Effective Monopole Potential for SU(2) Lattice Gluodynamics in Spatial Maximal Abelian Gauge

We investigate the dual superconductor hypothesis in finitetemperature SU(2) lattice gluodynamics in the Spatial Maximal Abelian gauge. This gauge is more physical than the ordinary Maximal Abelian gauge due to absence of non-localities in temporal direction. We show numerically that in the Spatial Maximal Abelian gauge the probability distribution of the abelian monopole field is consistent with the dual superconductor mechanism of confinement: the abelian condensate vanishes in the deconfinement phase and is not zero in the confinement phase. The dual superconductor hypothesis of color confinement [1] in gluodynamics has been confirmed by various lattice calculations [2] in the so-called Maximal Abelian (MaA) projection [3]. This hypothesis is based on a partial gauge fixing of a nonabelian group up to its abelian subgroup. After gauge is fixed abelian monopoles arise due to singularities in the gauge fixing conditions [4]. If monopoles are condensed then the vacuum of gluodynamics behaves as a dual superconductor and the electric charges (quarks) in such a vacuum are confined. The MaA projection on the lattice is defined by the condition [3]: max Ω RMaA[U ] , RMaA[U ] = ∑ l Tr[σ3U + l σ3Ul] , l = {x, μ} , (1) where the summation is over all lattice links and Ux,μ are the lattice SU(2) gauge fields. The gauge fixing condition (1) contains time components of the gauge fields, Ux,4, therefore abelian operators in the MaA projection correspond to nonlocal in time operators in terms of the original SU(2) fields Ux,μ. To show this let us consider the expectation value of the U(1) invariant operator O in the Maximal Abelian gauge [5, 6]: < O >MaA = 1 ZMaA ∫ DU exp{−S[U ] + λRMaA[U ]}∆FP [U ; λ]O(U) , λ → +∞ , (2) where ZMaA =< 1 >MaA is the partition function in the fixed gauge. ∆FP [U ; λ] is the Faddeev–Popov determinant: 1 = ∆FP [U ; λ] · ∫ DΩ exp{λRMaA[U]} , λ → +∞ . Shifting the fields U → UΩ+ in eq.(2) and integrating over Ω both in the nominator and denominator, we get: < O >MaA =< ÕMaA > , ÕMaA(U) = ∫ DΩ exp{λRMaA[U]}O(U) ∫ DΩ exp{λRMaA[UΩ]} , ÕMaA is the SU(2) invariant operator. Since λ → +∞ we can use the saddle point approximation to calculate ÕMaA: ÕMaA(U) = N(U) ∑ j=1 Det 1 2 MMaA[U Ω(j) ]O(UΩ(j))