  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.
