2, I start with a description of the combinatorics for the Abelian case in order to explain the procedure in a simpler setting, but also to show that the method proposed here does not change the perturbative behavior of the theory.
The layout of this work is as follows: in Sect. Any transitions between the two vacua happen in non-trivial backgrounds of finite temperature or fermion density.Īlthough the work presented here can be related to older phenomenological models, and can also be considered as an effective action that describes the properties of the strong interactions, it should be stressed that it is derived from first principles, namely the treatment of the constraints in the quantum theory, and is proposed as a complete and exact description of the Yang–Mills vacuum and associated features. Īs far as the Lorentz invariance of the theory is concerned, the two vacua admit a Lorentz invariant energy–momentum tensor, the one expected by the bag model, but they are completely stable for pure Yang–Mills theory at zero temperature, and there is no Lorentz invariant energy–momentum tensor that connects them (at least not with the effective action derived in this work, which concerns pure Yang–Mills at zero temperature).
Once the vacuum structure of the theory is better understood, several properties of the Yang–Mills theory, that were also expected to be related, can be easily seen: confinement, bag model, chiral symmetry breaking, as well as a possible solution to the strong-CP problem. They are solitonic solutions with a finite mass “gap” of order \(\mu /g^2\) (where g is the coupling constant of the non-Abelian theory). There are, however, stable Lorentzian solutions, “bubbles” of the chromoelectric field, “glueballs”, that connect the two vacua and can mediate the transitions between them. Also, two vacua emerge, a local minimum (the perturbative, Coulomb vacuum, \(\Omega _0\), at \(\lambda =0\)) and a maximum of the effective potential (the confining vacuum, \(\Omega _\mu \), at the generated mass scale \(\lambda ^2=\mu ^2\)), that are also shown to be quantum mechanically stable there are no finite action Euclidean solutions that mediate their decay. Here, however, \(\lambda \) has no kinetic term and no additional degrees of freedom, hence there is no symmetry breaking, and the effective potential term appears “inverted”, with the opposite sign.Īlthough this effective potential term here is unbounded below, because of the interplay of the gauge kinetic and gradient terms, as well as the constraint of Gauss’s law, stability is proven for all classical solutions. The present work also includes a scalar field, the Lagrange multiplier, and an associated effective potential term. In particular, the works of can be mentioned, where some interesting and intuitive phenomenological models of the confining mechanism have been proposed, with the addition of a scalar field and an associated effective potential term, that modify the dielectric and fermion condensate properties of the theory at its minimum. The symmetries and other properties of the perturbative and the confining vacuum are explored, and connections are made with older phenomenological models of the strong interactions. A discussion of the Euclidean action, the vacua and possible related vacuum transitions, confinement, and the strong-CP problem is also included. Here, I elaborate on the consequences of the procedure and proposed effective action, I derive an effective Hamiltonian, and examine the energy and stability of solutions with “bubbles” of the chromoelectric field. Yang–Mills Existence and Mass Gap.In a previous work, I considered the possibility of expressing the constraint of Gauss’s law in the perturbative expansion of gauge field theories via a Lagrange multiplier field, \(\lambda \), and argued for the generation of an effective potential term of the Coleman–Weinberg type for \(\lambda \), and itsS relation to the problems of the mass gap and confinement in the non-Abelian case. Yang–Mills existence and mass gap problem is officially stated by Clay Mathematics Institute:
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