Chiral dynamics of nuclear matter - TUM · Nuclear matter at ﬁnite temperatures: free energy per...

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


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


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



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:



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



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



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


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↑)


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


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


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��


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


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



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



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


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”


Chiral dynamics of nuclear matter - TUM · Nuclear matter at ﬁnite temperatures: free energy per...

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