The running coupling \(g^2(\mu)\) is the strong coupling of QCD expressed as a function of the renormalization scale \(\mu\). Its scale dependence — encoded in the beta function — is what makes QCD weakly coupled at high energy (asymptotic freedom) and strongly coupled at low energy (confinement). In thermal QCD one evaluates it at a scale set by the temperature.

The scale dependence obeys

\[ \mu\frac{dg}{d\mu}=\beta(g)=-\frac{g^3}{16\pi^2}\,b_0-\frac{g^5}{(16\pi^2)^2}\,b_1-\cdots, \]

with the one- and two-loop coefficients (for [[non-abelian|SU\((N_c)\)]] with \(N_f\) flavours)

\[ b_0=\frac{11}{3}N_c-\frac{2}{3}N_f,\qquad b_1=\frac{34}{3}N_c^2-\Big(\frac{13}{3}N_c-\frac{1}{N_c}\Big)N_f . \]

Both are positive for QCD, so \(g\) decreases as \(\mu\) grows. Integrating the one-loop equation,

\[ \alpha_s(\mu)=\frac{g^2(\mu)}{4\pi}=\frac{2\pi}{b_0\,\ln(\mu/\Lambda_{QCD})}, \]

with the intrinsic scale \(\Lambda_{QCD}\approx 200\) MeV generated by dimensional transmutation.

In finite-\(T\) applications the natural argument is the lowest non-static Matsubara scale, \(\mu\sim 2\pi T\) (up to a constant chosen by matching). Then \(g^2(T)\equiv g^2(\mu\sim2\pi T)\) decreases only logarithmically with \(T\).

The dimensional reduction prediction

\[ \sqrt{\sigma_s(T)}=c\,g_M^2\simeq c\,g^2(T)\,T,\qquad c=0.554(4), \]

is only as accurate as the definition of \(g^2(T)\). Using a two-loop, thermally matched coupling is what lets Cheng et al. (arXiv:0806.3264) reproduce the lattice spatial string tension down to \(T\approx1.5\,T_c\). Because \(g^2(T)\) falls slowly, \(\sqrt{\sigma_s}\propto g^2(T)T\) grows with temperature. See Why spatial string tension and dimensional reduction.