⚛️ phenomenology

Lepton flavor violating $\tau^- \to \ell_i^- \ell_i^- \ell_j^+$ ( $\ell_i\neq \ell_j$ ) decays induced by $S_1$ and $R_2$ scalar leptoquarks

This paper investigates one-loop induced lepton flavor violating tau decays ( $\tau^- \to \ell_i^- \ell_i^- \ell_j^+$ ) mediated by $S_1$ and $R_2$ scalar leptoquarks, demonstrating that within parameter spaces consistent with current experimental constraints, the predicted branching ratios could reach the sensitivity of near-future experiments.

Original authors: G. Hernández-Tomé, J. P. Hoyos-Daza, O. G. Miranda, R. Sánchez-Vélez

Published 2026-03-19

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: G. Hernández-Tomé, J. P. Hoyos-Daza, O. G. Miranda, R. Sánchez-Vélez

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Standard Model of physics as a massive, incredibly complex rulebook for how the universe's tiny building blocks (particles) interact. For decades, this rulebook has worked perfectly, but there are a few loose ends—mysteries it can't explain. One of the biggest mysteries is why certain particles seem to break the rules of "flavor."

In this paper, the authors act like cosmic detectives trying to solve a specific crime: Why would a heavy particle called a "Tau" suddenly decay into three lighter particles (like electrons or muons) in a way that shouldn't happen?

Here is the story of their investigation, told in everyday terms.

1. The Suspects: The "Leptoquarks"

In the Standard Model, there are two distinct families of particles: Quarks (which build protons and neutrons) and Leptons (like electrons and neutrinos). They usually don't talk to each other directly.

The authors are investigating a hypothetical new family of particles called Leptoquarks. Think of these as "universal translators" or "matchmakers." They are special particles that can shake hands with a Quark on one side and a Lepton on the other, allowing them to swap places or interact.

The paper focuses on two specific types of these matchmakers, named $S_1$ and $R_2$ . They are heavy (like a boulder compared to a pebble) and likely exist at a scale of "TeV" (Tera-electronvolts), which is the energy level of the Large Hadron Collider (LHC).

2. The Crime Scene: The "Tau" Decay

The crime in question is a rare event where a Tau particle (a heavy cousin of the electron) decays into three lighter particles: two of one type and one of another (e.g., two muons and one electron).

The Standard Model Rule: In our current rulebook, this specific crime is strictly forbidden. It's like a bank vault that has no cracks; the probability of it happening is zero.
The New Theory: If Leptoquarks exist, they could act as a secret tunnel. The Tau could turn into a Quark, swap places with a Leptoquark, and then turn back into three different Leptons. It's a "flavor violation"—a particle changing its identity in a way nature usually forbids.

3. The Investigation: Two Types of Tunnels

The authors calculated how these Leptoquarks would facilitate this crime. They found two main ways the "tunnel" could be built:

The "Penguin" Tunnel (The Usual Suspect): Most previous studies looked at processes where the particles interact via a photon or a Z-boson (like a messenger carrying a note). This is the "Penguin" diagram.
The "Box" Tunnel (The New Clue): This paper focuses on a different, more complex path called the "Box" diagram. Imagine a square box where four particles interact at the corners. In this scenario, the Tau doesn't just send a message; it goes through a complex loop involving heavy quarks (like the Top or Charm quark) inside the Leptoquark.

Why does this matter? The "Penguin" tunnels are often blocked by other strict rules. The "Box" tunnels are cleaner and harder to detect, but if we find them, they are a smoking gun for new physics.

4. The Constraints: The "Speed Bumps"

The detectives can't just invent any story; they have to fit the evidence. They checked three major "speed bumps" (constraints) that limit how big these Leptoquarks can be or how strongly they can interact:

The Muon's Magnetic Moment ( $g-2$ ): The muon is a particle that wobbles like a spinning top. Scientists measure how much it wobbles. Recently, the measurement got more precise, and the "gap" between the prediction and the measurement got smaller. This means the Leptoquarks can't be too strong, or they would have made the muon wobble too much.
Radiative Decays ( $\ell \to \ell \gamma$ ): This is when a particle decays into a lighter one and a flash of light (a photon). Experiments have looked for this and found nothing. This puts a tight leash on the Leptoquarks.
The Muon-to-Electron Conversion ( $\mu \to eee$ ): This is another forbidden decay. The fact that we haven't seen it yet means the Leptoquarks must be very careful about how they mix flavors.

5. The Verdict: Is the Crime Possible?

After running the numbers and checking the "speed bumps," the authors found some exciting possibilities:

The Sweet Spot: If the Leptoquarks are heavy (around 1.5 TeV, which is within reach of current or near-future colliders) and interact with the heaviest quarks (Top and Charm), they could cause the Tau to decay in this forbidden way.
The "Box" is Quiet: The "Box" diagrams (the focus of this paper) are generally quieter than the "Penguin" ones. However, the authors found that in certain scenarios, the "Box" contribution is strong enough to be seen by upcoming experiments.
Left vs. Right: The study looked at whether these particles interact with "left-handed" or "right-handed" versions of matter. They found that the "right-handed" interactions are less restricted by current rules, meaning there is more "wiggle room" for the Leptoquarks to hide there.

6. The Big Picture: Why Should We Care?

Think of the universe as a giant puzzle. We have most of the pieces, but the edges are blurry.

The "Tau" decay is a new piece. If we can catch a Tau decaying into three leptons in a way that breaks the rules, it proves that Leptoquarks exist.
It's a Complementary Test. Previous experiments looked for the "Penguin" path. This paper says, "Hey, let's also look at the 'Box' path." It's like checking the back door of a house when the front door is locked.
Future Experiments: The authors predict that if these Leptoquarks exist at the TeV scale, the next generation of particle detectors (like those at the Belle II experiment) might just be sensitive enough to catch this rare decay.

Summary Analogy

Imagine the Standard Model is a strict bouncer at a club who only lets Quarks and Leptons enter through separate doors.

Leptoquarks are secret passageways the bouncer doesn't know about.
The Tau Decay is a VIP guest (the Tau) trying to sneak out through the back door and reappear as three different people.
This Paper is a blueprint showing exactly how the "Box" passageway works, calculating how likely it is to be used, and checking if the bouncer (current experiments) would have noticed it yet.

The conclusion? The passageway is narrow and hidden, but if the VIPs are heavy enough (TeV scale), the bouncer might finally catch them in the act soon.

1. Problem Statement

The paper addresses the search for Charged Lepton Flavor Violation (CLFV) as a probe for physics Beyond the Standard Model (BSM). Specifically, it investigates the three-body tau decays of the form $\tau^- \to \ell^-_i \ell^-_i \ell^+_j$ (where $\ell_i \neq \ell_j$ , e.g., $\tau \to e^-e^-\mu^+$ or $\tau \to \mu^-\mu^-e^+$ ).

The Gap: While previous studies have extensively analyzed CLFV decays dominated by "penguin" diagrams (mediated by off-shell photons or $Z$ bosons), the "double flavor-changing" channels ( $\tau \to \ell^-_i \ell^-_i \ell^+_j$ ) are generated exclusively by box-type Feynman diagrams at the one-loop level. These modes have received comparatively little attention.
The Context: The study focuses on Scalar Leptoquarks (LQs), specifically the $S_1$ (lepton number violating, $|F|=2$ ) and $R_2$ (lepton number conserving, $F=0$ ) representations.
The Motivation: Recent updates to the muon anomalous magnetic moment ( $a_\mu$ ) have reduced the discrepancy between experimental measurements and Standard Model predictions. This necessitates a reassessment of BSM parameter spaces, as LQs can no longer be tuned solely to explain the $a_\mu$ anomaly but must satisfy stringent consistency constraints.

2. Methodology

Theoretical Framework

Model: The authors utilize a low-energy effective Lagrangian invariant under the SM gauge group $SU(3)_C \times SU(2)_L \times U(1)_Y$ .
Interactions: They consider the interaction of scalar LQs ( $S_1$ and $R_2$ ) with SM fermions via Yukawa couplings $\lambda^{jk}_{L,R}$ , where $j$ is the quark generation and $k$ is the lepton generation.
Flavor Structure: The analysis focuses on scenarios dominated by heavy quark contributions (Top and Charm), as first-generation contributions are suppressed. Two benchmark scenarios are defined:
1. Top-only: Only couplings to the third generation ( $\lambda_{3\ell}$ ) are non-zero.
2. Charm-only: Only couplings to the second generation ( $\lambda_{2\ell}$ ) are non-zero.

Computational Approach

Loop Calculation: The authors compute the one-loop box diagrams contributing to the decay amplitude. The calculation is performed in the limit of negligible external momenta ( $M_{LQ} \gg m_\tau$ ), reducing the loop amplitudes to local four-fermion operators.
Tools: Amplitudes were computed using the Package-X library and cross-checked with FeynCalc.
Analytical Derivation:
- The branching ratios are derived analytically for both $F=0$ ( $R_2$ ) and $|F|=2$ ( $S_1$ ) cases.
- The results are expressed in terms of scalar ( $S_{XY}$ ), vector ( $V_{XY}$ ), and tensor ( $T_{XY}$ ) form factors, which depend on Yukawa couplings and Passarino-Veltman loop functions ( $D_0, D_{00}$ ).
- Fierz rearrangements are applied to handle identical fermions in the final state.

Constraints and Parameter Space

The analysis imposes a multi-faceted set of constraints to define the viable parameter space:

Muon Anomalous Magnetic Moment ( $a_\mu$ ): Using the latest experimental data and SM theory updates, the authors derive constraints on the chirality-flipping couplings ( $\text{Re}(\lambda^L \lambda^{R*})$ ).
Radiative Decays: Bounds from $\ell_i \to \ell_j \gamma$ (e.g., $\mu \to e\gamma$ , $\tau \to \mu\gamma$ ).
Three-body Decays: The stringent bound from $\mu^- \to e^- e^- e^+$ .
Perturbativity: Yukawa couplings are restricted to $0 < \lambda < \sqrt{4\pi}$ .

3. Key Contributions

Systematic Analysis of Box-Induced Modes: The paper provides the first comprehensive analytical derivation and numerical analysis of the $\tau \to \ell^-_i \ell^-_i \ell^+_j$ channels specifically induced by $S_1$ and $R_2$ leptoquarks, isolating the pure box-diagram contribution.
Updated Phenomenology: The study incorporates the most recent $a_\mu$ data, which significantly alters the allowed regions for LQ couplings compared to older studies that assumed a larger anomaly.
Analytical Expressions: The authors provide explicit analytical formulas for the branching ratios of these double-flavor-violating decays, including the dependence on chiral ratios ( $k_{q\ell} = \lambda_L / \lambda_R$ ) and loop functions.
Comparison of $S_1$ vs. $R_2$ : The work explicitly compares the lepton-number-violating ( $S_1$ ) and lepton-number-conserving ( $R_2$ ) scenarios, demonstrating that current and future sensitivities cannot easily distinguish between them based solely on these decay rates.

4. Results

Branching Ratio Sensitivity: For leptoquark masses in the TeV range ( $M_{LQ} \sim 1.5$ TeV) and sizable couplings ( $\lambda \sim O(1)$ ), the predicted branching ratios for $\tau \to \ell^-_i \ell^-_i \ell^+_j$ can approach the sensitivities expected in near-future experiments (e.g., Belle II).
Dominant Contributions:
- The contributions from Top and Charm quarks are found to be of comparable size in the phenomenologically viable region, making both relevant for future tests.
- The Charm-only scenario often imposes tighter constraints on the coupling combinations than the Top-only scenario due to the specific behavior of the loop functions $D_{00}$ .
Chirality Dependence:
- The branching ratios are found to be largely insensitive to the left-handed couplings ( $\lambda_L$ ) distribution but are more sensitive to the right-handed couplings ( $\lambda_R$ ).
- Under the considered constraints, the observable effects are dominated by left-handed couplings, while right-handed couplings remain less restricted.
Channel Comparison:
- The channel $\tau \to \mu^- \mu^- e^+$ is less constrained by current experimental limits compared to $\tau \to e^- e^- \mu^+$ , allowing for a larger allowed parameter space.
- The predicted branching ratios generally lie one to two orders of magnitude below current experimental limits but are within reach of future sensitivities.
Indistinguishability: The numerical analysis shows that the branching ratios for $S_1$ and $R_2$ are nearly identical in magnitude for the same coupling strengths. Therefore, observing these decays alone would not distinguish between lepton-number-conserving and violating scenarios.

5. Significance

Complementary Probe: This work highlights that double-flavor-changing tau decays serve as a complementary and clean probe for scalar leptoquarks. Unlike penguin-dominated channels, these modes are free from photon/Z mediation, offering a direct window into box-diagram dynamics.
Future Physics Potential: The results suggest that upcoming high-luminosity experiments (like Belle II) have the potential to probe the TeV-scale parameter space of scalar leptoquarks through these specific rare tau decays, even if the muon $g-2$ anomaly is resolved.
Constraint Refinement: By updating the constraints based on the reduced $a_\mu$ discrepancy, the paper provides a more realistic map of the viable parameter space for LQ models, guiding future model-building and experimental searches.
Theoretical Utility: The derived analytical expressions for the branching ratios and the explicit form factors provide a valuable toolkit for other researchers studying CLFV in various BSM scenarios.