Linear response theory and neutrino mean free path in
infinite nuclear matter
A. Pastore1 , D. Davesne2 ,J. Navarro3 and J.W. Holt4
1
University
2
3
of York, Heslington YO10 5DD
Université de Lyon, F-69003 Lyon, France
IFIC (CSIC University of Valencia), Apdo. Postal 22085, E-46071 Valencia
4
Texas A&M, College Station, TX
Trento, May 2016
Outline
1
Introduction
2
Landau theory of Fermi liquids
3
Linear response theory with Skyrme functionals
4
Neutrino mean free path
5
Conclusions
A.Pastore (York)
Linear response
2 / 29
Outline
1
Introduction
2
Landau theory of Fermi liquids
3
Linear response theory with Skyrme functionals
4
Neutrino mean free path
5
Conclusions
A.Pastore (York)
Linear response
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Infinite nuclear matter (Hartree-Fock)
Main features
Pure Neutron Matter EoS
Ideal system made of neutron
and protons
35
No Coulomb
30
25
Iτ =
ρ1
ρ0
Iσ =
s0
ρ0
Iτ σ =
E/N [MeV]
plane-waves
s1
ρ0
20
15
BSk17
BSk18
BSk19
BSk20
BSk21
LNS1
SLy4
CBF
10
5
0
0
0.05
0.1
0.15
0.2
0.25
-3
ρ [fm ]
Effective interactions (functionals)
Correlations are hidden in the coupling constants!
A.Pastore (York)
Linear response
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RPA formalism
We act with an external field on an infinite medium
X
expiqr Θjα Θjα = 1, σ j , τ̂ j , σ j τ̂ j
j
The RPA correlated Green function is the solution of Bethe-Salpeter equation
(S,M,I)
GRP A (q, ω, k1 )
=
GHF (q, ω, k1 )
+
GHF (q, ω, k1 )
X Z d3 k2 S,M,I;S’,M’,I’
V
(q, k1 , k2 )GS’,M’,I’
RP A (q, ω, k2 )
(2π)3 ph
S’,M’,I’
The response function is now defined as
α
SRP
A (q, ω)
=
g
− =
π
Z
d3 k1 α
GRP A (q, ω, k1 )
(2π)3
g = 4 is the degeneracy of SNM.
Results
We have generalized the formalism to include temperature effects
[D. Davesne, A. P. and J. Navarro, Phys.Rev. C 89, 044302, (2014)];
[A. P., D. Davesne, Y. Lallouet, M. Martini , K. Bennaceur and J. Meyer, Phys. Rev. C 85, 054317 (2012)];
[A. P. , M. Martini, V. Buridon, D. Davesne, K. Bennaceur and J. Meyer, Phys. Rev. C 86, 044308 (2012)].
Excited States
Linear response theory to describe excited states in IM
The exchange term is the difficult one!
Different approximations
1
Landau (static) q1 = q2 = kF and q, ν ≈ ω/q = 0
2
Landau (dynamic) q1 = q2 = kF and q = 0
3
Landau (extended) q1 = q2 = kF and q = 0 (only for Vph )
4
RPA (complete) q1 , q2 , q (arbitrary asymmetry at T 6= 0)
A.Pastore (York)
Linear response
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Excited States
All the details/derivations....
A.Pastore (York)
Linear response
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Outline
1
Introduction
2
Landau theory of Fermi liquids
3
Linear response theory with Skyrme functionals
4
Neutrino mean free path
5
Conclusions
A.Pastore (York)
Linear response
8 / 29
Landau theory of Fermi liquids
Ground State
1
infinite nuclear system (symmetric or pure neutron matter)
2
weakly interacting quasi-particles long-lived around Fermi surface
Excited states
1
Landau parameters are universal (phenomenological or ab − initio)
2
(simple) residual interaction among quasi-particles as
X
Vph =
fl + fl0 τ1 τ2 + gl + gl0 τ1 τ2 σ1 σ2 Pl (k̂1 · k̂2 )
l
3
residual tensor interaction: S12 (k12 ), S12 (P12 ), A12 (k12 , P12 )
Extra Landau parameters
[A. Schwenck and B. Friman, Phys. Rev. Lett. 92, 8 (2004) ]
Coupling between tensor relative and center of mass motion: Hl , Kl , Ll
A.Pastore (York)
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Chiral Effective Field Theory (χ-EFT) (J.W. Holt)
The most general residual interaction reads
Vph
=
f (θ12 ) + g(θ12 ) σ1 · σ2 + h̃(θ12 )
+
k̃(θ12 )
k212
S12 (k̂12 )
kF2
P212
k12 · P12
l(θ12 )
S12 (P̂12 ) + ˜
A12 (k̂12 , P̂12 ).
kF2
kF2
Tensor terms
S12 (k̂12 ) = 3(k̂12 · σ1 )(k̂12 · σ2 ) − σ1 · σ2 .
S12 (P̂12 ) = 3(P̂12 · σ1 )(P̂12 · σ2 ) − σ1 · σ2 ,
A12 (k̂12 , P̂12 ) = (σ1 · P̂12 )(σ2 · k̂12 ) − (σ1 · k̂12 )(σ2 · P̂12 )
Question: how to interpret Kl , Ll ?
Hl , Kl , Ll have the same order of magnitude
They give important effects on the response function
Not considered in phenomenological interactions
A.Pastore (York)
Linear response
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Landau parameters in PNM
Landau parameters obtained with χ-EFT (Λ = 450 MeV).
1
(a)
Fl
0.5
0
-0.5
-1
1.2
0.8
(c)
~
Hl
0.3
(d)
~
Kl
0
(b)
-0.3
(e)
0
0.4
-0.2
0
-0.4
0.9
0.6
0.3
0
Gl
0.04
0.08
0.12
~
Ll
-0.4
0.16
0.2
0.24
0.04
-3
0.08
0.12
0.16
0.2
0.24
-3
ρ (fm )
ρ (fm )
Full line −→ 2NF+3NF; Dashed line −→ 2NF only
Remarks
Landau parameters are cut-off dependent (Λ)
3 body terms play a crucial role
A.Pastore (York)
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Static Landau with χ-EFT: effect of 2- and 3- body terms
Information on the static deformation of the Fermi surface
χHF (0)
= 1 + F0
(S=0)
χRP A (0)
T1
=
T2
=
/χFREE
RPA
(α)
χ
(S=1)
= 1 + G0 +
T1
T2
2
2
1
2
1
,
H̃0 − H̃1 + H̃2 + K̃0 + K̃1 + K̃2
3
5
3
5
1
7
2
3
7
2
3
2
6
1 + G2 −
H̃1 + H̃2 −
H̃3 +
K̃1 + K̃2 +
K̃3 + L̃1 −
L̃3 .
5
15
5
35
15
5
35
5
35
−2
3
2.5
χHF (0)
χRP A (0)
S=0
S=1
S=1 (Kl=Ll=0)
S=1 (Hl=Kl=Ll=0)
a)
2
b)
2NF (only)
2NF+3NF
x 0.25
1.5
1
0.5
0
0
0.04
0.08
0.12
0.16
0.2
0.24 0
0.04
-3
A.Pastore (York)
0.08
0.12
0.16
0.2
0.24
-3
ρ [fm ]
ρ [fm ]
Linear response
12 / 29
Static Landau q 6= 0, ω = 0
-3
PNM n=0.1 fm
Free
M.Buraczynski A. Gerzelis
SLy4
∆ρ
=48
1
SLy4 mod C
-1
-χ/n [MeV ]
0.12
5
MeVfm
0.08
0.04
0
0
1
2
3
4
5
6
q/qF
A simple formula for l = 0, 1
A simple formula to calculate the response in Landau limit
χHF (q, ω)
χS=0
(q, ω)
RP A
A.Pastore (York)
"
= 1 − f0 + f1
2
α2
1 /α0
1 − nd f1 (α0 α2 − α2
1 )/α0
Linear response
#
χHF (q, ω)
13 / 29
Extended Landau response function
PNM at kF = 1.68 fm−1 , q/kF = 0.1 for χ-EFT (Hl = Kl = Ll = 0)
S=0
S=1
HF
lmax=0
lmax=1
lmax=2
lmax=3
2
1
3
10 S
(1,M)
-1
-3
(q,ω) [MeV fm ]
3
0
2
4
6
8 10 12 14 2
ω [MeV]
4
6
8 10 12 14
ω [MeV]
Convergence
We have an infinite sets of Landau parameters, at which l shall we stop?
A.Pastore (York)
Linear response
14 / 29
Extended Landau response function
PNM at kF = 1.68 fm−1 , q/kF = 0.1 for χ-EFT
M=0
M=1
HF
lmax=0
lmax=1
lmax=2
lmax=3
2
1
3
10 S
(1,M)
-1
-3
(q,ω) [MeV fm ]
3
0
2
4
6
8 10 12 14 2
ω [MeV]
4
6
8 10 12 14
ω [MeV]
Convergence
Less clear convergence.... better results with Kl = 0
A.Pastore (York)
Linear response
15 / 29
Extended Landau response function
PNM at kF = 1.68 fm−1 , q/kF = 0.1 for χ-EFT
M=0
-1
2
1
3
10 S
(1,M)
M=1
Complete
L=0
L=K=0
L=K=H=0
-3
(q,ω) [MeV fm ]
3
0
2
4
6
8 10 12 14 2
ω [MeV]
4
6
8 10 12 14
ω [MeV]
Remarks
The Hl , Kl , Ll have a similar importance
Landau parameters do depend on cut off, interaction and model!!
Outline
1
Introduction
2
Landau theory of Fermi liquids
3
Linear response theory with Skyrme functionals
4
Neutrino mean free path
5
Conclusions
A.Pastore (York)
Linear response
17 / 29
From Landau theory to RPA (complete)
The particle-hole pair can be outside the Fermi sphere
Similarities
Bethe-Salpeter equations
Same formal method to obtain the solution (Green’s functions)
Differences
The residual interactions has a dependence on q
New coordinates k1 , k1 , q
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Skyrme functionals
The Skyrme functional can be written as
E = Ekin + ESkyrme + Epairing + ECoulomb + Ecorr.
where at second order (N1LO) reads
Skyrme functional
ESkyrme
=
X Z
n
d3 r Ctρ [ρ0 ] ρ2t + Ct∆ρ ρt ∆ρt + Ctτ ρt τt + Ctj j2t + Cts [ρ0 ] s2t
t=0,1
+Ct∇s (∇ · st )2 + Ct∆s st · ∆st + CtT st · Tt + CtF st · Ft + Ct∇J ρt ∇ · Jt
)
z
(0)
(1)
(2) X
(0)
(1)
(2) (2)
+Ct∇j st · (∇ × jt ) + CtJ (Jt )2 + CtJ (Jt )2 + CtJ
Jtµν Jtµν
µν=x
[E . Perlinska et al. Phys. Rev C 69, 014316 (2004)) ]
The terms in red do not contribute in the Landau limit
A.Pastore (York)
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Landau vs RPA (complete)
PNM at ρ = ρ0 using Skyrme-T44 (tensor).
I=0
1
3
k=0.1
k=0.5
2
(α)
1
3
3
-1
-3
10 S (k,ω) (MeV fm )
k=0.1 (q=0)
k=0.5 (q=0)
2
2
1
0
50
100
150
50
100
S=1 M=1 S=1 M=0 S=0 M=0
I=1
3
150
ω (MeV)
Results
The Landau approximation is very good for very low transferred momenta.
Possible extension: keeping q 2 dependence only on F0 /G0 (direct part)
A.Pastore (York)
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Temperature effect
Response function PNM at ρ = 0.16 fm−3 q = 0.1 fm−1 .
BSk21
BSk24
-3
10 S (q,ω) [MeV fm ]
4
-1
2
0
LNS1
SLy5
4
0
0
2
4
6
8 10 120
ω [MeV]
2
4
6
8 10 12
ω [MeV]
(0,0)
(1,0) BSk24
(1,1) (b)
BSk20
(a)
2
0
4
(α)
4
3
3 (α)
-1
-3
10 S (q,ω) [MeV fm ]
4
LNS1
(c)
SLy5
(d)
2
0
-10
-5
0
5
ω [MeV]
10
-5
0
5
10
ω [MeV]
Results
Thanks to the temperature we have negative energy excitations!
A.Pastore (York)
Linear response
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Outline
1
Introduction
2
Landau theory of Fermi liquids
3
Linear response theory with Skyrme functionals
4
Neutrino mean free path
5
Conclusions
A.Pastore (York)
Linear response
22 / 29
Astrophysical applications: NMFP
n(p) + νl
n + νl
→
→
n(p)0 + νl0
p+e
Towards an universal functional
The aim of the BSk functionals is to describe relevant process in both nuclei and stars.
A.Pastore (York)
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23 / 29
Landau parameters
For low-energy neutrinos Landau approximation is very good
2
1
n
0
F
BSk17
BSk18
BSk19
BSk20
BSk21
BSk24
LNS1
SLy5
CBF
0
-1
2
1
n
1
F
0
-1
0
0.1
0.2
0.3
0
G
n
0
G
n
1
0.1
-3
0.2
0.3
0.4
-3
ρ [fm ]
ρ[fm ]
Result
How to compare Landau parameters?
A.Pastore (York)
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Neutrino mean free path: n + νl → n0 + νl0
4
BSk17
BSk18
1×10
-3
2
ρ=0.08 fm
λRPA [m]
1×10
CBF
BSk19
BSk20
4
1×10
-3
2
ρ=0.16 fm
1×10
4
BSk21
BSk24
1×10
-3
2
ρ=0.24 fm
1×10
0
10
20
30
40
Eν [MeV]
d2 σ(Eν )
dΩk0 dω
=
G2F Eν2
16π 2
1
1
2
(1 + cos θ) S (0) (q, ω, T ) + gA
[3 − cos θ] S (1) (q, ω, T )
ρ
ρ
No charge exchange
We are now extending the formalism to include asymmetry and charge exchange
A.Pastore (York)
Linear response
25 / 29
Neutrino mean free path: n + νl → n0 + νl0
Calculations at T = 2 MeV
4
BSk17
BSk18
BSk19
BSk20
BSk21
BSk24
LNS1
SLy5
CBF
3
λRPA [m]
1×10
2
1×10
3
2
1
(b)
ρ=0.16 fm
4
(c)
3
2
1×10
1
0
1×10
(a)
-3
ρ=0.08 fm
1
λRPA/λFG
1×10
10
20
30
40
Eν [MeV]
50
60
0
ρ=0.24 fm
10
20
-3
30
40
Eν [MeV]
10
20
-3
30
40
50
BSk19 Eν[MeV]
BSk20
BSk21
BSk24
LNS1
SLy5
50
Results
We can use the comparison with ab − initio methods to discriminate between functionals
A.Pastore (York)
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Temperature effect
On the left Eν = 10 MeV and on the right Eν = 60 MeV
3
(a)
-3
ρ=0.08 fm
2
2
4
5 10 15 20 25 30
(c)
3
2
1
ρ=0.24 fm
0
0
-3
BSk19
BSk20
BSk21
BSk24
LNS1
SLy5
(a)
-3
ρ=0.08 fm
1
-3
ρ=0.16 fm
T [MeV]
λRPA/λFG
λRPA/λFG
1
3
(b)
4
-3
ρ=0.16 fm
5 10 15 20 25 30
(c)
3
2
1
0
0
5 10 15 20 25 30
T [MeV]
(b)
ρ=0.24 fm
-3
BSk19
BSk20
BSk21
BSk24
LNS1
SLy5
T [MeV]
5 10 15 20 25 30
T [MeV]
Results
A comparison with other methods is mandatory.
Very different results according to the adopted functional
A.Pastore (York)
Linear response
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Outline
1
Introduction
2
Landau theory of Fermi liquids
3
Linear response theory with Skyrme functionals
4
Neutrino mean free path
5
Conclusions
A.Pastore (York)
Linear response
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Conclusions and perspectives
The LR theory can be used to fit functionals
1
2
We use it to detect instabilities (S=1 channel still to be fully
investigated)
the LR method is now applied into the Saclay-Lyon protocol
Extensions of the LR formalism to treat Landau
1
2
3
Limit of stability of Landau theory
Possible extension of Landau q 2 dependence
Application to Neutron stars (NMFP, Specific heat...)
Extension of the LR formalism to asymmetric nuclear matter (soon
also charge exchange)
Extension to finite range interactions
THANK YOU!!
A.Pastore (York)
Linear response
29 / 29