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Neutrino-(anti)neutrino forward scattering potential for massive neutrinos at low energies

In this work, we calculate expression for the potential due to neutrino-(anti)neutrino forward scattering at low energies ($E<<m_{Z^0}$) for ultra-relativistic massive neutrinos ($E>>m_ν$), a representative regime within astrophysical scenarios. There is a broadly used expression for this potential in the literature, which, however, lacks an explicit derivation from basic principles of quantum field theory. Therefore, this paper has the intention to guide the reader through the steps and concepts to derive this potential, trying to be clear and pedagogical. Moreover, we used a rigorous approach concerning the massive nature of the neutrinos, using massive quantized neutrino fields throughout the entire process, while the usual approach is to consider massless neutrino fields at the interaction. In this context, we explicitly show the validity of the massless neutrino fields approximation at the ultra-relativistic regime, as expected. As the last step, we connect the potential expression to the density matrix formalism, which is a usual framework for works considering neutrino-neutrino interactions. We also discuss some theoretical details through the paper, such as the normal ordering of quantum operators and the implications of massive fields in the neutrino state at its production.