How is strong coupling constant related to cross section?

For comparison let us first look at Rutherford scattering, which is caused by the electromagnetic interaction. Its differential cross-section is given there by: $$frac{dsigma}{dOmega}= left(frac{Z_1Z_2alphahbar c}{4E_text{kin}sin^2frac{theta}{2}}right)^2$$ where

• $Z_1$ and $Z_2$ are the charges (in units of $e$) of the incoming and the target particle,
• $alphaapproxfrac{1}{137}$ is the dimensionless electromagnetic coupling constant (aka the fine structure constant),
• $E_text{kin}$ is the non-relativistic kinetic energy of the incoming particle,
• $theta$ is the angle of deflection.

The important piece here is $$frac{dsigma}{dOmega} propto alpha^2,$$ meaning that the cross-sections get larger when the coupling constant is large.

This is not special to the electromagnetic interaction. It will also hold for the strong interaction, except there you will have the strong interaction constant $alpha_text{s}approx 1$ instead of $alphaapproxfrac{1}{137}$. And of course, because the strong interaction doesn't obey Coulomb's law, you will also get a different dependency on $theta$.