We discuss the cooling of hybrid stars by considering the neutrino emission
from quark matter. As a current topic the appearance of various inhomogeneous
chiral phases have been studied near the chiral transition. Here we consider
the dual-chiral-density-wave (DCDW) specified by the spatially modulated quark
condensates. Since the DCDW state can be represented as a chirally rotated
state from the normal quark matter, the quark weak-current is accordingly
transformed to have an additional phase factor which modifies the
energy-momentum conservation at the vertex, and makes the quark direct Urca
process possible. The direct evaluation of the neutrino emissivity shows that
it is proportional to and the magnitude is comparable with the quark or pion
cooling. Since the DCDW phase develops only in the limited density region, this
novel mechanism may give an interesting scenario about cooling of hybrid stars
that lower-mass stars should be cooler than higher-mass ones.Comment: 6 pages, 3 figures, Proc. of XII Int.Sympo. on Nuclei in the Cosmo