With the advent of high luminosity accelerators, multi parton scattering (MPS)
physics at high-energy hadron colliders like the Tevatron at Fermilab and the
Large Hadron Collider (LHC) at CERN, in particular those observing double
parton scattering (DPS), has become one of the hottest topics of the modern
experimental particle physics. DPS results are usually interpreted in terms of
the so-called ``pocket formula''. This formula defines that the value of DPS
effective cross section should be universal, independent of the final states or
the phase space. However, there are some models that predict correlations
between partons inside a hadron. This leads us to a new class parton
distribution function, double PDF, that are not tantamount to the simple
multiplication of the standard single PDF.
Standard wisdom tells us that DPS should be more preferable at high energies.
This is why most of the phenomenological analysis of a possible impact of the
double PDF effect are focused on LHC energies. However, effects are mainly
expected at higher values of the Bjorken-$x$ or with a significant gap between
the $x$ values. Even for double $W$ boson production the typical value for the
Bjorken-$x$ is $\langle x \rangle \sim 0.01$.
In contrast to that, preferable conditions can be easily achieved at lower
energy experiments like the Spin Physics Detector at the NICA collider, where
the DPS contribution could be relatively small, but is still expected to be
far from zero.
In this talk we discuss the possible impact of the evolution of the double PDF
on the double $J/\psi$ production and the DPS effective cross sections at NICA
energies within the Gaunt--Stirling model.