@Neu3] are one such a case, and the goal of this article is to present the theoretical analysis and estimate of the effect of spontaneous collapses on the oscillatory behavior of neutrinos.

A previous analysis of this kind was proposed in [@Ch], based on the Penrose model of gravity induced collapse [@Pd1; @Pd2]. This model however lacks a fully consistent dynamical equation, and previous attempts to fill this gap [@Dg] have been criticized [@Gpd]. Moreover, as shown in [@Gpd], the model fails when applied to single constituents (protons, electrons, ...), since in this case its predictions are in conflict with known experimental data. Therefore, the application of gravity induced collapse models to neutrino oscillations is rather delicate.

Here we will compute the spontaneous collapse effect on neutrino oscillations using the mass proportional version [@Cslmass; @Fu] of the CLS model [@Csl], which is widely used in physical applications, together with the GRW model [@Grw]. Its dynamics is described by the following stochastic differential equation: $$\label{eq:csl-massa}
d|\phi_{t}\rangle=\left[-\frac{i}{\hbar}Hdt+\frac{\sqrt{\gamma}}{m_{0}}\int d\mathbf{x}\,\left(M(\mathbf{x})-\left\langle M(\mathbf{x})\right\rangle \right)dW_{t}(\mathbf{x})-\frac{\gamma}{2m_{0}^{2}}\int d\mathbf{x}\,\left(M(\mathbf{x})-\left\langle M(\mathbf{x})\right\rangle \right)^{2}dt\right]|\phi_{t}\rangle,$$ where the operator $H$ is the standard quantum Hamiltonian of the system and the other two terms induce the collapse of the wave function in space. The mass $m_0$ 