How to approach continuum physics in the lattice Weinberg-Salam model

We investigate the lattice Weinberg-Salam model without fermions numerically for the realistic choice of coupling constants correspondent to the value of the Weinberg angle ${\theta }_W\sim 30°$, and bare fine structure constant around $\alpha \sim \frac{1}{150}$. We consider the values of the scalar self-coupling corresponding to Higgs mass $M_H\sim 100$, 150, 270 GeV. It has been found that nonperturbative effects become important while approaching continuum physics within the lattice model. When the ultraviolet cutoff $\Lambda =\frac{\pi }{a}$ (where $a$ is the lattice spacing) is increased and achieves the value around 1 TeV, one encounters the fluctuational region (on the phase diagram of the lattice model), where the fluctuations of the scalar field become strong. The classical Nambu monopole can be considered as an embryo of the unphysical symmetric phase within the physical phase. In the fluctuational region quantum Nambu monopoles are dense, and therefore, the use of the perturbation expansion around the trivial vacuum in this region is limited. Further increase of the cutoff is accompanied by a transition to the region of the phase diagram, where the scalar field is not condensed (this happens at the value of $\Lambda$ around 1.4 TeV for the considered lattice sizes). Within this region further increase of the cutoff is possible, although we do not observe this in detail due to the strong fluctuations of the gauge boson correlator. Both above mentioned regions look unphysical. Therefore we come to the conclusion that the maximal value of the cutoff admitted within lattice electroweak theory cannot exceed the value of the order of 1 TeV.

Figure 1

The phase diagram of the model in the $(\gamma ,\lambda )$ plane at $\beta =12$. The dashed line is the tree level estimate for the line of constant physics correspondent to bare $M_H^0=270\mathrm{Gev}$. The continuous line is the line of phase transition between the physical Higgs phase and the unphysical symmetric phase (statistical errors for the values of $\gamma$ at each $\lambda$ on this line are about 0.005).

Figure 2

Susceptibility $\chi =⟨H^2⟩-⟨H{⟩}^2$ (for $H_x=\sum _y{Z}_{xy}^2$) as a function of $\gamma$ at $\lambda =0.001$ and $\beta =12$. Circles correspond to the lattice ${12}^3\times 16$. Crosses correspond to the lattice $8^3\times 16$.

Figure 3

The effective constraint potential at $\lambda =0.009$ and $\beta =12$. Solid squares correspond to ${\gamma }_c=0.273$. Empty squares correspond to $\gamma =0.29$. Triangles correspond to $\gamma =0.279$. The error bars are about the same size as the symbols used.

Figure 4

${\varphi }_m$ as a function of $\gamma$ at $\lambda =0.001$ and $\beta =12$. Circles correspond to lattice $8^3\times 16$. Crosses correspond to lattice ${12}^3\times 16$.

Figure 5

$H$ (points) vs $H_{\mathrm{fluct}}$ (stars) as a function of $\gamma$ at $\lambda =0.009$ and $\beta =12$. Statistical errors for $H_{\mathrm{fluct}}$ are about the same size as the symbols used.

Figure 6

Fluctuation $\delta \varphi$ as a function of $\gamma$ at $\lambda =0.0025$ and $\beta =12$. Circles correspond to the lattice $8^3\times 16$. Crosses correspond to the lattice ${12}^3\times 16$. Triangles correspond to the lattice ${16}^4$. The transition point is ${\gamma }_c=0.261\pm 0.001$; it is clear that the maximum of $\delta \varphi$ is shifted to larger values of $\gamma$.

Figure 7

Mean value of $|\varphi |$ as a function of $\gamma$ at $\lambda =0.0025$ and $\beta =12$. (Lattice $8^3\times 16$.)

Figure 8

The potential ($T=8$) for the right-handed leptons vs $1/R$ at $\gamma =0.262$, $\lambda =0.0025$, and $\beta =12$ (lattice ${16}^4$).

Figure 9

The inverse renormalized fine structure constant as a function of $\gamma$ at $\lambda =0.0025$ and $\beta =12$. It slowly approaches the tree level estimate $\sim 150$ when $\gamma$ and the lattice size are increased. Circles correspond to lattice ${12}^3\times 16$. Crosses correspond to lattice $8^3\times 16$. Triangles correspond to lattice ${16}^4$.

Figure 10

$Z$-boson mass in lattice units at $\lambda =0.009$ and $\beta =12$ as a function of $\gamma$. Solid triangles correspond to lattice ${12}^3\times 16$. Crosses correspond to lattice $8^3\times 16$. Circles correspond to lattice ${16}^4$. Square corresponds to lattice ${20}^3\times 24$. (The error bars for lattices ${16}^3\times 16$ and ${20}^3\times 24$ are about the same size as the symbols used.)

Figure 11

$Z$-boson mass in lattice units at $\lambda =0.001$ and $\beta =12$. Circles correspond to lattice $8^3\times 16$. Crosses correspond to lattice ${12}^3\times 16$.

Figure 12

$Z$-boson mass in lattice units at $\lambda =0.0025$ and $\beta =12$. Circles correspond to lattice ${12}^3\times 16$. Crosses correspond to lattice $8^3\times 16$. Triangles correspond to lattice ${16}^4$.

Figure 13

Nambu monopole density as a function of $\gamma$ at $\lambda =0.009$ and $\beta =12$. (Lattice $8^3\times 16$.)

Figure 14

Nambu monopole density as a function of $\gamma$ at $\lambda =0.0025$ and $\beta =12$. Circles correspond to lattice ${12}^3\times 16$. Crosses correspond to lattice $8^3\times 16$. Triangles correspond to lattice ${16}^4$.

Figure 15

Nambu monopole density as a function of $\gamma$ at $\lambda =0.001$ and $\beta =12$. Circles correspond to lattice ${12}^3\times 16$. Crosses correspond to lattice $8^3\times 16$.