Supersymmetric Wilsonian EFT action and threshold data

Use natural units $c=\mathrm{\hslash }=1$, flat spacetime $\mathcal{M}={\mathbb{R}}^{1,3}$, coordinates $x^0=ct$, and metric $\eta =\mathrm{diag}(-1,1,1,1)$. In these units $[x^{\mu }]=-1$, $[{\mathrm{\partial }}_{\mu }]=1$, $[{\theta }^{\alpha }]=[{\bar{\theta }}^{\dot{\alpha }}]=-1\mathrm{/}2$, $[d^2\theta ]=1$, and $[d^4\theta ]=2$. A local rigid four-dimensional $\mathcal{N}=1$ supersymmetric Wilsonian action datum uses finitely many smooth classical chiral superfields ${\mathrm{\Phi }}_i$ and, when a gauge sector is selected, a compact closed matrix subgroup $G\subset U(N)$ with its inherited embedded Lie-group structure, smooth finite-dimensional unitary representations ${\rho }_i:G\to U({\mathcal{H}}_i)$, and its specified vector-superfield data. Write $\mathfrak{g}=T_{I}G$ with the matrix commutator and ${\rho }_{i\ast }^{\mathbb{C}}:{\mathfrak{g}}_{\mathbb{C}}\to {\mathrm{End}}_{\mathbb{C}}({\mathcal{H}}_i)$ for the complex-linear extension of the differential of ${\rho }_i$. The dimensionless even Hermitian vector prepotential $V$ takes values in $i\mathfrak{g}$, and its represented value is $V_i={\rho }_{i\ast }^{\mathbb{C}}(V)$. The represented chiral gauge curvature is $W_{\alpha }=-\frac{1}{4}{\bar{D}}^{\dot{\beta }}{\bar{D}}_{\dot{\beta }}(e^V{D}_{\alpha }{e}^{-V})$ in the selected faithful matrix representation, where $D_{\alpha }$ and ${\bar{D}}_{\dot{\alpha }}$ are the covariant derivatives in the supplied chiral-superfield convention; ${\mathrm{Tr}}_{\mathrm{h}}$ is the supplied quadratic trace functional with fixed normalization and sign, invariant under ${\mathrm{Ad}}_k$ for every $k\in G$, including disconnected components. For multiple kinetic coefficients the displayed ${\tau }_{\lambda }{\mathrm{Tr}}_{\mathrm{h}}$ is shorthand for the full invariant quadratic polynomial ${\sum }_a{\tau }_{a,\lambda }{Q}_a$ specified by the supplied action data; coefficients on simple ideals permuted by disconnected components satisfy that invariance, and allowed Abelian mixing is included. A chiral field obeys ${\bar{D}}_{\dot{\alpha }}{\mathrm{\Phi }}_i=0$; when $[{\mathrm{\Phi }}_i]=1$, its physical scalar, Weyl-spinor, and auxiliary components have mass dimensions $1$, $3\mathrm{/}2$, and $2$, respectively, while $[W_{\alpha }]=3\mathrm{/}2$. If no gauge sector is selected, omit the gauge terms. The action has the superspace form

${\mathcal{L}}_{\lambda }=[{\sum }_i{\mathrm{\Phi }}_i^{†}{e}^{-V_i}{\mathrm{\Phi }}_i+G_{\lambda }{]}_D+2\mathrm{Re}[\frac{{\tau }_{\lambda }}{8\pi i}{\mathrm{Tr}}_{\mathrm{h}}(W^{\alpha }{W}_{\alpha })+F_{\lambda }(\mathrm{\Phi },W){]}_F,$

where the physical action brackets are $[U{]}_D=-I_D(U)\mathrm{/}4$, $[Y{]}_F=I_F(Y)\mathrm{/}2$ from Physical superspace action brackets and canonical chiral components, with physical chiral components ${\psi }_{\mathrm{r}\mathrm{a}\mathrm{w}}=\sqrt{2}\psi$ and $F_{\mathrm{r}\mathrm{a}\mathrm{w}}=2F$. The real local D-term integrand $G_{\lambda }$ has mass dimension two, and its integrated action is gauge invariant under the declared boundary conditions; it may have the local derivative dependence allowed for the D sector by the supplied action data. The holomorphic gauge-invariant F-term integrand $F_{\lambda }$ has mass dimension three and contains no superspace or spacetime derivatives, while $[{\tau }_{\lambda }]=0$. Thus $[{\mathcal{L}}_{\lambda }]=4$. The chiral and real terms are respectively called the Wilsonian F-term and D-term. The formal mode scales $\lambda$ and ${\mathrm{\Lambda }}_0$ have mass dimension one and satisfy $0<\lambda <{\mathrm{\Lambda }}_0$; the cutoff prescription supplies a coefficientwise local derivative expansion and asserts neither convergence nor a continuum functional measure. A connected perturbative graph is one-particle-irreducible if deleting any one internal propagator edge leaves it connected; the sum of such graphs is 1PI vertex data, separate from the Wilsonian action.

A finite supersymmetric EFT operator datum selects finitely many field species over $\mathbb{C}$, their coefficient spaces and Grassmann parities, their gauge and flavor actions, and a maximum spacetime-jet order $r$. A field jet is a component of a field or one of its spacetime derivatives ${\mathrm{\partial }}_I{\mathrm{\Phi }}_i$ for a multi-index $I\in {\mathbb{N}}_0^4$ with $\mathrm{\mid }I\mathrm{\mid }\le r$, where ${\mathrm{\partial }}_I={\mathrm{\partial }}_0^{I_0}{\mathrm{\partial }}_1^{I_1}{\mathrm{\partial }}_2^{I_2}{\mathrm{\partial }}_3^{I_3}$. It also selects a coefficient field, spurions, maximum engineering weight $D_{\max }$, field-degree cutoff $q$, loop cutoff $L_{\max }$, and a finite ordered basis ${\mathcal{O}}_{\alpha }$ for the declared quotient ${\mathcal{Q}}_D=V_D\mathrm{/}{R}_D$ from the supplied finite-order invariant operator-basis definition: $V_D$ is the invariant scalar subspace of finite local polynomials of engineering weight $D$, and $R_D$ is the supplied relation subspace restricted to $V_D$. Each ${\mathcal{O}}_{\alpha }$ is a Lorentz-scalar representative built from finitely many such jets and has the declared gauge/flavor transformation law and engineering weight ${\mathrm{\Delta }}_{\alpha }$. Each retained coefficient $C_{\alpha ,\mathrm{\ell }}$ is dimensionless in the convention

${\mathcal{L}}_{\mathrm{t}\mathrm{r}}={\sum }_{{\mathrm{\Delta }}_{\alpha }\le D_{\max }}{\sum }_{\mathrm{\ell }=0}^{L_{\max }}{\zeta }^{\mathrm{\ell }}{C}_{\alpha ,\mathrm{\ell }}({\mu }_W)\frac{{\mathcal{O}}_{\alpha }}{{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^{{\mathrm{\Delta }}_{\alpha }-4}},$

where $\zeta$ is a dimensionless formal loop-bookkeeping variable, ${\mathrm{\Delta }}_{\alpha }$ is the engineering weight of ${\mathcal{O}}_{\alpha }$, and ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}>0$ is its coefficient-suppression scale. The declared relations may include polynomial or Fierz identities, superspace integration by parts under stated compact-support or boundary conditions, gauge identities, and leading-superfield-equation relations. Every equation-of-motion or field-redefinition relation records its base action, parity-preserving local identity-leading map, inverse through the retained order, and all induced terms. This quotient records only the supplied relations and does not assert that it is a complete physical basis.

An optional on-shell amplitude-equivalence record for such a relation includes a local, parity-preserving formal field map with identity leading term and inverse through the retained order, an invariant action for the declared complex reductive symmetry, the exact finite-regulator change-of-variables density and its Jacobian, the correspondingly transformed source, and a common regulator limit with stable positive-mass simple poles, nonzero residues, and the stated LSZ pole factorization. Adopt full Axiom of Choice as a sufficient ambient assumption: every set-indexed family of nonempty sets has a choice function. The fixed-order equivalence theorem applies to the supplied record under its exact hypotheses; it requires no additional full-Choice premise. The quotient alone does not identify off-shell Green functions or prove that a regulator, its Jacobian, or an LSZ limit exists.

For one declared threshold, the high-energy and low-energy descriptions share their specified light fields and differ by a nonempty declared heavy-multiplet sector of threshold mass $M>0$. A finite matching record is a complete instance of the finite-order threshold-matching contract. It supplies the high- and low-energy coefficient spaces and bases, selected renormalized amplitudes or Green functions, their common infrared regulator, external states and kinematics, matching map, and retained loop, operator, and mass-expansion orders. In particular, separately supply the admitted external-kinematic domain and the range of validity assigned to the selected local expansion; every selected amplitude or Green-function comparison must lie in both. If the record is interpreted as an ordinary local heavy-particle expansion, supply an external-energy bound $0\le E_{\mathrm{e}\mathrm{x}\mathrm{t}}<M$ and restrict the selected kinematics accordingly. Other supplied finite-order expansion regimes must declare their own validity conditions; naming coefficient coordinates or an external-energy scale alone establishes no domain of physical applicability. These are conditions on using a matching record, not an existence or convergence assertion for an arbitrary expansion. For each sign $\sigma \in \{+,-\}$, let ${\mathcal{A}}_{\sigma }$ be the declared set of retained Wilsonian coefficient records and $a_{\sigma }({\mu }_W)\in {\mathcal{A}}_{\sigma }$ the record extracted from that theory's displayed action at $\lambda ={\mu }_W$ by physical component extraction, the declared quotient representatives, and the retained truncation. The matching record must supply coordinate identifications ${\iota }_{\sigma }({\mu }_W,\mu ):{\mathcal{A}}_{\sigma }\to U_{\sigma }\subseteq {\mathbb{C}}^{N_{\sigma }}$, bijective onto the declared scheme-coordinate domains at the scales used. These identifications specify the operator normalizations, suppression scales, field coordinates, and finite-order scheme conversion; they are additional input, not an equality of the two scales. With $t=\log (\mu \mathrm{/}{\mu }_0)$ for the matching record's reference scale ${\mu }_0>0$, its scheme vector is $c_{\sigma }={\iota }_{\sigma }({\mu }_W,\mu )(a_{\sigma }({\mu }_W))$. A supplied scheme matching map $M_t$ describes these actions only when $c_{-}=M_t(c_{+})$, equivalently $a_{-}={\iota }_{-}^{-1}\circ M_t\circ {\iota }_{+}(a_{+})$ at the fixed scale pair, and the selected amplitude conditions hold. Any claimed holomorphic component additionally specifies how these identifications act on the retained F-term coordinates; no preservation of that sector is inferred from arbitrary scheme coordinates. The Wilsonian scale ${\mu }_W$, renormalization scale $\mu$, heavy threshold mass $M$, external-energy bound $E_{\mathrm{e}\mathrm{x}\mathrm{t}}$, regulator cutoff ${\mathrm{\Lambda }}_{\mathrm{r}\mathrm{e}\mathrm{g}}$, and operator-suppression scale ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}$ all have mass dimension one and are distinct unless additional model data identify them. Matching equality concerns only the selected amplitudes and stated truncation; it supplies no discarded-term error bound and is not an equality of arbitrary one-particle-irreducible functionals.

Local rigid $N=1$ Wilsonian action and its formal cutoff data

Finite coefficient spaces, matching map, common infrared regulator, and separate scales

Finite field, jet, symmetry, and operator-basis data

Conditional on-shell equivalence under an exact regulated local change of variables

Full Axiom of Choice adopted as a sufficient ambient assumption

Weinberg, equations (1)–(3)

Martin, §§4.7 and 4.10

Manohar, §§5.5–5.7 and 7.1–7.2

Burgess, §§2.2.1–2.2.2 and 4.1.4

Criado–Pérez-Victoria, §2 and Appendix B

Physical superspace action brackets and canonical chiral components