Gauge-longitudinal kernel decoupling for explicit tree currents

Use $h=-\eta =\mathrm{diag}(1,-1,-1,-1)$ from Off-shell free Dirac Ward telescoping and tree currents for contractions: for $w\in {\mathbb{C}}^4$, set $w_a={\sum }_{b=0}^3{h}_{ab}{w}^b$, and sum repeated component indices over $0,1,2,3$. Let $p_1,p_1^{\mathrm{\prime }},p_2,p_2^{\mathrm{\prime }}$ be future-shell momenta for the same $\mu >0$ convention as that lemma, and suppose $k=p_1-p_1^{\mathrm{\prime }}=p_2^{\mathrm{\prime }}-p_2$. Define $J_1^{\mu }:=\bar{u}(p_1^{\mathrm{\prime }}){\gamma }^{\mu }u(p_1),J_2^{\nu }:=\bar{u}(p_2^{\mathrm{\prime }}){\gamma }^{\nu }u(p_2),$ with the on-shell Dirac equations in Off-shell free Dirac Ward telescoping and tree currents. If a finite rational covariant-gauge kernel changes by $\mathrm{\Delta }{D}_{\mu \nu }(k)=c(k)k_{\mu }{k}_{\nu }$, where $c(k)\in \mathbb{C}$ is finite at the chosen $k$, then $J_1^{\mu }\mathrm{\Delta }{D}_{\mu \nu }(k)J_2^{\nu }=0.$ Likewise, let $J^a=\bar{u}(p^{\mathrm{\prime }}){\gamma }^{a}u(p)$ with transfer $\mathrm{\ell }=p-p^{\mathrm{\prime }}$, or $J^a=\bar{v}(q){\gamma }^{a}u(p)$ with transfer $\mathrm{\ell }=p+q$, under the respective future-shell and Dirac-equation hypotheses of Off-shell free Dirac Ward telescoping and tree currents. If $k=\mathrm{\ell }$ or $k=-\mathrm{\ell }$, then for $ϵ\in {\mathbb{C}}^4$ and a finite $\alpha \in \mathbb{C}$, the contraction ${ϵ}_a{J}^a$ is unchanged by ${ϵ}^a\mapsto {ϵ}^a+\alpha k^a$. These statements concern only the displayed finite tree-current contractions; they assert no gauge independence for arbitrary loop diagrams or renormalization schemes.