Charged Lepton Flavour Violating Transitions

• Charged lepton flavour violating transitions are rare processes where a charged lepton changes flavor, providing a clear signal of new physics.
• They are systematically described using effective field theory, with key contributions from dipole, four-lepton, and semileptonic operators mediating decays like μ→eγ and μ→3e.
• Current experimental searches measure branching ratios and angular observables to set stringent new physics limits, probing scales up to 10⁵ TeV.

Charged lepton flavour violating (cLFV) transitions are processes in which a charged lepton (such as the muon, electron, or tau) changes flavour, i.e., a muon converts to an electron, or a tau to a muon, without emission of associated neutrinos. Such transitions are forbidden at any observable level in the Standard Model (SM) with massless neutrinos, and remain utterly negligible even after incorporating finite neutrino masses (e.g., $\mathrm{BR}(\mu\to e\gamma)\lesssim 10^{-54}$). As a result, any observation of cLFV constitutes incontrovertible evidence for new physics (NP). Searches for cLFV — especially in the muon sector — reach energy scales far beyond the direct kinematic frontier and test mechanisms of neutrino mass generation, the origin of flavour, and the existence of new heavy mediators (Davidson et al., 2022, Aoki et al., 28 Mar 2025, Abada, 2011).

1. Operator Structure and Theoretical Framework

The effective field theory (EFT) formalism underpins the paper of cLFV. Charged lepton flavour violating transitions are described by a tower of non-renormalisable local operators of mass dimension five and six, suppressed by the appropriate powers of a high mass scale, $\Lambda$. The most relevant operators are:

• Dipole Operator (dimension 5):

$$\mathcal{L}_\text{dipole} = \frac{e}{\Lambda_D^2}\, \bar{\ell}_j \sigma^{\mu\nu} P_{L,R} \ell_i F_{\mu\nu} + \mathrm{h.c.}$$

This operator directly mediates $\ell_i \to \ell_j\gamma$.

• Four-lepton Operators (dimension 6):

$$\mathcal{L}_\text{4L} = \frac{C_{LL}}{\Lambda_{4L}^2} \bar{\ell}_i \gamma^\mu P_L \ell_j\ \bar{\ell}_k \gamma_\mu P_L \ell_l + \cdots$$

These produce purely leptonic three-body decays like $\ell_i \to 3\ell_j$ at tree level.

• Semileptonic Operators (dimension 6):

$$\mathcal{L}_\text{SL} = \frac{C_{LQ}}{\Lambda_{LQ}^2} \bar{\ell}_i \gamma^\mu P_L \ell_j\ \bar{q}_k \gamma_\mu P_{L,R} q_l + \cdots$$

These mediate cLFV transitions involving hadrons, notably coherent $\mu\to e$ conversion in nuclei (Davidson et al., 2022, Feldmann, 2011, Chattopadhyay et al., 17 Jul 2025).

The full set of SMEFT $d=6$ 2-quark–2-lepton operators contributing to cLFV, as well as their matching onto low-energy LEFT coefficients, is systematically tabulated in (Chattopadhyay et al., 17 Jul 2025).

2. Benchmark Processes and Observables

The "golden channels" for cLFV searches are those that exhibit clean experimental signatures and minimal SM background:

Channel	Observable	Present Limit	Future Sensitivity	Key Experiment(s)
$\mu\to e\gamma$	$\mathcal{B}(\mu\to e\gamma)$	$<4.2\times 10^{-13}$ [MEG]	$6\times 10^{-14}$ [MEG II]	MEG II
$\mu\to 3e$	$\mathcal{B}(\mu\to 3e)$	$<1.0\times 10^{-12}$ [SINDRUM]	$10^{-15}-10^{-16}$ [Mu3e]	Mu3e
$\mu N\to e N$	$R_{\mu e}(N)$	$<7\times 10^{-13}$ (Au) [SINDRUM II]	$10^{-17}-10^{-18}$ [Muon facilities]	COMET, Mu2e
$\tau\to\mu\gamma$	$\mathcal{B}(\tau\to\mu\gamma)$	$<4.4\times 10^{-8}$ [BABAR/Belle]	$10^{-9}$ [Belle II]	Belle II
$\tau\to 3\mu$	$\mathcal{B}(\tau\to 3\mu)$	$<2.1\times 10^{-8}$	$10^{-9}$ [Belle II]	Belle II

Theoretical expressions for the branching ratios in terms of Wilson coefficients, neglecting lepton masses in the final state where appropriate, are (c.f. (Davidson et al., 2022, Calibbi et al., 2017)):

• Radiative decay:

$$\mathcal{B}(\ell_\beta \to \ell_\alpha \gamma) = \frac{3\alpha}{8\pi}(|C_{D,L}|^2 + |C_{D,R}|^2)$$

• Three-body decay:

$$\mathcal{B}(\ell_\beta \to 3\ell_\alpha ) \simeq \frac{1}{G_F^2\Lambda^4} \sum_{X,Y} |C_{V,XY}|^2 + \frac{\alpha}{3\pi} \left(\ln \frac{m_{\ell_\beta}^2}{m_{\ell_\alpha}^2} - \frac{11}{4}\right)(|C_{D,L}|^2+|C_{D,R}|^2)$$

• Coherent $\mu\to e$ conversion:

$$R_{\mu e}^N \simeq \frac{4G_F^2 m_\mu^5}{\Gamma_\text{capt}} \left| C_{D} D_N + \sum_{N=p,n}C_V^N V_N^N + ... \right|^2$$

Here, $C_D$ and $C_V$ denote Wilson coefficients of dipole and vector operators, and $D_N, V_N^N$ are nuclear overlap integrals (Signorelli, 2013, Davidson et al., 2022).

3. Sources of cLFV in Beyond Standard Model Theories

Charged lepton flavour violation arises naturally in a wide range of NP scenarios, often linked to the mechanism of neutrino mass generation:

a) Seesaw Models

• In type-I and type-III seesaws, heavy fermions generate $d=6$ operators $\sim (Y_\nu^\dagger Y_\nu)/M_N^2$, but cLFV rates scale as $(m_\nu/v^2)^2$ and are GIM suppressed for large $M_N$, rendering them unobservable unless $M_N \lesssim$ TeV (Abada, 2011, Romeri et al., 2017, Urquía-Calderón et al., 25 Nov 2025).
• In low-scale/inverse seesaw realizations, large mixings and/or approximate lepton-number symmetries can yield observable rates (Ilakovac et al., 2012, Romeri et al., 2017), with correlated signatures in radiative and three-body channels.

b) Supersymmetry

• Off-diagonal slepton mass insertions from RG running, $\sim (h_\nu^\dagger h_\nu)_{ij}\ln(M_{GUT}/M_N)$, drive dipole transitions via $\tilde{\ell}$–gaugino loops:

$$\mathcal{B}(\mu\to e\gamma)\sim \alpha^3 \tan^2\beta / (G_F^2 m_{\rm SUSY}^8) |(\delta m_{\tilde{L}}^2)_{ij}|^2$$

(Ilakovac et al., 2012, Feldmann, 2011).

• R-parity violating couplings and non-minimal Higgs contents can generate sizable four-fermion operators.

c) Scalar Triplet (Type-II Seesaw)

• No GIM suppression: Yukawa couplings $Y_\Delta$ can be $\mathcal{O}(1)$ for triplet masses at the TeV scale. Tensor and scalar operators can dominate $\mu\to 3e$ and $\mu N\to e N$, breaking dipole-dominated relations (Feldmann, 2011, Abada, 2011).

d) Leptoquarks and $Z'$ Models

• Tree-level 2-quark–2-lepton operators contribute directly to $\mu N\to e N$, meson decays, and semileptonic $\tau$ modes. Present $\mu N\to e N$ data constrain generic leptoquark or $Z'$ mass/coupling combinations above $10^3$–$10^5$ TeV for $\mathcal{O}(1)$ couplings (Chattopadhyay et al., 17 Jul 2025, Davidson et al., 2022).

e) Radiative/Radiative-Seesaw Models

• Radiative models involving additional scalars and/or heavy fermions (e.g., color octets (Li et al., 2016), doublets (Dūdėnas et al., 2022)) induce cLFV via loop diagrams. Future $\mu\to 3e$ and $\mu N\to e N$ measurements are expected to probe or exclude much of their viable parameter space.

4. Phenomenology: Correlated Patterns, CP Phases, and Angular Observables

Correlations among cLFV observables are sensitive to the nature of the NP mediators and the underlying flavour structure. For models dominated by a single dipole operator (as in many supersymmetry scenarios), approximate relations hold (Abada, 2011, Abada et al., 2021): $$\frac{\mathcal{B}(\mu\to 3e)}{\mathcal{B}(\mu\to e\gamma)} \approx \alpha \left[\ln\left(\frac{m_\mu^2}{m_e^2}\right) - \frac{11}{4} \right] \approx 6\times 10^{-3}$$

$$\frac{R_{\mu e}}{\mathcal{B}(\mu\to e\gamma)} \sim 0.1-1$$

However, the presence of sizable four-fermion, $Z$-penguin, or box contributions (as can arise in low-scale seesaws or type II models) can significantly break these correlations (Davidson et al., 2022, Kriewald et al., 2021).

Leptonic CP-violating phases (Dirac and Majorana) introduced via neutrino mixing with heavy neutral leptons can induce strong interference effects, leading to large enhancements or suppressions—even complete cancellations—of cLFV rates in selected channels (Abada et al., 2021, Kriewald et al., 2021, Darricau et al., 4 Dec 2025). Nontrivial CP phase structure may decorrelate signals in $\mu\to e\gamma$, $\mu\to 3e$, and $\mu N\to e N$, shifting the naive scaling by orders of magnitude.

Angular and polarization observables in three-body cLFV decays, such as parity-odd ($A_P$), transverse ($A_T$), and forward-backward asymmetries, are sensitive to the relative size and CP-phase structure of the dipole, penguin, and box amplitudes. Typical values up to several tens of percent are predicted in HNL scenarios, with specific patterns predicted for the allowed parameter space (Darricau et al., 4 Dec 2025). Measurement of nonzero T-odd asymmetry $A_T$ would signal CP-violating interference, indicative of loop-induced CP phases.

5. Experimental Techniques and Current Limits

Muon sector:

• $\mu\to e\gamma$ (MEG/MEG II): Detects coincidences of back-to-back positron and photon (each at ~53 MeV) from stopped $\mu^+$. Backgrounds dominated by accidental overlay events and radiative muon decay. Current bound: $\mathcal{B} < 3.1\times 10^{-13}$; MEG II aims for $6\times 10^{-14}$ (Aoki et al., 28 Mar 2025, Signorelli, 2013).
• $\mu\to 3e$ (Mu3e): Requires excellent vertexing, timing, and low-mass tracking to suppress accidental backgrounds from three overlapping Michel positrons. Mu3e targets $10^{-15}$ (Phase I) to $10^{-16}$ (Phase II) sensitivity.
• $\mu N\to e N$ (COMET, Mu2e): Search for a monoenergetic electron at 105 MeV in coincidence with the delayed muon-capture window in the presence of a stopped negative muon beam. Backgrounds include decay-in-orbit and radiative muon capture. Tight constraints on timing, momentum, and cosmic/beam-induced backgrounds. Sensitivity targets $10^{-15}$ to $10^{-18}$ (Aoki et al., 28 Mar 2025).

Tau sector:

• Belle II, LHCb are expected to probe $\tau\to\ell\gamma \sim 10^{-9}$ and $\tau\to 3\ell \sim 10^{-10}$. Semileptonic modes (e.g., $\tau \to \ell \rho$) are also of increasing interest (Urquía-Calderón et al., 25 Nov 2025).

Heavy bosons:

• $Z\to\ell_i\ell_j$ and $H\to\ell_i\ell_j$ decays are searched for at high-luminosity $e^+e^-$ colliders and the LHC. Present bounds: $\mathcal{B}(Z\to e\mu) < 7.5\times 10^{-7}$, $\mathcal{B}(H\to\mu\tau) < 2.5\times 10^{-3}$ (Davidson et al., 2022).

6. Current and Future Sensitivity to New Physics Scales

The present experimental constraints translate into exceedingly high lower limits on the effective scale $\Lambda$ of new flavour physics, depending on operator structure (Davidson et al., 2022, Chattopadhyay et al., 17 Jul 2025):

Observable	$\Lambda$ reach for $C=O(1)$
Dipole ($\mu\to e\gamma$)	$>10^4$ TeV ($>10^5$ TeV at future)
Four-fermion ($\mu\to 3e$)	$>10^3$ TeV
Semileptonic (mu-e conversion)	$>10^4$–$10^5$ TeV

Notably, $\mu N\to e N$ conversion, due to coherent enhancement, probes the four-fermion 2q–2$\ell$ operators most efficiently; present limits exclude many $\mathcal{O}(1)$-coupling NP mediators below $10^4$–$10^5$ TeV (Chattopadhyay et al., 17 Jul 2025). Future upgrades (AMF, Mu2e-II, PSI HIMB, COMET-II) will increase reach by another order of magnitude or more.

Tau-sector cLFV modes are less sensitive, but Belle II is predicted to probe $\Lambda \gtrsim 10^4$ TeV in $\tau\to\ell\rho$ and related channels, with semileptonic transitions competitive with leptonic channels in many seesaw scenarios (Urquía-Calderón et al., 25 Nov 2025).

Correlated measurements across $\mu\to e\gamma$, $\mu\to 3e$, $\mu N\to e N$, and tau/meson decays, together with angular observables and sensitivity to CP-odd phases, are identified as essential to discriminating among candidate NP mechanisms (Davidson et al., 2022, Darricau et al., 4 Dec 2025, Kriewald et al., 2021).

7. Open Challenges and Outlook

Key unresolved theoretical and phenomenological issues include:

• Operator mixing and RGE: QED/QCD running induces mixing among four-fermion, dipole, and tensor operators at the percent level, necessitating global, multi-operator fits (Davidson et al., 2022, Chattopadhyay et al., 17 Jul 2025).
• Nuclear uncertainties: Dominant in $\mu N\to e N$ conversion predictions, especially for spin-dependent operators and heavy targets; using multiple targets can disentangle operator contributions (Davidson et al., 2022).
• CP-phase reconstruction: Extraction of leptonic Dirac/Majorana phases via rate/rate and asymmetry/asymmetry correlations remains an ambitious goal (Abada et al., 2021, Darricau et al., 4 Dec 2025).
• Light new mediators: Scenarios involving light $Z'$, ALPs, or light scalar/pseudoscalar bosons require dedicated kinematical searches and analysis outside the standard SMEFT/LEFT framework (Davidson et al., 2022).
• Interplay with other probes: cLFV searches must be interpreted alongside limits from EDMs, neutrino oscillations, lepton universality tests, and rare meson decays to construct a coherent flavour-dynamics picture (Davidson et al., 2022).

In conclusion, charged lepton flavour violating transitions constitute one of the most sensitive and cleanest probes for physics beyond the Standard Model. The combination of upcoming experimental improvements, global analysis incorporating operator mixing and correlations, and detailed phenomenological work on angular observables and CP-violating signatures is poised to either discover cLFV or push the scale of lepton-flavour-violating new physics well beyond $10^5$ TeV (Davidson et al., 2022, Aoki et al., 28 Mar 2025, Chattopadhyay et al., 17 Jul 2025, Darricau et al., 4 Dec 2025).