Structure of Lepton (Remark II)

The first term is conventional electrical interaction.
To induce the second term, we consider the following equation.

d²V/dx² - M_Z²V = (e²/4)t^-2d(x)

This is an ordinary differential equation for the Green function V. Applying Fourier transformation on it, we get,

v(k) = -(2p)^-1òdx e^-ikx V(x) = -(e²/4)t^-2(2p)^-1(k² + M_Z²)^-1

Hence,

V(x) = -(e²/4)t^-2(2p)^-1ò_-¥^¥dk e^ikx(k² + M_Z²)^-1

There is only one pole on the upper half-plane for e^ikx(k² + M_Z²)^-1, and its residue is exp(-M_Zx)/(2iM_Z), therefore we calculate the above V(x) with Cauchy's residue theorem as,

V(x) = -(e²/4)t^-2(2p)^-1(2pi)exp(-M_Zx)/(2iM_Z) = -(e²/4)t^-2exp(-M_Zx)/(2M_Z)


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