Abstract
Using the background field method, we study in a general covariant gauge the renormalization of the 6-dimensional Yang-Mills theory. This requires background gauge invariant counterterms, some of which do not vanish on shell. Such counterterms occur, even off-shell, with gauge-independent coefficients. The analysis is done at one loop order and the extension to higher orders is discussed by means of the BRST identities. We examine the behaviour of the beta function, which implies that this theory is not asymptotically free.
Highlights
The background field formulation is a procedure which enables the calculation of quantum corrections, while preserving the gauge invariance of the background field
We have examined the renormalization of the sixdimensional Yang-Mills theory, which has a coupling with length dimension, as a model for the gauge theory of gravity
The YM theory was studied in a general covariant gauge which preserves the background field invariance
Summary
The background field formulation is a procedure which enables the calculation of quantum corrections, while preserving the gauge invariance of the background field. It has been shown that on mass shell, pure gravity is renormalizable to one-loop order, despite the fact that it contains a dimensional coupling which would make the theory nonrenormalizable by power counting This calculation has been done in particular gauges, by using a “topological invariant” which relates the scalars constructed from four derivatives of the gravitational field. Unlike the case of gravity, we find to one-loop order a counterterm which does not vanish on mass shell and appears with a gauge-independent coefficient This means that the six-dimensional YM theory is not renormalizable in the power-counting sense. IV, by means of a generalization of the BecchiRouet-Stora-Tyutin (BRST) identities This symmetry, together with the background gauge invariance, is sufficient to ensure the renormalizability of the theory to all orders, in the more general sense.
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