# Why are cross sections finite at a resonance?

The analytic structure of scattering amplitudes is such that they have a $\frac{1}{q^2-m^2+i\epsilon}$ pole at a particle resonances of mass $m$. Given this, one would expect that cross sections should be infinite at resonance. Why are cross sections not infinite at a resonance?

• because the iε takes care of that? It is under an integral after all see mit.edu/~levitov/8513/lec4.pdf – anna v Nov 25 '17 at 14:25
• because actual resonances have a non-zero width, $\frac{1}{q^2-m^2+i\Gamma}$... – AccidentalFourierTransform Nov 25 '17 at 15:18

In general near a resonance the cross-section has the form $\frac{Constant}{(E-E_R)^2+\Gamma^2/4}$ The quantity $\Gamma$ is the resonance width and $E_R$ the energy of resonance. The presence of the width factor in the denominator makes the cross-section finite. The absence of this factor would also mean that there is no way to detect the resonance.
• The origin of the $\Gamma$ term is the self-energy due to interactions (specifically, loop corrections to the particle's propagator). A free particle does indeed formally have an infinite cross-section at resonance, but as SAKhan said, this formally infinite cross-section isn't actually physically measurable if the particle doesn't interact with the detector. – tparker Nov 26 '17 at 20:03