Renormalization of the vector current in QED

It is commonly asserted that the electromagnetic current is conserved and therefore is not renormalized. Within QED we show (a) that this statement is false, (b) how to obtain the renormalization of the current to all orders of perturbation theory, and (c) how to correctly define an electron number operator. The current mixes with the four-divergence of the electromagnetic field-strength tensor. The true electron number operator is the integral of the time component of the electron number density, but only when the current differs from the $\overline{\mathrm{MS}}$-renormalized current by a definite finite renormalization. This happens in such a way that Gauss’s law holds: the charge operator is the surface integral of the electric field at infinity. The theorem extends naturally to any gauge theory.

Figure 1

(a) Graphs that are one-particle irreducible in the current channel for insertion of a current vertex in a Green function or matrix element; the standard nonrenormalization argument applies only to these. (b) These graphs, one-particle reducible in the current channel, also contribute to matrix elements of the current and to its renormalization. The two subgraphs that are cross hatched are irreducible in the photon line, while the other subgraph gives the full propagator corrections to the photon propagator. (c) Counterterm to (b). The filled square corresponds to an operator proportional to ${\partial }_{\nu }{F}^{\nu \mu }$.

Figure 2

One-loop wave-function and vertex corrections.

Figure 3

One-loop penguin graph contributing to the renormalization of the vector current. The cross denotes an insertion of the vector current.

Figure 4

Matrix element on the right of Eq. (a3) used in lowest-order BJL computation of the commutator. The circled cross is the vertex for the operator $j_{\mathrm{N}}^0$ and the simple cross is the vertex for $\overline{\psi }$. The external line of momentum $p$ is for the state $|1⟩$.

Figure 5

Graph for the one-loop vacuum polarization contribution to the BJL computation.

Figure 6

Graphs for the one-loop vacuum polarization contribution to the BJL computation, with cuts on the photon line.

Figure 7

Graphs for the one-loop vacuum polarization contribution to the BJL computation, with cuts on an electron-positron pair.