{title2=$\lim_{r\to0^+}I(r)=1$}

In <Feynman gauge>, use the auxiliary <photon propagator> $-ig_{\mu\nu}/(k^2-\lambda^2+i0)$ and $r=\lambda/m>0$. The projected one-loop <Feynman parameter> integral is
$$
F_2^{(1)}(0;\lambda)=\frac{\alpha}{2\pi}I(r),
\qquad
I(r)=\int_0^1\frac{2x(1-x)^2}{(1-x)^2+xr^2}\,dx.
$$
Here $x$ is the source's photon parameter $z$.<QED Vertex Correction (Vadim Kaplunovsky course notes)>[Eqs. (32), (35), (40), (68)]{ref}

To check the reduction, use $N_2=4z(1-z)m^2$ and $\Delta'=(1-z)^2m^2+z\lambda^2$. The three-parameter expression has prefactor $e^2/(16\pi^2)$ multiplying $N_2/\Delta'$. At fixed $z$, $\int_0^{1-z}du=1-z$, giving $e^2/(4\pi^2)\int_0^1 z(1-z)^2/[(1-z)^2+zr^2]\,dz$. Substituting $e^2=4\pi\alpha$ gives exactly the displayed integral. This starts from the projected <QED> numerator; it is not a derivation of the whole <QED> calculation from an arbitrarily chosen integral.

The regulator can then be removed by a short independent argument. Write its integrand as $f_r(x)$. For $0\le x<1$, $f_r(x)\to2x$, and
$$
0\le f_r(x)=2x\frac{(1-x)^2}{(1-x)^2+xr^2}\le2x,
\qquad\int_0^1 2x\,dx=1.
$$
The <dominated convergence theorem> therefore gives $I(r)\to1$ and $F_2^{(1)}(0)=\alpha/(2\pi)$. At $x=1$ the regulated integrand is zero, whereas the limiting function can be assigned value $2$; one point has zero measure and changes no integral. For every $r>0$, $0<I(r)<1$, and decreasing $r$ increases the integral. The visible suppression near $x=1$ narrows as the regulator shrinks; it is not a uniform pointwise approximation at that endpoint.

\Image[/qed-magnetic-anomaly-regulator-limit.png]
{title=A finite magnetic coefficient survives removal of the regulator}
{description=Original numerical evaluation of the reduced one-loop <Feynman parameter> integral, with $r=\lambda/m$. The left panel shows suppression near the endpoint; the right panel plots $I(r)=F_2^{(1)}(0;\lambda)/(\alpha/2\pi)$ against the auxiliary mass ratio. The dashed limit is the exact integral $\int_0^1 2x\,dx=1$. Finite $r$ values illustrate <regularization in quantum field theory>, not alternative measured <photon> masses.}

This auxiliary mass regulates an <infrared divergence>, not an <ultraviolet divergence>; the finite-mass prescription alone is not fully <gauge-invariant>.<QED Vertex Correction (Vadim Kaplunovsky course notes)>[p. 15, Eqs. (65)–(68)]{ref}

The ultraviolet problem belongs to the full <QED> amplitude. <Electron self-energy>, <wave-function renormalization> and the charge vertex require consistent <counterterms> and <renormalization conditions>. The <Ward identity for QED renormalization constants> ties $Z_1$ to $Z_2$, rather than allowing the vertex and external <Electron> normalization to be fitted independently. Minimal renormalizable <QED> has no independent <Pauli term> to adjust this finite one-loop $F_2(0)$: its dimension-five coefficient is predicted from the <mass> and <electric charge> inputs. Adding a free <Pauli term> would define a different theory with an extra parameter. Finiteness of the reduced integral alone proves neither all-order <renormalizability> nor finiteness of every <QED> observable.


 Regulator limit of the one-loop magnetic form factor (source code)