Light Nuclei from chiral chiral EFT interactions EFT interactions Light Nuclei from Petr Navratil Lawrence Livermore National Laboratory* Collaborators: V. G. Gueorguiev (UCM), J. P. Vary (ISU), W. E. Ormand (LLNL), A. Nogga (Julich), S. Quaglioni (LLNL), E. Caurier (Strasbourg) *This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. UCRL-PRES-236709 Program INT-07-3: Nuclear Many-Body Approaches for the 21st Century, 13 November 2007
Outline Outline • Motivation • Introduction to ab initio no-core shell model (NCSM) • Ab initio NCSM and interactions from chiral effective field theory (EFT) – Determination of NNN low-energy constants – Results for mid- p -shell nuclei • Beyond nuclear structure with chiral EFT interactions Photo-disintegration of 4 He within NCSM/LIT approach – – n+ 4 He scattering within the NCSM/RGM approach – Preliminary: n+ 7 Li scattering within the NCSM/RGM approach Talk by Sofia Quaglioni last week • Outlook 40 Ca within importance-truncated ab initio NCSM –
Motivation Motivation • Goal: – Describe nuclei from first principles as systems of nucleons that interact by fundamental interactions • Non-relativistic point-like nucleons interacting by realistic nucleon-nucleon and also three-nucleon forces • Why it has not been solved yet? – High-quality nucleon-nucleon (NN) potentials constructed in last 15 years • Difficult to use in many-body calculations New developments: – NN interaction not enough for A >2: chiral EFT NN+NNN interactions • Three-nucleon interaction not well known – Need sophisticated approaches & big computing power • Ab initio approaches to nuclear structure – A =3,4 – many exact methods • 2001: A=4 benchmark paper: 7 different approaches obtained the same 4 He bound state properties – Faddeev-Yakubovsky, CRCGV, SVM, GFMC, HH variational, EIHH, NCSM – A >4 - few methods applicable Green’s Function Monte Carlo (GFMC) • – S. Pieper, R. Wiringa, J. Carlson et al. • Effective Interaction for Hyperspherical Harmonics (EIHH) – Trento, results for 6 Li • Coupled-Cluster Method (CCM), Unitary Model Operator Approach (UMOA) – Applicable mostly to closed shell nuclei Presently the only method capable • Ab Initio No-Core Shell Model (NCSM) to apply chiral EFT interactions to A>4 systems
Ab Initio No-Core Shell Model (NCSM) No-Core Shell Model (NCSM) Ab Initio • Many-body Schrodinger equation H � = E � – A-nucleon wave function • Hamiltonian r A 2 A A p � � r r i 3 b H � � V ( r r ) � V � � = + � + NN i j ijk 2 m � � i 1 i j i j k = < < < – Realistic high-precision nucleon-nucleon potentials • Coordinate space – Argonne … Momentum space - CD-Bonn, chiral N 3 LO … • – Three-nucleon interaction Tucson-Melbourne TM’, chiral N 2 LO • • Modification by center-of-mass harmonic oscillator (HO) potential (Lipkin 1958) 2 r A A 1 1 m r � r r 2 2 2 2 2 � � Am R m r ( r r ) � = � � � i i j 2 2 2 A i 1 i j = < – No influence on the internal motion – Introduces mean field for sub-clusters r 2 2 A p 1 A m A � � � � � � r r r � r r i 2 2 2 3 b H � m r � V ( r r ) ( r r ) � V � � � = + � � + � � � + � � � i NN i j i j ijk 2 m 2 2 A � � � � � � i 1 i j i j k = < < <
Coordinates, basis and model space Coordinates, basis and model space Bound states (and narrow resonances): Square-integrable A -nucleon basis • NN (and NNN) interaction depends on relative Why coordinates and/or momenta HO basis? – Translationally invariant system – We should use Jacobi (relative) coordinates • However, if we employ: – i) (a finite) harmonic oscillator basis N =5 – ii) a complete N max h Ω model space N =4 • Translational invariance even when Cartesian N =3 coordinate Slater determinant basis used N =2 – Take advantage of powerful second N =1 quantization shell model technique N =0 – Choice of either Jacobi or Cartesian coordinates according to efficiency for the problem at hand This flexibility is possible only for harmonic oscillator (HO) basis. A downside: Gaussian asymptotic behavior.
Model space, truncated basis and effective interaction Model space, truncated basis and effective interaction • Strategy: Define Hamiltonian, basis, calculate matrix elements and diagonalize. But: • Finite harmonic-oscillator Jacobi coordinate or Cartesian coordinate Slater determinant basis Nucleon-nucleon interaction – Complete N max h Ω model space V NN Repulsive core and/or short-range correlations in V NN (and also in V NNN ) cannot be accommodated in a truncated HO basis Need for the effective interaction Need for the effective interaction
Effective Hamiltonian in the NCSM Effective Hamiltonian in the NCSM E 1 , E 2 , E 3 , K E d P , K E � H : H eff H (n) 0 eff E 1 , E 2 , E 3 , K E d P H eff : N max QXHX � 1 P = 0 model space dimension QXHX -1 Q 0 Q n X n H (n) X n -1 Q n H eff = PXHX � 1 P unitary X =exp[-arctanh( ω + - ω )] • n -body cluster approximation, 2 ≤ n ≤ A •Properties of H eff for A -nucleon system • H (n) eff n -body operator • A -body operator •Even if H two or three-body • Two ways of convergence: •For P → 1 H eff → H For P → 1 H (n) – eff → H For n → A and fixed P: H (n) – eff → H eff As difficult as the original problem
Effective interaction calculation in the NCSM Effective interaction calculation in the NCSM n -body approximation • For n nucleons reproduces exactly the full-space results (for a subset of eigenstates) – n =2, two-body effective interaction approximation – n =3, three-body effective interaction approximation Q n X n H ( n ) X n � 1 P n = 0
3 H and H and 4 4 He with He with chiral chiral N N 3 3 LO NN interaction LO NN interaction 3 • NCSM convergence test – Comparison to other methods N 3 LO NCSM FY HH NN 3 H 7.852(5) 7.854 7.854 4 He 25.39(1) 25.37 25.38 Short-range correlations ⇒ effective interaction Medium-range correlations ⇒ multi- h Ω model space Dependence on size of the model space ( N max ) HO frequency ( h Ω ) Not a variational calculation Convergence OK NN interaction insufficient to reproduce experiment
Nuclear forces Nuclear forces • Unlike electrons in the atom the interaction between nucleons is not known precisely and is complicated • Phenomenological NN potentials provide an accurate fit to NN data – CD-Bonn 2000 • One-boson exchange - π , ρ , ω + phenomenological σ mesons • χ 2 /N data =1.02 • But they are inadequate for A >2 systems – Binding energies under-predicted – N-d scattering: A y puzzle; n- 3 H scattering: total cross section – Nuclear structure of p -shell nuclei is wrong Dim=1.1 billion
Need to go beyond standard NN potentials Need to go beyond standard NN potentials • NNN forces? – Consistency between the NN and the NNN potentials – Empirical NNN potential models have many terms and parameters • Hierarchy? – Lack of phase-shift analysis of three-nucleon scattering data • Predictive theory of nuclei requires a consistent framework for the interaction Start from the fundamental theory of strong interactions QCD • QCD non-perturbative in the low-energy regime relevant to nuclear physics • However, new exciting developments due to Weinberg and others… – Chiral effective field theory (EFT) • Applicable to low-energy regime of QCD • Capable to derive systematically inter-nucleon potentials • Low-energy constants (LECs) must be determined from experiment
Chiral Effective Field Theory Effective Field Theory Chiral • Chiral symmetry of QCD ( m u ≈ m d ≈ 0), spontaneously broken with pion as the Goldstone boson • Systematic low-momentum expansion in (Q/ Λ χ ) n ; Λ χ ≈ 1 GeV, Q ≈ 100 MeV – Degrees of freedom: nucleons + pions – Power-counting: Chiral perturbation theory ( χ PT) • Describe pion-pion, pion-nucleon and inter-nucleon interactions at low energies – Nucleon-nucleon sector - S. Weinberg (1991) • Worked out by Van Kolck, Kaiser, Meissner, Epelbaum, Machleidt… • Leading order (LO) – One-pion exchange • NNN interaction appears at next-to-next-to-leading order (N 2 LO) NNNN interaction appears at N 3 LO order • • Consistency between NN, NNN and NNNN terms – NN parameters enter in the NNN terms etc. • Low-energy constants (LECs) need to be fitted to experiment • N 3 LO is the lowest order where a high-precision fit to NN data can be made – Entem and Machleidt (2002) N 3 LO NN potential Only TWO NNN and NO NNNN low-energy constants up to N 3 LO • Challenge and necessity: Apply chiral EFT forces to nuclei
Chiral N N 2 2 LO NNN interaction LO NNN interaction Chiral N 2 LO Two-pion exchange c 1 , c 3 , c 4 LECs appear in the chiral NN interaction • Determined in the A =2 system c 1 , c 3 , c 4 One-pion-exchange-contact New c D LEC New! c D Must be determined in A ≥ 3 system Contact To be used by Pisa group New c E LEC in HH basis New! c E Nontrivial to include in the NCSM calculations – Regulated with momentum transfer • local NNN interaction in coordinate space
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