Charged-Lepton Flavor Violation

Charged-lepton flavor violation (CLFV) is organized below the electroweak scale by dipole, four-lepton, and semileptonic operators. Radiative decay selects dipoles directly; three-body decay contains dipole conversion, contact terms, and interference; coherent nuclear conversion adds target-dependent scalar and vector responses. Correlations among these processes are model independent only after the operator basis, normalization, scale, running, matching, nuclear inputs, and restricted coefficient hypothesis are declared.

Required background. The Fermi Limit of Weak Interactions supplies the low-energy weak normalization. Flavor, Hermiticity, and CP Bookkeeping supplies flavor-index, conjugation, and Wilson-coefficient conventions.

Helpful background. Electroweak Currents in Few-Body Systems supplies the nuclear-current and shared-covariance interface for conversion.

Low-energy CLFV operators

For a representative $\mu\to e$ transition at a scale near $m_\mu$, write QED- and QCD-invariant operators as

$$\begin{aligned} \mathcal L_{\rm CLFV}=\frac{1}{\Lambda^2}\Bigg\{& e m_\mu\sum_{X=L,R}C_{D,X} (\overline e\sigma^{\mu\nu}P_X\mu)F_{\mu\nu}\\ &+\sum_{f,X,Y}C_{V,XY}^{f} (\overline e\gamma^\mu P_X\mu) (\overline f\gamma_\mu P_Y f)\\ &+\sum_{f,X,Y}C_{S,XY}^{f} (\overline eP_X\mu)(\overline fP_Yf)\\ &+\sum_{f,X}C_{T,X}^{f} (\overline e\sigma^{\mu\nu}P_X\mu) (\overline f\sigma_{\mu\nu}P_Xf) +\text{h.c.}\Bigg\}. \end{aligned}$$

Here the coefficients are dimensionless in the displayed normalization, the explicit $m_\mu$ makes the dipole a dimension-six interaction, and $f$ ranges over the light charged leptons and quarks allowed by the process. Other normalizations absorb $e$, $m_\mu$, $G_F$, or $\Lambda^{-2}$ into $C$; a numerical coefficient is meaningless without that map.

Above the electroweak scale this is not a complete gauge-invariant basis. The photon dipole descends from electroweak dipoles, and four-fermion coefficients descend from SMEFT operators with correlated charged and neutral components. Matching at $m_W$, threshold matching at heavy-quark masses, and QED/QCD running then produce the low-energy vector $C(\mu)$. The complete low-energy basis and one-loop anomalous dimensions are given by Jenkins, Manohar, and Stoffer 2018, §§ 2–5.

What each process measures

At a common low scale, the three classic muon channels probe different directions in coefficient space:

Observable	Leading operator sensitivity	Essential nuisance or qualification
$\mu\to e\gamma$	On-shell left- and right-chiral dipoles	Muon polarization and photon/electron angular acceptance if chirality is inferred
$\mu\to e\overline ee$	Four-lepton scalar/vector operators, an off-shell dipole photon, and their interference	Identical-particle phase space, cuts, polarization, and differential information
$\mu^-A\to e^-A$	Dipole plus semileptonic scalar/vector operators after nucleon matching	Isotope, capture normalization, nuclear density, overlap integrals, and nucleon form factors

Schematically,

$$\begin{aligned} \Gamma_{e\gamma}&\propto \frac{m_\mu^5}{\Lambda^4} (|C_{D,L}|^2+|C_{D,R}|^2),\\ \Gamma_{3e}&\propto\frac{m_\mu^5}{\Lambda^4} \left(|C_{4\ell}|^2+\alpha|C_D|^2 +2\operatorname{Re}[C_D C_{4\ell}^*]\,k_{\rm int}\right),\\ \Gamma_{\rm conv}^{A}&\propto\frac{m_\mu^5}{\Lambda^4} \left(|\mathcal A_L^A|^2+|\mathcal A_R^A|^2\right). \end{aligned}$$

The constants suppressed by $\propto$ depend on the displayed operator normalization. A conversion amplitude has the structure

$$\mathcal A_X^A=D_A C_{D,X} +\sum_{N=p,n}\left( V_A^N\widetilde C_{V,X}^N +S_A^N\widetilde C_{S,X}^N \right)+\cdots ,$$

where quark coefficients have first been matched to nucleon coefficients and $D_A,V_A^N,S_A^N$ are target-dependent overlaps. The ellipsis can contain spin-dependent, derivative, and higher-body responses. Changing target changes the observable direction; it does not simply repeat the same bound with more events. The operator and target discrimination problem is worked out in Cirigliano et al. 2009, §§ II–IV.

A dipole-dominance checkpoint

Suppose that at $m_\mu$ only one dipole chirality is appreciable and neglect non-dipole contact terms. The photon in $\mu\to e\gamma$ is on shell, while internal conversion produces $\mu\to e\gamma^*\to e\overline ee$. In the limit $m_e\ll m_\mu$,

$$\frac{\operatorname{Br}(\mu\to e\overline ee)} {\operatorname{Br}(\mu\to e\gamma)} \simeq \frac{\alpha}{3\pi} \left[ln\!\left(\frac{m_\mu^2}{m_e^2}\right)-\frac{11}{4}\right].$$

The logarithm comes from the nearly collinear virtual photon. This relation is a diagnostic of the restricted low-energy hypothesis, not a universal CLFV prediction. Four-lepton coefficients, interference, phase-space cuts, or running-induced contacts change it. A systematic operator treatment and the conditions behind this ratio are reviewed by Calibbi and Signorelli 2018, §§ 3–4, pp. 81–94.

The same caution applies to conversion-to-radiative ratios: even under dipole dominance they contain the isotope-specific overlap and capture rate. Under scalar or vector dominance they can change qualitatively.

Matching and running workflow

A reproducible CLFV analysis follows a fixed sequence:

1. Declare the ultraviolet statement. Name the SMEFT or mediator basis, renormalization scheme, input scale, flavor indices, and which coefficients may be nonzero together.
2. Run and match to the electroweak scale. Operator mixing can generate a dipole from tensor or four-fermion structures; threshold and finite matching terms must use the same scheme.
3. Run in the low-energy theory. Evolve QED and QCD coefficients to a common hadronic or muon scale. Quote $C_i(\mu)$ and $\mu$ together.
4. Match quarks to hadrons and nuclei. Propagate scalar charges, vector charges, spin responses, overlap integrals, and their correlations for each isotope.
5. Construct amplitudes before rates. Sum coefficients that reach the same external state with their complex phases, then square. Adding operator rates discards physical interference.
6. Attach the likelihood. Preserve channel efficiencies, correlated normalization, backgrounds, physical boundaries, and the exact experimental release. Current numerical limits belong in a dated evidence record, not in the operator definition.

At any common scale an observable can be represented schematically as

$$\Gamma_a\propto \sum_h|\boldsymbol v_{a,h}^\dagger\boldsymbol C(\mu)|^2,$$

where $h$ labels noninterfering helicity or nuclear channels. Parallel $\boldsymbol v_a$ probe the same coefficient combination; misaligned vectors are complementary; directions orthogonal to every measured $\boldsymbol v_a$ remain flat. This exposes why one-operator-at-a-time limits cannot be combined into a complete multi-coefficient constraint Davidson 2021, §§ 2–3.

Standard Model null baseline

If the Standard Model is extended only by the observed light-neutrino masses and a unitary PMNS matrix, the leading radiative amplitude is GIM suppressed:

$$\operatorname{Br}(\mu\to e\gamma) \simeq\frac{3\alpha}{32\pi} \left| \sum_iU_{\mu i}^*U_{ei}\frac{m_i^2}{M_W^2} \right|^2.$$

Unitarity removes any mass-independent term, so only tiny mass-squared differences remain. This provides a theoretical null baseline for laboratory CLFV, but it is not an exact zero once neutrino masses are included. Additional light states, nonunitarity, or new interactions change the loop and matching assumptions; observing CLFV would establish physics beyond this minimal massive-neutrino baseline, not identify a unique mediator.

Independent checks and handoffs

• Dimensions and chirality: every term in $\mathcal L$ has dimension four; the explicit mass multiplying a dipole and the placement of $P_L,P_R$ are recorded.
• Hermiticity: include the flavor-reversed Hermitian conjugate without double counting a coefficient.
• Scale closure: differentiate the matched amplitude with respect to $\ln\mu$; running coefficients and matrix elements must cancel to the calculated order.
• GIM limit: equal neutrino masses make the minimal light-neutrino amplitude vanish by $\sum_iU_{\mu i}^*U_{ei}=0$.
• Interference: turning on two coefficients must reproduce constructive and destructive phase limits; an always-positive term-by-term sum fails.
• Target dependence: the same quark-level coefficient vector propagated to two nuclei must use distinct, correlated response inputs.
• Likelihood identity: rates, branching ratios, and conversion ratios use their declared lifetime or capture-rate denominators.

Generic high-scale basis construction and renormalization belong to Flavor, Hermiticity, and CP Bookkeeping. Nuclear responses continue through Electroweak Currents in Few-Body Systems. Versioned limits, projections, and model rankings belong to Effective Field Theory and Tests of the Standard Model.

Common pitfalls

Calling a dipole pattern model independent. The ratio above follows only after restricting the low-energy coefficient vector. A mediator model can generate several operators, and running can populate new directions even when matching starts sparse.

Comparing coefficients at different scales. A low-energy dipole and a weak-scale SMEFT coefficient are not the same number. Specify basis, scheme, thresholds, and evolution before drawing a correlation.

Treating nuclear conversion as a free-muon decay. The coherent amplitude contains nucleon matching, isotope overlaps, and capture normalization. Omitting their covariance can create a false target preference or false operator discrimination.

References

• Calibbi, Lorenzo, and Giovanni Signorelli. “Charged Lepton Flavour Violation: An Experimental and Theoretical Introduction.” Rivista del Nuovo Cimento 41 (2018): 71–174. DOI.
• Cirigliano, Vincenzo, Ryuichiro Kitano, Yasuhiro Okada, and Paula Tuzón. “On the Model Discriminating Power of $\mu\to e$ Conversion in Nuclei.” Physical Review D 80 (2009): 013002. DOI.
• Davidson, Sacha. “Completeness and Complementarity for $\mu\to e\gamma$, $\mu\to3e$, and $\mu A\to eA$.” Journal of High Energy Physics 2021, no. 2 (2021): 172. DOI.
• Jenkins, Elizabeth E., Aneesh V. Manohar, and Peter Stoffer. “Low-Energy Effective Field Theory below the Electroweak Scale: Anomalous Dimensions.” Journal of High Energy Physics 2018, no. 1 (2018): 084. DOI.