Model-independent probes of CP violation in the heavy scalar sector at muon colliders

This paper proposes a model-independent method to detect CP violation in the heavy scalar sector at future muon colliders by observing the vector-boson-fusion production of a heavy neutral scalar decaying into a Z boson and the 125 GeV Higgs, demonstrating that the mere existence of this process confirms non-zero couplings necessary for CP violation.

The Gist

The Big Picture: Hunting for a Cosmic "Imbalance"

Imagine the universe as a giant, perfectly balanced scale. For a long time, physicists thought the rules of physics were perfectly symmetrical: if you swapped matter for antimatter, or flipped left for right, everything should work exactly the same. But we know the universe is made of matter, not antimatter. Something broke that perfect symmetry. This "breaking" is called CP violation.

We found a tiny crack in the symmetry in the 1960s, but it's too small to explain why our universe exists. We need to find a bigger crack. This paper proposes a new, clever way to hunt for that bigger crack, specifically looking at a mysterious, heavy particle that might be hiding in the "scalar sector" (the family of particles that includes the famous Higgs boson).

The Setting: A Muon "Smash-Up" Factory

The authors are proposing a test at a future Muon Collider. Think of this as a high-speed racetrack where tiny particles called muons zoom around and crash into each other.

• The Energy: They plan to smash them together with incredible force (3 to 10 TeV), which is like having a particle accelerator the size of a small city.
• The Goal: To see if a heavy, invisible particle (let's call it H2) exists and if it behaves in a way that breaks the symmetry rules.

The Detective Work: The "One-Process" Rule

The authors have a very specific, "model-independent" strategy. This means they aren't guessing about the details of a specific theory; they are looking for a smoking gun that proves the symmetry is broken, no matter what the underlying theory is.

Here is the analogy:
Imagine you are trying to prove that a secret handshake exists between two people, Alice and Bob. You can't see them talking, but you know that if they both do their part of the handshake, a specific light bulb will flash.

• The Light Bulb: The process where two force-carrying particles (W or Z bosons) smash together to create the heavy particle H2, which then immediately decays into a known Higgs boson (h1) and a Z boson.
• The Rule: The paper argues that for this light bulb to flash, both Alice and Bob must be present and active. In physics terms, this means two specific interaction strengths (called $c_2$ and $c_{12}$) must both be non-zero.
• The Conclusion: If you see this specific event happen even once, you have proven that CP violation exists in this sector. You don't need to know why it happens, just that it does happen.

The Obstacle: The "Beam-Induced Background" Noise

Muons are tricky. When they accelerate, they create a massive amount of "static noise" (beam-induced backgrounds).

• The Solution: The authors imagine building a giant "absorber" (like a thick soundproof wall) around the detector. This wall blocks the noise coming from the very front and back of the collision.
• The Trade-off: This means we can't see the particles that fly straight forward or backward. But that's okay! The signal they are looking for (the heavy H2 decaying) leaves a distinct "fingerprint" in the middle of the detector that doesn't rely on seeing those forward particles.

The Hunt: Finding the Needle in the Haystack

The team ran computer simulations to see if they could spot this signal against the background noise.

• The Signal: They are looking for a specific chain of events: A heavy particle decays into a Z boson (which turns into two electrons or muons) and a Higgs boson (which turns into two "bottom" quark jets).
• The Noise: There are many other processes that look similar, like two Z bosons colliding or random particles misbehaving.
• The Filter: They used a "sieve" (mathematical cuts) to filter out the noise. They looked at the mass of the particles produced. If the mass matches the heavy H2 they are looking for, they keep it. If not, they throw it away.

The Results: How Far Can We See?

The simulations showed that this method is very powerful, especially for heavy particles:

• At 3 TeV (a smaller collider): They could find this CP violation if the heavy particle is up to about 1,000 GeV (1 TeV) heavy.
• At 10 TeV (a massive collider): They could find it if the particle is up to 4,500 GeV (4.5 TeV) heavy.

Think of it like a lighthouse. The 10 TeV collider is a lighthouse with a much brighter beam, allowing them to see the "ghost" of the heavy particle much further away in the dark ocean of possibilities.

The Bottom Line

This paper doesn't claim to have found the new particle yet. Instead, it provides a blueprint for how to find it.

1. Build a muon collider with high energy.
2. Watch for a specific, rare collision where a heavy particle turns into a Higgs and a Z boson.
3. If you see it, you have proven that the universe has a fundamental asymmetry (CP violation) in its scalar sector, solving a major mystery about why we exist.

The authors emphasize that this is a "model-independent" test, meaning it works regardless of the specific complex theory physicists might invent to explain the universe. If the event happens, the symmetry is broken. Period.

Technical Summary

1. Problem Statement

The Standard Model (SM) contains a source of CP violation via the CKM matrix, but this is insufficient to explain the observed matter-antimatter asymmetry of the universe. While many Beyond the Standard Model (BSM) scenarios introduce new CP-violating sources in the scalar sector, these are often constrained by Electric Dipole Moment (EDM) measurements.

• The Gap: Existing collider studies often focus on the 125 GeV Higgs boson ($h_1$) or light new scalars. However, light new states are tightly constrained, and direct probes of CP violation in the pure bosonic sector involving heavy scalars ($h_2 \sim \mathcal{O}(\text{TeV})$) remain underexplored.
• The Challenge: Establishing CP violation requires demonstrating that a scalar state contains both CP-even and CP-odd components. In the bosonic sector, this manifests through specific interaction vertices.

2. Methodology

The authors propose a model-independent strategy to detect CP violation at future Muon Colliders (MuC) with center-of-mass energies $\sqrt{s} = 3$ TeV and $10$ TeV.

Theoretical Framework

The study assumes a heavy neutral scalar $h_2$ exists and interacts with the SM-like Higgs ($h_1$) and gauge bosons ($W^\pm, Z$) via the following effective Lagrangian at tree level:
$$\mathcal{L} \supset \left( \frac{g^2 v}{2} W^+_\mu W^{-\mu} + \frac{g^2 v}{4c_W^2} Z_\mu Z^\mu \right) (c_1 h_1 + c_2 h_2) + c_{12} \frac{g}{2c_W} Z_\mu (h_1 \partial^\mu h_2 - h_2 \partial^\mu h_1)$$

• $c_1 \approx 1$: Coupling of the SM-like Higgs to gauge bosons.
• $c_2$: Coupling of the heavy scalar $h_2$ to gauge bosons ($h_2 VV$). A non-zero $c_2$ implies $h_2$ has a CP-even component.
• $c_{12}$: Coupling of the $h_1 h_2 Z$ vertex. A non-zero $c_{12}$ implies $h_1$ and $h_2$ have opposite CP components (mixing).
• Criterion for CP Violation: The simultaneous observation of non-zero $c_2$ and $c_{12}$ is a sufficient condition to establish CP violation in the scalar sector.

Process Selection

The authors focus on the Vector Boson Fusion (VBF) production of $h_2$ followed by its decay:
$$VV \to h_2 \to Z h_1$$
where $V = W^\pm, Z$.

• Final State: $Z(\to \ell^+\ell^-) + h_1(\to b\bar{b}) + \text{Missing Energy}$.
  • The missing energy arises from undetected forward/backward muons (in $ZZ$ fusion) or neutrinos (in $WW$ fusion) due to the MuC detector geometry (absorbers block the very forward region to mitigate beam-induced backgrounds).
• Signal Topology: Two opposite-charge same-flavor leptons ($\ell = e, \mu$), two $b$-tagged jets, and large missing transverse energy.

Simulation and Analysis

• Tools: MadGraph5_aMC@NLO for event generation, PYTHIA 8.3 for parton showering, and Delphes 3.4.2 for detector simulation using a MuC configuration card.
• Benchmark Points:
  • Energies: $\sqrt{s} = 3$ TeV ($L = 0.9 \text{ ab}^{-1}$) and $10$ TeV ($L = 10 \text{ ab}^{-1}$).
  • Masses: $m_{h_2} \in \{700, 1000, 1500, 2000, 3000, 4500\}$ GeV.
  • Couplings: $c_2 = c_{12} = 0.2$ (conservative upper bounds allowed by current LHC data).
• Backgrounds:
  • BSM Background ($b_1$): $VV \to h_2 \to Z Z \to \ell\ell jj$.
  • SM Backgrounds ($b_2, b_3, b_4$): Processes involving $Z h_1$, $Z Z$, and $W W$ production with $b$-jets.
• Selection Strategy:
  1. Pre-selection: Exactly 2 leptons and 2 $b$-tagged jets.
  2. Reconstruction Cuts:
     • $100 \text{ GeV} \le m_{b\bar{b}} \le 140 \text{ GeV}$ (reconstructs $h_1$).
     • $m_{b\bar{b}\ell\ell}$ window centered on $m_{h_2}$ (reconstructs $h_2$).
  3. VBF Tagging: The invariant mass of the invisible system ($m_{/}$) is used as a tag but does not provide significant separation after kinematic cuts.

3. Key Contributions

1. Model-Independence: The method relies solely on the existence of the $h_2 VV$ and $h_1 h_2 Z$ vertices. It does not require assumptions about the specific UV-complete model (e.g., 2HDM, MSSM), provided the effective Lagrangian holds.
2. Heavy Scalar Focus: Unlike previous studies focusing on light scalars at Higgs factories, this work targets the TeV-scale regime accessible only at high-energy MuCs.
3. Sufficiency Condition: The paper rigorously demonstrates that observing the single process $VV \to h_2 \to Z h_1$ is sufficient to prove CP violation, as it necessitates $c_2 \neq 0$ and $c_{12} \neq 0$ simultaneously.
4. Robustness against Backgrounds: The analysis shows that the dominant background ($b_2$) can be effectively suppressed using invariant mass cuts, while the BSM background ($b_1$) is suppressed by the smallness of the branching ratio $\text{Br}(h_2 \to ZZ)$ relative to $\text{Br}(h_2 \to Zh_1)$ in the chosen parameter space.

4. Results

The study presents expected discovery sensitivities (statistical significance $\sigma$) for the $(c_2, c_{12})$ parameter space:

• At $\sqrt{s} = 3$ TeV ($L = 0.9 \text{ ab}^{-1}$):

  • CP violation can be discovered at the $5\sigma$ level for $m_{h_2} \lesssim 1$ TeV, assuming $c_2, c_{12} \approx 0.2$.
  • Sensitivity drops rapidly for higher masses due to the falling cross-section.

• At $\sqrt{s} = 10$ TeV ($L = 10 \text{ ab}^{-1}$):

  • The reach extends significantly. CP violation can be discovered at $5\sigma$ for $m_{h_2} \lesssim 4.5$ TeV (for $c_2, c_{12} \approx 0.2$).
  • Even for smaller couplings ($c_2, c_{12} \approx 0.1$), a $5\sigma$ discovery is possible up to $m_{h_2} \approx 3$ TeV.
  • The $W^+W^-$ fusion channel dominates over $ZZ$ fusion by an order of magnitude.

• Background Suppression: After selection cuts, the dominant background is the SM process $b_2$ ($\mu^-\mu^+ \to \nu\bar{\nu} Z h_1$). The BSM background $b_1$ is highly suppressed because the parameter $\xi \propto c_2^4$ is small, and $\text{Br}(Z \to b\bar{b})$ is small compared to $\text{Br}(Z \to \ell\ell)$.

5. Significance

• Complementarity: This approach complements EDM searches (which constrain CP phases indirectly) and Yukawa-sector probes (which rely on fermion couplings). It specifically targets the bosonic sector, which is crucial for models where CP violation is hidden from fermions.
• Muon Collider Advantage: The MuC is uniquely suited for this search due to:
  • High center-of-mass energy (access to TeV scalars).
  • Clean environment (low pile-up compared to hadron colliders).
  • VBF enhancement: The logarithmic enhancement of gauge boson radiation at high energies makes the MuC an efficient "gauge boson collider."
• Definitive Test: The observation of this channel provides a "smoking gun" for CP violation in the scalar sector without needing to reconstruct the full UV model. Conversely, non-observation sets robust, model-independent upper bounds on the coupling combination $(c_2, c_{12})$.

In conclusion, the paper establishes that future high-energy Muon Colliders offer a powerful, model-independent avenue to discover CP violation in the heavy scalar sector, capable of probing masses up to several TeV and couplings as small as 0.1.