Yukawa force versus scattering amplitude

[RedX]

When does the Yukawa potential apply and when does the scattering matrix apply?

Take QED for example. When calculating a scattering amplitude, you have an expansion in powers of the fine structure constant, 1/137. Where does Coulomb's law F=e^2/r^2 come in? As far as I can tell, all calculations in QED are done with plane wave initial and final states, so where does the distance they come towards each other even come in, since we're dealing with momentum eigenstates? Even if you create wave-packets and send the particles towards a head-on collision, ensuring that the two particles will get close to each other, the most likely scenario is that no interaction takes place at all and they just pass through each other! But according to Coulomb's law, there will always be an effect!

The Yukawa potential for nucleons is used to show that the nucleus is bound. If leptons are really close together, can they be bound by a Yukawa potential that involves the exchange of W and Z? But the exchange of spin 1 particles results in a repulsive force, so wouldn't the weak force cause a ball of neutrinos to explode rather than to glue together?

So say you try to slam two protons together to get them to interact strongly. If you try to aim it so good that it's head on, then does that mean r=0, and so by Yukawa's law then there will be a big force?What validity is there to Yukawa's potential to explain the weak force?

 

[Evalesco]

The Yukawa potential is a mathematical model used to describe the interactions between particles, and it is valid for both strong and weak nuclear forces. The Yukawa potential takes the form of an exponential decay, and is expressed as a function of the distance between two particles. In the case of the weak force, the Yukawa potential is modified by the exchange of W and Z bosons, which gives rise to the repulsive force that keeps neutrinos from binding together. The scattering matrix is a way of expressing the probability of a particle interaction, and is typically used to calculate cross sections in particle physics.