Fixed-order on-shell amplitude equivalence under local field redefinition

Fix $N\ge 0$ and put $R_N=\mathbb{C}[ϵ]\mathrm{/}({ϵ}^{N+1})$. For each regulator $\rho$, let $X_{\rho }=X_{\rho }^{\bar{0}}\oplus X_{\rho }^{\bar{1}}$ be the finite-dimensional graded field-coordinate space in the regulated perturbative LSZ amplitude datum. Its even coordinates are commuting, its odd coordinates generate the specified exterior algebra, and its source $J$ has commuting even components and anticommuting odd components paired in the declared order. Let ${\mathcal{I}}_{\rho }$ be the specified integration functional, consisting of the even-coordinate functional and ordered coefficient extraction over the odd coordinates, and let $S_{\rho ,ϵ}$ be an even action in its domain. Define

$$Z_{\rho ,ϵ}[J]={\mathcal{I}}_{\rho }\text{\hspace{0.17em}⁣}(\exp \text{\hspace{0.17em}⁣}(i{S}_{\rho ,ϵ}(\phi )+i⟨J,\phi ⟩\left)\right)\mathrm{.}$$

Let $F_{\rho ,ϵ}$ be the actual parity-preserving local formal coordinate map used at regulator $\rho$, with $F_{\rho ,0}=\mathrm{id}$ and an inverse modulo ${ϵ}^{N+1}$. Use the full pullback $S_{\rho ,ϵ}\circ F_{\rho ,ϵ}$, retaining every term through order $N$, and define

$$Z_{\rho ,ϵ}^{\mathrm{\prime }}[J]={\mathcal{I}}_{\rho }\text{\hspace{0.17em}⁣}(\exp \text{\hspace{0.17em}⁣}(i{S}_{\rho ,ϵ}(F_{\rho ,ϵ}(\psi ))+i⟨J,F_{\rho ,ϵ}(\psi )⟩){\mathcal{J}}_{F,\rho }(\psi )),$$

where the full graded Jacobian factor satisfies

$${\mathcal{I}}_{\rho }(h(\phi ))={\mathcal{I}}_{\rho }\text{\hspace{0.17em}⁣}(h(F_{\rho ,ϵ}(\psi )){\mathcal{J}}_{F,\rho }(\psi ))$$

for every admissible integrand $h$. Assume $Z_{\rho ,ϵ}[0]$ is a unit for each $\rho$, and that for the same ordered field components the connected Green functions defined by the graded source derivatives have a common regulator limit with stable external channels $a_j$, masses $m_{a_j}>0$, and nonzero simple-pole residues $Z_{a_j}$ at $s_j=m_{a_j}^2$, where $s_j=-{\eta }^{\mu \nu }{p}_{j,\mu }{p}_{j,\nu }$ and $\eta =\mathrm{diag}(-1,1,1,1)$. Use the same overlap matrix $U$, its channel-coordinate kernels $G_n$, and chosen roots $z_a^2=Z_a$ in both descriptions, with $Z_a^{1\mathrm{/}2}=z_a$. Assume the pole factorization and joint limits in the LSZ datum exist:

$$G_n(p_1,\dots ,p_n)=\left(\prod _{j=1}^n\frac{Z_{a_j}^{1\mathrm{/}2}}{s_j-m_{a_j}^2}\right){\mathcal{A}}_n(p_1,\dots ,p_n)+\text{terms missing at least one external pole},$$

with normalized joint residue

$${\mathcal{A}}_n=\left(\prod _{j=1}^n{Z}_{a_j}^{-1\mathrm{/}2}\right)\underset{s_j\to m_{a_j}^2}{\lim }(\prod _{j=1}^n(s_j-m_{a_j}^2))G_n\mathrm{.}$$

Then $Z_{\rho ,ϵ}^{\mathrm{\prime }}[J]=Z_{\rho ,ϵ}[J]$ coefficientwise in $R_N$ for every regulator and source, and the normalized on-shell LSZ amplitudes agree coefficientwise modulo ${ϵ}^{N+1}$. This equality follows from the actual regulated coordinate change, its full graded Jacobian, and the correspondingly transformed graded source; it does not assert equality of the two off-shell action expressions or that such regulated or LSZ data exist for a given quantum field theory.

Finite-order invariant operator-basis quotient

Fixed-order integration-by-parts and leading-equation field-redefinition relations

Regulated perturbative LSZ amplitude data

Criado–Pérez-Victoria, §2, equations (2.2)–(2.8), and Appendix B