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

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

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 $q^2 \ge 4m^2$?

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 $q^2 \ge 4m^2$?