(Todays) Progress in coupled cluster compuations of atomic nuclei

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1 (Todays) Progress in coupled cluster compuations of atomic nuclei Gaute Hagen Oak Ridge National Laboratory Progress in Ab Initio Techniques in Nuclear Physics TRIUMF, February 26 th, 2019

2 ORNL / UTK: G. R. Jansen, T. Morris, S. J. Novario, T. Papenbrock, W. Chalmers: A. Ekström, C. UNC: Jon TRIUMF: Peter Gysbers, Petr Navratil

3 Pragmatists view of interactions from chiral EFT NNLO sat : Accurate radii and BEs Simultaneous optimization of NN and 3NFs Include charge radii and binding energies of 3 H, 3,4 He, 14 C, 16 O in the optimization Harder interaction: difficult to converge beyond 56 Ni A. Ekström et al, Phys. Rev. C 91, (R) (2015). 1.8/2.0(EM): Accurate BEs Soft interaction: SRG NN from Entem & Machleidt with 3NF from chiral EFT K. Hebeler et al PRC (2011). T. Morris et al, arxiv: (2017).

4 The 1.8/2.0 (EM) interaction J. Simonis, et al, Phys. Rev. C 96, (2017). G. Hagen et al, Nat. Phys. 12, 186 (2016) 3.5 R p (fmd B R n (fmd 1.8/2.0 (EM) is great for binding energies and spectra. Nuclear matter saturates at too high density à Radii too small

5 Role of delta isobars on nuclear saturation A. Ekström, et al, Phys. Rev. C 97, (2018) Pion-nucleon LECs fixed from recent analysis based on Roy- Steiner equations Short ranged contacts fixed from NN scattering and 4 He BE and charge radius Estimate uncertainties at given order following Epelbaum, Krebs, Meissner (2015) and Furnstahl, Klco, Phillips (2015).

6 Role of delta isobars on nuclear saturation A. Ekström, et al, Phys. Rev. C 97, (2018)

7 Role of delta isobars on nuclear saturation A. Ekström, et al, Phys. Rev. C 97, (2018)

8 Optimization strategy Use the empirical saturation point of nuclear matter and constraints on symmetry energy and its slope Fit NN to phase shifts (up to 200MeV) and deuteron properties 3NF fixed to reproduce A = 3, 4 nuclei Informed by BE and radii in medium mass nuclei Ingo Tews, et al. Symmetry parameter constraints from a lower bound on neutron-matter energy. The Astrophysical Journal, 848(2):105, NNLO(450): S = 32MeV, L = 65MeV

9 Scattering phaseshifts

10 Light nuclei TABLE I. Binding energies (E) in MeV, charge radii (R ch ) in fm, for 2,3 H and 3,4 He with NNLO GO (450), compared to experiment. NNLO GO (450) Expt. E( 2 H) R ch ( 2 H) P D ( 2 H) Q( 2 H) E( 3 H) R ch ( 3 H) E( 3 He) R ch ( 3 He) E( 4 He) R ch ( 4 He)

11 Natural orbitals in many-body approaches A. Tichai, J. Müller, K. Vobig, R. Roth, arxiv: (2018). NCSM with natural orbitals show significant improvement in convergence wrt model-space We follow A. Tichai et al and construct natural orbitals from many-body perturbation theory Aim: explore convergence in higher order coupled-cluster approaches to nuclei Ch. Constantinou, M. A. Caprio, J. P. Vary, P. Maris, Nucl. Sci. Tech. 28, 179 (2017)

12 16-O with natural orbitals

13 16-O with natural orbitals

14 Equation-of-motion with perturbative energy correction for excited states Diagonalize H = e T H N e T via equation-of-motion technique: R " = % r ( ' p * ( n ' % r '/ (0 p * ( N * 0 N / n ' % r '/4 (05 p * ( N * 0 N * 5 N 4 N / n ' Diagonalize in the P-space: Ẽ pqr =ẽ p +ẽ q +ẽ r apple Ẽ3max ẽ p = N p N F Correct perturbativelyfor all excitations outside of P: E µ = h 0 L µ H PQ (E µ H QQ ) 1 H QP R µ 0i

15 3 state in 16-O

16 3 state in 16-O

17 Oxygen isotopes 28 O bound by 300keV

18 Calcium isotopes 2 + ( 48 Ca) : 3.8MeV : 3.6MeV : 3.8MeV 1.8/2.0 (EM) results from J. Simonis et al, PRC 96, (2017) S. R. Stroberg (private comm.) S. Michimasa et al. PRL 121, (2018)

19 Calcium isotopes

20 Nuclear/neutron matter at CCD(T) S = 32MeV, L = 65MeV 23.6 <= S <= 33.3 MeV

21 Neutrinoless ββ-decay of 48 Ca 0νββ Nuclear matrix element for neutrinoless double beta decay in 48 Ca using different methods. From Y. Iwata et al, PRL (2016). The NME for 0νββ differ by a factor two to six depending on the method Need to determine the NME more precisely with quantified uncertainties What does ab-initio calculations add to this picture?

22 Neutrinoless ββ-decay of 48 Ca h 48 Ti O 48 Cai 2 = h 48 Ti O 48 Caih 48 Ca O 48 Tii Closure approximation with Gamow-Teller, Fermi and Tensor contributions: M 0 GT + gv g A 2 M 0 F + M 0 T The ground-state of 48 Ca is computed in the CCSD approximation: H N 0i = E 0 0i, H N = e T H N e T, T = T 1 + T 2 The CC energy functional is expressed in term of left/right ground-states h 0 (1 + )H N 0i = E 0, h 0 (1 + ) 0i =1. = X ia i aa a a i X ijab ij ab a ba a a i a j

23 Neutrinoless ββ-decay of 48 Ca 48 Ti is computed using a double charge exchange equation of motion method with 2p2h and 3p3h excitations R µ = 1 4 L µ = 1 4 X ijab X ijab r ab ij p ap b n in j l ij ab p bp a n i n j H N R µ 0i = E µ R µ 0i h 0 L µ H N = h 0 L µ E µ X ijkabc X ijkabc The Nuclear matrix element for 0νββ in 48 Ca is given by: r abc ijk p ap b N c N k n i n j l ijj abc p ap b N c N k n i n j h 48 Ti O 48 Cai 2 = h 48 Ti O 48 Caih 48 Ca O 48 Tii = h 0 L 0 O N 0ih 0 (1 + )O N R 0 0i

24 ββ-decay of 48 Ca with GXPF1A shell-model interaction

25 Benchmark between CC and NCSM in light nuclei

26 Benchmark between CC and NCSM in light nuclei

27 Benchmark between CC and NCSM in light nuclei

28 48 Ti from CR-EOM-CCSD(T)

29 48 Ti from CR-EOM-CCSD(T)

30 ββ-decay of 48 Ca h0 + f O GT 1 + µ ih1 + µ O GT 0 + i i M 2 = X µ E µ E i + Q /2 = h0 + f O GT 1 H E i + Q /2 O GT 0 + i i = h 0 L 0 O GT 1 H E i + Q /2 O GT 0i

31 Lanczos continued fraction method M 2 = h 0 L 0 O GT 1 H E i + Q /2 O GT 0i Define left/right Lanczos pivots: M 2 = h 0 0 i 8 >< >: (a 0 Q /2) h 0 = h 0 L 0 O GT 1 (a 1 Q /2) 0 i = O GT 0i b 2 0 b 2 1 (a 2 Q /2) 9 >= >; Lanczos continued fraction method, see e.g. Engel, Haxton, Vogel PRC (1992), Haxton, Nollett, Zurek PRC (2005), Miorelli et al PRC (2016). Matrix element is converged to machine precision after ~10 iterations. Need more than states converged in 48 Sc ( Lanczos iterations) if we sum explicitly over intermediate states

32 ββ-decay of 48 Ca with GXPF1A shell-model interaction Horoiet al, PRL (2007). 56 Ni FCI EOM-CCSD EOM-CCSDT-1 M 2v (GT) Gap

33 ββ-decay of 48 Ca The role of 3p3h excitations in the ground-state of 48 Ti

34 ββ-decay of 48 Ca The role of 3p3h excitations in the intermediate 1 + states of 48 Sc

35 Gamow-Teller strengths in 48 Ca GT-strength computed using the Lanczosmethod and EOM-CCSDT-1 No 2BCs, strength function folded with a Lorentzian of width 0.5MeV.

36 Summary Optimized chiral interactions with explicit delta s show significant improvement in description of light- and medium mass nuclei and infinite matter Natural orbitals offers a promising route to include higher order correlations in coupled-cluster computations First NME for 0νββ and 2νββ in 48 Ca from coupled-cluster calcualtions