Charged lepton flavor violating decays with a pair of light dark matter and muonium invisible decay

This paper investigates charged lepton flavor violating interactions with a pair of light dark matter particles within an effective field theory framework, deriving kinematic distributions to distinguish operator structures and dark matter masses, establishing stringent limits from current experimental data, and demonstrating that future observations of muonium invisible decays could serve as a compelling signature for these flavored dark matter interactions.

The Gist

Imagine the universe as a giant, bustling city. For a long time, scientists have been mapping the "known citizens" of this city: the Standard Model particles like electrons, muons, and taus. They also know there's a massive, invisible fog called Dark Matter that makes up most of the city's mass, but they can't see it or touch it directly.

Usually, scientists think Dark Matter interacts with our visible world like a shy ghost: it might bump into a nucleus (like a ghost bumping into a wall), but it keeps to itself. It doesn't care about the "flavors" of the citizens (whether it's an electron or a muon).

This paper is about a new, exciting theory: What if Dark Matter isn't just a shy ghost, but a gossip?

What if Dark Matter has a "favorite" flavor? What if it prefers to hang out with muons and then suddenly switch to an electron, leaving a trail of invisible energy behind? This is called Lepton Flavor Violation (LFV).

Here is a breakdown of the paper's ideas using simple analogies:

1. The "Flavor-Changing" Party Crashers

In the Standard Model, a heavy muon (a "big" particle) can decay into an electron (a "small" particle) and two neutrinos (invisible ghosts). This happens all the time.

The authors ask: What if, instead of neutrinos, the muon decays into an electron and a pair of Dark Matter particles?

• The Analogy: Imagine a heavy bouncer (the muon) at a club. Usually, he kicks out a small patron (the electron) and two invisible security guards (neutrinos).
• The New Theory: What if the bouncer kicks out the small patron and two invisible Dark Matter ghosts instead?
• The Twist: The authors propose that these Dark Matter ghosts might be "flavorful," meaning they specifically cause the bouncer to switch from a muon to an electron.

2. The Detective Work: The "Missing Energy" Clue

Since Dark Matter is invisible, we can't see it in our detectors. We can only see the electron flying out.

• The Analogy: Imagine you are watching a magic trick. The magician (the muon) disappears, and a rabbit (the electron) appears. But the rabbit is lighter than the magician. Where did the rest of the weight go?
• The Clue: In a normal magic trick, the missing weight is the two invisible guards (neutrinos). In this new theory, the missing weight is the two Dark Matter ghosts.
• The Tool: The authors use a mathematical tool called Effective Field Theory (EFT). Think of this as a "rulebook" for low-energy physics. Instead of guessing exactly what the Dark Matter is made of (a scalar ball, a fermion cube, or a vector arrow), they write down every possible way these particles could interact using the rulebook.

3. The "Fingerprint" of the Invisible

One of the coolest parts of the paper is how they figure out what the Dark Matter is, even though they can't see it.

• The Analogy: Imagine you hear a car crash in the dark. You can't see the cars, but you can hear the sound of the impact. A heavy truck crash sounds different from a small sports car crash.
• The Science: The authors calculated the "sound" of the crash—the invariant mass distribution. By looking at how the energy is shared between the electron and the invisible Dark Matter, they can tell:
  1. How heavy the Dark Matter is.
  2. What "shape" (spin) the Dark Matter has (is it a scalar, a fermion, or a vector?).
  • Key Finding: If the electron comes out with a specific energy pattern, it's a "fingerprint" that tells us exactly which type of Dark Matter interaction caused it.

4. The "Muonium" Atom: The Ultimate Test

The paper also looks at Muonium, which is a tiny atom made of a positive muon and a negative electron. It's like a hydrogen atom, but with a muon instead of a proton.

• The Analogy: Think of Muonium as a delicate soap bubble.
• The Prediction: In our current understanding (Standard Model), this bubble can pop in a few ways, but one specific way (where it vanishes completely into two invisible particles) is practically impossible. It's like saying a soap bubble can vanish into thin air without a trace.
• The Breakthrough: The authors show that if these "flavorful" Dark Matter interactions exist, the Muonium bubble could vanish much more easily.
• The Smoking Gun: If future experiments (like the proposed MACE experiment) see a Muonium atom disappear into nothingness, it would be a smoking gun. It would prove that Dark Matter is interacting with our particles in this specific, flavor-changing way.

5. The Results: How Strong is the Evidence?

The authors checked all the current data from particle accelerators (like the MEG and Belle experiments).

• The Verdict: So far, no one has seen this "flavor-changing" decay yet. This means the "rulebook" (the effective scale) must be very strict. The interaction is either very weak or the Dark Matter is very heavy.
• The Limit: They calculated that if this interaction exists, it happens at energy scales up to the TeV range (trillions of electron volts). This gives future experiments a clear target: they need to look for these invisible decays with extreme precision.

Summary

This paper is a roadmap for a new kind of treasure hunt.

1. The Treasure: Light Dark Matter that talks to our particles.
2. The Map: A set of mathematical rules (EFT) covering all possibilities.
3. The Compass: The specific energy patterns of decaying particles (muons and taus).
4. The Goal: To find a "flavor-changing" decay where a muon turns into an electron and two invisible Dark Matter particles.

If we find this, it won't just tell us about Dark Matter; it will tell us that the universe has a secret language where particles can change their identity by swapping with the invisible dark sector. It's a potential revolution in how we understand the building blocks of reality.

Technical Summary

Here is a detailed technical summary of the paper "Charged lepton flavor violating decays with a pair of light dark matter and muonium invisible decay" by Sahabub Jahedi, Yi Liao, and Xiao-Dong Ma.

1. Problem Statement

The paper addresses two major gaps in Beyond the Standard Model (BSM) physics:

• Lepton Flavor Violation (LFV) with Dark Matter (DM): While DM-lepton interactions are often studied in the context of flavor-conserving direct/indirect detection, flavor-violating interactions involving a pair of light DM particles in the final state have received limited theoretical and experimental attention.
• Two-Body DM Production: Most existing studies focus on single invisible particle production (e.g., $\ell_i \to \ell_j + a$). However, many DM models require a stabilizing symmetry (e.g., $Z_2$) that forbids single DM production, necessitating the study of processes involving pairs of DM particles ($\ell_i \to \ell_j + \text{DM} + \text{DM}$).
• Muonium Invisible Decays: The potential for muonium ($M_\mu = \mu^+ e^-$) to undergo invisible decays into light dark sector particles is under-explored, particularly in the context of flavored DM interactions.

2. Methodology

The authors employ a Dark Sector Effective Field Theory (DSEFT) framework, valid at energy scales below the electroweak scale ($O(10 \text{ GeV})$).

• Effective Operators: They systematically construct all leading-order local operators of the form $(\bar{\ell}_j \Gamma \ell_i) \text{DM}^2$ for three DM spin scenarios:
  • Scalar DM ($\phi$): Dim-5 and Dim-6 operators.
  • Fermion DM ($\chi$): Dim-6 operators.
  • Vector DM ($X$): Dim-5, Dim-6, and Dim-7 operators. The vector case is further split into two parametrizations: Case A (using the four-vector potential $X_\mu$) and Case B (using the field strength tensor $X_{\mu\nu}$).
• Flavor Combinations: The analysis covers the three charged LFV combinations: $(e\mu)$, $(e\tau)$, and $(\mu\tau)$.
• Processes Analyzed:
  1. Three-body decays: $\ell_i \to \ell_j + \text{DM} + \text{DM}$.
  2. Four-body radiative decay: $\mu \to e + \text{DM} + \text{DM} + \gamma$ (specifically for the $e\mu$ channel).
  3. Muonium invisible decays: $M_\mu \to \text{DM} + \text{DM}$ (both para- and ortho-muonium).
• Constraints: The authors derive limits on the effective scale ($\Lambda$) by comparing their calculated branching ratios against current experimental upper limits for analogous SM processes involving neutrinos (e.g., $\mu \to e \nu \bar{\nu}$, $\tau \to \ell \nu \bar{\nu}$) and the ratio of tau leptonic widths ($R_{\mu/e}^\tau$).
• Kinematic Analysis: They calculate invariant mass ($q^2$) distributions for three-body decays and missing energy distributions for radiative decays to distinguish operator structures and determine DM mass.

3. Key Contributions

• Systematic Operator Classification: The paper provides the first comprehensive collection of leading-order DSEFT operators for LFV decays involving pairs of DM particles across scalar, fermion, and vector scenarios.
• Kinematic Discriminators:
  • Demonstrated that the $q^2$-distribution (invariant mass of the DM pair) in three-body decays is a powerful observable. It can distinguish between different Lorentz structures (Scalar vs. Pseudoscalar, Vector vs. Axial-Vector) and determine the DM mass, particularly in $\tau \to \mu + \text{DM} + \text{DM}$ where the muon mass is significant.
  • Showed that the missing energy distribution in the radiative decay $\mu \to e + \text{DM} + \text{DM} + \gamma$ provides a complementary probe, with the endpoint of the distribution sensitive to the DM mass.
• Reparametrization for Vector DM: Addressed the divergence issue in Vector DM Case A (where decay widths diverge as $m_X \to 0$) by introducing a mass-dependent reparametrization of the Wilson coefficients (Eq. 3.1), ensuring meaningful constraints for light DM.
• Muonium Sensitivity: Established that muonium invisible decays, particularly para-muonium, offer a unique "smoking gun" signature. Since the SM forbids two-body invisible decays for para-muonium, any observation would be a clear signal of new physics.

4. Key Results

• Constraints on Effective Scale ($\Lambda$):
  • Muon Decays ($\mu \to e + \text{DM} + \text{DM}$): Provide the most stringent constraints for scalar and fermion DM. For scalar operators ($O_S, O_P$), $\Lambda$ is constrained up to $\sim 3700$ TeV for $m_{\text{DM}} \sim 1$ MeV.
  • Tau Decays ($\tau \to \ell + \text{DM} + \text{DM}$): Provide comparable or slightly weaker constraints than muon decays but are crucial for distinguishing chiral structures due to the larger phase space and muon mass.
  • Vector DM: Constraints are generally weaker (tens of GeV to hundreds of GeV) due to the higher dimensionality of the operators and the specific mass-dependent parametrization required to avoid divergences.
• Branching Ratios:
  • For a fixed effective scale of $\Lambda_{\text{eff}} = 500$ GeV, the predicted branching ratios for vector DM Case A can reach $\sim 10^{-9}$ for $\tau$ decays and $\sim 10^{-10}$ for $\mu$ decays, depending on the operator.
  • Muonium: The branching ratios for muonium invisible decays can be significantly enhanced compared to the SM prediction ($B \sim 10^{-12}$ for ortho-muonium).
• Discrimination Power:
  • The process $\tau \to \mu + \text{DM} + \text{DM}$ is identified as the most effective channel for distinguishing between different Lorentz structures of the leptonic currents due to the non-negligible muon mass.
  • The $e\mu$ channel ($\mu \to e$) suffers from degeneracy between Scalar/Vector/Tensor and Pseudoscalar/Axial/Pseudotensor currents due to the negligible electron mass.

5. Significance

• New Experimental Targets: The paper identifies specific kinematic signatures (invariant mass and missing energy endpoints) that future experiments (MEG II, Mu2e, COMET, STCF, and the proposed MACE experiment for muonium) can use to search for light flavored DM.
• Complementarity: It highlights the complementarity between charged lepton decays and muonium decays. While charged lepton decays set bounds on the effective scale, muonium decays (especially para-muonium) offer a background-free environment for discovery.
• Model Independence: By using the EFT approach, the results are applicable to a wide range of UV-complete models (e.g., scotogenic models) without committing to a specific high-energy theory.
• Paradigm Shift: The work shifts the focus from single invisible particle searches to pair-production scenarios, which are theoretically motivated by DM stability symmetries, thereby expanding the search space for light dark matter.

In conclusion, this study establishes a robust theoretical framework for probing light, flavored dark matter through charged lepton flavor violation and muonium decays, providing critical benchmarks and kinematic observables for upcoming experimental campaigns.