Non-leptonic weak processes in spin-one color superconducting quark matter

The non-leptonic weak processes $s+u\to u+d$ and $u+d \to s+u$ are known to dominate the dissipation mechanism responsible for the viscosity of strange quark matter in its normal phase. The rates of such processes remain unknown for many color superconducting phases of quark matter. In this paper, we partially fill up the gap by calculating the difference of the rates of the two non-leptonic weak processes in four transverse spin-one color superconducting phases of quark matter (slightly) out of $\beta$-equilibrium. The four phases studied are the color-spin locked phase, the polar phase, the planar phase and the {\em A}-phase. In the limit of vanishing color superconducting gap, we reproduce the known results in the normal phase. In the general case, the rates are suppressed relative to the normal phase. The degree of the suppression is determined by the structure of the gap function in momentum space, which in turn is determined by the pairing pattern of quarks. At low temperatures, the rate is dominated by the ungapped modes. In this limit, the strongest suppression of the rate occurs in the color-spin-locked phase, and the weakest is in the polar phase and the {\em A}-phase.