Investigating the leptonic couplings of doubly charged scalars at the muon collider

1. Problem Statement and Motivation

The Standard Model (SM) fails to explain neutrino masses, leading to various Beyond the Standard Model (BSM) extensions such as the Higgs Triplet Model (HTM), Left-Right Symmetric Model (LRSM), and Zee-Babu models. These models often predict the existence of doubly charged scalars ($H^{\pm\pm}$).

• Current Limitations: The Large Hadron Collider (LHC) has searched for these particles primarily via pair production (Drell-Yan mode). However, the LHC faces limitations in probing high masses ($m_{H^{\pm\pm}} \gg \sqrt{s}$) or very small couplings due to the hadronic nature of the machine (parton distribution functions, high background noise) and energy constraints.
• The Gap: While flavor experiments constrain combinations of couplings, they are insensitive to individual coupling strengths at high masses. Furthermore, existing literature lacks dedicated analyses for probing $H^{\pm\pm}$ at a muon collider, specifically focusing on off-shell $t$-channel exchanges that allow probing masses beyond the collider's center-of-mass (c.o.m.) energy.
• The "Inverse Problem": A significant challenge in BSM physics is distinguishing between different new physics (NP) scenarios that produce identical final states. Specifically, a doubly charged scalar ($H^{\pm\pm}$) and a neutral scalar ($H^0$) can mediate the same lepton-flavor-violating (LFV) processes ($\mu^+\mu^- \to \ell^+\ell^-$), making them difficult to distinguish based on cross-sections alone.

2. Methodology

The authors adopt a model-independent approach using an effective Lagrangian to study the interaction of a doubly charged scalar with charged leptons:
$$\mathcal{L} \supset \sum_{i,j=e,\mu,\tau} h_{ij} \ell_{iL}^T C H^{++} \ell_{jL} + \text{h.c.}$$
where $h_{ij}$ are real, symmetric Yukawa couplings.

Simulation and Analysis Setup

• Collider Parameters: A future muon collider with $\sqrt{s} = 3$ TeV and an integrated luminosity of $L = 1 \text{ ab}^{-1}$.
• Signal Processes: Three specific final states are analyzed to probe individual couplings:
  1. $\mu^+\mu^- \to \mu^+\mu^-$ (probing $h_{\mu\mu}$)
  2. $\mu^+\mu^- \to e^+e^-$ (probing $h_{\mu e}$)
  3. $\mu^+\mu^- \to \tau^+\tau^-$ (probing $h_{\mu\tau}$)
• Tools:
  • Event Generation: MadGraph5@NLOv2.9.21 with FeynRules for model implementation.
  • Showering/Hadronization: PYTHIA8.
  • Detector Simulation: DELPHESv3.5.0 with a custom muon collider card.
• Backgrounds: Irreducible SM backgrounds ($\ell^+\ell^-$ via $\gamma/Z$) and reducible backgrounds ($W\ell\nu$, $Z\ell\ell$, $\tau\tau$).
• Selection Strategy:
  • Kinematic Cuts: Transverse momentum ($p_T > 50$ GeV) and pseudorapidity ($|\eta| < 2.4$).
  • Missing Energy ($E_T^{miss}$): Used to suppress backgrounds with neutrinos ($W, Z, \tau$ decays).
  • Angular Variables: For the $\tau\tau$ channel, $\Delta R_{\ell\ell}$ and $|\cos \theta_\mu|$ are used to distinguish signal from $W/Z$ backgrounds.
  • Significance Calculation: Uses the profile likelihood ratio approximation: $Z = \sqrt{2L [(\sigma_S + \sigma_B)\ln(1 + \sigma_S/\sigma_B) - \sigma_S]}$.

Addressing the Inverse Problem

To distinguish $H^{\pm\pm}$ from a neutral scalar $H^0$ (which can mediate similar $t$-channel processes), the authors propose an angular asymmetry variable.

• Logic: The $H^{\pm\pm}$ interaction involves a charge conjugation operator, leading to a "charge flip" in the $t$-channel leg relative to the beam, whereas $H^0$ does not.
• Observable: The distribution of $\cos(\theta_{\mu^\pm})$ (angle between the beam and the outgoing muon).
• Asymmetry Parameter ($A_\pm$): Defined as the difference in cross-sections between forward and backward hemispheres. The sign of $A_\pm$ differs for $H^{\pm\pm}$ vs. $H^0$, providing a robust discriminator even in the presence of backgrounds.

3. Key Contributions

1. Absolute Bounds on Individual Couplings: Unlike flavor experiments that constrain products of couplings (e.g., $h_{\mu e}h_{ee}$), this study demonstrates that a muon collider can set absolute bounds on individual couplings ($h_{\mu\mu}, h_{\mu e}, h_{\mu\tau}$) independently.
2. Extended Mass Reach: By utilizing the $t$-channel exchange mechanism, the study shows the muon collider can probe scalar masses significantly higher than the collider's c.o.m. energy (up to $\sim 10$ TeV), surpassing the LHC's pair-production limits ($\sim 1$ TeV).
3. Discrimination Variable: The proposal of the forward-backward asymmetry variable ($A_\pm$) based on angular distributions provides a concrete method to resolve the "inverse problem" between doubly charged and neutral scalars.
4. Interference Analysis: The authors explicitly calculate the interference effects between the SM $\gamma/Z$ exchange and the BSM $H^{\pm\pm}$ exchange, showing that while interference can be large (up to -77% for $\mu\mu$), the final sensitivity after cuts remains robust.

4. Results

The analysis yields the following sensitivity reach for a 3 TeV muon collider with $1 \text{ ab}^{-1}$ luminosity:

• $\mu^+\mu^- \to \mu^+\mu^-$ ($h_{\mu\mu}$):
  • Can probe masses up to $\sim 10$ TeV for couplings $|h_{\mu\mu}| \approx 0.15 - 0.65$.
  • This channel is unique as no direct limits on $h_{\mu\mu}$ currently exist in literature.
• $\mu^+\mu^- \to e^+e^-$ ($h_{\mu e}$):
  • Can probe masses in the 1.1 TeV to 10 TeV range for $|h_{\mu e}| \approx 0.1 - 0.41$.
  • Represents a substantial improvement over current LEP and LHC bounds.
• $\mu^+\mu^- \to \tau^+\tau^-$ ($h_{\mu\tau}$):
  • Due to $\tau$ reconstruction challenges, the reach is lower but still significant: $\sim 6$ TeV at $5\sigma$ and $\sim 8$ TeV at $2\sigma$ for $|h_{\mu\tau}| \approx O(1)$.
• Discrimination:
  • The asymmetry parameter $A_\pm$ successfully distinguishes the two scenarios. For $H^{\pm\pm}$, $A_+$ is negative (approx. -0.30 with backgrounds), while for $H^0$, it is positive (approx. +0.09). This distinction holds even with SM backgrounds included.

5. Significance

This paper establishes the muon collider as a premier facility for probing the scalar sector of BSM physics, specifically for doubly charged scalars.

• Complementarity: It complements LHC searches by accessing the high-mass, high-coupling regime via $t$-channel processes, a region inaccessible to hadron colliders.
• Precision: It offers the ability to disentangle individual flavor couplings, which is impossible with current flavor-violation experiments that only constrain coupling products.
• Model Discrimination: The proposed angular asymmetry variable solves a critical ambiguity in NP discovery, allowing physicists to determine the spin and charge of the mediator even when the final states are identical.
• Broader Impact: The production mechanism and discrimination strategies discussed are applicable to other BSM scenarios, such as $Z'$ bosons, leptoquarks, and spin-2 particles, reinforcing the scientific case for constructing a muon collider.