A dimension-six scalar operator reduced by integration by parts and the leading equation

Let $M={\mathbb{R}}^{1,3}$ have metric $\eta =\mathrm{diag}(-1,1,1,1)$ and use natural units $c=\mathrm{\hslash }=1$. Let $\varphi :M\to \mathbb{R}$ be a smooth compactly supported real scalar field of engineering dimension one. Let $m\ge 0$ and $\mathrm{\Lambda }>0$ have mass dimension one, let $g,C\in \mathbb{R}$ be dimensionless, and let $ϵ$ be a dimensionless formal parameter with ${ϵ}^2=0$. For dimension bookkeeping, adjoin a central formal scalar $M_s$ of engineering weight one and specialize $M_s=m$ only after forming homogeneous expressions. Let ${\widehat{W}}_6$ be the finite complex vector space of polynomials of weight six in $M_s$ and the scalar field jets, with derivative cutoff two and field-degree cutoff six; take the symmetry group to be trivial. Let ${\widehat{R}}_6$ be spanned by its total divergences and the density ${\varphi }^3{\widehat{\mathcal{E}}}_0$, where ${\widehat{\mathcal{E}}}_0=\mathrm{\square }\varphi -M_s^2\varphi -(g\mathrm{/}6){\varphi }^3$, and write ${\widehat{\mathcal{Q}}}_6={\widehat{W}}_6\mathrm{/}{\widehat{R}}_6$. For compactly supported densities define $\mathcal{I}[f]={\int }_{M}f{d}^{4}x$ and impose $\mathcal{I}({\mathrm{\partial }}_{\mu }{J}^{\mu })=0$ for every compactly supported current. Write $\mathrm{\square }={\eta }^{\mu \nu }{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }$ and set

$$S_0[\varphi ]=\mathcal{I}\text{\hspace{0.17em}⁣}[-\frac{1}{2}{\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi -\frac{1}{2}{m}^2{\varphi }^2-\frac{g}{4!}{\varphi }^4],{\mathcal{O}}_6={\varphi }^2{\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi ,$$

with $S_{ϵ}=S_0+ϵ(C\mathrm{/}{\mathrm{\Lambda }}^2)\mathcal{I}({\mathcal{O}}_6)$. After specializing $M_s=m$, the leading Euler derivative is ${\mathcal{E}}_0=\mathrm{\square }\varphi -m^2\varphi -(g\mathrm{/}6){\varphi }^3$. In ${\widehat{\mathcal{Q}}}_6$,

$$[{\mathcal{O}}_6]=-\frac{1}{3}[M_s^2{\varphi }^4]-\frac{g}{18}[{\varphi }^6]\mathrm{.}$$

The identity-leading local change of variables $\varphi \mapsto \varphi +ϵC{\varphi }^3\mathrm{/}(3{\mathrm{\Lambda }}^2)$ transforms the action through this order to

$$S_{ϵ}\text{\hspace{0.17em}⁣}[\varphi +ϵ\frac{C{\varphi }^3}{3{\mathrm{\Lambda }}^2}]=S_0[\varphi ]-ϵ\frac{C{m}^2}{3{\mathrm{\Lambda }}^2}\mathcal{I}({\varphi }^4)-ϵ\frac{Cg}{18{\mathrm{\Lambda }}^2}\mathcal{I}({\varphi }^6)\mathrm{.}$$

The mass spurion gives each term in the quotient identity weight six. Give the formal coefficient ${\mathrm{\Lambda }}^{-1}$ weight $-1$; multiplying the identity by $C\mathrm{/}{\mathrm{\Lambda }}^2$ then places every term in a finite weight-four action-density space, and $F[\varphi ]{\mathcal{E}}_0$ lies in that space for $F[\varphi ]=-C{\varphi }^3\mathrm{/}(3{\mathrm{\Lambda }}^2)$. The map uses the convention $\mathrm{\Psi }(\varphi )=\varphi -ϵF[\varphi ]$, and $ϵF$ has the units of $\varphi$. This is a formal scalar-action calculation under the stated boundary convention; it asserts neither a physical realization nor an on-shell scattering result.

Finite-jet local operators and engineering dimension

Finite-order invariant operator-basis quotient

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

Burgess, §2.1, equations (25)–(26)

Neubert, §2

Criado–Pérez-Victoria, §§3 and 5.1