= 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 $\lambda_{\rm bare}=\mu^{d_\lambda}\,Z_\lambda\lambda(\mu)$ 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>. \b[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.
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