Quantum Monte Carlo calculations for light nuclei using chiral - PowerPoint PPT Presentation

Quantum Monte Carlo calculations for light nuclei using chiral forces Joel Lynn Theoretical Division, Los Alamos National Laboratory March 19, 2014

Outline 1 Motivation Ab-initio calculations for nuclei Nuclear interactions Phenomenology Chiral Effective Field Theory - Standard approach Chiral Effective Field Theory - A new approach 2 Results A ≤ 4 binding energies A ≤ 4 radii Perturbative calculations Distributions 3 Conclusion Summary Future work Acknowledgments

Motivation Ab-initio calculations for nuclei - Quantum Monte Carlo (QMC) Nuclear structure methods seek to solve the many-body Schr¨ odinger equation H | Ψ � = E | Ψ � . Variational Monte Carlo (VMC) uses a Metropolis random walk to calculate an upper bound to the ground-state energy: E T = � Ψ T | H | Ψ T � ≥ E 0 . � Ψ T | Ψ T � Green’s function Monte Carlo (GFMC) uses propagation in imaginary time to project out the ground state. | Ψ( τ ) � = e − H τ | Ψ T � ⇒ lim τ →∞ | Ψ( τ ) � ∝ | Ψ 0 � .

Motivation Ab-initio calculations for nuclei - QMC GFMC enjoys a reputation as the most accurate method for solving the many-body Schr¨ odinger equation for light nuclei 4 < A ≤ 12. First: VMC. ◮ We begin with a trial wave function Ψ T and generate a random position: R = r 1 , r 2 , . . . , r A . ◮ Use the Metropolis algorithm to generate new positions R ′ based on the probability P = | Ψ T ( R ′ ) | 2 | Ψ T ( R ) | 2 . ◮ This gives us a set of “walkers” distributed according to the trial β c β | R β � . 3 A positions and 2 A � A � wave function: � spin/isospin Z states in the charge basis.

Motivation Ab-initio calculations for nuclei - QMC Second: GFMC. ◮ The wave function is imperfect: Ψ T = Ψ 0 + � i � =0 c i Ψ i . ◮ Propagate in imaginary time to project out the ground state Ψ 0 :   � Ψ( τ ) = e − ( H − E T ) τ Ψ T = e − ( E 0 − E T ) τ c i e − ( E i − E 0 ) τ Ψ i  Ψ 0 +  i � =0 ⇒ lim τ →∞ Ψ( τ ) ∝ Ψ 0 .

Motivation Ab-initio calculations for nuclei - QMC Second: GFMC. The Green’s function is calculated by introducing the short-imaginary time ∆ τ = τ/ n . Ψ( τ ) = [ e − ( H − E T )∆ τ ] n Ψ T � �� � G αβ ( R , R ′ ;∆ τ ) G αβ ( R , R ′ ; ∆ τ ) = � R α | e − ( H − E T )∆ τ | R ′ β � � Ψ( R n , τ ) = d R G ( R n , R n − 1 ) · · · G ( R 1 , R 0 )Ψ T ( R 0 ) n − 1 � d R = d R i i =0

Motivation Ab-initio calculations for nuclei - QMC Second: GFMC. ◮ We can calculate so-called “mixed estimates”: � d R Ψ † T ( R n ) G † ( R n , R n − 1 ) · · · G † ( R 1 , R 0 ) O Ψ T ( R 0 ) � Ψ( τ ) | O | Ψ T � = . � d R Ψ † � Ψ( τ ) | Ψ T � T ( R n ) G † ( R n , R n − 1 ) · · · G † ( R 1 , R 0 )Ψ T ( R 0 ) � O ( τ ) � = � Ψ( τ ) | O | Ψ( τ ) � ≈ � O ( τ ) � Mixed + [ � O ( τ ) � Mixed − � O � T ] . � Ψ( τ ) | Ψ( τ ) � ◮ For ground-state energies, O = H , and [ H , G ] = 0: � H � Mixed = � Ψ T | e − ( H − E T ) τ/ 2 He − ( H − E T ) τ/ 2 | Ψ T � � Ψ T | e − ( H − E T ) τ/ 2 e − ( H − E T ) τ/ 2 | Ψ T � τ →∞ � H � Mixed = E 0 . lim

Motivation Nuclear interactions - Nucleons A fundamental goal of low-energy nuclear physics is to describe and calculate properties of nuclei in terms of realistic bare nuclear interactions. Quantum chromodynamics (QCD) is the underlying theory, but nucleons are the relevant degrees of freedom for low-energy nuclear physics → nucleon-nucleon potentials. Figure 1: From www.scidacreview.org

Motivation Nuclear interactions - The Hamiltonian A A A p 2 � � � i H = + v ij + V ijk + · · · 2 m i i =1 i < j i < j < k The focus of this talk is on the two-body interaction. Until now, there were two broad choices for v ij . Local, real-space, phenomenological: Argonne’s v 181 - informed by theory, phenomenology, and experiment (well tested and very successful). Non-local, momentum-space, effective field theory (EFT): N 3 LO 2 - informed by chiral EFT and experiment (well liked and often used in basis-set methods, such as the no-core shell model). 1 R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C 51 , 38 (1995). 2 e.g. D. R. Entem and R. Machleidt, Phys. Rev. C 68 , 041001 (2003)

Motivation Nuclear interactions - Argonne’s v 18 Argonne’s v 18 consists of three parts. v ij = v γ ij + v π ij + v R ij . v γ ij includes one- and two-photon exchange Coulomb interactions, vacuum polarization, Darwin-Foldy, and magnetic moment terms with appropriate proton and neutron form factors. v π ij includes charge-dependent terms due to the difference in neutral and charged pion masses. v R ij is a short-range phenomenological potential.

Motivation Nuclear interactions - Argonne’s v 18 Operator form 18 � v p ( r ij ) O p v π ij + v R ij = ij . p =1 Charge-independent operators � 1 , σ i · σ j , S ij , L · S , L 2 , L 2 ( σ i · σ j ) , ( L · S ) 2 � O p =1 , 14 = ⊗ [1 , τ i · τ j ] . ij Charge-independence-breaking operators O p =15 , 18 = [1 , σ i · σ j , S ij ] ⊗ T ij , and ( τ zi + τ zj ) . ij Tensor operators S ij = 3( σ i · ˆ r ij )( σ j · ˆ r ij ) − σ i · σ j , T ij = 3 τ zi τ zj − τ i · τ j

Motivation Nuclear interactions - Argonne’s v 18 -20 1 + 4 + 2 + 7/2 − 2 + 0 + 2 + 0 + 5/2 − 0 + -30 0 + 3 + 0 + 1 + 4 He 2 + 5/2 − 1 + 6 He 4 + 8 He 2 + 7/2 + 7/2 − 4 + 6 Li 3 + 5/2 + 1/2 − 1 + 5/2 − -40 2 + 1 + 3 + 7/2 − 3/2 − 1/2 − 3 + 2 + 7/2 − 1 + 3/2 − 7 Li 2 + 4 + 2 + 3/2 − 3 + -50 9 Li Energy (MeV) 3 + 3/2 + 1 + 4 + 8 Li 2 + 1 + 5/2 + 3,2 + 2 + 0 + 1 + 0 + Argonne v 18 1/2 − 3 + -60 8 Be 5/2 − 2 + 2 + 2 + 1/2 + 1 + with Illinois-7 3/2 − 0 + 1 + -70 10 Be 3 + GFMC Calculations 9 Be 10 B 24 November 2012 -80 0 + AV18 -90 AV18 0 + +IL7 Expt. 12 C -100 Figure 2: Many excellent results using Green’s function Monte Carlo (GFMC) and phenomenological potentials. From http://www.phy.anl.gov/theory. This is great! But... Until now the nucleon-nucleon potentials used have been restricted to the phenomenological Argonne-Urbana/Illinois family of interactions.

Motivation Nuclear interactions - Chiral EFT Chiral EFT makes a more direct connection between QCD and the nuclear force. Weinberg prescription Start from the most general Lagrangian consistent with all symmetries of the underlying interaction... L = L ππ + L π N + L NN + · · · Define a power-counting scheme... ν = − 4+2 N +2 L + � i V i ∆ i , ∆ i = d i + 1 2 n i − 2. Figure 3: Hierarchy of the nuclear force in chiral EFT, from R. Machleidt and D. Entem, Phys. Rep. 503 , 1 (2011).

Motivation Nuclear interactions - Chiral EFT Chiral EFT makes a more direct connection between QCD and the nuclear force. Weinberg prescription An expansion in ( Q / Λ χ ). Q is a soft momentum scale. Λ χ ∼ 1 GeV is the chiral-symmetry-breaking scale. For example, the leading-order (LO) diagrams lead to NN ∝ ( σ 1 · q )( σ 2 · q ) V (0) τ 1 · τ 2 + · · · q 2 + M 2 Figure 3: Hierarchy of the nuclear force in π chiral EFT, from R. Machleidt and D. Entem, Phys. Rep. 503 , 1 (2011).

Motivation Nuclear interactions - Chiral EFT Chiral EFT makes a more direct connection between QCD and the nuclear force. Sources of non-locality in standard approach a b Regulator: f ( p , p ′ ) = e − ( p / Λ) n e − ( p ′ / Λ) n . Contact interactions ∝ k = ( p + p ′ ) / 2. F [ V ( p , p ′ )] → V ( r , r ′ ). a D. Entem and R. Machleidt, Phys. Rev. C 68 , 041001 (2003) b E. Epelbaum, W.Gl¨ ockle and U.-G. Meißner, Eur. Phys. J. A 19 , 401 (2004) Figure 3: Hierarchy of the nuclear force in chiral EFT, from R. Machleidt and D. Entem, Phys. Rep. 503 , 1 (2011).

Motivation Nuclear interactions - Chiral EFT Chiral EFT makes a more direct connection between QCD and the nuclear force. New approach a Regulator: f long ( r ) = 1 − e − ( r / R 0 ) 4 . Up to N 2 LO, V π = V π ( q ), q = p ′ − p . F [ V ( q )] → V ( r ) ⇒ Local! a A. Gezerlis et al., Phys. Rev. Lett. 111 , 032501 (2013) Figure 3: Hierarchy of the nuclear force in chiral EFT, from R. Machleidt and D. Entem, Phys. Rep. 503 , 1 (2011).

Motivation Nuclear interactions - Chiral EFT Chiral EFT makes a more direct connection between QCD and the nuclear force. New approach a V ( r ) = V C ( r ) + W C ( r ) τ 1 · τ 2 + ( V S ( r ) + W S ( r ) τ 1 · τ 2 ) σ 1 · σ 2 + ( V T ( r ) + W T ( r ) τ 1 · τ 2 ) S 12 . V C ( r ) = � ˜ 1 Λ 2 M π d µµ e − µ r ρ C ( µ ), etc. 2 π 2 r a A. Gezerlis et al., Phys. Rev. Lett. 111 , 032501 (2013) Figure 3: Hierarchy of the nuclear force in chiral EFT, from R. Machleidt and D. Entem, Phys. Rep. 503 , 1 (2011).

Motivation Nuclear interactions - Chiral EFT Local chiral EFT potential ∼ a v 7 potential 7 18 � � v p ( r ij ) O p v p ( r ij ) O p v ij = ij + ij . p =1 p =15 Charge-independent operators � 1 , σ i · σ j , S ij , L · S , L 2 , L 2 ( σ i · σ j ) , ( L · S ) 2 � O p =1 , 14 = ⊗ [1 , τ i · τ j ] . ij Charge-independence-breaking operators O p =15 , 18 = [1 , σ i · σ j , S ij ] ⊗ T ij , and ( τ zi + τ zj ) . ij Tensor operators S ij = 3( σ i · ˆ r ij )( σ j · ˆ r ij ) − σ i · σ j , T ij = 3 τ zi τ zj − τ i · τ j

Motivation Nuclear interactions - Chiral EFT Figure 4: Phase shifts for the np potential. From A. Gezerlis et al., Phys. Rev. Lett. 111 , 032501 (2013)