He explains that "a massless scalar field with a quartic interaction in [de Sitter] space develops a mass. . . . A self-interacting scalar field has the property to get mass by itself."

"de Sitter space" is a space-time in which special relativity applies, and is a background upon which the Standard Model of Particle Physics can be formulated that is more general than the usual Minkowski space (where only special relativity applies), but is symmetrical and lacks the mass-energy fields of general relativity; it is a particular vacuum solution of the equations of general relativity.

Instead, Frasca argues, the mathematics imply that there must also exist other, higher energy, excited states of the scalar field in addition to the Higgs boson observed so far. In other words, there must be higher energy versions of the Higgs boson in the kind of scalar field that it generates. He summarizes his argument by stating that:

[I]f we limit all the analysis to the coupling of the Higgs field with the other fields in the Standard Model, this is not the best way to say we have observed a true Higgs particle as the one postulated in the sixties. It is just curious that no other excitation is seen beyond the (eventually cloned) 126 GeV boson seen so far but we have a big desert to very high energies. Because the very nature of the scalar field is to have massive solutions as soon as the self-interaction is taken to be finite, this also means that other excited states must be seen.Frasca's observation goes beyond the canonical description of the Higgs boson, but isn't precisely beyond the Standard Model physics either. A better way to describe his observation would be to say that it is a non-canonical analysis of how Standard Model physics plays out that makes predictions that have not yet been observed and not yet achieved consensus status among physicists.

**Is The de Sitter Space Assumption An Important Loophole To Frasca's Conclusion?**

There is a loophole in Frasca's analysis, however. His analysis and that of the other paper he cites in support of his conclusion both assume a background of de Sitter space, as is natural and usually done without comment of any kind in quantum mechanics.

But, we don't live in de Sitter space. We live in a world where the entire background independent formulation of Einstein's theory of general relativity applies to a universe full of matter and energy.

Put another way, our world has stuff in it, while de Sitter space doesn't, and that might be relevant to a mechanism that has a fundamental role in giving rise to mass, which is a quantity upon which gravity acts that exists only outside de Sitter space which assumes away its existence, at least as a first order approximation. Some of the relevant distinctions are explored here.

It is possible that while a self-interacting massless scalar field does acquire mass and does imply the existence of excited states of the Higgs boson in de Sitter space, that this conclusion does not in fact hold in the space-time of the asymmetrical, mass filled universe version of general relativity in which we actually live. Indeed, the absence of of excited states of the Higgs boson could be a clue that could point physicists in the right direction when developing a coherent theory of quantum gravity.

I have no particularly good reason to think that Frasca's result shouldn't generalize, other than that we do not observe the Higgs field at its vacuum expectation value acquiring mass in real life (with the possible exception of dark energy which is many, many orders of magnitude too small for this conclusion to hold). So, if there is no flaw the mathematical reasoning that Frasca employs as he reaches his conclusion in de Sitter space, perhaps this counterfactual assumption about the nature of space-time does matter in some way. Given that the Higgs vev, by definition, permeates all of space-time and plays a fundamental role in giving rise to mass, this isn't such a far fetched possibility.

In general, no mathematically tractable theory of quantum gravity that can clear this kind of hurdle has been formulated, so it is impossible to say, in general, how a quantum mechanics based conclusion would play out in a context in which gravity could play an important role.

In similar situations (e.g. the quantum mechanics that take place at the event horizon of a black hole), physicists develop an ad hoc ansatz to deal with possible quantum gravity issues, and to determine where quantum gravity considerations might be relevant, on a case by case basis without the benefit of a full theory of quantum gravity.

**Could Excited Higgs Boson States Be Unattainably Heavy?**

It is also worth noting that since Frasca is discussing the qualitative properties of otherwise massless scalar fields that are self-interacting, his post, at least, does not predict any particular mass for an excited state of a Higgs boson. He simply predicts that they exist.

Thus, the loophole that he affords himself is that an excited state of the Higgs boson might exist, but it might be so heavy that it will never be realized outside of Big Bang conditions.

Indeed, if this were the case, excited Higgs bosons might play a role in making possible forms of baryongenesis and leptogenesis that cannot be achieved at sufficient rates to explain the presence of mass in the universe without them.

**Background On Scalar, Vector and Tensor Fields**

Spin-0 particles are either "scalar" or "pseudo-scalar" depending upon their parity and create "scalar fields" (i.e. fields described at any given point in space-time by a single number). Spin-1 bosons are called "vector bosons" and create "vector fields" (i.e. fields described at any given point in space-time by a directional arrow and the magnitude of that arrow, such as electromagnetic fields). And, spin-2 particles (such as the hypothetical graviton) are "tensor" bosons that create "tensor" fields (i.e. fields described at any given point in space-time by a matrix of numbers of the kind found in general relativity).

Since the Higgs boson is the only spin-0 particle in the Standard Model, as all other Standard Model bosons have spin-1, this issue doesn't crop up anywhere else in the Standard Model. Further, electromagnetic forces is not self-interacting. Only the Higgs boson generates a self-interacting scalar field.

Thus, Frasca's analysis does not imply that there must be a more massive excited version of the W or Z boson, a more massive excited version of the gluon, or a more massive excited version of the photon (although his reasoning is simply silent on these points and doesn't rule them out either).

## 2 comments:

There is no way that Frasca is right on this, and it underlines how unreliable he is as a theorist.

Hi ohwilleke,

Thanks a lot for this post. No, there is no loophole. A scalar field gets mass in a Minkowski space by self-interaction. The presence of a de Sitter background just provides an energy scale by the Hubble constant that would be arbitrary otherwise. It is the same result obtained in such different circumstances that makes the argument compelling.

Regards,

Marco

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