Scheme Transformations and RG Invariants

A renormalization scheme is a choice of finite coordinates for the same renormalized physics. When two mass-independent schemes are related by an analytic, locally invertible redefinition of couplings and fields, their beta functions are different component descriptions of one RG vector field. Coupling values, field anomalous dimensions away from fixed points, higher beta-function coefficients, and the conventional normalization of $\Lambda$ can change; consistently translated observables cannot.

This page derives those transformation laws and works a one-coupling map through three beta-function coefficients. It also marks the assumptions that are often suppressed: the map must be regular on the domain, all quantities must be re-expanded to the same order, and mass-dependent prescriptions require explicit threshold variables. These qualifications are essential before calling a fixed point, exponent, or transmuted scale “scheme independent.”

Required background. Beta Functions, Running Masses, and Field Anomalous Dimensions fixes the signs of $\beta$, $\gamma$, and mass running. Renormalization Conditions, Schemes, and Finite Parts explains how finite counterterms define a prescription.

Helpful background. Operator Anomalous-Dimension Matrices supplies the operator-basis conventions used below.

Finite prescriptions as coordinate maps

Let $g^i$ be dimensionless renormalized couplings in one scheme and let

$$g'^i=f^i(g), \qquad J^i{}_{j}(g) \equiv \frac{\partial f^i}{\partial g^j}.$$

Assume that $f$ is analytic near the perturbative point, preserves the declared coupling normalization, has no explicit $\mu$ dependence, and satisfies

$$\det J(g)\ne0$$

throughout the comparison domain. Differentiation at fixed bare data gives

$$\boxed{ \beta'^i(g') = J^i{}_{j}(g)\,\beta^j(g). }$$

Thus beta functions transform as a vector field. If $g(t)$ solves $d g^i/dt=\beta^i(g)$, then

$$g'(t)=f\bigl(g(t)\bigr)$$

solves the primed flow. The numerical coordinates differ, but the two curves are related point by point.

Finite local counterterms generate maps of this kind in perturbation theory. Their composition, identity, and local inverse form the physical finite-renormalization slice of the more general Stückelberg–Petermann description of renormalization freedom. The theorem-level construction is not needed here; the present application uses only the analytic map of a finite set of renormalized parameters and fields. Collins derives the chain-rule transformation directly from finite changes of prescription in Collins 1984/2023, § 7.8, pp. 200–202.

What makes a map admissible?

The conditions are substantive, not cosmetic:

Condition	What it protects	Failure mode
$f(g)=g+O(g^2)$ in the chosen coupling variables	the same weak-coupling normalization	a rescaled leading coefficient is mistaken for scheme dependence
analytic expansion near the reference point	order-by-order perturbative translation	fractional powers or essential singularities destroy the loop expansion
$\det J\ne0$ on the domain	a one-to-one coordinate patch	distinct physical points are collapsed or a false fixed point is created
consistent field and parameter translation	equality of complete predictions	only the running coupling is changed while amplitudes are left unconverted
common active degrees of freedom	comparison of the same theory	a threshold change is mislabeled as a finite scheme change

For example, $g'=g^3$ is not an admissible coordinate near the Gaussian point: $dg'/dg=0$ at $g=0$, and its inverse is nonanalytic. The formula $\beta'=J\beta$ still holds algebraically, but conclusions that require a regular perturbative coordinate map do not.

One-coupling coefficients through three orders

Take the asymptotically free convention

$$\beta(g) = -b_0g^3-b_1g^5-b_2g^7+O(g^9)$$

and an analytic finite redefinition tangent to the identity,

$$g' = g+a_1g^3+a_2g^5+O(g^7).$$

Its inverse is

$$g = g'-a_1g'^3 +(3a_1^2-a_2)g'^5 +O(g'^7).$$

The chain rule first gives

$$\beta'(g') = \left( 1+3a_1g^2+5a_2g^4+O(g^6) \right)\beta(g).$$

Re-expanding the right-hand side in $g'$ yields

$$\beta'(g') = -b'_0g'^3-b'_1g'^5-b'_2g'^7 +O(g'^9),$$

with

$$\boxed{ \begin{aligned} b'_0&=b_0, \\ b'_1&=b_1, \\ b'_2&= b_2-2a_1b_1 +(2a_2-3a_1^2)b_0. \end{aligned} }$$

The cancellation in $b'_0$ and $b'_1$ is the universal result for a single coupling whose beta function starts at $g^3$, compared between mass-independent analytic schemes with the same leading normalization. The next coefficient is a coordinate choice. Collins verifies the first-two-coefficient statement and its assumptions in Collins 1984/2023, § 7.8, p. 202.

As a concrete transformation, set $a_1=1/2$ and $a_2=0$. Then

$$b'_2 = b_2-b_1-\frac34b_0.$$

Nothing physical has acquired the extra term $-b_1-3b_0/4$. It is the third coefficient of the same vector field in the $g'$ coordinate.

The transformed scale parameter

For $b_0>0$, the leading transmuted scale in the two coordinates is

$$\Lambda = \mu\exp\left[-\frac{1}{2b_0g^2}\right] \bigl(b_0g^2\bigr)^{-b_1/(2b_0^2)} \left[1+O(g^2)\right],$$

with the same two-loop convention as on the preceding page. Since

$$\frac{1}{g'^2} = \frac{1}{g^2}-2a_1+O(g^2),$$

the consistently normalized scales satisfy

$$\boxed{ \frac{\Lambda'}{\Lambda} = \exp\left(\frac{a_1}{b_0}\right). }$$

For the example $a_1=1/2$, $\Lambda'/\Lambda=e^{1/(2b_0)}$. The scale is constant along either RG trajectory but changes by a fixed conversion factor between schemes. Collins derives this relation and explains what a measurement of $\Lambda$ means in Collins 1984/2023, § 7.9, pp. 205–206.

Fields, masses, and operator bases

Let the renormalized field coordinate change by a finite factor,

$$\Phi'_r=C_r(g)\Phi_r.$$

With the convention $d\Phi_r/dt=-\gamma_r\Phi_r$, differentiation gives

$$\boxed{ \gamma'_r(g') = \gamma_r(g) -\beta^i(g)\partial_i\ln C_r(g). }$$

If $\beta=O(g^3)$ and $C_r=1+c_rg^2+O(g^4)$, the leading $O(g^2)$ field anomalous dimension is unchanged, while higher coefficients can move. An elementary-field anomalous dimension can also depend on gauge fixing, so it should not be promoted to a physical observable merely because its first coefficient is stable under this narrower class of transformations.

For a multiplicatively running mass $m$ with

$$\eta_m \equiv \frac{1}{m}\frac{dm}{dt}, \qquad m'=D(g)m,$$

the sign is instead

$$\eta'_m = \eta_m +\beta^i\partial_i\ln D.$$

The difference follows because $m$ itself, rather than its inverse field normalization, is being rescaled.

For a column of renormalized operators satisfying

$$\frac{d\mathbf O}{dt} = -\boldsymbol\gamma\,\mathbf O,$$

let $\mathbf O'=R(g)\mathbf O$. Then

$$\boxed{ \boldsymbol\gamma' = R\boldsymbol\gamma R^{-1} -\left(\beta^i\partial_iR\right)R^{-1}. }$$

Wilson coefficients transform as $\mathbf C'=R^{-T}\mathbf C$, so the contraction $\mathbf C^T\mathbf O$ is unchanged. This is the matrix version of the same principle: neither an operator component nor a coefficient component is separately invariant under a finite basis change.

Fixed points and invariant flow data

If $g_\star$ is an exact fixed point and $f$ is regular there, then

$$\beta(g_\star)=0 \quad\Longrightarrow\quad \beta'\bigl(f(g_\star)\bigr)=0.$$

The coordinate location moves from $g_\star$ to $g'_\star=f(g_\star)$, but the zero survives. In several couplings, define the stability matrix

$$B^i{}_{j} \equiv \left. \frac{\partial\beta^i}{\partial g^j} \right|_{g_\star}.$$

At the fixed point, terms involving derivatives of $J$ multiply $\beta(g_\star)$ and vanish. Therefore

$$\boxed{ B'=J_\star B J_\star^{-1}. }$$

The matrix entries and eigenvectors are coordinate dependent; the eigenvalues are invariant under a regular map. For operators, the derivative term in the transformed anomalous-dimension matrix also vanishes at $\beta=0$, leaving a similarity transformation. Scaling dimensions in a closed physical operator sector are consequently invariant.

Two cautions prevent overstatement:

1. A zero of a truncated beta function is not an exact zero. Re-expansion after a scheme change can shift, create, or remove an apparent strong-coupling root by terms of the first omitted order.
2. A singular map does not preserve the fixed-point argument. If $J_\star$ is noninvertible, the similarity relation does not exist.

A reproducible calculation uses these as adversarial checks: a regular analytic map must preserve invariant data to the declared tolerance, while a singular map or a trajectory outside the perturbative box must fail explicitly.

RG-invariant combinations and effective charges

An invariant $I(\mu,g)$ satisfies

$$\left( \mu\frac{\partial}{\partial\mu} +\beta^i\frac{\partial}{\partial g^i} \right)I=0.$$

The transmuted $\Lambda$ parameter is one example. For a multiplicatively running mass,

$$\widehat m = m(\mu) \exp\left[ -\int^{g(\mu)} \frac{\eta_m(g')}{\beta(g')}\,dg' \right]$$

is constant along a one-coupling trajectory. Its conventional normalization can still change under $m'=D(g)m$, just as $\Lambda$ changes under a coupling redefinition. A physical mass expressed in the same inputs does not.

Another useful coordinate is an effective charge defined by an observable. Let $a=g^2$ and normalize an infrared-safe dimensionless observable so that

$$R(Q)=a(\mu)+r_1(Q/\mu)a^2(\mu)+\cdots.$$

Defining $a_R(Q)\equiv R(Q)$ makes the observable itself a coupling coordinate wherever $dR/da\ne0$. Its beta function is

$$\beta_R(a_R) = \frac{dR}{da}\,\beta_a(a),$$

which is again the vector-field chain rule. This removes an arbitrary intermediate coupling from the final statement, but $a_R$ remains observable specific and its finite-order extraction retains truncation and power-correction uncertainties. The effective-charge construction was introduced for perturbative QFT observables in Grunberg 1980, pp. 70–74.

Mass-dependent schemes and finite-order comparisons

The preceding coefficient claims require mass independence. If a scheme map depends on

$$\rho\equiv\frac{m}{\mu}, \qquad g'=f(g,\rho),$$

then

$$\frac{d\rho}{dt} = (\eta_m-1)\rho$$

and the transformed beta function is

$$\boxed{ \beta'(g',\rho) = \beta(g,\rho)\frac{\partial f}{\partial g} +(\eta_m-1)\rho \frac{\partial f}{\partial\rho}. }$$

For $\rho=m^2/\mu^2$, replace $\eta_m-1$ by $\eta_{m^2}-2$. The extra term is physical bookkeeping for the subtraction point moving relative to a mass threshold. One cannot compare its coefficients with a mass-independent beta function by applying the $b'_0=b_0$, $b'_1=b_1$ rule globally.

At finite perturbative order, translate in this order:

1. express the primed couplings, masses, fields, operators, and matching inputs in terms of the unprimed ones;
2. substitute into the complete truncated prediction;
3. re-expand consistently through the declared order;
4. compare only after using the same physical boundary data.

Keeping selected unexpanded pieces of the finite map inserts a subset of higher-order terms. That may define a resummation prescription, but it is not an order-by-order proof of scheme independence. The difference between consistently translated truncations begins at the first omitted order; its numerical size is a diagnostic, not a universal confidence level.

Coordinate-dependent data and invariant claims

The following comparison table is reused across this chapter. Each row separates what can move under a finite convention change from the claim that survives a complete translation.

Item	What may change	What survives a consistent translation	Required qualification or check
Renormalized $g^i$, masses, and field normalizations	Numerical values under finite scheme or basis changes	A prediction expressed in the same physical inputs	Translate every parameter and field factor through the retained order
Beta function away from a fixed point	Components and higher-order coefficients	The integral curves as geometric trajectories under a nonsingular coordinate map	Compare transformed vector fields, not coefficients at equal numerical coupling
Elementary-field anomalous dimension	Finite field rescaling; gauge parameter in a gauge theory	Scaling of a gauge-invariant observable after all factors are combined	Never identify a gauge-dependent elementary-field exponent with an observable
Exact fixed point	Coordinate location $g_\star^i$	Existence of the zero under a regular map	Exclude singular redefinitions and verify the fixed point lies in the method’s domain
Fixed-point stability data	Matrix representation and basis	Eigenvalues in a closed physical sector	Include operator mixing and redundant directions before diagonalizing
Transmuted scale	Its conventional normalization	Matched dimensionless ratios or predictions	State the scheme and reference condition defining the scale
Zero or singularity of a truncated beta function	Location and even apparent existence at insufficient order	Only the demonstrated breakdown of the stated approximation	Vary scheme/order and stop before couplings become large
Wilson coefficient versus power correction	Factorization scheme and, for an asymptotic series, summation prescription	Their consistently defined sum in an observable	Match the ambiguity of the perturbative term to the operator matrix element; developed on the renormalon page
Residual $\mu$ or scheme dependence	Numerical size at finite order	Vanishing in the exact consistently matched prediction	Treat the residual as a diagnostic, not a universal probability law

The broader classification of finite local ambiguities is developed in Microlocal Renormalization Ambiguities and Their Classification. That theorem-level result supplies a structural setting; the table above remains the practical contract for the perturbative calculations in this volume.

Analytic coordinate-map check

Take the exact benchmark from the preceding page,

$$\frac{dg}{dt}=-g^3, \qquad g(0)=0.4,$$

and apply the regular map

$$h=f(g)=g+\frac12g^3.$$

Since $dh/dg=1+3g^2/2>0$, the map is invertible on the entire positive real branch. The exact transformed equation is

$$\frac{dh}{dt} = -\left(1+\frac32g^2\right)g^3, \qquad g=f^{-1}(h),$$

while its perturbative re-expansion is

$$\beta_h(h) = -h^3+\frac34h^7+O(h^9).$$

The invariant in the new coordinate is not $1/h^2-2t$. It is

$$I_h(t) = \frac{1}{[f^{-1}(h(t))]^2}-2t =6.25.$$

Exact reference values expose the difference:

$t$	$g(t)$	$h(t)$	Corrected $I_h(t)$	Naive $1/h^2-2t$
$-2$	$0.666666666667$	$0.814814814815$	$6.25$	$5.50619834711$
$0$	$0.400000000000$	$0.432000000000$	$6.25$	$5.35836762689$
$2$	$0.312347523777$	$0.327583988352$	$6.25$	$5.31868577610$
$5$	$0.248069469178$	$0.255702375922$	$6.25$	$5.29433058588$

A numerical implementation should map the trajectory point by point and preserve the corrected invariant to relative tolerance $10^{-10}$. Comparing $g$ and $h$ at equal numerical values, or reusing the untransformed formula with $h$ substituted for $g$, is precisely the error this test is designed to catch.

Common pitfalls

Comparing coefficients at equal numerical coupling. Equal numbers $g=g'$ generally label different physical points. Convert the coordinate first, then compare the vector fields or predictions.

Calling all beta-function coefficients universal. For one mass-independent coupling with the stated normalization, $b_0$ and $b_1$ survive; $b_2$ and higher coefficients do not. Multiple couplings and mass-dependent schemes require separate analysis.

Using a singular redefinition to erase a fixed point. Fixed-point preservation assumes an invertible Jacobian. A map that fails this test is not evidence that the original zero was unphysical.

Transforming the running but not the observable. Finite parts of amplitudes, masses, fields, operator bases, and matching conditions must change with the coupling. A partial conversion manufactures scheme dependence.

Treating residual variation as a statistical interval. Scheme and scale scans probe selected higher-order directions. Without an explicit uncertainty model they are diagnostics, not probabilities.

Exercises

Starting from the general $g'$ map above, verify the expression for $b'_2$.

Solution

First multiply by the Jacobian:

$$\begin{aligned} \beta'(g') &= -b_0g^3 -(b_1+3a_1b_0)g^5 \\ &\quad -(b_2+3a_1b_1+5a_2b_0)g^7 +O(g^9). \end{aligned}$$

Use

$$\begin{aligned} g^3&=g'^3-3a_1g'^5+(12a_1^2-3a_2)g'^7+O(g'^9), \\ g^5&=g'^5-5a_1g'^7+O(g'^9), \\ g^7&=g'^7+O(g'^9). \end{aligned}$$

Collecting powers gives $b'_0=b_0$, $b'_1=b_1$, and

$$b'_2=b_2-2a_1b_1+(2a_2-3a_1^2)b_0.$$

Show that the slope of a one-coupling beta function at an exact fixed point is invariant under a regular redefinition.

Solution

For $g'=f(g)$,

$$\beta'(g')=f'(g)\beta(g).$$

Differentiate with respect to $g'$:

$$\frac{d\beta'}{dg'} = \frac{1}{f'(g)} \frac{d}{dg}\left[f'(g)\beta(g)\right] = \frac{f''(g)}{f'(g)}\beta(g) +\frac{d\beta}{dg}.$$

At $g=g_\star$, the first term vanishes because $\beta(g_\star)=0$. Hence $d\beta'/dg'\rvert_{g'_\star}=d\beta/dg\rvert_{g_\star}$, provided $f'(g_\star)\ne0$.

Where to continue

• Multiple Couplings and Coupled RG Flows treats vector fields, stability matrices, fixed points, and separatrices when coefficient universality is more limited.
• Large Logarithms and RG Improvement shows how scheme-consistent boundary data and evolution combine in a truncated prediction.
• Renormalons, OPE Ambiguities, and Power Corrections develops the paired prescription dependence summarized in the table.

References

• Collins, John C. Renormalization: An Introduction to Renormalization, the Renormalization Group, and the Operator-Product Expansion. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press, 1984; open-access digital edition, 2023. DOI. Open PDF.
• Grunberg, Georges. “Renormalization Group Improved Perturbative QCD.” Physics Letters B 95 (1980): 70–74; erratum 110 (1982): 501. DOI.