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Let's say we want to bosonize the contact interaction term: $:[\Psi^{\dagger}(x)\Psi(x)]^2:$, where $::$ denotes normal ordering. If one first writes $\Psi(x)={\rm e}^{-i k_F x}\psi_L(x) +{\rm e}^{i k_F x}\psi_R(x) $, then

$\Psi^{\dagger}(x)\Psi(x) = \rho_L +\rho_R + {\rm e}^{2ik_F x}\psi^{\dagger}_L\psi_R +{\rm e}^{-2ik_F x}\psi^{\dagger}_R \psi_L$,

where $\rho_{L/R}= \psi^{\dagger }_{L/R}\psi_{L/R}$. Then,

$:[\Psi^{\dagger}(x)\Psi(x)]^2:= :\rho_L^2+\rho_R^2 +{\rm cos}(2k_Fx)(\psi^{\dagger}_L \psi_R+\psi^{\dagger}_R \psi_L) : $.

Is this correct? In https://arxiv.org/abs/cond-mat/9805275 the authors do not consider the last term. What is the reason and how can it be dealt with?

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