TABLE II: Quantum numbers of the sources. TABLE I: Quantum numbers of the fields. 10 U ai µ M ai µ N ai µ V ai µ R ai µ T ai µ K a µ L a 2 2 2 2 2 2 3 4 − 1 0 1 0 0 -1 − 1 − 2 − 1 − 1 1 1 1 1 0 0 A a µ c a c a b a ϕ a i ϕ a i ω a i ω a i dimension 1 0 2 2 1 1 1 1 ghost number 0 1 − 1 0 0 0 1 − 1 Q$_{f}$ -charge 0 0 0 0 1 − 1 1 − 1 1. The Slavnov-Taylor identity is given by 2. The U ( f ) invariance is given by 3. The Landau gauge condition reads 4. The antighost equation yields 5. The linearly broken local constraints yield δ Σ ′ GZ δc a + ∂$_{µ}$ δ Σ ′ GZ δK a µ = 0 . (A16) δ Σ ′ GZ δb a = ∂$_{µ}$A a µ . (A15) U$_{ij}$ Σ ′ GZ = 0 , (A14) U$_{ij}$ = ∫ d $^{d}$x ( ϕ a i δ δϕ a j − ϕ a j δ δϕ a i + ω a i δ δω a j − ω a j δ δω a i − M aj µ δ δM ai µ − U aj µ δ δU ai µ + N ai µ δ δN aj µ + V ai µ δ δV aj µ + R aj µ δ δR ai µ + T aj µ δ δT ai µ ) . S (Σ $^{′}$$_{GZ}$) = ∫ d $^{4}$x ( δ Σ ′ GZ δK a µ δ Σ ′ GZ δA a µ + δ Σ ′ GZ δL a δ Σ ′ GZ δc a + b $_{a}$δ Σ ′ GZ δc a + ϕ a i δ Σ ′ GZ δω a i + ω a i δ Σ ′ GZ δϕ a i + M ai µ δ Σ ′ GZ δU ai µ + N ai µ δ Σ ′ GZ δV ai µ + R ai µ δ Σ ′ GZ δT ai µ ) . S (Σ $^{′}$$_{GZ}$) = 0 , (A13) S$_{ext}$ = ∫ d $^{4}$x ( − K a $_{µ}$( D$_{µ}$c ) a + 1 2 gL $^{a}$f $^{abc}$c$^{b}$cc ) , (A9) sK a µ = 0 , sL a = 0 . (A11) Σ ′ GZ = Σ$_{GZ}$ + S$_{ext}$ . (A12) δ Σ ′ GZ δϕ a i + ∂$_{µ}$ δ Σ ′ GZ δM ai µ + gf$_{dba}$T di µ δ Σ ′ GZ δK bi µ = gf $^{abc}$Ab $_{µ}$V ci µ , δ Σ ′ GZ δω a i + ∂$_{µ}$ δ Σ ′ GZ δN ai µ − gf $^{abc}$ωb i δ Σ ′ GZ δb c = gf $^{abc}$Ab $_{µ}$U ci µ . (A17) end, we have to set the sources equal to the following values: with It is exactly here that we have committed a mistake. In previous papers, we have forgotten to introduce the dou- blet ( T ai µ , R ai µ ), and therefore, the intended physical limit did not exactly reproduce the original action (A1). In the original article [4], these two sources were also not intro- duced. When taking the physical limit, an extra term was generated, which was then removed by doing a (non- local) shift in the ω field. Here, we have circumvented this unnecessary shift by introducing the doublet ( T ai µ , R ai µ ). with K a µ and L a two new sources which shall be put to zero at the end, Following the procedure of algebraic renormalization [41], we should try to find all possible Ward identi- ties. Before doing this, in order to be able to write the Slavnov-Taylor identity, we first have to couple all non- linear BRST transformations to a new source. Looking at (A4), we see that only A a µ and c a transform nonlin- early under the BRST s . Therefore, we add the following term to the action Σ$_{GZ}$, These sources are invariant under the BRST transforma- tion, The new action is therefore given by The next step is now to find all the Ward identities obeyed by the action Σ $^{′}$$_{GZ}$. We have enlisted all the iden- tities below: By means of the diagonal operator Q$_{f}$ = U$_{ii}$ , the i -valued fields and sources turn out to possess an additional quantum number. One can find all quan- tum numbers in Table I and Table II. K a µ ∣ phys = L $^{a}$|$_{phys}$ = 0 . (A10) U ai µ ∣ phys = N ai µ ∣ phys = T ai µ ∣ phys = 0 , M ab µν ∣ phys = V ab µν ∣ phys = − R ab µν ∣ phys = γ $^{2}$δ $^{ab}$δ$_{µν}$ . (A8) 2. The Ward identities