# Infrared Divergences in Gauge Theories

There are some confusions regarding infrared (IR) divergences in gauge theory:

1. What is the primary reason for the appearance of IR divergences in gauge theory? Anything other than the existence of massless particles in the spectrum of the theory.

2. Why there is no IR divergence for massive theories? The propagator is given by: $$K(p)\propto\frac{1}{p^2-m^2}$$ Therefore, it is expected that the internal momentum integration crosses the pole given by mass. If $i\varepsilon$-prescription is used to avoid this, why does it can not be used in the case of massless theories with propagator proportional to $\frac{1}{p^2}$?

3. Do IR divergences appear only at the loop-level?

4. The S-matrix of a QFT is defined by assuming that the interactions die off at spatial infinity. But in a theory with massless particles, this can not be assumed. Then how should we interpret the results of scattering amplitude computations in gauge theory?

1. The primary reason for IR divergences is the presence of massless states. There is nothing other than that, so there isn't much more to say. You have IR divergences if and only if you have massless states.

2. Divergences are much more transparent in the euclidean formulation. In the lorentzian one, integrations are over the complex $p^0$ plane which makes power-counting arguments much more counter-intuitive: you have to actually calculate the integrals in order to make sure they converge/diverge. In the euclidean formulation, you can more or less read off the convergence properties of integrals just by looking at the different powers of the momenta. Than being said, the euclidean propagator reads $$\frac{1}{p^2+m^2}$$ which is finite everywhere unless $m\equiv 0$, in which case it blows up at the origin. Therefore, you have an IR divergence if and only if $m\equiv 0$.

3. No. See for example Divergence of the tree level scattering amplitude in quantum field theory.

4. Gauge theories do not have a well-defined $S$ matrix. It simply doesn't exist.

For more details, see the aforementioned post and references therein.

• Thank you for the nice answer. Also, if S-matrix can not be defined for gauge theories, how do people compute the scattering amplitudes in gauge theories? – QGravity Sep 9 '17 at 18:04
• @QGravity I'm glad I could help :-) the typical "trick" is to introduce a small mass, so that nothing is massless, in which case the $S$ matrix is well-defined. Only when you have computed a particular scattering amplitude (taking into account the resolution of the detector, as described in any introductory text) can you take the massless limit, which is usually well-defined. – AccidentalFourierTransform Sep 15 '17 at 16:20

You find IR divergence by doing calculations with some finite mass $\mu$ and taking into account $i\epsilon$ prescription taking the limit $\mu\rightarrow 0$. The IR divergences appear also when you take into account that the collision of particles can produce arbitrarily high number of soft massless particles. When you sum over such processes you get a divergence... But it cancels the divergence coming from the loops.

From that you may understand the origin of the IR divergence which you already guessed in your last point. They appear because you can't fully turn off the interaction in the asymptotic states. Because of that the naive Fock states are actually not very good to represent them. If you could find the appropriate asymptotic space you would be able to write S-matrix without any IR divergence from the start. The classic example of such construction for QED was found in this paper of Faddeev and Kulish