Cool Pions Move at Less than the Speed of Light

In this proceeding we give a pedagogical explanation of recent work of ours. 1 In this section we show how the velocity of pions, or more generally Goldstone bosons, changes with temperature in a thermal distribution. This in itself is pretty trivial, and familiar from other contexts. For examples, the appropriate analogy to Goldstone bosons in a theory which is relativistically invariant at zero temperature are spin waves in an antiferromagnet. 2,3 This is because the dispersion relation for such spin waves is ω = vp, where ω is the frequency, p the magnitude of the spatial momentum, and so v is the velocity. (Spin waves in a ferromagnet behave as ω ∼ p.) In a relativistically invariant system, of course v = c, but in antiferromagnets, the velocity v is just some parameter that depends upon details of the couplings, etc. Thus for antiferromagnets, it is completely unremarkable that v changes with temperature, since the couplings change with temperature as well. This is in agreement with Goldstone’s theorem, which tells us that for the Goldstone modes, the inverse pion propagator vanishes at zero spatial momentum, p = 0. 4 Thus it is fine if ω vanishes like some constant v times p, where v changes with temperature. Less trivially, one can also understand the damping of pions or spin waves. Even without calculation, we expect that the damping rate vanishes at zero spatial momentum, since by Goldstone’s theorem the inverse pion propagator must vanish at zero momentum, for both the real and imaginary parts. How precisely the imaginary parts, and so the damping rate, vanishes is an interesting question which we face when we get into the gory details below. A similar problem is the propagation of light in a medium with an index of refraction greater than one. The analogy is imprecise, though, because in a thermal bath Debye screening generates a finite correlation length for photons, whereas in the broken symmetric phase, Goldstone bosons always have an infinite correlation length.