Two-loop three-gluon vertex from the Curci-Ferrari model and its leading infrared behavior to all loop orders

We evaluate the three-gluon vertex with one vanishing external momentum within the Curci-Ferrari (CF) model at two-loop order and compare our results to Landau-gauge lattice simulations of the same vertex function for the SU(2) and SU(3) gauge groups in four dimensions. The parameters of the model being adjusted by fitting the two-point functions to lattice data, our evaluation of the three-gluon vertex arises as a pure prediction. We find that two-loop corrections systematically improve the agreement between the model and the lattice data as compared to earlier one-loop calculations, with a better agreement in the SU(3) case, as already seen in previous studies [1,2]. We also analyze the renormalization scheme dependence of our calculation. In all cases, this dependence diminishes when two-loop corrections are included, which is consistent with the perturbative CF paradigm. In addition, we study the low momentum regime of the three-gluon vertex in relation with the possibility of zero-crossing. Within the CF model, we show that the leading infrared behavior of the exact vertex is given by the same linear logarithm that arises at one-loop order, multiplied by the all orders cubic ghost dressing function at zero-momentum (we provide similar exact results for other vertex functions). We argue that this property remains true within the FP framework under the assumption that the resummed gluon propagator features a decoupling behavior. This shows that the zero-crossing is a property of the exact three-gluon vertex function. Within the CF model, we find however that the scale of the zero-crossing is considerably reduced when going from one- to two-loop order. This seems consistent with some recent lattice simulations [3]. Our analysis also allows us to support recent claims about the dominance of the tree-level tensor component [4].