Generalization of Faddeev--Popov Rules in Yang--Mills Theories: N=3,4 BRST Symmetries

The Faddeev-Popov rules for quantization of theory with gauge group are generalized for case of nvariance of quantum actions, $S_N$, on N-parametric Abelian SUSY transformations with odd parameters ${\lambda }_p$, p=1,…,N and anticommuting generators $s_p$, for N=3,4 implying substitution of ghost fields N-plet, $C^p$ multipled on ${\lambda }_p$, instead of the parameter, $\xi$, of gauge transformations. Total configuration spaces for quantum theory of the same classical model coincide for N=3 ,4 cases. For N=3 transformations the superspace of irrep includes in addition 3 ghost $C^p$, 3 even $B^{pq}$ and odd $\widehat{B}$ fields for p,q=1-3. It is shown for quantum action $S_3$ the gauge-fixing by adding to classical action of N=3-exact term requires 1 antighost $\bar{C}$, 3 even $B^p$ 3 odd $\widehat{B}{}^p$ and Nakanishi–Lautrup fields. Action of N=3 transformations on the latter fields is found. The transformations appear by N=3 BRST ones for the vacuum functional, $Z_3(0)$. It is shown, the configuration space appears by irrep superspace for fields ${\mathrm{\Phi }}_4$ for N=4- transformations containing in addition to $A^{\mu }$: (4+6+4+1) ghost-antighost $C^r$, even $B^{rs}$, odd $\widehat{B}{}^r$ fields and B. Action $S_4$ is constructed by adding to classical action of N=4-exact with gauge boson $F_4$ as compared to gauge fermion ${\mathrm{\Psi }}_3$ for N=3 case. Procedure is valid for any admissible gauge. The equivalence with $N=1$ BRST-invariant quantization method is explicitly found. Finite N=3,4 BRST transformations are derived from algebraic transformations. Respective Jacobians for field-dependent parameters are calculated. They imply the presence of corresponding modified Ward identity to be reduced to new (usual) Ward identities for constant parameters and describe the problem of gauge-dependence. Introduction into diagrammatic Feynman techniques for N=3,4 cases is suggested.