I have a question regarding Weinberg's soft exponentiation result in his 1965 paper called 'Infrared Photons and Gravitons' (https://doi.org/10.1103/PhysRev.140.B516). When he tries to calculate the contribution of an arbitrary number of virtual soft photons to the elastic amplitude (which number then takes to infinity), he clearly states that

'We will define an infrared virtual photon or graviton as one which connects two external lines and carries energy less than $\Lambda$'

I am totally okay with the author calculating the effects of those specific photons or gravitons. However, a question regarding all the other virtual photons or gravtions naturally arises. How are we supposed to treat those? I have two guesses

  1. They do not diverge in the soft limit
  2. They do diverge, but the divergences occuring from those photons or gravitons (that are not connected to external fermions) are taken care by renormalization methods.

So, is one of the explanations I offer correct? Can someone shed some light please?? Also, I have a bonus question: when deriving the final result, Weinberg shows that the exponentiation of virtual soft photons results to the amlitude being $$S_{\beta\alpha}=\bigg(\frac{\lambda}{\Lambda}\bigg)^{\mathcal{B}_{\beta\alpha}} e^{i\phi_{\beta\alpha}}S_{\beta\alpha}^{(\Lambda)}$$ where $S_{\beta\alpha}^{(\Lambda)}$ is free of infrared divergences associated with virtual photons, $\lambda$ is an artificially introduced cutoff and $\mathcal{B}_{\beta\alpha}$ is a kinetic factor given by $$\mathcal{B}_{\beta\alpha}=-\frac{1}{16\pi^2}\sum_{i,j}\eta_i\eta_je_ie_j\frac{1}{\upsilon_{ij}}\ln\bigg(\frac{1+\upsilon_{ij}}{1-\upsilon_{ij}}\bigg)$$ where $\upsilon_{ij}$ is the relative velocity of particle $i$ in the reference frame of particle $j$. What happens to this quantity when the fermions reach the high energy/massless limit? Does it vanish due to charge conservation principle (in the sense that we will have a logarithmically divergent term multiplied with the sum of the charges, which is zero if the charge is conserved)? Any help will be appreciated.


1 Answer 1


Finally realized the answers for those subtleties. I post them here for completeness sake. All other contributions from virtual massless force carriers that are not attached onto external legs and external legs only are not infrared (IR) diverent, so its beyond the scope of Weinberg's analysis. The reason they are not IR divergent is because the internal fermion propagator that would have been connected with another propagator through a photon line has a denominator that does not vanish in the soft limit. This is because its momentum is far off its mass shell.

The kinematic factor $\mathcal{B}_{\beta\alpha}$ does not diverge in the high energy limit, despite the fact that it appears so. In fact, the logarithm diverges, but the divergence factors out from the sum, with the latter being zero due to charge conservation. This is what I suspect for the kinematic factor $\mathcal{B}_{\beta\alpha}$, but I am not 100% sure...


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