Class eos_had_rmf_delta (o2scl)

O2scl_eos : Class List

class o2scl::eos_had_rmf_delta : public o2scl::eos_had_rmf

Field-theoretical EOS with scalar-isovector meson, \( \delta \).

This essentially follows the notation in Kubis97, except that our definitions of b and c follow their \( \bar{b} \) and \( \bar{c} \), respectively.

Also discussed in Gaitanos04, where they take \( m_{\delta}=980 \) MeV.

The full Lagragian is:

\[ {\cal L} = {\cal L}_{Dirac} + {\cal L}_{\sigma} + {\cal L}_{\omega} + {\cal L}_{\rho} + {\cal L}_{\delta} \]

\[\begin{split}\begin{eqnarray*} {\cal L}_{Dirac} &=& \bar{\Psi} \left[ i {{\partial}\!\!\!{\slash}} - g_{\omega} {{\omega}\!\!\!{\slash}} - \frac{g_{\rho}}{2} {{\vec{\rho}}\!\!\!{\slash}}~ \vec{\tau} - M + g_{\sigma} \sigma - \frac{e}{2} \left( 1 + \tau_3 \right) A_{\mu} \right] \Psi \nonumber \\ {\cal L}_{\sigma} &=& {\textstyle \frac{1}{2}} \left( \partial_{\mu} \sigma \right)^2 - {\textstyle \frac{1}{2}} m^2_{\sigma} \sigma^2 - \frac{b M}{3} \left( g_{\sigma} \sigma\right)^3 - \frac{c}{4} \left( g_{\sigma} \sigma\right)^4 \nonumber \\ {\cal L}_{\omega} &=& - {\textstyle \frac{1}{4}} f_{\mu \nu} f^{\mu \nu} + {\textstyle \frac{1}{2}} m^2_{\omega}\omega^{\mu}\omega_{\mu} + \frac{\zeta}{24} g_{\omega}^4 \left(\omega^\mu \omega_\mu\right)^2 \nonumber \\ {\cal L}_{\rho} &=& - {\textstyle \frac{1}{4}} \vec{B}_{\mu \nu} \cdot \vec{B}^{\mu \nu} + {\textstyle \frac{1}{2}} m^2_{\rho} \vec{\rho}^{~\mu} \cdot \vec{\rho}_{~\mu} + \frac{\xi}{24} g_{\rho}^4 \left(\vec{\rho}^{~\mu}\right) \cdot \vec{\rho}_{~\mu} + g_{\rho}^2 f (\sigma, \omega) \vec{\rho}^{~\mu} \cdot \vec{\rho}_{~\mu} \nonumber \\ \end{eqnarray*}\end{split}\]
where the additional terms are

\[ {\cal L}_{\delta} = \bar{\Psi} \left( g_{\delta} \vec{\delta} \cdot \vec{\tau} \right) \Psi + \frac{1}{2} (\partial_{\mu} \vec{\delta})^2 - \frac{1}{2} m_{\delta}^2 \vec{\delta}^{~2} \]

The new field equation for the delta meson is

\[ m_{\delta}^2 \delta = g_{\delta} (n_{s,p} - n_{s,n}) \]

Idea for Future:

Finish the finite temperature EOS

Public Functions

int calc_e(fermion &ne, fermion &pr, thermo &lth)

Equation of state as a function of density.

int calc_eqd_p(fermion &neu, fermion &p, double sig, double ome, double rho, double delta, double &f1, double &f2, double &f3, double &f4, thermo &th)

Equation of state as a function of chemical potentials.

int calc_temp_eqd_p(fermion &ne, fermion &pr, double temper, double sig, double ome, double lrho, double delta, double &f1, double &f2, double &f3, double &f4, thermo &lth)

Finite temperature (unfinished)

int set_fields(double sig, double ome, double lrho, double delta)

Set a guess for the fields for the next call to calc_e(), calc_p(), or saturation()

int saturation()

Calculate saturation properties for nuclear matter at the saturation density.

This requires initial guesses to the chemical potentials, etc.

Public Members

double md

The mass of the scalar-isovector field.

double cd

The coupling of the scalar-isovector field to the nucleons.

double del

The value of the scalar-isovector field.

Protected Functions

int calc_e_solve_fun(size_t nv, const ubvector &ex, ubvector &ey)

The function for calc_e()

int zero_pressure(size_t nv, const ubvector &ex, ubvector &ey)

Compute matter at zero pressure (for saturation())

Private Functions

int set_fields(double sig, double ome, double lrho)

Forbid setting the guesses to the fields unless all four fields are specified.