Advances in Ab Initio Nuclear Structure Theory Robert Roth
Ab Initio Nuclear Structure Theory solve nuclear many-body problem based on realistic interactions using controlled and improvable truncations with quantified theoretical uncertainties Interactions chiral effective field theory for nuclear interactions (NN+N+ ) and electroweak operators physics-conserving unitary transformation to adapt Hamiltonian to finite low-energy model spaces Many-Body Method solve many-body Schrödinger equation numerically using only controlled truncations range of powerful methods for different mass ranges employing the same interaction Quantification of Theory Uncertainties Robert Roth - TU Darmstadt - March 15
Interactions
Chiral EFT for Nuclear Interactions low-energy effective field theory for relevant degrees of freedom (π,n) based on symmetries of QCD Weinberg, van Kolck, Machleidt, Entem, Meissner, Epelbaum, Krebs, Bernard, LO NN N 4N explicit long-range pion dynamics unresolved short-range physics absorbed in contact terms, low-energy constants fit to experiment hierarchy of consistent NN, N, 4N, interactions and current operators many ongoing developments improved NN up to N4LO N interaction up to NLO 4N interaction at NLO improved fits and error analysis NLO N LO N LO standard interaction: NN @ NLO (Entem&Machleidt, cutoff 5 MeV) N @ NLO (local, cutoff 4 or 5 MeV) Robert Roth - TU Darmstadt - March 15 4
Similarity Renormalization Group SRG Evolution in Three-Body Glazek, Wilson, Wegner, Perry, Bogner, Furnstahl, Hergert, Roth, E continuous unitary transformation driving Hamiltonian towards diagonal form 18 unitary transformation via flow equation H U! d d H = 6,H 8 Tint, H Robert Roth - TU Darmstadt - March 15 B-Jacobi HO matrix elements 4 E 18 (E, ) E 6 8 1 solve flow equation using matrix representation in two- and three-body space flow parameter α determines how far to go α= fm4 18 (E, ) = ( ) N - -4 SRG Evolution in Three-Body dynamic generator determines physics of transformation 4 4 6 8 α=16 fm4 E 18 4 (E, ) 6-6 -8 E [MeV] H =U E [MeV] (E, ) three-body Jacobi-HO matrix elements, Jπ=1/+, T=1/, ħω = 8 MeV B-Jacobi HO matrix elements 8 5 N - -4-6 -8
Similarity Renormalization Group SRG Evolution in Three-Body Glazek, Wilson, Wegner, Perry, Bogner, Furnstahl, Hergert, Roth, E pro: improves convergence of many-body calculations H =H +H [] +H [] +H [4] + 4 6 8 variation of flow parameter provides diagnostic for omitted many-body terms α= fm4 B-Jacobi HO matrix elements 4 E 18 (E, ) E 6 8 1 18 use initial NN, keep evolved NN+N Robert Roth - TU Darmstadt - March 15 4 6 NN+Nfull use initial NN+N, keep evolved NN+N - -4 8 α=16 fm4 E 18 4 (E, ) 6-6 -8 E [MeV] NN+Nind (E, ) truncations used in the following: N SRG Evolution in Three-Body need to truncate evolved Hamiltonian [1] E [MeV] (E, ) con: induces many-body interactions 18 three-body Jacobi-HO matrix elements, Jπ=1/+, T=1/, ħω = 8 MeV B-Jacobi HO matrix elements 8 6 N - -4-6 -8
Many-Body Methods I: Light Nuclei
No-Core Shell Model & Friends Barrett, Vary, Navrátil, Maris, Nogga, Roth, NCSM-type approaches are the most powerful and universal ab initio methods for the p- and lower sd-shell NCSM: solve eigenvalue problem of Hamiltonian represented in model space of HO Slater determinants truncated wrt HO excitation energy NmaxħΩ convergence of observables wrt Nmax is the only limitation and source of uncertainty Importance-Truncated NCSM: reduce NCSM model space to physically relevant basis states and extrapolate to full space a posteriori increases the range of applicability of NCSM significantly NCSM with Continuum: merge NCSM for description of clusters with Resonating Group Method for description of their relative motion explicitly includes continuum degrees of freedom more: Symplectic NCSM, Gamow NCSM, Robert Roth - TU Darmstadt - March 15 8
Spectrum of 9 Be Langhammer et al; PRC 91, 11(R) (15) 8 Be n 9 Be is excellent candidate to study continuum effects on spectra all excited states are resonances previous NCSM studies with NN interactions show clear discrepancies in spectrum: N or continuum effects? include n- 8 Be continuum in NCSMC use +, + NCSM states of 8 Be for n- 8 Be dynamics include 6 neg and 4 pos parity NCSM states of 9 Be use standard NN+N Hamiltonian NN @ NLO, Entem & Machleidt, cutoff 5 MeV N @ NLO, local, cutoff 5 MeV n 8 Be Robert Roth - TU Darmstadt - March 15 9
Phase Shifts for n- 8 Be Scattering Langhammer et al; PRC 91, 11(R) (15) [deg] 15 1 5-5 -1 15 [deg] 1 5 1 NN N-induced 7 5 N max 6 8 1 1 1-5 N max 7-1 9 11 1 4 5 E kin [MeV] 6 7 5 9 1 1 NN N-full 7 5 1 4 5 6 7 E kin [MeV] 5 9 negative parity positive parity negative parity phase-shifts are well converged, positive parity more difficult extract resonance parameters from inflection point and derivative Robert Roth - TU Darmstadt - March 15 α = 65 fm 4, ħω = MeV, Emax = 14 1
9Be: NCSM vs NCSMC Langhammer et al; PRC 91, 11(R) (15) Ethr [MeV] 8 7 6 5 4 1-1 - negative parity positive parity NCSM NCSMC NCSM NCSMC NCSM NCSMC 6 8 1 1 exp 1 1 8 6 (a) Ethr [MeV] 1 9 8 7 6 5 4 1 NCSM NCSMC 7 9 11 exp 11 9 7 Nmax Nmax Nmax Nmax (b) NN+Nfull α = 65 fm 4, ħω = MeV, Emax = 14 NCSMC shows much better Nmax convergence NCSM tries to capture continuum effects via large Nmax drastic difference for the 1/ + state right at threshold Robert Roth - TU Darmstadt - March 15 11
9Be: Spectrum Langhammer et al; PRC 91, 11(R) (15) Ethr [MeV] 7 6 5 4 1-1 - (a) chiral NN negative parity NCSM NCSMC NCSM NCSMC chiral NN+N exp chiral NN+N chiral NN Ethr [MeV] 9 8 7 6 5 4 1-1 (b) chiral NN positive parity chiral NN+N exp chiral NN+N chiral NN α = 65 fm 4, ħω = MeV, Emax = 14 continuum plays more important role than chiral N interaction NCSMC predictions for widths are in fair agreement with experiment Robert Roth - TU Darmstadt - March 15 1
Oxygen Isotopes oxygen isotopic chain has received significant attention and documents the rapid progress over the past years Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL 15, 51 (1) 1: shell-model calculations with N effects highlighting the role of N interaction for drip line physics Hagen, Hjorth-Jensen, Jansen, Machleidt, Papenbrock, PRL 18, 451 (1) 1: coupled-cluster calculations with phenomenological two-body correction simulating chiral N forces Hergert, Binder, Calci, Langhammer, Roth, PRL 11, 451 (1) 1: ab initio IT-NCSM with explicit chiral N interactions and first multi-reference in-medium SRG calculations Cipollone, Barbieri, Navrátil, PRL 111, 651 (1) Bogner, Hergert, Holt, Schwenk, Binder, Calci, Langhammer, Roth, PRL11, 1451(14) Jansen, Engel, Hagen, Navratil, Signoracci, PRL 11, 145 (14) since: self-consistent Green s function, shell model with valencespace interactions from in-medium SRG or Lee-Suzuki, Robert Roth - TU Darmstadt - March 15 1
Ground States of Oxygen Isotopes Hergert et al, PRL 11, 451 (1) -4 NN+N ind (chiral NN) 1 O NN+N full (chiral NN+N) 1 O -6 E [MeV] -8-1 -1-14 -16 4 6 8 1 1 14 16 18 N m x 14 O 16 O 18 O O O 4 O 6 O 4 6 8 1 1 14 16 18 N m x 14 O 16 O 18 O O O 6 O 4 O Λ N = 4 MeV, α = 8 fm 4, E m x = 14, optimal hω Robert Roth - TU Darmstadt - March 15 14
Ground States of Oxygen Isotopes Hergert et al, PRL 11, 451 (1) E [MeV] -4-6 -8-1 -1-14 NN+N ind (chiral NN) A O NN+N full (chiral NN+N) experiment IT-NCSM -16 parameter-free ab initio calculations -18 with explicit chiral 1 N 14 interactions 16 18 4 highlights 6 1 predictive 14 16 18 A power of chiral NN+N A 4 6 Λ N = 4 MeV, interactions α = 8 fm 4, E m x = 14, optimal hω Robert Roth - TU Darmstadt - March 15 15
Spectra of Oxygen Isotopes Hergert et al, PRL 11, 451 (1) & in prep E [MeV] E [MeV] 7 6 5 4 1 6 5 4 1 18 O 4 6 Exp 4 1 O 9 7 1 5 4 1 9 8 7 6 5 4 1 19 O 4 6 Exp 7 9 1 5 O 4 6 O 5 4 1 7 6 5 4 1 4 6 Exp 4 4 O 5 NN+N full (chiral NN+N) Λ N = 4 MeV, α = 8 fm 4, hω = 16 MeV 4 6 Exp N m x 5 4 6 Exp N m x 4 6 Exp N m x 1 5 Robert Roth - TU Darmstadt - March 15 16
Many-Body Methods II: Medium-Mass Nuclei
Medium-Mass Approaches advent of novel ab initio many-body approaches gives access to the medium-mass regime Hagen, Papenbrock, Dean, Piecuch, Binder, coupled-cluster theory: ground-state parametrized by exponential wave operator applied to single-determinant reference state truncation at doubles level (CCSD) plus triples corrections (Λ-CCSD(T)) equations of motion for excited states and near-closed-shell nuclei Bogner, Tsukiyama, Schwenk, Hergert, in-medium SRG: complete decoupling of particle-hole excitations from many-body reference state through SRG evolution normal-ordered evolving A-body Hamiltonian truncated at two-body level both closed- and open-shell ground states; excitations via EOM or SM controlling and quantifying the uncertainties due to various inherent truncations is a major task self-consistent Green s function approaches and others Barbieri, Soma, Duguet, Robert Roth - TU Darmstadt - March 15 18
Ground States of Oxygen Isotopes Hergert et al, PRL 11, 451 (1) E [MeV] -4-6 -8-1 -1-14 -16 NN+N ind (chiral NN) A O NN+N full (chiral NN+N) experiment IT-NCSM -18 1 14 16 18 4 6 A 1 14 16 18 4 6 A Λ N = 4 MeV, α = 8 fm 4, E m x = 14, optimal hω Robert Roth - TU Darmstadt - March 15 19
Ground States of Oxygen Isotopes Hergert et al, PRL 11, 451 (1) E [MeV] -4-6 -8-1 -1-14 -16 NN+N ind (chiral NN) A O NN+N full (chiral NN+N) experiment IT-NCSM MR-IM-SRG -18 1 14 16 18 4 6 A 1 14 16 18 4 6 A Λ N = 4 MeV, α = 8 fm 4, E m x = 14, optimal hω 1 Robert Roth - TU Darmstadt - March 15
Ground States of Oxygen Isotopes Hergert et al, PRL 11, 451 (1) E [MeV] -4-6 -8-1 -1-14 NN+N ind (chiral NN) A O NN+N full (chiral NN+N) experiment IT-NCSM MR-IM-SRG CCSD Λ-CCSD(T) -16 different many-body approaches -18 using the same chiral NN+N 1 14 interaction 16 18 give 4 6 1 14 16 18 4 6 consistent results A minor differences are A understood Λ N = 4 MeV, α = 8 fm 4 in terms of uncertainties due, E m x to truncations = 14, optimal hω Robert Roth - TU Darmstadt - March 15 1
Towards Heavy Nuclei - Ab Initio Binder et al, PLB 76, 119 (14) -6-7 CR-CC(,) NN+N ind (chiral NN) NN+N full (chiral NN+N) E/A [MeV] -8-9 -1 δe T /A 5-5 16 O 6 Ca 48 Ca 54 Ca 56 Ni 6 Ni 68 Ni 88 Sr 1 Sn 18 Sn 116 Sn 1 Sn 4 O 4 Ca 5 Ca 48 Ni 6 Ni 66 Ni 78 Ni 9 Zr 16 Sn 114 Sn 118 Sn 1 Sn Λ N = 4 MeV, α = 8 4 fm 4, E m x = 18, optimal hω Robert Roth - TU Darmstadt - March 15 8
Towards Heavy Nuclei - Ab Initio Binder et al, PLB 76, 119 (14) E/A [MeV] δe T /A -6-7 -8-9 -1 5-5 16 O 4 O CR-CC(,) 6 Ca 48 Ca 54 Ca 56 Ni 6 Ni 68 Ni 88 Sr 1 Sn 18 Sn 116 Sn 1 Sn 4 Ca 5 Ca 48 Ni 6 Ni 66 Ni 78 Ni 9 Zr 16 Sn 114 Sn 118 Sn 1 Sn NN+N ind (chiral NN) NN+N full (chiral NN+N) % residual uncertainty for ground-state energies due to truncations in many-body approach up to A~1 standard chiral NN+N interaction overbinds mediummass nuclei systematically SRG-induced interactions beyond the N level can pose severe problems; role of chiral 4N unclear Λ N = 4 MeV, α = 8 4 fm 4, E m x = 18, optimal hω Robert Roth - TU Darmstadt - March 15 9
Open-Shell Medium-Mass Nuclei Open-Shell Medium-Mass Nuclei Hergert et al, PRC 9, 41(R) (14) Hergert et al, PRC 9, 41(R) (14) E [MeV] E [MeV] -5 - -5-4 -45-5 -5 - -5-4 -45-5 E [MeV] -5 - -5 MR-IM-SRG -4 CR-CC(,) -45 E [MeV] -5-5 - -5-4 MR-IM-SRG CR-CC(,) A Ca NN+N ind (chiral NN) A Ca NN+N ind (chiral NN) A Ca A Ca NN+N full (chiral NN+N) NN+N full (chiral NN+N) -45 MR-IM-SRG MR-IM-SRG CR-CC(,) -5 CR-CC(,) 4 6 8 4 4 44 46 48 5 5 54 56 58 6 6 A 4 6 8 4 4 44 46 48 5 5 54 56 58 6 6 A Λ N α systematic MR-IM-SRG study systematic of even MR-IM-SRG Ca and Ni isotopes study of even Ca and Ni isotopes excellent agreement with study of even Ca and Ni isotopes best excellent available agreement coupledcluster with best results available coupled-cluster results systematic MR-IM-SRG excellent agreement with best available chiral N interaction coupled-cluster chiral N interaction results changes behavior at and changes beyond 54 behavior at and beyond 54 Ca Ca chiral N interaction changes behavior at and beyond 54 Ca Λ N α = 4 MeV = 4 fm 4 ( ) 8 fm 4 ( ) E m x = 14, 16 = 4 MeV = 4 fm 4 ( ) 8 fm 4 ( ) E m x = 14, 16 Robert Roth - TU Darmstadt - March 15 4
Open-Shell Medium-Mass Nuclei Open-Shell Medium-Mass Nuclei S n [MeV] S n [MeV] 5 4 1-1 4 1-1 S n [MeV] 4 Λ N = 4 MeV 1 Λ N = 5 MeV -1 Λ N = 4 MeV Λ N = 5 MeV A Ca NN+N ind (chiral NN) A Ca A Ca NN+N full (chiral NN+N) NN+N full (chiral NN+N) 4 6 8 44 6 4 8 4 444 464448 465 48 5 5 5 54 54 56 5658 6 6 A A all MR-IM-SRG α Hergert et al, PRC 9, 41(R) (14) Hergert et al, PRC 9, 41(R) (14) two-neutron separation energies hide overall energy shift compares well to updated Gro kov-gf results (not the published ones) [priv comm Soma] chiral N interaction chiral N interaction generates magicity of 54 Ca and defines dripline all MR-IM-SRG α two-neutron separation energies two-neutron hide separation overall energy energiesshift hide overall energy shift compares well to updated Gor'kov-GF compares well results to updated Gro kov-gf results (not the published ones) chiral N interaction [priv comm Soma] predicts flat "drip-region" from 56 Ca to 6 Ca generates magicity of 54 Ca and defines dripline = 4 fm 4 ( ) 8 fm 4 ( ) E m x = 14, 16 = 4 fm 4 ( ) 8 fm 4 ( ) E m x = 14, 16 [priv comm V Soma] Robert Roth - TU Darmstadt - March 15 1 1 5
Many-Body Methods III: Hypernuclei
Ab Initio Hypernuclear Structure precise data on ground states & spectroscopy of hypernuclei ab initio few-body (A 4) and phenomenological shell or cluster model calculations so far chiral YN & YY interactions at (N)LO are available time to transfer ab initio toolbox to hypernuclei Robert Roth - TU Darmstadt - March 15 7
Application: Λ 7 Li Wirth et al, PRL 11, 195 (14) - 6 Li 7 Λ Li IT-NCSM -5 NN @ NLO Λ NN = 5 MeV E [MeV] - 1 Entem&Machleidt N @ NLO Λ N = 5 MeV -5 7 5 Navratil A = fit E [MeV] -4 1-1 4 6 8 1 1 Exp N m x 1 4 6 8 1 1 Exp N m x 1 7 5 1 Jülich 4 Haidenbauer et al scatt & hypertriton α NN = 8 fm 4 α YN = fm 4 hω = MeV Robert Roth - TU Darmstadt - March 15 4 8
Application: Λ 7 Li Wirth et al, PRL 11, 195 (14) - 6 Li 7 Λ Li IT-NCSM -5 NN @ NLO Λ NN = 5 MeV E [MeV] - 1 Entem&Machleidt N @ NLO Λ N = 5 MeV -5 7 5 Navratil A = fit E [MeV] -4 1-1 4 6 8 1 1 Exp N m x 1 4 6 8 1 1 Exp N m x 1 7 5 1 YN @ LO Λ YN = 6 MeV Haidenbauer et al scatt & hypertriton α NN = 8 fm 4 α YN = fm 4 hω = MeV Robert Roth - TU Darmstadt - March 15 41 9
Application: Λ 7 Li Wirth et al, PRL 11, 195 (14) - 6 Li 7 Λ Li IT-NCSM -5 NN @ NLO Λ NN = 5 MeV E [MeV] - 1 Entem&Machleidt N @ NLO Λ N = 5 MeV -5 7 5 Navratil A = fit E [MeV] -4 1-1 4 6 8 1 1 Exp N m x 1 4 6 8 1 1 Exp N m x 1 7 5 1 YN @ LO Λ YN = 7 MeV Haidenbauer et al scatt & hypertriton α NN = 8 fm 4 α YN = fm 4 hω = MeV Robert Roth - TU Darmstadt - March 15 4
Application: Λ 9 Be Wirth et al, PRL 11, 195 (14) -4-45 8 Be 9 Λ Be NN @ NLO Λ NN = 5 MeV Entem&Machleidt E [MeV] E [MeV] -5-55 -6-65 4 1 5 1 5 N @ NLO Λ N = 5 MeV Navratil A = fit YN @ LO Λ YN = 6 MeV Haidenbauer et al scatt & hypertriton α NN = 8 fm 4 α YN = fm 4 hω = MeV 4 6 8 1 Exp N m x 4 6 8 1 Exp N m x 1 Robert Roth - TU Darmstadt - March 15 4 1
Application: Λ 1 C Wirth et al, PRL 11, 195 (14) E [MeV] -75-8 -85-9 -95-1 -15-11 6 E [MeV] 4 1 C 1 Λ C hypernuclear structure sets tight constraints on YN interaction 4 6 8 1 Exp N m x ready to explore the physics of p-shell hypernuclei 4 6 8 1 Exp N m x 1 1 NN @ NLO Λ NN = 5 MeV Entem&Machleidt N @ NLO Λ N = 5 MeV Navratil A = fit YN @ LO Λ YN = 6 MeV Haidenbauer et al scatt & hypertriton α NN = 8 fm 4 α YN = fm 4 hω = MeV Robert Roth - TU Darmstadt - March 15 44
Conclusions
A Look Back past few years have seen dramatic progress in ab initio many-body methods for nuclear structure (and reactions) extensions of NCSM, coupled-cluster theory, in-medium SRG, self-consistent Green's function, many-body perturbation theory, a number of important developments are in progress spectroscopy of open-shell nuclei, derivation of valence-space interactions, broad range of observables the reach of ab initio methods has grown tremendously medium-mass and heavy nuclei, continuum effects and reaction observables, hypernuclei Robert Roth - TU Darmstadt - March 15 4
A Look Ahead for the next few years the focus will move towards improvements of the chiral interactions improved fitting strategies, consistent higher orders, systematic study of order-by-order convergence, inclusion of consistent currents, rigorous quantification of theoretical uncertainties will play an important role propagation of uncertainties from chiral interaction to nuclear structure observables, full quantification of many-body uncertainties lots of relevant physics predictions for NUSTAR Robert Roth - TU Darmstadt - March 15 5
Epilogue Epilogue thanks to my group and my collaborators thanks J Braun, to E Gebrerufael, my group T & Hüther, my collaborators J Langhammer, S Schulz, H Spiess, C Stumpf, A Tichai, R Trippel, S Binder, J Braun, A Calci, S Fischer, K E Vobig, Gebrerufael, R Wirth H Spiess, J Langhammer, S Schulz, Technische Universität Darmstadt C Stumpf, A Tichai, R Trippel, R Wirth, K Vobig P Navrátil, A Calci Institut für Kernphysik, TU Darmstadt TRIUMF, Vancouver S P Binder Navrátil Oak TRIUMF Ridge Vancouver, National Laboratory Canada H J Hergert Vary, P Maris NSCL / Michigan State University Iowa State University, USA J Vary, P Maris Iowa S Quaglioni, State University G Hupin LLNL Livermore, USA S Quaglioni, G Hupin Lawrence Livermore National Laboratory P Piecuch H Feldmeier, T Neff E Michigan Epelbaum, State University, H Krebs USA GSI Helmholtzzentrum & the LENPIC Collaboration Universität Bochum, H Hergert Ohio State University, USA P Papakonstantinou IBS/RISP, Korea C Forssén Chalmers University, Sweden JUROPA LOEWE-CSC EDISON Robert Roth - TU Darmstadt - March 15 COMPUTING TIME 6