We study the momentum and frequency dependence of the dynamical spin
susceptibility in the superconducting state of bilayer cuprate superconductors.
We show that there exists a resonance mode in the odd as well as the even
channel of the spin susceptibility, with the even mode being located at higher
energies than the odd mode. We demonstrate that this energy splitting between
the two modes arises not only from a difference in the interaction, but also
from a difference in the free-fermion susceptibilities of the even and odd
channels. Moreover, we show that the even resonance mode disperses downwards at
deviations from Q=(Ï€,Ï€). In addition, we demonstrate that there
exists a second branch of the even resonance, similar to the recently observed
second branch (the Q∗-mode) of the odd resonance. Finally, we identify the
origin of the qualitatively different doping dependence of the even and odd
resonance. Our results suggest further experimental test that may finally
resolve the long-standing question regarding the origin of the resonance peak.Comment: 8 pages, 5 figure