Finite coordinate changes move coefficients while preserving predictions

Take a dimensionless four-dimensional marginal coupling g and define a new coordinate g′=g+c₁g²+c₂g³ plus higher powers, with no explicit scale dependence. Applying the chain rule gives β′(g′)=(dg′/dg)β(g). Express the right-hand side back in terms of g′ rather than leaving a mixture of variables. The inverse series begins g=g′−c₁g′²+(2c₁²−c₂)g′³. Without this inversion, even the apparent coefficient of the next power can be misidentified.

For β=b₁g²+b₂g³+b₃g⁴, the first two coefficients remain b₁ and b₂, while the third changes as shown below. This invariance assumes a single marginal coupling, an analytic near-identity redefinition, and compatible mass-independent conventions without an explicit μ-dependent finite transformation. Several couplings, threshold-dependent definitions, dimensionful parameters, or a nonstandard leading normalization require a more general analysis. It is therefore safer to quote the transformation assumptions than to say that two beta coefficients are universal without qualification.

Suppose a dimensionless physical quantity has expansion A=g+r₁g² through second order. Substituting the inverse transformation changes its next coefficient to r₁−c₁ in the primed scheme. This cancellation is the simplest example of scheme independence: the value assigned to a coupling and the coefficient multiplying its square are not separately observable. The same physical input must be used to determine g and g′. Comparing both formulas at the same numerical input generally violates that requirement.

Even after a consistent conversion, evaluating two truncated formulas as ordinary numbers can leave a difference of higher order. Such a difference is expected because the substitution itself generates unretained powers. It does not mean exact observables depend on arbitrary schemes. Conversely, the numerical size of a scheme variation is not a rigorous bound on all omitted terms. A pathological transformation can manufacture poor-looking coefficients or a misleading apparent convergence pattern without teaching anything about the underlying physical accuracy.

For an exact flow, an invertible smooth transformation maps a zero of β to a zero of β′. Its coordinate usually changes. At the fixed point, the derivative of β with respect to its coupling is invariant because the extra derivative of the coordinate map multiplies β itself and vanishes. This statement concerns an exact regular transformation near the fixed point. A zero produced by truncating a weak-coupling series at order-one coupling may move or disappear under allowed higher-order redefinitions and is not automatically a controlled physical fixed point.

The function lab illustrates the finite quadratic part of a coordinate change. Reset and set both controls to a=0.5 and x=0.20, predicting y=0.020. Reset and set a=1 and x=0.20, predicting y=0.040. Interpret y as δg=c₁g²; adding g=0.20 manually gives primed coordinates 0.220 and 0.240. This does not calculate an observable or beta coefficients. Its coefficient changes compare two independent coordinate definitions, not an actual scale trajectory or a change of the underlying interaction.