Gravitational Waves from Confining Dark Sectors with Self-Consistent Effective Potentials

TL;DR

This work develops a self-consistent finite-temperature EFT for confining dark SU(N) sectors with F flavors to predict gravitational waves from confinement-induced first-order phase transitions. By enforcing perturbativity, unitarity, and boundedness-from-below, and by incorporating Polyakov-loop confinement effects via PLM and CJT resummation, the authors compute the GW spectrum for N=3 and N=4 and map where the signal remains phenomenologically viable. The key finding is that theoretical consistency substantially narrows the viable parameter space and generally weakens the GW signal, often pushing detectable regions toward the edge of next-generation detector capabilities, though a subset remains potentially accessible to missions like BBO. The work highlights both the value and the current limits of EFT-based predictions for dark-sector phase transitions, and identifies avenues for refinement through higher-order resummations and expanded lattice inputs.

Abstract

In this work, we present a self-consistent prediction for the gravitational wave signal arising from confinement-induced phase transitions in hidden non-Abelian SU(N) gauge theories with F light flavors. To do this, we impose perturbativity and unitarity constraints on the thermal effective potential to identify the portion of parameter space that admits a reliable effective field theory description. We also include the Polyakov-loop-improved finite-temperature potential for both N=3 and N=4, where N is the number of dark colors, using an approximate computation of the mediating effects. We compute the resulting gravitational wave spectrum and delineate the regions of parameter space that remain phenomenologically viable after imposing theoretical consistency conditions. We find that these constraints make uncovering a stochastic background gravitational wave signal in this scenario more challenging, even for proposed future detectors.

Figures (15)

• Figure 1: Polyakov loop for $F = 4$ and $N_c = 3$: $\ell$ as a function of $T$. As the temperature decreases, $\ell$ drops from a deconfined value $\ell\neq 0$ to $\ell \simeq 0$, and the steep fall signals the confining phase transition. The vertical dashed line in black indicates the ratio of the critical temperature $T_c$ of the chiral phase transition to the critical temperature of the confining phase transition $T_\Lambda$, and in green the nucleation temperature $T_p$ to $T_\Lambda$, respectively. The benchmark point shown corresponds to $T_c = 833.21$ GeV, $T_p = 788.58$ GeV, $f_\pi = 2121.32$ GeV, and $T_\Lambda = \xi f_\pi$ with $\xi = 2$, together with $m_{\sigma}^2 = 1.0 \times 10^6\;(\text{GeV})^2$, $m_{\eta'}^2 = 1.54 \times 10^7\;(\text{GeV})^2$, and $m_{X}^2 = 5.8 \times 10^6\;(\text{GeV})^2$.
• Figure 2: Contours of the effective couplings $\lambda_{\text{eff},\sigma}$ and $\lambda_{\text{eff},\pi}$ defined in \ref{['eq:lambdaeffsigma']} and \ref{['eq:lambdaeffpi']} respectively, in the $(\sigma/f_\pi,\,T/f_\pi)$ plane. The benchmark shown corresponds to $N=3$, $F=3$, $T_c = 628.79~\text{GeV}$, $T_n = 608.82~\text{GeV}$, $f_\pi = 1102.27~\text{GeV}$, and $\xi = 1$. It is important to note that this point meets $49 \%$ of the perturbativity criteria. The dots indicate the position of the non-symmetric minimum as a function of temperature: blue dots denote that this minimum is metastable (false vacuum), while white dots denote that it is the true minimum at that temperature. Note the numerical noise associated with the convergence of the system of equations for the dressed masses when a particle mass passes through zero; see \ref{['app:cjt']} for details.
• Figure 4: The $|\Re(a_0)|$ vs $\sqrt{s}$. For $XX\rightarrow XX$ scattering in \ref{['fig:UnitarityXXXX']} and $\eta'\sigma\rightarrow \eta'\sigma$ in \ref{['fig:Unitarityeta2sigma2']} for $F=4$. The green dot-dashed line represents the minimum kinematically accessible centre of mass energy, and the dashed red lines represents (kinematically inaccessible) poles in the 4-point amplitude. Unitarity bounds are set by scanning from the minimum kinematically accessible energy to the EFT cutoff at $\Lambda_\text{cutoff}=4\pi f_\pi$. If the maximum exceeds $|\Re(a_0)|>0.5$, the point is non-unitary which is indicated by a horizontal dotted line in \ref{['fig:Unitarityeta2sigma2']}. Any poles in the kinematically accessible region are excluded from the scan by \ref{['eq:PoleCutout']}, which do not appear for the cases shown in this figure. Since the $|\Re(a_0)|$ does exceed 0.5 in the kinematically accessible region, this point does not obey tree-level perturbative unitarity. This benchmark is shown for $m_\sigma^2=1\times10^6\;\text{(GeV)}^2,\; m_{\eta'}^2=1.06\times10^7\;\text{(GeV)}^2,\; m_X^2=5.8\times10^6\;\text{(GeV)}^2$ and $f_\pi=1270\;\text{GeV}$. Equivalently in terms of Lagrangian parameters, $m_\Phi^2=5\times10^5\;(\text{GeV})^2,\,c=6.54,\lambda_\sigma=3.89$ and $\lambda_a=0.309$.
• Figure 5: The SNR for observing the GW signal in the BBO detector for different couplings in the potential that yield FOPTs. In all panels, $F=4$, $\xi=2$, and $v_w$ is calculated using the LTE method. The top set of plots shows $N=4$, while the bottom set shows $N=3$. The color scaling indicates SNR for each parameter point. In the top left panel for each set, we present $\lambda_\sigma$ vs. $c$. In the right column for each set, we present $\pm\lambda_a$ vs. $c$ (separated to show log scaling). In the bottom left panel, we plot the thermal parameters $\alpha$ vs. $\beta/H$. Filled points on the plot pass all EFT consistency requirements. Open circles are points that fail the perturbativity requirement and crosses represent points that fail the unitarity requirement. When a point fails both, the circle and cross are overlaid.
• Figure 6: The SNR for observing the GW signal in the BBO detector for different couplings in the potential that yield FOPTs. In all panels, $F=3$, $\xi=1$, and $v_w=1$. The top set of plots are for $N=4$, while the bottom set are the results when the Polyakov loop contributions are turned off. The coupling $c$ is in units of $\text{GeV}$ when $F=3$. Like in Fig. \ref{['fig:snr-f4']}, filled points on the plot pass all EFT consistency requirements; open circles are points that fail the perturbativity requirement and crosses represent points that fail the unitarity requirement; and when a point fails both, the circle and cross are overlaid.