Loophole in K --> pi nu nubar Search and New Weak Leptonic Forces

Weakly interacting K --> pi X^0 emission with m_{X^0} simeq m_{pi^0} is out of sight of the current K^+ --> pi^+ nu nubar study, but it can be sensed by the K_L --> pi^0 nu nubar search. This evades the usual Grossman-Nir bound of B(K_L --> pi^0 nu nubar) < 1.4 x 10^-9, thus the KOTO experiment is already starting to probe New Physics. An intriguing possibility is the Z gauge boson of a weak leptonic force that couples to L_mu - L_tau (the difference between the muon and tauon numbers), which may explain the long-standing muon g-2 anomaly, but is constrained by nu_mu N --> nu_mu N mu^+ mu^- scattering to m_{Z} lesssim 400 MeV. An explicit model for K --> pi Z is given, which illustrates the link between rare kaon and B --> K mu^+ mu^-, K^{(*)} nu nubar decays. Complementary to these searches and future lepton experiments, the LHC might discover the scalar boson phi responsible for light m_{Z} generation via phi --> Z Z --> 2(mu^+ mu^-).