  We study the multifield inflationary models where the cosmological
perturbation is sourced by light scalar fields other than the inflaton. The
corresponding perturbations are both scale invariant and special conformally
invariant. We exploit the operator product expansion technique of conformal
field theories to study the inflationary correlators enjoying the symmetries
present during the de Sitter epoch. The operator product expansion is
particularly powerful in characterizing inflationary correlation functions in
two observationally interesting limits, the squeezed limit of the three-point
correlator and the collapsed limit of the four-point correlator. Despite the
fact that the shape of the four-point correlators is not fixed by the
symmetries of de Sitter, its exact shape can be found in the collapsed limit
making use of the operator product expansion. By employing the fact that
conformal invariance imposes the two-point cross-correlations of the light
fields to vanish unless the fields have the same conformal weights, we are able
to show that the Suyama-Yamaguchi inequality relating the coefficients $f_{\rm
NL}$ of the bispectrum in the squeezed limit and $\tau_{\rm NL}$ of the
trispectrum in the collapsed limit also holds when the light fields are
intrinsically non-Gaussian. In fact, we show that the inequality is valid
irrespectively of the conformal symmetry, being just a consequence of
fundamental physical principles, such as the short-distance expansion of
operator products. The observation of a strong violation of the inequality will
then have profound implications for inflationary models as it will imply either
that multifield inflation cannot be responsible for generating the observed
fluctuations independently of the details of the model or that some new
non-trivial degrees of freedom play a role during inflation.
