  The color structure of the gluon field magnetic mass is investigated in the
lattice SU(2) gluodynamics. To realize that, the interaction between a
monopole-antimonopole string and external neutral Abelian chromomagnetic field
flux is considered. The string is introduced in the way proposed by Srednicki
and Susskind. The neutral Abelian field flux is introduced through the twisted
boundary conditions. Monte Carlo simulations are performed on 4D lattices at
finite temperature. It is shown that the presence of the Abelian field flux
weakens the screening of the string field. That means decreasing the gluon
magnetic mass for this environment. The contribution of the neutral Abelian
field has the form of "enhancing" factor in the fitting functions. This
behavior independently confirms the long-range nature of the neutral Abelian
field reported already in the literature. The comparison with analytic
calculations is given.
