Infrared-safe Minkowskian Curci-Ferrari model

We discuss the existence of Landau-pole-free renormalization group trajectories in the Minkowskian version of the Curci-Ferrari model as a function of a running parameter q2 associated to the four-vector q at which renormalization conditions are imposed, and which can take both spacelike (q2 < 0) and timelike (q2 > 0) values. We discuss two possible extensions of the infrared-safe scheme defined in [An infrared safe perturbative approach to Yang-Mills correlators, Phys. Rev. D 84, 045018 (2011)] for the Euclidean version of the model, which coincide with the latter in the spacelike region upon identifying Q2 ≡ −q2 with the square of the renormalization scale in that reference. The first extension uses realvalued renormalization factors and leads to a flow in the timelike region with a similar structure as the flow in the spacelike region (or in the Euclidean model), including a nontrivial fixed point and a family of trajectories bounded at all scales by the value of the coupling at this fixed point. Interestingly, the fixed point in the timelike region has a much smaller value of λ ≡ g2N=16π2 than the corresponding one in the spacelike region, a value closer to the perturbative boundary λ ¼ 1. However, in this real-valued infrared-safe scheme, the flow cannot connect the timelike and spacelike regions. Thus, it is not possible to deduce the relevant timelike flow trajectory from the sole knowledge of a spacelike flow trajectory. To try to cure this problem, we investigate a second extension of the Euclidean IR-safe scheme, which allows for complex-valued renormalization factors. We discuss under which conditions these schemes can make sense and study their ability to connect spacelike and timelike flow trajectories. In particular, we investigate onto which types of timelike trajectories the perturbative spacelike trajectories are mapped.