Phase Structure of Electroweak Vacuum in a Strong Magnetic Field: The Lattice Results

Using first-principle lattice simulations, we demonstrate that in the background of a strong magnetic field (around ${10}^{20}\mathrm{T}$), the electroweak sector of the vacuum experiences two consecutive crossover transitions associated with dramatic changes in the zero-temperature dynamics of the vector $W$ bosons and the scalar Higgs particles, respectively. Above the first crossover, we observe the appearance of large, inhomogeneous structures consistent with a classical picture of the formation of $W$ and $Z$ condensates pierced by vortices. The presence of the $W$ and $Z$ condensates supports the emergence of the exotic superconducting and superfluid properties induced by a strong magnetic field in the vacuum. We find evidence that the vortices form a disordered solid or a liquid rather than a crystal. The second transition restores the electroweak symmetry. Such conditions can be realized in the near-horizon region of the magnetized black holes.

Figure 1

(a) Normalized value $⟨{\varphi }^2{⟩}_n={\varphi }_r^2(B_Y)/{\varphi }_r^2(0)$ of the additively renormalized, volume-averaged, ${\varphi }^2\equiv (1/V)\int {\varphi }^{†}\varphi$, squared Higgs condensate ${\varphi }_r^2(B_Y)=⟨{\varphi }^2(B_Y)⟩-⟨{\varphi }^2(\infty )⟩$ (blue) and the likewise normalized action $⟨S{⟩}_n$ (green). The insets show the density plots of the $Z_{12}$ fluxes in the cross sections normal to the magnetic field axis in typical configurations (more details are provided in the description of Fig. 3). (b) The susceptibility of the Higgs field squared vs the hypermagnetic field $B_Y$. The right inset shows the fit of the Higgs condensate shown in (a) by the piecewise function (12). The left inset illustrates a typical 3D configuration in the inhomogeneous phase in the (hyper)magnetic field background $g^{\prime }{B}_Y=eB=1.1m_W^2$ (the total number of vortices is 24). The equipotential surfaces of the $W$ condensate (the Higgs condensate) are shown in blue and red (green). These quantities, which take their maximal values at the centers of the corresponding structures, are shown in complimentary regions. We used a cooling procedure to improve visibility of this 3D picture. The vertical red lines denote the transitions.

Figure 2

Expectation value of the $W$ condensate vs the (hyper)magnetic field $g^{\prime }{B}_Y=eB$. The right inset is the theoretical result based on the classical solution for the transverse $W$ condensate squared [21] around the first transition. The left inset shows the susceptibility of the transverse $W$ condensate.

Figure 3

Density plots of various quantities in the cross-sections normal to the axis of the (hyper)magnetic field. Theoretical results (a)–(d) are given for the classical solution of Ref. [21] at $B=1.01B_c$. The numerical results of the first-principle simulations, (e)–(m), are given for a typical lattice configuration in the background of the (hyper)magnetic field $g^{\prime }{B}_Y\equiv eB\simeq 1.1m_W^2\simeq 1.6e{B}_{c1}$ (the magnetic number $k=12$ for our lattices). (a) and (g): transverse $W$ condensate $W_{\perp }=\sqrt{|W_x{|}^2+|W_y{|}^2}$; (b) and (h): transverse $Z$ condensate $Z_{\perp }=\sqrt{|Z_x{|}^2+|Z_y{|}^2}$; (c) and (i) local excess of the Higgs expectation value over the condensate, $\mathrm{\Delta }\varphi (\mathit{x})=\varphi (\mathit{x})-⟨\varphi ⟩$; (d) and (k): local excess of the magnetic field value over the background, $\mathrm{\Delta }B(\mathit{x})=B(\mathit{x})-B_{\mathrm{ext}}$; (e) and (l): longitudinal $W$ and $Z$ condensates, $W_{\parallel }=|W_z|$ and $Z_{\parallel }=|Z_z|$, respectively (theoretically, $W_{\parallel }=Z_{\parallel }=0$ at the classical level); (f): the $Z$ flux; (m) the neutral Higgs currents ${\mathit{J}}_{\perp }^Z$. The red (blue) colors correspond to maxima (minima); absolute values are given for complex quantities. The same regions are circumvented by the red and green dashed lines (separately for the analytical solution and simulated configuration) to guide the eye. Each simulated picture is obtained by averaging data over a part of the Markov chain (last 5000 configurations). Averaging over the entire ensemble leads to blurring of the vortex structure.