Charged Lepton Flavor Violation: The Other Window on the Seesaw

Neutrino oscillation proved a striking thing about the lepton sector: lepton flavor is not conserved. The number of muon-type leptons or electron-type leptons in a population changes as neutrinos oscillate, even though total lepton number remains conserved. The discovery, secured by Super-Kamiokande and SNO around the turn of the millennium, has had a curious downstream consequence that experimentalists have spent the past two decades chasing: if lepton flavor is broken in the neutral lepton sector, it must also be broken — at some level, by some loop diagram — in the charged lepton sector. A muon should be able to decay to an electron plus a photon. A muon bound in an atom should be able to convert to an electron without producing neutrinos to balance the flavor count. And yet, despite half a century of increasingly sensitive searches, no one has ever seen these processes happen.

The reason is that, in the minimally extended Standard Model with just the observed tiny neutrino masses, the predicted rates are unobservably small — branching ratios of order $10^{-54}$, fifty orders of magnitude below the experimental frontier. Any beyond-Standard-Model physics that introduces new sources of lepton flavor breaking, however, would lift the rates dramatically. The current generation of experiments — MEG II at PSI, Mu2e at Fermilab, COMET at J-PARC — are designed to push the sensitivity by orders of magnitude into a regime where many realistic models, including the seesaw mechanism with right-handed neutrinos at the grand-unification scale, predict observable signals.

This post is about why charged lepton flavor violation is such a clean probe, what each of the three big channels measures, and why a discovery would tell us something specific about the seesaw.

The accidental symmetries

The Standard Model Lagrangian, when written down with massless neutrinos and the experimentally measured quark and lepton fields, happens to conserve three separate flavor numbers: an electron number $L_e$, a muon number $L_{\mu }$, and a tau number $L_{\tau }$. These conservation laws are not imposed by gauge invariance or by any obvious symmetry of the construction; they emerge automatically because no Standard Model interaction couples leptons of different flavors. This makes them accidental symmetries: properties of the theory that follow from its structure rather than from a fundamental principle.

Neutrino oscillation breaks these symmetries in the neutral lepton sector by allowing a ${\nu }_{\mu }$ to turn into a ${\nu }_e$ during propagation. The flavor changes; only total lepton number $L=L_e+L_{\mu }+L_{\tau }$ is preserved. Adding neutrino masses to the Standard Model therefore unavoidably breaks the separate flavor symmetries at the level of the full quantum theory.

But what about a charged lepton-flavor-violating process such as

${\mu }^{+}\to e^{+}+\gamma ?$

This process changes $L_{\mu }$ by one unit and $L_e$ by one unit, in opposite directions, while preserving total lepton number. It is therefore allowed once neutrino masses are included — but only at the loop level, through a diagram in which a virtual W boson and a virtual neutrino-mass-eigenstate run around a loop and connect the muon to the electron with the emission of a photon. Computing this loop with the measured neutrino mixing parameters and mass-squared differences gives

$\mathrm{BR}(\mu \to e\gamma {)}^{\mathrm{SM}}\approx \frac{3\alpha }{32\pi }{\left|{\sum }_i{U}_{\mu i}^{\ast }{U}_{ei}\frac{m_i^2}{m_W^2}\right|}^2\approx 10^{-54}$

The suppression comes from the GIM mechanism — the sum cancels almost exactly between the mass eigenstates because the differences in their squared masses are minuscule compared to $m_W^2$. The result is a rate that no conceivable experiment will ever measure. Charged lepton flavor violation in the minimal Standard Model is, for all practical purposes, not seen.

Why beyond the Standard Model lifts the rate

What breaks the GIM cancellation is the introduction of new heavy particles that couple to the lepton sector with their own flavor structure. The simplest example is the Type I seesaw: heavy right-handed neutrinos $N_i$ with masses near $10^{14}$ GeV. Their contribution to the $\mu \to e\gamma$ loop replaces the suppression by light-neutrino masses with a suppression by the heavy masses, but the mixing matrix that enters the rate is different — controlled by the heavy-light mixing $U_{lh}$ rather than by the light PMNS structure — and the cancellation does not happen.

In supersymmetric extensions the squarks and sleptons run in the loop, and any flavor mixing between sleptons of different generations leads to potentially large $\mu \to e\gamma$ rates. Models with additional Higgs doublets, with leptoquarks, with vector-like leptons, or with composite leptons all add new diagrams that can lift the rate to within experimental reach.

The general lesson: charged lepton flavor violation is a clean probe because the Standard Model expectation is essentially zero, so any observation is unambiguous new physics. The trade-off is that interpreting a signal requires a model — different mechanisms produce different patterns of CLFV rates in different channels, and disentangling the underlying physics requires combining multiple measurements.

The three big channels

Three processes dominate the experimental landscape, each with its own technique and its own systematic.

The first is $\mu \to e\gamma$, the radiative decay. Experimentally, one stops positive muons in a thin target and searches for the back-to-back coincidence of a 52.8 MeV positron and a 52.8 MeV photon — exactly the kinematics of a two-body decay, with the muon at rest. The dominant background is the accidental coincidence of a positron from ordinary muon decay $\mu \to e\nu \bar{\nu }$ with a photon from radiative muon decay or positron annihilation. The state-of-the-art experiment is MEG II at the Paul Scherrer Institut, which began full data-taking in 2021. Its current limit on the branching ratio is at the level of $\mathrm{BR}(\mu \to e\gamma )<4\times 10^{-13}$, with sensitivity expected to reach $6\times 10^{-14}$ by the end of the decade.

The second is $\mu \to eee$, the three-body decay to three electrons. Experimentally this involves identifying three light tracks with a common vertex, which is harder than the two-body signature but cleaner against accidentals. The Mu3e experiment at PSI, with a thin pixel-detector system optimized for low-momentum particle tracking, aims for $\mathrm{BR}(\mu \to eee)<10^{-16}$ — a four-orders-of-magnitude improvement over the previous SINDRUM limit.

The third is coherent muon-to-electron conversion in the field of an atomic nucleus. A negative muon is captured into an atomic orbital around a nucleus, where it usually either decays $\mu \to e\nu \bar{\nu }$ in flight or is absorbed by the nucleus through a weak interaction. A new physics contribution could convert the muon into an electron without emitting neutrinos, leaving the nucleus intact. The signature is a single monoenergetic electron at an energy slightly below the muon mass, roughly 105 MeV for aluminum. Because the final state is a single particle and the nucleus carries no excitation, the signal is much cleaner than the multi-particle decays. The experiments pursuing this channel are Mu2e at Fermilab and COMET at J-PARC, both starting data-taking in the mid-2020s, aiming for branching-ratio sensitivities at the $10^{-17}$ level — about four orders of magnitude beyond the previous SINDRUM II limit.

Three channels, complementary information

The three channels do not measure the same thing. They probe different combinations of effective operators that mediate the flavor change. In the language of an effective field theory below the new-physics scale, the relevant operators are penguin-like dipole operators that produce $\mu \to e\gamma$, four-fermion contact operators that mediate $\mu \to eee$ directly, and a combination of dipole and four-fermion operators for $\mu \to e$ conversion. Different models produce different relative weights for these operators, so the ratio of CLFV rates in the three channels is diagnostic.

A few rough rules of thumb:

• If new physics is dominantly dipole-mediated (a photon penguin), then $\mathrm{BR}(\mu \to e\gamma )$ and $\mathrm{BR}(\mu \to e\mathrm{conv})$ are both large, with a roughly fixed ratio set by atomic-physics factors.
• If new physics is dominantly four-fermion contact (a tree-level exchange of a heavy mediator), $\mu \to eee$ and $\mu \to e\mathrm{conv}$ are enhanced relative to $\mu \to e\gamma$.
• If new physics involves Higgs penguins (a scalar mediator), the relative rates depend strongly on the Higgs sector structure.

So if any one channel shows a signal, the next step is to measure the others and reconstruct the underlying operator structure. This is why the experimental program is multi-pronged rather than picking the single most sensitive channel.

The connection to the seesaw

In the Type I seesaw, heavy right-handed neutrinos with masses $M_R$ couple to the light leptons through Yukawa couplings $Y_{\nu }$. These couplings generate the light neutrino masses via the seesaw relation but also contribute to CLFV through loop diagrams. The induced branching ratio scales as

$\mathrm{BR}(\mu \to e\gamma )\sim {\alpha }^3\frac{m_{\mu }^4}{m_W^4}{\left|{\sum }_i(Y_{\nu }{)}_{ei}^{\ast }(Y_{\nu }{)}_{\mu i}\frac{m_W^2}{M_{R,i}^2}\right|}^2$

For grand-unification-scale heavy neutrinos, $M_R\sim 10^{14}$ GeV, this gives branching ratios around $10^{-30}$ — still far below experimental reach. But for low-scale seesaw variants with $M_R$ in the TeV range, the rates can sit right at the current experimental frontier. Inverse seesaw, linear seesaw and other low-scale realizations populate exactly the parameter space being explored.

Supersymmetric seesaw models add an additional contribution: slepton-flavor mixing induced by the neutrino Yukawas through renormalization-group running between the GUT scale and the electroweak scale. The induced slepton off-diagonal entries produce $\mu \to e\gamma$ at rates comparable to the upcoming experimental sensitivity over a wide range of SUSY parameter space.

The general lesson is that CLFV is a window on physics far above the electroweak scale, complementary to the direct collider searches that probe weaker but lower-scale new physics. The two approaches feed back: a positive CLFV signal would tell colliders where to look; a null result at a given level of sensitivity excludes specific patterns of high-scale physics.

What a discovery would mean

A clear $\mu \to e\gamma$ signal at MEG II’s reach would be one of the most consequential discoveries in particle physics since the Higgs. Combined with a near-simultaneous measurement in $\mu \to e$ conversion, the ratio would pin down the dipole versus four-fermion character of the underlying physics. A full picture would require tau-sector CLFV measurements as well — $\tau \to \mu \gamma$, $\tau \to e\gamma$, and various three-body decays — to constrain the inter-generational flavor structure. The Belle II experiment at SuperKEKB is the dominant probe in the tau sector, pushing its limits toward the $10^{-9}$ level.

A null result, on the other hand, would still be informative. Pushing the experimental limits another four orders of magnitude excludes large regions of seesaw parameter space, restricting low-scale realizations to corners where the relevant Yukawa structure produces strong cancellations or where the heavy mass exceeds tens of TeV. Either outcome — discovery or null — substantially constrains where the new physics behind neutrino masses might live.

Summary

Charged lepton flavor violation is the indirect window on the same beyond-Standard-Model physics that gives neutrinos their masses. The Standard Model contribution to processes like $\mu \to e\gamma$, $\mu \to eee$, and $\mu \to e$ conversion is so small that any signal at experimentally accessible rates is unambiguous new physics. The current generation of experiments — MEG II at PSI, Mu3e at PSI, Mu2e at Fermilab, and COMET at J-PARC — will push the sensitivity by three to four orders of magnitude over the next decade, reaching into the parameter space of low-scale seesaw models, supersymmetric extensions and many other beyond-Standard-Model frameworks. A discovery in any channel would be a generational result; multi-channel measurements would unwind the underlying operator structure. CLFV is, in this sense, the most powerful tool we have for probing physics far above any collider scale, and complements the direct seesaw signals that may or may not be in reach of the high-energy frontier.