Gravitational waves from CP domain wall collapse and electron EDM in a complex singlet model with dimension-five Yukawa interactions

This paper investigates the complementarity between gravitational waves from CP domain wall collapse and electron electric dipole moment measurements in a complex singlet model with dimension-five Yukawa interactions, demonstrating that future EDM experiments can probe parameter regions overlapping with detectable gravitational wave signals to constrain the model's vacuum structure and CP properties.

The Gist

Imagine the universe as a giant, complex machine. For decades, scientists have been trying to understand how this machine works using a "rulebook" called the Standard Model. But this rulebook has some missing pages; it can't explain why the universe is made of matter instead of antimatter, or what dark matter is.

This paper proposes a new chapter for that rulebook by adding a "ghostly" new particle called a complex singlet scalar. Think of this particle as a hidden dimension or a secret switch in the universe's machinery that we haven't found yet.

Here is the story of what happens when we flip that switch, explained through simple analogies:

1. The Two-Sided Coin (CP Domain Walls)

In this new model, the universe has a "vacuum" (its lowest energy state). Usually, a vacuum is like a flat floor. But in this model, the floor has two identical, deep valleys side-by-side.

• The Analogy: Imagine a coin that can land on Heads or Tails. In our universe, it always lands on Heads. But in this model, the universe could have settled on "Heads" in some places and "Tails" in others.
• The Wall: Where these two regions meet, a boundary forms. The authors call this a Domain Wall. It's like a fence separating a neighborhood where everyone faces North from a neighborhood where everyone faces South.
• The Collapse: These walls are unstable. Eventually, they crumble and collapse. When a giant wall collapses, it doesn't just disappear; it shakes the fabric of space-time, creating ripples. These ripples are Gravitational Waves (GWs).

2. The Invisible Ghost (Why we can't see the wall yet)

Here is the tricky part: The paper says that just seeing these gravitational waves isn't enough to prove the "Heads/Tails" mystery (which physicists call CP Violation).

• The Analogy: Imagine you hear a loud crash in a dark room. You know something fell, but you don't know what fell or why it fell. The gravitational wave is the sound of the crash, but it doesn't tell you the story behind it.
• To prove the "Heads/Tails" mystery, we need to see how this hidden particle interacts with things we can see, like electrons.

3. The New Connection (Dimension-Five Interactions)

The authors extend the model by connecting this hidden "ghost" particle to the electrons we know. They add a special bridge (called a dimension-five Yukawa interaction) that lets the ghost particle talk to the electron.

• The Result: Now, the "Heads/Tails" mystery leaves a fingerprint on the electron. Specifically, it makes the electron slightly lopsided.
• The Analogy: Imagine a perfectly round ball (the electron). If this hidden ghost particle interacts with it, the ball gets a tiny, invisible dent on one side. This dent is called the Electric Dipole Moment (EDM). If we can measure this dent, we prove the "Heads/Tails" mystery is real.

4. The Detective Work (Gravitational Waves vs. EDM)

The paper acts like a detective trying to solve a case using two different clues:

1. Clue A (Gravitational Waves): We listen for the "crash" of the collapsing walls using giant radio telescopes (like SKA) or space-based detectors (like THEIA).
2. Clue B (Electron EDM): We measure the "dent" on the electron using ultra-precise lab experiments.

The Findings:

• The Current Limit: Right now, our best electron experiments (like the JILA experiment) are so sensitive that they have already ruled out the "easy" cases. If the hidden particle's "ghostly" value (called the VEV) is too small, the electron dent would be huge, and we would have seen it by now. Since we haven't, we know the hidden particle must be "heavier" or "further away" than we thought.
• The Sweet Spot: The gravitational waves from the wall collapse are only loud enough to be heard if that hidden particle is quite heavy (around 10 to 100 times heavier than the Higgs boson).
• The Future Hunt: The paper calculates that if we build even better electron detectors in the future (sensitive enough to see a dent 100 times smaller than today), we will be able to find the exact same "heavy" hidden particle that creates the gravitational waves.

The Big Picture

The authors conclude that these two methods are complementary.

• Gravitational waves tell us that a cosmic event happened (the wall collapsed).
• Electron EDM tells us why it happened (the hidden particle has a specific "handedness" or CP violation).

If we hear the crash (GW) and measure the dent (EDM) at the same time, we will have a complete picture of this new hidden sector of the universe. It's like hearing a thunderstorm and then seeing the lightning; together, they confirm the storm is real and tell us exactly how it works.

In short: The paper shows that by combining the search for cosmic ripples (gravitational waves) with the search for tiny electron dents (EDM), we can finally catch a glimpse of a hidden particle that might explain why our universe exists the way it does.

Technical Summary

1. Problem Statement

The Standard Model (SM) fails to explain the baryon asymmetry of the universe and dark matter, suggesting the existence of new physics, particularly new sources of CP violation (CPV).

• The Gap: Complex Singlet Extensions of the SM (cxSM) can accommodate CP domain walls (CPDWs) formed by spontaneous CP violation (SCPV). The collapse of these walls generates a stochastic gravitational wave (GW) background, which serves as a probe for the vacuum structure of the singlet sector. However, the GW signal itself is not a direct CPV observable, and the singlet scalar in minimal cxSM does not couple directly to SM fermions, meaning it does not induce Electric Dipole Moments (EDMs).
• The Question: Can the parameter space where GWs from CPDW collapse are detectable by future observatories (like SKA and THEIA) be probed by current or future electron EDM ($d_e$) experiments? This requires a mechanism to link the singlet sector's CPV phase to observable fermion EDMs.

2. Methodology

The authors extend the cxSM by introducing dimension-five Yukawa interactions that couple the complex singlet scalar $S$ to SM fermions.

• Model Framework:

  • Scalar Sector: The scalar potential $V_0(H, S)$ includes terms allowing for degenerate CP-conjugate vacua ($v_S$ and $v_S^*$), leading to CPDW formation.
  • Yukawa Sector: An effective Lagrangian is added: $\mathcal{L}_{dim-5} \supset \bar{f}_L H (Y_f + C_f S/\Lambda) f_R + \text{H.c.}$, where $\Lambda$ is a cutoff scale. This coupling allows the singlet to induce CPV in fermion interactions.
  • CPV Invariants: The authors construct basis-invariant CP-odd quantities ($I_1, I_2, I_3$) to distinguish between explicit CPV and SCPV. They establish that SCPV (necessary for DWs) requires $I_{1,2,3}=0$ but non-zero rephasing invariants involving the VEV ($J_a \neq 0$).

• Analytical Derivations:

  • Domain Walls: They derived the equations of motion for the CPDW profiles and calculated the wall tension ($\sigma_{DW}$). They introduced a soft $Z_2$-breaking bias term ($\Delta V$) to ensure the walls collapse before Big Bang Nucleosynthesis (BBN).
  • Gravitational Waves: Using the scaling solution for DW networks, they derived the GW spectrum ($\Omega_{GW}$) and peak frequency ($f_{peak}$) as functions of the wall tension and bias term.
  • Electron EDM: They calculated the electron EDM contributions arising from two-loop Barr-Zee diagrams (involving top quarks and $W$ bosons) and kite diagrams. The dominant contribution comes from the Barr-Zee diagrams, which depend on the mixing angles between the Higgs and singlet scalars and the CPV phases in the Yukawa couplings.

• Numerical Analysis:

  • They scanned the parameter space defined by the singlet VEV imaginary part ($v_i^S$), mixing angles ($\alpha_1, \alpha_2$), scalar masses ($m_{h2}, m_{h3}$), and Yukawa coupling coefficients.
  • Constraints: The analysis imposed constraints from:
    • Current electron EDM limits ($|d_e| < 4.1 \times 10^{-30}$ e cm).
    • LHC Higgs coupling measurements ($\kappa_V, \kappa_t$).
    • Perturbative unitarity and vacuum stability.
    • Cosmological bounds (DWs must decay before BBN).
  • Detectability Criteria: They calculated the Signal-to-Noise Ratio (SNR) for GW detection using SKA (pulsar timing array) and THEIA (space-based detector), setting a benchmark of SNR = 20.

3. Key Contributions

1. Bridging Cosmology and Precision Physics: The paper establishes a direct theoretical link between the cosmological GW signal from CPDW collapse and the laboratory-measurable electron EDM within a unified framework (cxSM + dim-5 operators).
2. Basis-Invariant Formalism: The authors rigorously defined CP-odd invariants to characterize the CPV structure, clarifying how spontaneous CPV in the scalar sector translates into observable effects in the fermion sector via dimension-five operators.
3. Complementarity Strategy: They demonstrated that GWs and EDMs probe different but overlapping regions of the parameter space. GWs are sensitive to the energy scale of the phase transition (related to $v_i^S$ and scalar masses), while EDMs are sensitive to the CP-violating phases and mixing angles.

4. Results

• Current Constraints: The current electron EDM bound ($|d_e| < 4.1 \times 10^{-30}$ e cm) already excludes the parameter region where the imaginary part of the singlet VEV is below approximately 10 TeV (for maximal Higgs mixing allowed by current data).
• GW Detectability: The parameter regions where GWs are detectable by SKA and THEIA correspond to larger singlet VEVs, typically $v_i^S \gtrsim 10$ TeV.
• Future Probes:
  • The GW-detectable region (high $v_i^S$) is currently unconstrained by EDMs but will be accessible to future EDM experiments with sensitivities reaching $10^{-31}$ to $10^{-32}$ e cm.
  • The overlap between the future EDM sensitivity and the GW-detectable domain depends on the dimension-five Yukawa couplings. Specifically, if the couplings allow for "accidental cancellation" (structured cancellation) between top-loop and W-loop contributions, the EDM constraints are relaxed, allowing a wider parameter space to be probed.
• Mass Scale Dependence: For heavier scalar masses (e.g., $m_{h3} \sim 10$ TeV), the GW signal amplitude increases, expanding the detectable region. However, perturbative unitarity constraints limit the allowed mixing angles, shifting the viable parameter space to higher $v_i^S$.

5. Significance

This work provides a coherent picture of the singlet scalar sector by combining two distinct observational windows:

• Cosmological (GW): Probes the vacuum structure and the scale of symmetry breaking.
• Precision (EDM): Probes the CP-violating nature and the coupling to fermions.

The study concludes that the combined observation of a stochastic GW background from domain wall collapse and a non-zero electron EDM would provide compelling, complementary evidence for spontaneous CP violation in the singlet scalar sector. It suggests that next-generation EDM experiments are essential to fully test the parameter space predicted by GW observatories, offering a novel strategy to explore physics beyond the Standard Model.