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A modified Friedberg-Lee symmetry for the TM1 neutrino mixing

In this letter, we put forward a special neutrino mass matrix which is invariant under a modified Friedberg-Lee (FL) transformation $ν^{}_e \to ν^{}_e-2θ$ and $ν^{}_{μ, τ} \to ν^{}_{μ, τ}+θ$ with $θ$ being a space-time independent element of the Grassmann algebra. Compared to the original FL symmetry (with the transformation $ν^{}_{e, μ, τ} \to ν^{}_{e, μ, τ}+θ$) which results in the TM2 neutrino mixing, the modified FL symmetry will lead us to the TM1 mixing which has a better agreement with the experimental results. While the original FL symmetry has to be broken in order to produce a realistic neutrino mass spectrum, the modified FL symmetry is allowed to remain intact and give us a vanishing $m^{}_1$. A combination of the FL symmetry with the $μ$-$τ$ reflection symmetry is also discussed.