The Standard Model of Particle Physics is a quantum field theory model that describes the dynamics and interactions of fundamental particles. These particles possess different properties, the most important ones being their mass and their intrinsic angular momentum, called spin. The Standard Model contains matter fermions with spin 1/2 as well as gauge bosons with spin 1. The task of the bosons is to mediate the strong and electroweak forces between the fermions. Less than two years ago, the discovery of the Higgs boson with spin 0, which is responsible for electroweaksymmetry breaking and gives mass to particles, completed the picture: Fundamental particles with spin 0, 1/2 and 1 exist in nature and we know how to describe them in field theory, whether they possess a mass or not.

While giving a consistent description of the strong and electroweak interactions, the Standard Model does not take into account the fourth fundamental force: gravity. The theory that describes gravitational interactions and captures the interplay of energy and spacetime geometry is Einstein’s General Relativity. When viewed as a field theory, it is the unique classical description for a massless spin-2 field with nonlinear interactions. However, unlike the Standard Model, it cannot be promoted to a fundamental quantum field theory but can at most be treated in an effective framework at the quantum level which is valid only at low energies. This means that there must exist an underlying quantum theory which completes General Relativity at high energies. A popular candidate for this purpose is String Theory, but whether it can really act as a “theory of everything” and make observable predictions is still unknown.

But why not be a bit less ambitious? Let us leave the question of a quantum theory for gravity aside for the moment and focus on the classical descriptions of fields with different spin. Together, the field content of the Standard Model of Particle Physics and General Relativity exhausts all possibilities of spin less than or equal to two. The graviton of General Relativity is massless and, as it turns out, it is extremely difficult to change this property of the theory and introduce a mass for the spin-2 field. Until four years ago, the common belief was that it is in fact impossible to make the graviton massive. This is surprizing because for all fields with lower spin the theories for massless and massive degrees of freedom are well-known and there is no obvious reason why there should not exist an equivalent description for massive spin-2 fields. Nonetheless, before the year 2010, the situation was as summarized in this table:

The problem with constructing a theory for a massive spin-2 field is that, in general, including a mass term renders the theory inconsistent: The total energy will not be bounded from below and hence the theory becomes unstable. The degrees of freedom responsible for such an instability have negative kinetic energy and we refer to them as “ghosts”. At the quantum level, such ghost fields cause even greater problems because in their presence the probability interpretation of quantum mechanics breaks down. For these reasons, it is absolutely necessary to avoid ghosts in the theory. In the case of massive spin-2 fields it was known since 1939 how to accomplish this at the linearized level. But the nonlinear theory, i.e. the extension of General Relativity by a mass term, was believed to inevitably suffer from ghost instabilities.

The last four years, however, brought great progress: Conjectures from the 1970’s arguing against the existence of a nonlinear theory for massive spin-2 fields were disproven and a massive version of General Relativity without ghosts was indeed constructed. The resulting theory is often called “Nonlinear Massive Gravity” and it is one of the very few consistent modifications of our standard theory for gravitational interactions. We have come one step closer to understanding the structure of all field theories for spin 2 and lower. The only pieces that we are missing now are the quantum theories for massless and massive spin-2 fields:

You may wonder why we are not simply happy with General Relativity since, after all, there is no direct evidence for the existence of fundamental massive spin-2 particles in nature. This is a valid objection, but one should be aware of the fact that the phenomenology of a massless graviton leaves some rather fundamental questions unanswered, in particular when it comes to cosmology. It is therefore interesting to investigate the effects of massive spin-2 fields and see whether their presence can help with solving problems of the standard theory of gravity.

An example is the possibility of “self-acceleration”: In General Relativity, the accelerated expansion of the Universe is driven by vacuum energy (also referred to as “cosmological constant”), a scenario which comes with a lot of theoretical problems. In its massive version, there is no need for vacuum energy, the mass term itself is able to drive the cosmic acceleration. This does not solve all the problems of the standard massless theory but it slightly improves the situation and it is conceivable that giving mass to the graviton is part of the full solution.

Furthermore, massive spin-2 fields could also be of relevance to the dark matter problem: Only about 20% of all matter in the Universe is of the same form that makes up the sun, the earth and us humans; the rest is an unknown constituent which interacts at most very weakly with visible matter but feels the gravitational force. One could imagine this unknown component to be a massive spin-2 particle which couples to gravity but not (or only very weakly) to matter. Due to the lack of a consistent description of massive spin-2 fields and their gravitational interactions, this option had not been explored previously.

The mathematical structure of consistent Nonlinear Massive Gravity is quite complex and certainly still requires deeper understanding. And the next big step is clearly the development of a consistent quantum theory for massless and massive spin-2 fields in order to put them on the same footing as particles with lower spin. Because, ultimately, as theorists we still want to answer the question: Is there a fundamental theory that offers a unified description for all phenomena in our Universe?

Angnis Schmidt-May (angnis.schmidt-may@fysik.su.se) – PhD at the Physics Department