Surface tension of highly magnetized degenerate quark matter

We study the surface tension of highly magnetized three-flavor quark matter within the formalism of multiple reflection expansion. Quark matter is described as a mixture of free Fermi gases composed of quarks $u$, $d$, $s$ and electrons in chemical equilibrium under weak interactions. Due to the presence of strong magnetic fields the particles' transverse motion is quantized into Landau levels, and the surface tension has a different value in the parallel and transverse directions with respect to the magnetic field. We calculate the transverse and longitudinal surface tension for different values of the magnetic field and for quark-matter drops with different sizes, from a few fm to the bulk limit. For baryon number densities between 2 to 10 times the nuclear saturation density, the surface tension falls in the range of 2 to 20 $\mathrm{MeV}/{\mathrm{fm}}^2$. The largest contribution comes from strange quarks which have a surface tension an order of magnitude larger than the one for $u$ or $d$ quarks and more than two orders of magnitude larger than for electrons. Our results show that the total surface tension is quite insensitive to the size of the drop. We also find that the surface tensions in the transverse and parallel directions are almost unaffected by the magnetic field if $eB$ is below $\sim 5\times {10}^{-3}{\mathrm{GeV}}^2$. Nevertheless, for higher values of $eB$, there is a significant increase in the parallel surface tension and a significant decrease in the transverse one with respect to the unmagnetized case.

Figure 1

The density of states ${\rho }_{\text{MRE}}$ as a function of $k_z$ for the strange quark, using $V/S=10,100,1000$ fm. In panel (a) we show the results for the lowest Landau level $\nu =0$. In such case the curves are the same for any value of the magnetic field. In panel (b) we consider $\nu =1$ with $eB=0.05{\mathrm{GeV}}^2$ (thick curves) and $eB=0.5{\mathrm{GeV}}^2$ (thin curves).

Figure 2

Surface tension for quarks $s$ (a) in the parallel direction and (b) in the transverse direction. The results are shown for drops with $V/S=10\mathrm{fm}$, 100 fm and for the bulk limit with $V/S=\infty$. The magnetic-field intensities are $e{B}_1=5\times {10}^{-4}{\mathrm{GeV}}^2$, $e{B}_2=5\times {10}^{-3}{\mathrm{GeV}}^2$, and $e{B}_3=5\times {10}^{-2}{\mathrm{GeV}}^2$.

Figure 3

Surface tension for quarks $u$ and $d$ in the parallel direction [panels (a) and (b)] and in the transverse direction [panels (c) and (d)]. The values of $eB$ are the same as in the previous figure.

Figure 4

Parallel surface tension for electrons. The transverse surface tension is not shown because it is negligible.

Figure 5

Total surface tension obtained by summing the contributions of quarks $u$, $d$, $s$ and electrons (a) in the parallel direction and (b) in the transverse direction.