Regulator poles determine finite running through a fixed-bare derivative

Write the dimensionful bare coupling as με times a Laurent series F(g,ε) in a dimensionless renormalized coupling g. In a mass-independent minimal scheme F has no explicit μ dependence other than through g. Differentiating at fixed bare coupling gives zero equal to εF plus βε times its g derivative. Therefore βε=−εF/F′. Its leading term is −εg, the engineering contribution associated with the quartic interaction below four dimensions. This term must be included before expanding the poles.

Suppose the first correction in F is A g²/ε. Expanding the ratio consistently gives βε=−εg+A g² at this order. The finite interaction term comes from a cancellation between two contributions involving ε times a pole. It is not obtained by simply setting ε to zero in the original bare relation. The coefficient A depends on the coupling normalization and the definition of ε. A conventional four-dimensional scalar interaction λφ⁴/4! gives βλ=3λ²/(16π²) at one loop, while a rescaled coupling has a correspondingly rescaled coefficient.

At the next order the bare relation can contain both a double pole and a new simple pole. Substituting into the fixed-bare derivative shows that the double-pole coefficient is not arbitrary: it must equal A squared in this particular F convention. The new simple-pole coefficient C determines twice C in the four-dimensional two-loop beta function. These relations are useful bookkeeping checks on counterterm insertions. They are specific to the stated coupling relation; replacing F by a field or mass Z factor changes the algebra.

For scalar quartic theory define a=λ/(16π²). Its one-loop four-dimensional flow is da/dlnμ=3a². Positive coupling therefore grows toward the ultraviolet and decreases toward the infrared within the weak-coupling domain. The integrated one-loop solution has a finite-denominator obstruction at a sufficiently large ultraviolet scale. That perturbative Landau scale signals breakdown of the simple running description; it is not by itself a controlled prediction of a physical divergent scattering probability. Nonperturbative continuum questions require additional arguments.

A beta function defined using μ² differentiation differs by a factor of two from one using μ differentiation. Some statistical RG conventions evolve toward longer distance and reverse the flow parameter relative to increasing energy. Coupling redefinitions also change displayed coefficients. One should therefore compare the action normalization, dimensional continuation, derivative variable, and direction of flow before declaring two results inconsistent. For several couplings the fixed-bare differentiation becomes a coupled system, and masses can introduce additional ratios in a mass-dependent scheme.

The function lab shows a quadratic beta-function shape, without integrating it. Reset and set both controls to a=1 and x=0.20, predicting y=0.040. Reset and set a=2 and x=0.20, predicting y=0.080. Assign x to a dimensionless coupling and a to an illustrative one-loop coefficient. Doubling that coefficient doubles the local rate at fixed coupling. The coefficient control is not a computed counterterm residue, and playback does not evolve g with μ. The field-theory running still follows from the fixed-bare relation and its differential equation.