  We construct the soft-collinear effective Lagrangian which is manifestly
gauge invariant order by order. Field redefinitions of collinear gauge fields
and a proper decomposition of quark fields are necessary to make the Lagrangian
gauge invariant. We can obtain the effective Lagrangian order by order in
SCET_I and SCET_II by adopting appropriate power counting methods. Various
types of current operators can be investigated using this formalism, and we
choose the soft-collinear current as a specific example to present explicit
radiative corrections. The hard-collinear, collinear, and ultrasoft modes in
SCET_I and the collinear and soft modes in SCET_II reproduce the infrared
divergences of the full theory and the matching can be performed. We show our
results explicitly using two types of regularization schemes. The first scheme
is the dimensional regularization both for the ultraviolet and the infrared
divergences with on-shell particles, and the second scheme is regulating the
infrared divergence with the off-shellness of external particles. We discuss
other types of operators in terms of the two-step matching.
