Overview of SCGF methods in nuclear...

Correlations & π, ρ modes

Overview of SCGF methods in nuclear systems• SCGF first coined by... (as far as I know)

• Infinite matter --> ladders --> SRC

• Nuclear saturation problem

• (e,e’p) and propagators in finite nuclei

• Ab initio results --> limitation of shell model

• Understanding of proton properties in stable nuclei

• Asymmetric nuclear stuff and SRC

• Link with data for exotic nuclei

• Dispersive optical model --> drip lines

• Future

SCGF 2010 Workshop, Saclay, Sep 9-10, 2010

Wim DickhoffBob Charity

Lee SobotkaJonathan MorrisSeth WaldeckerWashington University in St. LouisCarlo Barbieri, SurreyArnau Rios, SurreyArturo Polls, BarcelonaDimitri Van Neck, Ghentand many earlier collaborators

Nucleon correlations

Some available applications:• Start from VNN (realistic) in infinite matter Rios et al.

– short-range correlations = SRC (ladders)

– fully self-consistent in nuclear and isospin asymmetric matter at finite temperature

• Start from VNN (realistic) in finite nuclei or Vee in atoms and molecules Barbieri et al.

– SRC ⇒ G-matrix (as effective interaction in model space)

– LRC : Faddeev-RPA (pp/hh and ph phonons)

– In progress: contribution of high-momenta to ground state and self-consistent Pauli principle (self-consistent model space)

• Start from DATA = Dispersive Optical Model St. Louis group

– Fit of complex potentials (self-energy) including nucleon asymmetry dependence ⇒predict more exotic systems

• Under construction: quasi-particle density functional theory Ghent

Correlations

SCGF in nuclear systems

• 1985 visit to Barcelona...

• 1988 Angels Ramos, U Barcelona, spectral functions in nuclear matter from SRC

• 1990 Mario Brand, Free U Amsterdam, spectral functions in nuclei from second-order G-matrix, Baym-Kadanoff -> excited states

• 1991 Brian Vonderfecht,Washington U, spectral functions from realistic interactions avoiding pairing instabilities

• 1992 Dimitri Van Neck, U Ghent, fully self-consistent second-order self-energy with Skyrme interaction

• 1993 Gustl Rijsdijk, Free U Amsterdam, inclusion of collective excitations (TDA/RPA either ph or pp/hh) in self-energy

• 1994 Chris Gearhart, Washinton U, towards self-consistency in matter, scattering in the medium of dressed nucleons

• 1996 Wouter Geurts, Free U Amsterdam, 2p1h/2h1p TDA insertion into self-energy

• 2000 Libby Roth-Stoddard, Washington U, full self-consistency in matter at T=0 and finite T

• 2000 Yves Dewulf, U Ghent, discrete approach to self-consistently dressed ladders in matter

• 2002 Carlo Barbieri, Washington U, Faddeev summation of RPA phonons (ph & pp/hh) into self-energy

• 2004 Tobias Frick, U Tübingen, fully self-consistent ladders at finite T

• 2007 Arnau Rios, U Barcelona, full exploration of thermodynamics derived from self-consistent ladders at finite T

• 2010 Jonathan Morris, Washington U, role of dynamic pions in nuclear matter and (p,2p) reactions

• 2011 Seth Waldecker, Washington U, extensions of the dispersive optical model

• and others I am less familiar with

• Senior people

– Arturo Polls

– Herbert Müther

– Piotr Bożek

(Effective)centralforce

Repulsivecore:500-600MeV

Attractivepocket:about30MeV

Bothofthesearesmallerthanweexpect.Thisisansweredbythequarkmassdependence.

The following diagram is contained.

This leads to the one pion exchangeat the large spatial separation.

N.Ishii, S.Aoki, T.Hatsuda, Phys.Rev.Lett.99,022001(’07).

Ishii talk at 3rd LACM-EFES-JUSTIPEN WorkshopOak Ridge 2009

Quarkmassdependenceofthecentralforce:(1) mπ=380MeV: Nconf=2034 [28 exceptional configurations have been removed](2) mπ=529MeV: Nconf=2000(3) mπ=731MeV: Nconf=1000

1S0

Strongquarkmassdependenceisfound.

Inthelightquarkmassregion,

therepulsivecoregrowsrapidly.

attractivepocketisenhancedmildly.

ThelatticeQCDcalculationatlightquarkmassregionisquiteimportant.

Ishii talk at 3rd LACM-EFES-JUSTIPEN WorkshopOak Ridge 2009

Correlations

Fetter & Walecka

Correlations

Old problem Figure adapted from Marcello Baldo

Nijmegen 1 <-- ONLY SRC: PRL 90, 152501 (2003)

Correlations

Pion collectivity: nuclei vs. nuclear matter

• Pion collectivity leads to pion condensation at higher density in

nuclear matter (including Δ-isobars) => Migdal ...

• No such collectivity observed in nuclei ⇒ LAMPF / Osaka data

• Momentum conservation in nuclear matter dramatically favors amplification of π-exchange interaction at fixed q

•In nuclei the same interaction is sampled over all momenta and exchange is more important Phys. Lett. B146, 1(1984)

NUCLEAR MATTER PROBLEM --> ill-posed

Vπ(q) = − f2π

m2π

q2

m2π + q2

Correlations

Inclusion of Δ-isobars as 3N- and 4N-force

⇒ No sensible convergence with Δ-isobars

Rings with Δ-isobars:

2N,3N, and 4N from B.D.Day, PRC24,1203(81)

Nucl. Phys. A389, 492 (1982)Prog. Part. Nucl. Phys. 12, 529 (1983)

But: central density --> SRC dominated --> proton occupation numbers in 208Pb

from (e,e’p)NIKHEF

Depletion of the nuclear Fermi sea

Repulsive core and wave function ⇒ high-momenta

Reid 1S0

k0 = 0.25 fm-1

~ few MeV

V(r)/100

sin(k0r)/k0r

correlated wffor a “hard” interaction

Depletion of the nuclear Fermi sea

SCGFself-consistentGreen’s functionsfor SRC and tensor effects

Interaction in the medium properly treating short-range and tensor correlations

Self-energy = complex potential in nuclear matter

Dyson equation ⇒Schrödinger equation for dressed nucleons

Arnau RiosArturo Polls (BCN)W.D.finite T to avoid pairing

self-consistency= thermodynamically consistent


2000s Verification oftwo effects associated with short-range correlations

• Depletion of the Fermi sea NIKHEF

• Admixture of high-momentum components Jefferson Lab

Recent data confirm both aspects (predicted by nuclear matter results)


M. van Batenburg & L. Lapikás from 208Pb (e,e´p) 207Tl NIKHEF still in preparation

Up to 100 MeV missing energy and270 MeV/c missing momentum

Covers the whole mean-field domainfor the FIRST time!!

Occupation of deeply-bound proton levels from EXPERIMENT

Confirms predictions for depletion

SRC LRC

n(0) ⇒ 0.85 Reid 0.87 Argonne V18 0.89 CDBonn

Nuclear matter

Green’s functions III

High-momentum protons have been seen in nuclei!Jlab E97-006 Phys. Rev. Lett. 93, 182501 (2004) D. Rohe et al.

12C

• Location of high-momentum components• Integrated strength agrees with theoretical prediction Phys. Rev. C49, R17 (1994)

⇒ ~0.6 protons for 12C ⇒ ~10%

Correlations at large asymmetry

Nuclei (e,e’p) reactionNIKHEF data, L. Lapikás, Nucl. Phys. A553, 297c (1993)

Wave functions as expected, except ….


Removal probability for valence protons

fromNIKHEF data

L. Lapikás, Nucl. Phys. A553,297c (1993)

Weak probe but propagation in the nucleus of removed proton

using standard optical potentials to generate

distorted waves --> associated uncertainty ~ 5-10%

Why: details of the interior scattering wave function

uncertain since nonlocality is not constrained (so far)

S ≈ 0.65 for valence protonsReduction ⇒ both SRC and LRC

(e,e’p)


Theory

NIKHEFG. Kramer, Thesis

Spectral function 48Ca (e,e´p) 47K (l =2)

Theory includes :low-energy mixing long-range correlations

Excludes:high-energy mixing short-range correlations

Brand et al.NPA531,253(1991)

G.Rijsdijk, K.Allaart, WDNucl.Phys.A550,159(1992)

d3/2

mostly d5/2


Propagator and Faddeev-RPA

Developed by Carlo Barbieri in his thesis at Washington U in St. LouisPRC 63, 034313 (2001); PRC 65, 064313 (2002)

Correlations

Ab initio results for spectroscopic factors 56Ni• 10 oscillator shells versus 1p0f shell-model space

• same effective interaction (soft N3LO --> G-matrix)

• FRPA 1p0f space already almost as good as full shell model

• FRPA in 10 oscillator shells necessary to describe experiment

HI knockout MSU

Carlo Barbieri PRL 103, 202502 (2009)


Location of single-particle strength in closed-shell

(stable) nuclei

SRC

SRC theory

For example:protons in 208Pb

NIKH

EF (e,e’p) dataL. Lapikás

Nucl. Phys. A

553,297c (1993)JLab E97-006

Phys. Rev. Lett. 93, 182501 (2004) D. Rohe et al.

Elastic nucleon scattering


We now essentially know what all the protons are doing in the ground state of a “closed-shell” nucleus !!!

• Unique for a correlated many-body system

• Information available for electrons in atoms (Hartree-Fock wave functions OK, occupation numbers still 1 or 0, some fragmentation of sp strenght)

• Not for electrons in solids • Not for atoms in quantum liquids• Not for quarks in nucleons

⇒ Demonstrates how interesting the nucleus is as a quantum many-body system!


F-RPA application to Neon atomC. Barbieri, D. Van Neck, and WD, Phys. Rev. A76, 052503 (2007)

Spectroscopic factors in brackets

*

* Consistent with results from Heidelberg group

One of the developers of the no-core shell model:“F-RPA/Green’s function method maybe the only way to do heavy nuclei”


Periodic table: Ne ionization energy

HF: 23.12 eV

Σ(2): 20.32 eVF-RPA: 21.73 eVExp: 21.57 eV


Also applied to molecules• Barbieri (Surrey) & Degroote and Van Neck (Ghent)

• Competitive with Coupled-cluster method!

Correlations at large asymmetry

Correlations for nuclei with N very different from Z?⇒ Radioactive beam facilities

• SRC about the same between pp, np, and nn • Tensor force disappears for n when N >> Z but …• Empirically p more bound for N>Z systems than n

• Any surprises?• Ideally: quantitative predictions based on solid foundation

Some pointers: one from theory and one from experiment

Nuclei are TWO-component Fermi liquids


Self-Consistent-Green’s-Function calculation for isospin-polarized nuclear matterincluding SRC ⇒ momentum distribution ⇒ n(k=0)

0.16 fm-3

T=10 MeVneutrons

protons

asymmetry = (N-Z)/A

Frick, Rios et al.PRC71,014313(2005)

CDBonnArV18Reid

n(k=

0)

SRCcan be handled!

{

small uncertainty


Temperature vs. correlations Rios, Polls, WD

arXiv:0904.2183


Fixed by phase shifts?!

arXiv:0904.2183


FRAMEWORK FOR EXTRAPOLATIONS BASED ON EXPERIMENTAL DATA

Combined analysis of protons/neutrons in 40Ca and 48CaCharity, Sobotka, & WD, Phys. Rev. Lett. 97, 162503 (2006)Charity, Mueller, Sobotka, & WD, Phys. Rev. C76, 044314 (2007).

Goal: Extract asymmetry dependence ⇒ δ/β = (N - Z)/A ⇒ Predict proton properties at large asymmetry ⇒ 60Ca ⇒ Predict(?) neutron properties … the dripline based on data!

Large energy window (> 200 MeV)

Recent extension of DOM


Dyson Equation and “experiment”Equivalent to …

�

− 2∇2

2mΨn

N−1 a r m Ψ0N + d

r ∫

m '∑ 'Σ'* ( r m, r 'm';En

−) ΨnN−1 a r 'm' Ψ0

N = En− Ψn

N−1 a r m Ψ0N

Self-energy: non-local, energy-dependent potentialWith energy dependence: spectroscopic factors < 1 ⇒ as observed in (e,e’p) �

En− = E0

N − EnN−1

Schrödinger-like equation with:

�

S = ΨnN−1 aαqh

Ψ0N 2

= 1

1−∂Σ'* αqh ,αqh;E( )

∂EEn

−

αqh solution of DE at En-

DE yields

�

ΨnN−1 a r m Ψ0

N =ψnN−1 r m( )

Ψ0N a r m Ψk

N +1 =ψkN +1 r m( )

ΨEc,N−1 a r m Ψ0

N = χcN−1 r m;E( )

Ψ0N a r m ΨE

c,N +1 = χcN +1 r m;E( )

Bound states in N-1 Bound states in N+1 Scattering states in N-1

Elastic scattering in N+1

Elastic scattering wave function for (p,p) or (n,n)


Dispersion relations (nuclear matter or nuclei)

Imaginary part both above and below εF

Above εF:required by elastic nucleon scattering data

Below εF:required by e.g. (e,e’p) and (p,2p) data

Consequences:Fermi sea is depletedand there is occupation of“particle-like” states (high-momenta e.g.)


DOM = Dispersive Optical Model

Vehicle for data-driven extrapolations / predictions to the drip lines Goal: extract “propagator”/”self-energy” from data

C. Mahaux and R. Sartor, Adv. Nucl. Phys. 20, 1 (1991)

Combined analysis of protons/neutrons in 40Ca and 48CaCharity, Sobotka, & WD, PRL 97, 162503 (2006)Charity, Mueller, Sobotka, & WD, PRC76, 044314 (2007)

Predict


Mougey et al., Nucl. Phys. A335, 35 (1980) 16O(e,e’p)

Energy

Moment

um

∝ Cross section


40Ca spectral functionRecent theoretical development: nonlocal self-energy below the Fermi energyVan Neck, Charity, Sobotka, Waldecker, WD 2009

Old (p,2p) data from Liverpool

Below εF


Spectroscopic factors as a function of δ

Protons more correlated with asymmetry

Modest asymmetry effectbut ...

Below εF


Gade et al. Phys Rev C77, 044396 (2008)

⇒ Spectroscopic factors become very small; way too small?

RS ≠ not spectroscopic factor

Reduction w.r.t. shell model

neutrons more correlated with increasing proton numberand accompanying increasing separation energy & vice versa


Transfer vs. HI Knockout Reactions

Completely different asymmetry dependencebetween transfer and HI knockout! (see next talk)

Lee et al. ArXiv:0911.4857

Different optical potentials --> different reduction factors


Transfer cross sections & DOM potentials

• Nunes, Waldecker, WD

• ADWA --> use DOM potentials for individual nucleons

• Nonlocality to be understood

Analyze transfer also with DOM potentials


Ab initio: Faddeev-RPA ⇒ Barbieri, WD IJMPA24, 2060 (2009)

Trend similar to HI knockout but magnitudevery, verydifferent ...


Recent DOM analysis -->

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Conclusions★ history of nuclear saturation interesting but unresolved★ pion-exchange in nuclei ≠ pion-exchange in nuclear matter★ RELEVANT FOR UNDERSTANDING NUCLEAR SATURATION★ pions should be allowed to propagate (not discussed)

★ traditional shell-model space too small for proper understanding★ FRPA successful but must be extended to include all particles★ Positive energy self-energy must be calculated properly ★ DOM allows data-driven extrapolation to the drip line★ reactions can be described in DOM framework (with dynamic π for (p,2p)


Future• Dynamic pions & nuclear saturation

• Nuclear saturation and the symmetry energy

• Neutron matter at higher density (ISOBARS)

• Pairing --> phase diagram of nuclear matter (T,ρ,α)

• Calculate ab initio self-energy at positive energy in nuclei

• Include all particles in Faddeev-RPA

• Excited states (Baym-Kadanoff)

• QP-DFT

• Microscopic link with DOM

• (p,pN) in inverse kinematics to replace (e,e’p) employing DOM

• Three-body forces

Overview of SCGF methods in nuclear...

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