On a Superfield Theoretical Treatment of the Higgs-Kibble Mechanism

Abstract

The superfield theoretical aspect of the Riggs-Kibble mechanism is discussed by realiz­ ing the supersymmetry of Becchi, Rouet and Stora (BRS) as translation and dilatation operations on the real elements of the Grassmann algebra. The Slavnov-Taylor identity becomes a statement on each connected Green's function separately in this formulation, and some of the pathological cases discussed in the literature can be avoided by imposing the manifest BRS covariance in terms of the superfields. This is applied to the discussion of non-linear (quadratic) gauge conditions. The manifestly BRS covariant treatment of non­ linear gauge conditions requires a quartic self-coupling of Faddeev-Popov fields to ensure the multiplicative renormalization, although the physical S-matrix is independent of this ghost coupling. The proof of unitarity (i.e., the ghost cancellation) follows the same mechanism as the divergence cancellation in the ordinary superfield theory. The gauge independence is proved by a simple classification of operators according to their BRS trans­ formation properties. We also briefly comment on the canonical treatment of superfields. The renormalization of the Yang-Mills fieldsn and the Riggs-Kibble mecha­ nism2> has been extensively discussed in the literature.s>.•> The Faddeev-Popov Lagrangian 5> provides a starting point for these discussions of the renormalizability and unitarity. In connection with the Faddeev-Popov Lagrangian, Becchi, Rcuet and Stora (BRS) a> introduced a supersymmetry pseudo-algebra which gives rise to a very neat way to derive the Slavnov-Taylor identityn (i.e., the Ward-Takahashi identity associated with the BRS symmetry). It was also recognized that a detailed study of the BRS symmetry is sufficient to investigate the unitarity of the S-matrix elements. 8> Recently, Kugo and Ojima9> further clarified the contents of the BRS symmetry and the structure of the physical Hilbert space. They made the following important observations: (i) The auxiliary Lagrangian multiplier field makes the BRS symmetry manifest, and (ii) the ghost number of the Faddeev-Popov fields should be properly associated with the scale transformation of the ghost fields. Although the latter point is not crucial in the perturbative treatmeneo> of the Yang-Mills fields, the structure of the physical space is much simplified by this interpretation. It is also nicer to keep the Lagrangian manifestly Hermitian at all the stages of the calculation. In this paper, we discuss the superfield theoryll)~J•> associated with the BRS symmetry. Some of the initial attempts 15> toward this direction have been made,