Probing radiative electroweak symmetry breaking with colliders and gravitational waves

This paper presents a comprehensive phenomenological study of radiative electroweak symmetry breaking, demonstrating how its characteristic logarithmic potential leads to a light scalar boson and first-order phase transitions, thereby enabling the exploration of conformal symmetry breaking scales up to $10^5\text{--}10^8$ GeV through combined future collider and gravitational wave observations.

The Gist

The Big Picture: Fixing a Broken Scale

Imagine the universe is like a giant, delicate scale. For a long time, physicists have been puzzled by why the "Higgs boson" (a particle that gives other particles mass) is so light. In the standard rules of physics, it should be incredibly heavy, like a bowling ball, but it's actually more like a feather. This mismatch is called the "hierarchy problem."

This paper proposes a solution called Radiative Symmetry Breaking. Think of it like this: instead of the scale being broken from the start (which would require fine-tuning), the scale is perfectly balanced at the beginning. However, tiny quantum "ripples" (like wind blowing on a still pond) eventually push the scale to tip over, creating the mass we see today. This process happens naturally without needing to manually adjust the settings.

The Main Characters: The Higgs and the New Scalar

The authors introduce a new character to the story: a "singlet scalar" particle (let's call it $\phi$).

• The Higgs ($h$): The famous particle we already know.
• The New Scalar ($\phi$): A mysterious, lighter cousin that mixes with the Higgs.

The paper claims that because of the way this new particle interacts with the Higgs, it creates a very specific shape for the energy landscape of the universe. Imagine a hill that is flat at the top but curves down into a valley. This shape is crucial because it leads to two major discoveries:

1. A Light Particle: We might find this new, light particle at particle colliders (like the Large Hadron Collider or a future Muon Collider).
2. Cosmic Ripples: In the early universe, this flat shape caused the universe to undergo a sudden "phase transition" (like water suddenly freezing into ice), which created gravitational waves (ripples in space-time).

The Cosmic Drama: Four Ways the Universe Cooled Down

The paper explores how the universe cooled down after the Big Bang. Because of the unique shape of the energy landscape, the universe didn't just cool smoothly; it might have "stalled" or "supercooled" before suddenly snapping into its current state.

The authors map out four different scenarios (like four different plotlines for a movie):

1. Normal Patterns: The universe breaks the symmetry (tips the scale) first, then the Higgs settles in.
2. Inverted Patterns: The universe cools down so much that other things (like the QCD transition, related to how protons form) happen before the main symmetry breaking.

A key finding here is that sometimes the universe gets "supercooled" (it stays in a high-energy state way longer than expected). You might think this would create a massive explosion of gravitational waves, but the authors found a twist: Sometimes, the transition happens so fast that the waves are actually weak. It's like a car accelerating incredibly quickly but for such a short time that it doesn't travel far.

The Detective Work: How We Can Find It

The paper acts as a roadmap for two types of detectives: Particle Physicists and Gravitational Wave Astronomers.

1. The Particle Detectives (Colliders):
They are looking for the new scalar particle ($\phi$).

• If it's heavy: They look for it decaying into pairs of other particles (like bottom quarks or Z bosons) at the LHC or a future 10 TeV Muon Collider.
• If it's light: It might live for a long time before decaying. They look for "long-lived particles" that travel a bit before disappearing.
• The Catch: The new particle mixes with the Higgs. The more they mix, the easier it is to spot. The paper calculates exactly how sensitive future machines need to be to catch a glimpse of it.

2. The Wave Detectives (Gravitational Waves):
They are listening for the "sound" of the universe freezing over.

• Space-based detectors like LISA (a future satellite mission) or BBO are the microphones.
• The paper predicts that if the universe went through one of these "supercooled" transitions, it would leave a specific signature in the gravitational waves.
• The Surprise: The authors found that even if the transition was incredibly violent (ultra-supercooled), the resulting gravitational waves might be too faint to hear if the transition happened too quickly. This means we can't just rely on listening; we need to look at the particles too.

The Grand Conclusion: Two Eyes Are Better Than One

The most important message of the paper is complementarity.

• Looking only at particle colliders might miss the story.
• Listening only to gravitational waves might miss the story (because some transitions are too fast to make loud waves).

But if we combine both methods, we can cover a massive range of possibilities. The paper shows that by using both particle colliders and gravitational wave detectors, we can probe energy scales up to $10^8$ GeV (a number so huge it's hard to imagine).

In summary: The paper suggests that the universe's mass-generating mechanism is a natural, quantum-driven process. To prove it, we need to hunt for a new, light particle in our labs and listen for the faint echoes of the universe's early phase transitions in space. If we find both, we solve the mystery of why the Higgs boson is so light.

Technical Summary

Technical Summary: Probing Radiative Electroweak Symmetry Breaking with Colliders and Gravitational Waves

Problem Statement
The Standard Model (SM) explains electroweak symmetry breaking (EWSB) via a negative mass squared term in the Higgs potential, a mechanism that lacks a fundamental origin and suffers from the hierarchy problem due to sensitivity to ultraviolet physics. Radiative symmetry breaking offers a solution by positing a classically scale-invariant (conformal) Lagrangian where EWSB is induced by quantum corrections (Coleman-Weinberg mechanism). However, direct application to the SM fails to reproduce the observed 125 GeV Higgs mass or vacuum stability. Consequently, models extend the SM with a BSM sector where radiative breaking occurs, transmitting symmetry breaking to the SM via Higgs-portal couplings. A comprehensive phenomenological study of this specific scenario—characterized by a logarithmic scalar potential, a light scalar boson, and distinct cosmological phase transitions—remains necessary to identify its unique signatures.

Methodology
The authors construct a benchmark model involving the SM Higgs doublet $H$ and a real scalar singlet $\phi$. The tree-level potential is classically conformal, with symmetry breaking driven by one-loop radiative corrections (Coleman-Weinberg potential).

1. Vacuum Structure and Parameterization: The authors derive exact analytical solutions for the vacuum expectation values (VEVs), scalar masses, and mixing angles. They introduce a novel parameterization scheme using the physical observables of the new scalar mass ($m_\phi$) and its mixing angle with the Higgs ($\theta$) as inputs, rather than bare Lagrangian parameters. This allows for a direct mapping to experimental constraints.
2. Collider Phenomenology: The study analyzes the production and decay of the singlet scalar $\phi$ at the LHC and a future 10 TeV muon collider. Production mechanisms include gluon-gluon fusion (LHC) and vector boson fusion (VBF) at muon colliders. Decay channels are categorized by mass: light scalars ($m_\phi \lesssim 2m_\tau$) are probed via long-lived particle (LLP) searches, while heavier scalars are studied via prompt decays into $b\bar{b}$, $\tau^+\tau^-$, $VV$, and $hh$. Parton-level simulations using MadGraph5_aMC@NLO are employed to estimate signal efficiencies and backgrounds.
3. Cosmological Dynamics: The thermal history of the early Universe is analyzed by incorporating finite-temperature corrections to the potential. The authors classify four distinct patterns of cosmic thermal history (Type-N1, N2, I1, I2) based on the timing of the conformal phase transition relative to the QCD and electroweak scales. They calculate the dynamics of first-order phase transitions (FOPTs), including bubble nucleation rates, percolation temperatures, and reheating effects.
4. Gravitational Wave (GW) Calculation: The stochastic GW background generated by FOPTs is computed, considering contributions from bubble collisions, sound waves, and turbulence. The spectrum is characterized by the transition strength $\alpha$ and the inverse duration $\beta/H_*$. The authors account for the effects of supercooling and potential thermal inflation on the GW signal.

Key Contributions

• Exact Analytical Solutions: The paper provides the first exact analytical solutions for the vacuum structure and scalar interactions in the limit $w \gg v$ (where $w$ is the BSM scale), enabling a robust $(m_\phi, \theta)$ parameterization.
• Thermal History Classification: A systematic classification of four thermal history patterns is presented, distinguishing between "normal" (conformal breaking before EW symmetry breaking) and "inverted" (QCD-EW breaking preceding conformal breaking) scenarios.
• Ultra-Supercooled FOPT Dynamics: A critical finding is that ultra-supercooled FOPTs (characterized by extremely large $\alpha$) do not necessarily yield strong GW signals. If the transition duration is extremely short (large $\beta/H_*$), the GW amplitude is suppressed, rendering even "super-strong" transitions undetectable.
• Complementarity of Probes: The study demonstrates the complementary nature of collider and GW searches. While colliders probe the scalar mass and mixing directly, GWs probe the scale of conformal symmetry breaking ($w$) and the dynamics of the phase transition.

Results

• Collider Reach:
  • For $m_\phi > m_h$, future HL-LHC and 10 TeV muon collider searches (via $ZZ$, $b\bar{b}$, $VV$, and $hh$ channels) can probe mixing angles down to $\theta \sim 10^{-2}$.
  • For $m_\phi < m_h$, LLP searches at LHCb, FASER, CODEX, and SHiP can probe $\theta$ as low as $10^{-6}$, corresponding to symmetry breaking scales $w$ up to $10^8$ GeV.
• GW Reach:
  • Future space-based interferometers (LISA, TianQin, Taiji, BBO) can probe the parameter space where $m_\phi > m_h$, reaching scales $w \sim 10^5$ GeV and $\theta \sim 10^{-5}$.
  • For $m_\phi < m_h$, the "inverted" thermal history leads to rapid transitions. Despite potentially huge $\alpha$ values (up to $10^{30}$), the short duration suppresses GW signals, making them detectable only by BBO in a small fraction of the parameter space.
• Cross-Verification: The paper identifies overlapping regions where both collider and GW experiments can detect signals. For instance, in the $m_\phi > m_h$ regime, the $\phi \to b\bar{b}$ and $\phi \to VV$ channels provide significant cross-verification for GW detections, contrasting with non-conformal singlet extensions where the $hh$ channel dominates.

Significance
The paper claims that the combination of collider and GW experiments offers a powerful tool to probe radiative EWSB up to scales of $10^5 - 10^8$ GeV. The significance lies in the ability to distinguish this mechanism from other BSM scenarios through the specific interplay of a light scalar, a logarithmic potential, and the unique thermal history patterns. The authors emphasize that the non-trivial relationship between the strength of the phase transition and the resulting GW signal (due to duration effects) is a crucial feature that must be accounted for to avoid false negatives in GW searches. Furthermore, the identified thermal history patterns provide a framework for addressing other cosmological puzzles, such as baryogenesis, dark matter production, and primordial black hole formation, within the context of radiative symmetry breaking.