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Green’s functions for dense matter
& mean-field connections
Arnau Rios Huguet
Lecturer in Nuclear Theory
Department of Physics
University of Surrey

IPN workshop 3rd October 2017

Complications
NN interaction is not unique

...but phase-shift equivalent!
δ [deg]

90

1

60

S0

30
0

δ [deg]

180

S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008)

Nij
CDBonn
Av18
N3LO
Nij93

120
60

3

S1

0
0

50

100

150

200

250

E [MeV]

2

Complications
NN interaction is not unique

...but phase-shift equivalent!
δ [deg]

90

1

60

S0

30
0

δ [deg]

180

S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008)

Nij
CDBonn
Av18
N3LO
Nij93

120
60

3

S1

0
0

50

100

150

200

250

E [MeV]

•Non-uniqueness of nucleon forces ✘
2

Complications
NN interaction is not unique

Strong short-range correlations

Carlson et al., Phys. Rev. C 68 025802 (2003)
S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008)

•Non-uniqueness of nucleon forces ✘
•Short-range core needs many-body treatment ✘
2

Complications
NN interaction is not unique

S. Aoki, et al. Comput. Sci. Dis. 1 015009 (2008)

Saturation point of nuclear matter

Li, Lombardo, Schulze et al. PRC 74 047304 (2006)

•Non-uniqueness of nucleon forces ✘
•Short-range core needs many-body treatment ✘
•Three-body forces needed for saturation ✘
2

NN forces from EFTs of QCD
ci

Chiral perturbation theory

π

ci
O

ˆ

Q


˙

•π and N as dof
•Systematic expansion
•2N at N3LO - LECs from πN, NN
•3N at N2LO - 2 more LECs
•(Often further renormalized)

⇤ „ 1 GeV

ci

Weinberg, Phys. Lett. B 251 288 (1990), NPB 363 3 (1991)
Entem & Machleidt, PRC 68, 041001(R) (2003)
Tews, Schwenk et al., PRL 110, 032504 (2013)
Epelbaum, Frebs & Meissner, PRL 115, 122301 (2015) 3

NN forces from EFTs of QCD
ci

Chiral perturbation theory

π

ci
O

ˆ

Q


˙

•π and N as dof
•Systematic expansion
•2N at N3LO - LECs from πN, NN
•3N at N2LO - 2 more LECs
•(Often further renormalized)

⇤ „ 1 GeV

ci

Weinberg, Phys. Lett. B 251 288 (1990), NPB 363 3 (1991)
Entem & Machleidt, PRC 68, 041001(R) (2003)
Tews, Schwenk et al., PRL 110, 032504 (2013)
Epelbaum, Frebs & Meissner, PRL 115, 122301 (2015) 3

SCGF Ladder approximation
In-medium interaction

Ladder self-energy
•Self-consistent resummation
•Energy and momentum integral
•@Finite T (Matsubara)

pp & hh Pauli blocking

Dyson equation

T-matrix at T=5 MeV
−5
−10
−15
−20

+

Re T(Ω ; P,q=0) [MeV fm3]

0

−25
−30
600
400

Spectral function
One-body properties
Momentum distribution
Thermodynamics & EoS
Transport

200

P [MeV]

0

−200

−100

0

100

200

Ω−2µ [MeV]

Ramos, Polls & Dickhoff, NPA 503 1 (1989)
Alm et al., PRC 53 2181 (1996)
Dewulf et al., PRL 90 152501 (2003)
Frick & Muther, PRC 68 034310 (2003)
Rios, PhD Thesis, U. Barcelona (2007)
Soma & Bozek, PRC 78 054003 (2008)
Rios & Soma PRL 108 012501 (2012)
4

Ladder approximation with 3BF
Effective interactions

Two-body interaction

1
2

In-medium T-matrix

=

+

In-medium T-matrix
GII

Self-energy

=

+

Self-energy

1

Dyson equation

12

Carbone et al., PRC 88 054326 (2013)
A. Carbone, PhD Thesis 5


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