⚛️ phenomenology

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

This paper investigates the complementarity between collider searches and gravitational-wave observations in a classically scale-invariant $U(1)_X$ extension of the Standard Model, demonstrating that gravitational waves from tachyonic phase transitions can probe regions of small gauge couplings and heavy $Z'$ bosons that are inaccessible to current and future collider experiments.

Original authors: Arindam Das, Katsuya Hashino, Yuta Orikasa, Masanori Tanaka

Published 2026-10-09

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CC BY 4.0

Original authors: Arindam Das, Katsuya Hashino, Yuta Orikasa, Masanori Tanaka

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is built on a set of rules that determine how particles interact and how forces behave. For decades, physicists have struggled with a specific puzzle regarding the mass of the Higgs boson, the particle responsible for giving other particles their weight. In the standard model of particle physics, the calculations for this mass suggest it should be enormous, yet observations show it is surprisingly light. This discrepancy, known as the hierarchy problem, implies that the universe's laws might be hiding a deeper symmetry. One compelling idea is that the universe began with no intrinsic mass scales at all; everything was massless, and the masses we see today were generated dynamically as the universe cooled. This concept, called classical scale invariance, suggests that the heavy particles we observe are the result of a spontaneous breaking of symmetry, much like a pencil balanced on its tip eventually falling to one side.

In this framework, the energy of the vacuum itself drives the creation of mass. A specific mechanism, known as the Coleman-Weinberg process, allows a field to acquire a non-zero value through quantum effects, even if it starts with no mass. This paper explores a model where this mechanism triggers the breaking of a new, hidden symmetry called U(1)X. This symmetry is linked to a new force and a new particle, a heavy version of the Z boson, which the authors call Z'. The model also includes three heavy, invisible particles known as right-handed neutrinos. These particles are essential for the math to work and are responsible for generating the tiny masses of the neutrinos we do observe. The researchers investigated how this new symmetry breaks as the early universe cooled, focusing on a specific, violent type of transition where the field does not settle down smoothly but rolls rapidly down a potential hill.

The study reveals that under certain conditions, this rolling motion creates a unique signature in the fabric of spacetime. When the new symmetry breaks in this rapid, unstable manner, the field fluctuates wildly, creating ripples in the gravitational field. These ripples form a background hum of gravitational waves, distinct from the signals produced by the more common, bubble-like phase transitions. The researchers simulated the dynamics of this event, calculating how the presence of the heavy neutrinos and the new Z' particle influences the timing and intensity of the transition. They found that if the new Z' particle is very heavy, with a mass between one million and ten million GeV, and the force it carries is relatively weak, this tachyonic transition becomes the dominant way the symmetry breaks.

The most significant finding is that these gravitational waves could be detectable by future observatories, specifically a mission called DECIGO. The simulations show that the signal would peak at a frequency that DECIGO is designed to hear, provided the new force is weak enough that traditional particle colliders cannot see it. This creates a powerful complementarity: while massive particle accelerators like the Large Hadron Collider are excellent at finding heavy particles with strong interactions, they struggle to detect particles that interact very weakly. The gravitational wave signal, however, thrives in this weak-coupling regime. The researchers mapped out the specific ranges of mass and force strength where this signal would be visible, showing that a large portion of the theoretical landscape remains hidden from colliders but is open to gravitational wave astronomy.

The team also examined how the mass of the heavy neutrinos affects the outcome. They found that if the neutrinos are light enough, the new field can decay into them, which changes the temperature of the universe after the transition and alters the gravitational wave signal. If the neutrinos are too heavy, this decay channel closes, and the signal shifts to lower frequencies and becomes weaker. This sensitivity means that observing the gravitational wave signal could tell us not just about the new force, but also about the mass spectrum of these invisible neutrinos. The study confirms that while the standard model works well for what we see, the universe likely contains a hidden sector that left a faint, but potentially audible, echo in the gravitational waves of the early cosmos.

By combining the constraints from existing experiments at CERN with the predictions for future gravitational wave detectors, the authors identified a clear path forward. They showed that the region where the new Z' particle is heavy and the force is weak is currently inaccessible to colliders, yet it is precisely where the tachyonic phase transition produces the strongest gravitational wave signals. This suggests that the next generation of gravitational wave observatories will not only listen to colliding black holes but could also probe the fundamental structure of particle physics, testing theories of scale invariance that have remained unverified for decades. The work demonstrates that the universe's history of symmetry breaking is written in the gravitational waves, offering a new way to read the laws of nature that governs the smallest scales of reality.

Technical Summary: Implications of Tachyonic Phase Transition in Classically Scale Invariant General U(1)X Models

Problem Statement
The Standard Model (SM) faces the gauge hierarchy problem, where quantum corrections to the Higgs mass are quadratically divergent. A proposed solution involves enforcing classical scale invariance (conformal symmetry), where the SM Lagrangian contains no explicit mass terms, with the electroweak scale generated radiatively via the Coleman-Weinberg (CW) mechanism. While applying CW directly to the SM fails due to the unbounded nature of the effective potential, extensions such as the general $U(1)_X$ model offer a viable framework. This model introduces a new gauge symmetry, three right-handed neutrinos (RHNs) for anomaly cancellation, and a scalar singlet $\Phi$ . The $U(1)_X$ scale is generated radiatively, inducing the electroweak scale via a Higgs portal.

A critical theoretical challenge in such models is understanding the dynamics of the phase transition (PT) associated with $U(1)_X$ symmetry breaking. While first-order phase transitions (FOPT) involving bubble nucleation are well-studied, recent work suggests that in strongly supercooled regimes, a "tachyonic phase transition" may occur. In this scenario, the scalar field rolls classically rather than tunneling, potentially generating a distinct stochastic gravitational wave (GW) background. The paper investigates whether this mechanism is realizable in the general $U(1)_X$ extension and whether the resulting GW signals offer a complementary probe to collider searches, particularly in parameter regions where collider constraints are weak (small gauge couplings, heavy $Z'$ bosons).

Methodology
The authors analyze the model under the assumption of classical scale invariance, where the scalar potential contains only quartic terms. The $U(1)_X$ charges are determined by solving gauge and gravitational anomaly cancellation conditions, expressing them as a linear combination of $U(1)_Y$ and $U(1)_{B-L}$ .

Effective Potential: The study employs the finite-temperature effective potential, $V_{\text{eff}}(\phi, T)$ , incorporating RG-improved zero-temperature terms and one-loop thermal corrections. The analysis considers the interplay between the $U(1)_X$ scalar $\Phi$ and the SM Higgs $H$ via a portal coupling $\lambda_{\text{mix}}$ .
Phase Transition Dynamics: The authors distinguish between two regimes:
- First-Order PT: Characterized by bubble nucleation, determined by the bounce action $S_3(T)$ .
- Tachyonic PT: Occurs when the percolation temperature $T_{\text{perc}}$ is lower than the temperature $T_{\text{roll}}$ at which the curvature of the potential at the origin becomes negative ( $m_{\text{eff}}^2 < 0$ ). In this regime, the symmetry breaking proceeds via classical rolling of the scalar field. The paper notes that $T_{\text{perc}}$ can fall below the QCD phase transition temperature ( $T_{\text{QCD}}$ ) specifically when the gauge coupling $g_X < 0.25$ , a condition that prevents the completion of the transition via bubble nucleation and necessitates the tachyonic mechanism.
Gravitational Wave Calculation: For the tachyonic regime, the authors estimate the GW spectrum generated by the inhomogeneous scalar field excitations. They calculate the peak frequency and amplitude based on the tachyonic mass scale and the subsequent reheating history, which depends on the decay width of the scalar field $\Gamma_\phi$ into RHNs and Higgs bosons.
Parameter Space Scan: The study scans the parameter space defined by the $U(1)_X$ gauge coupling $g_X$ , the $Z'$ boson mass $m_{Z'}$ , and the RHN masses ( $m_N$ ). They consider benchmark scenarios with degenerate RHN masses of 1 TeV and 10 TeV, and varying charge assignments ( $x_H$ ) corresponding to different model limits (e.g., $U(1)_R$ , $U(1)_{B-L}$ ).
Constraints: The viable parameter regions are constrained by:
- Perturbativity ( $|\lambda_\Phi| < 4\pi$ ).
- Collider limits from LEP (contact interactions) and LHC (dilepton and dijet searches).
- Theoretical consistency conditions for the tachyonic transition (e.g., $\Delta m_{\text{QCD}} > H(T_{\text{roll}})$ ).

Key Contributions and Results

Realization of Tachyonic PT: The authors demonstrate that the tachyonic phase transition is realizable in the classically scale-invariant $U(1)_X$ model within specific parameter regions (denoted as orange regions in Figures 1 and 2). This occurs primarily in the regime of small gauge couplings ( $g_X < 0.25$ ) and heavy $Z'$ bosons ( $10^6 \text{ GeV} < m_{Z'} < 10^7 \text{ GeV}$ ). In this range, the supercooling is sufficient such that the percolation temperature $T_{\text{perc}}$ drops below the QCD phase transition temperature ( $T_{\text{QCD}}$ ), preventing the transition from completing via bubble nucleation and forcing it to proceed via tachyonic instability.
GW Signal Characteristics:
- First-Order Regime ( $g_X > 0.25$ ): Produces GWs via bubble collisions, with peak frequencies accessible to LISA and DECIGO.
- Tachyonic Regime ( $g_X < 0.25$ ): Produces a stochastic GW background with a peak frequency and amplitude determined by the tachyonic mass scale and the reheating temperature. The authors find that for specific benchmarks ( $m_{Z'} \sim 10^6-10^7$ GeV, $g_X \sim 0.001-0.25$ ) falling within the valid tachyonic region, the signal falls within the sensitivity range of the DECIGO observatory.
Sensitivity to RHN Mass Spectrum: A significant finding is that the tachyonic GW signal is highly sensitive to the mass spectrum of the heavy neutrinos. If the scalar field cannot decay into heavy neutrinos (due to kinematic suppression, $m_\phi < 2m_N$ ), the reheating temperature changes, altering the redshift and dilution of the GW signal. This can shift the peak frequency and suppress the amplitude, potentially rendering the signal undetectable by DECIGO for heavy neutrino masses (e.g., 10 TeV) compared to lighter ones (1 TeV).
Complementarity: The study maps the exclusion limits from LEP and LHC against the projected sensitivity of GW observatories. The results show that GW observations can probe regions with small $g_X$ and large $m_{Z'}$ that are currently inaccessible to collider experiments. Conversely, colliders provide stringent bounds on larger couplings.

Significance and Claims
The paper claims to establish the complementarity between collider searches and gravitational wave observations in testing classically scale-invariant extensions of the Standard Model. Specifically, it argues that:

Probing the "Dark" Parameter Space: GW observations from tachyonic phase transitions offer a unique window into parameter regions (small $g_X$ , heavy $Z'$ ) where traditional collider searches are ineffective.
Distinctive Signatures: The tachyonic mechanism provides a distinct GW signature compared to standard first-order transitions, dependent on the scalar field's rolling dynamics and the specific decay channels of the scalar field.
Model Validation: The detection (or non-detection) of such signals by future observatories like DECIGO and LISA could validate or rule out the specific radiative symmetry breaking mechanism proposed in these $U(1)_X$ models, particularly regarding the role of heavy neutrinos in the reheating process.

The authors conclude that while collider constraints are tight for certain mass ranges, the synergy with GW astronomy is essential for a comprehensive test of the model, especially for the tachyonic regime which remains largely unexplored by current high-energy experiments.

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