1. Magnetic Field of a Neutron Star With Superconducting Quark Core in the CFL-phase
K.M.Shahabasyan, M. K. Shahabasyan,D.M.Sedrakyan
Yerevan State University, Armenia
Dubna, August 2007

2. Plan Introduction Ginzburg-Landau Equations
Solution for the potentials
Magnetic field components
Summary

3. Introduction
The possible existence of superdense quark matter in the cores of neutron stars has been examined over the last three decades. Because of the attraction between quarks in a color antitriplet channel owing to the exchange of a single
gluon, it is expected that a superconducting diquark condensate should develop in this material.
The quark pairs that form the condensate have zero total angular momentum .
It has been shown that in chiral quark models with a nonperturbative four-point interaction caused by instantons or with nonperturbative gluon propagators, the quark coupling amplitudes are on the order of 100 MeV. Thus, we might suppose that a diquark condensate would exist at densities above the deconfinement density and at temperatures below the critical temperature Tc (on the order of 50 MeV).

4. Introduction
In quark cores two types of condensates are possible`
1)2SC phase- where u and d quarks of two colors are coupling.
2)CFL phase with u, d, s quarks of all colors, which is the most stable in the limit of weak interaction near the
The existence of electric and color charges in the Cooper diquark pairs leads to the appearance of electrical and color superconductivity in the 2SC- and CFL-phases. These two phenomena are not independent because the photon and gluon gauge fields are coupled. One of the resulting mixed fields is massless, while the other field has mass (M. Alford, J. Berges, and K. Rajagopal, Nucl. Phys. B571, 269 (2000)).

5. Introduction
The Ginzburg-Landau free energy of a uniform superconducting CFL-phase has been found in T. Schäfer, Nucl. Phys. B575, 269 (2000) and K. Iida and G. Baym, Phys. Rev. D63, (2001).
In the work by D. M. Sedrakian, D. Blaschke, K. M. Shahabasyan, and D. N. Voskresenskii, Astrofizika 44, 443 (2001) it was shown that, in the absence of vortex filaments, the Meissner currents in the core will screen the external magnetic field almost completely.
Aim of this work is to study the magnetic field distribution in the quark CFL-phase and hadronic npe-phase of a neutron star.
We allow the generation of a magnetic field in the hadronic phase owing to the entrainment of superconducting protons by superfluid neutrons and sharp boundary between the quark and hadronic phases, since the thickness of the diffusive transition layer is small, on the order of the confinement radius l = 0.2 fm.
Following boundary conditions are taken into account: continuity of the components of the magnetic field and the condition for gluon confinement at the surface of quark core.

6. Ginzburg-Landau equation
Analysis of the gauge invariant derivative in E.V.Gorbar, Phys. Rev., D62 , , gives to the following expression for mixed fields
vector potentials of the magnetic and gluomagnetic fields
e, g strong and electromagnetic interaction constants and ,
so that

7. Ginzburg-Landau equation
Ginzburg-Landau free energy in the presence of an external field has the following form` (K.Iida, Phys. Rev., D71, , 2005 )
Where the functions and are given by

8. Ginzburg-Landau equation
By minimizing the free energy we can obtain the equations for the ordering parameter
and Maxwell’s equations for the mixed fields

9. Ginzburg-Landau equation
Equations for the magnetic and gluomagnetic field can be derived from Maxwell equations. For the CFL-phase , so Maxwell equations can be rewritten in the form
Here and
last one is a penetration depth of magnetic and gluomagnetic fields.

10. Solution for the potentials
Assuming that the CFL-condensate is a homogeneous type II superconductor, we rewrite equations for the potential in the form
where
After introducing the new vector potentials in the form
And assigning in spherical coordinates potentials will have only components, thus we can write
solution

11. Solution for the potentials
for electromagnetic potential
for gluomagnetic potential
Int. constant may be found from boundary condition on the surface of quark
core , thus
Final expressions for potentials`

12. Magnetic field components
In spherical coordinates expression for magnetic field components is
after substituting for r<a (radius of a quark core)

13. Magnetic field components
The magnetic field in the hadronic phase is found using the solution for and taking into account the fact that protonic vortical filaments generate a homogeneous average magnetic field of amplitude B, parallel to the axis of rotation of the star D. M. Sedrakian, K. M. Shahabasyan, and A. G. Movsisyan Astrofizika 19, 303 (1983) and D. M. Sedrakian, Astrofizika 43, 377 (2000)
In the hadronic phase
where

14. Magnetic field components
The external magnetic field of the neutron star (r>R), is dipole in character, with components
Where M is the total magnetic moment of the star.

15. Magnetic field components
Constants and M are determined from the continuity conditions and from
Let us consider magnetic field distribution at a distances r much greater than
and , then for a quark core`

16. Magnetic field components
For npe-phase `
For external field
As can be seen from eqs. the magnetic fields in the quark and hadronic phases depend on the coordinate r only near the phase boundary r = a. Since the penetration depths are small compared to a and r - a, the variable terms in these fields are nonzero only in a thin layer near the surface of the quark core, so that the magnetic field in both phases is constant and parallel to the axis of rotation.

17. Magnetic field components
finally
From the last equations follows
Thus, the constant in the expression for the magnetic field in the quark core is determined by constant magnetic field B in the hadronic phase,which is generated by protonic vortical filaments.
In this approximation the magnetic field B penetrates from the hadronic phase into the quark phase by means of quark vortical filaments. The transition region is of thickness on the order of

18. Summary
Ginzburg-Landau equations for the magnetic and gluomagnetic gauge fields in a color superconducting core of a neutron star containing a diquark CFL-condensate has been solved. The interaction of the diquark CFL-condensate with the magnetic and gluomagnetic gauge fields has been taken into account in these equations.
The problem was solved subject to the correct boundary conditions: continuity of the components of the magnetic field and the conditions for gluon confinement.
We have also determined the distribution of the magnetic field in the hadronic phase (the npe-phase), taking into account the fact that a magnetic field is generated by the “entrainment” effect of superconducting protons by superfluid neutrons. We have shown that this field penetrates into the quark core by means of quark vortical filaments owing to the presence of screening Meissner currents.