 Solution

A loop integral in a theory with dimensionless classical couplings can contain a logarithmic ultraviolet divergence. After subtraction, an expression such as $\log (-p^2/{\mu }^2)$ remains; one must specify the momentum scale at which the measured or renormalized coupling is fixed. A logarithm of a dimensionful momentum alone is not meaningful. In dimensional regularization, the same necessity appears through ${\lambda }_{\mathrm{bare}}={\mu }^{d_{\lambda }}{Z}_{\lambda }\lambda (\mu )$ in the continued dimension, with $d_{\lambda }$ tending to zero in the physical dimension.

Thus the renormalization prescription introduces an arbitrary renormalization scale, even when no mass or length occurs in the classical action. Physical quantities cannot depend on its arbitrary choice: explicit logarithms cancel the implicit change of the running coupling, governed by $\mu d\lambda /d\mu =\beta (\lambda )$. A nonzero renormalization-group beta function is a quantum scale anomaly. An integration constant may encode the same data as a dimensionful scale, which is dimensional transmutation. The scale labels a renormalization condition, not a new arbitrary measurable parameter. At a true fixed point the coupling need not run; the use of a subtraction scale is then a convention rather than an intrinsic generated physical scale.