Strongly first-order electroweak phase transition and classical scale invariance

In this work, we examine the possibility of realizing a strongly first-order electroweak phase transition within the minimal classically scale-invariant extension of the standard model (SM), previously proposed and analyzed as a potential solution to the hierarchy problem. By introducing one complex gauge-singlet scalar and three (weak scale) right-handed Majorana neutrinos, the scenario was successfully rendered capable of achieving a radiative breaking of the electroweak symmetry (by means of the Coleman-Weinberg mechanism), inducing nonzero masses for the SM neutrinos (via the seesaw mechanism), presenting a pseudoscalar dark matter candidate (protected by the $CP$ symmetry of the potential), and predicting the existence of a second $CP$-even boson (with suppressed couplings to the SM content) in addition to the 125 GeV scalar. In the present treatment, we construct the full finite-temperature one-loop effective potential of the model, including the resummed thermal daisy loops, and demonstrate that finite-temperature effects induce a first-order electroweak phase transition. Requiring the thermally driven first-order phase transition to be sufficiently strong at the onset of the bubble nucleation (corresponding to nucleation temperatures $T_N\sim 100–200\mathrm{GeV}$) further constrains the model’s parameter space; in particular, an $\mathcal{O}(0.01)$ fraction of the dark matter in the Universe may be simultaneously accommodated with a strongly first-order electroweak phase transition. Moreover, such a phase transition disfavors right-handed Majorana neutrino masses above several hundreds of GeV, confines the pseudoscalar dark matter masses to $\sim 1–2\mathrm{TeV}$, predicts the mass of the second $CP$-even scalar to be $\sim 100–300\mathrm{GeV}$, and requires the mixing angle between the $CP$-even components of the SM doublet and the complex singlet to lie within the range $0.2\lesssim \sin \omega \lesssim 0.4$. The obtained results are displayed in comprehensive exclusion plots, identifying the viable regions of the parameter space. Many of these predictions lie within the reach of the next LHC run.

Figure 1

The 95% C.L. experimental exclusion bounds in the $\sin \omega$-$m_{\sigma }$ plane. All colored regions are excluded. The LHC direct measurements of the properties of the 125 GeV $h$ scalar (solid vertical line) restrict the mixing angle to $\sin \omega \le 0.44$, whereas the electroweak precision tests (dot-dashed) and the LEP and LHC Higgs searches (dotted) further reduce the upper bound on the mixing depending on the $\sigma$ boson mass. (Figure taken from Ref. [6])

Figure 2

Schematic behavior of the (normalized) finite-temperature one-loop effective potential (20) as a function of the (rescaled) field value, $r_{\varphi }=\varphi /v_{\varphi }$ [cf. (17)], for various temperatures $T_0<T_1<T_c<T_2<T_3$. The finite-temperature contributions induce a first-order phase transition. At the temperature $T_3$, the potential has a global minimum at $\varphi =0$. As the temperature decreases to $T_2$, the potential exhibits an inflection point, where a secondary local minimum starts to develop at $\varphi \ne 0$. This minimum at $\varphi \ne 0$ becomes degenerate with the one at $\varphi =0$ once the temperature has reached the critical value $T_c$, with a barrier separating the two minima. At lower temperatures, $T_1<T_c$, the minimum at $\varphi \ne 0$ becomes global, whereas the minimum at $\varphi =0$ denotes a false vacuum with a diminishing separating barrier. The barrier disappears completely at zero temperature, $T_0=0$.

Figure 3

Formal and experimental constraints in the $\sin \omega$-$M_{\chi }$ plane, for ${\lambda }_{\chi }=0.1$ and representative values of ${\lambda }_m^{-}$ and $M_N$. All colored regions are excluded. The formal bounds arise due to perturbative unitarity (long-dashed line) and the stability of the one-loop effective potential (solid black bar). The 95% C.L. experimental bounds arise from the electroweak precision tests (dot-dashed line), direct measurements of the properties of the 125 GeV $h$ boson (solid vertical line), and the LEP and LHC Higgs searches (dotted line). The thick red band corresponds to the WIMP relic abundance within the $1\sigma$ uncertainty, quoted by the Planck Collaboration, assuming the $\chi$ pseudoscalar to constitute an $\mathcal{O}(1)$ fraction of the dark matter in the Universe. The constraints by the LUX dark matter direct detection experiment (short-dashed line) at 90% C.L. also rest on this assumption. Requiring the framework to accommodate a strongly first-order phase transition at the onset of the bubble nucleation (30) further excludes the horizontally shaded region, whereas no solution for $T_N$ (28) exists in the vertically shaded region. The plots illustrate the tension within the minimal scenario between realizing a strongly first-order electroweak transition and the constraints from a pseudoscalar-dominant dark matter component, leaving no viable parameter space.

Figure 4

Formal and experimental constraints in the $\sin \omega$-$M_{\chi }$ plane, for ${\lambda }_{\chi }=4\pi$ and representative values of ${\lambda }_m^{-}$ (rows) and $M_N$ (columns). All colored regions are excluded. As in Fig. 3, a simultaneous realization of a strongly first-order electroweak phase transition along with accommodating the entire dark matter in the Universe cannot be achieved within the considered minimal framework. (See the caption of Fig. 3 for the details of the plots.)

Figure 5

Formal and experimental constraints in the enlarged $\sin \omega$-$M_{\chi }$ plane, for ${\lambda }_{\chi }=0.1$ and representative values of ${\lambda }_m^{-}$ and $M_N$. All colored regions are excluded, with only the uncolored parts of the (center) wedge region unconstrained. The contribution of the $\chi$ pseudoscalar to the dark matter in the Universe is chosen to be 3%, $f_{\mathrm{DM}}=0.03$, mitigating the tension between the dark matter and strongly first-order phase transition constraints. (See the caption of Fig. 3 for the details of the plots.)

Figure 6

Formal and experimental constraints in the enlarged $\sin \omega$-$M_{\chi }$ plane, for ${\lambda }_{\chi }=4\pi$ and representative values of ${\lambda }_m^{-}$ (rows) and $M_N$ (columns). All colored regions are excluded, with only the uncolored parts of the (center) wedge region unconstrained. The contribution of the $\chi$ pseudoscalar to the dark matter in the Universe is chosen to be 3%, $f_{\mathrm{DM}}=0.03$, mitigating the tension between the dark matter and strongly first-order phase transition constraints. (See the caption of Fig. 3 for the details of the plots.)

Figure 7

Formal and experimental constraints in the $m_{\sigma }$-$M_{\chi }$ plane, for ${\lambda }_{\chi }=0.1$ and representative values of ${\lambda }_m^{-}$ and $M_N$. All colored regions are excluded, with only the uncolored parts of the (center) wedge region unconstrained. The contribution of the $\chi$ pseudoscalar to the dark matter in the Universe is now chosen to be 3%, $f_{\mathrm{DM}}=0.03$. (See the caption of Fig. 3 for the details of the plots.)

Figure 8

Formal and experimental constraints in the $m_{\sigma }$-$M_{\chi }$ plane, for ${\lambda }_{\chi }=4\pi$ and representative values of ${\lambda }_m^{-}$ (rows) and $M_N$ (columns). All colored regions are excluded, with only the uncolored parts of the (center) wedge region unconstrained. The contribution of the $\chi$ pseudoscalar to the dark matter in the Universe is now chosen to be 3%, $f_{\mathrm{DM}}=0.03$. (See the caption of Fig. 3 for the details of the plots.)

Figure 9

Feynman rules for the relevant quartic couplings, $i{\lambda }_{ijkl}$.