Ab initio lattice EFT from light to medium mass nuclei Nuclear Lattice EFT Collaboration

Transcription

1 Ab initio lattice EFT from light to medium mass nuclei Nuclear Lattice EFT Collaboration Evgeny Epelbaum (Bochum) Hermann Krebs (Bochum) Timo Lähde (Jülich) Dean Lee (NC State) Thomas Luu (Jülich/Bonn) Ulf-G. Meißner (Bonn/Jülich) Gautam Rupak (MS State) ECT* Workshop, Trento Three Body Forces: From Matter to Nuclei May 8,

2 Outline Introduction to lattice effective field theory Oxygen-16 structure and spectrum Ab initio calculations of light and medium mass nuclei along N = Z Physics insight from cluster effective field theory Symmetry sign extrapolation Beryllium-10 and carbon-10 ground state Summary and future directions 2

3 Lattice effective field theory n n p n p 3

4 Low energy nucleons: Chiral effective field theory 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) 4

5 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; 5

6 Leading order on the lattice p n p 6

7 Next-to-leading order on the lattice p p p p p p... 7

8 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 8

9 Three nucleon forces Two unknown coefficients at NNLO from three-nucleon forces. Determine c D and c E using 3 H binding energy and the weak axial current at low cutoff momentum. c E c D c 1, c 3, c 4 c D c 1, c 3, c 4 Park, et al., PRC 67 (2003) , Gårdestig, Phillips, PRL 96 (2006) , Gazit, Quaglioni, Navratil, PRL 103 (2009)

10 Neutrons and protons: Isospin breaking and Coulomb Isospin-breaking and power counting [Friar, van Kolck, PRC 60 (1999) ; Walzl, Meißner, Epelbaum NPA 693 (2001) 663; Friar, van Kolck, Payne, Coon, PRC 68 (2003) ; Epelbaum, Meißner, PRC72 (2005) ] Pion mass difference 10

11 Coulomb potential p p Charge symmetry breaking Charge independence breaking p p n p n n 11

12 Euclidean time projection p p 12

13 Auxiliary field method 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. 13

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15 Schematic of lattice Monte Carlo calculation Hybrid Monte Carlo sampling 15

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17 Particle clustering included automatically 17

18 Oxygen-16 ground state LO NLO EM & IB 3NF Epelbaum, Krebs, Lähde, D.L, Meißner, Rupak, PRL112, (2014) 18

19 Oxygen-16 spectrum and structure 19

20 A - Tetrahedral structure B,C - Square-like structure 20

21 NLO NNLO 21

22 tetrahedral square-like 22

23 (overbinding to be discussed shortly) Epelbaum, Krebs, Lähde, D.L, Meißner, Rupak, PRL112, (2014) 23

24 Epelbaum, Krebs, Lähde, D.L, Meißner, Rupak, PRL112, (2014) 24

25 Neon-20 ground state LO NLO EM & IB 3NF Lähde, Epelbaum, Krebs, D.L, Meißner, Rupak, PLB732, 110 (2014) 25

26 Magnesium-24 ground state LO NLO EM & IB 3NF Lähde, Epelbaum, Krebs, D.L, Meißner, Rupak, PLB732, 110 (2014) 26

27 Lattice results for nuclei along N = Z = even Lähde, Epelbaum, Krebs, D.L, Meißner, Rupak, PLB732, 110 (2014) 27

28 Physical insight from Cluster EFT Soft long-distance part of cluster interactions are good. But short-distance part probes higher nucleon momenta and are less accurate. The errors can be viewed as local counterterms in the cluster EFT. Think of pionless EFT at LO for bosons. 28

29 The short-distance part of the cluster interactions are probed more and more with increasing A. What can be done? The best solution is to work at high enough order and high enough cutoff momentum to fix the discrepancies to the desired accuracy. Lattice calculations at N3LO are underway. A less ambitious but pragmatic approach is to exploit the low-energy universality of cluster EFT. Include corrections from higher-order operators which leave the low-energy nucleon interactions the same but can fine tune the alpha binding energy, alpha-alpha contact interaction, and three-alpha contact interaction. Many ways to do this and the exact operators used are not essential. For example, a local 4N contact interaction; an order Q 4, S = I = 1, 2N contact interaction; and an S = I = 3/2, 3N contact interaction. 29

30 tune to the 1, 2, 3 alpha systems Work in progress 30

31 Preliminary results: Work in progress 31

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33 Sign oscillations There are different types of sign oscillation problems. In explicit fermion simulations such as diffusion or Green s function Monte Carlo, the sign problem is due to Fermi statistics. In implicit fermion simulations such as auxiliary-field Monte Carlo, the sign problem is predominantly due to repulsive interactions. 33

34 Theorem: Any fermionic theory with SU(2N) symmetry and two-body potential with negative semi-definite Fourier transform obeys SU(2N) convexity bounds. Corollary: System can be simulated without sign oscillations E weak attractive potential Chen, D.L. Schäfer, PRL 93 (2004) ; D.L., PRL 98 (2007) NK 2 N ( K +1) 2 N ( K +2) A A 2 NK 2 N ( K +1) 2 N ( K +2) E strong attractive potential 34

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36 Symmetry Sign Extrapolation Start with a physical Hamiltonian of interest that has a sign oscillation problem Suppose there exists some SU(2N)-invariant Hamiltonian that can be tuned to reproduce the overall size, binding, and other relevant observables for the system of interest Construct a one-parameter family interpolating between the two Hamiltonians 36

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38 We now demonstrate the method for lattice simulations of the ground state of beryllium-10 and carbon-10. For each energy observable we parameterize the asymptotic dependence of the projected energy as 38

39 Beryllium-10 LO NLO 39

40 Beryllium-10 EM & IB NNLO 40

41 Carbon-10 EM & IB 41

42 Preliminary results for ground state energies: Work in progress 42

43 3N Status and Needs We are currently working on including 2N and 3N forces up to N3LO on the lattice We need to improve the algorithmic efficiency for Monte Carlo sampling for higher-order operators and computing higher orders in perturbation theory We are also exploring the possibility of nonperturbative higher-order 2N terms and 3N forces in simulations for higher A as needed. 43

44 Summary A golden age for nuclear theory from first principles. Big science discoveries being made and many more around the corner. Lattice effective field theory is a relatively new and promising tool that combines the framework of effective field theory and computational lattice methods. May play a significant role in the future of ab initio nuclear theory. Topics being addressed now and in the near future Different lattice spacings, decoupling lattice spacing from ultraviolet regulator, N Z nuclei, elastic and inelastic reactions, neutron matter equation of state and superfluid transition from S-wave to P-wave, etc. 44