Speaker
Description
We investigate massive models of quantum field theory of scalar field in logarithmic dimensions in Euclidean space. The Schwinger-Dyson equation and non-trivial solution for mass are considered in the paper.
The Schwinger-Dyson equation has the form:
$$ D^{-1} = \Delta^{-1} - \Sigma $$
where $D$ is a full propagator, $\Delta$ is a bar propagator, $\Sigma$ is a self-energy operator.
In the minimal subtraction (MS) scheme it holds:
$$ \Delta (p) = \frac{1}{p^2} $$
where $p$ is a momentum.
The inverse full propagator has the following characteristic:
$$ \left\{ \begin{array}{l}
D^{-1} (p)|_{p^2=-m^2} = 0 \\
\left. \left( \frac{\partial}{\partial (p^2)} D^{-1} (p) \right) \right|_{p^2=-m^2} = \frac1A
\end{array} \right. . $$
In the main approximation of perturbation theory it holds:
$$ D(p) = \frac{A}{p^2+m^2} $$
where $A$ is an amplitude, $m$ is a mass.
We investigate the scalar models $\phi^3$, $\phi^4$ and $\phi^6$. For the theories $\phi^3$ and $\phi^4$ mass appears in the first order of perturbation theory whereas for the $\phi^6$-theory the mass does not appear in the first order.
