A thorny path of field theory: from triviality to interaction and confinement

Abstract

Summation of the perturbation series for the Gell-Mann--Low function \beta(g) of \phi^4 theory leads to the asymptotics \beta(g)=\beta_\infty g^\alpha at g\to\infty, where \alpha\approx 1 for space dimensions d=2,3,4. The natural hypothesis arises, that asymptotic behavior is \beta(g) \sim g for all d. Consideration of the "toy" zero-dimensional model confirms the hypothesis and reveals the origin of this result: it is related with a zero of a certain functional integral. This mechanism remains valid for arbitrary space dimensionality d. The same result for the asymptotics is obtained for explicitly accepted lattice regularization, while the use of high-temperature expansions allows to calculate the whole \beta-function. As a result, the \beta-function of four-dimensional \phi^4 theory is appeared to be non-alternating and has a linear asymptotics at infinity. The analogous situation is valid for QED. According to the Bogoliubov and Shirkov classification, it means possibility to construct the continuous theory with finite interaction at large distances. This conclusion is in visible contradiction with the lattice results indicating triviality of \phi^4 theory. This contradiction is resolved by a special character of renormalizability in \phi^4 theory: to obtain the continuous renormalized theory, there is no need to eliminate a lattice from the bare theory. In fact, such kind of renormalizability is not accidental and can be understood in the framework of Wilson's many-parameter renormalization group. Application of these ideas to QCD shows that Wilson's theory of confinement is not purely illustrative, but has a direct relation to a real situation. As a result, the problem of analytical proof of confinement and a mass gap can be considered as solved, at least on the physical level of rigor.