The formula given by Born approximation for differential cross section shows the photon energy $h\nu $ appearing in the denominator with cubic power; however most graphs show the cross section increasing with energy until it reaches a constant value. What is the correct energy dependence of cross section on photon energy in pair production?


The Born approximation is only the first term in an expansion that has higher-order corrections. These corrections are rated by the number of closed loops they have in their Feynman diagrams. I don't know how many loops are included, because the pedigree of this expression goes back to a textbook and not a paper that's available online, but this expression is the total cross section for electron/positron pair production [Gould and Schréder (1967)]:

$$\sigma = \frac{\pi r_0^2}{2} (1 - \beta^2) \left[(3 - \beta^4) \ln \frac{1+\beta}{1 - \beta} - 2\beta(2 - \beta^2)\right],$$

where $r_0 = \frac{e^2 }{ 4 \pi \epsilon_0 m_\text{e} c^2}$ is the classical electron radius, and $\beta c$ is the electron/positron velocity in the center of mass (CM) frame of the collision. I don't know how many loops were used to calculate this expression. The presence of the logarithm means they used at least one, but I've never done the calculation myself, so I don't know if higher-order loops will have a square of the logarithm or just more factors of $\beta$ times the logarithm.

The above expression can be reformulated in terms of the relativistic kinetic energy $E$ of the electron/positron in the CM frame:

$$\beta \equiv v /c = \sqrt{1 - \left( \frac{m_\text{e} c^2}{E + m_\text{e} c^2} \right)^2} \text{,}$$

where $m_\text{e} c^2$ is the rest mass-energy of the electron/positron. This satisfies the requirement that $v = 0$ gives $\gamma = 1$, which results in $E = 0$.

  • $\begingroup$ Thank you for answering this question. I've been looking high-and-low for a usable expression for $\sigma_\text{PP}$. $\endgroup$ – Joel DeWitt Jun 1 '18 at 14:55

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