
\section*{Acknowledgments}
We thank Gianluca Calcagni and Florian Nortier for valuable comments. AK is supported by FCT Portugal investigator project IF/01607/2015. 

\section*{Appendix: Confinement in local Yang-Mills theory}
\label{AppendixC}

Confinement in SU(N) Yang-Mills theory was investigated and proven in Ref. \cite{Chaichian:2018cyv}. Utilising the Kugo-Ojima formalism \cite{Kugo:1977zq,Kugo:1979gm}, we have
\begin{widetext}
\begin{eqnarray}
\int d^4xe^{ipx}\langle D_\mu\bar{c}^{a} (x),D_\nu c^{b} (0) \rangle&=&-\delta^{ab}\frac{p_\mu p_\nu}{k^2}\\
&+&\frac{(N^2-1)^2}{2N}g^2\delta^{ab}
\left(\delta_{\mu\nu} - \frac{p_{\mu} p_{\nu}}{p^{2}}\right)
\int\frac{d^4p'}{(2\pi)^4}K_2(p-p')G_2(p').\nonumber
\end{eqnarray}
\end{widetext}
The form of the propagators look like:
\begin{equation}
K_2(p)=-\frac{1}{p^2+i\epsilon}
\end{equation}
and that of the ghost field and
\begin{equation}
G_2(p)=\frac{\pi^3}{4K^3(i)}
	\sum_{n=0}^\infty\frac{e^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}(2n+1)^2\frac{1}{p^2-m_n^2+i\epsilon}
\end{equation}
and that of the gauge field, 
provided the mass spectrum
\begin{equation}
m_n=(2n+1)\frac{\pi}{2K(i)}\left(\frac{Ng^2}{2}\right)^\frac{1}{4}\mu,
\end{equation}
where $K(i)$ represents the complete elliptical integral of the first kind and $\mu$ is one of the integration constants of the theory. 
This is a good and natural approximation to the full propagator as long as we omit the mass shift induced due to quantum corrections. Then, the Kugo-Ojima confinement condition looks:
\begin{widetext}
\begin{equation}
\label{eq:NL}
u(0)=-\frac{(N^2-1)^2}{2N}g^2
\int\frac{d^4p}{(2\pi)^4}\frac{1}{p^2+i\epsilon}\frac{\pi^3}{4K^3(-1)}
	\sum_{n=0}^\infty\frac{e^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}(2n+1)^2\frac{1}{p^2-m_n^2+i\epsilon}=-1.
\end{equation}
\end{widetext}
Evaluating the integral one obtains the $\beta$-function in a closed form
\begin{equation}
\beta_{YM}=-\beta_0\frac{\alpha_s^2}{1-\frac{1}{2}\beta_0\alpha_s},
\end{equation}
with $\beta_0=(N^2-1)^2/8\pi N$. This beta function grants confinement of the theory with the coupling running to infinity at lower momenta with no Landau pole. In the ultraviolet we recover the asymptotic freedom as expected.

