Constraints on $\Delta L=2$ Vector Bosons with Tree Couplings to SM Particles

We investigate phenomenological implications of vector bosons $V$ transforming as (1, 2, -3/2) under the standard model (SM) product gauge group $SU(3)_C$, $SU(2)_L$ and $U(1)_Y$. These vector bosons can couple to two SM leptons at tree-level forming dimension-4 operators. These operators dictate $V$ to have two units of global lepton number, $\Delta L = 2$. The operators generated conserve the global lepton number but can violate generational lepton numbers. We study constraints on the couplings $Y$ of $V$ to SM particles using tree-level processes such as $l_\alpha^{-} \to l_\beta^{+} l_\rho^{-} l_\sigma^{-}$, muonium and antimuonium oscillation, neutrino trident scattering, inverse muon decay, $e^- e^+ \to l^- l^+$, and also one-loop level processes such as the magnetic dipole moment of a charged lepton and $l_i \to l_j \gamma$. Strong constraints are obtained from $l_\alpha^{-} \to l_\beta^{+} l_\rho^{-} l_\sigma^{-}$ with $\left|Y_{e e} Y^{*}_{\mu e}\right| < 3.29 \times 10^{-11}\left(m_{V}/ \mathrm{GeV}\right)^{2},\left|Y_{e e} Y^{*}_{e \mu}\right| < 3.29 \times 10^{-11}\left(m_{V}/ \mathrm{GeV}\right)^{2}$ and from $l_i \to l_j \gamma$ with $\left|Y_{\tau e}Y_{\mu\tau}^{*}\right|<3.46\times10^{-12}(m_{V}/ \mathrm{GeV})^2, \left|Y_{e \tau}Y_{\tau \mu}^{*}\right|<3.46\times10^{-12}\left(m_{V}/ \mathrm{GeV}\right)^2$, respectively. Interestingly, the imaginary part of the coupling constant in our model induces CP violation, which is constrained by experimental limits on the electric dipole moment.