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Let $S$ be a Fréchet differentiable functional on a Banach space. Its first-order expansion isfor fixed $f$ as $\eta \to 0$. Differentiability alone gives this little-o remainder; it does not imply the $O({\eta }^2)$ remainder displayed in the usual second-order-bounded form.

For that stronger bound, suppose the derivative is locally Lipschitz in operator norm with constant $L$. The fundamental theorem of calculus along the line segment givesso the remainder has magnitude at most $L\Vert h{\Vert }^2/2$. A locally bounded continuous second Fréchet derivative is a sufficient hypothesis. Where a functional derivative represents $D{S}_{\varphi }[f]$ by an integral, the corresponding coefficient is $\int f\delta S/\delta \varphi$.

Taking $h=-\eta f$ therefore produces the linear change $-\eta \int f\delta S/\delta \varphi$. That is the cancellation used in the equivalence theorem for field redefinitions. The functional Jacobian and transformed operator insertions still need consistent treatment; the Taylor estimate is a statement about the specified function space and admissible deformation.

At tree level and energies much greater than a gauge-vector mass, an amplitude with longitudinal external gauge bosons equals the corresponding amplitude with their would-be Goldstone bosons, up to conventional external-state phases and relative $O(m_V/E)$ corrections for a nonzero leading amplitude. In a renormalizable gauge, the broken-symmetry identities relate $k_{\mu }{\mathcal{M}}^{\mu }$ to $m_V{\mathcal{M}}_{\chi }$; inserting ${\epsilon }_L^{\mu }=k^{\mu }/m_V+O(m_V/E)$ explains the result. Loop-level versions include field-normalization factors. This is different from an equivalence theorem for field redefinitions.

Use the sign convention $E(\varphi )=\delta S_0/\delta \varphi$ from part b, and letIn the path integral, make the local field redefinition ${\varphi }_{\mathrm{old}}=\chi -\eta f(\chi )$. The functional Taylor expansion of the action givesThe two first-order terms cancel. Neglecting the functional Jacobian as permitted,Thus $fE$ is a redundant operator at first order, by the equivalence theorem for field redefinitions. The same argument works for a local derivative-dependent $f$, which is needed to remove $(\square \varphi {)}^2$. Insertions and sources must be transformed too: arbitrary untransformed off shell correlation functions need not be equal.

Sign qualification. If instead $E_{+}=\square \varphi +m^2\varphi +\lambda {\varphi }^3/6$ is used, then $E_{+}=-\delta S_0/\delta \varphi$. An added $+\eta f{E}_{+}$ is removed by ${\varphi }_{\mathrm{old}}=\chi +\eta f(\chi )$. The minus shift requested in the source therefore fixes which sign is meant by its otherwise unspecified $E$.

After the quadratic mixing cancellation, the vector inverse quadratic operator isInverting its two orthogonal projectors and multiplying by $i$ givesEquivalently $D^{(\xi )}=-i{P}^T/(k^2-m_A^2+i0)-i\xi P^L/(k^2-\xi m_A^2+i0)$. At fixed finite $\xi$, both sectors fall as $k^{-2}$ at large momentum in Euclidean space; $\xi =1$ gives $-i{g}_{\mu \nu }/(k^2-m_A^2+i0)$, and taking $\xi \to \infty$ at fixed $k$ recovers unitary gauge. The latter limit is not uniform in ultraviolet momentum.

The expected power counting conclusion is that this $k^{-2}$ behaviour, together with polynomial interactions of mass dimension at most four in Cartesian scalar variables and the gauge symmetry identities, permits renormalization of spontaneously broken gauge theories using a finite set of physical parameters. The propagator alone is not sufficient to establish that claim in the polar variables of this question. In fact, part b contains $h(\partial \pi {)}^2/f$ and $h^2(\partial \pi {)}^2/(2f^2)$, of dimensions five and six, and the nonlinear gauge fixing term contains $-\xi e^2{h}^4{\pi }^2/(2f^2)$, also of dimension six. Ordinary power counting is consequently not manifest in these variables.

For a representation where it is manifest, write $\varphi =(f+H+i\chi )/\sqrt{2}$ and choose the usual polynomial R-xi gauge function $\partial \cdot A-\xi ef\chi$. It yields the same free vector propagator, cancels the quadratic mixing, and leaves only dimension-at-most-four interactions, including its Faddeev-Popov ghosts. This supplies the missing vertex condition in the source's proposed inference; the equivalence theorem for field redefinitions relates physical predictions near the vacuum. Good propagator falloff is the necessary ingredient illustrated by the requested formula, rather than a complete proof for arbitrary parametrizations or nonlinear choices of gauge fixing.

On shell, the contact vertex is $4i{\lambda }^2{m}^2$. At a cubic vertex with two external on-shell legs and internal momentum $k$, the rule reduces to $2i\lambda (k^2-m^2)$. The propagator cancels one such factor, so the three exchange diagrams sum toFor equal-mass two-to-two scattering, $s+t+u=4m^2$, and the exchange sum is $-4i{\lambda }^2{m}^2$. It cancels the contact diagram exactly. Thus the connected on-shell amplitude vanishes, as required by the equivalence theorem for field redefinitions: an invertible local field redefinition cannot turn a free theory into a physically interacting one.