Quartic scalar contact jets descend through verified operator relations

Fix $m>0$, $\mathrm{\Lambda }>0$, integers $r\ge 1$, $D\ge 4$ and $N\ge 1$, and the metric $\eta =\mathrm{diag}(-1,1,1,1)$. Use the finite-jet operator quotient ${\mathcal{Q}}_D=V_D\mathrm{/}{R}_D$ for one commuting scalar $\varphi$ of weight one, field-degree cutoff four, derivative cutoff $r$, and complex orthogonal spacetime-index action preserving $\eta$. Let $U_D\subset V_D$ be the subspace of degree-exactly-four scalar polynomials whose indices are contracted using $\eta$ only, and let $Q_D^{(4)}$ be its image in ${\mathcal{Q}}_D$. Thus $Q_D^{(4)}=U_D\mathrm{/}(U_D\cap R_D)$; operators outside this image are not assigned a contact map. Fix the monomial basis in the commuting jet generators used to write these polynomials. Let $U_{D,\mathbb{R}}\subset U_D$ consist of the polynomials whose coefficients in that basis are real, and define $Q_{D,\mathbb{R}}^{(4)}$ to be the image of $U_{D,\mathbb{R}}$ in $Q_D^{(4)}$. This is the chosen real subspace of classes admitting a coefficientwise-real representative; no conjugation structure on the full complex quotient is presumed.

Require the following certificate for the actual declared relations: every $H\in U_D\cap R_D$ is a finite sum of identically zero jet polynomials, total divergences ${\mathrm{\partial }}_{\mu }{J}^{\mu }$ with $J$ quartic, and $F(\mathrm{\square }-m^2)\varphi$ with $F$ cubic, where $\mathrm{\square }={\eta }^{\mu \nu }{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }$. The summands may use a larger finite jet space and different engineering weights; their polynomial sum must equal $H$ before any on-shell substitution. This is an additional restriction on the selected $R_D$, not a claim about an arbitrary relation subspace or its completeness.

For $O\in U_D$ introduce commuting indeterminates $z_1,\dots ,z_4$ and complex four-vectors $k_1,\dots ,k_4$. Its polarized Fourier vertex is

$T_O(k_1,k_2,k_3,k_4)=[z_1{z}_2{z}_3{z}_4]O\text{\hspace{0.17em}⁣}({\mathrm{\partial }}_{{\mu }_1}\cdots {\mathrm{\partial }}_{{\mu }_a}\varphi \mapsto {\sum }_{j=1}^4{z}_j{i}^a{k}_{j,{\mu }_1}\cdots k_{j,{\mu }_a})\mathrm{.}$

The coefficient convention contains no $1\mathrm{/}4!$. Impose ${\sum }_j{k}_j=0$, $k_j^2=-m^2$, and set $s=-(k_1+k_2{)}^2$, $t=-(k_1+k_3{)}^2$, $u=-(k_1+k_4{)}^2$. Replace contracted products by $k_j^2=-m^2$, $k_1\cdot k_2=k_3\cdot k_4=m^2-s\mathrm{/}2$, $k_1\cdot k_3=k_2\cdot k_4=m^2-t\mathrm{/}2$, and $k_1\cdot k_4=k_2\cdot k_3=m^2-u\mathrm{/}2$. Put $t=0$, $s=2m^2+v$, $u=2m^2-v$, obtaining a polynomial $t_O(v)$. Define ${\nu }_0=2m^2$ and the linear contact-jet map

$M_{D,N}([O]{)}_k={\nu }_0^{2k+2}{\mathrm{\Lambda }}^{4-D}[v^{2k+2}]t_O(v),0\le k\le N\mathrm{.}$

Then this map $Q_D^{(4)}\to {\mathbb{C}}^{N+1}$ is well-defined and $t_O$ is even. Its restriction to $Q_{D,\mathbb{R}}^{(4)}$ takes values in ${\mathbb{R}}^{N+1}$. For $D=8$, $O_8=({\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi {)}^2$ satisfies, for every $C_8\in \mathbb{R}$,

$t_{O_8}(v)=8m^4+4v^2,M_{8,N}([C_8{O}_8])=(4{\nu }_0^2{C}_8\mathrm{/}{\mathrm{\Lambda }}^4,0,\dots ,0)\mathrm{.}$

In particular the coefficient $[v^2]{\mathrm{\Lambda }}^{-4}{t}_{C_8{O}_8}=4C_8\mathrm{/}{\mathrm{\Lambda }}^4$ is invariant under the certified relations. On $Q_{D,\mathbb{R}}^{(4)}$, the real target coordinates use precisely the normalized convention of the finite forward-amplitude bootstrap problem. This algebraic map describes one quartic contact insertion about the free massive scalar quadratic action. It does not equate the contact polynomial to an exact full amplitude, discard loop or other-operator contributions, or infer positivity of a perturbative term from positivity of a full amplitude.

Finite-jet local operators and engineering dimension

Finite-order invariant operator-basis quotient

Finite truncated forward-amplitude bootstrap data

Criado and Pérez-Victoria, §2 and Appendix B

For an exact even pole-subtracted amplitude $B$, any use of this contact result must also supply the matching remainder $R(v)=B(v)-{\mathrm{\Lambda }}^{4-D}{t}_O(v)$ near zero. The exact coordinates then satisfy $b_k=M_{D,N}([O]{)}_k+{\nu }_0^{2k+2}[v^{2k+2}]R(v)$. This identity retains all omitted contributions. A polynomial contact term growing like $v^2$ need not itself satisfy the dispersive high-energy hypotheses. Multiple insertions and interacting leading equations require their own matching calculation or the stronger regulator, Jacobian, source and stable-LSZ hypotheses of a field-redefinition theorem.