The $1/N$ expansion in the $O(N)\times O(N)$ scalar theory and the problem of the restoration of symmetry at high temperatures

The problem of symmetry restoration at high temperatures in the model (1) is considered. The large $N$ expansion method in space-time dimensions $d=3,4$ is used. In the case $d=4$ and $T=0$ the initial symmetry remains intact and the problem stays open. In the case $d=3$ the symmetry may be broken spontaneously at $T=0$ but it is always restored at $T\to\infty$.