  We present a microscopic theory of spin-dependent motive force ("spin motive
force") induced by magnetization dynamics in a conducting ferromagnet, by
taking account of spin relaxation of conduction electrons. The theory is
developed by calculating spin and charge transport driven by two kinds of gauge
fields; one is the ordinary electromagnetic field $A^{\rm em}_{\mu}$, and the
other is the effective gauge field $A^{z}_{\mu}$ induced by dynamical magnetic
texture. The latter acts in the spin channel and gives rise to a spin motive
force. It is found that the current induced as a linear response to
$A^{z}_{\mu}$ is not gauge-invariant in the presence of spin-flip processes.
This fact is intimately related to the non-conservation of spin via Onsager
reciprocity, so is robust, but indicates a theoretical inconsistency. This
problem is resolved by considering the time dependence of spin-relaxation
source terms in the "rotated frame", as in the previous study on Gilbert
damping [J. Phys. Soc. Jpn. {\bf 76}, 063710 (2007)]. This effect restores the
gauge invariance while keeping spin non-conservation. It also gives a
dissipative spin motive force expected as a reciprocal to the dissipative spin
torque ("$\beta$-term").

Mesoscale and Nanoscale Physics