15 we find We notice the close similarity between this counterterm and the one in expression (B3). check that equations (A39) and (A40) still hold, and also the Z -factors of Z 1 / 2 ϕ , Z 1 / 2 ϕ , Z 1 / 2 ω , Z 1 / 2 ω , Z$_{T}$ and Z$_{R}$ do not change. Only the renormalization of the sources U , V , M , N is different as they mix with respectively U $^{′}$, V $^{′}$, M $^{′}$, N $^{′}$. Indeed, we find that which again proves the renormalizability of the Gribov- Zwanziger action. The consequences of this mixing shall be explained in sections II.B and III.B. The last step is to find all the renormalization factors. Due to the close similarity with the output of Appendix A, many Z factors will be the same. One can indeed [24] Ph. Boucaud, J. P. Leroy, A. Le Yaouanc, J. Micheli, O. Pene and J. Rodriguez-Quintero, JHEP 0806 (2008) 099. [25] A. C. Aguilar, D. Binosi and J. Papavassiliou, Phys. Rev. D 78 (2008) 025010. [26] A. C. Aguilar, D. Binosi, J. Papavassiliou and J. Rodriguez-Quintero, Phys. Rev. D 80 (2009) 085018. [27] M. Q. Huber, R. Alkofer and S. P. Sorella, Phys. Rev. D 81 (2010) 065003 [28] T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. 66 (1979) 1. [29] T. Kugo, arXiv:hep-th/9511033. [30] D. Dudal, R. F. Sobreiro, S. P. Sorella and H. Verschelde, Phys. Rev. D 72 (2005) 014016. [31] D. Dudal, S. P. Sorella, N. Vandersickel and H. Ver- schelde, JHEP 0908 (2009) 110. [32] D. Zwanziger, Nucl. Phys. B 412 (1994) 657. [33] D. Dudal, S. P. Sorella, N. Vandersickel and H. Ver- schelde, Phys. Rev. D 79 (2009) 121701. [34] D. Dudal , V. E. R. Lemes, M. S. Sarandy, S. P. Sorella, M. Picariello, A. Vicini, J. A. Gracey and H. Verschelde, JHEP 0306 (2003) 003. [35] P. A. Grassi, T. Hurth and A. Quadri, Phys. Rev. D 70 (2004) 105014. [36] Ph. Boucaud, J. P. Leroy, A. Le Yaouanc, J. Micheli, O. Pene and J. Rodriguez-Quintero, Phys. Rev. D 80 (2009) 094501. [1] V. N. Gribov, Nucl. Phys. B 139 (1978) 1. [2] D. Zwanziger, Nucl. Phys. B 323 (1989) 513. [3] G. Dell’Antonio and D. Zwanziger, Nucl. Phys. B 326 (1989) 333. [4] D. Zwanziger, Nucl. Phys. B 399 (1993) 477. [5] D. Zwanziger, Phys. Rev. D 65 (2002) 094039. [6] J. A. Gracey, Phys. Lett. B 632 (2006) 282 [7] F. R. Ford and J. A. Gracey, J. Phys. A 42 (2009) 325402. [8] K. I. Kondo, arXiv:0909.4866 [hep-th]. [9] K. I. Kondo, arXiv:0907.3249 [hep-th]. [10] K. I. Kondo, Phys. Lett. B 678 (2009) 322. [11] A. Cucchieri and T. Mendes, PoS LAT2007 (2007) 297 [12] A. Cucchieri and T. Mendes, Phys. Rev. D 78 (2008) 094503. [13] V. G. Bornyakov, V. K. Mitrjushkin and M. Muller- Preussker, Phys. Rev. D 79 (2009) 074504. [14] I. L. Bogolubsky, E. M. Ilgenfritz, M. Muller-Preussker and A. Sternbeck, Phys. Lett. B 676 (2009) 69. [15] A. Maas, Phys. Rev. D 79 (2009) 014505. [16] A. Maas, arXiv:0907.5185 [hep-lat]. [17] A. Maas, J. M. Pawlowski, D. Spielmann, A. Sternbeck and L. von Smekal, arXiv:0912.4203 [hep-lat]. [18] C. S. Fischer, A. Maas and J. M. Pawlowski, Annals Phys. 324 (2009) 2408. [19] O. Oliveira and P. J. Silva, PoS LAT2007 (2007) 332. [20] P. J. Silva and O. Oliveira, AIP Conf. Proc. 892 (2007) 220. [21] D. Dudal, S. P. Sorella, N. Vandersickel and H. Ver- schelde, Phys. Rev. D 77 (2008) 071501. [22] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel and H. Verschelde, Phys. Rev. D 78 (2008) 065047. [23] Ph. Boucaud, J. P. Leroy, A. L. Yaouanc, J. Micheli, O. Pene and J. Rodriguez-Quintero, JHEP 0806 (2008) 012. [38] D. Dudal, O. Oliveira and N. Vandersickel, arXiv:1002.2374 [hep-lat]. [37] D. Dudal, S. P. Sorella, N. Vandersickel, H. Verschelde, work in progress . [39] M. A. L. Capri, V. E. R. Lemes, R. F. Sobreiro, S. P. Sorella and R. Thibes, Annals Phys. 323 (2008) 752. 4. The renormalization factors [ M$_{0}$ M ′ 0 ] = [ Z 1 / 2 g Z 1 / 4 A − a$_{1}$ 0 Z − 1 / 2 g Z − 1 / 4 A ][ M M ′ ] , [ U$_{0}$ U ′ 0 ] = [ Z 1 / 2 A − a$_{1}$ 0 Z − 1 g ] [ U U ′ ] , [ N$_{0}$ N ′ 0 ] = [ Z 1 g − a$_{1}$ 0 Z − 1 / 2 A ] [ N N ′ ] , [ V$_{0}$ V ′ 0 ] = [ Z 1 / 2 g Z 1 / 4 A − a$_{1}$ 0 Z − 1 / 2 g Z − 1 / 4 A ][ V V ′ ] , (B23) Σ (2) c GZ = 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 ai µ + ∂$_{µ}$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 ′ ia µ ϕ b $_{i}$∂$_{µ}$ c c − gf$_{abc}$V ′ ia µ ω b $_{i}$∂$_{µ}$ c c − gf$_{abc}$∂$_{µ}$ω $^{a}$ϕb $_{i}$∂$_{µ}$ c c − gf$_{abc}$R ai µ ∂$_{µ}$c $^{b}$ωc + gf$_{abc}$T ai µ ∂$_{µ}$c $^{b}$ϕc ) . (B22)