Nuclear Physics School 2009 Otranto, 1-5 June 2009

1. 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

2. 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

3. Effective fields theories EFT • 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 ).

4. 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 lowis defined withrespect 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.

5. Dimensions: • All physical parameters • Action in unit of , dimensionless. • Basicidea: 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.

6. 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.

7. 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.

8. 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 .

9. 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.

10. Cross section Factor because . • For the heavy propagatorgenerates 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 . • Marginal interactions • Examples: interactions, Yukawa coupling, gauge interactions. • Marginality: non equilibrium position. Generally quantum fluctuations change such operators to either relevant or irrelevant behaviour.

11. Ingredients of Effective Field Theories • 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. • Low-energy dynamics does not depend on details of high-energy dynamics. • Non-local heavy particle exchange are replaced by a set of local (generally non-renormalizable) interactions among light degrees of freedom. • The only relics of high energy dynamics are the symmetries and the low energy couplings of EFT.

12. 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 . • 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. • 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 .

13. Useful identity ( conservation of ends ): • Factor of vertices: denotes the various momenta running at the vertices. • Contributions of internal lines: is the generic momentum belonging to a line. • Number of momentum-conserving : ( one delta expresses the overall conservation of the external momenta ). • Number of integrations ( loops ):

14. 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 .

15. 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.

16. 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. • An additional criterion to construct a meaningful Lagrangian: Naive Dimensional Analysis (NDA) andnaturalness (H. Georgi andA. 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 .

17. Rules for NDA • Include a factor for each strongly interacting field. • 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 . • Multiply by factors to get dimension . Terms with derivatives are associated with powers of . • 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.

18. 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.

19. Walecka model: Quantum HadroDynamics-QHDJ.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.

20. 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.

21. 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 . 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.

22. The Dirac field equation: • shifts of nucleon mass and energy spectrum . • The nucleon number and the four-momentum operators annihilation operators for (quasi)nucleons and (quasi)antinucleons, spin-isospin index. contains the contribution from the Dirac sea with minus the contribution of the vacuum. is a dynamical quantity. The number operator is defined subtracting its vacuum expectation value: sum of the Dirac sea states, a non dynamical constant.

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

24. 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 • 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. NLC NLC

25. 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 . NLC EOS NLC • At high densities the EOS approaches the causal limit where . • 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

26. 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.

27. 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 ) which represents the sources in the equations for the meson fields • The equations for the nucleon wave functions are given by

28. 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 , 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 ( ).

29. 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 • 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 radiusof

30. and are in . For the compressibility modulus sets L2 and NLB give and respectively. The favored set NLC gives . Charge density distributions

31. Spectrum for single particle levels in • 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. • Fig.s taken from B.D.Serot, J.D. Walecka, Int. J. Mod. Phys. E6 (1997) 515 • To appreciate the features of the spin-orbit interaction one can perform the Foldy-Wouthuysen reduction of the Dirac equation.

32. 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.

33. 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 • where is the grand potential ( ) . • The mean field Hamiltonian is 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.

34. 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 correct identification

35. For a system at equilibrium with fixed the grand potential must be stationary, then this leads to • From we can obtain the energy density and the pressure as a function of the baryon density

36. 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 ( ). (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. • In addition determines a neutron-proton mass splitting ( - proton, + neutron), the component has the density as source.

37. Upper curves: Lower curves: Borders of the instabilty region: mechanical instability (vs density oscillations) for , mechanical+chemical (vs concentration oscillations) for

38. 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 Dyson equation for the nucleon propagator. is the non interacting propagator for a generic meson field. ( B.D. Serot, Rep. Prog. Phys. 55 (1992) 1855. )

39. 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 incorporates the propagation of virtual nucleons and antinucleons. describes the propagation of nucleons in the Fermi-sea and corrects for the Pauli principle

40. 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. • 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:

41. 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 . • 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.

42. 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 and are angular integrations with is the isospin degeneracy.

43. 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

44. 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. 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.

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

46. 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

47. 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.

48. 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

49. 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. • 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:

50. The equation of motion for the Wigner function can be derived from the field equation: • Assuming that the fields are slowly varying functions and retaining only terms up to first order in their expansion we obtain the following equation with acting on the first term of the products. • For the scalar and vector components of the Wigner function we get two coupled equations: