Counterterm domain

Builder/output: counterterm_domain.py, P14_counterterm_domain.json.

This is an order-bounded EFT constraint list, not an admission (XIV-J and XV-B are not opened). The one-loop ${\kappa }^n$ part has $n$ external legs and local dimension $4+n$, with $[\kappa ]=-1$; $[h]=[\varphi ]=[c]=[\bar{c}]=1$; and $[K^{\prime }]=[c^{\ast \prime }]=[{\varphi }^{\ast \prime }]=[{\bar{c}}^{\ast \prime }]=[b]=2$. The functionals have ghost number 0, are even under $\varphi \to -\varphi$ (with ${\varphi }^{\ast \prime }\to -{\varphi }^{\ast \prime }$), and are Lorentz covariant (parity-odd structures vanish for two-leg terms). The permitted species are the legs of loop-capable vertices; $b$, ${\bar{c}}^{\ast \prime }$ and $b^{\ast \prime }$ are excluded. The structures are generated mechanically from $\eta$ and the leg momentum and turned into densities, and their ranks are exact. Total derivatives vanish identically in the two-point representation.

• ${\kappa }^1$: empty (a one-leg term with derivatives is a total derivative, and there is no dimension-5 field).

• ${\kappa }^2$: 11 structures.

  • $hh$, 5: $h{\square }^{2}h$, $\tau {\square }^2\tau$, $(\partial \partial h)\square \tau$, $(\partial h)\square (\partial h)$, $(\partial \partial h{)}^2$;

  • $\bar{c}c$, 2: ${\bar{c}}_{\mu }{\square }^2{c}^{\mu }$, ${\bar{c}}_{\mu }{\partial }^{\mu }\square {\partial }_{\nu }{c}^{\nu }$;

  • $K^{\prime }c$, 3: $K^{\prime mn}{\partial }_m{\partial }_n{\partial }_r{c}^r$, $K^{\prime m}{}_m\square {\partial }_r{c}^r$, $K^{\prime mn}\square {\partial }_m{c}_n$;

  • $\varphi \varphi$, 1: $\varphi {\square }^2\varphi$.

Source dependence. $K^{\prime }$ enters only via $K^{\prime }c$. $c^{\ast \prime }$ and ${\varphi }^{\ast \prime }$ appear first in three-leg terms at ${\kappa }^3$, outside the order bound.

Equivalences. Apply ${\sigma }_0=({S^{(0)}}_{\Psi },\cdot )$ (frozen antibracket) to every generated ghost-number $-1$, dimension-5 bilinear $\Xi$. Antifield-linear $\Xi$ are field redefinitions (canonical transformations); field-only $\Xi$ are $s$-exact. The domain $\cap \mathrm{Im}{\sigma }_0$ is given in the summary; the ${S^{(0)}}_{\Psi }$ density used for this is checked against the verified Hessians. Eight candidate directions modulo the computed exact equivalences, before imposing the relevant identity constraints. This is not a cohomological classification, which needs $D\cap \ker {\sigma }_0$ (the ${\sigma }_0$-closedness constraints) and the quotient $(D\cap \ker {\sigma }_0)/(D\cap \mathrm{Im}{\sigma }_0)$. $D\cap \ker {\sigma }_0$ has not been computed, so the count 8 is an upper bound only. The enumeration itself is unchanged.

Normalisation. $\overline{\mathrm{MS}}$ (pure poles at $\mu$). There is no ${\kappa }^2$ two-derivative counterterm, so neither $\kappa$ nor the $h$ normalisation is renormalised at this order. This is the imposed normalization condition N4 of M3, not a consequence of masslessness. There is no ${\kappa }^1$ counterterm. Finite adjustments stay inside the domain.