What is the dependence of the beta function of pure $SU(N_\text{c})$ Yang-Mills theory on the number of colors? I guess
$$\mu\frac{dg_\text{YM}}{d\mu}=-\beta_0N_\text{c}g_\text{YM}^3-\beta_1N_\text{c}^2g_\text{YM}^5-\beta_2N_\text{c}^3g_\text{YM}^7-\cdots\,,$$
with the appropriate constants $\beta_0,\beta_1,…$, as has been computed in QCD (including quarks) at four loops [arXiv:hep-ph/9701390]. Is this correct? How could it be proved (to all orders)?
Then the 't Hooft coupling $\lambda=g^2_\text{YM}N_\text{c}$ runs independently of $N_\text{c}$:
$$\frac{\mu}{2}\frac{d\lambda}{d\mu}=-\beta_0\lambda^2-\beta_1\lambda^3-\beta_2\lambda^4-\cdots\,.$$
The goal is to assure that in the large $N_\text{c}$ limit the coupling runs independently of $N_\text{c}$, so that the confinement scale is held fixed. So the series in $\lambda$ could be truncated for $N_\text{c}\rightarrow\infty$ (there could be negative powers of $N_\text{c}$).
Best Answer
Based on the exact result for a supersymmetric Yang-Mills
$$ \beta(\alpha_s)=-\frac{\alpha_s^2}{4\pi}\frac{3N}{1-\frac{N\alpha_s}{2\pi}} $$
these guys arXiv:0711.3745 postulated the following exact result for a non-supersymmetric Yang-Mills
$$ \beta(\alpha_s)=-\alpha_s^2\frac{11N}{12\pi}\frac{1}{1-\frac{17N}{11}\frac{\alpha_s}{2\pi}} $$
Here the beta-function is defined as $$ \mu^2\frac{d\alpha_s}{d\mu^2}=\beta(\alpha_s) $$