Probing muon anomaly and lepton flavor violation with scalar leptoquarks in the 331LHN model

This paper extends the 331LHN model by introducing scalar leptoquarks to demonstrate that a singlet leptoquark with a mass between 1.8 and 6 TeV can explain the muon anomalous magnetic moment discrepancy while satisfying constraints from lepton flavor violation and LHC bounds, with a predicted normal hierarchical Yukawa coupling pattern and suppressed current collider signals.

The Gist

Imagine the universe is a giant, incredibly complex Lego set. For decades, scientists have been building a model of how everything works using a specific instruction manual called the Standard Model. It's a brilliant manual that explains almost everything we see, from atoms to stars.

But recently, scientists noticed a few "glitches" in the model. The pieces just don't fit perfectly in a few specific spots. One of the biggest glitches involves a tiny particle called the muon (a heavy cousin of the electron). When scientists measured how much the muon "wobbles" in a magnetic field, the result didn't match the prediction from the manual. It was like measuring a car's speed and finding it's going 5 mph faster than the engine manual says it should.

This paper proposes a new set of "extra Lego pieces" to fix the glitch. Here is the story of how they did it, explained simply:

1. The New "Bridge" Pieces: Leptoquarks

The authors suggest adding a new type of particle called a Leptoquark.

• The Analogy: Think of the Standard Model as a city with two separate neighborhoods: the Quark District (where heavy particles like protons live) and the Lepton District (where light particles like electrons live). In the old model, these neighborhoods had strict borders; you couldn't easily walk from one to the other.
• The Fix: A Leptoquark is like a magic bridge or a universal translator that can talk to both neighborhoods at once. It carries a bit of "quark" and a bit of "lepton" in its pocket. By building these bridges, the authors hope to explain why the muon is wobbling more than expected.

2. The Specific Bridge: The "Singlet" Leptoquark

The paper looks at many different types of bridges, but they focus on one specific, simple type called a Singlet Leptoquark.

• Imagine you have a toolbox full of complex, multi-tool gadgets. The authors found that a simple, single-purpose screwdriver (the Singlet) is actually the perfect tool to fix the muon wobble.
• They calculated that if this "screwdriver" exists and weighs about 1.8 to 6 times heavier than a proton (which is incredibly heavy for a particle), it would perfectly explain the muon's extra wobble.

3. The "Too Good to Be True" Problem

Here is the catch: If you build a bridge, you have to make sure it doesn't cause traffic jams elsewhere.

• The Glitch: In the world of particles, if you connect the muon to other particles too strongly, you might accidentally cause forbidden events, like a muon spontaneously turning into an electron and a flash of light (a process called Lepton Flavor Violation).
• The Solution: The authors ran the numbers and found a "Goldilocks Zone." The bridge (Leptoquark) must be heavy enough to not cause these forbidden traffic jams, but light enough to still fix the muon wobble.
• The Result: They found that the "couplings" (how strongly the bridge connects to different particles) must follow a very specific pattern. It's like a hierarchy: The bridge connects very strongly to the heavy "top" particles but barely touches the light "first-generation" particles. This keeps the universe stable while fixing the muon problem.

4. The Future Hunt: Can We Find It?

Now, the big question: Where is this bridge?

• The LHC (Current Collider): The Large Hadron Collider (LHC) is like a giant particle smashing machine. The authors say that if this Leptoquark exists, it's likely too heavy for the current LHC to spot easily. It's like trying to find a specific grain of sand on a beach using a net that has holes too big to catch it. The signal is too faint.
• The Future: However, if we build a bigger, more powerful collider in the future (a "Super Collider"), we will have a finer net. We might finally catch this heavy particle.

5. The "2025 Update" Twist

The paper also mentions a very recent update to the "manual" (the Standard Model calculations).

• In 2021, the gap between the prediction and the experiment was huge (a 4.2σ discrepancy).
• In 2025, scientists updated their math using supercomputers (Lattice QCD). The gap got smaller, but it didn't disappear.
• The Consequence: Because the gap is smaller, the "magic bridge" (Leptoquark) needs to be even heavier (around 6 TeV) to fit the new, tighter rules. This makes it even harder to find, but it doesn't rule the idea out.

Summary

This paper is a detective story.

1. The Crime: The muon is wobbling too much.
2. The Suspect: A new particle called a Singlet Leptoquark (a bridge between particle families).
3. The Alibi Check: The suspect is heavy and connects to particles in a very specific, hierarchical way so it doesn't break other laws of physics.
4. The Verdict: The suspect is likely hiding in the "multi-TeV" weight class. We probably can't see it with our current tools (LHC), but if we build bigger tools in the future, we might finally catch it and solve the mystery of the wobbling muon.

In short: The universe might be missing a heavy, invisible bridge that connects different families of particles, and this paper maps out exactly where to look for it.

Technical Summary

1. Problem Statement

The Standard Model (SM) of particle physics faces several unresolved anomalies, most notably the discrepancy in the muon anomalous magnetic moment ($a_\mu$). While the 2021 experimental results indicated a $4.2\sigma$ deviation from SM predictions, recent updates (2025) based on improved lattice QCD calculations have reduced this tension, though significant theoretical uncertainties remain. Additionally, the SM cannot explain the origin of neutrino masses, dark matter, or the replication of fermion families.

The authors investigate whether Scalar Leptoquarks (LQs) embedded within the $SU(3)_C \times SU(3)_L \times U(1)_X$ (331LHN) model can simultaneously:

1. Explain the muon $g-2$ anomaly ($\Delta a_\mu$).
2. Satisfy stringent constraints from Charged Lepton Flavor Violation (CLFV), specifically $\mu \to e\gamma$, $\tau \to \mu\gamma$, $\tau \to e\gamma$, and $\mu-e$ conversion in nuclei.
3. Remain consistent with current Large Hadron Collider (LHC) bounds.

2. Methodology

Theoretical Framework

The authors extend the 331LHN model (which already includes neutral leptons to form lepton triplets) by introducing scalar leptoquarks. They analyze the gauge-invariant Yukawa interactions allowed by the model's symmetry structure.

• Leptoquark Representations: They identify possible LQ representations arising from fermion bilinears: singlets ($S, \tilde{S}$), triplets ($S_3, \tilde{S}_3$), sextets ($S_6$), and octets ($S_8$).
• Focus: The analysis focuses on the singlet scalar leptoquark ($S$) with quantum numbers $(3, 1, -1/3)$. This specific state is chosen because it induces chirality-flipping interactions, which are essential for generating a large contribution to the muon magnetic moment.

Calculation of Observables

1. Muon Anomalous Magnetic Moment ($\Delta a_\mu$):

   • Calculated at the one-loop level involving the singlet LQ and quarks (dominated by the top quark due to mass enhancement).
   • The contribution scales as $\Delta a_\mu \propto \frac{m_t}{m_S} y_L y_R$, where $y_{L,R}$ are the Yukawa couplings and $m_S$ is the LQ mass.
   • The authors evaluate the parameter space required to match both the 2021 ($4.2\sigma$) and 2025 (updated) experimental values.

2. Charged Lepton Flavor Violation (CLFV):

   • Radiative Decays ($l_i \to l_j \gamma$): Calculated via one-loop dipole diagrams. The branching ratios depend on the product of couplings involving different generations (e.g., $y_{32}y_{31}$).
   • $\mu-e$ Conversion in Nuclei: Calculated at the tree level via LQ exchange. This process is highly sensitive to the product of couplings involving first-generation quarks and leptons ($y_{11}y_{12}$), providing much stronger constraints than loop-induced processes.

3. Collider Phenomenology:

   • Analyzed the pair production of singlet LQs ($pp \to SS^\dagger$) via QCD interactions (gluon fusion and $q\bar{q}$ annihilation).
   • Calculated the signal cross-section for the decay channel $S \to t\mu$, considering the branching ratios constrained by low-energy observables.

3. Key Contributions

• Model Extension: Successfully integrated scalar leptoquarks into the 331LHN framework, defining the full particle content and Yukawa sector.
• Dual Constraint Analysis: Performed a comprehensive global fit combining the muon $g-2$ anomaly with low-energy CLFV limits (MEG, BaBar, Belle II, SINDRUM II) and $\mu-e$ conversion limits (Mu2e, COMET projections).
• Hierarchy Discovery: Demonstrated that satisfying all constraints requires a specific normal hierarchical pattern in the Yukawa couplings:
  • Third-generation couplings ($y_{32}$) must be relatively large to explain $g-2$.
  • First-generation couplings ($y_{11}, y_{12}$) must be heavily suppressed to avoid violating $\mu-e$ conversion limits.
• Updated Projections: Provided constraints based on the latest 2025 lattice QCD results for $a_\mu$, which significantly tighten the lower bound on the LQ mass compared to the 2021 analysis.

4. Results

Muon $g-2$ and Mass Bounds

• 2021 Data: To explain the $4.2\sigma$ discrepancy, the singlet LQ mass must be $m_S \gtrsim 1.8$ TeV, with the product of couplings $y_{32}^L y_{32}^R$ in the range $(3.3 - 9.3) \times 10^{-3}$.
• 2025 Data: Incorporating the updated SM prediction (which reduces the experimental discrepancy), the required contribution is smaller. However, to remain consistent with the 1$\sigma$ band of the new data, the lower bound on the mass increases to $m_S \gtrsim 6$ TeV.

Lepton Flavor Violation Constraints

• $\mu \to e\gamma$: The branching ratio constraint forces the first-generation coupling $y_{31}$ to be extremely small ($\sim 10^{-6}$) if $y_{32} \sim 10^{-2}$. This implies a hierarchy $h_{21} = y_{32}/y_{31} \sim 10^4$.
• $\tau \to \mu\gamma$ and $\tau \to e\gamma$: Similar hierarchical suppression is required for third-to-second and third-to-first generation transitions.
• $\mu-e$ Conversion: This is the most stringent constraint. The tree-level nature of the process requires the product $y_{11}^L y_{11}^R y_{12}^L y_{12}^R \lesssim 1.2 \times 10^{-11}$.
• Coupling Pattern: The viable parameter space exhibits a "normal hierarchy" where couplings to the third generation are $\mathcal{O}(10^{-2})$, while couplings to the first generation are suppressed to $\mathcal{O}(10^{-6})$ or lower.

Collider Phenomenology

• LHC Reach: For LQ masses in the multi-TeV range (required by the 2025 $g-2$ analysis), the QCD pair production cross-section at the LHC ($\sqrt{s}=14$ TeV) drops rapidly (from $\sim 10$ fb at 1 TeV to $\sim 10^{-4}$ fb at 3 TeV).
• Signal Suppression: Even with high luminosity (HL-LHC), the signal rates for $pp \to SS^\dagger \to t\bar{t}\mu^+\mu^-$ are suppressed due to the small branching ratios required by CLFV constraints.
• Future Prospects: Direct detection at the current LHC is challenging. Future hadron colliders (e.g., $\sqrt{s}=27$ TeV) are necessary to probe the multi-TeV mass range effectively.

5. Significance

This paper provides a critical test of the 331LHN model extended with scalar leptoquarks. It demonstrates that while the model can theoretically accommodate the muon $g-2$ anomaly, the simultaneous requirement of satisfying CLFV and $\mu-e$ conversion limits forces the leptoquark mass into the multi-TeV regime (specifically $>6$ TeV under 2025 assumptions).

The work highlights a "tension" between explaining the muon anomaly and evading flavor constraints:

1. Explaining $g-2$ requires large couplings to the muon and top quark.
2. Avoiding $\mu-e$ conversion requires tiny couplings to the electron and light quarks.
3. The resulting hierarchy implies that the leptoquark is likely too heavy to be discovered via direct production at the current LHC, making indirect probes (precision $g-2$ measurements and next-generation CLFV experiments like Mu2e and COMET) the primary tools for testing this scenario.

The study underscores the importance of future high-energy colliders and precision low-energy experiments in distinguishing between different New Physics scenarios.