Finite renormalization schemes and Wilson coefficient vectors

Work in four spacetime dimensions and natural units $c=\mathrm{\hslash }=1$. Fix an engineering weight $d\in \mathbb{Q}$ and take as input a finite-dimensional complex quotient $V={\mathcal{Q}}_d$ of local operators at that fixed weight, with an ordered basis of representatives $O_1,\dots ,O_N$ and $N={\dim }_{\mathbb{C}}V\ge 1$. The quotient and its declared relations are supplied data; this definition does not assert that every relation relevant to an EFT, observable, or infinite operator tower has been included. Each $O_i$ has the same engineering weight $d$. For a fixed cutoff $\mathrm{\Lambda }>0$ of mass dimension one, a coefficient vector $C=(C_1,\dots ,C_N)\in {\mathbb{C}}^N$ determines the formal coefficient expression in $V$ ${\sum }_{i=1}^N\frac{C_i{O}_i}{{\mathrm{\Lambda }}^{d-4}}\mathrm{.}$ Each $C_i$ is dimensionless, and each summand has mass dimension four. This is a formal expression in the supplied quotient, not a constructed action. No Hermiticity or reality condition is imposed; a real-action condition is additional conjugation data.

A finite scheme record consists of a nonempty set $B$ of input labels, an open interval $J\subset (0,\mathrm{\infty })$ of renormalization scales, and supplied coefficient-coordinate maps ${\mathcal{R}}_{\mu }:B⟶{\mathbb{C}}^N(\mu \in J)\mathrm{.}$ The maps encode the chosen subtraction and normalization convention and are not constructed here. For $b\in B$, the scale-dependent Wilson coefficient vector is $C_{\mathcal{R}}(\mu ;b):={\mathcal{R}}_{\mu }(b)$. The scale $\mu$ and cutoff $\mathrm{\Lambda }$ are distinct inputs. Choose a reference scale ${\mu }_0\in J$ and set $t:=\log (\mu \mathrm{/}{\mu }_0),I:=\{\log (\mu \mathrm{/}{\mu }_0):\mu \in J\};$ then $t$ is dimensionless and $I$ is an open interval containing $0$.

Use the Euclidean norm $\mathrm{\parallel }C{\mathrm{\parallel }}_2^2={\sum }_{i=1}^N\mathrm{\mid }{C}_i{\mathrm{\mid }}^2$ on ${\mathbb{C}}^N$. Define the realification ${\rho }_N:{\mathbb{C}}^N\to {\mathbb{R}}^{2N}$ by ${\rho }_N(C_1,\dots ,C_N)=(\mathrm{Re}{C}_1,\mathrm{Im}{C}_1,\dots ,\mathrm{Re}{C}_N,\mathrm{Im}{C}_N)\mathrm{.}$ With the Euclidean norm on ${\mathbb{R}}^{2N}$, ${\rho }_N$ is a real-linear isometry. A supplied finite-dimensional truncated RG datum for this scheme is an open set $\mathrm{\Omega }\subset I\times {\mathbb{R}}^{2N}$ and a supplied continuous map $\beta :\mathrm{\Omega }\to {\mathbb{R}}^{2N}$. A truncated RG trajectory is a continuously differentiable curve $C:I^{\mathrm{\prime }}\to {\mathbb{C}}^N$, on an interval $I^{\mathrm{\prime }}\subset I$ for which $(t,{\rho }_N(C(t)))\in \mathrm{\Omega }$, satisfying $\frac{d}{dt}{\rho }_N(C(t))=\beta (t,{\rho }_N(C(t)))\mathrm{.}$ The components of $\beta$ are the beta functions under this logarithmic-scale convention. If the supplied flow is complex-linear, with a supplied matrix function $\mathrm{\Gamma }:I^{\mathrm{\prime }}\to {\mathbb{C}}^{N\times N}$, its equation can be written $C^{\mathrm{\prime }}=\mathrm{\Gamma }(t)C$, where the real-coordinate vector field is $\beta (t,{\rho }_N(C))={\rho }_N(\mathrm{\Gamma }(t)C)$. In this column-vector convention, $\mathrm{\Gamma }$ is called the anomalous-dimension matrix. A scheme record does not assert that its coefficient-coordinate maps solve a supplied RG equation; that relation, the entries of $\beta$, and any matching map are separate inputs. This definition calculates no QFT beta function and supplies no bound on omitted perturbative or higher-dimension terms.

Finite-jet local operators and engineering dimension

Finite-order invariant operator-basis quotient

Manohar, full EFT lecture notes

Burgess, full EFT review

The current batch-16 item carriers define the finite jet operator space and a quotient by an explicitly specified invariant relation subspace. They are listed in the current batch-16 page manifest. This definition uses only their fixed-weight finite complex quotient and basis interface. It does not claim that the relation space captures every physical integration-by-parts, equation-of-motion, or field-redefinition equivalence.