First-order QCD transition in a primordial magnetic field

Recalling the expectation of an extremely strong primordial magnetic field $H$, we recheck transitions among the phases of chiral symmetry restoration ($\chi SR$), chiral symmetry breaking ($\chi SB$), and pion superfluidity ($\pi SF$) in the QCD epoch of the early universe. For homogeneous phases in a finite $H$, a sensible scheme is adopted to determine the phase boundaries of $\pi SF$, which is also the superconductivity phase itself. In the first part, the QCD phase diagrams are studied in detail within the chiral effective Polyakov-Nambu–Jona-Lasinio model, and the transitions involving $\pi SF$ are found to be of first order at relatively small $H$. As expected from the Meissner effect, the regime of $\pi SF$ shrinks with increasing $H$ and completely vanishes beyond a threshold value. In the second part, the bubble dynamics is illuminated for the stronger first-order transition, $\chi SR\to \pi SF$, in the more convenient Polyakov-quark-meson model. The coupled equations of motion of pion condensate and magnetic field are solved consistently to give the bubble structure. Then, based on bubble collisions, we explore gravitational wave (GW) emission by developing a simple toy model in advance; and the characteristic frequency of the relic GW is estimated to be of the order 0.1–1 K or ${10}^9–{10}^{10}\mathrm{Hz}$ in our galaxy.

Figure 1

Upper panel: the $T-(l_{\mathrm{e}}^{\mathrm{M}}+l_{\mu }^{\mathrm{M}})$ phase diagrams for the magnetic fields $eH=0$, 0.01, 0.02, and $0.022{\mathrm{GeV}}^2$ (colors: blue, red, green, and cyan) with the bullets the critical end points. Lower panel: the $T-eH$ phase diagram for the lepton flavor-entropy ratio $l_{\mathrm{e}}^{\mathrm{M}}+l_{\mu }^{\mathrm{M}}=-0.2$. The shadows correspond to the pion superfluidity phase, and the blue dashed line and other colored lines denote second- and first-order transition boundaries, respectively.

Figure 2

The temperature $T_{\mathrm{CEP}}$ (upper panel) and lepton flavor-entropy ratio $(l_{\mathrm{e}}^{\mathrm{M}}+l_{\mu }^{\mathrm{M}}{)}_{\mathrm{CEP}}$ (lower panel) as functions of magnetic field $eH$ at the critical end points.

Figure 3

The quark masses $m_{\mathrm{f}}$, pion condensate ${\mathrm{\Delta }}_{\pi }$, and Polyakov loop $L$ as functions of the temperature $T$, decreasing with the expansion of the early universe, for the case $eH=0.01{\mathrm{GeV}}^2$ and $l_{\mathrm{e}}^{\mathrm{M}}+l_{\mu }^{\mathrm{M}}=-0.2$. Here, $m_{\mathrm{u}}$ and $m_{\mathrm{d}}$ are almost the same with each other and thus can be consistently presented as $m_{\mathrm{l}}$.

Figure 4

Upper panel: the cosmic trajectories of the chemical potentials of electric charge (${\mu }_{\mathrm{Q}}$), baryon (${\mu }_{\mathrm{B}}$), and lepton flavors (${\mu }_{\mathrm{e}},{\mu }_{\mu }$, and ${\mu }_{\tau }$) as functions of the temperature $T$; lower panel: the entropy density ($s$) and density-entropy ratios ($q,b,l_{\mathrm{e}},l_{\mu }$, and $l$) as functions of the temperature $T$. The same as that in Fig. 3, we consider the case $eH=0.01{\mathrm{GeV}}^2$ and $l_{\mathrm{e}}^{\mathrm{M}}+l_{\mu }^{\mathrm{M}}=-0.2$.

Figure 5

The free energy differences of strong interaction sector $\mathrm{\Delta }{\mathrm{\Omega }}_{\mathrm{C}}$ (dotted line) and electroweak interaction sector $\mathrm{\Delta }{\mathrm{\Omega }}_{\mathrm{EW}}$ (dashed line) as functions of temperature $T$ at $eH=0.01\text{Ge}{\mathrm{V}}^2$. After taking the constraints $n^{\mathrm{Q}}=0,b^{\mathrm{M}}=8.6*{10}^{-11},l^{\mathrm{M}}=-0.012,l_{\mathrm{e}}^{\mathrm{M}}=0$, and $l_{\mu }^{\mathrm{M}}=-0.2$ to fix the chemical potentials, we compare the cases with $H$-dependent (blue line) and $H=0$ (red line) fermion loops. At the low temperature $T=0.05\mathrm{GeV}$, the value $\mathrm{\Delta }{\mathrm{\Omega }}_{\mathrm{C}}=5.45\times {10}^{-4}\text{Ge}{\mathrm{V}}^4$ actually equals the energy density of the background magnetic field $H^2/2$.

Figure 6

For homogeneous phases, the pure QCD part of the free energy ${\mathrm{\Omega }}_{\mathrm{QCD}}$ is depicted as a function of the order parameters ${\mathrm{\Delta }}_{\pi }$ and $L$ at temperature $T=0.2\mathrm{GeV}$.

Figure 7

The solution $L$ as a function of $\tilde{\pi }$ from Eq. (60) within the range where the homogeneous solutions Eq. (58) (red bullets) are covered.

Figure 8

The solutions $\tilde{\pi }(r)$ (yellow solid line), $L(r)$ (green dotted line), and $B(r)$ (black dashed line) as functions of the radius $r$ from Eqs. (60), (68), and (72). The Polyakov loop and pion condensate are normalized by their maximal values to $L^{\mathrm{r}}$ and ${\mathrm{\Delta }}_{\pi }^{\mathrm{r}}$, and the magnetic field $B$ by its background value $H$ to $B^{\mathrm{r}}$, respectively. The red baseline is $f(r)=1$.

Figure 9

The spacetime evolution of the reduced fields ${\tilde{\pi }}^{\mathrm{r}}$ (upper panel) and $B^{\mathrm{r}}$ (lower panel).

Figure 10

A toy model for bubble collision: the system and nucleated bubbles are all chosen to be cylinders with the height equal to the diameter and their axes along the direction of the background magnetic field, and the two bubbles are assumed to be generated simultaneously at the locations $x=\pm d/4,y=z=0$. The system size is $d=200{\mathrm{GeV}}^{-1}$ and the bubble radius was found to be $R_{\mathrm{b}}=9.60{\mathrm{GeV}}^{-1}$ according to the calculation in Sec. 3a.

Figure 11

The spectra of the GW strain: the real (red solid line) and imaginary (blue dashed line) parts of $H_{+}$ with frequency $\omega$ in the upper panel and $h_{+}$ with time $t$ in the lower panel.