I am trying to study the canonical formulation of Yang-Mills theories so that I have direct access to the $n$-particle of the theory (i.e. the Hilbert Space). To that end, I am following Kugo and Ojima's (1978) 3-part paper.
At the outset, I am confused by their Lagrangian, and their two differences from the conventional one (I write the Lagrangian 2.3 of their paper):
$$\mathcal{L}=-\frac{1}{4}F^a_{\mu\nu}F^{a\,\mu\nu}-i\partial^\mu\bar{c}D_\mu^{ab}c^b-\partial^\mu B^a A_\mu^a+\alpha_0 B^a B^a/2$$
- They have rescaled the Ghost field so that its kinetic term has a factor of $i$ in front.
- They have integrated by parts on $B^a\partial_\mu A^{a\,\mu}$, effectively making $B$ dynamical.
The authors chose these two differences are so that (1) BRS variations (eq 2.15 in their paper) preserve Hermiticity of the Ghost fields, and (2) to make the Lagrangian BRS invariant.
I am totally confused by their second point. I thought the standard BRS Lagrangian appearing in standard texts, for example, in Peskin and Schroeder was already BRS invariant. Why the $\partial B.A$ term?
Best Answer
Kugo and Ojima's work was one of the major breakthroughs in understanding the role of BRST in the quantization of gauge theories. Historically BRST was discovered in the path integral formalism. The understanding of this theory as a cohomology theory started from the Kugo and Ojima's work.
Now, the action is BRST invariant with and without the Gaussian integration over the auxiliary field $B^a$ (called the Lautrup-Nakanishi multipliers). They are introduced in order to not have explicit dependence on the gauge parameter in the Ward identities (please see a recent review by Becchi). The BRST invariance Ward identities is a crucial step in the unitarity proof.
Kugo and Ojima actually solved the BRST cohomology problem of the Yang-Mills theory. They actually identified the physical and unphysical states of the theory (in terms of the BRST operator) as follows: The physical states correspond to the states annihilated by the BRST operator and in addition of positive norm.
The unphysical states are arranged in degenerate quartets. This is called the Kugo - Ojima quartet mechanism. One quartet corresponds to a ghost, anti-ghost, longitudinal and temporal gluons. In their formalism these states can be generated from the vacuum by the action of the ghost antighost operators as well as by the field $B^a$ and its conjugate momentum. They also conjectured that since colored quark operators and transversal gluon operators belong to the quartet sector, then these states must be confined.