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Chiral dynamics of nuclear matter
N. Kaiser
Physics Department T39, Technical University Munich
3. Japanese-German workshop on Nuclear Structure and Astrophysics,Frauenwörth im Chiemsee, 29.9.2007
Introduction: Scales in nuclear matter
Chiral expansion of nuclear matter equation of state
Inclusion of chiral πN∆-dynamics
Single-particle potential, finite temperatures, isospin properties
Nuclear spin-orbit coupling strength
Nuclear mean-fields for Λ and Σ hyperons
In-medium chiral condensate beyond linear ρ-approximation
N. Kaiser Chiral dynamics of nuclear matter
Introduction
Problem of nuclear binding central in nuclear physics
1. step: Infinite nuclear matter: N/Z = 1, e→ 0
Empirical saturation point:
ρ0 ≃ 0.16 fm−3
E0 ≃ −16 MeV
K = (260± 30) MeV
dNN ≃ 1.8 fm = 1.3m−1π
ρ =2k3
f
3π2, kf 0 ≃ 263 MeV ≃ 2mπ
K = k2f 0
∂2E(kf )
∂k2f
˛
˛
˛
kf 0
(compressibility)
σ-ω mean field approach of Walecka et al.
Realistic NN-potentials + sophisticated many-body techniques
Role of explicit pion-dynamics: 2π-exchange
Chiral effective field theory
Simple but realistic parametrization of nuclear matter equation of state:
E(kf ) =3k2
f
10MN− α
k3f
M2N
+ βk4
f
M3N
N. Kaiser Chiral dynamics of nuclear matter
Introduction
Fit α = 5.3, β = 12.2 tosaturation point (ρ0, E0):→ K = 236 MeV
- - - - - - sophisticatedmany-body calculation ofFriedman-Pandharipande
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0ρ [fm
-3]
-20
0
20
40
60
80
100
120
140
160
180
200
E [M
eV]
Loop expansion inChiral Perturbation Theory ←→
systematic expansion of E(kf ) in powersof kf modulo functions fn(kf /mπ),
Basic ingredient: in-medium nucleon propagator
(/p + MN )
ip2 −M2
N + iǫ− 2πδ(p2 −M2
N)θ(p0)θ(kf − |~p|)ff
Organize many-body calculation in number of ”medium insertions”→ 1 medium insert.: Renormalization of nucleon mass MN in vacuum→ two- and three-body interaction terms including Pauli blocking
N. Kaiser Chiral dynamics of nuclear matter
Result up to order k4f
Contributions to energy per particle E(kf ):
kinetic energy: O(k2f )
1π-exchange: O(k3f )
iterated 1π-exchange: O(k4f )
one adjusted contact term ∼ k3f
E0 = −15.3 MeV, ρ0 = 0.17 fm−3 , K = 252 MeV
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5ρ [fm
−3]
−20
−15
−10
−5
0
5
10
15
20
25
30
E [M
eV]
Saturation mechanism: take chiral limit mπ = 0
β =3
70
“ gπN
4π
”4(4π2 + 237− 24 ln 2)−
356≃ 13.5, βfit = 12.2
saturation mainly from Pauli blocking on iterated 1π-exchange
unresolved short-distance dynamics encoded in one single contact term
N. Kaiser Chiral dynamics of nuclear matter
Inclusion of chiral πN∆-dynamics
however: p-dependent single-particle potential U(p, kf ) and isospinproperties A(kf ), En(kn) not well reproduced in this approximation
include next important long-range dynamics: 2π-exchange withexcitation of virtual ∆(1232)-isobars (→ isoscalar central NN-attraction)
mass splitting ∆ = 293 MeV comparable to Fermi momentum kf 0 ≃ 2mπ
additional two-body terms:
additional three-body terms:
N. Kaiser Chiral dynamics of nuclear matter
Inclusion of chiral πN∆-dynamics
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
ρ [fm-3]
-20
-10
0
10
20
30
40
50
E [
MeV
]
Nuclear matter saturation curve E(kf ):
one single term linear in ρ adjusted
E0 = −16 MeV
ρ0 = 0.157 fm−3
K = 300 MeV (somewhat high)
0 50 100 150 200 250 300 350 400 450 500p [MeV]
-75
-50
-25
0
25
50
MeV
U(p,k f0
)
U(p
,k f0 )
+ T ki
n(p
)
p =
kf,
0 = 2
61.6
MeV
Real single-particle potential U(p, kf 0):
p-dependence of U(p, kf ) improved
effective nucleon mass at Fermisurface: M∗(kf 0) = 0.88 MN
Hugenholtz-van-Hove theorem:Tkin(kf ) + U(kf , kf ) = E(kf ) + kf
3∂E∂kf
N. Kaiser Chiral dynamics of nuclear matter
Inclusion of chiral πN∆-dynamics
0 50 100 150 200 250 300 350 400 450p [MeV]
-40
-30
-20
-10
0
10
20
30
W(p
,kf,
0 )
[MeV
]
p =
kf,
0 = 2
61.6
MeV
Imaginary single-particle potential W (p, kf 0):
spreading width of hole (p < kf ) andparticle states (p > kf )
completely parameterfree:only iterated 1π-exchange contributes
Luttingers theorem:W (p, kf ) ∼ ±(p − kf )
2 near p = kf
0 0.05 0.1 0.15 0.2
ρ [fm-3]
-1
0
1
2
3
4
P [M
eV/f
m3 ]
T=0MeV
T=5MeV
T=10MeV
T=15MeV
T=20MeVT=25MeV
Nuclear matter at finite temperatures:
free energy per particle: F (ρ, T )
θ(kf − p)→h
1 + exp p2/2MN−µT
i−1
pressure: P(ρ, T ) = ρ2 ∂F (ρ,T )∂ρ
liquid-gas phase transition with criticalpoint: Tc ≃ 15 MeV, ρc ≃ 0.053 fm−3
N. Kaiser Chiral dynamics of nuclear matter
Inclusion of chiral πN∆-dynamics
0 0.05 0.1 0.15 0.2 0.25
ρ [fm-3]
0
10
20
30
40
50
A [
MeV
]
Asymmetry energy A(kf ):
isospin-asymmetric nuclear matter:E(kp, kn) = E(kf ) + δ2A(kf ) + ...,kp,n = kf (1∓ δ)1/3
at saturation: A(kf 0) = 34 MeV
downward bending for ρ > 0.2 fm−3
eliminated by chiral πN∆-dynamics
0 0.05 0.1 0.15 0.2 0.25
ρ [fm-3]
0
10
20
30
40
50
60
UI(p
,kf)
[MeV
]
p=k f
p=0
Isovector single-particle potential UI(p, kf 0):
different mean-fields for p and n:U(p, kf )− UI(p, kf )τ3δ + . . .
UI(kf 0, kf 0) = 40.4 MeV, in goodagreement with optical potential
generalized HvH theorem:
UI(kf , kf ) = 2A(kf )int −kf3
∂U(p,kf )∂p
˛
˛
p=kf
N. Kaiser Chiral dynamics of nuclear matter
Spin-stability of nuclear matter
Energy per particle of spin-polarized nuclear matter:
Epol(k↑, k↓) = E(kf ) + η2 S(kf ) + ..., k↑,↓ = kf (1± η)1/3
Spin-asymmetry energy S(kf ) > 0 must be positive
1π- and iterated 1π-exch. only:
0 0.05 0.1 0.15 0.2 0.25ρ [fm
−3]
−35
−30
−25
−20
−15
−10
−5
0
5
S(k
f) [M
eV]
Chiral πN∆-dynamics included:
0 0.05 0.1 0.15 0.2 0.25ρ [fm
−3]
0
20
40
60
80
100
120
140
S(k
f) [M
eV]
S5=0
S5=−7.45
S5=−30
adjustable terms linear in ρ satisfy: 3S(kf )lin + 3A(kf )lin + 2E(kf )lin = 0
2π-exch. with ∆-excitation guarantees spin-stability: 2~l1 ·~l2 − i ~σ · (~l1 ×~l2)
spin-isospin stability is a generic feature: (p↑, n↓)↔ (p↓, n↑)
N. Kaiser Chiral dynamics of nuclear matter
Nuclear spin-orbit coupling strength
Dynamical origin of strong spin-orbit coupling? Success of RMFTEnergy density functional: density-matrix expansion of Negele-Vautherin
Eso[ρ,~J ] = Fso(kf ) ~∇ρ · ~J , ~J(~r ) =X
Ψ†α(~r )i ~σ × ~∇Ψα(~r )
Skyrme phenomenology Fso(kf ) = 3W0/4 ≃ 90 MeVfm5: Compare withshort-range spin-orbit contact term extracted from realistic NN-potentials
SIII SkM SkP Sly4-7 MSk1-6 SkI1-590.0 97.5 75.0 93.2 87.6 92.7
Bonn-B CD-Bonn Nijm-I Nijm-II AV-18 Vlow−k80.3 89.6 82.4 87.7 88.9 89.4
0 0.05 0.1 0.15 0.2ρ [fm
−3]
0
20
40
60
80
100
[MeV
fm5 ]
Fso(kf)
Gso(kf)
Spin-orbit from long-range 2π-exchange issmall, more pronounced effect in isovectorchannel Gso(kf )~∇ρv · ~Jv : Gso/Fso << 1/3
O. Pohl and C. Fuchs, PRC74(06)034325:In relativistic Brueckner calculations largescalar and vector mean fields (of oppositesign) are closely connected to short-rangespin-orbit part of the NN-interaction
Nuclear spin-orbit = short-range dynamics,not medium-modified
N. Kaiser Chiral dynamics of nuclear matter
Λ-hyperons in nuclear medium
Spectroscopy of Λ-hypernuclei: Λ-nucleus potential−28 MeV about halfas deep N-nucleus potential, Λ spin-orbit interaction is very small
Leading long-range ΛN interaction from K -exchange and 2π-exchangewith intermediate Σ-hyperons
�� K �
�� ��Λ-nuclear mean field (~pΛ=0)
0 0.05 0.1 0.15 0.2 0.25ρ [fm
−3]
−40
−30
−20
−10
0
UΛ(k
f) [M
eV]
Cut-off scale Λ = 0.7 GeV represents allshort-distance dynamics UΛ(kf )
(sd) ∼ ρ
UΛ(kf 0) = (4.2K−39.82πΣ+7.5Pauli) MeV= −28.1 MeV
Exceptionally small mass splittingMΣ −MΛ = 77 MeV figures prominently
N. Kaiser Chiral dynamics of nuclear matter
Λ-hyperons in nuclear medium
Λ-nuclear spin-orbit coupling:
Spin-dep. part of Λ-selfenergy in weakly inhomogeneous nuclear matter
Σspin =i2
~σ · (~q × ~p ) UΛls(kf ) , Hls = UΛls(kf 0)12r
df (r)dr
~σ · ~L
2π-exch. with intermediate Σ-hyperons generates ”wrong-sign” spin-orbitterm: ~σ · (~l + ~q
2 )~σ · (~l − ~q2 ) = i ~σ × ~q ·~l + ..., ~p ·~l from energy denominator
0 0.05 0.1 0.15 0.2 0.25ρ [fm
−3]
−24
−20
−16
−12
−8
−4
0
4
UΛ
ls(k
f) [M
eVfm
2 ]
Cls=2/3
Cls=1/2Second order 1π-exchange tensor force
Long-range plus short-range pieces:
UΛls(kf )tot = UΛls(kf )
2πΣ + ClsM2
NM2
Λ
UNls(kf )
Almost complete cancellation: UΛls(kf 0)= (24.8Cls − 16.72πΣ + 1.6Pauli) MeVfm2
For nucleons: “wrong-sign” spin-orbit interaction fromiterated 1π-exchange exists also, but canceled by three-body contribution from 2π-exchange with ∆-excitation N
N���
N. Kaiser Chiral dynamics of nuclear matter
Σ-hyperons in nuclear medium
Earlier linear ρ-fits to Σ− atoms gave attraction UΣ(kf 0) ≃ −27 MeV
(π−, K +) reaction on medium and heavy nuclei: substantial repulsion
Leading long-range contributions to the complex Σ-nuclear mean field:K -exchange + iterated 1π-exch. with intermediate Σ and Λ-hyperons
�� K �
��;� ��Complex Σ-nuclear mean field:
0 0.05 0.1 0.15 0.2 0.25 ρ [fm
−3]
−30
−20
−10
0
10
20
30
40
50
60
70
80
90
100
[MeV
]
UΣ(kf)
WΣ(kf)
WΣ(kf 0)=(7.5−28.9)MeV=−21.4 MeV,consistent with −16 MeV from Σ− atomsand −20MeV in SU(3) quark models
UΣ(kf 0)=((40.9+16.1)2πΣ+(8.2−6.6)2πΛ
+0.4K ) MeV = 59 MeV
Moderate repulsion from “genuine” long-range 2π-exchange dynamics
U(2π)Σls + i W (2π)
Σls = (−20− 12 i) MeVfm2
N. Kaiser Chiral dynamics of nuclear matter
In-medium chiral condensate
〈0|qq|0〉 is order parameter of spontaneous chiral symmetry breaking
Feynman-Hellmann theorem: mq-derivative of energy density, mq ∼ m2π
〈qq〉(ρ)
〈0|qq|0〉= 1−
ρ
f 2π
σN
m2π
+ D(kf )
ff
, D(kf ) =1
2mπ
∂E(kf , mπ)
∂mπ
nucleon sigma-term σN = mq(∂MN/∂mq) ≃ 45 MeV determines lineardecrease of quark condensate |〈qq〉| with density ρ = 2k3
f /3π2
interaction contributions from π-exchange: corrections to linear ρ-approx.
D(kf )˛
˛
mπ=0=
g2A kf
(4πfπ)2
„
k2f
M2N
−94
«
+g4
A MN k2f
5(4πfπ)4
“
8π2 + 36 ln 2− 3”
+(g2
A − 1)k3f
(4πfπ)4(7g2
A + 1) lnkf
Λ+
g4A k4
f
∆(4πfπ)4
„
12π2
35− 25
«
+ ...
chiral limit mπ → 0 exists: singular ln(mπ/Λ)-terms from renormalizedπN-coupling and irreducible 2π-exchange cancel each other exactly
N. Kaiser Chiral dynamics of nuclear matter
In-medium chiral condensate
Dropping quark condensate in chiral limit mπ = 0:
versus density ρ
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
density [fm-3
]
0
0.2
0.4
0.6
0.8
1
versus temperature T
0 25 50 75 100 125 150 175 200T [MeV]
0
0.2
0.4
0.6
0.8
1
compare with ”melting” condensate at finite temperature:
〈qq〉(T )
〈0|qq|0〉= 1−
T 2
8f 2π
−T 4
384f 4π
−T 6
288f 6π
lnΛq
T
attractive interactions accelerate tendency towards chiral restoration
Recent lattice-QCD result: Y. Aoki et al., Phys.Lett.B643(2006)46,Chiral phase transition (crossover) in the region Tc = (151± 28) MeV
at finite quark/pion mass mπ = 135 MeV effects are markedly different
N. Kaiser Chiral dynamics of nuclear matter
In-medium chiral condensate
Quark condensate at physical pion mass mπ = 135 MeV:
0 0.1 0.2 0.3 0.4 0.5
ρ [fm-3
]
0
0.2
0.4
0.6
0.8
1
mπ=135 MeV
mπ=0
linear ρ-approx.
Nonlinear effects of mπ 6= 0 sizeable
In agreement with Brueckner calcul.,estimates of ∂Mσ,ω/∂mq etc.
Error band due to nucleon σ-termσN = (45± 8) MeV (?) and linearmq-dependence of unresolvedshort-range NN-interaction
Isoscalar central NN-amplitude from 2π-exchange with single ∆-excitation
VC(q) =3g4
A
32πf 4π∆
»
(2m2π + q2)2
2qarctan
q2mπ
+ mπ q2 + 4m3π
–
An accelerated tendency towards chiral symmetry restoration wouldundermine chiral approach to nuclear matter: Goldstone bosons gone
Explicit treatment of 2π-exch. with ∆-excit. “saves” conventional nuclearphysics up to densities (2...3)ρ0, active degrees of freedom: N and π
N. Kaiser Chiral dynamics of nuclear matter
Summary
Chiral expansion of nuclear matter equation of state, explicit pions:
E(kf ) =
5X
n=2
knf fn(kf /mπ , ∆/mπ)
Saturation mainly from Pauli blocking on iterated 1π-exchange
Substantial improvement by inclusion of chiral πN∆-dynamics:U(p, kf ), M∗(kf 0) = 0.88MN , Tc = 15 MeV, A(kf ), En(kn),...
Chiral πN∆-dynamics guarantees spin-stability
Nuclear spin-orbit coupling is predominantly of short-range origin:Skyrme phenom. ↔ realistic NN-potentials↔ scalar/vector mean fields
Λ and Σ hyperons in nuclear medium, weak Λ spin-orbit interaction fromcancellation between long-range “wrong-sign” term (from 2π-exchangewith intermediate Σ) and short-range term, Σ nuclear potential repulsive
In-medium chiral condensate: No accelerated tendency towards chiralsymmetry restoration at mπ = 135 MeV, πN∆-dynamics is important
Open questions: “Convergence” of kf -expansion,Interpretation of adjusted short-distance parameters, Connection toVlow−k approach (A. Schwenk et al.): “Nuclear matter is perturbative”
N. Kaiser Chiral dynamics of nuclear matter