# Understanding Weinberg's soft-photon theorem

The soft-photon theorem is the following statement due to Weinberg:

Consider an amplitude ${\cal M}$ involving some incoming and some outgoing particles. Now, consider the same amplitude with an additional soft-photon ($\omega_{\text{photon}} \to 0$) coupled to one of the particles. Call this amplitude ${\cal M}'$. The two amplitudes are related by $${\cal M}' = {\cal M} \frac{\eta q p \cdot \epsilon}{p \cdot p_\gamma - i \eta \varepsilon}$$ where $p$ is the momentum of the particle that the photon couples to, $\epsilon$ is the polarization of the photon and $p_\gamma$ is the momentum of the soft-photon. $\eta = 1$ for outgoing particles and $\eta = -1$ for incoming ones. Finally, $q$ is the charge of the particle.

The most striking thing about this theorem (to me) is the fact that the proportionality factor relating ${\cal M}$ and ${\cal M}'$ is independent of the type of particle that the photon couples to. It seems quite amazing to me that even though the coupling of photons to scalars, spinors, etc. takes such a different form, you still end up getting the same coupling above.

While I can show that this is indeed true for all the special cases of interest, my question is: Is there a general proof (or understanding) that describes this universal coupling of soft-photons?

• I'm asking why the amplitude factorizes universally? As in, why is the form of the factor always $\frac{p\cdot\epsilon}{p\cdot p_\gamma}$?? – Prahar Dec 9 '13 at 15:16
• @Prahar : There is a nice discussion in this paper, Chapter $4$ pages $17-24$, while the interesting formula is $109$, page $20$ – Trimok Dec 9 '13 at 18:47
• @Trimok the epsilon in the denominator is different from the epsilon in the numerator (look closely). The epsilon in the denominator has to do with the Feynman prescription for integrating around poles. (I would guess you were being facetious.) – Brian Moths Dec 9 '13 at 20:07
• @joshphysics - One can look this up in Weinberg Vol. I Eq (13.1.3) – Prahar Dec 9 '13 at 22:09

## 2 Answers

The universality of the coupling of the photon to charged particles exhibited by this formula is only valid in the limit of ultrasoft photons. This is also known as the eikonal approximation, in which the photon couples only to the charge x velocity of the charged particle.

Since it is valid only in the low energy limit you can think as it is the leading term of an expansion. An expression of this is that the photons with long wavelengths cannot resolve the internal structure of the source. But as you go higher energies ( next to leading order terms ) it starts to reveal the internal structure of the target.

• He's addressing why the coupling is independent of the type of particle and only depends on the charge. That's the core of OP's question. – octonion Mar 8 '17 at 3:46