with B the nilpotent linearized Slavnov-Taylor op- erator, Here we should add that due to the presence of the sources T ai µ and R ai µ , the ghost Ward identity [41] is bro- ken, and we are unable to restore this identity. For the standard Yang-Mills theory, this identity has the follow- ing form a linear breaking. However, it shall turn out that this is not a problem for the renormalization procedure being undertaken. The next step in the algebraic renormalization is to translate all these symmetries into constraints on the counterterm Σ c $_{GZ}$, which is an integrated polynomial in the fields and sources of dimension four and with ghost number zero. The classical action Σ ′ GZ changes under quantum corrections according to whereby h is the perturbation parameter. Demanding that the perturbed action (Σ $^{′}$$_{GZ}$+ h Σ c $_{GZ}$) fulfills the same set of Ward identities obeyed by Σ $^{′}$$_{GZ}$, see [41], it follows that the counterterm Σ c GZ is constrained by the following identities. Now we can write down the most general counterterm Σ c GZ of d = 4, which obeys the linearized Slavnov-Taylor and with 11 B = ∫ d $^{4}$x ( δ Σ ′ GZ δK a µ δ δA a µ + δ Σ ′ GZ δA a µ δ δK a µ + δ Σ ′ GZ δL a δ δc a + δ Σ ′ 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 + R ai µ δ δT ai µ ) , (A26) R$_{ij}$ = ∫ d $^{4}$x ( ϕ a i δ δω a j − ω a j δ δϕ a i + V ai µ δ δN aj µ − U aj µ δ δM ai µ + T ai µ δ δR aj µ ) . (A19) ∫ d $^{4}$x ( c a δ Σ ′ GZ δω a i + ω a i δ Σ ′ GZ δc a + U ai µ δ Σ ′ GZ δK a µ ) = 0 . (A20) B 2 = 0 . (A27) U$_{ij}$ Σ c GZ = 0 . (A28) δ Σ c GZ δb a = 0 . (A29) δ Σ c GZ δc a + ∂$_{µ}$ δ Σ c GZ δK a µ = 0 . (A30) G $^{a}$( S$_{YM}$ + S$_{gf}$ ) = Δ a $_{cl}$, (A21) G a = ∫ d $^{d}$x ( δ δc a + gf $^{abc}$cb δ δb c ) , (A22) Δ a cl = g ∫ d $^{4}$xf abc ( K b $_{µ}$A c µ − L $^{b}$cc ) , (A23) ( δ δϕ a i + ∂$_{µ}$ δ δM ai µ + ∂$_{µ}$ δ δM ai µ + gf$_{abc}$T bi µ δ δK ci µ ) Σ c GZ = 0 , (A31) with and ( δ δω a i + ∂$_{µ}$ δ δN ai µ − gf $^{abc}$ωb i δ δb c ) Σ c GZ = 0 . (A32) R$_{ij}$ Σ c GZ = 0 , (A33) ∫ d $^{4}$x ( c a δ Σ c GZ δω a i + ω a i δ Σ c GZ δc a + U ai µ δ Σ c GZ δK a µ ) = 0 . (A34) Σ ′ GZ → Σ ′ GZ + h Σ c $_{GZ}$, (A24) B Σ c GZ = 0 , (A25) with R$_{ij}$ given in (A19). 6. The exact R$_{ij}$ symmetry reads 7. Finally, the integrated Ward identity becomes 1. The linearized Slavnov-Taylor identity yields 5. The linearly broken local constraints yield 4. The antighost equation 3. The Landau gauge condition 2. The U ( f ) invariance reads 7. The integrated Ward identity is given by 3. The counterterm 6. The exact R$_{ij}$ symmetry reads R$_{ij}$ Σ ′ GZ = 0 , (A18)