Integrating out fields and the Wilsonian effective theory

Let $(M_d,g)$ be a smooth time-oriented Lorentzian $d$-manifold with signature $(-,+,\dots ,+)$ and a declared metric convention (for example, Einstein or string metric). On each region used for a momentum comparison, choose a smooth orthonormal frame $(e_0,\dots ,e_{d-1})$ with $e_0$ future-directed, $g(e_0,e_0)=-1$, and $g(e_i,e_j)={\delta }_{ij}$ and $g(e_0,e_i)=0$ for spatial indices. Thus the observer field is $e_0$; existence of a global frame is not assumed. For a real momentum covector $p\in T_x^{\ast }{M}_d$, put $p_a:=p(e_a)$ and define its observer energy by $E_{e_0}(p):=-p(e_0)$ for $p=0$ or for a nonzero covector satisfying $g^{-1}(p,p)\le 0$ and $p(e_0)<0$ (the future causal covectors in this signature). Declare a momentum domain $\mathcal{P}\subseteq T^{\ast }{M}_d$ over the stated region, containing all momentum arguments of the propagators being expanded, including channel sums when relevant. Choose a finite bound $E\ge 0$ such that $\mathrm{\mid }{p}_a\mathrm{\mid }\le E$ for every $p\in \mathcal{P}$ and every $a$. This bounds energy and spatial momentum, including off-shell momenta; energy alone is not a bound on arbitrary spacelike covectors. Use $\mathrm{\hslash }=c=1$, so $E$ and mass thresholds have units of inverse length. Field configurations and tree elimination below do not themselves require a momentum domain. Let $C_{<},C_{>}$ be specified $C^1$ Banach manifolds of retained fields $\varphi$ and omitted fields $\chi$ (sections of declared bundles with fixed regularity, boundary conditions, and gauge treatment), and let $S:C_{<}\times C_{>}\to \mathbb{R}$ be a specified $C^1$ action. Define the omitted variational equation by ${\mathcal{E}}_{>}(\varphi ,\chi ):=D_{\chi }S{\mathrm{\mid }}_{(\varphi ,\chi )}\in T_{\chi }^{\ast }{C}_{>}\mathrm{.}$ A tree-elimination branch is an open set $U\subseteq C_{<}$ and a $C^1$ map ${\chi }_{\ast }:U\to C_{>}$ such that ${\mathcal{E}}_{>}(\varphi ,{\chi }_{\ast }(\varphi ))=0$ for every $\varphi \in U$, against every admissible omitted variation. It defines $S_{\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{e}}[\varphi ]:=S[\varphi ,{\chi }_{\ast }(\varphi )]\mathrm{.}$ The branch, its domain, and its boundary and gauge prescriptions are part of the definition. If a quadratic operator $K$ is inverted, its domain, codomain, gauge fixing, boundary prescription, and inverse on the specified source space must be declared. Any zero or gauge modes on which $K$ is not invertible must first be retained, constrained by an explicitly stated condition, or gauge-fixed.

For a stable omitted quadratic sector in a common external frame, let ${\mathrm{\Sigma }}_{\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}\subseteq [0,\mathrm{\infty })$ be the declared spectrum of physical mass-squared values, including the support of any continuous spectrum. Define its mass threshold by $M_{\mathrm{g}\mathrm{a}\mathrm{p}}:=\inf \{\sqrt{\lambda }:\lambda \in {\mathrm{\Sigma }}_{\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}\}\mathrm{.}$ This has units of inverse length. If there is no omitted sector, set $M_{\mathrm{g}\mathrm{a}\mathrm{p}}=+\mathrm{\infty }$; if the spectral threshold is zero, then $M_{\mathrm{g}\mathrm{a}\mathrm{p}}=0$ and there is no positive-gap derivative expansion. An omitted instability with negative mass-squared lies outside the stable-sector definition and also supplies no positive gap.

A regulated Wilsonian integration is a different operation. In $\mathrm{\hslash }=1$ units, continue to a declared Euclidean problem, choose a regulator and finite-volume or finite-dimensional regularization, fix the gauge and boundary conditions, choose a matching scale $\mathrm{\Lambda }>0$ of inverse-length dimension, and declare a high-energy subspace ${\mathcal{H}}_{>}(\mathrm{\Lambda })$ of the regulated omitted-field space (for example, by a spectral projector of a specified gauge-fixed quadratic operator). Declare which modes remain in the retained theory and a regulated measure ${\mathcal{D}}_{\mathrm{\Lambda },\mathrm{r}\mathrm{e}\mathrm{g}}{\chi }_{>}$ on the integrated subspace, including any gauge-fixing determinant, and define, when the integral exists and is nonzero, $e^{-S_{W,\mathrm{\Lambda }}[\varphi ]}:={\int }_{{\mathcal{H}}_{>}(\mathrm{\Lambda })}{\mathcal{D}}_{\mathrm{\Lambda },\mathrm{r}\mathrm{e}\mathrm{g}}{\chi }_{>}{e}^{-S_{E,\mathrm{r}\mathrm{e}\mathrm{g}}[\varphi ,{\chi }_{>}]}\mathrm{.}$ This operation includes fluctuation and loop contributions; it is not the tree saddle. No functional measure or integral is asserted to exist without the stated regulator and measure data.

The tree-eliminated action is generally nonlocal. A local derivative expansion is a separate low-momentum approximation. Specify the momentum representation of each omitted-sector propagator, its domain and expansion about zero momentum, and a finite Taylor truncation with a remainder estimate valid uniformly on $\mathcal{P}$. Require that the propagators are analytic on a neighborhood of that expansion domain, with $0<M_{\mathrm{g}\mathrm{a}\mathrm{p}}<\mathrm{\infty }$ and a declared bound $E\mathrm{/}{M}_{\mathrm{g}\mathrm{a}\mathrm{p}}\le \delta <1$; the notation $E\ll M_{\mathrm{g}\mathrm{a}\mathrm{p}}$ means a family for which this ratio tends to zero with the other declared bounds fixed. Replace the propagators by those Taylor polynomials, retaining the supplied remainder estimates. In a translation-invariant flat problem the representation is the ordinary Fourier multiplier. On a curved or nontranslation-invariant background an application must additionally specify its local coordinate or symbol representation, its rule converting polynomial momenta to differential operators, and the corresponding remainder controls; none is inferred from the gap or from pointwise analyticity alone. If $M_{\mathrm{g}\mathrm{a}\mathrm{p}}=0$ there is no positive-gap expansion, and if $M_{\mathrm{g}\mathrm{a}\mathrm{p}}=+\mathrm{\infty }$ because the omitted sector is empty there is no omitted propagator to expand. All comparisons use the chosen orthonormal frame for the same declared metric convention. Neither this approximation nor setting $\chi =0$ is the tree-elimination operation. An exact tree-elimination branch gives a consistent truncation in the solution-uplift sense only when its reduced stationary solutions uplift to solutions of every full equation under the separately defined consistent-truncation criterion.

Solution-preserving consistent truncation