Gauge-invariant Lagrangian of the soft-collinear effective theory and its application to soft-collinear currents

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.