F-Muon: Hypothetical Heavy Lepton

• F-muon is a hypothetical heavy charged lepton defined in extended Standard Models that addresses the muon g–2 discrepancy and flavor universality violations through new symmetry structures.
• It obtains mass via a Yukawa-like interaction with a new scalar field and contributes at one loop to muon g–2, requiring coupling strengths around 0.01–0.1.
• Collider searches and precision observables constrain its mass to be ≳200 GeV, making the F-muon a viable target for testing in current and future experiments.

An F-muon is a hypothetical heavy charged lepton, denoted $F_\mu$, introduced in extensions of the Standard Model that address the observed anomalies in the muon sector, such as the muon anomalous magnetic moment (muon $g$–2) and lepton flavor universality violations in $B$-meson decays. These models, such as the one proposed by Dhargyal (Dhargyal, 2017), embed $F_\mu$ in a new symmetry structure and particle content designed to resolve both flavor anomalies and small neutrino masses, while satisfying collider, electroweak, and flavor constraints. The F-muon distinctively couples to Standard Model muons through new Yukawa and gauge interactions, and participates in loop diagrams generating the required deviations in precision muon observables.

1. Field Content and Charge Assignments

The $F_\mu$ is part of a pair of vector-like heavy charged leptons, $(F_{\mu L}, F_{\mu R})$, both of which are color-singlets and isosinglets. Its quantum numbers under the Standard Model and the new $U(1)_F$ family symmetry, with discrete $\mathbb{Z}_2$ symmetry, are:

• $F_{\mu L}$: $(1, 1, -1 ; -1)_{-}$
• $g$0: $g$1 where the numbers denote representations under $g$2. All Standard Model fields are $g$3-even and uncharged under $g$4 (Dhargyal, 2017).

A new scalar field $g$5 is introduced to break $g$6 and generate masses for both the F-muon and the new $g$7 gauge boson ($g$8). The discrete $g$9 ensures the stability and decay pattern of new fields.

2. Mass Generation Mechanism

The mass of the F-muon arises from a renormalizable Yukawa-like interaction involving the $B$0-breaking scalar:

$B$1

Once $B$2 acquires a vacuum expectation value,

$B$3

the F-muon obtains a Dirac mass,

$B$4

This mechanism is analogous to the Higgs mechanism for Standard Model fermion masses, but is confined to the new sector (Dhargyal, 2017).

3. Gauge and Yukawa Interactions

F-muon's left-handed component couples to the $B$5 gauge boson via the covariant derivative:

$B$6

with $B$7. This yields the interaction term:

$B$8

and, after chiral recombination,

$B$9

The F-muon additionally couples to the Standard Model muon via the inert-Higgs doublet $F_\mu$0:

$F_\mu$1

where $F_\mu$2 is the SM lepton doublet. This interaction is central to generating loop corrections to $F_\mu$3 and $F_\mu$4 processes (Dhargyal, 2017).

4. Contributions to Muon $F_\mu$5–2 and Flavor Physics

The F-muon's dominant phenomenological impact is at one loop, correcting the muon anomalous magnetic moment via:

$F_\mu$6

with loop function:

$F_\mu$7

In the limit $F_\mu$8:

$F_\mu$9

Matching the observed discrepancy $F_\mu$0 for $F_\mu$1–$F_\mu$2 GeV requires $F_\mu$3–$F_\mu$4 (Dhargyal, 2017).

In $F_\mu$5 transitions, F-muon and the new leptoquark $F_\mu$6 appear in box diagrams, generating new effective operators:

$F_\mu$7

consistent with global fits to $F_\mu$8, $F_\mu$9, and related observables at $(F_{\mu L}, F_{\mu R})$0 (Dhargyal, 2017).

5. Experimental and Theoretical Constraints

Collider searches: Because $(F_{\mu L}, F_{\mu R})$1 is odd under $(F_{\mu L}, F_{\mu R})$2 and decays via $(F_{\mu L}, F_{\mu R})$3 with prompt width, direct LHC vector-like lepton searches constrain $(F_{\mu L}, F_{\mu R})$4 GeV, weaker than bounds for stable heavy leptons (excluded up to $(F_{\mu L}, F_{\mu R})$5 GeV).

Electroweak precision: $(F_{\mu L}, F_{\mu R})$6 is an SU(2) singlet, so its contributions to $(F_{\mu L}, F_{\mu R})$7 and $(F_{\mu L}, F_{\mu R})$8 are negligible; corrections to $(F_{\mu L}, F_{\mu R})$9 are $U(1)_F$0, well within uncertainties.

$U(1)_F$1 searches: The new $U(1)_F$2 gauge boson $U(1)_F$3 predominantly couples to F-muons; with $U(1)_F$4 TeV, it is consistent with collider constraints.

Flavor: $U(1)_F$5 symmetry forbids tree-level $U(1)_F$6–SM lepton mixing, suppressing lepton flavor violating (LFV) decays.

Dark matter and neutrino masses: The scalar $U(1)_F$7, in particular its neutral component $U(1)_F$8, is a dark matter candidate, though a small relic density is predicted for viable $U(1)_F$9. Neutrino mass is generated radiatively via the addition of heavy Majorana singlets (scotogenic mechanism).

6. Phenomenological Consequences and Prospects

• Low-energy flavor data: Further measurement of muon $\mathbb{Z}_2$0–2 (FNAL E989), $\mathbb{Z}_2$1, $\mathbb{Z}_2$2, and related observables at Belle II and LHCb Upgrade will probe the parameter space where $\mathbb{Z}_2$3 explains current anomalies.
• Collider signatures: Pair production $\mathbb{Z}_2$4 (via Drell–Yan or $\mathbb{Z}_2$5 exchange), with decays $\mathbb{Z}_2$6 and $\mathbb{Z}_2$7, yields $\mathbb{Z}_2$8. Associated production with leptoquarks provides $\mathbb{Z}_2$9 + jet + MET signatures.
• Discrimination and exclusion: The model is testable by the next generation of LHC multi-muon + MET searches up to mass scales of several hundred GeV. Improved $F_{\mu L}$0 and $F_{\mu L}$1 measurements will also constrain the viability of $F_{\mu L}$2.

7. Summary Table of F-muon Key Properties

Property	Value/Description	Source
SM quantum #'s	$F_{\mu L}$3, SU(3)$F_{\mu L}$4, SU(2)$F_{\mu L}$5, U(1)$F_{\mu L}$6	(Dhargyal, 2017)
$F_{\mu L}$7 charge	$F_{\mu L}$8 (left), $F_{\mu L}$9 (right)	(Dhargyal, 2017)
$(1, 1, -1 ; -1)_{-}$0	odd	(Dhargyal, 2017)
Mass	$(1, 1, -1 ; -1)_{-}$1, $(1, 1, -1 ; -1)_{-}$2 GeV	(Dhargyal, 2017)
Dominant decays	$(1, 1, -1 ; -1)_{-}$3 (prompt)	(Dhargyal, 2017)
Collider bounds	$(1, 1, -1 ; -1)_{-}$4 GeV (prompt), $(1, 1, -1 ; -1)_{-}$5 GeV (stable)	(Dhargyal, 2017)
$(1, 1, -1 ; -1)_{-}$6 role	1-loop, $(1, 1, -1 ; -1)_{-}$7–$(1, 1, -1 ; -1)_{-}$8 for $(1, 1, -1 ; -1)_{-}$9	(Dhargyal, 2017)
$g$00	$g$01	(Dhargyal, 2017)

The F-muon provides a testable mechanism for linking several anomalies in the muon sector and flavor observables, with distinct experimental signatures and constrained parameter space. Ongoing and future experiments in both low-energy flavor physics and high-energy colliders are positioned to fully probe or exclude its existence (Dhargyal, 2017).