Where the fourier transform of the potential is given

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where the Fourier transform of the potential is given by V ˜ ( k ) = integraldisplay d 3 xe i k x V ( x ) (36) 6 Section 3
and k is the difference in the electron momentum before and after scattering, sometimes called the momentum transfer . For example, if this is a Coulomb potential V ( x ) = e 2 4 π | x | , then V ˜ ( k ) = e 2 k 2 so parenleftbigg d Ω parenrightbigg born = m e 2 4 π 2 parenleftbigg e 2 k 2 parenrightbigg 2 (37) Let’s check the dimensions in these formulas: [ V ( x )] = M 1 thus [ V ˜ ( k )] = M 2 and so bracketleftBigparenleftBig d Ω parenrightBig born bracketrightBig = M 2 , which is the correct dimension for a cross section. For the field theory version, the center-of-mass frame is the proton rest frame to a good approximation and E cm = m p . Also the scattering is elastic so | p i | = | p f | . Then the prediction is parenleftbigg d Ω parenrightbigg cm = 1 64 π 2 m p 2 |M| 2 (38) What dimension should M have? Since bracketleftBig d Ω bracketrightBig = M 2 and [ m p 2 ] = M 2 , it follows that M should be dimensionless. If we ignore spin, we will see in Lecture II-2 that the Lagrangian describing the interaction between the electron, proton and photon has the form L = 1 4 F μν 2 φ e ( square + m e 2 ) φ e φ p ( square + m p 2 ) φ p ieA μ ( φ e μ φ e φ e μ φ e ) + ieA μ ( φ p μ φ p φ p μ φ p ) + O ( e 2 ) (39) with φ e and φ p representing the electron and proton respectively. (This is the Lagrangian for scalar QED.) In the non-relativistic limit the momentum p μ = ( E, p ) is close to being at rest ( m, 0) . So, E m , that is t φ imφ and | p | ≪ m . Let us use this to factorize out the leading order time dependence, φ e φ e e im e t and φ p φ p e im p t . Then the Lagrangian becomes L = 1 4 F μν + φ e 2 φ e + 2 e m e A 0 φ e φ e + φ p 2 φ p 2 e m p A 0 φ p φ p + (40) with higher order in 2 m 2 . We have removed all the time dependence, which is appropriate because we are trying to calculate a static potential. Although we don’t know exactly how to calculate the matrix element, by now we are capable of guessing the kinds of ingredients that go into the calculation. The matrix element must have a piece proportional to 2 em p from the interaction between the proton and the photon, a factor of the propagator 1 k 2 from the photon kinetic term, and a piece proportional to 2 em e from the photon interacting with the electron. Thus M∼ ( 2 em p ) 1 k 2 (2 em e ) (41) Then from Eq. (38) parenleftbigg d Ω parenrightbigg cm = 1 64 π 2 m p 2 |M| 2 e 4 m e 2 4 π 2 1 k 4 (42) which agrees with Eq. (37) and so the non-relativistic limit works. We will perform this calcula- tion again carefully and completely, without asking you to accept anything without proof, once we derive the perturbation expansion and Feynman rules. The answer will be the same. The factors of m in the interaction terms are unconventional. It is more standard to rescale φ 1 2 m φ so that L = 1 4 F μν + 1 2 m e φ e 2 φ e + 1 2 m p φ p 2 φ p + eA 0 φ e φ e eA 0 φ p φ p (43) which has the usual p 2 2 m for the kinetic term and an interaction with just a charge e and no mass m in the coupling. Of course, the final result is independent of the normalization, but it is still

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