7 this, but this was achieved by also using anti-BRST in- variance. We refer to the literature [34, 35] for details. In the presence of the restriction to the Gribov region Ω, we even lack the concept of the anti-BRST symmetry as far as we know. We shall therefore present a slightly different argument here. Using the correspondence (30), we may also write with V $^{′}$. Taking the inverse of this matrix yields We can do the same for the other operator. Only some care has to be taken as the source V ′ ai µ couples to a sum of two operators ( gf$_{akb}$A k $_{µ}$ϕ b i − gf$_{abc}$D bd µ c $^{d}$ω c $_{i}$). We therefore have to subtract the operator coupled to the source R ai µ , namely As in the previous section we have that from which the renormalizability immediately follows, see equation (42). If one wishes to introduce bare quantities, one can write down but this is just a formal relation. It nevertheless teaches us that one cannot simply speak about the (multiplicative) renormalization of u ( p $^{2}$) in terms of a bare Kugo-Ojima function u$_{0}$ ( p $^{2}$). As a corollary, using the identification (38), we de- rive the following correspondence between the horizon function and the Kugo-Ojima parameter u (0), We can thus conclude that the operators gf$_{akb}$A $^{k}$ϕb i and gf$_{akb}$A $^{k}$ϕ b i are not multiplicatively renormalizable and mix with the operators ∂$_{µ}$ϕ a i and ∂$_{µ}$ϕ a $_{i}$, respectively. Therefore, one should keep in mind that the limit θ → 0 has to be taken as the final step, and that it can only be taken at the local level. Furthermore, the mixing we have found tells us that one should always leave the covariant derivative of a field “in one piece”. As a consequence, much care has to be taken at the nonlocal level when de- riving all kinds of results, as we shall now explain in the next section. As explained in [33], there exists a close link between the Gribov-Zwanziger formalism and the Kugo-Ojima analysis of gauge theories. An important role in the Kugo-Ojima work is played by the parameter u (0), which is the zero momentum limit of the function u ( p $^{2}$), which can be extracted from This relation was also discussed in [4, 8–10, 26, 36]. Us- ing this relation, one can again derive a formal correspon- dence between bare and finite quantities, G ( p $^{2}$) = 1 p $^{2}$(1 + u ( p $^{2}$) + w ( p $^{2}$)) . (61) − 〈 h$_{3}$ 〉 = ( N 2 − 1) [( d − 1) u (0) − 1] , (59) − 1 N$_{c}$ ∫ d $^{d}$xeipx 〈 D ad µ ω ds λ ( x ) D be ν ω es λ (0) 〉$_{0}$ = δ ab [( δ$_{µν}$ − p$_{µ}$p$_{ν}$ p 2 ) u ( p $^{2}$) − p$_{µ}$p$_{ν}$ p 2 ] 0 ⇔ − Z − 1 c N$_{c}$ ∫ d $^{d}$xeipx 〈 D ad µ ω ds λ ( x ) D be ν ω es λ (0) 〉 = Z − 1 c δ ab [( δ$_{µν}$ − p$_{µ}$p$_{ν}$ p 2 ) u ( p $^{2}$) − p$_{µ}$p$_{ν}$ p 2 ] 0 , (58) ( gf$_{akb}$A k $_{µ}$ϕ b $_{i}$)$_{0}$ = ( gf$_{akb}$A k $_{µ}$ϕ b i − gf$_{abc}$D bd µ c $^{d}$ωc $_{i}$)$_{0}$ − ( − gf$_{abc}$D bd µ c $^{d}$ωc $_{i}$)$_{0}$ = a$_{1}$ ( ∂$_{µ}$ϕ a $_{i}$) + Z − 1 / 2 g Z − 1 / 4 A ( gf $^{akb}$Ak $_{µ}$ϕ b i − gf$_{abc}$D bd µ c $^{d}$ωc $_{i}$) − Z 1 / 2 g Z 1 / 4 A ( − gf$_{abc}$D bd µ c $^{d}$ωc $_{i}$) = a$_{1}$ ( ∂$_{µ}$ϕ a $_{i}$) + Z − 1 / 2 g Z − 1 / 4 A ( gf $^{akb}$Ak $_{µ}$ϕ b $_{i}$) . (55) [1 + u ( p $^{2}$) + w ( p $^{2}$)]$_{0}$ = Z − 1 c [1 + u ( p $^{2}$) + w ( p $^{2}$)] . (62) ∫ d $^{d}$xeipx 〈 D ad µ c $^{d}$( x ) D be ν c $^{e}$(0) 〉 = δ ab [( δ$_{µν}$ − p$_{µ}$p$_{ν}$ p 2 ) u ( p $^{2}$) − p$_{µ}$p$_{ν}$ p 2 ] . (56) which hereby rigourously proves the relation which was one of the starting points of the paper [33]. In [29], it was shown that one can parametrize the ghost propagator, defined as follows but as already noticed in [36], it does not allow to separately speak about (1 + u ( p $^{2}$))$_{0}$ and w$_{0}$ ( p $^{2}$). In order to have a finite quantity u ( p $^{2}$), the previous Green function should of course be renormalizable. In the case of pure Yang-Mills theories, one can indeed prove − 1 N$_{c}$ ∫ d $^{d}$x e ipx 〈 D ad µ ω ds λ ( x ) D be ν ω es λ (0) 〉 = δ ab [( δ$_{µν}$ − p$_{µ}$p$_{ν}$ p 2 ) u ( p $^{2}$) − p$_{µ}$p$_{ν}$ p 2 ] , (57) [ M M ′ ] = [ Z − 1 / 2 g Z − 1 / 4 A a$_{1}$ 0 Z 1 / 2 g Z 1 / 4 A ] [ M$_{0}$ M ′ 0 ] , [ V V ′ ] = [ Z − 1 / 2 g Z − 1 / 4 A a$_{1}$ 0 Z 1 / 2 g Z 1 / 4 A ] [ V$_{0}$ V ′ 0 ] . (53) ( gf $^{akb}$Ak $_{µ}$ϕ b $_{i}$)$_{0}$ = a$_{1}$ ( ∂$_{µ}$ϕ a $_{i}$) + Z − 1 / 2 g Z − 1 / 4 A ( gf $^{akb}$Ak $_{µ}$ϕ b $_{i}$) . (54) V. THE KUGO-OJIMA GREEN FUNCTION(S) AND THE LINK WITH THE HORIZON CONDITION in terms of[45] 〈 c $^{a}$( − p ) c $^{b}$( p ) 〉 = δ $^{ab}$G ( p $^{2}$) , (60)