Consider the QED Lagrangian
L=ˉψ0(iγμ∂μ−e0γμA0μ−m0)ψ0−14(∂μA0ν−∂νA0μ)2
where the 0 subscript denotes bare fields. The bare fields are related with the renormalized fields via
ψ0=√Z2ψA0μ=√Z3Aμ
m0=Zmme0=Zee
with these redefinitions the Lagrangian takes the form
L=Z2ˉψiγμ∂μψ−eZeZ2√Z3ˉψγμAμψ−mZ2Zmˉψψ−14Z3(∂μAν−∂νAμ)2
it is customary to define
Z1≡ZeZ2√Z3
leaving
L=Z2ˉψiγμ∂μψ−eZ1ˉψγμAμψ−mZ2Zmˉψψ−14Z3(∂μAν−∂νAμ)2.
Moreover, the Z renormalization constants are defined to be
Z1=1+δ1Z2=1+δ2Z3=1+δ3Zm=1+δm
it is often said that this leaves the QED Lagrangian in the following form (see page 2 of these notes http://isites.harvard.edu/fs/docs/icb.topic1146665.files/III-5-RenormalizedPerturbationTheory.pdf).
L=ˉψiγμ∂μψ+δ2ˉψiγμ∂μψ−eˉψγμAμψ−eδ1ˉψγμAμψ−mˉψψ−m(δm+δ2)ˉψψ−14(∂μAν−∂νAμ)2−14δ3(∂μAν−∂νAμ)2.
Nonetheless, the carefull reader will notice that one term coming from mZ2Zmˉψψ is completely ignored, namely
mδ2δmˉψψ.
Why?