Why can we approximate massive particles as massless or vice versa?

Question

Our descriptions of massless and massive particles are very different. For example:

• Massless particles have only two polarizations, which we call helicities. Spin projection on axes different than that of momentum is not defined.
• Yang-Mills fields come equipped with a gauge symmetry group, which would be ruined if we added mass.

However, when considering processes at energy scales much larger than the rest masses of particles involved, we often treat particles as if they were massless, with good results. How is this possible if the massive and massless cases are so qualitatively different? Is there really a continuous transition between $m=0$ and $m= \epsilon$? If so, how is it reconciled with the facts above?

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For example, consider the process $\pi^- \to e^- + \bar{\nu}_e$. Since the pion mass is much greater than the electron's or neutrino's, we can assume them to be massless. Due to properties of weak interaction, the electron will be left-handed and the neutrino will be right-handed. Since they are massless, chirality is the same as helicity. So, in the center of mass frame, where their momenta are antiparallel, their spins will be parallel. Therefore the total angular momentum is $1$, which is impossible because $\pi^-$ is a pseudoscalar. On the other hand, the $\mu ^-$ is much more massive (mass comparable to that of pion), so this argument doesn't work. Decay to muon is allowed and experimentally verified to be much more probable than decay to electron.

Answer 1

At the level of representation theory, massive and massless particles are fundamentally different. As you mentioned, massless particles have 2 d.o.f. (helicities $\pm h$) and massive particles have $2j+1$ d.o.f. This fundamental difference is quite important and leads to many structures - such as gauge invariance. It is also not possible for this reason to define an $S$-matrix (in the same way that we do now) for massless particles.

However, despite this, there is a reason that in the high energy limit, massive particles can be taken to be approximately massless. Take the example of a spin-1 particle with momentum $(E,0,0,p)$. This has 3 polarizations $$ \epsilon^{\pm,\mu} = (0,1,\pm i,0)~, \qquad \epsilon^{L,\mu} = (p,0,0,E)~. $$ The spin summed amplitude squared is $$ \big| \epsilon^{+,\mu} {\cal M}_\mu + \epsilon^{-,\mu} {\cal M}_\mu + \epsilon^{L,\mu} {\cal M}_\mu \big|^2 $$ Now, in the high energy limit $p \to E$ in which case $\epsilon^{L,\mu} \to p^\mu$. Then, since $p^\mu {\cal M}_\mu = 0$ (coming from the fact that $\partial^\mu A_\mu = 0$ for massive fields), the last term simply drops out. This, in the high energy limit all contributions from the longitudinal polarization drops out and only the two photon polarizations contribute to the amplitude. Finally, the amplitude itself ${\cal M}_\mu$ reduces to the massless one since the massive propagator $\sim \frac{g_{\mu\nu} }{ p^2 + m^2 } \to \frac{g_{\mu\nu} }{p^2}$ reduces to the massless propagator in the high energy limit.

Thus, purely from the standpoint of amplitudes, at high energies all massive particles are effectively massless. Fundamentally though, the particles still have longitudinal polarizations. However, at high energies the particles are highly boosted and none of the longitudinal polarizations interact - only the transverse ones.

Answer 2

IF the number of degrees of freedom in a theory are the same in the massive and massless limit then you can approximate particles as massless at high energies.

This is because position space $ n $-point functions in quantum field theory do not have IR divergences (i.e., they don't blow up as you probe larger and larger distances). This means that physical quantities are not going to depend on long distance physics but only at physics at around the energy scale you are probing. This is a familiar property from other aspects of physics and transfers over the QFT. For example if you measure the spectra of hydrogen in a gas the details of the gas don't effect the spectra you find (to a good approximation).

For spin-$ 0 $ and spin-$ 1/2 $ particles the number of degrees of freedom are indeed the same in the massive and massless cases. Thus you can approximate the massive correlations functions with the massless ones at high energies. However, for spin-$ 1 $ particles the situation is more subtle. Massless spin-$ 1 $ particles have only $ 2 $ degrees of freedom while massive spin-$1$ particles have $3$ degrees of freedom. Depending on the type of current it couples to, one may or may not be able to drop the gauge mass when computing amplitudes at high energies. If it couples to a conserved current then it is still true that the massless limit is applicable since $p_\mu {\cal M}^\mu = 0$ (see other answer). If it couples to a non-conserved current this is no longer the case and there can be amplitudes that grow as (vector mass)$^{-1}$. Clearly in such theories it's not possible to use the massless limit to approximate the massive contributions at high energies.