Helical ordering in the ground state of spin-one color superconductors as a consequence of parity violation

We investigate spin-one color superconductivity of a single quark flavor using the Ginzburg-Landau theory. First we examine the classic analysis of Bailin and Love and show that by restricting to the so-called inert states, it misses the true ground state in a part of the phase diagram. This suggests the use of the more general, noninert states in particular within three-flavor quark matter where the color neutrality constraint imposes stress on the spin-one pairing and may disfavor the symmetric color-spin-locked state. In the second part of the paper we show that, in analogy to some ferromagnetic materials, lack of space-inversion symmetry leads to a new term in the Ginzburg-Landau functional, which favors a spatially nonuniform long-range ordering with a spiral structure. In color superconductors, this new parity-violating term is a tiny effect of weak-interaction physics. The modified phase diagram is determined and the corresponding ground states for all the phases constructed. At the end, we estimate the coefficient of the new term in the free energy functional, and discuss its relevance for the phenomenology of dense quark matter.