= Calibrated one-loop QCD evolution
{title2=$\alpha_s^{-1}(Q)=\alpha_s^{-1}(Q_0)+\frac{\beta_0}{2\pi}\ln(Q/Q_0)$}
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 $\beta_0=23/3$ gives
$$
\alpha_s^{-1}(Q)=6.25+\frac{23}{6\pi}\ln\frac{Q}{20\,\mathrm{GeV}},\qquad
\Lambda=Q_0\exp\!\left(-\frac{2\pi}{\beta_0\alpha_0}\right)\simeq0.1193\,\mathrm{GeV}.
$$
Consequently $\alpha_s(10\,\mathrm{GeV})\simeq0.1850$, $\alpha_s(40\,\mathrm{GeV})\simeq0.1409$, and $\alpha_s(100\,\mathrm{GeV})\simeq0.1217$. Substituting $Q=Q_0$ returns the input, while differentiating returns the stated <beta function (physics)>. 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 approximations
$$
R_1(\mu)=1+a(\mu),\qquad
R_2(\mu)=1+a(\mu)+b\,a(\mu)^2\ln\frac{\mu^2}{Q_h^2}.
$$
Using the one-<loop order> equation $da/d\ln\mu^2=-ba^2$ gives
$$
\frac{dR_1}{d\ln\mu^2}=-ba^2,\qquad
\frac{dR_2}{d\ln\mu^2}=-2b^2a^3\ln\frac{\mu^2}{Q_h^2}.
$$
The explicit <logarithm> cancels the order-$a^2$ scale drift. At $\mu=10,20,40\,\mathrm{GeV}$, respectively, $R_1\simeq1.0589,1.0509,1.0449$, whereas $R_2\simeq1.0497,1.0509,1.0502$. 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>.
\Image[/qcd-calibrated-inverse-coupling.png]
{title=Calibrating a running coupling and cancelling leading auxiliary-scale dependence}
{description=Left: the original five-<quark flavor> calibration written as an inverse <QCD coupling>. Right: varying the <renormalization scale> at fixed $Q_h$ in the original toy $R_1,R_2$. The reduced variation illustrates cancellation between running and an explicit <logarithm>; it is not an experimental uncertainty band.}
Back to article page
