Effective potential in Yang-Mills theory

In [22] these results were applied to study the effective action in the Yang-Mills theory in flat space of any dimension with arbitrary compact simple gauge group and arbitrary matter. I assumed a covariantly constant gauge field strength of the most general form, (Savvidy type chromomagnetic vacuum) and covariantly constant scalar fields as the background and calculated explicitly the one-loop low-energy effective action. For groups of higher rank and spacetimes of higher dimensions such field configurations have many independent color components taking values in Cartan subalgebra of the gauge algebra and many ``magnetic fields'' in each color component. A new method to study the stability of the vacuum based on the behavior of the corresponding heat kernels is proposed and a simple criterion of stability is explicitly formulated. It is shown that the background field configurations with covariantly constant chromomagnetic fields can be stable only in the case when more than one independent ``magnetic fields'' are present and their amplitudes do not differ much from each other. Moreover, I showed that this is possible only in Euclidean spaces of dimension not less than four and in space-times of Lorentzian signature with dimensions not less than five. An explicit example of stable background field configurations is also found.

In [23] I exactly calculated the relevant zeta-functions in the case of equal amplitudes of ``magnetic fields''. For two ``magnetic fields'' with equal amplitudes the behavior of the effective action is studied in detail. It is shown that in dimensions d=4,5,6,7 $({\rm mod}\, 8)$, the perturbative vacuum is metastable, whereas in dimensions d=9,10,11 $({\rm mod}\, 8)$ the perturbative vacuum is absolutely stable. In dimensions d=8 $({\rm mod}\, 8)$ the perturbative vacuum is stable for small values of the coupling constant but becomes unstable for large coupling constant leading to the formation of a non-perturbative stable vacuum with nonvanishing ``magnetic fields''. The critical value of the coupling constant and the amplitudes of the vacuum ``magnetic fields'' are evaluated exactly.