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Light Stückelberg bosons deported to the swampland

Conjecture would also imply that photons have to be strictly massless

I am rather happy about the following new hep-th preprint that adds 21 pages of somewhat nontrivial thoughts to some heuristic arguments that I always liked to spread. Just to be sure, Harvard's Matt Reece released his paper

Photon Masses in the Landscape and the Swampland
What's going on? Quantum field theory courses usually start with scalar fields and the Klein-Gordon Lagrangian. At some moment, people want to learn about some empirically vital quantum field, the electromagnetic field, whose Lagrangian is\[

{\mathcal L}_\gamma = -\frac 14 F_{\mu\nu} F^{\mu\nu}.

\] The action is invariant under the \(U(1)\) gauge invariance which is why 3+1 polarizations of the \(A_\mu\) field are reduced to the \((D-2)\) i.e. two transverse physical polarizations of the spin-1 photon. Are there also massive spin-one bosons?



Yes, there are, e.g. W-bosons and Z-bosons that were discovered at CERN more than 30 years ago. The addition of masses naively corresponds to a simple mass term\[

{\mathcal L}_{\rm mass} = \frac {m^2}{2} A_\mu A^\mu.

\] A problem is that this term isn't gauge-invariant. So the theory must be defined without the gauge invariance and we can't consistently reduce the 3+1 polarizations (including one, time-like polarization that has the wrong sign of the norm so it would lead to negative probabilities) to 3 (for a massless photon, 2) polarizations.

However, the Standard Model allows massive spin-1 bosons by the Higgs mechanism. The fundamental Lagrangian actually is gauge-invariant and the gauge-invariance-violating mass term above isn't included directly. Instead, it is generated from the Higgs field's vacuum expectation value \(\langle h\rangle = v\) through the interactions of the gauge field \(W_\mu\) or \(Z_\mu\) with the Higgs field that is included in the Higgs boson's kinetic term \(\partial_\mu h \cdot\partial^\mu h\) once the partial derivatives are replaced with the covariant derivatives. These covariant derivatives \(D_\mu=\partial_\mu - i g A_\mu\) are not only allowed but needed to construct gauge-invariant kinetic terms



So the W-bosons and Z-bosons get their masses via the interaction with the Higgs boson (that's also true for the fermions – leptons and quarks). This is the pretty way to generate masses of spin-1 bosons. It is exploited by the Standard Model and the Higgs mechanism is the last big clear discovery of experimental particle physicists. So massive gauge bosons automatically point to the Higgs mechanism.

But then there's the "ugly" way – and I've always considered it an ugly way – to make spin-1 bosons massive, the Stückelberg mechanism. The mass term for the photons is rewritten as\[

{\mathcal L}_{\rm mass} = \frac 12 f_{\theta}^2 (\partial_\mu \theta - eA_\mu)^2.

\] We added a new scalar field \(\theta\) and preserved the gauge invariance \(A_\mu\to A_\mu +(1/e)\partial_\mu \alpha\) but the new scalar field must also transform under it, \(\theta\to \theta+\alpha\). Because we have the same "amount" of gauge invariance as we have in the massless photons, but there is one scalar field added, we end up with 3 physical polarizations of the massive particle instead of the massless photon's two polarizations. They're the ordinary three spatial or transverse polarizations of a massless vector particle, \(x,y,z\).

One may gauge-fix the Stückelberg action by setting \(\theta=0\) which reduces the system to the Proca action for the "regular" massive spin-1 boson. But the advantage of the Stückelberg form is that you know how to write down the field's interactions with others in a gauge-invariant way.

The mass of the (Swiss) Ernst Stückelberg's boson is \(m_A = ef_\theta\). You may send it zero either by sending the gauge coupling \(e\to 0\) or sending \(f_\theta\to 0\) or some combination of both. Note that \(e\to 0\) is something that the weak gravity conjecture labels dangerous and, under certain assumptions, forbidden. OK, this kind of a description of a massive spin-1 boson doesn't seem to be exploited by the Standard Model. It's ugly because the scalar field transforms in a suicidal way and the theory doesn't point to any non-Abelian gauge symmetry and other pretty things.

In principle, people would always say that the photon that we know and love (and especially see) can in principle be massive, thanks to a Stückelberg mechanism. Well, I always protested when someone presented it as a real possibility. If a photon were massive, we still know that the mass must be much smaller than the inverse radius of the Earth – because we know that the magnetic fields around the Earth behave as those in the proper massless electromagnetism, not in some Proca-Yukawa way. And if the photon were massive but this light, it would at least amount to a new, unsubstantiated fine-tuning. It's more likely and we are encouraged to assume that the photon is exactly massless.

Reece places this "negative sentiment" of mine into a potentially axiomatic if not provable framework. He argues that the limit of the very light photon is "very far in the configuration space" and in consistent theories of quantum gravity, the swampland reasoning implies the existence of some light enough particles (well, a whole tower of them) and/or other reasons why the effective field theory has to break at relatively low energy scales. Quantitatively, Reece claims that the effective field theory has to break above\[

\Lambda_{UV} = \sqrt{ \frac{m_\gamma M_{\rm Planck}}{e} }.

\] Well, the theory would have to break down earlier, at the scale \(e^{1/3} M_{\rm Planck}\), if the latter scale were even lower. At any rate, using the scale in the displayed equation above, we know that the photon mass is rather tiny (recall my comments about the geomagnetic field etc.) and the geometric average with the Planck mass sends us to an atomic physics scale where QED still seems OK, and that's how the massive photon hypothesis could be strictly refuted.

We're not quite sure about any of these swampland-based principles but I tend to think that many of them, when properly formulated, are right and powerful. I find this picture intriguing. Lots of the constructions in effective field theory, like the Stückelberg masses, looked ugly and heuristically "less consistent" to the people who had as good a taste as your humble correspondent. Finally, we may be becoming able to clearly articulate the arguments showing that this "feeling of reduced consistency" is not just some emotion. When coupled to quantum gravity, these ugly scenarios could indeed be strictly forbidden.

Quantum gravity and/or string theory could only allow the solutions that seemed "more pretty" than their ugly competitors. And you could stop issuing politically correct disclaimers such as "we are assuming that the photon mass is exactly zero; if it had a nonzero mass, we would have to revise the whole analysis".

Reece paper has no direct relationship to the de Sitter vacua and the cosmological controversies. But if it's right or at least accepted, it clearly strengthens the Vafa Team in that dispute. There are really different sketches of the general spirit of the stringy research in the future. In Team Stanford's plan, we're satisfied with some Rube Goldberg-style construction, we don't know which one (or which class) is the right one, we get used to it, and we train ourselves to be happy that we won't learn anything new.

On the other hand, in Team Vafa's plan for the future, string theory research continues to do actual progress, trying to answer well-defined questions about the world around us that weren't previously answered, such as "Can some massive bosons we will produce have Stückelberg masses? Is our photon allowed by string theory to be massive?" Truly curious physicists simply want new answers like that to be found. It may be impossible to answer some of these questions, especially if our vacuum is a relatively random one in a set of vacua that have different properties. But this possibility is not a proven fact and even if it is true for some properties, it is not true for all questions.

We can't ever accept the belief that all questions that haven't been answered so far will remain unanswered forever! That would be a clear religious attitude that stops progress in science – and that could have stopped it at every moment in the past. Harvard's Reece sketched some arguments that may prohibit Stückelberg masses in quantum gravity and you – I am primarily talking about you, dear reader in Palo Alto – should better think about it and decide whether he's right or not.

In some technical questions within the de Sitter controversy, I am uncertain, and so are others. But I am certain about certain principles of the scientific method. The real pleasure of science is to find ways to answer questions – to discriminate between possible answers. Many people in Northern California (which includes Palo Alto) may have adopted a non-discrimination approach to society and science (all people and answers and vacua are equally good) – but without discrimination, there is no science.

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