Neutrino Oscillations in Presence of Diagonal Elements of Scalar NSI: An Analytic Approach

Scalar Non-Standard Interactions (SNSI) in neutrinos can arise when a scalar mediator couples to both neutrinos and standard model fermions. This beyond the Standard Model (BSM) scenario is particularly interesting as the SNSI contribution appears as a density-dependent perturbation to the neutrino mass, rather than appearing as a matter-induced potential, and the neutrino oscillation probabilities uniquely depend on the absolute neutrino masses. In this work, we show the complex dependence of the SNSI contributions on the neutrino masses and discuss how the mass of the lightest neutrino would regulate any possible SNSI contribution in both mass ordering scenarios. We derive the analytic expressions for neutrino oscillation probabilities, employing the Cayley-Hamilton theorem, in the presence of diagonal elements of SNSI. The expressions are compact and shows explicit dependence on matter effects and the absolute neutrino masses. The analytic expressions calculated here allow us to obtain the dependence of the SNSI contribution on mass terms of the form $m_1 + m_2$, $m_2 - m_1$, $m_1c_{12}^2 + m_2s_{12}^2,$ $ m_1s_{12}^2 + m_2c_{12}^2$, and $m_3$. We then explore the non-trivial impact of neutrino mass ordering on the SNSI contribution. The dependence of the SNSI contribution on the 3$\nu$ parameters is then thoroughly explored using our analytic expressions.