Static and covariant meson-exchange interactions in nuclear matter

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1 Workshop on Relativistic Aspects of Two- and Three-body Systems in Nuclear Physics - ECT* /10/2009 Static and covariant meson-exchange interactions in nuclear matter Brett V. Carlson Instituto Tecnológico de Aeronáutica, São José dos Campos Brazil and Daisy Hirata The Open University, Milton Keynes, United Kingdom

2 Why use the Dirac equation? The nucleus is usually not considered a relativistic system: However, the four component Dirac spinor is momentumdependent with The momentum-dependent interaction that naturally results can produce saturation, through the difference between the scalar and vector densities, and and can produce a spin-orbit interaction of the expected magnitude.

3 Dirac mean fields The Dirac equation for a nucleon of momentum k in nuclear matter is The Dirac mean field can be decomposed as In terms of these, we can define the effective mass and momentum, and the single-particle energy of the state, u(k,s),

4 Self-consistent approximations to the mean field Dyson equation for the single-particle propagator Hartree includes direct meson term with all nucleons Hartree-Fock includes direct meson and nucleon exchange terms Brueckner-Hartree-Fock iterates meson exchange to take into account the repulsive 'hard' core of the interaction

5 The Bethe-Salpeter equation The Brueckner G-matrix satisfies a Bethe-Salpeter equation. We calculate it in the ladder approximation, using the anti-symmetrized bare nucleon-nucleon interaction in the nn, np, pn and pp channels. In free space, the G-matrix reduces to the two-nucleon scattering Tmatrix. nn or pp

6 The reduced Bethe-Salpeter equation We simplify the Bethe-Salpeter equation by 1) neglecting holes and antiparticle states in the intermediate propagator; 2) projecting onto a set of positive-energy single-particle states; 3) and reducing the integral to a three-dimensional one. nn or pp where and v are the reduced G-matrix and antisymmetrized bare interaction, Q is an angle-averaged Pauli blocking factor and g is a reduced propagator of two particles with on-shell energies. This equation is solved by reducing it to coupled equations for the principal parts of 5 (or 6) independent helicity amplitudes, expanding these in partial waves and solving them as matrix equations. (K. Erkelenz, Phys. Reports, C13 (1974) 191.)

7 The Brueckner mean field Dyson equation for the single-particle propagator Quasi-particle approximation Re[ ], Re[ ] :

8 Reconstructing the G-matrix To calculate the Dirac mean field, we need the complete G-matrix, not just its projection. To reconstruct it, we solve for the component functions of a covariant expansion, projected on positive-energy states with distinct masses, (C.J. Horowitz and B. Serot, Nucl. Phys. A464 (1987) 613.) Due to the ambiguity between the pseudoscalar and pseudovector pion- and eta-nucleon coupling, we reconstruct the difference between the G-matrix and the bare interaction V and use the unprojected expression for the latter. (E. Schiller and H. Müther, Eur. Phys. J. A 11 (2001) 227.)

9 Brueckner Calculations We show calculations as a function of the density and the asymmetry, = 0 nuclear matter = 1 neutron matter We represent the density by an effective Fermi momentum, given by We use the Bonn A, B, C interactions (B. Brockmann and R. Machleidt, Phys. Rev, C42 (1990) 1965.) We take into account the tensor coupling of the meson and use pseudovector coupling for the and We use the Thompson form of the reduced two-nucleon propagator and a static interaction.

10 Brueckner calculations Bonn A0, B0, C0 constant mean fields Bonn A, B, C momentumdependent mean fields Calculations stop where instability dominates. Neutron matter same result for all interactions. Momentum dependence: - softens the equation of state of neutron matter; - increases the Fermi momentum for saturation; - increases the binding energy per nucleon and incompressibility and decreases the symmetry coefficient at saturation.

11 Momentum dependence of the mean fields Neutron fields - Momentum dependence becomes stronger as asymmetry grows Proton fields momentum dependence becomes weaker The momentum dependence of HF mean fields obtained with the bare interaction is very similar to that of the Brueckner mean fields. The effects of the ladder sum can be approximated by a densitydependent zero-range contribution in the interaction, (E. Schiller and H. Müther, Eur. Phys. J. A 11 (2001) 227.) The signs of the terms are opposite of what would be expected of the exchange of infinite mass mesons.

12 Covariance and retardation A one-boson exchange interaction has a covariant vertex-propagator-vertex form Static interaction in propagators and vertices: Covariance lost The Bonn interactions were obtained using a static interaction and the Thompson form of the reduced propagator, where An alternative form of the reduced propagator is the BlankenbeclerSugar one,

13 Retardation low energy observables With retardation terms, rs ~ 7% too low and rt ~ 15% too high. Other parameters are fairly well fit.

14 Retardation adjusted coupling constants Retardation terms requires stronger repulsion g ~ 5-10% larger.

15 Retardation phase shifts BBS without retardation terms - 1P1 amplitude?

16 Retardation phase shifts Fits with retardation terms - discrepancies in 3D1 and 3 P2-3F2 amplitudes

17 Retardation symmetric nuclear matter Calculations performed with constant mean fields. No saturation seen for interactions including retardation terms, before divergence of calculations. Effective mass is about 5% smaller when retardation effects are included. -Three-body interaction? -Pseudoscalar / pseudovector ambiguity? -Antiparticles?

18 So now what? A few things we ve left out: 1) Intermediate hole-hole and antiparticle-antiparticle propagation and full Dirac structure; 2) Quasi-particle approximation Re[ ] Re[ ] spectral function A(k, ); 3) Three-body interactions / modified off-shell interactions; 4) RPA/particle-hole correlations, vertex corrections; 5) Pairing instability density-dependent single-particle energy gap; Still others: 6) meson dynamics correlated s and dynamics; 7) s; 8) Meson and nucleon structure chiral symmetry restoration? 9)... The results serve only as a guide to the effects of different approximations to the description of nuclear matter.

19 Brueckner calculations Bonn C HF with Bonn C parameters varied (g and g ) to saturate at kf=270 MeV/c Eb= MeV. M* - e ffective mass at the Fermi momentum Neutrons little difference between M* s in nuclear and neutron matter Protons higher M* than neutrons at high density but lower M* at low density (?)