/ee/lecceq.tex

Nuclei beyond the independent particle description:

The importance of correlations

Ingo Sick

Historical development of nuclear physics

strongly influenced by mean-field ideas

till today main approach to describe nuclei

Main assumption: nucleons move in mean field

field created by average interaction with all other nucleons

individual nucleon moves independently of others

residual interactions incorporated perturbatively

Consequences:

nuclei have many similarities to atoms

appearance of shells, magic numbers, ..

existence of quasi-particle (QP) orbits

Mean field description

reasonably successful

when use fitted effective interactions, fitted mean field

can explain many features of nuclei

amazing that works, given properties of N-N-interaction

But: fitted parameters, limited to selected observables

for new observable need different parameters

connection to underlying N-N interaction remote

”easy” calculations, but not very satisfactory

More fundamental approach: start from N-N interaction

solve Schrodinger equation for A nucleons interacting with VNN

Difficulty:

N-N interaction very complicated

spin- and (angular)-momentum dependent

has strong short range repulsion

⇒ solution of n-body Schrodinger equation very difficult

Exact (numerical) solution for known VNN :

feasible for infinite nuclear matter (NM)

Bethe-Bruckner-Goldstone theory

Correlated Basis Function (CBF) theory

feasible for light nuclei (A<12)

Faddeev, Hyperspherical, Variational MC

gradually also feasible for heavier nuclei

Fermi Hypernetted Chain calculations

Advantage:

basis = N-N interaction known from N-N scattering

applicable to a priori all nuclei

no free parameters

decisive for predictions at higher densities as occur e.g. in stars

Main difference to mean-field

account for short-range N-N correlations

short-range interactions leads to scattering of N to orbits E ≫ EFshort-range interactions leads to scattering of N to orbits k ≫ kFpartial depletion for E < EF , k < kF

for quantitative understanding

correlations absolutely crucial

Ideal approach to expose correlations: CBF theory

appear explicitly as variational functions f(rij) in wave function

|N) = G|N ], G = S∏

j>i

F (i, j), F (i, j) =∑

nfn(rij)O

n(i, j)

Effect of correlations

on components of potential on f(rij)

|N ] = MF state

O = operators of VNNf = correlation

functions

variationally det.

correlation hole for some components, short-range enhancements for others

Exact calculations explain light nuclei amazingly well

large effect of correlations

.... a real success story that takes Nuclear Physics to a new level

Important difference quasi-particle ↔ correlated strength

QP wave functions: R(k) falls quickly at large k

correlated strength: has long tail towards large k

example: 4He from Variational Monte Carlo

i.e. exact calculation for realistic NN-interaction

2222 QP orbital, observed e.g. in 4He(e, e′p)3H

drops off rapidly at large k

3333 correlated strength in continuum at large E

falls off much less quickly, dominates large-k totally

Perhaps more typical for nuclei: nuclear matter

Effect of correlations for momentum distribution and spectral function S(k,E):

• correlations give strength at both large k and E

• strength very spread out, hard to identify experimentally

• correlated N have ∼20% probability (NM),

but give 37% of removal energy

47% of kinetic energy

(CBF calculation of Benhar, Fabrocini, Fantoni 1989)

⇒ mean-field approach cannot work

exception(?): differences of energies, spect. factors

insight for time being lost on shell-model community

calculations of ever increasing sophistication

ignoring 20% of nucleons

e.g. review Talmi, ”50 years of shell model”, 275pages

not one word on 20%, 37%, 47%

correlated N even more crucial for:

2N-dependent processes, MEC

enhancement of integrated transverse (e,e’) strength

NM study of A. Fabrocini

transverse sum rule in 3He, 4He, J.Carlson et al.

effect of MEC with correlations 8 times larger

only with correlations can explain data

Analogous studies of liquid Helium

calculation easier, no spin-dependence

Lennard-Jones potential, r−12 − r−6, even more repulsive at small r

⇒ even stronger correlations

But

find quasi-particle states for k < kFwith much reduced occupation

occupation of MF states ∼ 30%

= size of discontinuity at kFrest moved to k > kF mainly

”depopulation” of MF states

= main consequence for MF

Shown by Moroni et al.

via calculations for L3He and L4He

depopulation similar for bosonic/fermionic systems

mainly consequence of short-range VNN , not Pauli principle

Finite systems

more complicated

”occupation” requires concept of ”orbit”

not a priori obvious which one:

mean field, overlap, natural, ....?

Studies of L3He drops

Variational Monte Carlo

drops with A, A–1 atoms

deduce difference

find for A=70

corresponds to 3s-state

quasi-hole orbital close to MF orbitals +LDA ( )

ψQH = ψMF

√

z(ρ(r)) z=renormalization

= quantities observable in transfer, (e,e), (e,e’p)

main message

in nuclei and LHe find orbits ∼ quasi-particle states

R(r), R(k) ± as given by mean-field models

observed in transitions to E < 10 MeV

but

single particle states have partial occupation

spectroscopic factors (= overlap with IPSM wave functions) <1

rest of strength at very large k,E

large effect of correlated strength on Ekin, Eint

experimental observation of depopulation?

long in coming

suitable experiments not available

Traditional tools used to measure spectroscopic factors s:

(d,3He), (p,d), ..., (p,2p), ..., (e,e’p)

most extensive source of information: transfer reactions

What do they really measure?

consider radial sensitivity of probes, for Pb

sensitivity of (e,e) flat, (p,2p) further outside than (e,e’p) [more absorption],

transfer reactions measure asymptotic norm, not s

should really only quote this quantity! ... but somehow people want s

Why want s?

better intuitive meaning than asymptotic norm

(typical) HO-based calculations only give s

If want s from transfer: suffer from strong dependence on R(r)

s typically changes 10% for a 1% change in rms-radius of R(r)

since rms-radius not known → s rather arbitrary

Past prejudice

∑

i

si, summed over final states i, gives occupation np,h (pickup, transfer)

nparticle + nhole = 1, i.e. 2j+1 particles in state j

chose R(r) such as to get 1

Result

find small values for nhfind np close to 1

Problem

cannot determineE=∞∑

E=0

si , data restricted tofew MeV

∑

si∑

does not contain all the strength

rest not identifiable as in continuum

Doubts in n∼1

• elastic M5 – M9 form factors

• form factors of high-multipole transitions E12, E14

little subject to configuration mixing

sensitive where R(r) large

relative to MF F 2 reduced by factor ∼1.5÷2.

indicates occupation of ∼75%

Pandharipande et al. PRL 53(84)1133

• ∆ρ Pb-Tl (R2(r)3s): same conclusion

Better information on orbits: from (e,e’p)

measures R(k) (hence R(r)), no need for (arbitrary) input

measures absolute strength as sensitivity large where R(r) large

easier to treat as only one strongly interacting particle

no composite particles subject to strong absorption

Data

early results from Saclay:

find orbits with R(k) ∼ MF

MF also applicable to nuclear interior

find occupation n ∼0.7

n not taken very seriously

doubts about DWBA treatment

... which however was OK

Systematic studies of (e,e’p)

mainly done at NIKHEF

measure R(k) and absolute strength

use sophisticated interpretation FRDWBA, e-distortion, ...

10-2

10-1

100

101

102

103

1p3/2

1p1/2

16O

10-2

100

102

104

1d3/2

2s1/2

-100 0 20010-2

10-1

100

101

102

p [MeV/c]

mom

entu

m d

istr

ibut

ion

[(G

eV/c

) s

r ]

2p1/2

1f5/2

10-4

10-2

100

102

208Pb

3s1/2

2d3/2

90Zr

40Ca

100 -100 0 200100

-100 0 200100 -100 0 200100

m

-3-1

produces quantitative measurement of momentum distributions

find systematic reduction of occupation of QP states

Now have convincing data that QP strength z<1

occupation of ∼ 0.7 (outer shells)

∼ 0.8 without surface + LRC effects

(note: peculiar behavior of 12C, see below)

Striking example for MF R(r) despite depopulation:

3s-state in nuclei206Pb−205 T l

(e,e) sensitive to interior

radial distribution ∼ MF orbit

even in high-density region

but: occupation = 0.7

Consistency with transfer reactions?

transfer data reanalyzed by Kramer et al. NPA 679(01)267

use R(r) from (e,e’p)

use modern DWBA

find good agreement with (e,e’p)

main change: due to input (measured) R(r)

emphasizes importance to use good R(r) !!

Unsatisfactory situation of experiment:

have identified missing strength

have fair theoretical understanding

have not seen correlated strength

measured: (1–correlated strength) → large uncertainty

blows up uncertainty by factor of 5

Past attempts to identify strength at large k

• reactions of type (x,p)

low momentum transfer from x, observe high momentum p

e.g. (γ,p), (p,p) with high-momentum backward p

problem: Amado+Woloshyn, 1977

• in limit q→0 FSI cancels IA term from high-k component (orthogonality!)

no quantitative interpretation possible

(applies also to (p,2p), .. )

• (e,e’) at large q, low ω

~k

~q~k + ~q

idea: small ω ∼ (~k + ~q)2/2m, large ~q → ~k ∼ −~q, large

problem: FSI (e.g. 3He(e,e’), PWIA = )

• see Benhar, Fabrocini, Fantoni,... (1991)

provide first treatment of FSI using Glauber, find that dominates

but: not entirely satisfactory, still in the works

for either case: cannot address large k anyway as strength is at large E

... an insight that has yet to sink in

Best tool to measure strength at large k and large E:

(e,e′p) at large q

can minimize effect FSI

can treat reliably via Glauber

complications:

• must look at large E

not large k and small E as done initially and ± all existing experiments

• strength spread out over 100-200MeV

small in given (k,E)-bin

very hard to observe

• reaction mechanism (see diagram)

p rescatters, reappears at lower kp′

simulates large missing energy E, large k

covers small genuine strength

• similar for (e, e′∆), with ∆ → p+ π(undetected)

Study of all available data

compare experimental and calculated dσ/dΩdω

in IA, using realistic S(k,E)

use R(k)MF + ScorrNM(ρ) in LDA

look if data ≃ or >> theory

find

• most experiments give σexp ≫ σIA• standard perpendicular kinematics worst, // kinematics best

studies of kinematics of rescattering processes:

understand how (p, p′N) and (e, e′pπ) move strength

identify optimal kinematics: parallel (standard: perpendicular!)

confirmed by MC calculations of Barbieri (see below)

Pro memoria: perpendicular kinematics

reasons for choice

experimentally convenient

need to change only angle of p-spectrometer

can keep q and spectrometer-momenta constant

convenient for DWIA-analysis

± same optical potential for all kinematics

However

maximizes multi-step contribution

maximizes MEC

Optical potential anyway wrong approach

must describe protons that have inelastically interacted

and not been ”swallowed up” by Im(V)

need Glauber-type approach to describe dominating (p,pN)

Detector stack:

2 drift chambers 4 scintillator planes 1 Cerenkov

HMS

SOS

Q Q Q D

D

D

e

e′

p′

3.2 GeV

0.85-1.7 GeV/c

2.05-2.75 GeV/c

Target spectrometer

(e,e’p) experiment

JLab hall C:

Daniela Rohe et al.

Can achieve with JLab:

large q → low FSI, treatable with Glauber

acceptable true/accidental ratio

despite unfavorable (//) kinematics

Tests

in single-particle region, kinematics with same Ep′ as production runs

use: T=0.6... (Benhar+Pieper), integrate over E <80MeV

find: occupation agrees with CBF S(k,E)

Results for correlated region

0.1 0.2 0.3 0.4E

m (GeV)

1e-13

1e-12

1e-11

1e-10

S(E

m,p

m)

[MeV

-4 s

r-1]

0.1700.2100.2500.2900.3300.3700.4100.4500.4900.5300.1700.2100.5700.6100.650

Spectral function for C using ccparallel: kin3, kin4, kin5

pm

(GeV/c)Em

= p2

m2 M

p

main observation on E-dependence

maximum of S(k,E) of theories at too large E

understood by recent calculation of Muther+Polls?

selfconsistent GF, ladder approximation, finite T

Momentum dependence

0.2 0.4 0.6p

m [GeV/c]

10-3

10-2

10-1

n(p m

) [f

m3 s

r-1]

parallel

CBF theoryGreens function approachexp. using cc1(a)exp. using cc

⇒ theory and experiment ± agree

how about standard perpendicular kinematics?

used for overwhelming majority of experiments

find

(distorted) S(k,E) >> S from parallel kinematics

ckp1p2_e01trec16n_cc1on_nice.agr

confirmed by calculation of Barbieri

includes multistep (p,p′N) via Glauber

0 0.1 0.2 0.3 0.4 0.5E

m (GeV)

1e-13

1e-12

1e-11

1e-10

S(E

m,p

m)

[MeV

-4 s

r-1]

input S(k,E) distorted S(E

m,p

m) in parallel kin.

distorted S(Em

,pm

) in perpendicular kin.

Parallel vs. perpendicular kinematics for 12

C

330 MeV/c

410 MeV/c

490 MeV/c

570 MeV/c

C. Barbieri,preliminary

lesson

in parallel kinematics multi-step = manageable correction

in perpendicular kinematics multi-step reactions dominate

(standard) perpendicular kinematics useful only to check multistep calculation

How much correlated strength??

cannot integrate over entire correlated region

∆-excitation and QP strength cover part of correlated strength

integrate over ’clean’ region, both data and theory

80%

250

80 300

1.5%

5%

700

0

4.5%

8% 1%

Em(MeV)

pm(MeV/c)used region

# of correlated protons used total

integral over S from experiment 0.59

integral over S from CBF 0.64 1.32

integral over S from SGGF 0.61 1.27

→ good agreement

→ can believe total from theory

→ 20%, integrated over k,E

heavier nuclei

experiment performed for C, Al, Fe, Au

interest in A>>

→ nuclear matter

ratio to C of correlated strength

0 50 100 150 200mass number A

11.21.41.61.8

22.22.4

spec

tral

rat

io

parallel kinematicsperpendicular

Ratio Al, Fe, Au to C spectral functionintegrated over correlated region

∆-resonance!

enhancement for Au

not yet understood

consequence of n-p correlations as N > Z??, rescattering ??

would like to get S(k,E) for N6=Z

overall

• have now experiment with optimized kinematics

to minimize multi-step contributions

• have identified strength at large k,E

• theory produces S(k,E) with ± correct strength

SGFT, CBF+LDA

E-dependence does not entirely agree

strength at too low E

enhancement for large A not understood

• would want kinematics more strictly parallel

rather restrictive kinematics

unfavorable true/accidental ratio

but it’s worth it!

for details:

see habilitation work of Daniela Rohe

Strength at large k,E: physical origin

correlated nucleon pairs

large, opposite momenta ~k, −~k

electron hits proton, transfers ~q

correlated N leaves nucleus with −~k

kinetic energy k2/2M → ridge in S(k,E)

direct observation: (e,e’2p), |kp| >400MeV/c

Shneor et al., PRL

find: anti-correlation of p-momenta, γ ∼ 180

a) ratio strength12C(e,e′pp)12C(e,e′p)

∼ 0.1

(extrapolated to full kinematical coverage)

b) add result of Schiavilla et al.

ratio strength pp+nppp

∼ 6 for 4He and k >400MeV/c

c) add transparency for high-k proton of ∼0.7

⇒12C(e,e′pN)12C(e,e′p)

∼ 0.1 · 6 / 0.7 ∼ 0.9

⇒ high-k protons in (e,e’p) ± all from correlated pairs, i.e. (e,e’p)N; nice confirmation

Basic insight: need S(k,E) to describe nuclei

n(k) not sufficient!

Example: role of interaction in DIS

use only momentum distribution

parton distribution functions

i.e. ignore energy of initial state parton

also ignore final-state interaction of recoiling parton

Importance of interaction:

binding in case of DIS must be important

nuclear binding explains EMC-effect!

E for correlated N much larger than EMF

final state interaction plays role

claimed erroneously to disappear for Q2 = ∞

Brodsky et al. show that this is wrong

→ response functions are not distribution functions

will need to be improved upon

what is spectral function of quarks in nucleons?

Orthogonal look:

where correlated strength in r-space?

motivation: difficulties with QP-R(r)

• QP radial wave functions fitted to ρ(r)

poorly explain F(q) of QP-dominated transitions

• QP wave functions poorly explain ρ(r) at small r

reason: ρ(r) contains correlated contribution

presumably correlated radial shape 6= QP shape

⇒ question: radial distribution of correlated strength = ?

Two opposing tendencies:

• large E pulls correlated strength to small r

• higher (angular) momenta tend to shift it to larger r

which wins?

2 independent answers:

• study via selfconsistent Green’s function theory SGFT

H. Muther

• determine from (e,e) and (e,e′p)

S(k,E) from Green’s function method (Muther, Polls, ..)

split S into QP plus correlated piece

ρ(r) =∑

lj

SQPlj (r, r) +∑

lj

∫ ∞

ε2h1p

dE Scontlj (r, r;E)

= ρQP (r) + ρcorr(r) ,

CD-Bonn NN interaction → 1.0 correlated protons (low?)

observations

ρcorr concentrated much more towards small r

does not contribute at large r

there tail of QP dominates completely

ρcorr at small r despite contributions of large l

31% l=0, 37% l=1, rest large l

large E of states pulls R(r) to small r

at small r ρcorr contributes ∼30% of ρ(r)

explains failure of QP wave functions

ρcorr from (e,e)+(e,e′p) data

ρcorr(r) = ρ(r)point −∑

QP−orbits

FBT (RQP (k))2

point density of C

have very precise (e,e) data up to large q

have µ-X-ray data

do modelindependent analysis (SOG)

→ charge density with small δρ

unfold nucleon size to get point density

QP wave functions from (e,e′p)

extensive set of (e,e′p) data

• low-q from NIKHEF, Saclay

analyzed with DWBA

optical potentials from (p,p)

• high-q data from SLAC, JLAB

analyzed with theoretical transparencies

confirmed by data

compilation: L. Lapikas et al., PRC61, 64325

problem

results apparently not consistent

low q: 3.4 uncorrelated protons

low as compared to other nuclei

high-q: 5 – 5.6 uncorrelated protons

discrepancy

embarrassing for practitioners of (e,e′p)!

deathblow to (e,e′p) as quantitative tool?

question: which is true occupation? is q-dependent? (Strikman et al.)

issues: quality optical potentials, role MEC’s,

coupled-channel effects, value of T

one reason for difference obvious: sloppy interpretation

low-q data:

E ≤ 50MeV , k ≤ 180MeV/c

high-q data:

E ≤ 80MeV , k ≤ 300MeV/c

integrate over part of correlated strength, must remove before comparing!

correction of high-q result

use analysis of QP region of Rohe et al.

agrees with previous SLAC/JLAB data, uses most reliable T

calculate correlated contribution using theory, remove

result:

high-q: 4.5 uncorrelated protons

low-q: 3.4 uncorrelated protons

⇒ discrepancy much reduced

but: still larger than desirable, larger than believed ±10%

my choice: (corrected) high-q result

low-q for 12C anomalously low

(remember figure occupations)

low-q have significant MEC effects

Boffi: up to 20%

low-q would imply unrealistic correlated strength

would disagree with Rohe et al.

use high-q result

choice confirmed by consistency check, see below

further adjustment needed

1s fitted to data E < 50MeV

this region contains already some correlated strength

R(k)corr falls less quickly than R(k)QPmust correct shape of Lapikas R(k)1s

correction of shape:

• can do using QP and correlated S(k,E) from theory

• can analyze Rohe data with QP+correlated parts

find same result:

R(k) compressed by 11%

ρcorr from (e,e)+(e,e′p)

start with point density

subtract QP contribution, Fourier-Bessel-transformed R(k)

using high-q (corrected) occupation

result

observations

ρcorr concentrated towards small r

as was seen in theory

ρcorr gives ∼30% contribution at small r

explains failure of QP models

reasonable agreement with theory

(uncertainty of ρcorr ∼20%)

in exp. density perhaps more l > 1 strength

important consistency check: large r

perfect agreement ρQP ... ρpointshould occur as ρcorr cannot contribute

large-r = the region where MF ± OK

conclusions of r-space study

shape of ρcorr differs strongly from shape of ρQP

ρcorr gives 30% contribution in nuclear interior

explains failure of QP models, cannot be ’compensated’ using eeff , etc.

reasonable agreement with Green’s function theory

Conclusions

have finally data on correlated strength

... some 15 years after CBF calculation

± agrees with modern many-body theories

... which were amazingly good!

for quantitative understanding of nuclei:

must go beyond mean-field, include correlated nucleons

for good S(k,E) of finite nuclei

look forward to results from Fermi Hypernetted Chain calculations