One-loop evolution sums an infinite tower of logarithms

Take an illustrative asymptotically free coupling with β(g)=−bg² and b positive. Integrating from μ to a hard scale Q gives g(Q)=g(μ)/(1+bg(μ)L), where L=ln(Q/μ). Expanding this solution gives g−bg²L+b²g³L² and higher terms. At the order g to the power n+1, the highest logarithm L to the power n is fixed by repeated one-loop running. These terms can compete when bgL is order one even though both endpoint couplings remain perturbatively small.

The logarithm appears because widely separated momentum scales contribute to the same fixed-order expression. Choosing μ close to Q removes that particular explicit large logarithm from a hard matching function, while evolving from a reference scale relates the input coupling to g(Q). Exact μ independence is the reason these two operations fit together. It does not imply the observable is independent of Q. If a process has several independent large ratios, a single choice of μ cannot remove all logarithms, and an appropriate factorization with multiple evolution equations may be needed.

Let a Wilson coefficient obey dlnC/dlnμ=cg in the same coupling convention. Dividing by the beta equation gives dlnC/dg=−c/(bg), so the coefficient ratio is a power of the running-coupling ratio with exponent −c/b. Expanding that power generates logarithms determined by the one-loop anomalous dimension and beta function. The sign follows the specified coefficient equation; an operator whose product with C is scale independent evolves with the opposite anomalous dimension.

The initial coefficient must still be found by matching at a suitable high scale. One-loop running alone does not determine its nonlogarithmic one-loop constant, and solving an evolution equation exactly does not make the whole calculation exact. Leading-log, next-to-leading-log, and fixed-order accuracy describe related but distinct information. Threshold matching must be applied when active fields change. Near an infrared strong-coupling scale, the same formal one-loop expressions cease to be perturbatively controlled even if a compact formula continues to return a number.

A useful check expands the resummed answer back to the fixed order that was explicitly calculated. The logarithmic coefficients must agree. One can then add finite matching terms without double counting the already-resummed expansion, using a consistent matched prescription. Residual renormalization-scale dependence starts beyond the retained accuracy, but an accidentally flat scale curve is not proof that every omitted constant or power correction is small. Perturbative, logarithmic, and EFT power-counting errors should be assessed in their respective domains.

The function lab displays the local negative beta function behind this example. Reset and set both controls to a=−1 and x=0.20, predicting y=−0.040. Reset and set a=−1 and x=0.10, predicting y=−0.010. Assign x to g and y to β=−g²: the flow rate is negative and its magnitude decreases as the coupling weakens. The lab draws a quadratic rate rather than integrating over scale. It has no logarithm or matching control, so the resummed coupling must be obtained from the differential equation above.