Euclidean scalar field theory in the bilocal approximation

The blocking step of the renormalization group method is usually carried out by restricting it to fluctuations and to local blocked action. The tree-level, bilocal saddle point contribution to the blocking, defined by the infinitesimal decrease of the sharp cutoff in momentum space, is followed within the three dimensional Euclidean ${\varphi }^6$ model in this work. The phase structure is changed, new phases and relevant operators are found, and certain universality classes are restricted by the bilocal saddle point.

Figure 1

The graphical representation of the terms, appearing in the right-hand side of the evolution, (27). The solid line denotes the propagator $D^{(k)}$ within the thin shell of modes, being eliminated, the dashed line stands for the bilocal potential, $V_{x-y}({\chi }_x,{\chi }_y)$ and the black square represents $U^{\prime }(\chi )$.

Figure 2

The graphical representation of the terms, appearing in the right-hand side of the evolution, (27), the external legs suppressed. The notation is the same as in Fig. 1 except that the cross represents the local vertex, $U^{\prime \prime }(\chi )$.

Figure 3

The phase structure on the $({\overline{g}}_2,{\overline{g}}_4)$ plane at ${\overline{g}}_6(\mathrm{\Lambda })=10$. The tree- and the loop-level phase boundaries are depicted by thin and thick lines, respectively. The vertical dotted lines indicate a second order phase transition for $g_4>0$. The bilocal theory is stable, $c>0$, within the horizontal parabola, $|h|=1$, shown by a solid line. The dashed lines represent the tree-level first order transition. The fluctuations restore the symmetry for weakly negative $g_2$ and induce a small shift by decreasing $g_4$ for positive $g_2$.

Figure 4

(a): The projection of few renormalized trajectories with $g_6(\mathrm{\Lambda })=0.1$ on the plane $({\tilde{g}}_2,{\tilde{g}}_4)$ of the local model, (b): zoom around the Gaussian fixed point. The solid curves run within the symmetric phase and the dashed curves belong to the symmetry broken phase, generated by a second order phase transition. The evolution is stopped at a finite value of the cutoff where a symmetry breaking saddle point with zero mode is generated. The trajectories, shown by the dotted lines, stop at the instability, $c=0$.

Figure 5

The same as Fig. 4 except for the bilocal theory.