 Calibrated one-loop QCD evolution

A useful original example fixes five active quark flavors and chooses $Q_0=20\mathrm{GeV}$ and ${\alpha }_0={\alpha }_s(Q_0)=0.16$. These are illustrative inputs, not a measured determination. Integrating $d(1/{\alpha }_s)/d\ln Q={\beta }_0/(2\pi )$ with ${\beta }_0=23/3$ givesConsequently ${\alpha }_s(10\mathrm{GeV})\simeq 0.1850$, ${\alpha }_s(40\mathrm{GeV})\simeq 0.1409$, and ${\alpha }_s(100\mathrm{GeV})\simeq 0.1217$. Substituting $Q=Q_0$ returns the input, while differentiating returns the stated beta function. Rescaling all dimensional inputs by the same factor leaves the QCD coupling unchanged: only ratios enter. The reciprocal QCD coupling is a straight line against $\ln (Q/Q_0)$, so the calibration sets the intercept and $n_f$ sets the slope.

Changing the physical probe energy and changing an auxiliary renormalization scale are different operations. To isolate the latter, hold a hypothetical hard scale $Q_h=20\mathrm{GeV}$ fixed and define $a(\mu )={\alpha }_s(\mu )/\pi$, $b={\beta }_0/4$, and the toy approximationsUsing the one-loop order equation $da/d\ln {\mu }^2=-b{a}^2$ givesThe explicit logarithm cancels the order-$a^2$ scale drift. At $\mu =10,20,40\mathrm{GeV}$, respectively, $R_1\simeq 1.0589,1.0509,1.0449$, whereas $R_2\simeq 1.0497,1.0509,1.0502$. These are values of an invented dimensionless observable chosen to expose the cancellation; they are not a cross section prediction. Smaller residual scale variation does not bound the unknown constant at the next loop order.