  We derive the spectral indices and their runnings of single inflation models
by a new approach. We perform a dilatation transformation to the linear
cosmological perturbations and derive a current (non-)conservation law. Using
it, we construct a dilatation charge and a Ward-Takahashi identity for the
two-point correlators, and derive two \textit{exact} expressions for the
tree-level spectral indices. First, we apply the slow roll expansion to one of
the exact expressions. We calculate the spectral indices and their runnings up
to the second and the third order of slow roll parameters respectively, with
use of the "horizon crossing formalism". By construction, our results are more
rigorous and generic than the previous works. Then, we analyze another exact
expression to understand how the perturbations and the slow roll parameters
contribute to the spectral indices. By a numerical calculation, we confirm that
only the behaviors of the slow roll parameters during a few e-folds around the
horizon crossing affect significantly to the values of spectral indices. The
analysis in this article indicates that if one cannot use the slow roll
parameters, regardless of their values, as the expansion parameters around the
horizon crossing, then one can no longer apply the slow roll expansion to the
spectral indices, and it is thus necessary to apply the more generic method
introduced here.
