Numerical solution of the color superconductivity gap in a weak coupling constant

I. Zakout; H. R. Jaqaman; W. Greiner

doi:10.1103/PhysRevC.68.034901

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Numerical solution of the color superconductivity gap in a weak coupling constant

I. Zakout, H. R. Jaqaman, W. Greiner

DOI 10.1103/PhysRevC.68.034901 · Physical Review C

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Abstract

We study the numerical solution of hard dense loop (HDL) color superconductivity gap equation in the momentum space. We investigate the validity of various approximations for the HDL gap equation with a weak coupling constant g. We find that the standard approximations to derive the leading gap equation to g-leading order are essential for a secure numerical evaluation of the logarithmic singularity with a small g. The leading integral gap equation with a small g should be inverted to a soft integral gap equation to smear the logarithmic singularity near the Fermi surface. The HDL gap equation is solved for a rather large coupling constant g>~2.0, while the leading and soft integral gap equations are solved for a small coupling constant g<~1. When their solutions near the Fermi surface are extrapolated to larger g values, they coincide with the HDL gap equation. Furthermore, the analytical solution matches the numerical one up to order O(1). Our results confirm the previous estimates derived by other approximations that the gap energy is of the order of 10–100 MeV for a chemical potential μ<~1000MeV. They also support the validity of leading approximations applied to the HDL gap equation near the Fermi surface and a small g to derive the soft integral gap equation and its analytical solution.