What the Heeck: How Stable is the Photon?

I am not a particle physicist or a cosmologist but I would like to talk about a particle cosmology paper I read a while back that I found quite interesting and readable. It's an older paper, but I just started this blog.

The paper "How Stable is the Photon?" by Julian Heeck was published in Physical Review Letters (PRL), one of the top physics journals, in 2013. A free version can be found on arXiv. The first sentence of the abstract is "Yes, the photon." which suggested to me that it was going to be my kind of paper.

As far as we know, photons do not decay. This is related to the fact that light has no mass, and there's nothing it can decay into: heavier particles decay into lighter particles, but nothing is lighter than massless light. Heeck points out that the masslessness and stability of light need not only be assumptions, they can be tested empirically.

What would massive decaying light imply experimentally? The electrostatic interaction, that decays with a 1/r Coulomb potential, can be thought of as mediated by the exchange of virtual photons, and if those photons had mass and decayed, the force would fall more quickly (I believe it would be described as a Yukawa potential, which decays as e^-r/r but I could be wrong here). So, one could do an experiment measuring the force between two charges at varying distance, plot a graph of force vs distance, and perform a power-law fit to measure if the exponent is consistent with -2 or not. If it isn't, that hints at a photon mass, and if it is, the error bars on that exponent put limits on the photon mass. A lot of physics doesn't ask the question "how big is this thing" but rather "how big could this thing be, and still not be detected by our experiment?" Williams et al. did a fancier version of this experiment, and found that the exponent is within 3x10^-16of 2, or to write it out in a silly manner they found that the Coulomb force scaling exponent was between -1.99999999999999996 and -2.00000000000000027. This put a bound on the photon mass of less than 10^-50 kg or 10^-15 eV. More recent measurements push that down a few orders of magnitude, but needless to say the photon mass is very small, if it exists.

Theoretically, a photon mass implies that electromagnetism is described by the Proca rather than Maxwell equations. It also has the strange implication that light does not actually travel at the speed of light, which seems tautologically false. It would imply that the-speed-that-light-travels and the-speed-that-is-fixed-in-special-relativity are not the same, and that light now has a rest frame.

If light could decay, what would it decay into? Heeck mentions that, without introducing new particles beyond the Standard Model, it could decay into the lightest neutrino without violating what is known about their oscillations (which depend on the differences in mass between the three types). The upper bound of the sum of three masses is known (about 0.1 eV), but the lower bound and the breakdown of each mass are not known. In Heeck's decaying photon scenario, the photon can decay into two of the lightest neutrinos without violating conservation of energy. This decay would be very unlikely to occur, to which Heeck declares "Still, unmeasurable small SM [Standard Model] rates never stopped anyone from looking for a signal, as it would be a perfect sign for new physics."

So if photons had mass and could decay into neutrinos, how would we check if that were actually happening? Light is so light that it barely ever decays, and it's so fast that its lifetime would be extended further by time dilation. Heeck's solution is to look at the light that has been travelling the longest: the cosmic microwave background (CMB). Four hundred thousand years after the big bang, the universe became cool enough for hydrogen to form, which made it transparent to light. At the time, the gas in the universe was hot enough to emit visible thermal radiation, which and 13 billion years later some of that reaches the Earth from all directions, having been shifted into the microwave part of the spectrum by the expansion of the universe. The spectrum of the CMB is still consistent with thermal radiation (before the COBE satellite confirmed this in the 1990s, a number of PhD theses were written explaining why it was not thermal, or so my undergraduate relativity professor told me), and possible deviations from this could be signs of photon decay.

The bulk of the "theory" in Heeck's paper is a re-derivation of the Planck spectrum that includes the photon mass and the expansion of the universe. The new spectrum deviates from the classic Planck spectrum in the lower-energy part of the spectrum, where fewer photons are now expected (my understanding of the paper is that the Lorentz factor of the lower-energy photons is smaller, so the lifetime in the low-energy regime isn't as dilated). Heeck fits the CMB spectral data to his new equation with the mass and decay rate as free parameters. Unsurprisingly, he finds that both are consistent with zero. However, much like the deviations from 2 in the Coulomb exponent, fitting to the CMB data can put constraints on the photon lifetime. Heeck finds that if photons can decay, they must have a lifetime in their rest frame exceeding three years. Because they are going so fast, this is extended quadrillion times (10¹⁵) by time dilation, making their mean lifetime much longer than the age of the universe.

Heeck summarizes: "In conclusion, a massive photon sounds crazy and exotic, but it really is not. A massless photon is neither a theoretical prediction nor a necessity, but rather a phenomenological curiosity." I liked this paper because it used a clever analysis to test something everybody had been taking for granted and gets a concrete result, and in general it was a pleasure to read and I learned a lot. Reading it makes me wish I knew more about particle physics so I could play around with stuff like this.