Scalar derivative operator reduction at fixed EFT order

Assume countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$, meaning that every countable family of nonempty sets has a choice function, for the supplied gauge datum. Use $M={\mathbb{R}}^{1,3}$, coordinates $x^0=ct$, metric $\eta =\mathrm{diag}(-1,1,1,1)$, and natural units $c=\mathrm{\hslash }=1$. Let $SU(2)=\{u\in G{L}_2(\mathbb{C}):u^{†}u=I,\det u=1\}$ act on the trivial right principal bundle $P=M\times SU(2)$ with projection $(x,u)\mapsto x$ and right action $(x,u)v=(x,uv)$. Supply $\mathrm{\Sigma }=\{[P]\}$ and $P$ as its representative, with product connection ${\omega }_{(x,u)}(v,W)=u^{-1}W$ and global section $x\mapsto (x,I_2)$. Let $\varphi \in C_c^{\mathrm{\infty }}(M,\mathbb{R})$ be an even real Lorentz-scalar of engineering weight one in the trivial $SU(2)$ representation. Let the global group ${\mathbb{Z}}_2=\{1,-1\}$ act by $\varphi \mapsto -\varphi$. The finite-jet module is the complexification of this real scalar field with conjugation fixing $\varphi$.

For the finite-jet quotient use $G_{\mathrm{a}\mathrm{l}\mathrm{g}}=SO(\eta ,\mathbb{C})\times SL(2,\mathbb{C})\times {\mu }_2$, where $SO(\eta ,\mathbb{C})=\{L\in G{L}_4(\mathbb{C}):L^T\eta L=\eta ,\det L=1\}$, $SL(2,\mathbb{C})=\{g\in G{L}_2(\mathbb{C}):\det g=1\}$, and ${\mu }_2=\{z\in \mathbb{C}:z^2=1\}$. The Lorentz factor acts on derivative indices, $SL(2,\mathbb{C})$ acts trivially on $\varphi$, and ${\mu }_2$ acts by sign. Here $S{O}^{+}(1,3)=\{L\in G{L}_4(\mathbb{R}):L^T\eta L=\eta ,\det L=1,L^0{}_0\ge 1\}$. At the finite field-representation level, the physical group $S{O}^{+}(1,3)\times SU(2)\times {\mathbb{Z}}_2$ maps to $G_{\mathrm{a}\mathrm{l}\mathrm{g}}$ by $(L,u,\sigma )\mapsto (L,I_2,\sigma )$, where $L$ is included as the same real matrix in $SO(\eta ,\mathbb{C})$, the gauge factor maps trivially, and $\sigma =\pm 1$ is included as the same element of ${\mu }_2$. Thus each scalar jet transforms by the Lorentz action on its derivative indices, trivially under $SU(2)$, and by sign under ${\mathbb{Z}}_2$.

Let $m\ge 0$ be a fixed physical mass of mass dimension one, let $C\in \mathbb{R}$ be a fixed dimensionless scalar parameter, and let ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}>0$ be a fixed coefficient-suppression scale of engineering weight one. Treat $m,C,{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}$ as constant Lorentz scalars and $SU(2)$ gauge singlets that transform trivially under the physical ${\mathbb{Z}}_2$ and under the full $G_{\mathrm{a}\mathrm{l}\mathrm{g}}$-action. Define $\mathrm{\square }={\eta }^{\mu \nu }{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }$ and adopt the base action

$S_0[\varphi ]=-\frac{1}{2}{\int }_M({\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi +m^2{\varphi }^2)d^{4}x$

and add the dimension-six correction

$S_6[\varphi ]=\frac{C}{{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^2}{\int }_M{\varphi }^2{\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi d^{4}x\mathrm{.}$

For compactly supported $h$, define the leading Euler derivative by $D{S}_0[\varphi ](h)={\int }_{M}h{\mathcal{E}}_0(\varphi )d^{4}x$; the calculation below verifies ${\mathcal{E}}_0(\varphi )=(\mathrm{\square }-m^2)\varphi$. For the graded quotient, adjoin an even central indeterminate $\mathfrak{m}$ to the complex scalar jet algebra, with engineering weight one, trivial $G_{\mathrm{a}\mathrm{l}\mathrm{g}}$-action, and horizontal derivatives ${\mathrm{\partial }}_{\mu }\mathfrak{m}=0$. It is a formal mass insertion, not a complex coefficient. Take derivative cutoff $r=2$, maximum dynamical-field degree four, maximum total polynomial degree $q=6$ including $\mathfrak{m}$, retained engineering dimension $D_{\max }=6$, and loop order $L_{\max }=0$. Define ${\mathcal{E}}_0^{\mathrm{g}\mathrm{r}}=(\mathrm{\square }-{\mathfrak{m}}^2)\varphi$. In the weight-six invariant subspace of this extended finite polynomial space, declare the relation subspace to be the complex span of ${\mathrm{\partial }}_{\mu }({\varphi }^3{\mathrm{\partial }}^{\mu }\varphi )$ and ${\varphi }^3{\mathcal{E}}_0^{\mathrm{g}\mathrm{r}}$. Evaluation ${\mathrm{ev}}_m$ fixes every field jet and sends $\mathfrak{m}$ to the numerical mass $m$; it maps this relation space into its evaluated relation space and hence induces a map of quotients. It need not preserve engineering grading after specialization. This quotient is a formal algebraic selection only; no equality with the invariant subspace for the specified physical group is asserted. All fields and variations have compact support, so the displayed integrations by parts have no boundary term. Retain terms through formal order ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^{-2}$, with $O({\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^{-4})$ denoting terms at least quadratic in the formal bookkeeping parameter ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^{-2}$, not a numerical error bound. Then ${\varphi }^2{\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi$ represents $-\frac{{\mathfrak{m}}^2}{3}{\varphi }^4$ in the graded quotient; applying ${\mathrm{ev}}_m$ gives the corresponding numerical-mass relation, and the local field substitution ${\varphi }_{\mathrm{o}\mathrm{l}\mathrm{d}}=\varphi +\frac{C}{3{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^2}{\varphi }^3$ changes $S_0+S_6$ to $S_0-\frac{C{m}^2}{3{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^2}{\int }_M{\varphi }^4{d}^{4}x+O({\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^{-4})$. This is an action-level statement at the retained order only.

C. P. Burgess, Introduction to Effective Field Theory, §2.1, equations (21)–(26)

Juan Carlos Criado and Manuel Pérez-Victoria, Field redefinitions in effective theories at higher orders, §§3 and 5.1

Brian Conrad, Reductive Group Schemes, Definition 1.1.11 and Example 1.1.12

Physical EFT operator data and power-counting contract

Countable-choice assumption inherited by the non-Abelian gauge datum

Compact non-Abelian gauge datum with its specified bundle and representation

Finite-jet local operator and engineering-weight convention

Declared invariant operator quotient