Finite-order EFT threshold matching map

Use natural units $c=\mathrm{\hslash }=1$ and Minkowski signature $\eta =\mathrm{diag}(-,+,+,+)$. Fix a reference renormalization scale ${\mu }_0>0$ and a common open interval $J\subset (0,\mathrm{\infty })$ of renormalization scales $\mu$, with ${\mu }_0\in J$. Define $I:=\{\log (\mu \mathrm{/}{\mu }_0):\mu \in J\}$. Because $\mu \mapsto \log (\mu \mathrm{/}{\mu }_0)$ is an increasing homeomorphism from $(0,\mathrm{\infty })$ to $\mathbb{R}$, $I$ is an open interval containing $0$. For $\mu \in J$, write $t=\log (\mu \mathrm{/}{\mu }_0)\in I$; $t$ is dimensionless. A threshold datum consists of a high-energy description, denoted $+$, and a lower-energy effective description, denoted $-$. The descriptions have a common declared light-field sector and differ by the degrees of freedom retained across a specified threshold of mass $M>0$. The external-energy domain and the range of validity of the local expansion are supplied separately; in a heavy-particle expansion they ordinarily require external energy $E_{\mathrm{e}\mathrm{x}\mathrm{t}}<M$. The Wilsonian matching scale ${\mu }_W$, the renormalization scale $\mu$, the threshold mass $M$, the external-energy bound $E_{\mathrm{e}\mathrm{x}\mathrm{t}}$, and the regulator cutoff ${\mathrm{\Lambda }}_{\mathrm{r}\mathrm{e}\mathrm{g}}$ (denoted $\mathrm{\Lambda }$ in the finite-regulator model data) have mass dimension one and are distinct data; no equality between ${\mu }_W$ and $\mu$ is part of this definition.

For each sign $\sigma \in \{+,-\}$, supply a finite nonempty ordered set $D_{\sigma }\subset \mathbb{Q}$ of retained engineering weights. For every $d\in D_{\sigma }$, supply an instance of the fixed-weight scheme contract: a finite complex quotient ${\mathcal{Q}}_{\sigma ,d}$ of operators of weight $d$, an ordered basis $O_{\sigma ,d,1},\dots ,O_{\sigma ,d,N_{\sigma ,d}}$ with $N_{\sigma ,d}\ge 1$, and scheme maps ${\mathcal{R}}_{\sigma ,d,\mu }:B_{\sigma }\to {\mathbb{C}}^{N_{\sigma ,d}}$ on the common nonempty input-label set $B_{\sigma }$ and common scale interval $J$. The quotient relations and power-counting truncation are declared for each block. Set $N_{\sigma }={\sum }_{d\in D_{\sigma }}{N}_{\sigma ,d}$ and concatenate these bases and maps in the supplied weight order to obtain the ordered basis of ${⨁}_{d\in D_{\sigma }}{\mathcal{Q}}_{\sigma ,d}$ and the scheme map ${\mathcal{R}}_{\sigma ,\mu }(b)=({\mathcal{R}}_{\sigma ,d,\mu }(b){)}_{d\in D_{\sigma }}\in {\mathbb{C}}^{N_{\sigma }}$. The common label $b$ allows correlations between blocks; no independence of their coefficients is assumed. The fixed-weight case is recovered by taking a singleton $D_{\sigma }$. Any identification of the Wilsonian coefficient data at ${\mu }_W$ with these scheme coordinates is supplied input; this definition does not identify ${\mu }_W$ with $\mu$. The vector $C_{\sigma }(t)\in {\mathbb{C}}^{N_{\sigma }}$ is the column of dimensionless Wilson coefficients in that basis. The corresponding finite-order operator expression uses the supplied convention $C_{\sigma ,i}{O}_{\sigma ,i}\mathrm{/}{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p},\sigma }^{d_i-4}$, with each $O_{\sigma ,i}$ of engineering dimension $d_i$ and ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p},\sigma }>0$ the coefficient-suppression scale for that theory, denoted ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}$ in its physical operator contract and $\mathrm{\Lambda }$ in the generic scheme convention. It is distinct from ${\mathrm{\Lambda }}_{\mathrm{r}\mathrm{e}\mathrm{g}}$, $M$, $E_{\mathrm{e}\mathrm{x}\mathrm{t}}$, ${\mu }_W$, and $\mu$ unless additional model data identify some of them. Write ${\rho }_{N_{\sigma }}$ for the realification that lists the real and imaginary parts of each coefficient. The supplied full RG vector field may couple different weight blocks; no block-diagonal restriction is imposed. A supplied RG vector field is a continuous map ${\beta }_{\sigma }:{\mathrm{\Omega }}_{\sigma }\to {\mathbb{R}}^{2N_{\sigma }}$ on a supplied open domain ${\mathrm{\Omega }}_{\sigma }\subset I\times {\mathbb{R}}^{2N_{\sigma }}$; an RG trajectory satisfies $\frac{d}{dt}{\rho }_{N_{\sigma }}(C_{\sigma }(t))={\beta }_{\sigma }(t,{\rho }_{N_{\sigma }}(C_{\sigma }(t)))\mathrm{.}$ For each sign, the physical operator datum is an instance of the selected physical EFT operator contract. It specifies finitely many field species and jet generators, finite-dimensional graded field spaces, smooth field domains and regularity, reality data, and rational engineering weights. For a chosen right principal bundle $P\to M$ and gauge representation ${\rho }_i$ on the component space ${\mathrm{\Phi }}_i$, fields are smooth sections of $E_i=P{\times }_{{\rho }_i}{\mathrm{\Phi }}_i$, with coefficients of the appropriate parity in the supplied finite Grassmann algebra. Here $[pg,v]=[p,{\rho }_i(g)v]$; if local principal sections obey $s_{\beta }=s_{\alpha }{g}_{\alpha \beta }$, the field components obey ${\psi }_{i,\beta }={\rho }_i(g_{\alpha \beta }^{-1}){\psi }_{i,\alpha }$. Smooth maps on all of $M$ describe these sections only after a global trivialization is chosen; with no gauge sector use $E_i=M\times {\mathrm{\Phi }}_i$.

Select the proper orthochronous Lorentz group $S{O}^{+}(1,3)=\{L\in G{L}_4(\mathbb{R}):L^T\eta L=\eta ,\det L=1,L^0{}_0\ge 1\}$, or its spin double cover $\mathcal{L}=SL(2,\mathbb{C})$ with projection $p:\mathcal{L}\to S{O}^{+}(1,3)$ and kernel $\{I,-I\}$ from the physical operator contract. In its convention $X(p(a)x)=aX(x)a^{†}$, where $X(x)=\left(\begin{array}{cc}{x}^0+x^3& x^1-i{x}^2\\ x^1+i{x}^2& x^0-x^3\end{array}\right)$. Specify the parity-preserving component representations of the selected group; spacetime and derivative indices transform through $p$ and its dual (with $p=\mathrm{id}$ in the tensor-only choice), and require each Lagrangian density to be a scalar. A global action on gauge-bundle sections additionally requires chosen lifts covering the spacetime action. The datum gives the full gauge and flavor actions on fields, indices, and spurions. A spurion is a fixed non-dynamical parameter used to record a declared symmetry-breaking insertion; absent flavor symmetry is recorded as trivial, and broken symmetry requires its spurions and weights.

The datum fixes finite derivative, field-degree, dimension, and loop cutoffs, the relation subspace and representatives, and a complex reductive affine algebraic group with maps relating its representations to the selected Lorentz, physical gauge and flavor groups. Here reductive means there is no nontrivial closed normal unipotent subgroup; unipotent means every nonzero finite-dimensional rational module has a nonzero fixed vector. On each retained jet space the data either verify that the algebraic invariant subspace equals the intended physical invariant space or label the quotient as a formal algebraic selection only.

Each relation generator is recorded as a polynomial or Fierz identity, a total divergence with compact-support or stated boundary conditions, or a leading-equation/field-redefinition relation tied to a base action and retained order. The latter records its variational derivative, multiplier, inverse through the retained order, and induced higher-order terms. This quotient alone does not establish action, off-shell Green-function, quantum-measure, or scattering equivalence.

If a non-Abelian gauge sector is selected, its actual compact real global group (compact meaning every open cover has a finite subcover), smooth right principal bundle (with smooth free and transitive right action and equivariant local trivializations), admitted bundle sectors, and unitary representations (actions preserving specified positive-definite Hermitian forms) are part of the data. A selected smooth principal connection uses the right-action convention $Z_P(p):={\frac{d}{dt}\mid }_{0}p{\exp }_G(tZ)$, ${\omega }_p(Z_P(p))=Z$, and $R_g(p)=pg$, $R_g^{\ast }\omega ={\mathrm{Ad}}_{g^{-1}}\omega$, where ${\mathrm{Ad}}_g=d(C_g{)}_e$ and $C_g(h)=gh{g}^{-1}$. Locally, for $A=s^{\ast }\omega$ and a representation $\rho$ with ${\rho }_{\ast }:=d{\rho }_e$, its curvature and associated covariant derivative are $F_A=dA+\frac{1}{2}[A\wedge A]$ and $D_{\mu }={\mathrm{\partial }}_{\mu }+{\rho }_{\ast }(A_{\mu })$, using the supplied graded Lie-bracket convention. Each finite basis represents only this supplied fixed-order quotient; no completeness claim about all physical relations or an infinite operator tower is made. These field and counting inputs are not inferred from coefficient coordinates. For $x\in {\mathbb{R}}^{2N_{\sigma }}$, write ${\mathrm{\Phi }}_{\sigma }(t,s)x\in {\mathbb{R}}^{2N_{\sigma }}$ for the value at $t$ of the unique realified coefficient solution with value $x$ at $s$, whenever a unique such solution exists on the interval between $s$ and $t$.

Let $S$ be a finite index set whose element $j$ specifies selected external states, kinematics, and, for a Green-function datum, an ordered field component together with any test-function pairing needed to define its value. For each $j\in S$, supply a common complex vector space ${\mathcal{Y}}_j$ of outputs and a finite-dimensional subspace ${\mathcal{Y}}_j^{\mathrm{r}\mathrm{e}\mathrm{t}}\subseteq {\mathcal{Y}}_j$, with a supplied linear projection $P_j:{\mathcal{Y}}_j\to {\mathcal{Y}}_j$ satisfying $P_j^2=P_j$ and $\mathrm{im}{P}_j={\mathcal{Y}}_j^{\mathrm{r}\mathrm{e}\mathrm{t}}$. This projection selects the declared retained loop, operator-dimension, and mass-expansion orders. Also supply a set ${\mathcal{R}}_{\mathrm{I}\mathrm{R}}$ of infrared-regulator labels and choose one common ${\lambda }_{\mathrm{I}\mathrm{R}}\in {\mathcal{R}}_{\mathrm{I}\mathrm{R}}$ for each comparison. For each $\sigma \in \{+,-\}$ and $j\in S$, a matching calculation supplies a map $A_{\sigma ,j}:D_{\sigma ,j}\to {\mathcal{Y}}_j$, where $D_{\sigma ,j}\subset I\times U_{\sigma }\times {\mathcal{R}}_{\mathrm{I}\mathrm{R}}$ is a specified domain. The supplied renormalization and normalization prescription for each theory is part of this data; in this definition, a renormalized amplitude or Green-function value means the resulting element of ${\mathcal{Y}}_j$ for the selected external-state, kinematic, and component data at the stated regulator. No regularity of $A_{\sigma ,j}$ is required beyond being defined at the inputs used below, and this definition neither constructs those maps nor asserts that they exist for every model.

Let $U_{\sigma }\subset {\mathbb{C}}^{N_{\sigma }}$ be open coefficient domains for $\sigma \in \{+,-\}$. The supplied matching family is a jointly continuously differentiable real map $\widehat{\mathcal{M}}:\mathcal{V}\to {\rho }_{N_{-}}(U_{-})$ on an open domain $\mathcal{V}\subset I\times {\rho }_{N_{+}}(U_{+})$. Its admissible domain is the open subset $\mathcal{U}=\{(t,x)\in \mathcal{V}:(t,x)\in {\mathrm{\Omega }}_{+},(t,\widehat{\mathcal{M}}(t,x))\in {\mathrm{\Omega }}_{-}\}$. It defines $M_t(c):={\rho }_{N_{-}}^{-1}(\widehat{\mathcal{M}}(t,{\rho }_{N_{+}}(c)))$ for each admitted $(t,c)$, and ${\widehat{M}}_t(x):=\widehat{\mathcal{M}}(t,x)$ on the corresponding slice. Thus the scale derivative and coefficient derivative in the compatibility condition below both exist and are continuous. It is a matching map through the declared loop order, operator dimension, and mass-expansion order when, for every selected $j\in S$ and every admitted pair $(t,c)$ with $c\in U_{+}$, both $(t,c,{\lambda }_{\mathrm{I}\mathrm{R}})\in D_{+,j}$ and $(t,M_t(c),{\lambda }_{\mathrm{I}\mathrm{R}})\in D_{-,j}$ and $P_j\text{\hspace{0.17em}⁣}(A_{+,j}(t,c;{\lambda }_{\mathrm{I}\mathrm{R}}))=P_j\text{\hspace{0.17em}⁣}(A_{-,j}(t,M_t(c);{\lambda }_{\mathrm{I}\mathrm{R}}))\mathrm{.}$ The retained orders and projection are part of the supplied data. This equality records only the selected outputs and orders; it gives no estimate on discarded terms and asserts no matching calculation exists for every model.

A matching map is RG-compatible on these domains when, in realified coefficient coordinates, ${\mathrm{\partial }}_t{\widehat{M}}_t(x)+D_x{\widehat{M}}_t(x){\beta }_{+}(t,x)={\beta }_{-}\text{\hspace{0.17em}⁣}(t,{\widehat{M}}_t(x))$ for every admitted $(t,x)$. Here $D_x$ is the ordinary real derivative with respect to the coefficient coordinates. RG-compatibility is an additional condition to be checked from the supplied beta functions and matching calculation; it does not follow merely from defining $M_t$ by amplitude equality.

For several thresholds, let $t_1>\cdots >t_k$ be their logarithmic scales, let ${\mathrm{\Phi }}_j$ be the realified evolution in the theory active between thresholds $t_j$ and $t_{j+1}$, and let ${\widehat{M}}_j(t_j)$ map realified coefficient data from the theory above threshold $j$ to the theory below it. The ordered high-to-low transport is ${\mathrm{\Phi }}_k(t_f,t_k)\circ {\widehat{M}}_k(t_k)\circ {\mathrm{\Phi }}_{k-1}(t_k,t_{k-1})\circ \cdots \circ {\widehat{M}}_1(t_1)\circ {\mathrm{\Phi }}_0(t_1,t_i),$ whenever all displayed flows and matching maps are defined. This definition specifies supplied finite-order data and their conventions; it computes no beta-function entry or matching coefficient and asserts no continuum limit.

Finite-regulator EFT model and scale data

Physical EFT operator data and power-counting contract

Finite renormalization schemes and Wilson coefficient vectors

Manohar, full EFT lecture notes

Burgess, full EFT review