[Paper Review] Lattice study of monopoles in the Electroweak theory

B. L. G. Bakker, A.I. Veselov+1 more|ArXiv.org|Aug 21, 2007

Particle physics theoretical and experimental studies3 references3 citations

TL;DR

This lattice study investigates Nambu monopoles in the Electroweak theory at realistic coupling constants ($\alpha \sim 1/128$, $\theta_W = \pi/6$) with a heavy Higgs boson ($M_H > 2M_W$). Using numerical simulations on $8^4$, $16^4$, and $24^4$ lattices, it finds that monopole density is not determined by standard parameters but must be treated as a new external parameter, while monopole action excess confirms their physicality as stable, finite-energy objects.

ABSTRACT

We investigated numerically properties of Nambu monopoles in lattice Electroweak theory at realistic values of $α$ and $θ_W$. Our choice of parameters of lattice Lagrangian corresponds to large values of the Higgs boson mass $M_H > 2 M_W$. We find that the density of Nambu monopoles cannot be predicted by the choice of the initial parameters of Electroweak theory and should be considered as the new external parameter of the theory. We also investigate the difference between the versions of Electroweak theory with the gauge groups $SU(2)\otimes U(1)$ and $SU(2)\otimes U(1)/Z_2$. We do not detect any difference at $α\sim {1/128}$. However, such a difference appears in the strong coupling region and is related to the properties of monopoles constructed of the hypercharge field.

Motivation & Objective

• To investigate the existence and properties of Nambu monopoles in the Electroweak theory at realistic coupling constants.
• To test whether the choice of gauge group ($SU(2)\times U(1)$ vs. $SU(2)\times U(1)/Z_2$) leads to observable differences in monopole behavior.
• To determine whether the monopole density is fixed by initial parameters or requires an additional external input.
• To evaluate the physical scale and ultraviolet cutoff achievable in the lattice formulation with a London-limit Higgs potential.

Proposed method

• Numerical simulations of the quenched lattice Weinberg-Salam model with $SU(2)\times U(1)$ and $SU(2)\times U(1)/Z_2$ gauge groups.
• Use of a lattice action with plaquette terms for $SU(2)$ and $U(1)$ fields, and a fermion-like term to generate $W$ and $Z$ bosons.
• Fixing the unitary gauge and employing the lattice Landau gauge to study $W$ boson propagators.
• Defining monopole worldlines on the dual lattice and measuring excess action ($\Delta S_p$, $\Delta S_l$) near monopole trajectories to assess magnetic energy.
• Evaluating the $W$-boson mass to set the physical scale and lattice spacing, with simulations along a line of constant $\alpha_R = 1/128$.
• Analyzing phase diagrams in the $(\beta, \gamma)$-plane to locate the symmetric and broken phases and the transition points.

Experimental results

Research questions

• RQ1 Does the Nambu monopole density in the Electroweak theory depend on the initial parameters of the Lagrangian, or is it an independent physical parameter?
• RQ2 Are there observable differences between the $SU(2)\times U(1)$ and $SU(2)\times U(1)/Z_2$ versions of the Electroweak theory at realistic coupling constants?
• RQ3 Can the action excess near monopole worldlines confirm that Nambu monopoles are physical, finite-energy objects?
• RQ4 What is the maximum achievable ultraviolet cutoff in the lattice formulation when the Higgs self-coupling is infinite (London limit)?
• RQ5 Is the percolation probability of monopoles an order parameter for the electroweak phase transition?

Key findings

• The density of Nambu monopoles cannot be predicted from the initial parameters of the Electroweak theory and must be treated as a new external parameter.
• The monopole density decreases with increasing $\gamma$ (for $\gamma > 1$) and is well approximated by $\rho \sim e^{2.08 - 4.6\gamma}$, contrary to naive expectations.
• The action excess $\Delta S_p$ near monopole worldlines exceeds the average plaquette action, confirming that monopoles carry finite magnetic energy and are physical objects.
• The maximum ultraviolet cutoff achievable along the line of constant $\alpha_R = 1/128$ is $350 \pm 30$ GeV, limited by the London-limit Higgs potential.
• No significant difference was found between the $SU(2)\times U(1)$ and $SU(2)\times U(1)/Z_2$ models at $\alpha \sim 1/128$, but differences emerge in the strong coupling regime related to hypercharge monopoles.
• The percolation probability of monopoles is found to be an order parameter for the transition between the symmetric and broken phases.