12 renormalization of the fields and sources. If we try to ab- sorb the counterterm into the original action, we easily find, identity, has ghost number zero, and vanishing Q$_{f}$ num- ber, with a$_{0}$, . . . , a$_{11}$ arbitrary parameters. Now we can un- leash the constraints on the counterterm. Firstly, al- though the the ghost Ward identity (A21) is broken, we know that this is not so in the standard Yang-Mills case. Therefore, we can already set a$_{3}$ = 0 as this term is not allowed in the counterterm of the standard Yang- Mills action, which is a special case of the action we are studying[47]. Secondly, due to the Landau gauge condi- tion (3.) and the antighost equation (4.) we find, The results (A39) are already known from the renormal- ization of the original Yang-Mills action in the Landau gauge. Further, we also obtain Next, the linearly broken constraints (5.) give the fol- lowing relations The R$_{ij}$ symmetry does not give any new information, while the integrated Ward identity relates the two previ- ous strings of parameters: Taking all this information together, we obtain the fol- lowing counterterm This concludes the proof of the renormalizability of the action (A1) which is the physical limit of Σ ′ $_{GZ}$. We can also write down an alternative proof of the renormalization of the Gribov-Zwanziger action. Look- ing at (20), we are tempted to treat gf$_{akb}$A k $_{µ}$ϕ bc ν and gf$_{akb}$A k $_{µ}$ϕ bc ν as the relevant composite operators. How- ever, we shall see that these operators mix with ∂$_{µ}$ϕ ac ν and ∂$_{µ}$ϕ ac ν , respectively. A similar observation holds for gf$_{akb}$A k $_{µ}$ω bc ν and gf$_{akb}$A k $_{µ}$ω bc ν . Such conclusion is ev- idently not unexpected, as we have learnt already from Appendix A that e.g. D ab µ ϕ bc ν = ∂$_{µ}$ϕ ab ν + gf$_{akb}$A k $_{µ}$ϕ bc ν is multiplicatively renor- malizable. As a final step, we have to show that the countert- erm (B3) can be reabsorbed by means of a multiplicative Appendix B: Renormalization of the Gribov-Zwanziger action, option 2 4. The renormalization factors a$_{1}$ = − a$_{8}$ = − a$_{9}$ = a$_{10}$ = a$_{11}$ = − b$_{3}$ = b$_{4}$ ≡ a$_{3}$ = a$_{4}$ = − a$_{5}$ = a$_{6}$ . (A38) Σ c = a$_{0}$S$_{Y M}$ + a$_{1}$ ∫ d $^{d}$x ( A a µ δS$_{Y M}$ δA a µ + ∂$_{µ}$c $^{a}$∂$_{µ}$ c a + K a $_{µ}$∂$_{µ}$ c a + M ai µ ∂$_{µ}$ϕ a i − U ai µ ∂$_{µ}$ω a i + N ai µ ∂$_{µ}$ω a i + V ai µ ∂$_{µ}$ϕ a i + ∂$_{µ}$ϕ a $_{i}$∂$_{µ}$ ϕ a i + ∂$_{µ}$ω a i ∂$_{µ}$ω a i + V ai µ M ai µ − U ai µ N ai µ − gf$_{abc}$U ai µ ϕ b $_{i}$∂$_{µ}$ c c − gf$_{abc}$V ai µ ω b $_{i}$∂$_{µ}$ c c − gf$_{abc}$∂$_{µ}$ω a i ϕ b $_{i}$∂$_{µ}$ c c − gf$_{abc}$R ai µ ∂$_{µ}$c $^{b}$ωc i + gf$_{abc}$T ai µ ∂$_{µ}$c $^{b}$ϕc i ) . Z 1 / 2 ϕ = Z 1 / 2 ϕ = Z − 1 / 2 g Z − 1 / 4 A = 1 − h a$_{1}$ 2 , Z 1 / 2 ω = Z − 1 / 2 A , Z 1 / 2 ω = Z − 1 g , Z$_{M}$ = 1 − a$_{1}$ 2 = Z − 1 / 2 g Z − 1 / 4 A , Z$_{N}$ = Z − 1 / 2 A , Z$_{U}$ = 1 + h a$_{0}$ 2 = Z − 1 g , Z$_{V}$ = 1 − h a$_{1}$ 2 = Z − 1 / 2 g Z − 1 / 4 A , Z$_{T}$ = 1 + h a$_{0}$ 2 = Z − 1 g , Z$_{R}$ = 1 − h a$_{1}$ 2 = Z − 1 / 2 g Z − 1 / 4 A . (A41) a$_{1}$ = a$_{2}$ . (A36) a$_{1}$ = − a$_{8}$ = − a$_{9}$ = a$_{10}$ = a$_{11}$ = − b$_{3}$ = b$_{4}$ , a$_{4}$ = a$_{5}$ = − a$_{6}$ = a$_{7}$ , b$_{1}$ = b$_{2}$ = b$_{5}$ = b$_{6}$ = 0 . (A37) Z$_{g}$ = 1 − h a$_{0}$ 2 , Z 1 / 2 A = 1 + h ( a$_{0}$ 2 + a$_{1}$ ) , (A39) Z 1 / 2 c = Z 1 / 2 c = Z − 1 / 4 A Z − 1 / 2 g = 1 − h a$_{1}$ 2 , Z$_{b}$ = Z − 1 A , Z$_{K}$ = Z 1 / 2 c , Z$_{L}$ = Z 1 / 2 A . (A40) and Σ c GZ = a$_{0}$S$_{YM}$ + B ∫ d $^{d}$x {[ a$_{1}$K a $_{µ}$A a $_{µ}$+ a$_{2}$∂$_{µ}$c $^{a}$Aa $_{µ}$+ a$_{3}$ L $^{a}$ca + a$_{4}$U ai µ ∂$_{µ}$ϕ a i + a$_{5}$ V ai µ ∂$_{µ}$ω a i + a$_{6}$ ω a $_{i}$∂ $^{2}$ϕ a i + a$_{7}$ U ai µ V ai µ + a$_{8}$ gf $^{abc}$U ai µ ϕ b $_{i}$A c µ + a$_{9}$ gf $^{abc}$V ai µ ω b $_{i}$A c µ + a$_{10}$ gf $^{abc}$ωa $_{i}$A c µ ∂$_{µ}$ϕ b i + a$_{11}$ gf $^{abc}$ωa $_{i}$( ∂$_{µ}$A c $_{µ}$) ϕ b i + b$_{1}$R ai µ U ai µ + b$_{2}$T ai µ M ai µ + b$_{3}$gf$_{abc}$R ai µ ω b $_{i}$A c µ + b$_{4}$gf$_{abc}$T ai µ ϕ b $_{i}$A c µ + b$_{5}$R ai µ ∂$_{µ}$ω a i + b$_{6}$T ai µ ∂$_{µ}$ϕ a i ]} , (A35)