Lepton-flavor violating decays induced by Lorentz violation in the Yukawa sector of the Standard Model Extension

This paper investigates tree-level lepton-flavor-violating decays induced by Lorentz violation in the Yukawa sector of the Standard Model Extension, deriving significantly more restrictive experimental bounds on the associated Lorentz-violating parameters than previously reported.

The Gist

Imagine the universe is a giant, perfectly symmetrical dance floor. For decades, physicists have believed that this dance floor follows strict rules: no matter which way you spin, jump, or look, the laws of physics stay exactly the same. This is called Lorentz symmetry. It's the idea that the universe doesn't have a "preferred direction."

However, some scientists suspect that at the very deepest, tiniest levels of reality (like the Planck scale), this dance floor might actually have a subtle tilt or a hidden groove. Maybe the universe does have a "North" and a "South" that we just haven't noticed yet. This is called Lorentz Violation (LV).

This paper is a detective story where the authors try to find evidence of this hidden tilt by looking at a very specific, rare, and forbidden dance move: Lepton-Flavor Violation (LFV).

The Characters: The Leptons

In the Standard Model (our current best rulebook for particles), there are three types of "lepton" dancers:

1. The Electron (the lightest, most common).
2. The Muon (a heavier, unstable cousin).
3. The Tau (the heavyweight champion, very unstable).

According to the old rulebook, these dancers are strictly segregated. An electron stays an electron; a muon stays a muon. They can never magically turn into each other. If a Muon were to suddenly turn into an Electron and shoot out a flash of light (a photon), that would be a "flavor violation." In the standard universe, this is so incredibly rare it's basically impossible.

The Culprit: The "Yukawa" Sector

The authors of this paper are looking at a specific part of the "Standard Model Extension" (SME). Think of the SME as a massive, expanded rulebook that includes the possibility of Lorentz Violation.

They focus on the Yukawa sector, which is the part of the rulebook that deals with how particles get their mass. In this paper, they introduce a new "suspect": a mathematical object called a tensor (let's call it the Y-tensor).

• The Analogy: Imagine the Y-tensor is like a wind blowing through the universe. This wind doesn't blow randomly; it has a specific direction and strength. If this wind exists, it could push a Muon to change its costume into an Electron, something it's not supposed to do.

The Investigation: The "Forbidden" Decays

The authors calculated what would happen if this "wind" (the Y-tensor) existed. They looked at two specific scenarios:

1. The Two-Body Decay ($l_B \to l_A \gamma$): A heavy lepton (like a Tau) decays into a lighter one (like a Muon) and a photon.
   • The Metaphor: Imagine a heavy dancer (Tau) spinning and suddenly transforming into a lighter dancer (Muon) while shooting a laser beam (photon).
2. The Three-Body Decay ($l_B \to l_A l_C \bar{l}_C$): A heavy lepton decays into a lighter one and a pair of other particles.
   • The Metaphor: The heavy dancer transforms, but this time they also create a new pair of dancers (a particle and its anti-particle) out of thin air.

The Twist: The "Background" Vectors

The Y-tensor is complex, but the authors simplified it by breaking it down into two "vectors" (arrows):

• The "Electric" Vector ($e$): Like an electric field.
• The "Magnetic" Vector ($b$): Like a magnetic field.

They assumed these vectors are either purely real numbers or purely imaginary numbers (mathematical concepts, but think of them as two different "flavors" of the wind) and that they are perpendicular to each other (like the X and Y axes on a graph).

The Results: Catching the Culprit

The authors took the current experimental limits (the strictest rules we have from real-world experiments) and asked: "How strong can this 'wind' be before we would have already seen the forbidden dance moves?"

They found that the wind must be incredibly weak.

• The Bounds: They calculated the maximum possible strength of these vectors. For example, the "Mu-Tau" connection (how much a Muon can be influenced by a Tau) must be smaller than $10^{-11}$ or $10^{-12}$.
• The Comparison: Previous studies had set limits around $10^{-15}$. The authors' new limits are seven orders of magnitude tighter.
  • Analogy: If previous studies said, "The wind can't be stronger than a hurricane," this paper says, "Actually, the wind can't be stronger than a gentle breeze from a fan."

Why This Matters

1. New Physics: If we ever detect a Muon turning into an Electron, it would prove that the universe has a preferred direction (Lorentz Violation) and that our current rulebook (Standard Model) is incomplete.
2. Tighter Constraints: Even though they didn't find the violation, they have made the "search area" much smaller. Any future theory trying to explain new physics must now fit within these incredibly tight boundaries.
3. Methodology: The authors also checked if this "wind" would create "dipole moments" (like a tiny magnetic compass inside the particle). They found that, surprisingly, it doesn't create the expected magnetic effects, but instead creates new, weird types of currents. This tells us that if we find Lorentz violation, it will look very different than we thought.

The Bottom Line

This paper is like a super-precise ruler. The authors measured the "forbidden dance moves" of particles and found that the "wind" causing them must be almost non-existent. They have tightened the screws on our understanding of the universe, telling us that if Lorentz symmetry is broken, it's broken in a way that is far more subtle and restricted than anyone previously thought.

In short: The universe is still looking very symmetrical, but we now know exactly how small the cracks in that symmetry can be.

Technical Summary

1. Problem Statement

The Standard Model (SM) of particle physics assumes exact Lorentz symmetry. However, various theoretical frameworks (e.g., string theory, quantum gravity) suggest that Lorentz symmetry might be broken at fundamental scales (e.g., the Planck scale). The Standard Model Extension (SME) provides a model-independent effective field theory framework to parameterize these Lorentz-violating (LV) effects.

While neutrino oscillations confirm lepton-flavor violation (LFV), the SM predicts LFV processes involving charged leptons (e.g., $\mu \to e\gamma$) to be vanishingly small due to the GIM mechanism and the assumption of massless neutrinos. Experimental searches have set stringent upper bounds on these processes. The paper addresses the specific problem of how LV effects in the Yukawa sector of the SME induce tree-level LFV decays, specifically:

• Two-body decays: $l_B \to l_A \gamma$
• Three-body decays: $l_B \to l_A l_C \bar{l}_C$

The goal is to derive new, tighter constraints on the LV parameters associated with the Yukawa sector using current experimental data.

2. Methodology

Theoretical Framework

The authors focus on the Minimal Extended Yukawa (MEY) sector of the SME. The LV effects are parameterized by a complex, antisymmetric rank-2 tensor $(Y_f)_{AB}^{\mu\nu}$, where:

• $\mu, \nu$ are Lorentz indices.
• $A, B$ are flavor indices.
• The tensor is antisymmetric under Lorentz index interchange: $(Y_f)_{AB}^{\mu\nu} = -(Y_f)_{AB}^{\nu\mu}$.

This tensor is decomposed into two complex 3-vectors, $\mathbf{e}_{AB}^f$ and $\mathbf{b}_{AB}^f$, analogous to the electric and magnetic fields in electromagnetism:

• $(Y_f)_{0i} = (\mathbf{e}_{AB}^f)_i$
• $(Y_f)_{ij} = \epsilon_{ijk}(\mathbf{b}_{AB}^f)_k$

The authors analyze two specific scenarios for these parameters:

1. Pure Real Parameters: The antisymmetric part of the tensor vanishes.
2. Pure Imaginary Parameters: The symmetric part vanishes.
   They also assume the vectors $\mathbf{e}$ and $\mathbf{b}$ are mutually orthogonal in the rest frame of the decaying particle to simplify phase-space integration.

Calculations

1. Tree-Level Amplitudes:

   • The authors calculate the first-order LV corrections to the $f_B f_A \gamma$ vertex and the $f_B \to f_A f_C \bar{f}_C$ process.
   • They derive the invariant amplitudes using Feynman rules derived from the MEY Lagrangian.
   • Crucially, they perform a full Dirac algebraic manipulation to check for dipolar transition moments (magnetic/electric). They find that at tree level, there are no contributions to transition magnetic or electric dipole moments; the amplitudes are purely vector/axial in nature.

2. Second-Order Corrections:

   • To ensure no dipolar structures were missed, they computed second-order LV corrections to the $f_B f_A \gamma$ vertex.
   • Result: Even at second order, no dipolar-transition structures ($\sigma_{\mu\nu}q^\nu$) appear. The corrections only induce generalized vector and axial flavor-changing neutral currents and new Lorentz structures.

3. Decay Rates and Branching Ratios (BR):

   • The authors calculate the decay rates $\Gamma$ for $l_B \to l_A \gamma$ and $l_B \to l_A l_C \bar{l}_C$.
   • They perform phase-space integration in a specific reference frame defined by the unit vectors of $\mathbf{e}$, $\mathbf{b}$, and $\mathbf{e} \times \mathbf{b}$.
   • They express the branching ratios as functions of the magnitudes $|\mathbf{e}_{AB}^f|$ and $|\mathbf{b}_{AB}^f|$.

4. Constraint Derivation:

   • The calculated theoretical BRs are compared against experimental upper bounds (e.g., from MEG, Belle, BaBar) listed in Table I.
   • By imposing $BR_{SME} \leq BR_{Exp}$, they derive upper bounds on the magnitudes of the LV parameters.

3. Key Contributions

• Inclusion of Momentum Dependence: Unlike previous studies (e.g., Ref. [48]) which neglected Lorentz contractions between the LV tensors and particle momenta, this work explicitly includes these terms. The authors note that these contractions contribute approximately 60% to the partial decay widths, significantly affecting the resulting bounds.
• Tree-Level LFV: The paper demonstrates that the MEY sector induces LFV at the tree level, making these processes highly sensitive probes for LV, unlike loop-suppressed SM predictions.
• Second-Order Analysis: The explicit calculation of second-order corrections confirms the absence of dipolar transitions, clarifying that the dominant effects are vector/axial currents.
• Orthogonality Assumption: The introduction of a specific reference frame where $\mathbf{e}$ and $\mathbf{b}$ are orthogonal allows for analytical solutions to the phase-space integrals, which are otherwise cumbersome.

4. Results

The study establishes significantly tighter constraints on the LV parameters compared to previous literature.

A. Two-Body Decays ($l_B \to l_A \gamma$):
The bounds on the magnitudes of the vectors $|\mathbf{e}_{AB}^l|$ and $|\mathbf{b}_{AB}^l|$ (in the rest frame of the decaying particle) are:

• $\tau \to \mu \gamma$: $|\mathbf{e}_{\mu\tau}^l| < 1.51 \times 10^{-11}$, $|\mathbf{b}_{\mu\tau}^l| < 1.95 \times 10^{-11}$
• $\tau \to e \gamma$: $|\mathbf{e}_{e\tau}^l| < 1.34 \times 10^{-11}$, $|\mathbf{b}_{e\tau}^l| < 1.73 \times 10^{-11}$
• $\mu \to e \gamma$: $|\mathbf{e}_{e\mu}^l| < 3.65 \times 10^{-18}$, $|\mathbf{b}_{e\mu}^l| < 4.71 \times 10^{-18}$

B. Three-Body Decays ($l_B \to l_A l_C \bar{l}_C$):
For the process $\tau \to \mu e \bar{e}$, assuming purely real LV parameters:

• $|\mathbf{e}_{\mu\tau}^l| < 3.05 \times 10^{-12}$
• $|\mathbf{b}_{\mu\tau}^l| < 4.31 \times 10^{-12}$

Note: For purely imaginary parameters in the three-body decay, the region is unbounded. However, by combining the results with the two-body decay constraints, a bounded region is established.

C. Comparison with Literature:

• Previous bounds on the combination $||\text{Re}(\mathbf{e}_{\mu\tau}^l)|^2 - |\text{Re}(\mathbf{b}_{\mu\tau}^l)|^2|$ were of the order $10^{-15}$.
• This work derives bounds of the order $10^{-22}$, representing a restriction seven orders of magnitude tighter.
• The authors attribute this improvement to the inclusion of momentum-dependent Lorentz contractions which were previously neglected.

D. Lorentz Invariants:
The paper also provides bounds on the Lorentz-invariant scalars $(Y_l)_{\alpha\beta}^{AB} (Y_l)^{*BA \alpha\beta}$, which are valid in any reference frame (Table V).

5. Significance

• Stringent Constraints: The results provide the most restrictive bounds to date on the Yukawa sector of the SME regarding lepton-flavor violation.
• Methodological Advancement: By explicitly accounting for momentum-dependent terms in the LV tensors, the paper corrects previous underestimations of the decay rates and demonstrates the necessity of full kinematic treatment in SME phenomenology.
• New Physics Search: The derived bounds serve as critical targets for future high-sensitivity experiments (like MEG II or Mu3e). If future experiments observe LFV at levels consistent with these bounds, it would strongly point toward Lorentz violation in the Yukawa sector.
• Theoretical Clarity: The confirmation that tree-level LV effects do not generate dipole moments helps distinguish SME signatures from other New Physics models (like Supersymmetry) that typically predict large dipole contributions.

In summary, this paper significantly advances the phenomenological study of Lorentz violation by leveraging precise experimental data on charged lepton decays to place unprecedented limits on the SME's Yukawa sector parameters.