Exploring Higgs Inflation After Moriond & Planck

[fzero]

The authors write as if this finding falsifies this version of Higgs inflation. But doesn't it just mean that the Higgs-Ricci coupling ζ is going to be a prediction, rather than a free parameter? The authors note (top of page 4) that Higgs inflation, for a critical Higgs, can work with quite a small value of ζ; one might view this as improving the model's prospects... I should also note that for the simpler case where there's just a Higgs, and no fermions, ζ is relevant and therefore a tunable parameter. It's possible that this is also true for the full standard model, or some extension of it. Further calculation is required.

But what intrigues me most is the possibility that there is a connection between Higgs criticality and finetuning of Higgs inflation, and that for some reason the unknown UV physics actually determines that the Higgs-Ricci coupling is appropriately tuned.

I can explain a bit of the generalities of RG flows that are assumed by their discussion and not explicitly spelled out. Let's say we have some conformal fixed point. Usually we think of the fixed point as an explicit point in the coupling constant space of the theory. If we have a Lagrangian description of the theory, then we could specify an action $S$ at the fixed point. In addition we would have a collection of local operators $\mathcal{O}_i$ constructed from the fields in the theory. Usually the RG flows are generated by deforming the theory by adding a local operator to the action, resulting in a new theory
$$S' = S + \int g_i \mathcal{O}_i.$$

The RG equations will tell us how the parameters $g_i$ transform with the scale. The resulting trajectory in the coupling constant space is called the RG flow. The place to start is for infinitesimal $g_i$, so we start in a small neighborhood of the fixed point and the RG equations will specify critical exponents for the scale dependence. We also need to have these flows start in a small neighborhood if we want to discuss flows that are smoothly connected to the fixed point. This leads to the picture of critical surfaces as discussed in the paper.

The deformation above will generate a one-dimensional flow, where, of course, we are always flowing from the UV to the IR. For an irrelevant deformation, the coupling constant is getting smaller as we flow to the IR, so the theory is actually returning to the fixed point, with $g_i=0$ . Conversely, for a relevant deformation, the coupling constant grows as we move to longer scales, so we get a new theory.

If we consider a two-dimensional deformation by adding both an relevant and irrelevant deformation, $g_r \mathcal{R} + g_i \mathcal{I}$, then we can think of the local geometry of the fixed point in the following way. We have axes for the coupling constants $(g_r,g_i)$ with the fixed point at the origin and we consider the quadrant between the axes (since we'll ignore the actual sign of the coupling constants). The RG equations kind of dictate a "potential energy" function (in 2d this would be the $c$-function) that pushes the flows. In the irrelevant direction $g_i$, this potential function is increasing, so we are pushed back to the origin. In the $g_r$ direction, the function is decreasing and some RG flow is allowed. In between, the function must be sloping down from the $g_i$ axis to the $g_r$ axis, so at least for small deformations, the flow from a given point $(g_r,g_i)$ is to a point $(g_r',0)$ and then further flows will take place along the $g_r$ axis.

So what the paper is saying is that the theories in the IR with nonzero nonminimal coupling cannot be smoothly connected to the UV fixed point.