Fetter & Walecka
Correlations

• NN interaction requires summation of ladder diagrams
• Study influence of SRC on sp propagator
• Formulate in self-consistent form
⇥ k m m | pphh
(
K , E
) | k m
⇥ m
⇥
⇤
= ⇥ k m m | V | k m
⇥
= ⇥ k m m | V | k m
⇥ m
⇥
⇤ + ⇥ k m m | ⇥ m
⇥
⇤ +
1
2 m m
⇥ d 3 pphh q
(2 ) 3
( K , E ) | k m
⇥ m
⇥
⇤
⇥ k m m | V | q m
⇤ m
⇤
⇤
G f pphh
( K , q ; E ) ⇥ q m
⇤ m
⇤
| pphh
( K , E ) | k m
⇥ m
⇥
⇤
• employs noninteracting but dressed convolution of sp propagators
G f pphh
( K , q ; E ) =
⇤ dE
⇥
F
F dE ⇥
⇤
⇤ dE
⇥⇥
S p
( q + K / 2; E ⇥ ) S p
( K / 2 q ; E ⇥⇥ )
E E ⇥ E ⇥⇥ + i
F
F dE ⇥⇥
⇤
S h
( q + K / 2; E ⇥ ) S h
( K / 2 q ; E ⇥⇥ )
E E ⇥ E ⇥⇥ i
QMPT 540

• Dispersion relation (arrows for location of poles in energy plane)
⇤ k m m | ⇥
= pphh
⇥
1
( K , E ) | k
⌅ m
⇥ m
⇥
⌅
⇧
2 ⇤
F dE
⌅
Im ⇤ k m m | ⇥ pphh
(
K
E E ⌅ + i
, E ⌅ ) | k ⌅ m
⇥ m
⇥
⌅
+
1
⇥
⇥ ⇤ k m m | ⇥
2 ⇤
F dE
⌅
Im
⇧
⇤
(
K , E
) | k
⇤ k m m | ⇥
E pphh
E ⌅
( K i
, E
⌅ m
⇥ m
⇥
⌅ + ⇤ k m m | ⇥
⌅ ) |
⇥ k
(
⌅ m
K
⇥ m
, E
)
⇥
| k
⌅
⌅ m
⇥ m
⇥
⌅
• Corresponding self-energy
⇤
⇥
( k
;
E
) = i
1
⇥ d 3 k
(2 ⇥ ) 3
⇥ dE m m
2 ⇥
G
( k
;
E
)
⇥ ⇤ 1
2
( k k ) m m | ⇥ pphh
( K , E + E ) | 1
2
( k k ) m m ⌅
• Energy integral can be performed (note arrows)
⇤
⇥
( k ; E ) =
1
1 m m m m
⇥
⇥ d 3
(2 ⇥ ) 3 d 3 k k ⌅
⌅
(2 ⇥ ) 3
⇥
⇥
⇥
F
⇧
⇧
⇥
F dE dE
⌅
⌅
⇤
⇤
1
2
1
2
( k k
⌅ ) m m | ⇥
( k k
⌅ ) m m | ⇥
⇥
⇤
( E + E
⌅ ) | 1
2
( E + E
⌅ ) | 1
2
( k k
⌅ ) m m ⌅ S h
( k k
⌅ ) m m ⌅ S p
( k
⌅
, E
⌅ )
( k
⌅
, E
⌅ )
⇥ ⇥⇤
⇤
( k ; E ) + ⇥⇤
⇥
( k ; E ) QMPT 540

• Total self-energy
⇥ ( k ; E ) = ⇥
V
( k )
1
⇥
⇧ dE
⌅
Im ⇥ ( k ; E ⌅ )
E E ⌅ + i
+
F
= ⇥
V
( k ) + ⇥
⇤
• includes HF-like term
V
( k ) = d 3 k
(2 ⇥ ) 3
1 m m
⇥ 1
2
( k ; E ) + ⇥
⇥
( k ; E )
( k k ) m m | V | 1
2
1
⇥
F
⇧ dE
⌅
Im ⇥ ( k ; E ⌅ )
E E ⌅ i
( k k ) m m ⇤ n ( k )
• Practical calculations first done with mean-field sp propagators in scattering equation
⇤ (0)
⇥
( k ; E ) =
1
⇥
1
⇥ m m
⇥
⇥ d 3 k ⇤
(2
⇤
) 3 d 3 k ⇤
(2
⇤
) 3
⇥ 1
2
( k k
⇤ ) m m | ⇥
⇥ 1
2
( k k
⇤ ) m m | ⇥
(0)
⇥
( E + ⌅ ( k
⇤ )) | 1
2
( k k
⇤ ) m m ⇤ ( k
F
(0) ( E + ⌅ ( k
⇤ )) | 1
2
( k k
⇤ ) m m ⇤ ( k
⇤ k
⇤ ) k
F
) m m
QMPT 540

• Realistic interactions generate pairing instabilities
• Avoid with “gap” is sp spectrum according to
( k ) =
( k ) = dE E S h
( k, E ) dE S h
( k, E ) dE E S
Q
( k, E ) dE S
Q
( k, E
)
= E
Q
( k ) k < k
F k > k
F
• 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
QMPT 540

• Wave vectors 0.1 (solid), 0.51 (dotted), and 2.1 fm -1 (dashed)
• k
F
= 1.36 fm -1
• Im part
• Spectral functions with peaks at
• QP spectral function
S
Q
( k, E ) =
1
E
Q
( k ) =
Z 2
Q
2 k
2 m
2
+ Re (
( k ) | W ( k ) | k ; E
Q
( k ))
( E E
Q
( k )) 2 + ( Z
Q
( k ) W ( k )) 2
Z
Q
( k ) =
⇤
1
Re ( k ; E )
⇥
E
⌅
1
E = E
Q
( k ) QMPT 540

• Near k
F
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 k
F
QMPT 540

• Wave vectors 0.79 (dotted), 1.74 (solid), and 3.51 fm -1 (dashed)
• Common distribution -> SRC
• For 0.79 -> 17%
• ε
F
+ 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 k
F
)
• Solid: self-consistent
QMPT 540

• 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
• Z
F
from 0.72 to 0.75
QMPT 540

• Energy sum rule
E N
0
N
⇤
=
2 ⇤ d 3 k
(2 ⇥ ) 3
⇤
F
⇥ dE
2 k 2
2 m
+ E
⇥
S h
( k ; E )
• Contribution from high momenta after energy integral for different densities for Reid potential
QMPT 540

• Colorful and continuing story
• Initiated by Brueckner: proper treatment of SRC in medium -> ladder diagrams but only include pp propagation
⇥ k m m
+
1
2 m m
|
⇥
G ( K d
(2
3
⇤ q
)
, E
3
⇥
) k
| k m
⇥ m m m
|
⇥
V
⇤
|
= q
⇥ m k
⇤ m m m
⇤
⇤
⇥ (
|
|
V q
|
+ k m
K /
⇥
2 m
|
⇥
⇤ k
F
) ⇥ ( | K / 2 q | k
E ⌅ ( q + K / 2) ⌅ ( K / 2 q ) + i
F
)
⇥ q m
⇤ m
⇤
| G ( K , E ) | k m
⇥ m
⇥
⇤
• Brueckner G-matrix but Bethe-Goldstone equation…
• Dispersion relation
⇤ k m m | G ( K , E ) | k
⇥ m
⇥ m
⇥
⌅ = ⇤ k m m | V | k
⇥ m
⇥ m
⇥
⌅
1
⇥
⇤ dE
⇥
2 ⇤
F
Im ⇤ k m m | G ( K , E
E E ⇥ + i
⇥ ) | k ⇥ m
⇥ m
⇥
⌅
• Include HF term in “BHF” self-energy
BHF
( k ; E )) = d 3 k
(2 ⇤ ) 3
1
⇥
⇥ ⇤ k m m | V | k
⇥
( k k ) ⇥ 1
2 m
⇥ m
⇥
⌅ + ⇤ k m m | G ( K , E ) | k
⇥
( k k ) m m | G ( k + k ; E + ⌅ ( k ) | 1
2 m
⇥ m
⇥
⌅ m m
F
( k k ) m m ⇤
• Below Fermi energy: no imaginary part
QMPT 540

• DE for k < k
F
yields solutions at
BHF
• with strength < 1
( k ) =
2 k 2
2 m
+
BHF
( k ;
BHF
( k ))
• Since there is no imaginary part below the Fermi energy, no momenta above k
F
can admix -> problem with particle number
• Only sp energy is determined self-consistently
• Choice of auxiliary potential
U ( k ) = ( k ; ( k ))
F
(0 above)
U ( k ) = ( k ; ( k ))
• Only one calculation of G-matrix for standard choice
• Iterations for continuous choice
QMPT 540

• Propagator
G
BHF ( k ; E ) =
⇥ ( k k
F
)
E ⇤
BHF
( k ) + i
+
⇥ ( k
F k )
E ⇤
BHF
( k ) i
• Energy
E A
0
A
=
⇥
2
⌅
⇤ d 3 k
(2
⇤
) 3
2 k 2
2 m
+
⇧
BHF
⇥
( k k
F
)
• Rewrite using on-shell self-energy
E A
0
A
=
4
⇤
⇥ d 3 k
(2 ⇥ ) 3
2 k
2 m
2
+
1
2 ⇤
⇥ d 3 k
(2 ⇥ ) 3
⇥ d 3 k
(2 ⇥ ) 3 m m
( k
F k ) ( k
F k )
⇥ 1
2
( k k ) m m | G ( k + k ; ⌅
BHF
( k ) + ⌅
BHF
( k )) | 1
2
( k k ) m m ⇤
• First term: kinetic energy free Fermi gas
• Compare
E
HF =
1
2
⇤ p
( p
F p
) p 2
2 m
+
⇥
HF ( p
)
⇥
= T
F G
+
1
2
⇤
( p
F p ) ( p
F p ) ⇥ pp | V | pp ⇤ pp
• so BHF obtained by replacing V by G
QMPT 540

• 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

• First attempt using scattering in the medium Brueckner 1954
• Formal development (linked cluster expansion) Golds to ne 1956
• Reorganized perturbation expansion (60s) Be th e & student s involving ordering in the number of hole lines BBG expansion
• Variational Theory vs. Lowest Order BBG (70s) Clark (also crisis paper)
Pandharipand e
• Variational results & next hole-line terms (80s) Day, Wiring a
• 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

• 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

• 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

• 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 3 He)
QMPT 540

• 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 208 Pb charge density: 0.16 nucleons/fm 3
• 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!!
QMPT 540

Prog.Part.Nucl.Phys.12,529(83)
NPA389,492(82)
2N,3N, and 4N from
B. D. Day, PRC24,1203(81)

•
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)
V
π
( q )
= − f
π
2 m 2
π m 2
π q
2
+ q 2
Needs further study!
Exclude collective pionic contributions to nuclear matter binding energy

•
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 selfconsistency of spectral functions => more reasonable saturation density and binding energy acceptable
•
Neutron matter: pionic contributions must be included ( Δ )

•
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?

•
Saturation depends on
NN σ -coupling in medium but underlying correlated two-pion exchange behaves differently in medium
• m * 0 with increasing
ρ opposite in liquid 3 He appears unphysical
•
Dirac sea under control?
• sp strength overestimated too many nucleons for k<k
F
•
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