Quantum Field Theory

where we’ve introduced the dimensionless parameter λ = g/2M. We can trivially invertthis to find,U(⃗r) = −λ 2 ∫d 3 p(2π) 3e i⃗p·⃗r⃗p 2 + m 2 (3.72)But this is exactly the integral (3.66) we just did in the classical theory. We haveU(⃗r) = −λ24πr e−mr (3.73)This is the Yukawa potential. The force has a range 1/m, the Compton wavelength ofthe exchanged particle. The minus sign tells us that the potential is attractive.Notice that quantum field theory has given us an entirely new perspective on thenature of forces between particles. Rather than being a fundamental concept, the forcearises from the virtual exchange of other particles, in this case the meson. In Section 6of these lectures, we will see how the Coulomb force arises from quantum field theorydue to the exchange of virtual photons.We could repeat the calculation for nucleon-anti-nucleon scattering. The amplitudefrom field theory is given in (3.59). The first term in this expression gives the sameresult as for nucleon-nucleon scattering with the same sign. The second term vanishes inthe non-relativisitic limit (it is an example of an interaction that doesn’t have a simpleNewtonian interpretation). There is no longer a factor of 1/2 in (3.70), because theincoming/outgoing particles are not identical, so we learn that the potential betweena nucleon and anti-nucleon is again given by (3.73). This reveals a key feature offorces arising due to the exchange of scalars: they are universally attractive. Noticethat this is different from forces due to the exchange of a spin 1 particle — such aselectromagnetism — where the sign flips when we change the charge. However, forforces due to the exchange of a spin 2 particle — i.e. gravity — the force is againuniversally attractive.3.5.3 φ 4 TheoryLet’s briefly look at the Feynman rules and scattering amplitudes for the interactionHamiltonianH int = λ 4! φ4 (3.74)The theory now has a single interaction vertex, which comes with a factor of (−iλ),while the other Feynman rules remain the same. Note that we assign (−iλ) to the– 69 –