Introduction DFT Interaction The model Results Conclusions Appendix

Relativistic nuclear energy density functionalconstrained by low-energy QCD

[Nucl. Phys. A735 (2004) 449 and nucl-th/0509040]

Paolo Finelli1,2, N. Kaiser2, D. Vretenar3, W. Weise2

1Physics Department of Bologna, Italy2T39 group, TUM

3Physics Department of Zagreb, Croatia

DPG, Munich, 20 march 2006

Introduction DFT Interaction The model Results Conclusions Appendix

Signatures of QCD at low energy

Goldstone modes Quark condensate

Pions as low-energy degrees of freedom Long-range physics should beexplicitly treated

At T ' 0, 〈qq〉ρ is large (' 1.8 fm−3) andmodifies nucleon properties Large scalarand vector self-energies arise

[Figures borrowed from Vogl and Weise, Prog. Part. Nucl. Phys. 27 (1991) 195 ] More

Introduction DFT Interaction The model Results Conclusions Appendix

Working hypothesis

The nuclear many-body system is governedby pionic fluctuations on top of large fieldsgenerated by the in-medium changes of thequark condensates

Steps:

1. calculate an effective N-N interaction in homogeneous nuclear matter

2. introduce an effective lagrangian for inhomogeneous systems

3. evaluation of ground state properties

4. evaluation of excited state properties

improve the calculation at step 1 and repeat to solve the nuclear many bodyproblem.

Introduction DFT Interaction The model Results Conclusions Appendix

Working hypothesis

In detail

• Relativistic nuclear Kohn-Sham functional

E0[jµ] = Efree [j

µ] + E (0)[jµ] + Eexc [jµ] + Ecoul [j

µ]

• E (0): Large scalar and vector fields, arising from thein-medium changes of the chiral condensate 〈qq〉 and thequark density 〈q†q〉, act as background fields.

• Eexc E(π)exc : The exchange correlation term is described in

terms of pion-nucleon interaction (within the framework ofin-medium ChPT).

Introduction DFT Interaction The model Results Conclusions Appendix

Nuclear interactionBackground fields More

[Cohen, Furnstahl, Griegel, and Jin, Prog. Part. Nucl. Phys. 35 (1995) 221; Drukarev and Levin, 27 (1991) 77]

Σ(0)S ' −

σNMN

m2πf 2

π

ρ+ . . .

Σ(0)V '

4(mu + md )MN

m2πf 2

π

ρ+ . . .

Of approximately equal magnitude(300 MeV) but of opposite sign origin of the spin-orbit interaction

Exchange correlation term[Fritsch, Kaiser and Weise, Nucl. Phys. A750 (2005) 259]

The nuclear matter energy density iscalculated at three loop order, includingmedium and ∆-insertions.

ππ

π

NN

+ + ...

N N

N N

+

(a) (b)

(c)

Introduction DFT Interaction The model Results Conclusions Appendix

Exchange correlation term: in-medium ChPT

Key ingredient: In-medium propagator

GF = (/p + MN)h

ip2−M2

N+iε

−2πδ(p2 −M2N)θ(kf − |~p2|)θ(p0)

ifull vacuum medium

• Expansion of E(kf ) = E(kf )/A in powers ofq

Qwith

q = kf ,mπ ,M∆ −MN small momentum ≤ 300 MeVQ = MN , 4π, fπ large momentum ' 1 GeV

• E(kf ) can be written as:

E(kf ) =3k2

f

10MN− α

„kf

mπ

«k3f

M2N

+ β

„kf

mπ

«k4f

M3N

+ . . .

Introduction DFT Interaction The model Results Conclusions Appendix

Exchange correlation term: in-medium ChPT

First step: no ∆ intermediate states[Fritsch, Kaiser and Weise, Nucl. Phys. A697 (2002) 255; A700 (2002) 343; A724 (2003) 47]

1π − exchange O(q3) :

Iterated 1π − exchange O(q4) :

Irreducible 2π − exchange O(q5) :

Regularization: a single contact term (or an equivalent cut-off Λ ' 2πfπ) at orderk3f Encodes unresolved short-distance information.

Introduction DFT Interaction The model Results Conclusions Appendix

Exchange correlation term: in-medium ChPT

0 0.1 0.2 0.3 0.4 0.5ρ [fm-3]

-20

0

20

40 [M

eV]

0 0.05 0.1 0.15 0.2ρ [fm-3]

0

20

40

60

[MeV

]

0 0.05 0.1 0.15 0.2ρ [fm-3]

0

10

20

30

[MeV

]

0 100 200 p [MeV]

-60

-40

-20

0

[MeV

]

E/A

E/AN

U(p,kf0)

Asym

ρsat = 0.178 fm−3, E = −15.3 MeV, K = 255 MeV,Asym = 33.8 MeV, U(0, kf 0) = −53.2 MeV

Introduction DFT Interaction The model Results Conclusions Appendix

Exchange correlation term: in-medium ChPT

Inclusion of ∆ intermediate states[Fritsch, Kaiser and Weise, Nucl. Phys. A750 (2005) 259]

Irreducible 2π-exchange:

Divergent momentum space loop integrals are regularized by

∆E(kf )(ct) = b3

k3f

Λ2+ b5

k5f

Λ4+ b6

k6f

Λ5

∆S2(kf )(ct) = a3

k3f

Λ2+ a5

k5f

Λ4+ a6

k6f

Λ5

(few) subtraction constants with Λ = 2πfπ ' 0.58 MeV

Introduction DFT Interaction The model Results Conclusions Appendix

Exchange correlation term: in-medium ChPT

0 0.1 0.2 0.3 0.4 0.5ρ [fm-3]

-20

0

20

40 [M

eV]

0 0.05 0.1 0.15 0.2ρ [fm-3]

0

20

40

60

[MeV

]

0 0.05 0.1 0.15 0.2ρ [fm-3]

0

10

20

30

[MeV

]

0 100 200 300 400 p [MeV]

-80

-40

0

40

[MeV

]

E/A

E/AN

U(p,kf0)

Asym

[With ∆ - Without ∆]

ρsat = 0.157 fm−3, E = −16.0 MeV, K = 304 MeV,Asym = 34.0 MeV, U(0, kf 0) = −78.2 MeV

Introduction DFT Interaction The model Results Conclusions Appendix

From infinite systems to finite nuclei

• In-medium ChPT calculations not feasible for finite systems

• introduce an effective Lagrangian

L = . . .+1

2GV (ρ)(ψγµψ)2 + . . .

with density dependent couplings Gi (ρ)

Gi (ρ) = G(0)i (ρ) + G

(π)i (ρ)

(G

(0)i (ρ) : In −medium QCD sum rules

G(π)i (ρ) : In −medium ChPT

• Additional dynamics Rearrangement terms[Literature for meson-exchange models: Fuchs, Lenske and Wolter, Phys. Rev. C 52 (1995) 3043; Typel

and Wolter, Nucl. Phys. A 656 (1999) 331 ]

Introduction DFT Interaction The model Results Conclusions Appendix

Energy functional

Functional : E0[ρ] = Efree [ρ] + E (0)[ρ] + E(π)exc [ρ] + Ecoul [ρ]

Efree [ρ] =

Zd3x 〈φ0|ψ[−iγ ·∇ + MN ]ψ|φ0〉 Free

E (0)[ρ] =1

2

Zd3x [〈φ0|G (0)

S (ψψ)2|φ0〉+ 〈φ0|G (0)V (ψγµψ)2|φ0〉] BG

E(π)exc [ρ] =

1

2

Zd3x [〈φ0|G (π)

S (ρ)(ψψ)2|φ0〉+ 〈φ0|G (π)V (ρ)(ψγµψ)2|φ0〉]

+

Zd3x [〈φ0|G (π)

TS (ρ)(ψ~τψ)2|φ0〉+ 〈φ0|G (π)TV (ρ)(ψγµ~τψ)2|φ0〉]

−Z

d3x [〈φ0|D(π)S (∇ψψ)2|φ0〉]

ffPions

Ecoul [ρ] =1

2

Zd3x 〈φ0|Aµe

1 + τ3

2(ψγµψ)|φ0〉 Coulomb

Equation of motion

hDψk = [−iβα ·∇ + MN + γ0Σ0V + γ0τ3Σ

0TV + γ0Σ

0R + ΣS + τ3ΣTS ]ψk = εkψk

Jump to Self-energies

Introduction DFT Interaction The model Results Conclusions Appendix

Density-dependent couplings More details

Coupling structure:

QCD Sum Rules| z high order| z G(ρ) = g3 + g4ρ

1/3 + g5ρ2/3 + g6ρz |

ChPTz | ChPT

z | ChPT

z | ChPT

polynomials in fractional powers of the baryon density

Introduction DFT Interaction The model Results Conclusions Appendix

Ground state properties: lead isotopes

Binding energies and isotopic shifts Neutron skin thickness

170 180 190 200 210 220-0.4

-0.2

0.0

0.2

0.4

δ E

(%

)

FKVW [2005]

190 195 200 205 210 215 220A

-0.2

0.0

0.2

0.4

0.6

0.8

∆rch

2 - ∆

r LD2 (

fm2 )

Exp. values

190 200 210 220A

0

0.05

0.1

0.15

0.2

0.25

0.3

r n -

r p [f

m]

FKVWExp. data

δE = [Bth − Bexp ]/Bexp

∆r2ch−∆r2

LD = [r2ch(A)− r2

ch(208Pb)]− [r2LD (A)− r2

LD (208Pb)]

Successful description at the same level of well knownphenomenological approaches (Skyrme, Gogny, RMF,. . .)

Introduction DFT Interaction The model Results Conclusions Appendix

Ground state properties: form factors

10-4

10-2

100 FKVW

Exp. data

10-4

10-2

100

10-4

10-2

100

10-4

10-2

100

0 0.5 1 1.5 2 2.5

10-4

10-2

100

0 0.5 1 1.5 2 2.5

10-4

10-2

100

48Ca

90Zr

92Mo

94Zr

144Sm

208Pb

q (fm-1

)

|F(q

)|

Charge density profiles are reproduced with great accuracy

Introduction DFT Interaction The model Results Conclusions Appendix

Ground state properties: β2 deformations

130 140 150 160

0

0.2

0.4

140 150 160 170 150 160 170 180 190

130 140 150 160

0

0.2

0.4

β 2

150 160 170

Exp. dataFKVW [2005]

160 170 180 190

140 150 160 170

0

0.2

0.4

150 160 170 180A

160 170 180 190 200

Nd

Sm

Gd

Dy

Er

Yb

Hf

Os

W

Global behaviour correctly reproduced

Introduction DFT Interaction The model Results Conclusions Appendix

208Pb - Electric excitations

0

10

20

30

R(E

) [fm

2 MeV

-1]

0 10 20Exc [MeV]

FKVWFree

0

2

4

6

8

10

R(E

) [fm

4 MeV

-1] x

100

0

0 10 20 30 40Exc [MeV]

IVGDR [Eexp = 13.30 MeV] IVGQR [Eexp = 22.0 MeV]

Isovector interactions correctly described by 1π and 2π-exchange

(with ∆-isobars) and a single contact term

Introduction DFT Interaction The model Results Conclusions Appendix

208Pb - Spin-isospin excitations Landau-ChPT

0 10 20 30E

xc [MeV]

0

5

10

15

20

25

30

R(E

) [M

eV-1

]

0 10 20 30E

xc [MeV]

0

10

20

30

40

50

60

70

R(E

) [M

eV-1

]

IAS [Eexp

= 18.83 MeV] GT [Eexp

= 19.2 MeV]

Lis = − fπmπ

ψγ5γµ∂µ~π~τψ − 1

2Gστ (ρ)(ψγ5γµ~τψ)2

one more counterterm bστ5 at order k5f .

Introduction DFT Interaction The model Results Conclusions Appendix

Conclusions

Now:

• Linking to the low-energy phenomenology of QCD

• Successful description of ground-state properties

• Investigation of excited-state properties in progress

Future:

• Hypernuclei

• Collective motions M1 transitions

• 2-body correlations

Thanks to:Stefan Fritsch, Nils Paar (GSI) and Tamara Niksic (TUM)

Introduction DFT Interaction The model Results Conclusions Appendix

Safety slides

• Introduction

• Background fields: in-medium QCD sum rules

• Spin-orbit from in-medium QCD sum rules

• Details about density-dependent couplings

• Self-energies

• Influence of the three-body term

• Gradient terms

• Landau-Migdal terms

Introduction DFT Interaction The model Results Conclusions Appendix

Building an energy functional Back

• Starting point: E(ψ†ψ,∇(ψ†ψ), . . .) constrained by symmetries

• Find a controlled power expansion [Furnstahl and Serot, Nucl. Phys. A 671 (2000) 447]

• Energy density can be expanded in powers of density:

1. E1 ∼ ρ2/3 (Fermi gas kinetic energy)

2. E2 ∼ ρ (2-Body term)

3. E3 ∼ ρ2 (3-Body term??)

4. . . .

The scheme is modified if we consider that Pauli blocking effects modify the 2-Bodyeffective interaction: E3 ∼ ρ4/3

An expansion in kf could be a better choice. . .

Introduction DFT Interaction The model Results Conclusions Appendix

Background fields: in-medium QCD sum rules

In-medium nucleon self-energies

Σ(0)S = −

8π2

Λ2B

[〈qq〉ρ − 〈qq〉0] = −8π2

Λ2B

σN

mu + mdρs

Σ(0)V =

64π2

3Λ2B

〈q†q〉ρ =32π2

Λ2B

ρ

Strong scalar and vectormean fields generatedby in-medium changesof QCD condensates

• Scalar self-energy

Σ(0)S = MN(ρ)−M∗

N(ρ) = −σNMN

m2πf 2

π

ρs

• Vector self-energy

Σ(0)V =

4(mu + md )MN

m2πf 2

π

ρ

Σ

(0)S

Σ(0)V

=G

(0)S

ρs

G(0)V

ρ= − σN

4(mu+md )ρsρ' −1

Ioffe’s sum rule

MN = −8π2

Λ2B

〈qq〉0

GMOR relation

(mu + md )〈qq〉0 = −m2π f 2

π

Introduction DFT Interaction The model Results Conclusions Appendix

Spin-orbit from in-medium QCD sum rules Back

[Σ

(0)S + Σ

(0)V ] −iσ ·∇

−iσ ·∇ [−Σ(0)S + Σ

(0)V ]

! „φk

χk

«=

„εk 00 2MN + εk

« „φk

χk

«

Non-relativistic reduction

0BB@−∇1

2MeffN (r)

∇| z + Σ(0)V + Σ

(0)S| z +

1

r

d

dr

1

2MeffN (r)

!l · σ| z

1CCAφk = εkφk

Kinetic Central Spin − orbit

If Σ(0)S /Σ

(0)V ' −1

8>>>><>>>>:Central Σ

(0)V + Σ

(0)S ' 0

Spin − orbit MeffN (r) ' MN − 1

2(Σ

(0)V − Σ

(0)S )| z

' 600/700 MeV

Introduction DFT Interaction The model Results Conclusions Appendix

Density-dependent couplings back

QCD Sum Rules| z high order| z G(ρ) = g3(b3, a3,G

(0)S ,G

(0)V ) + g4ρ

1/3 + g5(b5, a5)ρ2/3 + g6(b6, a6)ρz | ChPT

z | ChPT

z | ChPT

z | ChPT

Background termsFit FKVW QCD SR

G(0)S [fm2] - 11.5 -11.0

G(0)V [fm2] 11.0 11.0

Counter terms / Pionic fluctuationsFit FKVW ChPT

b3 -2.93 -3.05a3 2.20 2.16b5 0 0a5 0 -3.5b6 -5.68 -2.83a6 -0.13 2.83

D(π)S [fm4] -0.76 -0.7

• nice agreement for the QCD SRestimates

• truncation at order k6f could not

be adequate for finite nuclei,expand to k7

f ?

• Jump to Power expansion

• Jump to Gradient term

Introduction DFT Interaction The model Results Conclusions Appendix

Self-energies Back

From the variation of the vertex functionals

δLint

δψ=∂Lint

∂ψ+∂Lint

∂ρ

δρ

δψwith

δρ

δψ=∂ρ

∂ψ= γµuµψ and ρuµ = jµ

the self-energies are defined as

ΣV = [G(0)V + G

(π)V (ρ)]ρ+ eA0 1 + τ3

2

ΣTV = G(π)TV (ρ) ρ3

ΣS = [G(0)S + G

(π)S (ρ)]ρS + D

(π)S ∇2ρS

ΣTS = G(π)TS (ρ) ρS3 ,

ΣR =1

2

(∂G

(π)V (ρ)

∂ρρ2 +

∂G(π)S (ρ)

∂ρρ2

S +

∂G(π)TV (ρ)

∂ρρ2

3 +∂G

(π)TS (ρ)

∂ρρ2

S3

)

Introduction DFT Interaction The model Results Conclusions Appendix

Influence of the three-body term Back

Central potential

U(r) = U3ρ(r)

ρ(0)+ U4

„ρ(r)

ρ(0)

«4/3

+ U5

„ρ(r)

ρ(0)

«5/3

+ U6

„ρ(r)

ρ(0)

«2

-100

0

100

US

+ U

V [M

eV]

kf3

kf4

kf5 kf

6

• Almost no contribution from backgroundfields

U(0) = G(0)S ρS + G

(0)V ρ ' 0

• Convergence still to be achieved

• Terms of order ρ4/3 and ρ5/3 arefree-parameters

Introduction DFT Interaction The model Results Conclusions Appendix

Gradient terms Back

Inhomogeneous systems: using a density-matrix expansion we can derive

E(ρ,∇ρ) = ρ E(kf ) + (∇ρ)2 F∇(kf ) + . . .

A non relativistic reduction permits to estimate D(π)S if F∇ is treated as a constant

D(π)S = −2F∇ = −0.7 fm4 ' −140 MeVfm5

0 0.05 0.1 0.15 0.2

ρ [fm-3]

0

25

50

75

100

125

150

Fgr

ad [M

ev fm

5 ]

no-∆ with ∆

Sly

SIII

MSk

• Anoumalousbehaviour close tozero density due tochiral singularities

limρ→0

F∇ = 340 Mevfm5

• The shaded arearepresents a regionwhere calculationsare under control

This estimate −0.7 fm4 is in good agreement with the fitted value −0.76 fm4

Introduction DFT Interaction The model Results Conclusions Appendix

Landau-Migdal terms Back

Diagrammatic calculation of nuclear Fermi liquid parameters:[N. Kaiser, Nucl. Phys. A768 (2006) 99]

1π and 2π-exchange (with ∆-isobars)

0 50 100 150 200 250 300 350kf [MeV]

−5

−4

−3

−2

−1

0

1

2

3

4

5

[fm2 ]

f0(kf)

g0(kf)

g0’(kf)

f0’(kf)

F0(kf ) =1

4π2

ZdΩ1dΩ2〈~p1~p2|Veff |~p1~p2〉 =

f0(kf ) + g0(kf )~σ1 · ~σ2 + f ′0 (kf )~τ1 · ~τ2 + g ′0(kf )~σ1 · ~σ2~τ1 · ~τ2