QFT — Michael Luke Lectures · Problem V.3

Interacting Fields — Michael Luke

Physics — QFT · yukawa-potential-nonrelativistic-limit

Problem

Consider the nonrelativistic limit of the ``nucleon''-``nucleon'' scattering amplitude. Using the Born approximation from NRQM,

\mathcal{A}_{\rm NR}(\vec{k} \to \vec{k}^{\,\prime}) = -i \int d^3r\, e^{-i(\vec{k}^{\,\prime}-\vec{k})\cdot\vec{r}}\, U(\vec{r}), \tag{V.100}

compare with the field-theory amplitude from the first diagram in Fig. (V.8),

i\mathcal{A} = \frac{ig^2}{|\vec{p}_1 - \vec{p}_1^{\,\prime}|^2 + \mu^2}, \tag{V.101}

and by inverting the Fourier transform show that the corresponding nonrelativistic potential is the Yukawa potential

U(r) = -\frac{\lambda^2}{4\pi r}e^{-\mu r}, \tag{V.104}

where

\lambda = g/2m

. Comment on the range and sign (attractive or repulsive) of this potential.