Implications of tachyonic phase transition in classically scale invariant general U(1) X models

The gist: Gravitational waves from tachyonic phase transitions in classically scale-invariant general U(1)X models can probe regions with small gauge couplings and heavy Z′ bosons that are difficult to access through collider searches, demonstrating the complementarity of these probes.

Model Setup and Symmetry Breaking

The investigation focuses on the classically conformal U(1)X extension of the Standard Model, which supplements SM content with three generations of right-handed neutrinos (RHNs) and a U(1X Higgs field <ref:2610.11192#pg5>). This model is anomaly-free when generation-independent fermion charges are required, identifying the U(1X symmetry as a linear combination of – 1 – U(1)Y and U(1)B−L <ref:2610.11192#pg4>. Radiative symmetry breaking via the Coleman-Weinberg mechanism generates masses for both the Z′ boson and the right-handed Majorana neutrinos <ref:2610.11192#pg4>. A portal quartic coupling between the U(1X scalar and the SM Higgs doublet then induces a negative mass squared for the SM Higgs, driving electroweak (EW) symmetry breaking <ref:2610.11192#pg4>.

Phase Transition Dynamics

The phase transition dynamics are thoroughly discussed, and it is possible that the U(1X phase transition can proceed after the QCD phase transition <ref:2610.11192#pg4>. In such a case, all quarks do not acquire their current masses, implying the QCD phase transition is expected to be first-order <ref:2610.11192#pg4>. A new type of phase transition called the tachyonic phase transition may occur if the U(1B−L and the EW symmetry breaking are not induced even if the QCD phase transition occurs <ref:2610.11192#pg4>. During this tachyonic phase transition, an inhomogeneous scalar field excitation can form, and its dynamics may leave a characteristic stochastic gravitational wave (GW) signal <ref:2610.11192#pg4>.

Gravitational Wave Signal

The connection to GWs follows from the tensor perturbation carried out by the inhomogeneous scalar field formed by the tachyonic phase transition <ref:2610.11192#pg6>. The spatial part of the scalar energy-momentum tensor is expressed by Tij = ∂iφ ∂jφ + δij 1/2 φ˙2 − 1/2a2(∇φ)2 − Veff(φ) <ref:2610.11192#pg6>. Since ΠTTij is quadratic in terms of δϕ which is exponentially amplified during the tachyonic phase transition, the GW source is enhanced much more rapidly than the scalar mode functions themselves <ref:2610.11192#pg6>. The peak frequency at production f∗ is set by the characteristic scale of the scalar inhomogeneities, estimated as f∗ ∼ k∗ a∗ ∼ ∆mQCD <ref:2610.11192#pg6>.

Complementarity and Observational Reach

The study identifies regions of the U(1X gauge coupling and Z′ mass accessible to future GW observatories, including LISA and DECIGO, and compares them with existing LEP and LHC constraints <ref:2610.11192#pg4>. The results find that GWs from tachyonic phase transitions can probe regions with small gauge couplings and heavy Z′ bosons that are difficult to access through collider searches, demonstrating the complementarity of these probes <ref:2610.11192#pg4>. Specifically, if the mass of the new gauge boson Z′ and the U(1X gauge coupling are within the range 106 GeV < mZ′ < 107 GeV and 0.001 < gX < 0.25, the GW signal from the tachyonic phase transition may be tested by DECIGO <ref:2610.11192#pg4>.

Conclusion on Observational Prospects

The parameter region with gX > 0.25 can be explored by observing GWs produced by the dynamics of vacuum bubbles nucleated by first-order phase transitions <ref:2610.11192#pg5>. This finding indicates that the complementarity of colliders and GW observations is useful in testing our model thoroughly <ref:2610.11192#pg8>. The parameter region with the heavy Z′ boson and the small gX where the collider exploration is challenging can be tested via GW observations <ref:2610.11192#pg8>.

How it works

The effective potential at finite temperatures, including RG improvement, is given by Veff(φ) = V0(φ) + VT (φ, T) <ref:2610.11192#pg4>. The thermal contribution VT (φ, T) is calculated using the Matsubara summation over various degrees of freedom <ref:2610.11192#pg4>. The mass eigenvalues for longitudinal components of the U(1) gauge fields are given by M2 Xi L(φ, T) = 1/2 M2 X(φ, T) + ΠBB(T) ± q M2 X(φ, T) − ΠBB(T) 2 + 4Π2BB′(T)<ref:2610.11192#pg4>.

Key Findings and Constraints

The analysis shows that the percolation temperature Tp can be lower than the QCD phase transition temperature TQCD if gX. This implies that in this range, the tachyonic phase transition plays an important role in completing the U(1X phase transition in addition to the subsequent EW symmetry breaking <ref:2610.11192#pg8>. The dark gray region shows constraints by current collider searches from LEP and LHC <ref:2610.11192#pg5>. Furthermore, the peak frequency and amplitude of the GWs can be sensitive to the RHN mass spectrum through its impact on reheating <ref:2610.11192#pg8>.

Parameter Exploration

Figure 3 illustrates the parameter region explored by GW observatories and colliders for different benchmark points <ref:2610.11192#pg8>. The colored regions indicate where GWs can be detectable at each observatory, while the dark gray region shows constraints from LEP and LHC <ref:2610.11192#pg8>. The results show that we can explore wide parameters using GW observations even if the value of U(1X gauge coupling is relatively small <ref:2610.11192#pg8>.

Benchmark Comparison

Table 2 compares representative benchmarks with degenerate and nondegenerate RHN mass spectra at fixed (mZ′, gX, xH) <ref:2610.11192#pg8>. For the tachyonic transitions, the RHN mass spectrum substantially affects both the peak frequency and amplitude <ref:2610.11192#pg8>. Replacing lighter RHNs with heavier ones reduces the total scalar decay width, changing the reheating temperature and the scale-factor ratio rRH <ref:2610.11192#pg8>. This results in a shift of the peak to lower frequencies and suppression of its amplitude <ref:2610.11192#pg8>.

Final Summary

The paper concludes that if the mass of the new gauge boson Z′ and the U(1X gauge coupling are within the range 106 GeV < mZ′ < 107 GeV and 0.001 < gX < 0.25, the GW signal from the tachyonic phase transition may be detectable at DECIGO <ref:2610.11192#pg8>. The parameter region with gX > 0.25 can be explored by observing GWs produced by the dynamics of vacuum bubbles nucleated by first-order phase transitions <ref:2610.11192#pg8>. This finding indicates that the complementarity of colliders and GW observations is useful in testing our model thoroughly <ref:2610.11192#pg8>.

Acknowledgments

This work is supported by Grant-in-Aid for Early-Career Scientists 25K17398 (KH) <ref:2610.11192#pg9>.

RGE Equations

The RGEs for the U(1X gauge coupling are given by dgX dt = gX3 (4π)2 [−7] <ref:2610.11192#pg9>.

1. A computational framework for parameter space exploration can be built by implementing Eq. (5.1) to map out regions where the requirement for the tachyonic phase transition given in Eq. (5.1) is satisfied, allowing researchers to identify specific combinations of the U(1)X gauge coupling and Z′ mass accessible to future GW observatories.

2. The AI system can perform sensitivity analysis using the provided Figure 3 and 4, enabling it to determine which observational targets—LEP, LHC, LISA, or DECIGO—are best suited to probe specific model parameters like the U(1)X gauge coupling for different heavy neutrino mass spectra.

3. The system can be trained on the numerical results in Table 2 to predict the present-day peak frequency and amplitude of gravitational waves (as shown in Figure 5) based on the input values of mN1, mN2, mN3 and whether the transition is first-order or tachyonic.

4. The AI can automatically calculate the reheating temperature using Eq. (5.16), which is crucial for determining the present-day peak amplitude, thus providing a quantitative estimate of the resulting GW signal for specific parameter choices like gX = 0 and various heavy neutrino masses.

We investigate the complementarity between collider searches and gravitational-wave (GW) observations in the classically scale-invariant general U(1) X extension of the Standard Model. The U(1) X scale is generated radiatively through the Coleman-Weinberg mechanism, while the electroweak scale is induced through the Higgs portal, without explicit mass terms in the scalar potential. Three right-handed neutrinos are introduced to ensure anomaly cancellation and acquire Majorana masses upon U(1) X symmetry breaking, which also generates the mass of the neutral gauge boson Z'. In the strongly supercooled regime, a tachyonic U(1) X phase transition can produce a stochastic GW background. Assuming three heavy Majorana neutrinos, we study their effects on the phase-transition dynamics and reheating, and estimate the resulting GW signals. We identify the regions of the U(1) X gauge coupling and Z' mass accessible to future GW observatories, including LISA and DECIGO, and compare them with existing LEP and LHC constraints. We find that GWs from tachyonic phase transitions can probe regions with small gauge couplings and heavy Z' bosons that are difficult to access through collider searches, demonstrating the complementarity of these probes.