Photon Decay: Can High Energy Photons Transform?

[PAllen]

Ok, I will try out a non-perturbative argument that photon decay/splitting is forbidden, based on many of the ideas raised in this thread. See what you think:

1) From the SR argument Bcrowell first implied, and presented explicitly in the paper I posted, we know that Lorentz invariance required decay time to be proportional to energy.

2) However large the constant of proportionality, for any photon, there exists a frame of reference where the photon energy can be made arbitrarily small, and the required decay time arbitrarily fast.

3) Yet QED, which we take to be a consistent, Lorentz invariant theory, requires the decay to be mediated by virtual charged fermions of substantial mass. This further requires that the probability interaction is small (because the available energy can be made arbitrarily small), and the lifetime cannot be arbitrarily fast.

This is a contradiction unless the overall probability is exactly zero.

 

[bcrowell]

Here is a paper that seems relevant: Gelmini et al., "On photon splitting in theories with Lorentz invariance violation," CAP 0506 (2005) 012, http://arxiv.org/abs/hep-ph/0503130

In standard QED the photon splitting process γ → 3γ does not occur because, due to energy and momentum conservation, the three momenta must all be parallel and the amplitude vanishes in this configuration.

The paper is about using the nonobservation of γ → 3γ to put upper limits on Lorentz violation. She has a parameter with units of mass that measures the amount of Lorentz violation, by affecting the momentum-energy relation for photons. The motivation is that you might get this kind of effect at the Planck scale. The predicted half-life is of the form \tau=(\text{unitless})E^{-14}\mu^5m_e^8, where E is the energy of the photon, and my \mu is her M/\xi, the parameter that measures Lorentz violation. The result is not of the form \tau\propto E that is required by Lorentz invariance, presumably because she gets it by assuming Lorentz violation. Because her half-life gets shorter rather than longer with increasing energy, she doesn't get the decay of a photon into an infrared jet in finite time as described in my #35. She takes the tightest empirical bound on the rate to be from TeV gammas from the Crab pulsar. The result is that \mu is constrained to be at least about 10^4 times the Planck mass, whereas you'd kind of expect that if there was vacuum dispersion due to quantum gravity effects, you'd get something on the same order as the Planck mass.

For googling, a good phrase seems to be "photon splitting." You get a ton of hits that aren't vacuum effects, though, such as photons being split by interaction with matter, and photons splitting in strong magnetic fields.

 

[bcrowell]

Another paper: Jacobson et al., "Threshold effects and Planck scale Lorentz violation: combined constraints from high energy astrophysics," Phys.Rev. D67 (2003) 124011, http://arxiv.org/abs/hep-ph/0209264

p. 20:

In the Lorentz invariant case the equations of motion imply that k_a is a null vector and k_a\epsilon^a=0. Energy-momentum conservation then implies that these 4-momenta are all parallel, so being null they are orthogonal to each other and to all the polarization vectors. The rate thus vanishes for two different reasons. First, since the momenta are necessarily all parallel, the phase space has vanishing volume. Second, the rate must be a scalar formed by contracting these four field strengths using only the metric. Any such contraction vanishes since it must involve contractions of the momenta with each other or with the polarizations. Hence the amplitude vanishes. In the case of an odd number of photons, another reason for vansihing of the amplitude is Furry’s theorem, which states that the sum over loops with an odd number of electron propagators vanishes.

 

[bcrowell]

Ok, I will try out a non-perturbative argument that photon decay/splitting is forbidden, based on many of the ideas raised in this thread. See what you think:

1) From the SR argument Bcrowell first implied, and presented explicitly in the paper I posted, we know that Lorentz invariance required decay time to be proportional to energy.

2) However large the constant of proportionality, for any photon, there exists a frame of reference where the photon energy can be made arbitrarily small, and the required decay time arbitrarily fast.

3) Yet QED, which we take to be a consistent, Lorentz invariant theory, requires the decay to be mediated by virtual charged fermions of substantial mass. This further requires that the probability interaction is small (because the available energy can be made arbitrarily small), and the lifetime cannot be arbitrarily fast.

This is a contradiction unless the overall probability is exactly zero.

This sounds right to me. One thing to check about this argument is that it doesn't prove something false in cases where we know that decay of massless particles is possible. In the Fiore and Modanese paper, the only explicit example they offer of a field theory in which massless particles decay is quantum gravity with a positive cosmological constant. This is a theory with only massless particles in it (they allow other massless bosons besides the graviton), so the premise of your point #3 doesn't hold. This is as it should be, and it encourages me to believe that your argument is right.

If we look at the arguments given in the Gelmini and Jacobson papers, they both seem to say that phase space arguments, based on the collinearity of the decay products, are sufficient to prove that photon decay can't happen in standard QED. They make it sound like this argument is sufficient in and of itself, and since the argument doesn't appeal to any features of QED as opposed to quantum gravity, this would seem to contradict Fiore and Modanese's use of quantum gravity as a counterexample. I think what's happening is that in quantum gravity, there is another factor that blows up to infinity so fast that it overwhelms the vanishing phase-space factor. At least, that's my interpretation of the Fiore paper, which I don't claim to understand in detail.

One other thing that seems odd to me about all this is that in a Feynman diagram there's supposed to be a distinction between internal lines and external legs. The external legs are what you can actually observe. They represent particles that fly off and get far enough away from the others that they no longer interact, and they can be detected individually. In the case of massless-particle splitting, it's not clear to me how you make such a distinction. The three decay products fly off in the same direction at the same speed, so in some sense those three "external" legs never actually separate; you could argue that they are internal legs. In fact, splitting, unlike every other form of radioactive decay, can occur and then spontaneously reverse itself. So actually in a field theory that allows splitting, the process of formation of the infrared jet I described in #35 will not run to completion in a finite time in a vacuum, because there will be a tendency for the particles to reverse-decay just as frequently as they decay.

 

[bcrowell]

It looks like I may be the only one still interested in this thread, but anyway I thought I'd post a couple more thoughts and see if anyone has any comments.

There are some odd thermodynamic aspects to all this. Consider a one-dimensional gas of massless particles that can split and unsplit spontaneously. This is different from the usual blackbody problem, because there doesn't have to be a heat bath; the particles can change their frequencies spontaneously, without interacting with the walls of the box. So let's say the box is of length L, is perfectly insulating, and doesn't interact thermally with the particles. We're not going to get the standard blackbody results, because there's no heat bath. Let the total energy of the gas be E=nE_o, where E_o=h/2L is the ground-state energy (c=1). The entropy is S=ln p(n,k), where k is the number of particles, and p(n,k) is the number of partitions of the integer n into k nonzero terms, ignoring order. (Boltzmann's constant=1.) Playing around with Stirling's formula I convinced myself that the entropy was maximized for k\sim\sqrt{n}. So the result is that if you introduce a single short-wavelength particle of energy E into the box, it ends up in a state with a temperature of something like T\sim\sqrt{hE/L}. Letting L approach infinity, T approaches zero, so I think my original idea about the particle spontaneously decaying to an infrared jet was actually correct.

Another strange thing is that if you start off with, say, a Gaussian wavepacket, splitting causes the wavelengths to increase, so the Gaussian has to broaden over time. (The thermodynamic argument above shows that there really is a net broadening. In free space, you never reach an equilibrium between splitting and unsplitting.) That means that the corresponding classical wave equation is dispersive. But unless I'm missing something, you can't have dispersion of the vacuum for massless particles without violating Lorentz invariance. This is of course exactly what is happening in the Gelmini and Jacobson papers: they start by assuming Lorentz violation. This makes me wonder whether Fiore and Modanese's treatment, with Lorentz invariance, is even self-consistent.

For the particle in a box, assuming Lorentz invariance so that \tau\propto E, the rate of decay increases geometrically as the splitting continues, so you reach thermal equilibrium in only a few times the original particle's lifetime. This thermal equilibration time is independent of the size of the box. But this assumes that the initial particle is inserted into the box in an eigenstate of energy, so that its wavepacket fills the entire box. In free space, it's impossible to prepare a particle in an initial state such that it fills the whole infinite "box." An initial wavepacket with finite length will disperse at a rate limited by c. But even so, this seems to go against the third law of thermodynamics. As the jet's temperature approaches zero, its entropy blows up to infinity.

 

[bcrowell]

This recent paper seems relevant: Julian Heeck, How stable is the photon?, http://arxiv.org/abs/1304.2821