LFV decays in a 3-3-1 model with singlet leptoquarks

Technical Summary: LFV decays in a 3-3-1 model with singlet leptoquarks

Problem and Motivation
The paper addresses the persistent discrepancy between Standard Model (SM) predictions and experimental measurements of the anomalous magnetic moments (AMMs) of charged leptons, particularly the muon ($g-2)_\mu$. While recent data suggests a deviation of approximately $10^{-9}$, explaining this within a unified framework that also accounts for lepton-flavor-violating (LFV) processes remains a significant challenge. Many Beyond the Standard Model (BSM) scenarios that enhance AMMs via new interactions often induce charged LFV (cLFV) decays (e.g., $\mu \to e\gamma$) at levels that conflict with stringent experimental bounds. The authors aim to investigate whether the 3-3-1 model, specifically when supplemented with a singlet scalar leptoquark (denoted as 331LQ), can simultaneously accommodate the observed muon AMM anomaly and satisfy current constraints on cLFV decays, while predicting observable rates for LFV decays of the SM-like Higgs boson ($h$) and the $Z$ boson.

Methodology
The study extends the 331LQ framework, which incorporates a singlet scalar leptoquark $S$ interacting with both leptons and quarks. The authors perform a comprehensive phenomenological analysis including:

1. Model Construction: They define the particle content, including the gauge group $SU(3)_C \otimes SU(3)_L \otimes U(1)_X$, and the Yukawa interactions generating tree-level masses for quarks and leptons. The leptoquark $S$ couples to quarks and leptons, with specific attention paid to the couplings involving the third quark family (top quark) in the physical basis, which are derived from the CKM mixing matrix and unknown leptoquark coupling parameters.
2. Analytical Derivations: The authors derive general analytical expressions for one-loop contributions to:
   • Charged lepton AMMs ($\Delta a_e, \Delta a_\mu$).
   • cLFV radiative decays ($e_b \to e_a \gamma$).
   • LFV decays of the Higgs boson ($h \to e_a e_b$).
   • LFV decays of the $Z$ boson ($Z \to e_a e_b$).
     These calculations utilize Passarino-Veltman functions and consider contributions from both fermionic (quark) and scalar (leptoquark) loops in the unitary gauge.
3. Numerical Analysis: A numerical scan is performed over the model's free parameters (leptoquark mass $m_S$, Higgs-leptoquark coupling $\lambda_{hSS}$, and Yukawa couplings $g^{LQ}_{3i}, h^{LQ}_{3i}$) within perturbative limits. The analysis enforces all current experimental constraints on AMMs and cLFV branching ratios, specifically focusing on regions that allow for sizable deviations in $\Delta a_\mu$ and potentially $\Delta a_e$.

Key Contributions and Results

• Correlation between AMMs and cLFV: The study confirms that the dominant one-loop contributions to $\Delta a_e$ and $\Delta a_\mu$ are proportional to the real part of the product of leptoquark couplings involving the top quark and the respective charged lepton ($\Delta a_e \propto \text{Re}[m_t g^{LQ*}_{3e} h^{LQ}_{3e}]$). Consequently, satisfying the experimental bound on $Br(\mu \to e\gamma)$ imposes severe constraints on the product of couplings involving different generations, effectively suppressing $\Delta a_e$ to negligible levels ($< 10^{-19}$) in regions where $\Delta a_\mu$ is large.
• Linear Correlation of LFV Branching Ratios: A central finding is the nearly linear correlation between the branching ratios of LFV decays of the Higgs and $Z$ bosons and their corresponding cLFV radiative decays: $Br(h, Z \to e^\pm_b e^\mp_a) \propto Br(e_b \to e_a \gamma)$. This relationship arises because the underlying form factors for these processes are dominated by the same coupling combinations ($f_{ba}$).
• Phenomenological Implications for $\Delta a_\mu$:
  • In parameter regions where $|\Delta a_\mu| \ge 10^{-11}$, the model predicts that $Br(h, Z \to \tau^\pm \mu^\mp)$ can reach current experimental upper limits. However, this comes at the cost of suppressing $Br(h, Z \to \tau^\pm e^\mp)$ and $Br(h, Z \to \mu^\pm e^\mp)$ to levels far below current sensitivities due to the tight constraint on $Br(\mu \to e\gamma)$.
  • Conversely, regions allowing for a measurable electron AMM ($10^{-14} \le |\Delta a_e| \le 8 \times 10^{-13}$) require $|\Delta a_\mu|$ to be suppressed to the order of $O(10^{-16})$. In this scenario, only the $\tau \to e\gamma$, $h \to \tau e$, and $Z \to \tau e$ channels approach current or future experimental sensitivities.
• Suppression of Light Quark Contributions: The analysis demonstrates that contributions from the two light up-type quarks are negligible compared to the top quark contributions, validating the approximation that only the top quark loop dominates the LFV amplitudes.

Significance
The paper claims that the 331LQ framework provides a consistent theoretical structure to explore the interplay between charged lepton AMMs and LFV processes. Its primary significance lies in revealing specific, testable correlations:

1. It establishes that the 331LQ model cannot simultaneously explain a large $\Delta a_\mu$ and a large $\Delta a_e$ without violating current cLFV bounds.
2. It predicts that if the muon AMM anomaly is explained by this model, the most promising experimental signatures for LFV will be in the $\tau-\mu$ sector (specifically $h \to \tau\mu$ and $Z \to \tau\mu$), while $\tau-e$ and $\mu-e$ channels remain highly suppressed.
3. It highlights that future improvements in the sensitivity of cLFV radiative decays (e.g., $\mu \to e\gamma$) will directly constrain the predicted rates for LFV Higgs and $Z$ decays, offering a pathway to test the model's viability.

The authors conclude that while the model offers a mechanism to address the muon $g-2$ anomaly, it does so by predicting a specific hierarchy of LFV signals that are tightly constrained by existing data, particularly the non-observation of $\mu \to e\gamma$.