Lorentz symmetry is the invariance of physical interactions under Lorentz transformations consisting of boosts and rotations. Lorentz symmetry is a feature of the theories of particle physics and gravitation. Other symmetry in the particle physics is CPT symmetry that formed from the product of charge conjugation C, parity P, and time reversal T. However, various experiments confirm Lorentz and CPT symmetry to high precision, but we expect Lorentz and CPT violation can provide signals of Planck scale physics. In fact, the original motivation in the subject, arose from spontaneous symmetry breaking mechanism that exist in unified theories at the Planck scale. The Standard Model Extension, or SME, provides a description of possible of Lorentz and CPT violations in the context of effective field theory. In this work, the regard is on using a useful SME limit to investigate low-energy experiments. This limit is called the minimal SME (mSME) that contains all known particle fields in the Standard Model (SM) as well as observer-independent terms that break Lorentz and CPT symmetry but maintain SU(3) SU(2) U(1) gauge invariance and power-counting renormalizability. The mSME consist of the lepton, quark, Yukawa, Higgs, and gauge sectors. In this thesis, we develop the neutral weak interaction in the mSME. From this extended Lagrangian by Lorentz violation terms in lepton sector, we derive vertices interaction photon-fermion and Z boson-lepton. The mSME coefficients moreover Lorentz symmetry break, can violate the lepton flover conservation. In particular, we focus on the lepton flavor violation effect on the decay and calculate the decay rate and branching rate of . At the end, we compare this calculations to experimental data and obtain a bound on the mSME coefficient c zx .