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Let the bare fields obey the wave-function renormalization definitions ${\psi }_0=Z_2^{1/2}\psi$, $A_{0\mu }=Z_3^{1/2}{A}_{\mu }$, and put $m_0=Z_{m}m$. Define the vertex renormalization factor by $e_0={\mu }^{ϵ/2}e{Z}_1{Z}_2^{-1}{Z}_3^{-1/2}$. Substitution into the bare QED Lagrangian density givesEquivalently, write $Z_i=1+{\delta }_i$ and ${\delta }_m=Z_2{Z}_m-1$. The counterterm Lagrangian isFor the subsequent Feynman gauge calculation include $-(\partial \cdot A{)}^2/2$ in the gauge-fixed action. The Ward identity for QED renormalization constants implies $Z_1=Z_2$ when the regularization preserves gauge invariance; this is a consequence, rather than an extra independent definition of the renormalization constants.

The integral printed in the text can be evaluated without reconstructing Figure 1. Put $A=l^2-m^2+i0$ and $B=(l+k{)}^2-m^2+i0$, and let $I^{\mu }$ denote that integral without its factor $\xi$. The numerator contraction responsible for its covariant gauge dependence isWith the scalar electrodynamics three-point vertex $-ie(p+p^{\prime }{)}^{\mu }$, three such vertices, two scalar propagators and one photon propagator have overall factor $(-ie{)}^3{i}^2(-i)=-e^3$. Thus the supplied integrand gives $V_{\mathrm{supplied}}^{\mu }=-e^3\xi I^{\mu }$.

First isolate the numerator term $l^2$. Its contribution isUnder $l\mapsto -l-k$ the two denominators exchange and the numerator changes sign. A shift-invariant dimensional regularization therefore gives $\boxed{I_0^{\mu }=0}$, including the finite part.

For the remaining numerator $2(l\cdot k)(2l+k{)}^{\mu }$, only its term $4l^{\mu }(l\cdot k)$ contributes to the ultraviolet pole. Corrections from expanding the denominators, and the numerator $2k^{\mu }(l\cdot k)$, are ultraviolet finite by power counting. Use auxiliary-mass extraction of ultraviolet poles with any $M>0$ rather than literally integrating a massless scaleless integral. Lorentz covariance then givesTODO (source gap): The algebra of the supplied expression is complete, but its graphical identification with Figure 1(b), including the routing at each internal vertex, cannot be visually verified from the unreadable panel.

The complete renormalization constant can nevertheless be obtained independently. Here is a short scalar QED wave-function pole calculation, to avoid assuming either missing panel's value. For an external four-momentum $p$, the exchange contribution to the scalar self-energy has numeratorIts denominator is $l^2[(p-l{)}^2-m^2+i0]$. The necessary ultraviolet poles, with the measure $d^{d}l/(2\pi {)}^d$, are respectivelyThe first two follow by a Feynman parameter and a shift of $l$; the third is a tadpole integral; the last follows by the same large-$l$ extraction used above. The seagull vertex contributes only a photon tadpole integral, which is a scaleless integral, and in any case cannot affect the coefficient of $p^2$. Multiplication by the exchange graph's factor $-e^2$ gives the scalar self-energy insertionThe kinetic counterterm $i{\delta }_2{p}^2$ therefore requires ${\delta }_2=e^2(3-\xi )C_{ϵ}$. The scalar QED Ward identity sets ${\delta }_1={\delta }_2$, soIndeed the full unrenormalized scalar electrodynamics three-point vertex has ultraviolet pole $i{e}^3(3-\xi )C_{ϵ}{k}^{\mu }$, which is cancelled by $-ie{\delta }_1{k}^{\mu }$. Subtracting the supplied contribution gives the gauge-independent sum stated in part (ii), without assigning either unreadable panel an integrand.

The scalar QED Ward identity equates the vertex renormalization and wave-function renormalization constants. Consequently, using part (iv),The renormalization constants describe gauge-dependent fields and off shell vertices, so their dependence on $\xi$ is permitted. A physical scattering amplitude includes all Feynman diagrams, the counterterms, and the external-state factors from LSZ reduction. The Ward identities ensure that the dependence on $\xi$ cancels for physical external states. Infrared-sensitive probabilities also require the usual inclusive treatment of unobserved soft photons. Thus no dependence on $\xi$ is implied for consistently calculated physical observables.

A renormalization constant relates bare parameters or fields to renormalized ones, for example ${\varphi }_0=Z_{\varphi }^{1/2}\varphi$. Its regulator-dependent terms cancel ultraviolet divergences; its finite part depends on the renormalization scheme. Field and vertex constants may depend on the chosen gauge fixing even when physical observables do not.

In scalar electrodynamics with photon propagator $-i[g_{\mu \nu }-(1-\xi )l_{\mu }{l}_{\nu }/l^2]/(l^2+i0)$ and dimensional regularization $d=4-2ϵ$, the minimal subtraction scheme gives $Z_{\varphi }=1+e^2(3-\xi )/(16{\pi }^2ϵ)+O(e^4)$. For an external four-momentum $p$, expand the exchange numerator as $4p^2-4\xi p\cdot l+\xi l^2-4(1-\xi )(p\cdot l{)}^2/l^2$. The logarithmic ultraviolet poles of the scalar, vector and tensor integrals give the $p^2$ coefficient $4-2\xi -(1-\xi )=3-\xi$. A massless photon tadpole integral does not change it. The scalar QED Ward identity gives the same vertex renormalization constant.