Functional-RG Truncations and Projection Methods

The Wetterich equation defines an exact vector field on an infinite-dimensional space of functionals. A finite calculation requires three additional choices: an ansatz for the effective average action, a projection of the functional flow onto its running parameters, and a closure for structures outside the ansatz. Those choices are logically independent, and changing any one of them can change a truncated result.

This page makes that approximation contract explicit. It compares local-potential, derivative, vertex, polynomial, and field-grid representations; derives projections at fixed and moving expansion points; and tests polynomial and grid projections against an independently integrated zero-dimensional scalar benchmark. The result is a bounded stability statement, not a claim that either truncation is exact.

Required background. Effective Average Actions and the Wetterich Equation supplies the exact flow, regulator endpoints, inverse-Hessian condition, and scalar local-potential example used here.

Helpful background. The Polchinski Exact RG Equation shows how the same exact-versus-projected distinction appears in a Wilson-action formulation.

Ansatz, projection, and closure

Write the exact flow abstractly as

$$\partial_t\Gamma_k=\mathcal F_k[\Gamma_k], \qquad \mathcal F_k[\Gamma] = \frac12\operatorname{STr} \left[ \left(\Gamma^{(2)}+\mathcal R_k\right)^{-1} \partial_t\mathcal R_k \right].$$

A finite ansatz is a map from finitely many coordinates into theory space. For a fixed operator basis,

$$\Gamma_{k,N}[\varphi] = \sum_{i=1}^{N}g_i(k)\,\mathcal O_i[\varphi], \qquad \partial_t\Gamma_{k,N} = \sum_{i=1}^{N}\beta_i\,e_i, \qquad e_i\equiv\frac{\partial\Gamma_{k,N}}{\partial g_i}.$$

Choose linear functionals $\mathcal P_i$ dual to the tangent vectors, $\mathcal P_i[e_j]=\delta_{ij}$. The projected flow is then

$$\beta_i = \mathcal P_i\!\left[\mathcal F_k[\Gamma_{k,N}]\right], \qquad P_N X \equiv \sum_{i=1}^{N}e_i\,\mathcal P_i[X].$$

The equality is only between the tangent part of the two sides. The discarded functional residual,

$$\mathcal R_N[\varphi] = (1-P_N)\mathcal F_k[\Gamma_{k,N}],$$

need not be small merely because every projected beta function is numerically well resolved.

The closure rule says how the right-hand side is evaluated when it requests information outside the ansatz. Substituting a finite $\Gamma_{k,N}$ into the inverse Hessian is one closure. In a vertex hierarchy, setting $\Gamma_k^{(n>N)}=0$, replacing it by a classical vertex, or reconstructing it from a symmetry identity are different closures even if the retained vertices and projectors are unchanged.

If the basis itself runs, the coordinate flow contains a connection term:

$$\partial_t\Gamma_{k,N} = \sum_i\beta_i\mathcal O_i + \sum_i g_i\,\partial_t\mathcal O_i.$$

Consequently, $\beta_i=\mathcal P_i[\mathcal F_k-\sum_jg_j\partial_t\mathcal O_j]$. A scale-dependent field normalization, moving expansion point, or rebosonization map cannot be inserted after the projection without this chain rule.

Truncation families and their omissions

No representation is uniformly best. The useful question is which structures control the observable and regime under study.

Representation	Typical ansatz or data	Natural projection	Structures omitted first
Local-potential approximation (LPA)	A full or polynomial $U_k(\rho)$ with fixed kinetic term	Constant fields	Wave-function, momentum, and higher-derivative dependence
Derivative expansion	$U_k$, $Z_k$, $Y_k$, and successively higher derivative operators	Small external momenta and selected field values	Higher powers of momentum and tensor structures at the next derivative order
Vertex expansion	Momentum-dependent $\Gamma_k^{(n)}$ through a declared $n$	Functional derivatives at chosen momenta	Higher vertices and unresolved momentum channels
Field grid or spectral basis	Values or basis coefficients over a finite field domain	Collocation or weighted residuals	Behavior outside the domain and unresolved field-space oscillations
Hybrid expansion	Selected vertices with resolved momentum and field dependence	Mixed momentum, field, and tensor projectors	Channels not included in the hybrid basis

For an $O(N)$ scalar with $\rho=\varphi^a\varphi^a/2$, a derivative expansion can begin as

$$\Gamma_k[\varphi] = \int d^dx\, \left[ U_k(\rho) + \frac12 Z_k(\rho)\, \partial_\mu\varphi^a\partial^\mu\varphi^a + \frac14Y_k(\rho)\, \partial_\mu\rho\,\partial^\mu\rho + O(\partial^4) \right].$$

LPA retains $U_k$ and fixes $Z_k$; LPA-prime also runs a field-independent $Z_k$. A polynomial potential resolves local derivatives near its expansion point, while a grid can resolve a wider field interval but introduces domain and interpolation errors. Neither choice determines whether the derivative expansion itself is adequate.

A vertex expansion instead writes

$$\Gamma_k[\varphi] = \sum_{n=0}^{\infty}\frac1{n!} \int_{p_1\cdots p_n} (2\pi)^d\delta\!\left(\sum_jp_j\right) \Gamma_k^{(n)}(p_1,\ldots,p_n) \prod_{j=1}^{n}\varphi(p_j).$$

Cutting this series at $n=N$ does not by itself specify the momentum dependence retained in each $\Gamma_k^{(n)}$, nor how $\Gamma_k^{(N+1)}$ and $\Gamma_k^{(N+2)}$ are closed when the flow differentiates the inverse propagator. A complete method statement includes both decisions.

Projection points and moving coordinates

For a potential expanded about a fixed $\rho_\star$,

$$U_k(\rho) = \sum_{n=0}^{M} \frac{\lambda_n(k)}{n!} (\rho-\rho_\star)^n,$$

the projector is $\mathcal P_n=\partial_\rho^n|_{\rho_\star}$ after a declared normalization. Expanding about the running minimum $\kappa_k$ can be more efficient in a broken phase, but the minimum condition must flow:

$$U_k'(\kappa_k)=0, \qquad \dot\kappa_k = - \frac{ \left.\partial_tU_k'(\rho)\right|_{\rho=\kappa_k} }{ U_k''(\kappa_k) }.$$

The coupling derivatives then obey

$$\dot\lambda_n = \left. \partial_\rho^n\partial_tU_k \right|_{\rho=\kappa_k} + \lambda_{n+1}\dot\kappa_k.$$

These formulas require $U_k''(\kappa_k)\ne0$. Near a flat minimum, the moving-coordinate chart becomes ill conditioned even when the underlying potential is regular; a grid or a fixed expansion point provides an important cross-check.

Momentum projectors need equally explicit kinematics. For example, a wave-function factor extracted from a two-point function may use

$$Z_k(\rho_\star) = \left. \frac{\partial\Gamma_k^{(2)}(p,-p;\rho_\star)} {\partial p^2} \right|_{p^2=p_\star^2}.$$

The choices $p_\star^2=0$, $p_\star^2=k^2$, and a symmetric nonzero momentum configuration are distinct truncated projections. Tensor projectors must also remove longitudinal, trace, flavor, or gauge components with a stated normalization. Symmetrizing over equivalent external legs before projection prevents the numerical procedure from selecting one channel accidentally.

Regulator dependence as an approximation diagnostic

At the exact infrared endpoint, admissible regulator profiles describe different paths to the same physical effective action. A finite ansatz generally leaves residual profile dependence because projection and exact evolution do not commute. It is therefore meaningful to vary the shape and normalization of $R_k$ for physical observables, but not to demand that nonuniversal running couplings agree.

Exact large-$N$ solutions make that distinction especially clean: in the models studied there, critical exponents and qualitative fixed-point properties are regulator independent, whereas fixed-point coupling coordinates are generically regulator dependent even without truncation error. This rules out using agreement of raw coupling coordinates as a universal regulator test. Knorr 2021, preprint §§ II.F, III.D, and IV, pp. 9, 13–14

Regulator smoothness can also change the behavior of a derivative expansion. In a perturbative scalar test, Morris and Tighe found rapid two-loop convergence for the Legendre flow with a smooth exponential cutoff, slow or failed momentum expansions for a sharp cutoff in some channels, and divergent higher-derivative coefficients for certain power-law cutoffs. This is evidence for those flows and operators, not a theorem that one profile is optimal in every theory. Morris and Tighe 1999, preprint § 7, pp. 17–18

For a regulator family $R_k^{(\alpha)}$, the principle of minimal sensitivity chooses a stationary region,

$$\left. \frac{\partial O_N(\alpha)}{\partial\alpha} \right|_{\alpha=\alpha_{\mathrm{PMS}}} =0.$$

Stationarity is a diagnostic, not proof of convergence: a plateau can move, split, or disappear at the next truncation order. Quantitative arguments connecting regulator choice, momentum analyticity, and successive derivative orders have been developed for scalar derivative expansions; their conclusions should not be transferred unchanged to fermionic or gauge systems. De Polsi and Wschebor 2022, pp. 024111-1–024111-2 and 024111-8

A zero-dimensional polynomial–grid benchmark

Consider the independently integrable $\mathbb Z_2$-symmetric fixture

$$\frac{Z}{Z_G} = \frac{ \displaystyle\int_{-\infty}^{\infty}dx\, e^{-x^2/2-x^4/24} }{ \displaystyle\int_{-\infty}^{\infty}dx\, e^{-x^2/2} }, \qquad F\equiv-\ln\frac{Z}{Z_G}.$$

Adaptive quadrature with absolute tolerance $10^{-13}$ gives

$$F_{\mathrm{quad}} = 0.08455740217041746,$$

with a raw integral error estimate $3.49\times10^{-14}$. This value is not used to tune the flow; it is revealed afterward as an independent endpoint check.

In zero dimensions the regulator is a positive number $r$. The exact effective-average-action equation becomes

$$\frac{\partial\Gamma_r(\phi)}{\partial r} = \frac{1}{2\left[\Gamma_r''(\phi)+r\right]}.$$

We start at $r_{\mathrm{UV}}=10^7$ with the same microscopic action $S(\phi)=\phi^2/2+\phi^4/24$ in every interacting run and subtract the identically evolved Gaussian vacuum term. Varying $r_{\mathrm{UV}}$ measures the finite-start error. Because there is no momentum in zero dimensions, any monotone scalar profile $r=r(k)$ merely reparameterizes this equation; matched endpoints have exactly zero regulator-shape spread. The fixture tests ansatz, projection, closure, and numerics, but it cannot test momentum-profile dependence.

The polynomial ansatz is

$$\Gamma_r^{(2M)}(\phi) = c_r + \sum_{m=1}^{M} \frac{g_{2m,r}}{(2m)!}\phi^{2m}, \qquad 2M\in\{4,6,8\},$$

with higher even coefficients set to zero when the reciprocal Hessian is re-expanded at $\phi=0$. The field-grid calculation instead evolves the potential at 29 Chebyshev nodes on $|\phi|\le4$, evaluates $\Gamma_r''$ with the spectral differentiation matrix, and projects every right-hand side back to its even part. Polynomial flows use DOP853 with relative tolerance $2\times10^{-12}$; the central grid uses Radau with relative tolerance $10^{-9}$. Every ansatz retains the running constant $c_r$.

Approximation	Endpoint $F$	$F-F_{\mathrm{quad}}$	Diagnostic not using the residual
Exact adaptive quadrature	$0.084557402170$	—	Absolute tolerance $10^{-13}$
Polynomial through $\phi^4$	$0.083405246187$	$-1.1522\times10^{-3}$	First retained order
Polynomial through $\phi^6$	$0.084702231728$	$+1.4483\times10^{-4}$	Change from $\phi^4$: $1.2970\times10^{-3}$
Polynomial through $\phi^8$	$0.084548469582$	$-8.9326\times10^{-6}$	Change from $\phi^6$: $1.5376\times10^{-4}$
Chebyshev grid, 29 nodes	$0.084557396058$	$-6.1125\times10^{-9}$	Declared variation envelope: $1.1\times10^{-6}$

The polynomial residual alternates in sign, so these three points do not establish monotone convergence. The last nested change supports only the conservative statement

$$F_{\mathrm{poly}}=0.08455\pm0.00016$$

within this polynomial family. For the grid, lowering $r_{\mathrm{UV}}$ to $10^6$ changes $F$ by $1.69\times10^{-7}$; at $|\phi|\le5$, changing from 25 to 29 nodes changes it by $8.76\times10^{-7}$; and tightening the ODE tolerance changes it by $2.76\times10^{-10}$. Adding those diagnostic magnitudes and rounding upward gives

$$F_{\mathrm{grid}}=0.0845574\pm0.0000011.$$

That interval is deliberately much wider than the observed exact residual. It is an explored-variation envelope, not a statistical confidence interval. The accepted runs keep the sampled regularized Hessian above $1.32$ and the grid’s $\mathbb Z_2$ asymmetry below $2.5\times10^{-14}$. A result from a resolution that creates a false Hessian pole is rejected rather than averaged into the envelope.

Minimum FRG validation checklist

The following table is the minimum information needed to interpret a finite functional-RG result. A row may be inapplicable—for example, momentum-profile variation in the zero-dimensional fixture—but it may not be silently omitted.

Record	Declare before solving	Required check	Failure that blocks the claim
Regulator and endpoints	Kernel for every field species, shape parameters, normalization, ultraviolet data, and infrared removal limit	Repeat with admissible regulator choices and verify both endpoint conditions	A singular trace, unmatched endpoints, or an observable that moves beyond the stated range
Ansatz and omitted structures	Fields, operators, derivative order, vertex order, field domain, and every deliberately omitted channel	Enlarge the ansatz in at least one physically relevant direction	No explicit account of what the next enlargement adds
Projection and coordinates	Field values, external momenta, tensor normalization, running basis, and expansion point	Change projection point or representation and include all chain-rule terms	An unsupported dependence on projector kinematics or a singular coordinate chart
Closure	Treatment of higher vertices, operators, and momentum dependence requested by the flow	Compare a distinct closure or bound the discarded functional residual	A hidden replacement of an omitted structure by zero or by classical data
Nested truncations	At least three successive orders when available, with identical inputs and observables	Report signed values and successive differences rather than only the final order	A digit retained beyond the largest relevant nested change
Symmetry identity	Exact, modified, or broken identity appropriate to the regulator and approximation	Evaluate the identity residual independently of the projected flow equations	A residual comparable to the retained physical signal without a qualified claim
Convexity and invertibility	Domain on which $\Gamma_k^{(2)}+R_k$ must be invertible and the expected infrared convexity behavior	Monitor its smallest relevant eigenvalue and field-domain dependence	An unhandled Hessian pole, unstable branch, or claimed endpoint before convexification
Independent benchmark	Analytic limit, perturbative coefficient, solvable model, alternative method, or external data	Reproduce the benchmark without tuning inputs to its answer	Unmatched inputs, circular calibration, or unexplained discrepancy
Numerics and justified digits	Solver, tolerances, grid or basis, convergence criterion, precision, and uncertainty prescription	Vary resolution and tolerance and retain only digits stable under all larger effects	Solver convergence alone or more printed digits than the uncertainty supports

Turning variations into a bounded claim

Keep the diagnostic components separate:

$$\boldsymbol\delta = \left( \delta_{\mathrm{num}}, \delta_{\mathrm{UV}}, \delta_{\mathrm{domain}}, \delta_{\mathrm{trunc}}, \delta_{\mathrm{reg}}, \delta_{\mathrm{proj}}, \delta_{\mathrm{sym}}, \delta_{\mathrm{bench}} \right).$$

They are usually systematic variations, not independent Gaussian errors, so adding them in quadrature is rarely justified. A conservative envelope can use a linear sum of independently bounded components or the largest observed excursion across a declared family. The report must say which rule was used.

A defensible conclusion has the form: “For observable $O$, within ansätze $A_N$, regulators $R_\alpha$, projectors $P_j$, and the stated field and momentum domain, all accepted variations lie in $O_\star\pm\Delta$.” It does not say that the exact functional flow has been solved. Independent perturbation theory, a solvable limit, or a second nonperturbative method is what turns internal stability into stronger evidence.

Common pitfalls

A converged ODE is not a converged truncation. Solver tolerances control the finite equations that were supplied. They say nothing about the discarded functional residual.

A small regulator spread is not an error bar by itself. Several profiles can agree because they probe the same restricted ansatz. Regulator variation must accompany ansatz and projection enlargement.

Polynomial order is not derivative order. Adding $\rho^n$ terms improves field dependence inside a local potential; it does not add momentum dependence or wave-function operators.

Raw couplings are not automatically physical comparisons. Field normalization, regulator coordinates, and redundant operators can move coupling values while invariant observables remain fixed.

Exercises

1. Starting from $U_k'(\kappa_k)=0$, derive the flow of $\kappa_k$ and identify the condition under which this coordinate choice fails.

Solution

Take a total $t$ derivative:

$$0 = \left.\partial_tU_k'(\rho)\right|_{\rho=\kappa_k} + U_k''(\kappa_k)\dot\kappa_k.$$

Solving gives the formula above when $U_k''(\kappa_k)\ne0$. If the curvature vanishes, the minimum is not a regular local coordinate and the projected $\dot\kappa_k$ can diverge even though $U_k$ remains finite.

2. Let $r=f(k)$ be positive and monotone. Show that two zero-dimensional regulator profiles with the same endpoint values give the same flow when $\Gamma$ is regarded as a function of $r$.

Solution

The scale flow is

$$\partial_k\Gamma_k = \frac12 \frac{\partial_kr} {\Gamma_k''+r}.$$

Where $\partial_kr\ne0$, divide by $\partial_kr$ to obtain $\partial_r\Gamma_r=1/[2(\Gamma_r''+r)]$. Thus the shape of $f$ only changes the speed along the same trajectory. This argument does not apply to momentum-dependent profiles $R_k(q)$, whose shapes are distinct functions rather than one scalar coordinate.

3. The exact residual of the 29-node grid is about $6.1\times10^{-9}$, while its declared envelope is $1.1\times10^{-6}$. Explain why reporting the smaller number as the method uncertainty would be circular.

Solution

The residual uses the exact answer, which is normally unknown and is reserved here for validation. A prospective uncertainty must be constructed from information available without that answer: UV-start, domain, resolution, tolerance, regulator, projection, and truncation variations. The larger envelope is therefore the honest prediction-stage statement even though this benchmark later shows that it is conservative.

The next page applies this validation structure to regulator dependence, modified symmetry identities, convexity, and reliability criteria in realistic functional flows: Symmetry, Regulator Dependence, and Functional-RG Error Control.

References

• De Polsi, Gonzalo, and Nicolás Wschebor. “Regulator Dependence in the Functional Renormalization Group: A Quantitative Explanation.” Physical Review E 106 (2022) 024111. DOI. Open PDF
• Knorr, Benjamin. “Exact Solutions and Residual Regulator Dependence in Functional Renormalisation Group Flows.” Journal of Physics A: Mathematical and Theoretical 54 (2021) 275401. DOI. Open PDF
• Morris, Tim R., and John F. Tighe. “Convergence of Derivative Expansions of the Renormalization Group.” Journal of High Energy Physics 1999, no. 08 (1999) 007. DOI. Open PDF