Matter degrees of freedom and string breaking in Abelian projected quenched $\mathrm{SU}\left(2\right)\mathrm{}$ QCD

In the Abelian projection the Yang-Mills theory contains Abelian gauge fields (diagonal degrees of freedom) and the Abelian matter fields (off-diagonal degrees) described by a complicated action. The matter fields are essential for the breaking of the adjoint string. We obtain numerically the effective action of the Abelian gauge and the Abelian matter fields in quenched $\mathrm{SU}\left(2\right)\mathrm{}$ QCD and show that the Abelian matter fields provide an essential contribution to the total action even in the infrared region. We also observe the breaking of an Abelian analogue of the adjoint string using Abelian operators. We show that the adjoint string tension is dominated by the Abelian and the monopole contributions similarly to the case of the fundamental particles. We conclude that the adjoint string breaking can successfully be described in the Abelian projection formalism.