The dispersive optical model as an interface between FRIB ab initio - PowerPoint PPT Presentation

The dispersive optical model as an interface between FRIB ab initio calculations and experiment 6/20/2018 •Motivation •Green’s functions/propagator method W i m D i c k h o •vehicle for ab initio calculations —> matter f f B o b C h a r i t y •as a framework to link data at positive and L e e S o b o t k a H o s s negative energy (and to generate predictions for e i n M a h z o o n ( P h . D . M 2 0 exotic nuclei) a c 1 5 k ) A t k i n s o n N a t a l y a C a l l e y -> dispersive optical model (DOM <- Claude Mahaux) a M i c h a e l K e i m B l a k e B • Recent DOM extension to non-local potentials o r d e l o n • Revisit (e,e’p) data from NIKHEF & outlook (p,pN) • Neutron skin in 48 Ca (importance of total xsections) Recent DOM review: • Preliminary 208 Pb results WD, Bob Charity, Hossein Mahzoon J. Phys. G: Nucl. Part. Phys. 44 (2017) 033001 • Outlook for transfer reactions • Conclusions reactions and structure

Motivation • Rare isotope physics requires a much stronger link between nuclear reactions and nuclear structure descriptions • We need an ab initio approach for optical potentials —> optical potentials must therefore become nonlocal and dispersive • Current status to extract structure information from nuclear reactions involving strongly interacting probes unsatisfactory • Intermediate step: dispersive optical model as originally proposed by Claude Mahaux —> recent extensions discussed here reactions and structure

Problems with ab initio optical potentials • angular momentum constraints (illustrated here) • configuration space & density of low-lying states PHYSICAL REVIEW C 95 , 024315 (2017) Optical potential from first principles J. Rotureau, 1,2 P . Danielewicz, 1,3 G. Hagen, 4,5 F. M. Nunes, 1,3 and T. Papenbrock 4,5 • multiple scattering T x rho cannot be systematically improved • consistency requires simultaneous description of particle removal which determines the density reactions and structure

Comparison with ab initio FRPA calculation • Volume integrals of imaginary part of nonlocal ab initio (FRPA) self-energy compared with DOM result for 40 Ca • Ab initio S. J. Waldecker, C. Barbieri and W. H. Dickhoff 
 Microscopic self-energy calculations and dispersive-optical-model potentials. 
 reactions and structure Phys. Rev. C84, 034616 (2011), 1-11.

Ab initio calculation of elastic scattering n+ 40 Ca • Dussan, Waldecker, Müther, Polls, WD PRC84, 044319 (2011) • Also generates high-momentum nucleons below the Fermi energy • ONLY treatment of short-range and tensor correlations DOM & data DOM l ≤ 4 CDBonn l ≤ 4 reactions and structure

Propagator / Green’s function | a † • Lehmann representation h Ψ A 0 | a k ` j | Ψ A +1 i h Ψ A +1 k 0 ` j | Ψ A 0 i m m X G ` j ( k, k 0 ; E ) = E � ( E A +1 � E A 0 ) + i η m m 0 | a † h Ψ A k 0 ` j | Ψ A � 1 i h Ψ A � 1 | a k ` j | Ψ A 0 i n n X + 0 � E A � 1 E � ( E A ) � i η n n • Any other single-particle basis can be used & continuum integrals implied • Overlap functions --> numerator • Corresponding eigenvalues --> denominator 1 • Spectral function S ` j ( k ; E ) = π Im G ` j ( k, k ; E ) E  ε − F 2 � � X � h Ψ A − 1 | a k ` j | Ψ A δ ( E � ( E A 0 � E A − 1 = 0 i )) � � n n � n • Spectral strength in the continuum Z ∞ dk k 2 S ` j ( k ; E ) S ` j ( E ) = 0 • Discrete transitions q ` j φ n ` j ( k ) = h Ψ A − 1 | a k ` j | Ψ A S n 0 i n • Positive energy —> see later reactions and structure

Propagator from Dyson Equation and “experiment” Equivalent to … Schrödinger-like equation with: n = E A 0 − E A − 1 E − n Self-energy : non-local, energy-dependent potential With energy dependence: spectroscopic factors < 1 ⇒ as extracted from (e,e’p) reaction k 2 Z dq q 2 Σ ∗ 2 m φ n n ) φ n n φ n ` j ( k ) + ` j ( k, q ; E − ` j ( q ) = E − ` j ( k ) Z � 2 < 1 Spectroscopic factor S n � h Ψ A − 1 | a k ` j | Ψ A dk k 2 � � 0 i ` j = n ⇤ ∗ = h Ψ A +1 elE | a † χ elE r ` j | Ψ A Dyson equation also yields for positive energies ⇥ ` j ( r ) 0 i Elastic scattering wave function for protons or neutrons Dyson equation therefore provides: Link between scattering and structure data from dispersion relations reactions and structure

Propagator in principle generates • Elastic scattering cross sections for p and n • Including all polarization observables • Total cross sections for n • Reaction cross sections for p and n • Overlap functions for adding p or n to bound states in Z+1 or N+1 • Plus normalization --> spectroscopic factor • Overlap function for removing p or n with normalization • Hole spectral function including high-momentum description • One-body density matrix; occupation numbers; natural orbits • Charge density • Neutron distribution • p and n distorted waves • Contribution to the energy of the ground state from V NN reactions and structure

Dispersive optical potential <--> nucleon self-energy • e.g. Bell and Squires --> elastic T-matrix = reducible self-energy • e.g. Mahaux and Sartor Adv. Nucl. Phys. 20 , 1 (1991) – relate dynamic (energy-dependent) real part to imaginary part – employ subtracted dispersion relation – contributions from the hole (structure) and particle (reaction) domain General dispersion relation for self-energy: Z E − Z 1 Re Σ ( E ) = Σ HF − 1 dE 0 Im Σ ( E 0 ) + 1 dE 0 Im Σ ( E 0 ) T π P π P E − E 0 E − E 0 E + �1 T Calculated at the Fermi energy ( E A +1 − E A 0 ) + ( E A 0 − E A − 1 � ⇥ ε F = 1 ) 0 0 2 Z E − Z 1 Re Σ ( ε F ) = Σ HF − 1 dE 0 Im Σ ( E 0 ) ε F − E 0 + 1 dE 0 Im Σ ( E 0 ) T π P π P ε F − E 0 E + �1 T Subtract HF ( ε F ) g Re Σ ( E ) = Re Σ Z E − Z 1 1 ( E − E 0 )( ε F − E 0 ) + 1 Im Σ ( E 0 ) Im Σ ( E 0 ) T dE 0 dE 0 π ( ε F − E ) P π ( ε F − E ) P − ( E − E 0 )( ε F − E 0 ) E + �1 T reactions and structure

Functional form and fitting • Choice of potentials based on empirical knowledge • Volume absorption —> WS • Surface absorption —> WS’ • Coulomb • Spin-orbit • Hartree-Fock —> WS & WS’ • non-locality —> Gaussian • E-dependence imaginary part <—> some theory • Many parameters have canonical values reactions and structure

Nonlocal DOM implementation PRL112,162503(2014) • Particle number --> nonlocal imaginary part • Ab initio FRPA & SRC --> different nonlocal properties above and below the Fermi energy Phys. Rev. C84, 034616 (2011) & Phys. Rev.C84, 044319 (2011) • Include charge density in fit • Describe high-momentum nucleons <--> (e,e’p) data from JLab Implications • Changes the description of hadronic reactions because interior nucleon wave functions depend on non-locality • Consistency test of interpretation (e,e’p) reaction (see later) reactions and structure

Differential cross sections and analyzing powers 33 10 25 40 40 n+ Ca p+ Ca 40 40 p+ Ca n+ Ca E > 100 lab 26 20 10 40 < E < 100 lab 20 < E < 40 lab [mb/sr] 10 < E < 20 lab 0 < E < 10 15 19 10 lab A Ω /d 10 σ 12 10 d 5 5 10 0 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 [deg] θ cm [deg] θ cm reactions and structure

Critical experimental data—> charge density Local version Charge density 40 Ca radius correct… Non-locality essential PRC82,054306(2010) PR PRL 112,162503(2014) PRC82, 054306 (2010) 0.12 experiment 0.1 calculated 0.08 -3 ] ! ch (r) [fm 0.06 0.04 0.02 ρ 0 0 2 4 6 r [fm] High-momentum nucleons —> JLab can also be described —> E/A reactions and structure

Do elastic scattering data tell us about correlations? • Scattering T-matrix (neutrons) Z dqq 2 Σ ⇤ ` j ( k, q ; E ) G (0) ( q ; E ) Σ ` j ( q, k 0 ; E ) Σ ` j ( k, k 0 ; E ) = Σ ⇤ ` j ( k, k 0 ; E ) + 1 • Free propagator G (0) ( q ; E ) = E − ~ 2 q 2 / 2 m + i η • Propagator G `j ( k, k 0 ; E ) = δ ( k − k 0 ) G (0) ( k ; E ) + G (0) ( k ; E )Σ `j ( k, k 0 ; E ) G (0) ( k ; E ) k 2 • Spectral representation i ⇤ i ⇤ h h dE 0 χ cE 0 χ cE 0 φ n + φ n + ` j ( k ) ` j ( k 0 ) Z 1 ` j ( k ) ` j ( k 0 ) G p X X ` j ( k, k 0 ; E ) = + E − E 0 + i η E − E ⇤ A +1 + i η n T c n c • Spectral density for E > 0 `j ( k, k 0 ; E ) = i h i ⇤ ⇤ S p G p `j ( k, k 0 ; E + ) − G p X χ cE χ cE ⇥ `j ( k, k 0 ; E � ) `j ( k 0 ) = `j ( k ) 2 π c • Coordinate space ⇤ ⇤ S p X χ cE χ cE ⇥ ` j ( r, r 0 ; E ) = ` j ( r 0 ) ` j ( r ) c • Elastic scattering also explicitly available � 1 / 2 ⇢  2 mk 0 � Z χ elE dkk 2 j ` ( kr ) G (0) ( k ; E ) Σ ` j ( k, k 0 ; E ) ` j ( r ) = j ` ( k 0 r ) + π ~ 2 reactions and structure