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We consider a non-Abelian Lee--Wick gauge theory and discussBecchi-Rouet-Stora-Tyutin (BRST) invariance. It contains fourth-orderderivative as extensions of the kinetic term, leading to massive ghosts in thetheory upon quantization. We particularly provide essential clues towardsconfinement conditions in strongly-coupled regimes, using the Kugo-Ojimaapproach, and obtain the $\beta-$functions in the non-perturbative regimes.This is achieved using a set of exact solutions of the corresponding localtheory in terms of Jacobi elliptical functions. We obtain a similar$\beta-$function just as for the ordinary Yang-Mills theory but the maindifferences are that now, the cut-off arises naturally from the Lee-Wick heavymass scales (M). We show that the fate of the ghosts is fixed in these regimes:they are no more the propagating degrees of freedom in the infrared (IR)-limit.As it also happens for the ordinary case, confinement is due to the non-Abeliannature of the theory. In the limit $M\rightarrow\infty $, one recovers thestandard results for the local non-Abelian Yang-Mills theory.

Non-perturbative Lee-Wick gauge theory: Towards Confinement and RGE with strong couplings