$q^2=4m^2$ as the threshold for creation of a real electron-positron pair

Question

On page 252, Peskin & Schroeder remark that the branch cut of the quantity

$$\widehat \Pi_2(q^2) \equiv \Pi_2(q^2)-\Pi_2(0) = -\frac{2\alpha}{\pi}\int_0^1dx\,x(1-x)\log\left(\frac{m^2}{m^2-x(1-x)q^2}\right)\tag{7.91}$$

beginning at

$$q^2=4m^2$$

is at the threshold for creation of a real electron-positron pair. Here, $q$ is the 4-momentum of a virtual photon.

I see that $q^2 \ge 2m^2$ is a necessary condition for the creation of a real electron-positron pair, but are there other necessary conditions and are they sufficient? If not, why do they imply that pair creation only happens at $q^2 \ge 4m^2$?

Answer 1

$q^2 \geq 4m_e^2$ or equivalently in the center of mass frame $q^0 \geq 2m_e$ is a sufficient condition in the sense that $$ (\Gamma(\gamma^*(q) \rightarrow e^- e^+))_{\text{CoM}} \propto \Theta(q^0 - 2m_e^2) f(q^0) $$ where $f(q^0)$ is a non-zero function, $\gamma^*$ denotes a virtual photon and $\Gamma$ is the decay rate (verifying this a straightforward exercise).

You can see in your one-loop expression for the photon self energy that this function has a branch cut starting at $q^2 > 4m_e^2$, if this is not clear look here Branch cut singularity in photon propagator at one loop. That this is true also for the full self energy follows from the optical theorem, which states $$ \text{Im }\Pi(q^2) \propto \sum_X \Gamma(\gamma^*(q^2) \rightarrow X), $$ where the sum goes over all states $X$ in the Hilbertspace. This implies that the photon self energy $\Pi(q^2)$ acquires an imaginary part if $\gamma^*(q^2)$ can decay into some state (i.e. if some decay rate is non-zero), and this is possible in QED for only if $q^2 > 4m_e^2$. Since we have the identity $$ \lim_{\epsilon \rightarrow 0}(\Pi(q^2+i\epsilon)-\Pi(q^2-i\epsilon)) = 2i \text{Im } \Pi(q^2) $$ you can see that we have branch cut on the real line starting at $q^2 = 4m_e^2$.