Correlations

Fetter & Walecka

QMPT 540

Nucleons in nuclear matter• NN interaction requires summation of ladder diagrams• Study influence of SRC on sp propagator

• Formulate in self-consistent form

• employs noninteracting but dressed convolution of sp propagators

⇥km�m�� | �pphh(K, E) |k�m⇥m⇥�⇤= ⇥km�m�� | V |k�m⇥m⇥�⇤ + ⇥km�m�� | ⇥�pphh(K, E) |k�m⇥m⇥�⇤

= ⇥km�m�� | V |k�m⇥m⇥�⇤ +12

�

m�m��

⇥d3q

(2�)3⇥km�m�� | V |qm⇤m⇤�⇤

� Gfpphh(K, q;E) ⇥qm⇤m⇤� | �pphh(K, E) |k�m⇥m⇥�⇤

Gfpphh(K, q;E) =

� ⇤

�F

dE⇥� ⇤

�F

dE⇥⇥Sp(q + K/2;E⇥)Sp(K/2� q;E⇥⇥)E � E⇥ � E⇥⇥ + i�

�� �F

�⇤dE⇥

� �F

�⇤dE⇥⇥Sh(q + K/2;E⇥)Sh(K/2� q;E⇥⇥)

E � E⇥ � E⇥⇥ � i�

QMPT 540

Lehmann representation• Dispersion relation (arrows for location of poles in energy plane)

• Corresponding self-energy

• Energy integral can be performed (note arrows)

⇤km�m�� | ⇥�pphh(K, E) |k⌅m⇥m⇥�⌅

=�1⇥

� ⇧

2⇤F

dE⌅ Im ⇤km�m�� | ⇥�pphh(K, E⌅) |k⌅m⇥m⇥�⌅E � E⌅ + i�

+1⇥

� 2⇤F

�⇧dE⌅ Im ⇤km�m�� | ⇥�pphh(K, E⌅) |k⌅m⇥m⇥�⌅

E � E⌅ � i�

⇥ ⇤km�m�� | ⇥�⇤(K, E) |k⌅m⇥m⇥�⌅+ ⇤km�m�� | ⇥�⇥(K, E) |k⌅m⇥m⇥�⌅

⇤⇥�(k;E) = �i1�

�

m�m��

⇥d3k�

(2⇥)3

⇥dE�

2⇥G(k�;E�)

⇥ ⇤ 12 (k � k�)m�m�� | ⇥�pphh(K, E + E�) | 1

2 (k � k�)m�m��⌅

⇤⇥�(k;E) =1�

�

m�m��

⇥d3k⌅

(2⇥)3

⇥ ⇥F

�⇧dE⌅ ⇤ 1

2 (k � k⌅)m�m�� | ⇥�⇤(E + E⌅) | 12 (k � k⌅)m�m��⌅Sh(k⌅, E⌅)

� 1�

�

m�m��

⇥d3k⌅

(2⇥)3

⇥ ⇧

⇥F

dE⌅ ⇤ 12 (k � k⌅)m�m�� | ⇥�⇥(E + E⌅) | 1

2 (k � k⌅)m�m��⌅Sp(k⌅, E⌅)

⇥ ⇥⇤⇤(k;E) + ⇥⇤⇥(k;E)

QMPT 540

Final steps and implementation• Total self-energy

• includes HF-like term

• Practical calculations first done with mean-field sp propagators in scattering equation

⇥(k;E) = ⇥V (k)� 1⇥

� ⇧

�F

dE⌅ Im ⇥(k;E⌅)E � E⌅ + i�

+1⇥

� �F

�⇧dE⌅ Im ⇥(k;E⌅)

E � E⌅ � i�

= ⇥V (k) + �⇥⇤(k;E) + �⇥⇥(k;E)

�V (k) =⇥

d3k�

(2⇥)31�

�

m�m��

⇥ 12 (k � k�)m�m�� | V | 1

2 (k � k�)m�m��⇤n(k�)

⇤(0)⇥�(k;E) =

1⇥

�

m�m��

⇥d3k⇤

(2⇤)3⇥ 1

2 (k � k⇤)m�m�� | ⇥�(0)⇥ (E + ⌅(k⇤)) | 1

2 (k � k⇤)m�m��⇤ �(kF � k⇤)

� 1⇥

�

m�m��

⇥d3k⇤

(2⇤)3⇥ 1

2 (k � k⇤)m�m�� | ⇥�(0)� (E + ⌅(k⇤)) | 1

2 (k � k⇤)m�m��⇤ �(k⇤ � kF )

QMPT 540

Results for mean-field input• Realistic interactions generate pairing instabilities• Avoid with “gap” is sp spectrum according to

• Hole strength spread so “sp energy” below QP energy --> gap

• Requires full knowledge of self-energy • Self-consistent determination of sp spectrum

• Subsequent determination of self-energy constrained by sp spectrum -> imaginary part exhibits momentum dependent domains

�(k) =�

dE E Sh(k, E)�dE Sh(k, E)

k < kF

�(k) =�

dE E SQ(k, E)�dE SQ(k,E)

= EQ(k) k > kF

QMPT 540

Self-energy below Fermi energy• Wave vectors 0.1 (solid), 0.51 (dotted), and 2.1 fm-1 (dashed)• kF = 1.36 fm-1

• Im part

• Spectral functions with peaks at• QP spectral function

• with and

EQ(k) =�2k2

2m+ Re�(k;EQ(k))

SQ(k, E) =1�

Z2Q(k) | W (k) |

(E � EQ(k))2 + (ZQ(k)W (k))2

W (k) = Im �(k;EQ(k))ZQ(k) =

⇤1�

��Re �(k;E)

�E

⇥

E=EQ(k)

⌅�1

QMPT 540

Spectral functions• Near kF about 70% of sp strength is contained in QP peak• additional ~13% distributed below the Fermi energy

• remaining strength (~17%) above the Fermi energy• Note “gap”

• Distribution narrows• towards kF

QMPT 540

Spectral functions above the Fermi energy• Wave vectors 0.79 (dotted), 1.74 (solid), and 3.51 fm-1 (dashed)• Common distribution -> SRC

• For 0.79 -> 17%• εF + 100 MeV -> 13%

• above 500 MeV -> 7%• Without tensor force:

• strength only 10.5%• Momentum distribution

• n(0) same for different methods• and interactions (not at kF)

• Solid: self-consistent

QMPT 540

Self-consistent results• Wave vectors 0.0 (solid), 1.36 (dotted), and 2.1 fm-1 (dashed)• Limited set of Gaussians --> self-consistent

• Spectral functions• Main difference:

– common strength

• Slightly less correlated• ZF from 0.72 to 0.75

QMPT 540

SRC and saturation• Energy sum rule

• Contribution from high momenta after energy integral for different densities for Reid potential

EN0

N=

�

2⇤

⇤d3k

(2⇥)3

⇤ �F

�⇥dE

��2k2

2m+ E

⇥Sh(k;E)

QMPT 540

Saturation properties of nuclear matter• Colorful and continuing story• Initiated by Brueckner: proper treatment of SRC in medium ->

ladder diagrams but only include pp propagation

• Brueckner G-matrix but Bethe-Goldstone equation…• Dispersion relation

• Include HF term in “BHF” self-energy

• Below Fermi energy: no imaginary part

⇥km�m�� | G(K, E) |k�m⇥m⇥�⇤ = ⇥km�m�� | V |k�m⇥m⇥�⇤

+12

�

m�m��

⇥d3q

(2⇤)3⇥km�m�� | V |qm⇤m⇤�⇤ ⇥(|q + K/2| � kF ) ⇥(|K/2 � q| � kF )

E � ⌅(q + K/2) � ⌅(K/2 � q) + i�⇥qm⇤m⇤� | G(K, E) |k�m⇥m⇥�⇤

⇤km�m�� | G(K, E) |k⇥m⇥m⇥�⌅ = ⇤km�m�� | V |k⇥m⇥m⇥�⌅ � 1⇥

� ⇤

2⇤F

dE⇥ Im ⇤km�m�� | �G(K, E⇥) |k⇥m⇥m⇥�⌅E � E⇥ + i�

⇥ ⇤km�m�� | V |k⇥m⇥m⇥�⌅+ ⇤km�m�� | �G�(K, E) |k⇥m⇥m⇥�⌅

�BHF (k;E)) =⇥

d3k�

(2⇤)31⇥

�

m�m��

�(kF � k�) ⇥ 12 (k � k�)m�m�� | G(k + k�;E + ⌅(k�) | 1

2 (k � k�) m�m��⇤

QMPT 540

BHF• DE for k < kF yields solutions at

• with strength < 1• Since there is no imaginary part below the Fermi energy, no

momenta above kF can admix -> problem with particle number

• Only sp energy is determined self-consistently• Choice of auxiliary potential

– Standard only for k < kF (0 above)

– Continuous all k

• Only one calculation of G-matrix for standard choice• Iterations for continuous choice

�BHF (k) =�2k2

2m+ �BHF (k; �BHF (k))

Us(k) = �BHF (k; �BHF (k))

Uc(k) = �BHF (k; �BHF (k))

QMPT 540

BHF• Propagator• Energy

• Rewrite using on-shell self-energy

• First term: kinetic energy free Fermi gas• Compare

• so BHF obtained by replacing V by G

GBHF (k;E) =⇥(k � kF )

E � ⇤BHF (k) + i�+

⇥(kF � k)E � ⇤BHF (k)� i�

EA0

A=

⇥

2⌅

⇤d3k

(2⇤)3

��2k2

2m+ ⇧BHF

⇥�(k � kF )

EA0

A=

4⇤

⇥d3k

(2⇥)3�2k2

2m+

12⇤

⇥d3k

(2⇥)3

⇥d3k�

(2⇥)3�

m�m��

�(kF � k)�(kF � k�)

⇥ 12 (k � k�) m�m�� | G(k + k�; ⌅BHF (k) + ⌅BHF (k�)) | 1

2 (k � k�) m�m��⇤

EHF =12

⇤

p

�(pF � p)�

p2

2m+ ⇥HF (p)

⇥

= TFG +12

⇤

pp�

�(pF � p)�(pF � p�) ⇥pp�| V |pp�⇤

QMPT 540

BHF• Binding energy usually

within 10 MeV from empirical volume term in the mass formula even for very strong repulsive cores

• Repulsion always completely cancelled by higher-order terms

• Minimum in density never coincides with empirical value when binding OK -> Coester band

Location of minimum determined by deuteron D-state probability

QMPT 540

Historical perspective• First attempt using scattering in the medium Brueckner 1954

• Formal development (linked cluster expansion) Goldstone 1956

• Reorganized perturbation expansion (60s) Bethe & students involving ordering in the number of hole lines BBG-expansion

• Variational Theory vs. Lowest Order BBG (70s) Clark (also crisis paper) Pandharipande

• Variational results & next hole-line terms (80s) Day, Wiringa

• New insights from experiment & theory NIKHEF Amsterdam about what nucleons are up to in the nucleus (90s)

• 3-hole line terms with continuous choice (90s) Baldo et al.

• Ongoing...

QMPT 540

Some remarks• Variational results gave more binding than G-matrix calculations• Interest in convergence of Brueckner approach

• Bethe et al.: hole-line expansion• G-matrix: sums all energy terms with 2 independent hole lines

(noninteracting ...)

• Dominant for low-density• Phase space arguments suggests to group all terms with 3

independent hole lines as the next contribution

• Requires technique from 3-body problem first solved by Faddeev -> Bethe-Faddeev summation

• Including these terms generates minima indicated by✴in figure

• Better but not yet good enough

QMPT 540

More• Variational results and 3-hole-line results more or less in

agreement• Baldo et al. also calculated 3-hole-line terms with continuous

choice for auxiliary potential and found that results do not depend on choice of auxiliary potential, furthermore 2-hole-line with continuous choice is already sufficient!

• Conclusion: convergence OK for a given realistic two-body interaction for the energy per particle

• Very recently: Monte Carlo calculations may modify this assessment

• Also: for other observables no such statements can be made• Still nuclear matter saturation problem!

QMPT 540

Possible solutions• Include three-body interactions: inevitable on account of isobar

– Simplest diagram: space of nucleons -> 3-body force

– Inclusion in nuclear matter still requires phenomenology to get saturation right

– Also needed for few-body nuclei; there is some incompatibility

• Include aspects of relativity– Dirac-BHF approach: ad hoc adaptation of BHF to nucleon spinors

– Physical effect: coupling to scalar-isoscalar meson reduced with density

– Antiparticles? Dirac sea? Three-body correlations?

– Spin-orbit splitting in nuclei OK

– Nucleons less correlated with higher density? (compare liquid 3He)

QMPT 540

Personal perspective• Based on results from (e,e’p) reactions as discussed here

– nucleons are dressed (substantially) and this should be included in the description of nuclear matter (depletion, high-momentum components in the ground state, propagation w.r.t. correlated ground state <--> BHF?)

– SRC dominate actual value of saturation density• from 208Pb charge density: 0.16 nucleons/fm3

• determined from s-shell proton occupancy at small radius• occupancy determined mostly by SRC (see next chapter)

– Result for SCGF of ladders• Ghent discrete approach• St. Louis gaussians (Libby Roth)

• ccBHF --> SCGF closer to box• do not include LRC!!

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

NPA389,492(82)

Prog.Part.Nucl.Phys.12,529(83)

2N,3N, and 4N from

B. D. Day, PRC24,1203(81)

Pion collectivity: nuclei vs. nuclear matter

Vπ (q) = −fπ2

mπ2

q2

mπ2 + q2

• 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-exhange interaction at fixed q

• Nuclei: interaction is sampled over all momenta Phys. Lett. B146, 1(1984)

Needs further study!

⇒ Exclude collective pionic contributions to nuclear matter binding energy

Nuclear Saturation without π-collectivity • Variational calculations treat LRC (on average) and SRC

simultaneously (Parquet equivalence) so difficult to separate LRC and SRC

• Remove 3-body ring diagram from Catania hole-line expansion calculation ⇒ about the correct saturation density

• Hole-line expansion doesn’t treat Pauli principle very well• Present results improve treatment of Pauli principle by self-

consistency of spectral functions => more reasonable saturation density and binding energy acceptable

• Neutron matter: pionic contributions must be included (Δ)

Two Nuclear Matter Problems

• With π-collectivity and only nucleons

• Variational + CBF and three hole-line results presumed OK

• NOT OK if Δ-isobars are included

• Relevant for neutron matter

• Without π-collectivity • Treat only SRC• But at a sophisticated level by

using self-consistency• Confirmation from Gent results ⇒ PRL 90, 152501 (2003)

• 3N-forces difficult ⇒ π ...

• Relativity?

The usual one The relevant one

Comments

• Saturation depends on NNσ-coupling in medium but underlying correlated two-pion exchange behaves differently in medium

• m* ⇒ 0 with increasing

ρopposite in liquid 3He appears unphysical

• Dirac sea under control?• sp strength overestimated

too many nucleons for k<kF

• Microscopic models yield only attraction in matter and more so with increasing ρ

• Microscopic background of phenomenological repulsion in 3N-force (if it exists)?

• 4N-, etc. forces yield increasing attraction with ρ

• Needed in light nuclei!• Mediated by π-exchange

Relativity Three-body forces