Doubly Charged Scalar in Extended Models

• The doubly charged scalar is a spin-0 particle with electric charge ±2, emerging in extended models to address neutrino masses and the Higgs hierarchy.
• It interacts via Yukawa couplings that induce lepton flavor violating decays, offering clear signals such as same-sign lepton pairs or W boson pairs.
• High-energy colliders can probe its existence through pair and associated production, providing essential tests for new physics frameworks.

A doubly charged scalar is a spin-0 particle with electric charge $Q = \pm 2$, predicted in numerous extensions of the Standard Model (SM). Its presence is a characteristic prediction in a variety of frameworks seeking to address the origin of neutrino masses, the naturalness problem in the Higgs sector, and other phenomena beyond the SM. Models that contain such states include the littlest Higgs model, extended Higgs sectors with scalar doublets or triplets, and mechanisms with lepton number violation, such as the type-II seesaw. Owing to its exotic charge, the doubly charged scalar (often denoted $\phi^{++}$, $H^{++}$, or similar) is associated with distinctive signatures at colliders and in rare processes, providing an unambiguous probe for new physics.

1. Genesis in Extended Scalar Sectors

In the littlest Higgs model, the SM is extended by a global SU(5) symmetry spontaneously broken to SO(5) at a characteristic scale $f \sim 1$ TeV. This breaking yields 14 Goldstone bosons, with the scalar sector comprising both the SM Higgs doublet and an additional complex SU(2) triplet containing a doubly charged scalar ($\phi^{++}$), a singly charged scalar, a neutral scalar, and a neutral pseudoscalar. The presence of such a triplet is directly linked to the cancellation of quadratic divergences in the Higgs mass and forms part of the model’s solution to the hierarchy problem (Cagil, 2011).

The underlying breaking pattern is: $$[SU(2)_1 \otimes U(1)_1] \otimes [SU(2)_2 \otimes U(1)_2] \longrightarrow SU(2)_L \otimes U(1)_Y$$ with both doubled gauge and scalar sectors. Upon further electroweak symmetry breaking, the triplet’s components become massive and physical, with the doubly charged scalar ($\phi^{++}$) emerging as a discrete eigenstate.

2. Gauge Interactions and Lepton Flavor Violation

The triplet scalar’s phenomenology is shaped by its Yukawa interactions with leptons. A crucial term in the Lagrangian takes the Majorana form: $$\mathcal{L}_{\text{LFV}} = i\,Y_{ij}\,L_i^T\,\phi\,C^{-1} L_j + \mathrm{h.c.}$$ where $L_i$ are lepton doublets, $Y_{ij}$ is a symmetric Yukawa matrix (with diagonal elements $Y$ and off-diagonal $Y'$). This operator breaks lepton number by two units, enabling lepton flavor violating (LFV) decays of $\phi^{++}$, and is responsible for the triplet's connection to radiative neutrino mass mechanisms. The decay widths are then governed by the formula: $$\Gamma_{\phi^{++}} \approx \frac{v^{\prime 2} M_\phi^3}{2\pi v^4} + \frac{3}{8\pi} |Y|^2 M_\phi + \frac{3}{4\pi} |Y'|^2 M_\phi$$ where $v'$ denotes the triplet vacuum expectation value (typically small, to satisfy electroweak $\rho$-parameter constraints), and $v$ is the SM Higgs VEV.

The branching ratios between leptonic and gauge boson final states depend sensitively on the relative sizes of $Y$ and $Y'$. Large $Y$ enhances LFV decays, making collider signatures starkly dependent on the underlying flavor physics.

3. Production Mechanisms in High-Energy Colliders

At $e^+ e^-$ colliders, $\phi^{++}$ is accessible through two principal processes:

• Pair production: $e^+ e^- \to \phi^{++} \phi^{--}$, driven by $s$-channel gauge boson exchange.
• Associated production: $e^+ e^- \to Z_L \phi^{++} \phi^{--}$, where $Z_L$ is a neutral gauge boson eigenstate close to the SM $Z$ for the appropriate mixing.

For model parameters $s/s' = 0.8/0.6$, the direct pair production cross section at $\sqrt{s} > 1.7$ TeV is $\sim 3 \times 10^{-2}$ pb, yielding approximately 3000 events per $100~\mathrm{fb}^{-1}$ of integrated luminosity at a high-energy $e^+e^-$ collider. Associated production cross sections are an order of magnitude lower ($0.4$–$0.8\times10^{-3}$ pb at $\sqrt{s}=3$ TeV). The relative magnitude of these cross sections is sensitive to the mixing angles in the extended gauge sector, as well as the symmetry breaking scale $f$ (Cagil, 2011).

4. Collider Signatures and Signal Identification

The phenomenology bifurcates based on the strength of LFV:

• Large LFV Couplings ($Y\sim 1$): The doubly charged scalar decays predominantly into same-sign, possibly flavor-violating lepton pairs (e.g., $e^+e^+$, $\mu^+\mu^+$, or $e^+\mu^+$). This leads to characteristic four-lepton final states with explicit flavor violation, such as $e^+ e^+ \mu^- \mu^-$. The signal has negligible Standard Model (SM) background due to the explicit LFV and doubly charged signature.

The branching fraction for $l_i l_i\bar{l}_j\bar{l}_j$ final states reaches $66\%$ for $Y\sim 1$, corresponding to $\sim 1800$ events per year for $100~\mathrm{fb}^{-1}$.

• Small LFV Couplings ($Y<0.01$): $\phi^{++}$ decays primarily into same-sign $W$ boson pairs, with the final state $W^+W^+W^-W^-$. Identification then relies on reconstructing the invariant and transverse masses of the $W$ pairings. In this regime, careful background subtraction from SM multi-boson production is essential.

The presence of four high-$p_T$ leptons with specific flavor combinations, or of events with multiple same-sign $W$ bosons, provides robust handles for discovery. The absence of SM backgrounds in the LFV-rich channels means that observation in these channels would point directly to physics beyond the SM.

5. Model-Building Implications and Theoretical Significance

The detection of a doubly charged scalar at a collider would constitute decisive evidence for an extended scalar sector, as predicted by the littlest Higgs framework and related models. Such a discovery would corroborate the mechanism of collective symmetry breaking as a solution to the hierarchy problem.

Moreover, the coupling structure of the doubly charged scalar mediates Majorana mass terms for neutrinos and facilitates LFV, providing experimental access to aspects of neutrino mass generation mechanisms absent in the minimal SM. The direct measurement of LFV decay rates, along with the reconstruction of Yukawa matrix elements $Y$ and $Y'$, could illuminate the flavor structure underlying the observed pattern of neutrino masses and mixing.

A conclusive observation of four-lepton, LFV-rich final states would signal lepton number violation and underwrite further theoretical development of Majorana neutrino scenarios.

6. Experimental Strategies and Future Prospects

Achieving high sensitivity to the doubly charged scalar regime requires electron-positron colliders with $\sqrt{s} > 1.7$ TeV and integrated luminosities upwards of $100~\mathrm{fb}^{-1}$. The detailed measurement of lepton flavor compositions, event kinematics, and invariant mass distributions is paramount.

Further analysis should focus on:

• Scanning the $Y$, $Y'$ parameter space to map the transition between purely $W^+ W^+$ decays and LFV-dominated decays.
• Associated production channels (such as those involving the $Z_L$) for cross-checking model predictions and constraining the scalar multiplet structure.
• Complementary studies of extra gauge boson resonances and heavy top partners that might be associated with the symmetry structure of the littlest Higgs model.

A discovery or robust exclusion of doubly charged scalars in this regime would have profound implications for the viability of the various mechanisms that protect the Higgs mass and generate neutrino masses without the need for fine-tuning.

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In summary, the doubly charged scalar is a theoretically and phenomenologically distinct signature of extended symmetry breaking in models like the littlest Higgs. Its presence at high-energy colliders—particularly in LFV-dominated decays—is directly linked to both the mechanism of neutrino mass generation and the solution to the hierarchy problem, with experimental observation providing crucial tests of these fundamental ideas (Cagil, 2011).