Class eos_had_rmf_delta (o2scl)

O2scl : Class List

class eos_had_rmf_delta : public o2scl::eos_had_rmf

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

See also [Kubis97] and [Gaitanos04].

This essentially follows the notation in Kubis et al. (1997), except that our definitions of b and c follow their \( \bar{b} \) and \( \bar{c} \), respectively.

Also discussed in Gaitanos et al. (2004), where they take \( m_{\delta}=980 \) MeV.

The full Lagragian is:

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

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

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

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

Equation of state as a function of density.

virtual 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)

inline virtual 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()

virtual 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

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

The function for calc_e()

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

Compute matter at zero pressure (for saturation())

Private Functions

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

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