14 5. The linearly broken local constraints 6. The exact R$_{ij}$ symmetry can be adapted to 7. The integrated Ward identity is now linearly broken as follows 6. The exact R$_{ij}$ symmetry 7. Finally, the integrated Ward identity becomes 2. The U ( f ) invariance 3. The Landau gauge condition 4. The antighost equation ∫ d $^{4}$x ( c $_{a}$δ Σ (2) c GZ δω a i + ω a i δ Σ (2) c GZ δc a + U ′ ai µ δ Σ (2) c GZ δK a µ ) = 0 . (B21) R$_{ij}$ Σ (2) c GZ = 0 . (B20) ∫ d $^{4}$x ( c $_{a}$δ Σ (2) GZ δω a i + ω a i δ Σ (2) GZ δc a + U ′ ai µ δ Σ (2) GZ δK a µ ) = U ai µ ∂$_{µ}$c a − U ′ ai µ ∂$_{µ}$c a . (B13) δ Σ (2) c GZ δM ai µ = 0 , δ Σ (2) c GZ δN ai µ = 0 , δ Σ (2) c GZ δU ai µ = 0 , δ Σ (2) c GZ δV ai µ = 0 . (B19) ( δ δϕ a i + ∂$_{µ}$ δ δM ai µ + ∂$_{µ}$ δ δM ′ ai µ + gf$_{abc}$T bi µ δ δK ci µ ) × Σ (2) c GZ = 0 , ( δ δω a i + ∂$_{µ}$ δ δN ai µ + + ∂$_{µ}$ δ δN ′ ai µ − gf $^{abc}$ωb i δ δb c ) × Σ (2) c GZ = 0 , (B18) R$_{ij}$ Σ (2) GZ = 0 , (B11) R$_{ij}$ = ∫ d $^{4}$x ( ϕ a i δ δω a j − ω a j δ δϕ a i + V ai µ δ δN ai µ + V ′ ai µ δ δN ′ aj µ − U aj µ δ δM ai µ − U ′ ai µ δ δM ′ ai µ + T ai µ δ δR aj µ ) . (B12) and with We again translate all the identities into identities for the counterterm Σ (2) c GZ Now we can write down the most general counterterm Σ (2) c GZ of d = 4, which obeys the linearized Slavnov-Taylor identity, has ghost number zero, and vanishing Q$_{f}$ num- ber, with B (2) the nilpotent linearized Slavnov-Taylor operator, 1. The linearized Slavnov-Taylor identity: 3. The counterterm B $^{(2)}$Σ (2) c GZ = 0 , (B14) Σ (2) c GZ = a$_{0}$S$_{YM}$ + B (2) ∫ 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 ∂ω a i + b$_{6}$T ai µ ∂ϕ a i + 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 (2) = ∫ d $^{4}$x ( δ Σ (2) GZ δK a µ δ δA a µ + δ Σ (2) GZ δA a µ δ δK a µ + δ Σ (2) GZ δL a δ δc a + δ Σ (2) GZ δc a δ δL a + b a δ δc a + ϕ a i δ δω a i + ω a i δ δϕ a i + M ai µ δ δU ai µ + N ai µ δ δV ai µ + M ′ ai µ δ δU ′ ai µ + N ′ ai µ δ δV ′ ai µ + R ai µ δ δT ai µ ) . U$_{ij}$ Σ (2) c GZ = 0 . (B15) δ Σ (2) c GZ δb a = 0 . (B16) δ Σ (2) c GZ δc a + ∂$_{µ}$ δ Σ (2) c GZ δK a µ = 0 . (B17) Notice that the part in a and b parameters is exactly the same as in the previous Appendix A, see equation (A35). We shall now impose all the constraints induced by the Ward identities. We keep in mind that the argument con- cerning the broken ghost Ward identity still holds. Also, the 4 constraints (B19) invoke the counterterm to be in- dependent of the sources U $^{′}$, V $^{′}$, M ′ and N $^{′}$. Ultimately,