The observed Higgs boson mass is at an interesting place in parameter space, placing the standard model electroweak vacuum right at the edge of metastability. Among the proposed explanations of this value is the existence of a "shift symmetry" in the Higgs sector.
Today we have the possible detection of gravitational waves produced during inflation, of an amplitude suggesting that the inflationary potential was flat right up to near the Planck scale. Various expert commentaries (McAllister, Reece) say that one would expect interactions between the inflaton and the ultraheavy degrees of freedom to appear, and their absence might, once again, imply the existence of a shift symmetry.
One of the many many approaches to inflation is "Higgs inflation": the Higgs field also serves as the inflaton field, the source of inflation. In discussions of Higgs inflation, I've been told that it's an unlikely model for exactly the same reason as mentioned above - effective field theory ought to break down near the Planck scale, the flatness of the potential should be disturbed by ultraheavy interactions, Higgs inflation would require finetuning.
But now we may have evidence of such finetuning in inflation! Or perhaps, evidence of a protective symmetry. So my question is, Could the same symmetry be finetuning both the Higgs mass and the inflaton's interactions? Could the same shift symmetry make the Higgs mass critical and protect the inflaton from ultraheavy interactions?
These questions might be asked first in the context of basic Higgs inflation - only one fundamental scalar - and later in the context of a multi-scalar theory, in which the Higgs is part of a larger scalar sector with a single big potential.
edit: I have found a discussion of shift symmetry in the context of Higgs inflation, but it was written prior to the measurement of the Higgs mass.