Probing classically conformal $B-L$ model with gravitational waves

TL;DR

The work investigates a classically conformal $B-L$ gauge extension of the SM in which all mass scales arise from radiative breaking, focusing on the thermal breaking of $U(1)_{B-L}$ and the consequent gravitational waves. Owing to ultra-supercooling and the weak temperature dependence of the nucleation rate, the first-order phase transition can be exceptionally strong, producing sizable GW signals. By deriving the nucleation and completion criteria and computing the GW spectrum using established envelope-approximation results, the authors show that $\Omega_{\rm GW}$ can reach about $10^{-8}$ at frequencies $f \sim 0.01$--$1$ Hz for viable parameter choices, within reach of future detectors such as eLISA, LISA, DECIGO and BBO. The GW signal is mapped in the $(M,\alpha_{B-L})$ plane, highlighting broad regions where observations could test classically conformal constructions and noting potential eternal-inflation effects in false-vacuum patches.

Abstract

We study the cosmological history of the classical conformal $B-L$ gauge extension of the standard model, in which the physical scales are generated via the Coleman-Weinberg-type symmetry breaking. Especially, we consider the thermal phase transition of the U$(1)_{B-L}$ symmetry in the early universe and resulting gravitational-wave production. Due to the classical conformal invariance, the phase transition tends to be a first-order one with ultra-supercooling, which enhances the strength of the produced gravitational waves. We show that, requiring (1) U$(1)_{B-L}$ is broken after the reheating, (2) the $B-L$ gauge coupling does not blow up below the Planck scale, (3) the thermal phase transition completes in almost all the patches in the universe, the gravitational wave spectrum can be as large as $Ω_{\rm GW} \sim 10^{-8}$ at the frequency $f \sim 0.01$-$1$Hz for some model parameters, and a vast parameter region can be tested by future interferometer experiments such as eLISA, LISA, BBO and DECIGO.

Figures (13)

• Figure 1: Plot of the finite-temperature effective potential $V_{\rm eff}$. Parameters are taken to be $\alpha_{B-L}(0) = 0.01$ and $T/M = 0.1$ (blue), $0.2$ (red) and $0.25$ (yellow). The origin is the true vacuum for high enough temperature, while it is a false one after the temperature drops.
• Figure 2: Blow-up of Fig. \ref{['fig_pot']}. Solid line is the same as in the blue line of Fig. \ref{['fig_pot']}, while dashed and dotted lines show the contributions from the thermal potential $V_T$ and the zero-temperature potential $V_0$, respectively. Because of the classical conformal requirement, the thermal contribution dominates the zero-temperature one near the origin (see Eq. (\ref{['eq_Veff_approx']})).
• Figure 3: Plot of $S$ as a function of $T/M$. The blue, red, yellow and green lines correspond to $\alpha_{B-L} = 10^{-1.8}(\simeq 0.016),~10^{-1.9}(\simeq 0.013), ~10^{-2}(= 0.01)$ and $10^{-2.1}(\simeq 0.008)$, respectively. As the temperature decreases, $S$ first drops due to the $S_3/T$ contribution, while it starts to increase at some point because of the $\ln (T/M)$ contribution.
• Figure 4: Plot of $I$ defined in Eq. (\ref{['eq_I']}) for $\alpha_{B-L} = 10^{-2.12} \simeq 0.0076$ (blue), $10^{-2.14} \simeq 0.0072$ (red), $10^{-2.16} \simeq 0.0069$ (yellow), $10^{-2.18} \simeq 0.0066$ (green). $M$ is fixed to be $10^4$GeV.
• Figure 5: $I'(t = \infty)$ as a function of $\alpha_{B-L}$.