Test of vector portal with dark fermions in the charge-exchange reactions in the NA64 experiment at CERN SPS

We consider an experiment to search for dark sector particles in dark photon kinetic mixing model by analyze invisible and semi-invisible decays of neutral mesons \(M^0 = \pi^0\), \(\eta\), \(\eta'\), \(\omega\), \(f_2(1270)\), produced in the NA64 experiment at the CERN SPS. The approach proposed in Ref.~\cite{Gninenko:2014sxa} is to use the charge-exchange reactions \(\pi^- + (A, Z) \to M^0 + (A,Z-1); M^0 \to\) invisible or semi-invisible of high-energy pions (or kaons) at a nuclei target as a source of \(M^0\)s, which subsequently decay invisibly into dark sector. This reaction chain would lead to a striking signature of the signal event - the complete disappearance of the beam energy in the setup. Using data obtained from the study of charge-exchange reactions at IHEP (Protvino) and Fermilab (Batavia) we show that the integral cross sections \(\sigma\) for production of the neutral mesons \(M^0\) are slightly deviate from phenomenological formula \(\sigma \sim Z^{2/3}\), where \(Z\) is the nuclei charge. In particular, we present the formulas for the differential and integral sections that explicitly depend on the Mandelstam and \(Z\) variables. Derived formulas are used to predict the cross sections as a function of beam energy for several target nuclei, and to estimate the projection sensitivity for the proposed search for the \(M^0\to\) semi-invisible and \(M^0\to\) invisible decays through the vector portal to dark sector. Sensitivity to different semi-invisible decay modes of neutral pseudoscalar mesons is studied.