# Derivation of Yukawa potential: quandry

## Main Question or Discussion Point

Can someone provide some help with a derivation in Peskin and Schroeder (equation 4.126, p.122):

$$V(\bold{x}) = \int \frac{d^3q}{(2\pi)^3} \frac{-g^2}{|\bold{q}|^2+m^2}e^{i\bold{q}\cdot\bold{r}}$$

$$= \frac{-g^2}{4\pi^2}\int_0^\infty dq\; q^2\; \frac{e^{iqr}-e^{-iqr}}{iqr}} \frac{1}{q^2+m^2}$$

They derive the position-space Yukawa potential by Fourier-transforming the Feynman amplitude for the process. Perhaps I'm just being obtuse, but the simplifications from the first to the second line of the equation (once they've done the angular integration) don't seem clear to me.

In particular, what is the justification for this term: $$\frac{e^{iqr}-e^{-iqr}}{iqr}}$$

Any help would be appreciated.

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malawi_glenn
Homework Helper
Use $$\bold{q}\cdot\bold{r} = qrcos(\theta )$$

$$d^3q = q^2sin(\theta ) d\phi dq$$

$$q \text{ from } 0 \text{ to} +\infty$$

$$\theta \text{ from } 0 \text{ to}\pi$$

$$\phi \text{ from } 0 \text{ to}2\pi$$