 2 ] J 10/17

23 Comparison with exact matrix elements 16 O c E =1 (rest LECs zero) x=y 1 <V> approx <V> exact 11/17

24 Comparison with exact matrix elements 16 O c E =1 (rest LECs zero) x=y 1 <V> approx 0.1 source of discrepancies? <V> exact 11/17

25 Inclusion of center-of-mass degrees of freedom Get rid of the approximation P = 0. Simplified expression feasible for an S-wave interaction (c E term of the Hamiltonian) pp [(LS)j rel L cm]jt V eff p P [(L S )j rel L cm]jt ce 2J + 1 2T + 1 d(cos θ P )P Lcm (cos θ P ) 2S + 1 2T + 1 c J T s 3 s 3 d 3 k 3 R nclc (k 3 )R nclc (k 3 ) 2lc + 1 4π P l c (cos(ˆk 3 ˆk 3 )) p, 2 3 k 3 P, α V 3N p, k P P, α effective two-body matrix elements (relative and center-of-mass quantum numbers). center-of-mass angular contribution. occupation number regulated by harmonic oscillator wave function. three-body matrix elements in Jacobi basis as a function of k c, P and P. 12/17

26 Normal ordered matrix elements Extra steps towards single-particle basis: Transformation to harmonic oscillator basis Nn[(LS)jrel L cm]j V eff N n [(L S )j rel L ]J cm = dp P 2 R NLcm (P ) dp P 2 R N L cm (P ) dpp 2 R nl (p) dp p 2 R nl (p ) P p[(ls)jrel L cm]j V eff P p [(L S )j rel L cm]j. (Modified) Talmi-Moshinsky transformation including center-of-mass degrees of freedom as input. nn[(ls)jrel L cm]jt V eff n [(L S )j rel L ]JT cm ] [ ] [ 1 n 1 n 2 (l 1 2 )j 1 1(l 2 2 )j 2 J V eff n 1 n 2 (l )j 1 (l 2 2 )j 2 J 13/17

27 Results for S-wave Hamiltonians rs: 4 He, h _ ω= ,4 Reference state: 4 He. energy of particles in reference state: e rs = 0 e Max =6. Perfect agreement with reference matrix elements. <V JB > 0,2 0-0,2-0,4-0,6-0,8-0,8-0,6-0,4-0,2 0 0,2 0,4 <V ref > 14/17

28 Motivation Normal ordering P=0 Exact P Summary Results for S-wave Hamiltonians 16 _ rs: O, hω= Reference state: 16 O. ea+eb<12 ea+eb=12 Energy of particles in reference state: ers = 0, 1 1 For the 2BNO taken as reference, the condition E3N = ea + eb + ec 12 is applied. Perfect agreement with reference matrix elements with energy up to 12. <VJB> em ax = <Vref> /17

29 Inclusion of center-of-mass degrees of freedom: general pp [(LS)j rel L cm] J V [ p P (L S )j rel cm] L J c α,α M cm,m j,m J T J,M J m j,m j C JM J j rel M j L cmm cm C JM J j rel M j L cmm cm C J M J j rel M j jm C jm j C J M J j lm l 1/2m sc j rel M j j m C j m j l j m l 1/2msc 1 + 2L cm 4π dθ P sin θ P Y L cm M cm (θ P, 0)Ylm l (ˆq)Y l m (ˆq ) l dk3 (2π) 3 R nclc (k 3 )R nclc (k 3 ) 2l c + 1 4π P l c (k 3 k ) 3 pqα V 3N p q α n c,l c effective two-body matrix elements (relative and center-of-mass quantum numbers). center-of-mass angular contribution. occupation number regulated by harmonic oscillator wave function. three-body matrix elements in Jacobi basis as a function of k c, P and P. 16/17

30 Summary Current limitations in treatment of three-body forces will lead to significant truncation effects in calculations for heavy nuclei. The approximtion P=0 is not applicable to finite nuclei calculations. Purely S-wave interaction normal ordered matrix elements including center-of-mass degrees of freedom agree with reference calculations. Generalization of the input Hamiltionian in progress... Thank you for your attention! 17/17

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