We have done a phenomenological study on the neutrino mass matrix $M_ u$ favoring two zero texture in the framework of left-right symmetric model (LRSM) where type I and type II seesaw naturally occurs. The type I seesaw mass term is considered to be following a trimaximal mixing (TM) pattern. The symmetry realizations of these texture zero structures has been realized using the discrete cyclic abelian $Z8times Z2$ group in LRSM. We have studied six of the popular texture zero classes named as A1, A2, B1, B2, B3 and B4 favoured by neutrino oscillation data in our analysis. We basically focused on the implications of these texture zero mass matrices in low energy phenomenon like neutrinoless double beta decay (NDBD) and lepton flavour violation (LFV) in LRSM scenario. For NDBD, we have considered only the dominant new physics contribution coming from the diagrams containing purely RH current and another from the charged Higgs scalar while ignoring the contributions coming from the left-right gauge boson mixing and heavy light neutrino mixing. The mass of the extra gauge bosons and scalars has been considered to be of the order of TeV scale which is accessible at the colliders.
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