Why does pair production cross section increase with photon energy?

Question

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?

Answer 1

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 $E = 0$.