[Paper Review] The anomalous lepton magnetic moment, LFV decays and the fourth generation

W. J. Huo, Tong Feng|ArXiv.org|Jan 20, 2003

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper investigates the sequential fourth generation Standard Model with a heavy fourth neutrino ($\nu^\prime$) to explain the muon anomalous magnetic moment ($g-2$) anomaly and lepton flavor violation (LFV) decays ($\tau\to l\gamma$, $\mu\to e\gamma$). Using experimental bounds on LFV and the $1.6\sigma$ $g-2$ deviation, it shows that the $4\times4$ leptonic mixing matrix elements $|V_{1\nu^\prime}V_{2\nu^\prime}|^2$, $|V_{1\nu^\prime}V_{3\nu^\prime}|^2$, and $|V_{3\nu^\prime}V_{2\nu^\prime}|^2$ constrain the fourth neutrino mass to $m_{\nu^\prime} < 80\,\text{GeV}$, excluding most of the parameter space and imposing stringent limits on the existence of a fourth generation.

ABSTRACT

We investigate the lepton flavor violation (LFV) decays, $τ o lγ$ ($l=μ, e$) and $μ o eγ$, and the newly observed muon $g-2$ anomaly in the framwork of a squential fourth generation model with a heavy fourth neutrino, $ν'$. By using the recent experimental bounds, we take the constraints of the $4 imes 4$ leptonic mixing matrix element factors, $|V_{1ν'} V_{2ν'}|^2$, $|V_{1ν'} V_{3ν'}|^2 $ and $|V_{3ν'} V_{2ν'}|^2$. We find that LFV decays and $g_μ-2$ can exclude most of the parameter space of the 4th generation neutrino mass $m_{ν'}$ and give stringent constraints on the existence of the fourth generation.

Motivation & Objective

• Address the persistent $1.6\sigma$ discrepancy in the muon anomalous magnetic moment ($g-2$) observed in E821 experiments, which may signal new physics beyond the Standard Model.
• Investigate whether a heavy fourth-generation neutrino ($\nu^\prime$) in the sequential fourth generation model can simultaneously explain the $g-2$ anomaly and lepton flavor violation (LFV) decays such as $\tau\to\mu\gamma$, $\tau\to e\gamma$, and $\mu\to e\gamma$.
• Use current experimental bounds on LFV decays and the $g-2$ deviation to constrain the $4\times4$ leptonic mixing matrix elements involving $\nu^\prime$, particularly $|V_{l\nu^\prime}|^2$ for $l = e, \mu, \tau$, and derive limits on $m_{\nu^\prime}$.
• methodical analysis of the $\nu^\prime$-mediated one-loop diagrams for $g-2$ and LFV decays, using the effective Lagrangian and the $f(x)$ function to compute decay rates and $g-2$ contributions as functions of $m_{\nu^\prime}$ and mixing parameters.
• Apply unitarity constraints on the $4\times4$ MNS mixing matrix to narrow down the allowed parameter space for $m_{\nu^\prime}$ and $V_{l\nu^\prime}$, ensuring consistency with observed limits.

Proposed method

• The paper computes the decay width for $\tau\to\mu\gamma$ as $\Gamma(\tau\to\mu\gamma) = \frac{\alpha G_F^2 m_\tau^5}{96} |V_{2\nu^\prime}V_{3\nu^\prime}|^2 f^2(x)$, with $x = m_{\nu^\prime}^2 / m_W^2$ and $f(x)$ defined via a logarithmic and rational function.
• Similarly, the widths for $\tau\to e\gamma$ and $\mu\to e\gamma$ are derived as $\Gamma(\tau\to e\gamma) = \frac{\alpha G_F^2 m_\tau^5}{96} |V_{1\nu^\prime}V_{3\nu^\prime}|^2 f^2(x)$ and $\Gamma(\mu\to e\gamma) = \frac{\alpha G_F^2 m_\mu^5}{96} |V_{1\nu^\prime}V_{2\nu^\prime}|^2 f^2(x)$.
• The contribution to the muon anomalous magnetic moment is calculated as $a^{\text{SM4}}_\mu = \alpha \left(\frac{g}{\sqrt{2}} V_{2\nu^\prime}\right)^2 \frac{2m_\mu^2}{m_W^2} f(x)$, with $f(x)$ as in the decay amplitudes.
• The parameter space for $f(x)$ and $V_{2\nu^\prime}$ is derived from the observed $\delta a_\mu = 26(16)\times 10^{-10}$, leading to $|V_{2\nu^\prime}|^2 = \frac{\sqrt{2} \cdot \delta a_\mu}{8 G_F \alpha m_\mu^2 f(x)}$.
• The function $f(x)$ is evaluated numerically, revealing that it is negative except in a narrow range $58\,\text{GeV} < m_{\nu^\prime} < 80\,\text{GeV}$, where it yields a positive contribution to $a_\mu$.
• Unitarity of the $4\times4$ MNS matrix is applied to constrain the mixing elements, further narrowing the allowed range of $m_{\nu^\prime}$ and $V_{l\nu^\prime}$.

Experimental results

Research questions

• RQ1 Can the $1.6\sigma$ discrepancy in the muon anomalous magnetic moment ($g-2$) be explained by a heavy fourth-generation neutrino ($\nu^\prime$) in the sequential fourth generation model?
• RQ2 What are the constraints on the $4\times4$ leptonic mixing matrix elements $|V_{l\nu^\prime}|^2$ derived from current experimental bounds on $\tau\to l\gamma$ and $\mu\to e\gamma$ decays?
• RQ3 What is the allowed mass range for the fourth-generation neutrino $\nu^\prime$ when simultaneously satisfying the $g-2$ anomaly and LFV decay constraints?
• RQ4 How does the unitarity of the $4\times4$ MNS mixing matrix further restrict the viable parameter space for $m_{\nu^\prime}$ and $V_{l\nu^\prime}$?
• RQ5 Is the $\nu^\prime$ contribution to $a_\mu$ positive only in a narrow mass window, and does this window align with current experimental limits on new physics?

Key findings

• The fourth-generation neutrino mass $m_{\nu^\prime}$ is constrained to be less than 80 GeV by the combined constraints of the $g-2$ anomaly and LFV decay bounds, excluding most of the parameter space.
• The function $f(x)$, which governs the $\nu^\prime$ contribution to $a_\mu$, is negative except in a narrow range $58\,\text{GeV} < m_{\nu^\prime} < 80\,\text{GeV}$, where it yields a positive contribution to $a_\mu$, matching the observed $\delta a_\mu > 0$.
• The upper bound of $m_{\nu^\prime} < 80\,\text{GeV}$ from $g-2$ and LFV decays is in tension with current experimental limits that suggest no new physics below several hundred GeV, suggesting the fourth generation may be heavier or excluded.
• The unitarity of the $4\times4$ MNS matrix, combined with small mixing elements, further narrows the allowed range for $m_{\nu^\prime}$, making the model more constrained and less viable.
• The product $|V_{1\nu^\prime}V_{2\nu^\prime}|^2$ is constrained by $\mu\to e\gamma$ bounds, and the resulting $m_{\nu^\prime}$ upper limit is consistent with the $g-2$ constraint but conflicts with the absence of signals in direct searches.
• The paper concludes that LFV decays and the $g-2$ anomaly together exclude most of the parameter space for $m_{\nu^\prime}$, imposing a stringent constraint on the existence of a fourth generation.