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The relativistic many-body problem and effective hadronic theories: EOS of high density nuclear

matter and collective motions.

Francesco Matera Dipartimento di Fisica Firenze

Nuclear Physics School 2009

Otranto, 1-5 June 2009

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Outline

• Effective Field Theories• Quantum Hadrodynamics (QHD): Mean Field approximation• Mean Field approximation at finite temperature• QHD: exchange ( Fock ) terms• Relativistic Wigner function• Fock exchange terms in nonlinear QHD• Mean Field approximation with derivative couplings• Collective modes in nuclear matter

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Only hadronic degrees of freedom will be considered. In particular, nucleons interacting among themselves by means of boson fields, and with electromagnetic or leptonic external probes.

Need of relativity in particular kinematic conditions, i.e. high energy nucleus-nucleus collisions, and/or in extreme conditions of density for nuclear matter, i.e. neutron stars.

More generally, relativistic effects can survive in non relativistic regime: spin, spin-orbit splitting in atomic physics, for instance.

A relativistic theory for microscopic systems is a quantum field theory. If well constructed, it is, in some sense, more fundamental than non relativistic theories based on phenomenological potentials. Thus a relativistic theory can give a deeper comprehension for the ingredients of non relativistic approaches ( even if relativity, to a large extent, does not need for many investigations in nuclear physics ).

Effective fields theories EFT

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At present, Quantum Chromodynamics ( QCD ) of quarks and gluons represents the fundamental theory of strongly interactions. Many difficulties,primarly because the confinement property: at distance scales relevant for nuclear processes, predictions of QCD are not yet available, particularly with regard to many nucleon systems.

Effective field theories ( EFT) based on hadronic degrees of freedom can circumvent this wall.

EFT: a tool to describe low-energy physics, where low is defined with respect to an appropriate energy scale.

Theoretical basis of EFT ( S. Weinberg, Physica 96 A (1979) 327; H. Leutwyler,

Ann. Phys. 325 (1994) 165) :

For a given set of asymptotic states, perturbation theory with the most general Lagrangian containing all terms allowed by the assumed symmetries will yield the most general S-matrix elements consistent with analyticity, perturbative unitarity and the assumed symmetries.

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Dimensions:

All physical parameters

Action in unit of , dimensionless.

Basic idea: a physical process typified by some energy can be described in

in terms of an expansion, , physical scale with dimension and

Example: Rayleigh scattering Low energy scattering of photons with neutral atoms.

excitation energy

(“Bohr radius”) , atomic mass, ,

nonrelativistic description of the atom.

1==c

[ ]nlenergymass 1,,

∫ L= )(4 xxdS [ ] 4=L

[ ] nME

E

M 1 EM >

,10 AMaEE <<<<<< ,≈ 2

emE ce 2=

ema ×≈10- 1-

AM 1<<AME

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Effective Lagrangian

Atoms in their ground state , no recoil :

Gauge invariance: only coupled to .

Dimensions:

The lowest dimensional : ( and dimensionless )

Low energy photons cannot probe the internal structure of the atom. Cross section

depends only on the size of the scatterer:

But , . Therefore and the sky looks blue.

Correct energy dependence without specific calculations, once the relevant degrees of freedom have been determined.

00=E 1<<AME

),( BEF =

[ ] [ ] [ ] .2

3,2,1∂ === F

intL 1c 2c 1

60≈ a

[ ] 2= [ ] 660 =a ,60

4aE ≈

≈

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Example: low energy neutrino interactions

(a) Tree level and exchange between fermions. (b) The vertex in the Fermi effective interaction. scattering of fermions, or decays.

and propagators

For low momentum transfer , we cannot get physical bosons. No need

to include them in the model. Therefore

fermionic fields, dimensions: and .

Cross section , square of the total energy in c.m.f.

W Z22→ 3→1

W Z ,≈2,

2ZWMq

g.90,80≈, GeVM ZW

2,

2ZWMq <<

,4321 Gweak =L i [ ]2

3=i [ ] 2=G

sG 2≈ s

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Relevant, Irrelevant , Marginal operators

EFT characterized by effective Lagrangians

: operators containing light degrees of freedom. The heavy degrees of freedom

are hidden in the couplings . typified by their dimensions

: high energy scale.

Three types of operators (in four-dimensional space):

Relevant , Marginal , Irrelevant .

Irrelevant, but important, operators contain powers of , then suppressed at low

energies.

Relevant operators become more and more important at lower and lower .

∑ iiAa=LiA

ia iA id

[ ]4

1≈→=

idiiiM

adAM

4<id 4=id 4>idME

E

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Few possible relevant operators:

A part the unit operator , boson mass terms , fermion mass terms .

Finite mass effects negligible at very high energy , but important for ,

: mass scale of light degrees of freedom.

Example

Real scalar fields with the Lagrangian

. Scalar-scalar interaction relevant .

scattering at the tree level

Scattering amplitude times the

propagator.

0=d 2=d 3=d

mE>> mE ≤

m

,

Mm <<≈ [ ] 1= →

2

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Cross section

Factor because .

For the heavy propagator generates a contact interaction with

the effective coupling:

For the Fermi effective interaction the irrelevant coupling gives a

neutrino cross-section , irrelevant at very low energy. In contrast the relevant

( ) interaction produces a sizable behaviour ( ) when .

E1 [ ] 2=

MEm <<<< 4

22 M

6=d2≈ E

3=d 2 21 E 0→E

Marginal interactions

Examples: interactions, Yukawa coupling, gauge interactions.

Marginality: non equilibrium position. Generally quantum fluctuations change such operators to either relevant or irrelevant behaviour.

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Ingredients of Effective Field Theories

(a) The appropriate degrees of freedom for the physics at the considered scale should be defined. If there are large energy gaps, the light scales are put to zero, and finite corrections can be taken into account as perturbations.

(b) Low-energy dynamics does not depend on details of high-energy dynamics.

(c) Non-local heavy particle exchange are replaced by a set of local (generally non-renormalizable) interactions among light degrees of freedom.

(d) The only relics of high energy dynamics are the symmetries and the low energy couplings of EFT.

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Effective Lagrangians potentially involve an infinite number of interactions.

For a real utility in practical calculations it is necessary an identification of terms in

, required in order to calculate observables at a given order in . effLM

1

Power counting

Terms of an effective Lagrangian for a light-boson field can be written as

[generalization to heavy fermions (nucleons) and heavy (non Goldstone) bosons will be given shortly] : constants with dimension of mass; : index wich labels the effective interactions ( having dimension ); : dimensionless coefficients.

Since is dimensionless , dimensions are carried by derivatives ; counts the number of derivatives.

vf , kkO kd kc

v ∂ kd

Consider a Feynmann diagram involving external lines, with four-momenta collectively denoted by , internal lines and vertices with lines converging into the vertex, alternatively is the power of appearing in the terms of .

)(qAE Eq I kV kn

kn effL

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Useful identity ( conservation of ends ): ∑2 kkVnEI =+

Factor of vertices:

denotes the various momenta running at the vertices.p

Contributions of internal lines:

is the generic momentum belonging to a line.p

Number of momentum-conserving : ( one delta expresses the overall conservation of the external momenta ).

Number of integrations ( loops ):

)(4 p ∑ 1kV

)1(= ∑ kVIL

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Dimensionally regulated integrals:

with a dimensionless factor depending on the dimension , wich may be singular for

, is the dominant scale, for external momenta .

Size of the momentum integration:

link of the contributions of the terms of EFT with the dependence of observables on

Contributions of more and more complexes diagrams ( and larger and larger ) are suppressed if .

n

4→n m mq≥ mq →

Mq

kV LfMq ,<<

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Including heavy-particles ( nucleons and heavy-bosons ) we get for the propagator

and , numbers of nucleon and heavy-boson fields of the k-th term of the interaction, if a boson field couples to two nucleon fields.

The index characterizing a given term is :

( S. Weinberg, Nucl. Phys. B 363 (1991) 3; R.J. Furnsthal, B.D. Serot, Hua-Bin Tang, Nucl. Phys. A 615 (1997) 441. )

The index allows to organize the Lagrangian in increasing powers of the fields and their derivatives: fix an order , then take only the terms with < .

However, it is only an heuristic criterion to check the relevance of the various terms of the Lagrangian.

To complete the Lagrangian the coupling constants should be determined by using the full underlying theory. If this step proves to be impossible, the coefficients of the various terms should be regarded as unknown parameters and are to be determined from experiments.

kbk

fkkkk VnndVd qq )22()2(

)()(

→ ++ )()( , bk

fk nn

)()( 2 bk

fkkk nnd

kMax k Max

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Renormalizability

Field theories with irrelevant operators ( dimensions >4 ) generally are not renormalizable: one needs an infinite number of counterterms to get finite results.

However, for a given order in only a finite number of terms in contribute and these terms appear in a finite number of loops only. Then, only a finite number of renormalizations are required to make finite predictions to any fixed order . Thus, although an effective Lagrangian is not normalizable, it nevertheless can be predictive.

Mq effL

nMq )(

An additional criterion to construct a meaningful Lagrangian: Naive Dimensional Analysis (NDA) and naturalness ( H. Georgi and A. Manohar, Nucl. Phys. B 234 (1984) 189; H. Georgi, Phys. Lett. B 298 (1993) 187 ) .

For strong interactions two relevant scales: the pion decay constant

and the mass scale of physics beyond the non-Glodstone bosons .

MeVf 93≈

GeV1≈

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Rules for NDA

(1) Include a factor for each strongly interacting field.

(2) Assign a factor as an overall normalization, for instance the mass term of a scalar heavy field can be put as

, extracting the dimensionless coefficient .

(3) Multiply by factors to get dimension . Terms with derivatives are associated with powers of .

(4) Finally extract combinatorial factors for terms containing powers of boson fields.

Then the naturalness assumption implies that any dimensionless coefficient should be of order unity.

f122f

2

2

2

22222

2

1

2

1

f

mfm s

s =

≈sm 22f 1≈

1 4mass1

!1 n

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Until one can derive the effective Lagrangian from QCD the naturalness should be checked by fitting to experimental data.

According to NDA a generic term containing scalar and vector meson fields coupled to the nucleon field can be written as

The coupling constant is dimensionless and if naturalness holds.

References: D.B. Kaplan, nucl-th/9506035; A. Pich, Les Houches Summer School 1997, hep-ph/9806303; C.P. Burgess, Ann. Rev. Nucl. Part. 57 (2007) 329.

g

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Walecka model: Quantum HadroDynamics-QHD

J.D. Walecka, Theoretical nuclear and subnuclear Physics, Oxford Univ. Press (1995); B.D. Serot, J.D. Walecka, Int. J. Mod. Phys. E 6 (1997) 515;and Refs. Quoted therein.

Effective theory for interacting nucleons in nuclei or in nuclear matter, well below the phase transition to QGP.

Nucleons as point-like particles interacting by means of boson fields.

Simplest version: only two isoscalar boson fields, a Lorentz scalar and a Lorentz four-vector .

The effective Lagrangian:

: generic counterterms ( when necessary ). Masses and coupling constants to be determined from experimental data. Motivation: scattering described in terms of Lorentz covariants contains large isoscalar, scalar and four-vector terms.

,∂∂ VVF = LNN

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Field equations for the model:

Conservation of the barion current:

and of the canonical energy-momentum tensor:

For a uniform system the expectation value of must have the form:

: pressure, : energy density,

: four-velocity of the fluid.

Non linear quantum field equations, exact solutions ( if they exist ) very complicate, in addition coupling constants expected to be large, then perturbative solutions are not useful.

T

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Relativistic Mean Field Theory ( RMFT )

If the sources ( baryon densities ) are large, the meson field operators can be approximated by their expectation values ( classical fields ), quantum fluctuations are neglected.

For stationary, uniform systems, and independent of space-time coordinates. For matter at rest .

0 0V0=iV

Mean Field Lagrangian

Energy-momentum tensor

No need of symmetrizing the tensor for uniform matter, because additional terms enter as a total four-divergence, whose expectation value vanishes.

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The Dirac field equation:

shifts of nucleon mass and energy spectrum .

The nucleon number and the four-momentum operators

0* SgMM = 0VgV

annihilation

operators for (quasi)nucleons and (quasi)antinucleons, spin-isospin index. contains the contribution from the Dirac sea with

H

*→ MMminus the contribution of the vacuum. H is a dynamical quantity. The number operator is defined subtractingits vacuum expectation value: sum of the Dirac sea states, a non dynamical constant.

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The ground state is given by filling energy levels with degeneracy up to the Fermi momentum . The nucleon density is related to by:

Fk Fk

the vector field is given by:

The energy density and the pressure take the forms:

The scalar field is determined by minimizing the energy density. We get the

self-consistency condition

0

with

Note . BS <

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To analyze these equations we initially put the equation for the energy density shows a unbound system at either low or high densities. The system saturates at intermediate densities.

Nuclear matter at equilibrium with and

Is obtained with the couplings

0==

NLC

NLC

NLC set includes non linear coupling terms and is obtained by a more complete and quantitative fit of parameters to nuclear properties. Without non linear terms the compressibility modulus takes a value of .

Hartree-Fock estimates in the non- relativistic potential limit (Yukawa) of the interaction, yield a collapse of such a system: the relativistic properties of the scalar and vector fields are responsible of the saturation. The Lorentz structure of the interaction provides a different saturation mechanism.

MeV545

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decreases with the density, less than unity at the saturation. Consequence of the large scalar field, , at the saturation. A sensitive cancellation between the large scalar attraction and vector repulsion .

MM *

MeVgS 400≈0

MeVgV 350≈V0

NLC

EOS

NLC

At high densities the EOS approaches the causal limit where .

lightsound cc =

Simple two-parameter model consistent with the saturation properties of nuclear matter and allowing for a covariant, causal extrapolation to any density. However, the model predicts a too small value for the bulk symmetry energy ( there is only the kinetic part ). This can be corrected by introducing a mean field for the isovector meson.

Fig.s taken from B.D.Serot, J.D. Walecka, Int. J. Mod. Phys. E6 (1997) 515

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Finite Nuclei

Spherical nuclei: the meson fields depend only on the radius and the spatial part of the vector field vanishes ( since the nucleon current is conserved ).

The mean field Lagrangian becomes

The Dirac equation is

The normal modes of the nucleon field are given by the eigenvalue equation:

The positive-energy solutions can be written as

is a spin- spherical harmonic and a two-component isospinor.

km 21t

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We assume that the nuclear system is given by filled shells up to some values of the principal quantum number and the integer . This is appropriate for spherical nuclei.

We assume that the bilinear products of nucleon fields are normal ordered, thus the contributions from negative-energy spinors are removed. This amounts to neglect the Dirac sea.

The nucleon densities are given by (with )

n ))(12( jljk +=

which represents the sources in the equations for the meson fields

The equations for the nucleon wave functions are given by

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These results can be derived in a different way: the ground-state energy can be calculated by means of the energy-momentum tensor, like the case of infinite nuclear matter, and is given by

This quantity can be interpreted as an energy functional for the Dirac-Hartree ground state. Extremizing with respect to meson fields and the nucleon spinors, subject to the constraint

introduced by Lagrange multipliers , E

reproduces the field equations for mesons and the Dirac equation for nucleons.

The isoscalar meson fields play the most important role in describing the general features of nuclear matter, but for a quantitative comparison with the properties of nuclear matter and actual nuclei some additional dynamics should be introduced. Introduction of isovector mesons, and , which couple differently to protons and neutrons, provides a sensible improvement of the model, mainly for asymmetric systems ( ).

ZN≠

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Spherical nuclei

and mesons fields and the Coulomb potential are added. However the nuclear ground state has well-defined charge and parity. Then, only the neutral rho meson ( ) enters and the expectation value of the pion field vanishes.

The mean-field Lagrangian

0A

0b

We have to add two further equations: for the rho field with a source term given by the difference between the proton and neutron densities, and for the Coulomb potential where the source term is given by the proton density.

For and the experimental values are taken, besides the values of and . The parameters are chosen in order to reproduce the equilibrium density ( ),energy/nucleon ( ) symmetry energy ( ) of infinite nuclear matter and the rms charge radius of

MeVmmV 783== MeVm 770=M 2e ,,,,, gmgg SVS

10 3.1 fmkF = MeV75.15MeV35 Ca40

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and are in . For the compressibility modulus sets L2 and NLB give

Sm MeV

and respectively. The favored set NLC gives . MeVK 545≈ MeVK 420≈ MeVK 225≈

Charge density distributions

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Spectrum for single particle levels in Pb208

Level orderings and the major shell closures of the shell model correctly reproduced. Spin-orbit splitting occurs naturally for a Dirac particle moving in non uniform classical fields

Note that no parameter is adjusted to reproduce such interaction.

To appreciate the features of the spin-orbit interaction one can perform the Foldy-Wouthuysen reduction of the Dirac equation.

Fig.s taken from B.D.Serot, J.D. Walecka, Int. J. Mod. Phys. E6 (1997) 515

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To order one finds: with

where and

Note that whereas and tend to cancel in the central potential, they add constructively in the spin-orbit potential.

21 M

0Sg 0VgV

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Thermodynamics

Grand canonical ensemble

Homogeneous nuclear matter in the thermodynamic limit finite .

Formalism is reported for the simplest case of RMFT: symmetric nuclear matter with only isoscalar meson fields. More general results for a richer Lagrangian will be next discussed.

The key quantity is the grand partition function

VAB = ∞→,VA

where is the grand potential ( ) .

The mean field Hamiltonian is

),,( VT 1=Bk

and it is equivalent to that of a system of non interacting fermions with an effective mass

and an effective chemical potential . The classical fields play the role of simple parameters.

0* SgMM = 0

* VgV =

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The grand potential can be easily calculated:

The ensemble average of the baryon density is given by

with

Moreover, we notice that

and its ensemble average is

Then we make the correctidentification

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For a system at equilibrium with fixed the grand potential must be stationary, then

,,VT

this leads to

From we can obtain the energy density and the pressure as a function of the baryon density

),,( VT

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Results with a Lagrangian which includes isovector mesons, a Lorentz scalar and a Lorentz four-vector with Yukawa couplings to nucleons. A non linear potential for the field is added ( ). 43 , (B.Liu,V. Greco, M. Colonna, M. Di Toro, Phys. Rev. C 65 (2002) 045201).

The inclusion of the gives relevant contributions to the slope and the curvature of the symmetry energy

where is the energy per nucleon and is the asymmetry parameter.

E

In addition determines a neutron-proton mass splitting

( - proton, + neutron), the component has the density as source. 3

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)( +

Upper curves: 0=T

Lower curves: MeVT 8=

0=

0=T

Borders of the instabilty region: mechanical instability (vs density oscillations) for , mechanical+chemical (vs concentration oscillations) for

)( +

0=

0≠

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Relativistic Hartree-Fock ( HF ) approximation

In RMFT the quantum nature of boson fields is neglected.

The simplest, but powerful, way to introduce quantum fluctuations is given by the HF approximation.

Here we consider infinite symmetric nuclear matter. The approximation is illustrated only for isoscalar fields, scalar (or ) and vector (or ) fields. The inclusion of isovector fields will be shortly discussed later. The meson-nucleon couplings are of Yukawa type.

Diagrammatically the approximation is illustrated in figures

V

Dyson equation for the nucleon propagator. is the non interacting propagator for a generic meson field.

0

( B.D. Serot, Rep. Prog. Phys. 55 (1992) 1855. )

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Generic meson propagator used in the calculation of energy density.

The term does not contribute when the vector meson couples to the conserved baryon current.

Two contributions fo the non interacting nucleon propagator

kk

incorporates the propagation of virtual nucleons and antinucleons. )(0 kGF )(0 kGD

describes the propagation of nucleons in the Fermi-sea and corrects for the Pauli principle

)(0 kGF

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In its rest frame the nuclear matter shows translational and rotational invariances, then the nucleon self-energy may be written quite generally as

the tensor part does not contribute to the HF self-consistent field for boson-nucleon Yukawa couplings.

t

The Dyson equation includes the effects of interactions to all the orders

the inverse is

By defining the following quantities:

one obtains for the propagator:

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The single-particle energy is determined by the equation

It is assumed that the nucleon propagator has simple poles with unit residue and that the nucleons fill levels up to . Fkk=

The preceding results are valid for any approximation to .

In the present model the vertices entering the Feynmann diagrams for are specified by the interaction Lagrangian

For scalar meson interactions we get:

The first term comes from the tad pole diagram (direct) and the second term gives the exchange contribution. The vector-meson contributions have a similar form. This is an integral equatioin for to be solved self-consistently.

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The integrals are divergent due to the contribution of to . In principle a regularization of the integrals and a renormalization procedure adding appropriate counterterms to the Lagrangian are possible to get finite results. Generally, a simple shortcut is used ( no-Dirac-sea approximation ): is replaced by , i.e. only contributions from real nucleons of the Fermi sea are included. Thi is equivalent to a truncation of the Fock space of intermediate states ( in the following this point will be briefly discussed ).

The self-energy is given by three coupled non linear equations

FG G

G DG

and

are angular integrations with

is the isospin degeneracy.

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The energy density can be calculated from the energy-momentum tensor

is given by bilinear forms of the field operators and their derivatives, thus its expectation value can be expressed by the Green functions and their derivatives with an appropriate choice of time ordering:

Retardation terms 2)( qk EE∝

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Remark: one can see that the RMFT results may be recovered by summing only the tad pole diagrams ( Hartree approximation ) and retaining only the contributions from nucleons of the Fermi sea ( no-Dirac-sea approximation ).

Parameters of the present model: two coupling constants and the mass of the scalar boson field, the mass of the vector boson is that of the meson. They are determined from the equilibrium properties of nuclear matter like in RMTF.

With a difference: in RMFT for infinite nuclear matter, results depend only on the ratio

, in HF masses and coupling constants are to be specified separately.

We remark that the values of the two sets of parameters ( for RMFT and HF ) differ only by using the same ingredients.

mg

%15≈

The fits to the binding energy per nucleon for the equilibrium Fermi momentum

are obtained with two different sets of parameters for the Hartree (or RMFT) and the HF cases.

MeV75.15

10 42.1= fmkF

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The large discrepancy between the HF and Hartree approximations is due to the larger bulk symmetry energy for HF ( ) compared to the RMFT result ( ), in RMFT only the kinetic term contributes.

MeV35≈MeV22≈

A quite similar procedure is followed when charged mesons and are included. The meson interacts with nucleons with a pseudovector coupling with

1)2( ≈= Mmgf fixed.

We remark that the direct (Hartree) contribution from to the nucleon self-energy vanishes. Since the range of interaction is sensibly larger than the range of exchange correlations of nucleons in nuclear matter, the contribution of meson to exchange self-energy is rather low compared to the contributions of heavier mesons. This can be appreciated in the following figures.

N

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The charged meson contributes to the symmetry energy also in RMFT. The discrepancy between the Hartree and HF approximations observed in the ( ) is quite filled.

+

Figs. from B.D. Serot, Rep. Prog. Phys. 55 (1992) 1855

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Difficulties of QHD

Difficulties arise from summations over nucleon and antinucleon intermediate states (Loop contributions).

Problems result when one attempts to describe short-range dynamics using effective heavy QHD degrees of freedom.

In principle such divergences may be cured by renormalization procedures at a given order of approximation ( also in the case where the effective theory is not globally renormalizable).

The finite contributions from loops (for instance contributions to nucleon self-energy) generally results in corrections of the scalar or vector contributions, if separately considered. But nuclear quantities are obtained by sensitive cancellations between large quantities. The loop corrections degrade the agreement with experiment.

One needs a well founded scheme to truncate the effective Lagrangian at a given order of approximation. In addition a robust criterion is necessary to identify the relevant terms of actual calculations for an admitted approximation. At present, this problem is not yet solved.

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However, people have developed, refined and extended the original Walecka model in many fields of nuclear physics with remarkable successes, simply neglecting the loop problem. A similar point of view has been adopted for other relativistic approaches to the nuclear many-body problem (e.g. The Dirac-Brueckner approach).

For recent applications and extensions of QHD (including the relativistic generalization of the Hartree-Bogoliubov approach) see the reviews:

D. Vretenar, A.V. Afanasjev, G.A. Lalazissis, P. Ring, Phys. Rep. 409 (2005) 101;

N. Paar, D. Vretenar, E. Khan, G. Colò, Rep. Prog. Phys. 70 (2007) 691.

The reliance of people in QHD models can be to some extent justified when one considers the models in the framework of the Density Functional Theory.

An additional (technical) difficulty concerns the inclusion of the meson self-interaction terms in a consistent HF scheme. The problem arises from the nonlinearity of the equatios of motion for the meson fields. A perturbative expansion in the coupling constants is not satisfactory, in RMFT these terms can be introduced exactly, at least in the case of nuclear matter. A non perturbative approximation is necessary

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Wigner function for fermions

the brackets denote statistical averaging and the double dots denote normal ordering.

and are spin-isospin indices. is a matrix. For simplicity we consider only isoscalar bosons interacting with nucleons and symmetric nuclear system. The Wigner function is degenerate in the isospin indices. a matrix in spin space.

88×

44×

The various densities are related to the integrals of

For instance scalar density and

current density. Traces are taken over the spin states.

I assume that nucleons interact with classical neutral fields, the fermion field equation is:

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The equation of motion for the Wigner function can be derived from the field equation:

2Rxx ±=±

Assuming that the fields are slowly varying functions and retaining only terms up to first order in their expansion

we obtain the following equation

For the scalar and vector components of the Wigner function we get two coupled equations:

with acting on the first term of the products. x∂

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Defining the four-velocity

One obtains the equation

is the effective mass.

which coincides with the relativistic one-body Liouville equation for particles subject to external forces due to the vector, , and scalar, , fields. The time-space varying mass gives rise to the third term in the brackets.

( S.R. De Groot, W.A. van Leewen, Ch.G. van Weert “ Relativistic Kinetic Theory” )

52

The kinetic equation together with the relations

allows one to identify the Wigner functions with the classical phase-space distributions in the classical limit ( long-wavelength and low frequency limit ). The Wigner functions can represent a useful tool to make semiclassical approximations. Semiclassical: spin quantization and quantum statistics should be preserved.

53

Fock exchange terms in nonlinear QHD

In order to illustrate a procedure to include self-interactions of mesons in a HF scheme i limit myself to a simple model sufficient to reproduce basics properties of nuclear matter: model with self-interacting terms for the scalar meson .

The Lagrangian is

The Compton wavelengths of the mesons are sensibly smaller than the average interparticle spacing. Systems slowly varying in space-time are considered, then, in the equations for the meson fields, derivatives are neglected with respect to mass terms.

,≡

Non linear equation for the field operator .

V≡

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A non expensive way to handle the problem is that of linearizing the equation for ,

, equivalent to introduce an effective mass for .

Here i present a procedure whose result is an expansion of the Fock terms in a N1

series, is the multiplicity of the nucleon intrinsic degrees of freedom. The approach is based on the Wigner function formalism.

4=N

Now the meson fields are operators. The equation for the Wigner function is:

.2Rxx ±=± The term with the vector field contains the expectation value of four field- operators . In a mean-field approximation (HF), which neglects correlations, but preserves statistics, the Wick’s theorem prescribes:

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The scalar field operator can be expanded in powers of the scalar density

Again the Wick’s theorem gives:

The averages of and its derivatives are given by series of which in turn can be expanded as

Then for one obtains

where and its derivatives are given by series of powers of .

)(x

The parameter fixing the order of approximation is the number of factors

broken in (a). For each breaking there is a trace less giving a quenching factor . 81

(a)

This scheme implies that the contributions of the Dirac sea are neglected.

56

Exchange corrections only up to the next-to-leading term, with respect the Hartree contribution.

The energy-momentum tensor is given by

and is solution of .

Remark: the traces of powers of may give contributions to the energy-momentum tensor from all the densities having tensorial properties consistent with the symmetries of the system.

EOS for symmetric nuclear matter:

57

Density dependent constants:

The parameters from fits to reproduce equilibrium properties of symmetric nuclear matter. Solid lines: full calculations. Dashed lines: neglecting second derivatives. Up to densities the contributions of the second derivatives can be considered as a corrections.

BAff VS ,,,

eq4≈

58

The relation between pressure and energy density, and the Hugenholtz-Van Hove theorem are satisfied:

Consistency of the truncation scheme used:

( V. Greco et al., Phys. Rev. C 63 (2001) 035202 )

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RMFT with derivative couplings ( DC )

( S. Typel, T.V. Chossy, H.H. Wolter, Phys. Rev. C 67 (2003) 034002 )

For infinite nuclear matter RMFT predicts that nucleon self-energy does not depend on the momentum. Effective mass (contribution from scalar fields) and the shift of single-particle energy levels (contribution from vector fields) result constant quantities only depending on density and asymmetry.

A momentum dependence of self-energies is suggested by the behaviour of optical potentials for elastic p-Nucleus scattering.

Derivative coupling (DC) models lead to momentum dependent self-energies already in the mean field approximation.

DC model: isoscalar mesons and , in addition isovector mesons (scalar) and (vector). Couplings to the nucleon field linear or quadratic for and mesons ,only linear for and mesons.

)( )( V

60

The Lagrangian

with the covariant derivative

Instead of the matrices of the standard QHD the DC model contains

with

The terms describe the coupling of the meson fields to the derivative of the nucleon field.

XTW ,,

61

The couplings are parametrized up to quadratic terms in the isoscalar fields

and linear terms in the isovector fields

The vector and tensor terms are essential to give a momentum dependence to the scalar and vector self-energies, respectively.

Seven new coupling constants in addition to the four coupling constants specifying the minimal nucleon-meson coupling.

Mean field approximation: meson field operators replaced by the expectation values of the fields, classical fields.

No-Dirac-sea approximation: only positive energy states of the nucleons are taken into account.

62

Dirac equation:

scalar self-energy vector self-energy

The self-energies and are differential operators acting on the nucleon field. Then, self-energies contain a state dependence in addition to a medium dependence.

One can derive a continuity equation for the current density . The usual vector density is no longer a conserved quantity. Similarly

for the isospin density operator .

The equations for the meson fields contain as source terms the usual scalar densities (isoscalar and isovector) and the expectation values of the new current densities. In addition the source terms contain derivative densities.

0=

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Infinite nuclear matter

Infinite nuclear matter in its ground state: a homogeneous, isotropic and stationary system. Densities do not depend on space-time coordinates and only the time-like component of the currents survives, then the meson fields are constant and the spatial components vanish. Moreover, only the third component in isospin of the various quantities remains.

EOS:

Energy-momentum tensor

2))(1( 3 +=P 1±=

with

Energy density

with , +1 protons,-1 neutrons. and are independent

of momentum. The mass depends on the momentum, density and isospin. The

dispersion relation connects energy and mass, with the effective momentum

0VS

W is indipendent of the momentum.,

64

Thermodynamical consistency for pressure

dd )(2

and chemical potentials ,

± are the nucleon energies at the Fermi surface (Hugenholtz-van Hove) theorem.

65

Results

All the parametrizations are fitted to the properties of symmetric nuclear matter at saturation and to an effective nucleon mass of .

DC1: without momentum-dependent self-energies.

DC2/DC3: parameters chosen to reproduce the relevant features of the optical potential for the nucleon in nuclear matter. In DC3 the further coupling constant ( for the

MM eff 6.0=

meson ) is adjusted with the derivative of the symmetry energy with respect the density.

66

NL3: minimal meson-nucleon coupling and self-interactions for the field.

TM1: NL3+self-interactions for the field.

VDD: meson-nucleon coupling constants depending on density.

Fig.1

Fig.2

Fig.2: the effective constants are normalized to unit at saturation density.

tHM,BMB: coupling constants extracted from Dirac-Brueckner calculations.

The derivative couplings generate a density and energy dependence of self-energies that lead to asofter EOS.

67

: asymmetry.

for at saturation density of symmetric nuclear matter.

The curves for in DC2 and DC3 are identical

MM eff 6.0= Fpp=

0=

68

Collective modes This topic will be illustrated in a particular formalism based on the Wigner function.

In this formalism many ingredients may be included: concerning both meson fields with their interactions and appropriate approximation strategies, although with some simplifications.

Wigner function allows one to treat both equilibrium properties and dynamics of nuclear matter self-consistently, using a single equation to be solved for different physical conditions.

This appoach is very similar to the Relativistic Random Phase Approximation (RRPA). Different approaches, with recent applications, can be found e.g. in:

S.S. Avancini, L. Brito, D.P. Menezes, C. Providencia, Phys. Rev. C 71 (2005) 044323; A.M. Santos, L. Brito, C. Providencia, Phys. Rev. C 77 (2008) 045805, (for symmetric and asymmetric nuclear matter) and Zhong-yn Ma, Nguyen Van Giai, A. Wandelt, D. Vretenar, P. Ring, Nucl. Phys. A 686 (2001) 173; D. Vretenar, A.V. Afanasjev, G.A. Lalazissis, P. Ring, Rhys. Rep. 409 (2005) 101, (for finite nuclei).

69

The model includes isoscalar scalar and vector mesons, and isovector scalar and vector mesons, with cubic and quartic self-interactions for the field.

Basic approximation: in the equations of motion terms containing derivatives of meson fields are neglected with respect to the mass terms. This implies that retardation and finite-range effects are neglected. Thanks to the small Compton wave-lengths of the mesons this approximation can be considered quite reasonable.

Remark: in a RMFT approach, i.e. neglecting exchange contributions, such limitation can be missed.

In the semiclassical limit (long wavelengths and low frequencies) the equation for the Wigner function is given by :

70

x∂

np,=

with acting only on the first term of the products.

Effective masses

Kinetic momentum

Isovector scalar and vector densities

Effective coupling constants

Effective masses embody an isospin contribution from Fock terms also without the direct inclusion of the meson.

><= g

is the coupling constant

g N

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Linear response in the scalar and vector channels

Small oscillations around the equilibrium

In the linear approximation the equations for the fluctuations are

for protons and

for neutrons.

The scalar and vector self-consistent fields: Derivatives of effective coupling constants come from the exchange contributions of the self- interactions of the field.

72

Two coupled equations for fluctuations of scalar and current densities:

with the interaction matrices

=iN densities of quasinucleon states at the Fermi surface.

73

The RMFT (Hartree) approximation (i.e. neglecting the exchange contributions) is recovered substituting the effective coupling constants with the meson-nucleon couplings, but

due to the self-interaction of the field.

The physical content is more transparent in the Hartree approximation. The solutions for the eigen-modes correspond to longitudinal waves and depend only on the ratio

Isovector modes

For symmetric nuclear matter the equation for the sound velocity is given by

Interactions, which give the same value of at the saturation, can give different isovector reponse. Once the symmetry energy is fixed their effect on the dynamical response depends on the strength of the coupling of each isovector field.

Analogous effect for the isoscalar modes with the bulk modulus instead of the symmetry energy and the isoscalar fields instead of the isovector fields.

4a

74

The isoscalar meson parameters are fixed from nuclear matter properties at the saturation and assuming an effective mass . The isovector meson parameters are fitted to the coefficient of the symmetry energy, the coupling constant is chosen according to the corresponding term of the Bonn potential.

MM 7.0*=N

HF

HF

HH

+

+

H H

+

+

Results differ already for 0=at the saturation even if the coefficient is the same in the two cases.

4a

Quite similar results for H and HF calculations, at the saturation. This because the effective couplings for the isovector channels in HF are tuned to roughly reproduce the corresponding coupling constants of H.

75

Isospin distillation in dilute matter

Isoscalar-like ( neutrons and protons oscillating in phase ) unstable modes can be found for dilute nuclear matter. They have imaginary sound velocity, that gives rise to an exponential growth of fluctuations. Mechanism for the spinodal decomposition.

HF

HH--

full circles in (b). Then, proton oscillations are relatively larger than neutron oscillations leading to a more symmetric liquid (higher density) phase and to a more neutron rich gas (lower density): the so-called isospin distillation.

>np np

( V. Greco et al., Phys. Rev. C 67 (2003) 015203 )

eq 4.0