Thermodynamic pressure for massless QCD and the trace anomaly

From statistical mechanics the trace of the thermal average of any energy-momentum tensor is $⟨T_{\mu }^{\mu }⟩=T\mathrm{\partial }P\mathrm{/}\mathrm{\partial }T-4P$. The renormalization group formula $⟨T_{\mu }^{\mu }⟩=\beta (g_M)\mathrm{\partial }P\mathrm{/}\mathrm{\partial }{g}_M$ for QCD with massless fermions requires the pressure to have the structure \begin{equation} P=T^{4}\sum_{n=0}^{\infty} φ_{n}(g_{M})\big[\ln\big({M\over 4πT}\big)\big]^{n},\end{equation} where the factor $4\pi$ is for later convenience. The functions ${\phi }_n(g_M)$ for $n\ge 1$ may be calculated from ${\phi }_0(g_M)$ using the recursion relation $n{\phi }_n(g_M)=-\beta (g_M)d{\phi }_{n-1}\mathrm{/}d{g}_M$. This is checked against known perturbation theory results by using the terms of order $(g_M{)}^2,(g_M{)}^3$, $(g_M{)}^4$ in ${\phi }_0(g_M)$ to obtain the known terms of order $(g_M{)}^4,(g_M{)}^5$, $(g_M{)}^6$ in ${\phi }_1(g_M)$ and the known term of order $(g_M{)}^6$ in ${\phi }_2(g_M)$. The above series may be summed and gives the same result as choosing $M=4\pi T$, viz. $T^4{\phi }_0(g_{4\pi T})$.