Nuclear Density Functional Theory constrained by low-energy QCD

Dario Vretenar

Paolo Finelli, Norbert Kaiser, Wolfram Weise

P. Finelli, N. Kaiser, D. Vretenar, W. Weise, Eur. Phys. J. A17 (2003) 573Nucl. Phys. A 735 (2004) 449nucl-th/0509040

D. Vretenar, W. Weise, Lecture Notes in Physics (Springer) 641 (2004) 65

The Nuclear Many-Body Problem

A~12 A~60

Ab initio few-bodycalculations (GFMC)No-core Shell Model

0 ħω Shell Model

Self-co

nsist

ent m

ean-f

ield m

ethod

s

-use of global effective nuclear interactions-description of arbitrarilyheavy systems-intuitive picture of intrinsic shapes

Nuclear Density Functional Theory

Hohenberg-Kohn:

1) The ground-state expectation value of any observable is a unique functional of the exact ground-state density

2) Variational principle for the ground-state energy of a system of interacting fermions

Universal functional => depends only on the local density

Kohn-Sham DFT

Consider an auxiliary system of N non-interacting particles

For any interacting system, there exists a local single-particle (Kohn-Sham)potential Vs(r), such that the exact ground-state density of the interactingsystem equals the ground-state density of the auxiliary problem.

Kinetic energy of the non-interacting N-particle system

Hartree term

Exchange-correlation energywhich, by definition, includeseverything else!

The practical usefulness of the Kohn-Sham scheme depends entirelyon whether accurate approximations for Exc can be found.

Mean absolute error of the atomization energies for 20 molecules, evaluated by various approximations:

0.05Desired “chemical accuracy”

0.3 (mostly overbinding)GGA

1.3 (overbinding)LSD

3.1 (underbinding)Unrestricted Hartree-Fock

Mean abs. error (eV)Approximation

Hint: multiply by 106 and compare with the nuclear case!

A microscopic nuclear energy density functional must go beyondthe mean-field approximation and systematically include the exchange-correlation part of the energy functional, starting from the relevantactive degrees of freedom at low energy.

Use an effective field theory (EFT) of low-energy in-mediumnucleon-nucleon interactions to construct accurate approximations to the exact exchange-correlation functional.

The description of nuclear many-body systems must eventually berelated to and constrained by low-energy, non-perturbative QCD.

1) The long-range behavior of the underlying theory must be known, and it must be built into the effective theory.

2) Introduce an ultraviolet cutoff to exclude high-momentum states,which are sensitive to the unknown short-distance dynamics.

3) Add local correction terms to the effective Lagrangian. These mimicthe effects of the high-momentum states excluded by the cutoff. Eachcorrection term consists of a theory-specific constant multiplied by atheory-independent local operator. The correction terms systematicallyremove dependence on the cutoff.

How to build an effective theory ?

Low-energy QCD

At low energies characteristic for nuclei, QCD is realized as a theory of pions coupled to nucleons. Their dynamics is governed by the chiralSU(2) x SU(2) symmetry and its spontaneous breakdown at low energy.

CHIRAL EFFECTIVE FIELD THEORY

The dynamics of the active light particles and the heavy (almost) static sources is described by an effective Lagrangian which includes all relevant symmetries of the underlying fundamental theory.

Low-energy expansion => CHIRAL PERTURBATION THEORY

--

0

100

200T[MeV]

01

23

4ρ/ρ0

0.2

0.6

1

<qq>

Non-trivial vacuum: THE CHIRAL (QUARK) CONDENSATE

-order parameter of spontaneouslybroken chiral symmetry-hadrons and nuclei are excitationsof the chiral condensate

Changes of the condensate structure in the presence of baryonic matterare a source of strong fields experienced by the nucleons.

> 10x nuclear matterdensity at saturation!

CONJECTURES:

1) The nuclear ground state is characterized by strong scalar and vector mean fields which have their origin in the in-medium changes of the scalar quark condensate (the chiral condensate) and of the quark density.

2) Nuclear binding and saturation arise primarily from chiral (pionic) fluctuations (reminiscent of van der Waals forces) in combination with Pauli blocking effects, superimposed on the condensate background fields and calculated according to the rules of in-medium chiral perturbation theory (ChPT).

Nuclear energy density functional:

The large scalar and vector mean fields that have their origin in the in-medium changes of the chiral condensate and of the quark density, determine the Hartree energy functional E0[ρ].

The chiral (pionic) fluctuations including one- and two-pion exchangewith single and double virtual ∆(1232)-isobar excitations, plus Pauli blocking effects, determine the exchange-correlation energy functional Exc[ρ].

QCD Sum Rules at Finite Density

relate the leading changes of the scalar quark condensate and of the quark density at finite baryon density, with the scalar and vector self-energies of a nucleon in the nuclear medium.

To first order in the scalar and baryon densities:

…use Ioffe’s formula

and Gell-Mann, Oakes, Renner relation

Scalar self-energy:

Vector self-energy:

The large scalar and vector self-energies are about equal in magnitude (≈350 MeV) but of opposite sign => generate the large effectivespin-orbit potential in nuclei.

Chiral Dynamics and the Nuclear Many-Body Problem

Relevant scale: Fermi momentum

PIONS introduced as EXPLICIT degrees of freedom

In-medium Chiral Perturbation Theory

pion-exchange processes in the presence of a filled Fermi sea

short-distance dynamics:

IN-MEDIUM nucleon propagator

expansion of the ENERGY DENSITY of nuclear matter in powersof the Fermi momentum

1st step: CHIRAL “one-parameter” approach to NUCLEAR MATTER

2nd step: inclusion of ∆(1232) degree of freedom

S. Fritsch, N. Kaiser, W. Weise, Nucl. Phys. A 750 (2005) 259

mass difference

large SPIN-ISOSPIN POLARIZABILITY of the nucleon.

Irreducible 2-pionexchange diagramsin nuclear matter:

FINITE NUCLEI

The density distribution and the energy of the nuclear ground state areobtained from self-consistent solutions of the relativistic generalizationsof the linear single-nucleon Kohn-Sham equations.

Construct an equivalent effective Lagrangian with DENSITY-DEPENDENT four-point couplings:

Matching at the level of nucleon self-energies

The couplings Gi(ρ) (i=S,V,TS,TV) are decomposed as follows:

density-dependent couplingsgenerated by one- and two-pion exchange dynamics

contribution of the strongisoscalar scalar and vectorcondensate fields

QCD SUM RULES estimates for the condensate background self-energies:

The exchange-correlation term: In-medium chiral perturbation theory

1st step: Local Density Approximation

The momentum and density-dependent single-nucleon potential in asymmetric nuclear matter:

ChPT self-energies

For the ChPT isoscalar and isovector self-energies a polynomialfit up to order kf

6 is performed:

dimensionless short-distance regularization constants appearing in the counter terms

derived directly from the in-medium ChPT calculations of the finite (regularization-independent) parts of one- and two-pion-exchangecontributions to the energy

general form of the density-dependent couplings:polynomial in fractional powers of the baryon density

2nd step: second-order gradient correction to the LDA

ChPT calculations for inhomogeneous nuclear matter

S. Fritsch, N. Kaiser, W. Weise, Nucl. Phys. A724 (2003) 47

… density-matrix expansion method isoscalar energy density:

…determines the coefficient of the derivative term in the equivalentpoint-coupling model

… approximate F(kf) with a constant in the region of relevant densities

Adjusting the parameters:

The parameters of the point-coupling model are fine tuned simultaneouslyto nuclear matter and to ground-state properties of spherical nuclei.

reference scale:

The total number ofadjustable parametersis SEVEN.

parameters related to the three-body contact interaction terms

CONVERGENCE?

Isoscalar single-nucleon potential: expansion in powers of kf or

Coefficients Un in theexpansion for

-0.2

0.0

0.2

δE (

%)

FKVW[2005]DD-ME1 [2002]

100 110 120 130A

4.5

4.6

4.7

r ch (

fm)

Exp. values

Sn50

The deviations (in percent) of the calculated binding energies from the experimental values:

and charge radii in comparison with data for even-A Sn isotopes.

170 180 190 200 210 220-0.5

0.0

0.5 δ

E (

%)

FKVW [2005]DD-ME1 [2002]

190 195 200 205 210 215 220A

-0.2

0.0

0.2

0.4

0.6

0.8

Δrch

2 - Δ

r LD2 (

fm2 )

Exp. values

Pb82

The deviations (in percent) of the calculated binding energies from the experimental values:

and charge isotope shifts in comparison with data for Pb isotopes.

100 110 120 130 140-0.1

0

0.1

0.2

0.3

Exp. dataFKVW [2005]

180 190 200 210 220A

0.1

0.2

0.3r n - r

p (fm

)Sn

Pb

FKVW plus Gogny RHB-model predictions for the differences betweenneutron and proton rms radii:

10-4

10-2

100

Exp. dataFKVW [2005]

10-4

10-2

100

|F(q

)|

0 0.5 1 1.5 2 2.5

q (fm-1

)

10-4

10-2

100

48Ca

90Zr

208Pb

Charge form factors of48Ca, 90Zr and 208Pbcalculated in the point-coupling model with theFKVW density-dependent effective interaction, in comparison with the experimental form factors.

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

δεls (

%)

FKVW [2005]DD-ME1 [2002]

16O

(ν1

p)

16O

(π1

p) 40C

a (ν

1d)

40C

a (π

1d)

48C

a (ν

1f)

48C

a (π

1f)

132 Sn

(ν2

d)

132 Sn

(π1

g)13

2 Sn (

π2d)

208 Pb

(ν2

f)

208 Pb

(ν1

i)

208 Pb

(π2

d)

208 Pb

(ν3

p)

208 Pb

(π1

h)

The deviations (in percent) between the theoretical and experimental values of the energy spacings between spin-orbit partner-states in doubly closed-shell nuclei.

130 140 150 160-0.5

0

0.5

140 150 160 170 150 160 170 180 190

140 150 160-0.5

0.0

0.5

δE (

%)

150 160 170 170 180 190 200

140 150 160-0.5

0

0.5

150 160 170 180

A170 180 190 200

Nd

Sm

Gd

Dy

Er

Yb

Hf

Os

Pt

130 140 150 1604.8

5.0

5.2

150 160 170

5.0

5.2

150 160 170 180 190

5.2

5.4

140 150 1604.8

5.0

5.2

r ch (

fm)

150 160 1705.0

5.2

Exp.data

FKVW [2005]

160 170 180 190 2005.2

5.4

140 150 160 170

5.0

5.2

150 160 170 180

A5.0

5.2

5.4

160 170 180 190 2005.2

5.4

Sm

Nd

Gd

Dy

Er

Yb

Hf

Os

Pt

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]

160 170 180 190 200

140 150 160 170

0

0.2

0.4

150 160 170 180

A160 170 180 190 200

Nd

Sm

Gd

Dy

Er

Yb

Hf

Os

Pt

0

2

0

2

0

2

0

2

0

2

0

2

-0.2 0 0.2 0.4β2

0

2

-0.2 0 0.2 0.4β2

0

2

Ebi

nd -

Egs

[M

eV]

182Pb

184Pb

186Pb

188Pb

190Pb

192Pb

194Pb

196Pb

Shape coexistence

Constrained RMF calculationwith the FKVW interaction.

CONCLUDING REMARKS

…a relativistic nuclear energy density functional constrained withbasic features of low-energy QCD:

a) IN-MEDIUM CHANGES OF VACUUM CONDENSATES…at the origin of the large spin-orbit splitting in nuclei

b) SPONTANEOUS CHIRAL SYMMETRY BREAKING…the exchange-correlation functional (binding and saturation!)is deduced from the long- and intermediate-range one- and two-pion exchange processes.

The density functional involves an expansion of nucleon self-energiesin powers of the Fermi momentum.

Up to SIXTH order the equivalent point-coupling model has SEVENparameters: -four are related to counter terms in ChPT calculations

-one fixes a surface (derivative) term -two represent the strengths of Hartree fields.

FURTHER DEVELOPMENTS

…an accurate generalized gradient approximation for the nuclearexchange-correlation energy!

…detailed ground-state properties and spin-isospin excitations

…relation with the universal low-momentum potential, deducedfrom phase-shift equivalent nucleon-nucleon interactions.

CHIRAL EFFECTIVE FIELD THEORY provides a consistent microscopicframework in which both the isoscalar and isovector channels of a universal nuclear energy density functional can be formulated.

Paolo Finelli, Norbert Kaiser, Wolfram Weise, D.V.Thanks to Peter Ring, Stefan Fritsch, Tamara Nikšić, Nils Paar.