  We study the magnetic excitation spectra in L-edge resonant inelastic x-ray
scattering (RIXS) from undoped cuprates. We analyze the second-order dipole
allowed process that the strong perturbation works through the intermediate
state in which the spin degree of freedom is lost at the core-hole site. Within
the approximation neglecting the perturbation on the neighboring sites, we
derive the spin-flip final state in the scattering channel with changing the
polarization, which leads to the RIXS spectra expressed as the dynamical
structure factor of the transverse spin components. We assume a spherical form
of the spin-conserving final state in the channel without changing the
polarization, which leads to the RIXS spectra expressed as the 'exchange'-type
multi-spin correlation function. Evaluating numerically the transition
amplitudes to these final states on a finite-size cluster, we obtain a sizable
amount of the transition amplitude to the spin-conserving final state in
comparison with that to the spin-flip final state. We treat the itinerant
magnetic excitations in the final state by means of the 1/S-expansion method.
Evaluating the higher-order correction with 1/S, we find that the peak arising
from the one-magnon excitation is reduced with its weight, and the continuous
spectra arising from the three-magnon excitations come out. The interaction
between two magnons is treated by summing up the ladder diagrams. On the basis
of these results, we analyze the L_3-edge RIXS spectra in Sr_2CuO_2Cl_2 in
comparison with the experiment. It is shown that the three-magnon excitations
as well as the two-magnon excitations give rise to the intensity in the high
energy side of the one-magnon peak, making the spectral shape asymmetric with
wide width, in good agreement with the experiment.
