  In a series of recent papers Kallosh, Linde, and collaborators have provided
a unified description of single-field inflation with several types of
potentials, ranging from power law to supergravity, in terms of just one
parameter $\alpha$. These so-called $\alpha$-attractors predict a spectral
index $n_{s}$ and a tensor-to-scalar ratio $r$, which are fully compatible with
the latest Planck data. The only common feature of all $\alpha$-attractors is a
non-canonical kinetic term with a pole, and a potential analytic around the
pole. In this paper, starting from the same Einstein frame with a non-canonical
scalar kinetic energy, we explore the case of non-analytic potentials. We find
the functional form that corresponds to quasi-scale invariant gravitational
models in the Jordan frame, characterised by a universal relation between $r$
and $n_{s}$ that fits the observational data but is clearly distinct from the
one of the $\alpha$-attractors. It is known that the breaking of the exact
classical scale-invariance in the Jordan frame can be attributed to one-loop
corrections. Therefore we conclude that there exists a class of non-analytic
potentials in the non-canonical Einstein frame that are physically equivalent
to a class of models in the Jordan frame, with scale-invariance softly broken
by one-loop quantum corrections.
