Doubly Charged Scalars in BSM Models

• Doubly charged scalars are exotic bosons with charge ±2, appearing in SM extensions to generate neutrino masses and induce lepton number violation.
• They yield distinctive collider signatures, notably same-sign dilepton events, enabling experimental probes of new electroweak sectors.
• Precise measurements of their decay patterns provide insights into the neutrino mass matrix and help differentiate between theoretical models.

Doubly charged scalars are exotic bosons with electric charge $Q = \pm 2$, predicted in a broad class of Standard Model (SM) extensions addressing the origin of neutrino masses, lepton number violation, and new electroweak sectors. Their defining feature—quantum numbers with $Q = \pm 2$—gives rise to characteristic collider signatures, especially same-sign, high-momentum dilepton pairs. These scalars appear as members of various $SU(2)_L$ multiplets, such as triplets in the type II seesaw mechanism, but also as singlets and higher representations in more general frameworks. A central theoretical and experimental motivation is the direct connection between the couplings and decay modes of doubly charged scalars and the underlying structure of the neutrino mass matrix.

1. Theoretical Origin and Model Realizations

Doubly charged scalars ($H^{\pm\pm}$, $\Delta^{\pm\pm}$, $S^{\pm\pm}$, etc.) can arise in a variety of SM extensions:

• Type II Seesaw Mechanism: The SM is extended by an $SU(2)_L$ triplet scalar ($\Delta \sim \mathbf{3}, Y=2$), with components

$$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$

A small triplet vev $v_T = \langle H^0 \rangle \ll v_{\text{SM}}$ induces neutrino masses: $Q = \pm 2$0, where $Q = \pm 2$1 are Yukawa couplings to lepton doublets (Guedes et al., 31 Dec 2025, Zhou et al., 15 Aug 2025).

• Left-Right Symmetric Models (LRSMs): Both $Q = \pm 2$2 and $Q = \pm 2$3 triplets ($Q = \pm 2$4, $Q = \pm 2$5) are present, generating heavy right-handed neutrino masses and accommodating parity restoration at high scales. The masses of doubly charged scalars can be decoupled from the $Q = \pm 2$6-breaking scale by "hidden" symmetries in the scalar potential (Akhmedov et al., 2024, Belfkir et al., 2023, Dev et al., 2018).
• Singlet and Multiplet Extensions: Doubly charged $Q = \pm 2$7 singlets (e.g., $Q = \pm 2$8) can appear in two-loop neutrino mass models (Zee–Babu mechanism) or other frameworks. Isospin $Q = \pm 2$9, $SU(2)_L$0, $SU(2)_L$1, $SU(2)_L$2 representations can also yield doubly charged states (Aguila et al., 2013, Crivellin et al., 2018).
• Composite Higgs and Little Higgs Models: $SU(2)_L$3 can be present as pseudo-Nambu–Goldstone bosons in composite Higgs sectors or as parts of extended scalar triplets in little Higgs constructions (Flacke et al., 2023, Cagil, 2011).

2. Gauge Interactions, Couplings, and Decay Patterns

The gauge quantum numbers and couplings of doubly charged scalars are model-dependent:

• Yukawa Interactions: In type II seesaw and many related models, the main renormalizable interaction is with left-handed leptons through

$SU(2)_L$4

yielding $SU(2)_L$5 with partial width

$SU(2)_L$6

$SU(2)_L$7 for $SU(2)_L$8, $SU(2)_L$9 for $H^{\pm\pm}$0 (Guedes et al., 31 Dec 2025).

• Gauge-Boson Decays: For $H^{\pm\pm}$1 GeV, decays $H^{\pm\pm}$2 become relevant. The partial width to $H^{\pm\pm}$3 boson pairs scales as $H^{\pm\pm}$4 (Aguila et al., 2013). This mode is suppressed compared to leptonic decays for small $H^{\pm\pm}$5 or large $H^{\pm\pm}$6.
• Cascade and Multi-Scalar Decays: In models with multiple triplets, e.g., two-triplet type II seesaw, the heavier $H^{\pm\pm}$7 may dominantly decay into a lighter singly-charged scalar and a $H^{\pm\pm}$8, $H^{\pm\pm}$9, if kinematically accessible. This can reach branching fractions above 99% (Chaudhuri et al., 2013).
• Flavor Structure: Decay patterns are tightly linked to the origin of neutrino mass. The various $\Delta^{\pm\pm}$0 correspond directly to entries in the neutrino mass matrix, so branching ratio measurements can reconstruct underlying mass and mixing parameters, subject to charged-lepton mass suppressions in singlet scenarios (Chen et al., 2010).

3. Collider Production and Signature Phenomenology

3.1 Production Mechanisms

• Drell–Yan Pair Production: The dominant production at hadron colliders is $\Delta^{\pm\pm}$1. The cross section at $\Delta^{\pm\pm}$2 TeV is $\Delta^{\pm\pm}$3 fb for $\Delta^{\pm\pm}$4 GeV, dropping below $\Delta^{\pm\pm}$5 fb for $\Delta^{\pm\pm}$6 TeV (Guedes et al., 31 Dec 2025).
• Photon–Photon Fusion: At higher masses, $\Delta^{\pm\pm}$7 is relevant. While sub-dominant below $\Delta^{\pm\pm}$8 TeV (few-percent effect), the contribution can rise to $\Delta^{\pm\pm}$9–$S^{\pm\pm}$0 at $S^{\pm\pm}$1–$S^{\pm\pm}$2 TeV, especially as widths grow (Guedes et al., 31 Dec 2025, Zhou et al., 15 Aug 2025).
• Associated Production: For non-singlet $S^{\pm\pm}$3 (e.g., triplet or doublet), $S^{\pm\pm}$4 is possible. For further details on cross section scaling with isospin, see (Aguila et al., 2013).
• Gluon Fusion Cascades: In models where a heavy neutral scalar with significant gluon fusion production can decay to a pair of $S^{\pm\pm}$5, additional cross section gains are possible. For instance, $S^{\pm\pm}$6 can exceed Drell–Yan in parts of parameter space (Akeroyd et al., 2011).
• Lepton Collider Production: At $S^{\pm\pm}$7 or $S^{\pm\pm}$8 colliders, pair production proceeds via $S^{\pm\pm}$9-channel $SU(2)_L$0, with cross sections scaling as

$SU(2)_L$1

where $SU(2)_L$2 (Cagil, 2011, Belfkir et al., 2023, Ghosh et al., 1 Jun 2025).

3.2 Experimental Signatures

• Prompt Leptonic Decays: The canonical signature is two same-sign, same-flavor (or mixed-flavor) isolated leptons with high $SU(2)_L$3 and narrow invariant-mass peaks at $SU(2)_L$4, resulting in four-lepton final states (Guedes et al., 31 Dec 2025).
• Long-Lived and Displaced Signatures: For small Yukawa couplings ($SU(2)_L$5), $SU(2)_L$6 may be long-lived, giving rise to high-$SU(2)_L$7 ionization tracks. Dedicated long-lived particle searches can probe $SU(2)_L$8–$SU(2)_L$9 GeV depending on the production mechanism (Akhmedov et al., 2024).
• Multi-lepton plus Missing Energy: Cascade decays in multiple-triplet models or via singly-charged scalars lead to $\Delta \sim \mathbf{3}, Y=2$0 bosons and further leptons, yielding higher lepton multiplicities, missing energy, and non-standard $\Delta \sim \mathbf{3}, Y=2$1 spectra (Chaudhuri et al., 2013, Akeroyd et al., 2011).
• Flavor Violating Channels: Observing mixed-flavor pairs (e.g., $\Delta \sim \mathbf{3}, Y=2$2 from the same scalar) or lepton-number-violating decays (e.g. $\Delta \sim \mathbf{3}, Y=2$3 without associated missing energy) points to new physics in the underlying Yukawa sector, potentially distinguishing between singlet, triplet, or other origins (Zhou et al., 15 Aug 2025, Cagil, 2011).
• High-Mass Reach at Future Colliders: The reach extends to $\Delta \sim \mathbf{3}, Y=2$4 TeV at $\Delta \sim \mathbf{3}, Y=2$5 TeV FCC-hh with $\Delta \sim \mathbf{3}, Y=2$6 for favorable branching scenarios (Guedes et al., 31 Dec 2025). Muon and $\Delta \sim \mathbf{3}, Y=2$7 colliders can probe masses up to the kinematic limit ($\Delta \sim \mathbf{3}, Y=2$8) in pair production, and beyond via $\Delta \sim \mathbf{3}, Y=2$9-channel exchange in off-shell channels (Belfkir et al., 2023, Ghosh et al., 1 Jun 2025).

4. Experimental Constraints and Current Bounds

LHC Run II (13 TeV, $$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$0 fb$$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$1):

• For $$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$2, $$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$3 TeV.
• For $$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$4, $$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$5 TeV.
• For $$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$6, $$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$7 TeV.
• For mixed $$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$8 $$\Delta = \begin{pmatrix} H^+/\sqrt{2} & H^{++} \ H^0 & -H^+/\sqrt{2} \end{pmatrix}$$9, $v_T = \langle H^0 \rangle \ll v_{\text{SM}}$0 TeV (Guedes et al., 31 Dec 2025).

Photon Fusion Limits: At 14 TeV, forward proton detectors enable bounds up to $v_T = \langle H^0 \rangle \ll v_{\text{SM}}$1–$v_T = \langle H^0 \rangle \ll v_{\text{SM}}$2 TeV, depending on flavor structure (Zhou et al., 15 Aug 2025).

Projections at Hadron Colliders:

• FCC-hh (100 TeV, 3 ab$v_T = \langle H^0 \rangle \ll v_{\text{SM}}$3): up to $v_T = \langle H^0 \rangle \ll v_{\text{SM}}$4 TeV (Guedes et al., 31 Dec 2025).
• HL-LHC (14 TeV, 3 ab$v_T = \langle H^0 \rangle \ll v_{\text{SM}}$5): up to $v_T = \langle H^0 \rangle \ll v_{\text{SM}}$6 TeV (Melo et al., 2019).
• HE-LHC (27 TeV, 3 ab$v_T = \langle H^0 \rangle \ll v_{\text{SM}}$7): up to 3.1 TeV; with 15 ab$v_T = \langle H^0 \rangle \ll v_{\text{SM}}$8, up to 4.3–4.8 TeV depending on flavor (Melo et al., 2019).

Lepton Colliders: Muon colliders with $v_T = \langle H^0 \rangle \ll v_{\text{SM}}$9–$Q = \pm 2$00 TeV and high integrated luminosity ($Q = \pm 2$01–$Q = \pm 2$02 ab$Q = \pm 2$03) probe $Q = \pm 2$04 masses up to $Q = \pm 2$05 TeV (discovery reach for $Q = \pm 2$06). Off-shell t-channel exchange offers sensitivity to even heavier masses given sufficiently large leptonic couplings (Ghosh et al., 1 Jun 2025, Belfkir et al., 2023).

Low-Energy and Precision Constraints:

• Lepton-flavor violating decays (e.g., $Q = \pm 2$07, $Q = \pm 2$08) constrain products $Q = \pm 2$09–$Q = \pm 2$10 GeV$Q = \pm 2$11 for $Q = \pm 2$12 TeV (Geib et al., 2015, Crivellin et al., 2018).
• Parity-violating Møller scattering can probe right-handed doubly charged scalars up to $Q = \pm 2$13 TeV in the mass–coupling plane (Dev et al., 2018).

5. Phenomenological Connections: Neutrino Masses and Model Diagnostics

Doubly charged scalar decay branching ratios and coupling measurements offer a diagnostic for the origin and structure of neutrino masses:

• Direct Mapping to Neutrino Mass Matrix: In the type II seesaw, $Q = \pm 2$14 ratios encode the flavor structure of neutrino masses, and, with sufficient collider data, may allow extraction of absolute neutrino masses and Majorana phases (Chen et al., 2010). Closed analytical expressions link observed event rates to PMNS parameters in the tribimaximal limit.
• Multiplet and Charge Assignments: Event rates and associated production ($Q = \pm 2$15), the observation of gauge boson modes, and the absence/presence of charge–current production channels can be used to determine the $Q = \pm 2$16 multiplet assignment of the new state (singlet, doublet, triplet, etc.), using cross section ratios and event-type discriminants (Aguila et al., 2013).
• Discrimination from Neutral Scalars and Alternative Explanations: Angular distributions and polarization asymmetries in lepton-collider final states can distinguish t-channel exchange of doubly charged scalars from that of neutral (e.g., $Q = \pm 2$17) exotics (Ghosh et al., 1 Jun 2025).
• Complementarity with Low-Energy Observables: LFV decays and $Q = \pm 2$18–$Q = \pm 2$19 conversion, together with collider searches, constrain overlapping but not identical regions of the mass–coupling parameter space. In some scenarios (e.g., vanishing off-diagonal couplings), only high-energy colliders can probe the model (Geib et al., 2015, Crivellin et al., 2018).

6. Outlook and Future Directions

• High-Energy Colliders: Next-generation machines (HL-LHC, FCC-hh, muon/electron colliders) will either exclude or discover doubly charged scalars in the multi-TeV range, decisively testing type II seesaw models and related frameworks (Guedes et al., 31 Dec 2025, Belfkir et al., 2023, Ghosh et al., 1 Jun 2025).
• Multi-Scalar and Exotic Decays: More complex triplet-seesaw and extended multiplicity models predict distinctive cascade and multi-lepton signatures, which require tailored search strategies beyond canonical four-lepton analyses (Chaudhuri et al., 2013, Flacke et al., 2023).
• Long-Lived Particle Searches: Parameter regimes yielding long-lived $Q = \pm 2$20 (due to tiny $Q = \pm 2$21 or $Q = \pm 2$22) motivate continued development of high-$Q = \pm 2$23, timing, and dedicated LLP detector technologies (Akhmedov et al., 2024).
• Precision Measurements and Parity Violation: Low-energy experiments (MOLLER, $Q = \pm 2$24 conversion, $Q = \pm 2$25) serve as essential complementary probes, especially where high-energy production cross sections are suppressed (Dev et al., 2018, Geib et al., 2015).
• Model Discrimination and Coupling Extraction: Should a doubly charged scalar be observed, flavor-resolved event rates and kinematic distributions will allow distinction between models of different multiplet structure, extraction of leptonic couplings, and ultimately direct interrogation of the neutrino mass-generation mechanism (Chen et al., 2010, Aguila et al., 2013).

Doubly charged scalars thus provide a uniquely incisive probe of physics beyond the Standard Model, linking high-energy collider, precision low-energy, and flavor-violating phenomena within a common theoretical framework. The next decade of experiments is poised to either discover these states or place stringent limits on the charge structure of possible new scalar sectors.