---
title: 'F-Muon: Hypothetical Heavy Lepton'
url: https://www.emergentmind.com/topics/f-muon
type: topic
---

# F-Muon: Hypothetical Heavy Lepton

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 [1711.09772], 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:
- $F_{\mu L}$: $(1, 1, -1 ; -1)_{-}$
- $F_{\mu R}$: $(1, 1, -1 ;  0)_{-}$
where the numbers denote representations under $(\mathrm{SU}(3)_C, \mathrm{SU}(2)_L, \mathrm{U}(1)_Y; \mathrm{U}(1)_F)_{\mathbb{Z}_2}$. All Standard Model fields are $\mathbb{Z}_2$-even and uncharged under $U(1)_F$ [1711.09772].

A new scalar field $\phi\sim(1, 1, 0; +1)_{+}$ is introduced to break $U(1)_F$ and generate masses for both the F-muon and the new $U(1)_F$ gauge boson ($X_\mu$). The discrete $\mathbb{Z}_2$ 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 $U(1)_F$-breaking scalar:
$$
\mathcal{L}_\mathrm{Yuk} \supset y_\phi\, \overline{F_{\mu L}}\,F_{\mu R}\,\phi + \text{h.c.}
$$
Once $\phi$ acquires a vacuum expectation value,
$$
\langle\phi\rangle = v_\phi,
$$
the F-muon obtains a Dirac mass,
$$
m_{F_\mu} = y_\phi\, v_\phi.
$$
This mechanism is analogous to the Higgs mechanism for Standard Model fermion masses, but is confined to the new sector [1711.09772].

## 3. Gauge and Yukawa Interactions

F-muon's left-handed component couples to the $U(1)_F$ gauge boson via the covariant derivative:
$$
D_\mu F_{\mu L} = \left(\partial_\mu + i g_F n_\mu X_\mu\right) F_{\mu L}
$$
with $n_\mu = -1$. This yields the interaction term:
$$
\mathcal{L}_\mathrm{int} \supset g_F n_\mu\, \overline{F_\mu}\gamma^\mu P_L F_\mu\, X_\mu
$$
and, after chiral recombination,
$$
\mathcal{L}_\mathrm{int} \simeq g_F q_{F_\mu} \overline{F_\mu}\gamma^\mu F_\mu\, X_\mu,\qquad q_{F_\mu} = -1.
$$

The F-muon additionally couples to the Standard Model muon via the inert-Higgs doublet $\phi_l$:
$$
\mathcal{L}_\mathrm{Yuk} \supset y_l\, \overline{L_\mu} P_R F_\mu\, \phi_l + \mathrm{h.c.}
$$
where $L_\mu$ is the SM lepton doublet. This interaction is central to generating loop corrections to $(g-2)_\mu$ and $b \to s \mu^+\mu^-$ processes [1711.09772].

## 4. Contributions to Muon $g$–2 and Flavor Physics

The F-muon's dominant phenomenological impact is at one loop, correcting the muon anomalous magnetic moment via:
$$
\Delta a_\mu = \frac{|y_l|^2}{8\pi^2}\frac{m_\mu^2}{m_{F_\mu}^2} F\!\left(\frac{m_{\phi_l}^2}{m_{F_\mu}^2}\right)
$$
with loop function:
$$
F(r) = \int_0^1 dz\, \frac{z^2(1 - z)}{1 - z + r z}
= \frac{1 - 6r + 3r^2 + 2r^3 - 6 r^2 \ln r}{6 (1-r)^4}
$$
In the limit $m_{\phi_l}\simeq m_{F_\mu}\gg m_\mu$:
$$
\Delta a_\mu \simeq \frac{m_\mu^2 |y_l|^2}{12\pi^2 m_{F_\mu}^2}
$$
Matching the observed discrepancy $\Delta a_\mu \sim 2.8 \times 10^{-9}$ for $m_{F_\mu}\sim 150$–$200$ GeV requires $|y_l| \sim 10^{-2}$–$10^{-1}$ [1711.09772].

In $b\to s\mu^+\mu^-$ transitions, F-muon and the new leptoquark $\phi_Q$ appear in box diagrams, generating new effective operators:
$$
C_9^\text{NP} = -C_{10}^\text{NP} \simeq -0.6
$$
consistent with global fits to $R_K$, $R_{K^*}$, and related observables at $1\,\sigma$ [1711.09772].

## 5. Experimental and Theoretical Constraints

**Collider searches:** Because $F_\mu$ is odd under $\mathbb{Z}_2$ and decays via $F_\mu\to\mu\,\phi_l$ with prompt width, direct LHC vector-like lepton searches constrain $m_{F_\mu}\gtrsim 200$ GeV, weaker than bounds for stable heavy leptons (excluded up to $620$ GeV). 

**Electroweak precision:** $F_\mu$ is an SU(2) singlet, so its contributions to $S$ and $T$ are negligible; corrections to $Z\to\mu\bar{\mu}$ are $<10^{-5}$, well within uncertainties.

**$Z'$ searches:** The new $U(1)_F$ gauge boson $X_\mu$ predominantly couples to F-muons; with $m_X / g_F \gtrsim \mathcal{O}(4)$ TeV, it is consistent with collider constraints.

**Flavor:** $\mathbb{Z}_2$ symmetry forbids tree-level $F_\mu$–SM lepton mixing, suppressing lepton flavor violating (LFV) decays.

**Dark matter and neutrino masses:** The scalar $\phi_l$, in particular its neutral component $H^0$, is a dark matter candidate, though a small relic density is predicted for viable $y_l$. 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 $g$–2 (FNAL E989), $R_K$, $P_5'$, and related observables at Belle II and LHCb Upgrade will probe the parameter space where $F_\mu$ explains current anomalies.
- **Collider signatures:** Pair production $pp\to F_\mu^+ F_\mu^-$ (via Drell–Yan or $X_\mu$ exchange), with decays $F_\mu\to\mu \phi_l$ and $\phi_l\to H^0 + \text{soft}$, yields $\mu^+\mu^- + \not{E}_T$. Associated production with leptoquarks provides $\mu$ + 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 $b\to s\mu^+\mu^-$ and $(g-2)_\mu$ measurements will also constrain the viability of $F_\mu$.

## 7. Summary Table of F-muon Key Properties

| Property         | Value/Description                                           | Source      |
|------------------|------------------------------------------------------------|-------------|
| SM quantum #'s   | $(1, 1, -1)$, SU(3)$_C$, SU(2)$_L$, U(1)$_Y$               | [1711.09772]|
| $U(1)_F$ charge  | $-1$ (left), $0$ (right)                                   | [1711.09772]|
| $\mathbb{Z}_2$   | odd                                                        | [1711.09772]|
| Mass             | $m_{F_\mu}=y_\phi v_\phi$, $m_{F_\mu}\gtrsim 200$ GeV      | [1711.09772]|
| Dominant decays  | $F_\mu\rightarrow\mu\,\phi_l$ (prompt)                     | [1711.09772]|
| Collider bounds  | $m_{F_\mu}\gtrsim200$ GeV (prompt), $>620$ GeV (stable)    | [1711.09772]|
| $g-2$ role       | 1-loop, $|y_l|=0.01$–$0.1$ for $\Delta a_\mu\sim2.8\cdot10^{-9}$| [1711.09772]|
| $b\to s\mu^+\mu^-$ | $C_9^\text{NP}=-C_{10}^\text{NP}\simeq-0.6$                | [1711.09772]|

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 [1711.09772].

Source: https://www.emergentmind.com/topics/f-muon