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.
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