The gravitational waves from the first-order phase transition with a dimension-six operator

TL;DR

This paper analyzes gravitational waves from first-order phase transitions in the Standard Model extended by a dimension-six operator, developing a unified description for slow and fast PTs grounded in percolation-based temperatures and a mean bubble-size scale. It performs a full one-loop effective potential calculation with both fixed and running RG scales, finding that RG running can raise the GW peak amplitude by an order of magnitude and shift the peak to lower frequencies, altering detectability. A unified prescription for key GW parameters—total released vacuum energy $\Delta\rho_{\mathrm{vac}}^{\mathrm{tot}}(T_*)$ and the length scale via $HL_*=H_*R_{\mathrm{mean}}$—is proposed to interpolate between regimes. The results indicate that RG effects substantially impact predictions and that viable detection may require different sextic-term cutoffs than previously thought, motivating more non-perturbative studies in future work.

Abstract

We investigate in details the gravitational wave (GW) from the first-order phase transition (PT) in the extended standard model of particle physics with a dimension-six operator, which is capable of exhibiting the recently discovered slow first-order PT in addition to the usually studied fast first-order PT. To simplify the discussion, it is sufficient to work with an example of a toy model with the sextic term, and we propose an unified description for both slow and fast first-order PTs. We next study the full one-loop effective potential of the model with fixed/running renormalization-group (RG) scales. Compared to the prediction of GW energy density spectrum from the fixed RG scale, we find that the presence of running RG scale could amplify the peak amplitude by amount of one order of magnitude while shift the peak frequency to the lower frequency regime, and the promising regime of detection within the sensitivity ranges of various space-based GW detectors shrinks down to a lower cut-off value of the sextic term rather than the previous expectation.

Figures (13)

• Figure 1: The pedagogical introduction of the bounce equation and bounce solution for the first-order PT. The upper left panel presents a schematic illustration of the effective potential at some characteristic temperatures, for example, the inflection temperature $T_\mathrm{inf}$ when a second minimum is about to appear, the degeneration temperature $T_\mathrm{deg}$ when the second minimum is degenerated with the first one, the transition temperature $T_\mathrm{tra}$ when the exit point of bounce solution is exactly sitting at the true vacuum, the nucleation temperature $T_\mathrm{nuc}$ when there are enough nucleated bubbles for the unbroken phase to be transited to the broken phase, and the vanishing temperature $T_\mathrm{van}$ when the potential barrier separating the two vacuums disappeares. The upper right panel presents a schematic illustration of an equivalent particle moving in the inverse of field potential $-V(\phi,T)$ with Hubble friction, where its position is labeled by the field value and its time is labeled by the radial coordinate of bounce solution. The bottom left panel presents a schematic illustration of the shooting algorithm, namely the particle is released from a finely adjusted exit point of bounce solution, above/below which the particle overshoots/undershoots and oscillates around the inverse potential barrier. Only when the appropriate exit point of bounce solution is found could the particle stand still at the origin. The bottom right panel presents a schematic illustration of various field values evolving with the decreasing temperature, for example, the field value $\phi_\mathrm{false}/\phi_\mathrm{true}$ where the false/true vacuum sits, the field value $\phi_\mathrm{zero}$ where the potential crosses zero, the field value $\phi_\mathrm{max}$ where the potential barrier lies, and the field value $\phi_\mathrm{exit}$ where the field penetrates from the other side of potential barrier. The first three panels are also used in Cai:2017cbj.
• Figure 2: Left: Each curve presents the approximated bounce action $S(T)\approx\min[S_4(T),S_3(T)/T]$ at different temperatures $T$ and cut-off scale $\Lambda$. Right: Each curve presents the exit point $\phi_\mathrm{exit}$ of bounce solution at different temperature $T$ and cut-off scale $\Lambda$. Both panels use the cut-off scales between $530\,\mathrm{GeV}$ and $750\,\mathrm{GeV}$ with interval $1\,\mathrm{GeV}$ from top-left to bottom-right. With decreasing cut-off scale in the left panel, the first non-monotonic curve has cut-off scale $\Lambda=593\,\mathrm{GeV}$, below which there is a near-flat platform for each choice of $\Lambda\leq593\,\mathrm{GeV}$ below some characteristic temperature $T<1/R_0$.
• Figure 3: Left: The ratio of bubble nucleation rate with respect to Hubble expansion rate at given temperature. The nucleation temperature is defined by the moment when the bubble nucleation rate first catches up with the Hubble expansion rate. Right: The accumulated number of bubbles within our casual Hubble volume at given temperature since the beginning of PT at transition temperature. The nucleation temperature is defined by the moment when the number of bubbles within our causal Hubble volume first reaches the order of unity. In both panels, the nucleation temperature is found by the intersection point of each curve with the black horizontal line. Each curve is drawn from certain choice of cut-off scale $\Lambda$, ranging from $750\,\mathrm{GeV}$ at top-right to $530\,\mathrm{GeV}$ at bottom-left in both panels. However, both definitions of nucleation temperature break down for $\Lambda\leq582\,\mathrm{GeV}$.
• Figure 4: The probability $P(T)$ of staying at the false vacuum at given temperature $T$ with certain cut-off scale $\Lambda$ for the fast first-order PT (left) with $\Lambda\gtrsim583\,\mathrm{GeV}$ and slow first-order PT (right) with $\Lambda\lesssim582\,\mathrm{GeV}$. The percolation temperature $T_\mathrm{per}$ is given by the intersection point of $P(T)$ with the black horizontal line $P(T)\equiv0.7$. The slow first-order PT is only drawn for $\Lambda\gtrsim578\,\mathrm{GeV}$ to meet the BBN bound.
• Figure 5: The characteristic temperatures of first-order PT at a given cut-off scale. The orange solid line presents the degeneration temperature $T_\mathrm{deg}$, below which the newly emerged second minimum becomes the global minimum. However, the first-order PT can only possibly happen below the transition temperature $T_\mathrm{tra}$ presented as the blue dashed line, and the potential barrier disappears below the vanishing temperature $T_\mathrm{van}$ presented as the red dashed line, therefore only the blue-shaded region could possibly commit a first-order PT. The nucleation temperature is provided by $T_{\Gamma/H^4}/T_\mathrm{num}$ presented as the red/green solid line in the regime of fast first-order PT, and by $T_\mathrm{min}$ presented as the green dashed line in the regime of slow first-order PT. The percolation temperature $T_\mathrm{per}$ presented as the blue solid line is well-defined in both regimes of slow and fast first-order PTs, of which the regime of slow first-order PT is presented in the zooming windows.