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mitchell porter

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- TL;DR Summary
- Maybe things are renormalizable up to the Planck scale after all

After the news of Peter Higgs's death, I was thinking of the Higgs field and boson, how central they are to physics now, and the remaining mysteries associated with them.

One such is the meaning of the mass actually observed for the Higgs boson. In the mainstream of theoretical physics, the significance of that mass is that it is light (relative to the Planck scale), which should make it vulnerable to corrections produced by massive virtual particles. The old expectation was that there is a symmetry, such as supersymmetry, which makes most of these corrections cancel out. The combination of a light Higgs boson with the apparent absence of the partner particles implied by such a symmetry, was a big shock to theorists (as well as being the worst-case scenario for experiment - no clear guidance for what lies beyond the standard model), and is arguably still the puzzle that most vexes the theorists (for whom it is often paired with the mystery of the "unnaturally" small-but-nonzero cosmological constant).

Meanwhile, if we look at the specific value of the Higgs mass, it famously also has the consequence that it places the vacuum of the standard model right at the threshold of being a false vacuum. There has inspired speculation that maybe the Higgs mass is somehow a form of self-organized criticality (the prototype of this is the sandpile which has reached maximum height, and in which little avalanches occur when you try to add more sand grains than the maximum will allow). But as far as I know, no one has proposed an actual mechanism for this.

If there's any place where this aspect of the Higgs is illuminated, it might be the early universe. And this brings us to "Higgs inflation" - the idea that the Higgs is also the inflaton, the scalar field required by the theory of cosmic inflation. While this would be economical, it requires a new coupling constant (the coupling between the Higgs and the Ricci scalar curvature) that must also be tuned just right; and in any case, there should also be new self-couplings of the gravitational field (the higher-derivative interactions which make gravity non-renormalizable) that spoil the model.

Or at least, that's what the paradigm of "effective field theory" tells us. The standard model, by itself, is renormalizable to high energies. But the effective field theory paradigm says that we shouldn't just consider these renormalizable terms, we should consider all combinations of the field operators, including non-renormalizable terms that are only meaningful up to a particular energy. These non-renormalizable terms are to be considered as approximations to unknown physics, in the same way that the four-fermion interaction of Fermi's original theory of beta decay, actually arises from the exchange of W bosons. As a result, theorists now routinely consider the "standard model effective field theory", which is the standard model Lagrangian augmented with arbitrarily complex, non-renormalizable multi-particle interactions, each of which has a coefficient that in principles can be measured, to provide information about physics beyond the standard model.

In the same way, it is usually supposed that perturbative quantum gravity involves, not just the interaction terms of the original Einstein field equations, but an infinite series of non-renormalizable interactions involving arbitrary numbers of gravitons, each such term having its own coefficient, whose value derives from the unknown details of the true complete theory of quantum gravity.

Now, effective field theory has proven itself in other contexts, like the interactions of hadrons. But in using EFT to reason about unknown physics, it is often assumed that the coefficients of the new terms are "natural" (of order 1) and independent of each other; and this is precisely the reasoning that led to the expectation of new symmetries and new particles (besides the Higgs) at the electroweak scale. The swampland research program in string theory, is a minor step beyond this philosophy, in that it looks for unexpected constraints on the allowed coefficients of the EFT. Such a program might possibly produce arguments for a light Higgs mass; but I don't see how it could explain the criticality of the Higgs mass.

In any case, until recently I had nothing to say in reply to the EFT argument against critical Higgs inflation. But now I have learned about an old technical result in perturbative quantum gravity, due to K. Stelle, that is receiving attention again. The result is that if you add just R^2 terms to the Einstein field equations, terms that are quadratic in the Ricci scalar, the resulting modified theory of gravity

This modified gravity is now known as "quadratic gravity", and it has had a minor revival of interest as a candidate for quantum gravity, after many years in which Stelle's result was generally ignored, because EFT thinking implies that there will be numerous other non-renormalizable quantum-gravitational interactions too. In particular, John Donoghue, who has done more than anyone else to demonstrate the usefulness of perturbative quantum gravity as an effective theory at low energies, has recently written a few papers on quadratic gravity; maybe he'll talk about it, during his forthcoming appearance on the popular Youtube series, "Theories of Everything".

So this is my first thought in a post-Higgs world: what happens if you consider Higgs inflation in the framework of quadratic gravity? Could the same unknown serendipity that somehow protects the partnerless Higgs from massive quantum corrections, also protect Higgs inflation from non-renormalizable quantum-gravitational corrections? Remember that EFT and the perturbation series are not fundamental, they're an approximation framework that dominates people's thinking because it's hard to calculate any other way.

I have already found one paper that consider Higgs inflation and quadratic gravity together, so that's a start. :-)

"Higgs Inflation as Nonlinear Sigma Model and Scalaron as its σ-meson" by Yohei Ema, Kyohei Mukaida, Jorinde van de Vis (2020)

One such is the meaning of the mass actually observed for the Higgs boson. In the mainstream of theoretical physics, the significance of that mass is that it is light (relative to the Planck scale), which should make it vulnerable to corrections produced by massive virtual particles. The old expectation was that there is a symmetry, such as supersymmetry, which makes most of these corrections cancel out. The combination of a light Higgs boson with the apparent absence of the partner particles implied by such a symmetry, was a big shock to theorists (as well as being the worst-case scenario for experiment - no clear guidance for what lies beyond the standard model), and is arguably still the puzzle that most vexes the theorists (for whom it is often paired with the mystery of the "unnaturally" small-but-nonzero cosmological constant).

Meanwhile, if we look at the specific value of the Higgs mass, it famously also has the consequence that it places the vacuum of the standard model right at the threshold of being a false vacuum. There has inspired speculation that maybe the Higgs mass is somehow a form of self-organized criticality (the prototype of this is the sandpile which has reached maximum height, and in which little avalanches occur when you try to add more sand grains than the maximum will allow). But as far as I know, no one has proposed an actual mechanism for this.

If there's any place where this aspect of the Higgs is illuminated, it might be the early universe. And this brings us to "Higgs inflation" - the idea that the Higgs is also the inflaton, the scalar field required by the theory of cosmic inflation. While this would be economical, it requires a new coupling constant (the coupling between the Higgs and the Ricci scalar curvature) that must also be tuned just right; and in any case, there should also be new self-couplings of the gravitational field (the higher-derivative interactions which make gravity non-renormalizable) that spoil the model.

Or at least, that's what the paradigm of "effective field theory" tells us. The standard model, by itself, is renormalizable to high energies. But the effective field theory paradigm says that we shouldn't just consider these renormalizable terms, we should consider all combinations of the field operators, including non-renormalizable terms that are only meaningful up to a particular energy. These non-renormalizable terms are to be considered as approximations to unknown physics, in the same way that the four-fermion interaction of Fermi's original theory of beta decay, actually arises from the exchange of W bosons. As a result, theorists now routinely consider the "standard model effective field theory", which is the standard model Lagrangian augmented with arbitrarily complex, non-renormalizable multi-particle interactions, each of which has a coefficient that in principles can be measured, to provide information about physics beyond the standard model.

In the same way, it is usually supposed that perturbative quantum gravity involves, not just the interaction terms of the original Einstein field equations, but an infinite series of non-renormalizable interactions involving arbitrary numbers of gravitons, each such term having its own coefficient, whose value derives from the unknown details of the true complete theory of quantum gravity.

Now, effective field theory has proven itself in other contexts, like the interactions of hadrons. But in using EFT to reason about unknown physics, it is often assumed that the coefficients of the new terms are "natural" (of order 1) and independent of each other; and this is precisely the reasoning that led to the expectation of new symmetries and new particles (besides the Higgs) at the electroweak scale. The swampland research program in string theory, is a minor step beyond this philosophy, in that it looks for unexpected constraints on the allowed coefficients of the EFT. Such a program might possibly produce arguments for a light Higgs mass; but I don't see how it could explain the criticality of the Higgs mass.

In any case, until recently I had nothing to say in reply to the EFT argument against critical Higgs inflation. But now I have learned about an old technical result in perturbative quantum gravity, due to K. Stelle, that is receiving attention again. The result is that if you add just R^2 terms to the Einstein field equations, terms that are quadratic in the Ricci scalar, the resulting modified theory of gravity

*is*renormalizable.This modified gravity is now known as "quadratic gravity", and it has had a minor revival of interest as a candidate for quantum gravity, after many years in which Stelle's result was generally ignored, because EFT thinking implies that there will be numerous other non-renormalizable quantum-gravitational interactions too. In particular, John Donoghue, who has done more than anyone else to demonstrate the usefulness of perturbative quantum gravity as an effective theory at low energies, has recently written a few papers on quadratic gravity; maybe he'll talk about it, during his forthcoming appearance on the popular Youtube series, "Theories of Everything".

So this is my first thought in a post-Higgs world: what happens if you consider Higgs inflation in the framework of quadratic gravity? Could the same unknown serendipity that somehow protects the partnerless Higgs from massive quantum corrections, also protect Higgs inflation from non-renormalizable quantum-gravitational corrections? Remember that EFT and the perturbation series are not fundamental, they're an approximation framework that dominates people's thinking because it's hard to calculate any other way.

I have already found one paper that consider Higgs inflation and quadratic gravity together, so that's a start. :-)

"Higgs Inflation as Nonlinear Sigma Model and Scalaron as its σ-meson" by Yohei Ema, Kyohei Mukaida, Jorinde van de Vis (2020)