293. Power counting semi-classical inflation models and Higgs-Inflation

This is a guest post by Michael Trott from Perimeter Institute. Dmitry.

Thanks Dmitry for offering me the chance to explain the results of a recent paper (arXiv:0902.4465) by Cliff Burgess, Hyun Min Lee and myself. 

In this paper, we developed a power counting formalism to study the (often unspoken) limits of validity of various inflation models and then applied this tool to the examples of Higgs-Inflation and higher curvature inflation. To keep this post  brief,  I will simply set the stage, explain in some detail how the main result of the paper is obtained, and then describe the application to Higgs-Inflation, leaving all other details to the paper.

First the basic ideas. Inflation is now a well accepted paradigm in cosmology. During inflation, the universe underwent a period of accelerated expansion when a Lorentz invariant energy density dominated the equation of state. Inflation is a good theory, it efficiently explains away the flatness, isotropy, homogeneity, horizon and undesired relic problems of the early universe. The growth of quantum fluctuations during inflation seeds the large scale structure that we observe in the universe, and inflation leads qualitatively and quantitatively to the properties of the CMB that are observed by WMAP. One can be rather confident, given this set of facts, that inflation occurred in the early universe.

However, one can also be relatively confident that we really have no singularly good idea as to what underlying physics lead to the inflationary epoch. Every theorist has their favourite model (interestingly, this is usually the particular model that a particular theorist came up with), and broadly speaking the inflationary energy density is generally the energy density of a scalar as it undergoes a classical “slow roll” due to a very flat potential. This scalar is coupled to gravity, and what we first studied was the general power counting of a Lagrangian of such scalars coupled to gravity. 

When power counting, one determines the way that the scales of an effective Lagrangian will enter into a general amplitude. For the Lagrangian describing scalars   coupled to gravity 

we have the following scales; the reduced Plank mass , a potential scale , and the “cutoff scale” . This later scale is present as this is a nonrenormalizable effective Lagrangian that comes about by integrating out some new physics that comes in at the cutoff scale. Generally we don’t know what this scale is. What matters here is that any underlying theory that leads to an effective Lagrangian of this form, and hence an effective theory of inflation, can be studied consistently in this low energy theory. The nonrenormalizability does NOT mean that the theory is meaningless and not predictive, it simply means that the theory can only be used to predict things to a finite accuracy. If you haven’t met this thinking before, this post is not the right forum to discuss this, but many reviews of effective field theory exist in the literature.

From this Lagrangian one can derive that the following power counting formula for an amplitude with  loops and  external legs

Here  is the largest physical scale that appears in the propagators or vertices in the calculation and we have split up contribution to the amplitude from interactions with no derivatives  from the terms with two derivatives  and the terms with four or greater derivatives . This is an unassuming formula, but it is very useful and powerful when studying an inflation model!

The first thing one notices is that for the theory to have a will defined loop expansion one must have 

It is common in the literature to study inflation models at tree level and not worry too much about all the loop corrections that one’s model necessarily leads too. It is dangerous at times to neglect constraints such as (3). Some quantum corrections that are simpler to include can (and are) incorporated in inflation models, especially when the help make the case for the model! However, a systematic study of loop corrections and all quantum effects can be boring and requires either a large number of post docs and students or one concise and powerful power counting formula.

Lets see how this can help one appreciate the strengths or limitations of a model by studying a particular example. Our interest in undertaking this work was sparked in part by the Higgs-Inflation model that was recently put forth by Bezrukov and Shaposhnikov (arXiv:0710.3755). This model is potentially very exciting as it economically uses the Higgs boson non-minimally coupled to gravity through the dimension four term to drive inflation. It is important to note that although this dimension four term is usually neglected when one studies the SM, it certainly exists as it is required when you renormalize the theory is curved space.  The only argument is what is the coefficient. For the Higgs to be the scalar leading to inflation, requiring that the amplitude of primordial fluctuations agree with what is found by WMAP gives

First of all this is very exciting. This model has a beautifully testable consequence in that once the Higgs is found and its mass determined, one can examine the CMB to see if it is consistent that the Higgs has lead to inflation as well as EW symmetry breaking. It is very remarkable that qualitatively the required mass lies within the stability and triviality bounds on the Higgs mass (due to entirely different physics) and is remarkably consistent with global fits to electroweak precision data. This did not have to be and is quite a conspiracy! However, one also notices that for a typical mass   GeV that . That is a troubling coupling to imagine putting in scattering diagrams. Does this theory really make sense? Perhaps it just violates unitarity for the energies of interest and all of this excitement is misplaced.

The power counting formula allows one to  check this unitarity worry directly and efficiently. One can bound the cutoff scale  of the theory by examining energetic graviton-higgs scattering (gh -> gh) or higgs-higgs scattering through graviton exchange in flat space. We use the power counting result to examine when these amplitudes saturate the bound as a function of the loop order . Here  is the center of mass energy of the scattering and the relevant interactions come from the nonminimal coupling term, so . One easily finds that the most problematic amplitude for unitarity for (gh -> gh) scales as 

Demanding that the cross section not saturate the unitarity bound  gives an  dependent upper bound on  (that in turn gives an upper bound on ). This bound is 

Now as the Hubble scale in this theory is  this means that for this model to make sense as an effective theory, the cutoff scale lies in the uncomfortably small window

.

The upper bound comes from the usual constraint of an adiabatic inflationary expansion. This is a small window of validity, although it is logically possible that such a cut off scale exists. This window gets even more uncomfortable  if one has any other new physics coupled to the Higgs, as this scale is extremely unstable when the Higgs couples to any new physics particles.

Notice that this analysis is accomplished without the painful explicit calculation of partial wave scattering amplitudes of graviton Higgs or Higgs-Higgs scattering. If one actually calculates this explicitly, and some brave souls have calculated Higgs-Higgs scattering through graviton exchange (Han and Willenbrock  hep-ph/0404182) before,the unitarity cut off constraint they find agrees with the scaling one obtains directly with the power counting directly and virtually instantaneously.

In summary, if one has a viable inflation model of some scalar(s) couples to gravity through some potential, and you want to check the validity of such a model through examining its quantum corrections and unitarity constraints, power counting is the right way to go about such a check if you don’t want to suffer too much. Eqn. (2) provides a powerful tool to check the quantum consequences of  your inflationary theory. We have used this tool to examine the Higgs-Inflation theory and have found that the required cut off scale must lie in a rather small window for this to be a sensible low energy theory with underlying physics that is unitary. We hope we have provided a useful tool to those who might propose interesting and exciting semi-classical theories of inflation in the future to easily check some quantum consequences of their theory.

Update: Since our paper came out, a follow up paper by Barbon and Espinosa appeared (2 days later!) that agrees with some of our findings but makes the stronger claim that the cutoff scale rules out the higgs inflation scenario entirely (0903.0355). People sure are excited about Higgs inflation!