Acoustic phonons in a magnetized vacuum? First-principle lattice results on the mass spectrum of the electroweak model in a strong magnetic field

TL;DR

This work uses first-principles lattice simulations to map the bosonic electroweak sector in a strong background field at zero temperature, confirming a three-phase structure with crossovers at $B_{c1}$ and $B_{c2}$ and detailing the mass spectrum of Higgs, $W$, and $Z$ bosons across phases. The $W$ boson with spin projection $s_z=+1$ becomes an almost massless excitation in the intermediate vortex phase, which the authors interpret as a phonon-like mode of the magnetized vortex lattice that hybridizes with $W$ fluctuations; Higgs and $Z$ masses remain nonzero throughout. Analytical Landau-level expectations are juxtaposed with numerical results, revealing quantitative shifts due to quantum corrections and lattice effects. The findings illuminate how a magnetized EW vacuum supports lattice-vibration-like collective modes and refine understanding of phase structure under extreme fields with potential implications for cosmology and high-field astrophysical environments.

Abstract

We use numerical Monte Carlo simulations to determine the mass spectrum of the bosonic sector of the electroweak model in an external magnetic field of the electroweak-scale strength ($10^{20}\,{\rm T}$) at zero temperature. It is known that as the magnetic field gets stronger, the electroweak vacuum undergoes two consecutive crossover-type transitions, passing from (i) the conventional symmetry-broken homogeneous phase to (ii) an intermediate inhomogeneous vortex phase characterized by a (superconducting) condensate of electrically charged $W$ bosons and then to (iii) a homogeneous phase with a restored electroweak symmetry. We show that the spin component of the $W$ boson aligned with the direction of the magnetic field is the lightest excitation in all three phases. Its mass continuously decreases in the low-field broken phase and becomes very small in the intermediate phase. We argue that this nearly massless excitation corresponds to a Goldstone acoustic phonon mode associated with vibrations of the lattice of electroweak vortices. In the high-field symmetry-restored phase, where the vortices disappear, the lightest $W$ mass rises again. Neither Higgs nor $Z$ boson masses vanish across all studied phases and crossover transitions.

Figures (2)

• Figure 1: Masses of $Z$ and $W$ bosons with $s_z = +1$ (top) and $s_z = 0$ (bottom) projections of the spin on the magnetic field axis as a function of magnetic field strength given in terms of the Higgs (the lower axis) and $W$ (the upper axis) masses for different lattice sizes. The mass of the Higgs field is shown on both plots for reference. Two vertical solid lines mark the pseudocritical transitions \ref{['eq_Bcs_lat']} obtained in Ref. PhysRevD.110.030001. The yellow solid line in each plot represents the best fit of the Higgs mass by function \ref{['eq_Higgs_high_fit']}.
• Figure 2: The masses of the $s_z = +1$ and $s_z = 0$ spin states of the $W$ boson field in the broken and superconducting phases. The theory-inspired fit \ref{['eq_W12_fit']} of the $s_z = +1$ mass is represented by the solid brown line, which hits zero at the point $B = B_{c1}$ of the first phase transition \ref{['eq_Bc2_lat']}. The dashed magenta line shows the parabolic fit \ref{['eq_fit_mz0']} of the mass of the $s_z = 0$ state. The plateau in the mass of the $s_z = + 1$ state in the intermediate, superconducting phase ($B_{c1} < B < B_{c2}$) appears at value \ref{['eq_W_plateau']}, shown in the inset by the horizontal solid line.