Overview from Nuclear Lattice Effective Field Theory Serdar - PowerPoint PPT Presentation

Overview from Nuclear Lattice Effective Field Theory Serdar Elhatisari Nuclear Lattice EFT Collaboration HISKP , Universität Bonn Workshop on Polarized light ion physics with EIC Ghent University, Belgium February 5-9, 2018

Nuclear Lattice Effective Field Theory collaboration Serdar Elhatisari (Bonn) Ning Li (MSU) Evgeny Epelbaum (Bochum) Bing-Nan Lu (MSU) Nico Klein (Bonn) Thomas Luu (Jülich) Hermann Krebs (Bochum) Ulf-G. Meißner (Bonn/Jülich) Timo Lähde (Jülich) Gautam Rupak (MSU) Dean Lee (MSU) Gianluca Stellin (Bonn)

Outline � Introduction � Lattice effective field theory � Adiabatic projection method : scattering and reactions on the lattice � Degree of locality of nuclear forces � Nuclear clusters : probing for alpha clusters � Pinhole Algorithm : density profiles for nuclei � Summary

4/4/2017 nuchart1.gif (1054×560) Ab initio nuclear structure and nuclear scattering ✷ nuclear structure : Source: rarfaxp.riken.go.jp/ gibelin/Nuchart/ ✷ nuclear scattering : ... processes relevant for stellar astrophysics ✄ scattering of alpha particles : 4 He + 4 He → 4 He + 4 He ✄ triple-alpha reaction : 4 He + 4 He + 4 He → 12 C + γ ✄ alpha capture on carbon : 4 He + 12 C → 16 O + γ . . . http://rarfaxp.riken.go.jp/~gibelin/Nuchart/nuchart1.gif 1/1

Progress in ab initio nuclear structure and nuclear scattering Unexpectedly large charge radii of neutron-rich calcium isotopes. Garcia Ruiz et al. , Nature Phys. 12, 594 (2016). Structure of 78 Ni from first principles computations. Hagen, Jansen, & Papenbrock, PRL 117, 172501 (2016). A nucleus-dependent valence-space approach to nuclear structure. Stroberg et al. , PRL 118, 032502 (2017). Ab initio many-body calculations of the 3 H(d, n) 4 He and 3 He(d, p) 4 He fusion. Navratil & Quaglioni, PRL 108, 042503 (2012). 3 He( α , γ ) 7 Be and 3 H( α , γ ) 7 Li astrophysical S factors from the no-core shell model with continuum Dohet-Eraly, J. et al. PLB B 757 (2016) 430-436. Elastic proton scattering of medium mass nuclei from coupled-cluster theory. Hagen & Michel PRC 86, 021602 (2012). Coupling the Lorentz Integral Transform (LIT) and the Coupled Cluster (CC) Methods. Orlandini, G. et al. , Few Body Syst. 55, 907-911 (2014).

Nuclear LEFT: ab initio nuclear structure and scattering theory ✷ Lattice EFT calculations for A = 3, 4, 6, 12 nuclei, PRL 104 (2010) 142501 ✷ Ab initio calculation of the Hoyle state, PRL 106 (2011) 192501 ✷ Structure and rotations of the Hoyle state, PRL 109 (2012) 252501 ✷ Viability of Carbon-Based Life as a Function of the Light Quark Mass, PRL 110 (2013) 112502 ✷ Radiative capture reactions in lattice effective field theory, PRL 111 (2013) 032502 ✷ Ab initio calculation of the Spectrum and Structure of 16 O, PRL 112 (2014) 102501 ✷ Ab initio alpha-alpha scattering, Nature 528, 111-114 (2015). ✷ Nuclear Binding Near a Quantum Phase Transition, PRL 117, 132501 (2016). ✷ Ab initio calculations of the isotopic dependence of nuclear clustering. ���������������������� PRL 119, 222505 (2017). �� �� E � – E α � A /4 ���������� � αα ���� � � αα ���� ∞ � � λ �� λ �� λ �� λ � � �� ��� λ λ ���� λ ���� ������������ � �� � ���� ! � � ���� � ����� ! � ����� ! � � ���� � ����� ! ���� ��������� �������������� � � � � � �� �� �� �� �� �� �� �� �� �� � 25 ������������������������������ � ���������������������� � �����������������������������

Lattice effective field theory Lattice effective field theory is a powerful numerical method formulated in the framework of chiral effective field theory. Accessible by Lattice QCD early quark-gluon 100 universe plasma heavy-ion collisions T [MeV] Accessible by 10 gas of light Nucleons Lattice EFT nuclear nuclei liquid excited nuclei superfluid neutron star crust neutron star core 𝑏 𝑀 1 -3 -2 -1 1 10 10 10 r r -3 [fm ] N Fig. courtesy of D. Lee

Chiral EFT for nucleons: nuclear forces Chiral effective field theory organizes the nuclear interactions as an expansion in powers of momenta and other low energy scales such as the pion mass ( Q / Λ χ ) . 2N force 3N force 4N force LO NLO N LO 2 3 N LO Fig. courtesy of E.Epelbaum Ordonez et al. ’94; Friar & Coon ’94; Kaiser et al. ’97; Epelbaum et al. ’98,’03,’05,’15; Kaiser ’99-’01; Higa et al. ’03; ...

Chiral EFT for nucleons: NN scattering phase shifts 80 180 10 10 70 8 8 160 60 6 6 δ ( 1 S 0 ) [degrees] δ ( 3 S 1 ) [degrees] 140 ε 1 [degrees] ε 2 [degrees] 50 4 4 120 40 2 2 100 30 0 0 80 20 NPWA -2 -2 LO 60 10 -4 -4 NLO,N2LO N3LO 0 40 -6 -6 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 p CM [MeV] p CM [MeV] p CM [MeV] p CM [MeV] 10 20 5 14 12 5 15 0 10 δ ( 1 P 1 ) [degrees] δ ( 3 P 0 ) [degrees] δ ( 3 P 1 ) [degrees] δ ( 3 P 2 ) [degrees] 8 0 10 -5 6 -5 4 5 -10 2 -10 0 0 -15 -15 -2 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 p CM [MeV] p CM [MeV] p CM [MeV] p CM [MeV] 10 2 16 5 14 0 4 8 12 δ ( 3 D 1 ) [degrees] δ ( 3 D 2 ) [degrees] δ ( 3 D 3 ) [degrees] δ ( 1 D 2 ) [degrees] -2 3 6 10 -4 2 8 4 6 -6 1 2 4 -8 0 2 0 -10 -1 0 -2 -12 -2 -2 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 p CM [MeV] p CM [MeV] p CM [MeV] p CM [MeV]

Lattice Monte Carlo calculations M = : exp ( − H a t ) : Transfer matrix operator formalism Microscopic Hamiltonian H = H free + V Z ( L t ) = Tr ( M L t ) = � Dc Dc ∗ exp [ − S ( c , c ∗ )] Creutz, Found. Phys. 30 (2000) 487. The exact equivalence of several different lattice formulations. Lee, PRC 78:024001, (2008); Prog.Part.Nucl.Phys., 63:117-154 (2009) e − E 0 a t = lim L t → ∞ Z ( L t + 1 ) / Z ( L t ) These amplitudes are computed with the Hybrid Monte Carlo methods. Phys. Lett. B195, 216-222 (1987), Phys. Rev. D35, 2531-2542 (1987).

Lattice Monte Carlo calculations Nuclear forces posses approximate SU ( 4 ) symmetry. H SU ( 4 ) acts as an approximate and inexpensive low energy filter at few first/last time steps. Significant suppression of sign oscillations. Chen, Lee, Schäfer, PRL 93 (2004) 242302 τ ′ = L ′ | ψ I ( τ ′ ) � = [ M SU ( 4 ) ] L ′ t | ψ I � M SU ( 4 ) = : e − a t H SU ( 4 ) : t a t For time steps in midsection, the full H LO Hamiltonian is used. | ψ I ( τ ) � = [ M LO ] L t | ψ I ( τ ′ ) � M LO = : e − a t H LO : τ = L t a t The ground state energy at LO can be extracted from Z ( L t + 1 ) � ψ I ( τ /2 ) | M LO | ψ I ( τ /2 ) � e − E LO a t = lim LO = lim Z ( L t ) � ψ I ( τ /2 ) | ψ I ( τ /2 ) � L t → ∞ L t → ∞ LO

Lattice Monte Carlo calculations Higher order calculations (perturbative) ho = NLO, NNLO, · · · M ho = : e − a t ( H LO + V ho ) : where the potential V ho is treated perturbatively. The higher order correction to the ground state energy can be extracted from e − ∆ E ho a t = Z ( L t + 1 ) = � ψ I ( τ /2 ) | M ho | ψ I ( τ /2 ) � ho Z ( L t + 1 ) � ψ I ( τ /2 ) | M LO | ψ I ( τ /2 ) � LO

Lattice Monte Carlo calculations Compute observable O The observable O at LO � ψ I | [ M LO ] L t O [ M LO ] L t | ψ I � � O � 0, LO = lim � ψ I | [ M LO ] 2 L t + 1 | ψ I � L t → ∞ The observable O at (NLO, NNLO, · · · ) � ψ I | [ M LO ] L t − 1 M ho O [ M LO ] L t | ψ I � � O � 0, ho = lim � ψ I | [ M LO ] L t M ho [ M LO ] L t | ψ I � L t → ∞

Lattice EFT: (Euclidean time) projection Monte Carlo e − H τ τ = L t a t 𝛒 𝛒 Euclidean time 𝑀 ✷ evolve nucleons forward in Euclidean time 𝑏 ✷ allow them to interact

Auxiliary field Monte Carlo Use a Gaussian integral identity √ � − s 2 � � 2 � � �� − C 1 � � � N † N N † N exp = ds exp 2 + C s 2 2 π s is an auxiliary field coupled to particle density. Each nucleon evolves as if a single particle in a fluctuating background of pion fields and auxiliary fields. 𝒕 𝑱 Euclidean time 𝒕 𝝆 𝛒 𝛒 𝒕

Adiabatic projection method Scattering and reactions on the lattice The first part use Euclidean time projection to construct an ab initio low-energy cluster Hamiltonian, called the adiabatic Hamiltonian. � � The second part compute the two-cluster scattering phase shifts or reaction amplitudes using the adiabatic Hamiltonian. −�/2 �/2 Rupak, Lee., PRL 111 (2013) 032502. 1 1 1 interacting free Pine, Lee, Rupak, EPJA 49 (2013) 151. SE, Lee, PRC 90, 064001 (2014). 0.5 0.5 0.5 R w 0 ( r ) 0 ( r ) 0 ( r ) Rokash, Pine, SE, Lee, Epelbaum, Krebs, PRC 92,054612 (2015) R ( p ) R ( p ) R ( p ) SE, Lee, Meißner, Rupak, EPJA 52: 174 (2016) 0 0 0 -0.5 -0.5 -0.5 0 0 0 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 r (fm) r (fm) r (fm)