Diagrammatic approach to pair production in slowly varying and constant fields

We consider the problem of pair creation in slowly varying and essentially constant fields using directly momentum-space Feynman diagrams for the pair production amplitude. When the production occurs via many interactions with small energy transfer, the problem can be viewed as an integration on paths in energy space. An estimate via the integral over the optimal path given by Hamilton's principle, a fuller Lagrangian equation, is introduced and discussed. The existence of such a classical path reflects in an asymptotic behavior such as $n!c^n$ of high-order Feynman diagrams, which in a formal Borel-type summation yields the essentially singular tunneling behavior in constant fields.