  Attention has focussed recently on models of inflation that involve a second
or more fields with a mass near the inflationary Hubble parameter $H$, as may
occur in supersymmetric theories if the supersymmetry-breaking scale is not far
from $H$. Quasi-single-field (QsF) inflation is a relatively simple family of
phenomenological models that serve as a proxy for theories with additional
fields with masses $m\sim H$. Since QsF inflation involves fields in addition
to the inflaton, the consistency conditions (ccs) between correlations that
arise in single-clock inflation are not necessarily satisfied. As a result,
correlation functions in the squeezed limit may be larger than in single-field
inflation. Scalar non-Gaussianities mediated by the massive isocurvature field
in QsF have been shown to be potentially observable. These are especially
interesting since they would convey information about the mass of the
isocurvature field. Here we consider non-Gaussian correlators involving tensor
modes and their observational signatures. A physical correlation between a
(long-wavelength) tensor mode and two scalar modes (tss), for instance, may
give rise to local departures from statistical isotropy or, in other words, a
non-trivial four-point function. The presence of the tensor mode may moreover
be inferred geometrically from the shape dependence of the four-point function.
We compute tss and stt (one soft curvature mode and two hard tensors) bispectra
in QsF inflation, identifying the conditions necessary for these to "violate"
the ccs. We find that while ccs are violated by stt correlations, they are
preserved by the tss in the minimal QsF model. Our study of primordial
correlators which include gravitons in seeking imprints of additional fields
with masses $m\sim H$ during inflation can be seen as complementary to the
recent "cosmological collider physics" proposal.
