Wilson Loops for U(4) and Su(4) Lattice Gauge Theories in Four Dimensions

In previous studies, we examined pure U(4) [1] and SU(4) [2] lattice gauge theories in four spacet ime dimensions. Since first-order phase transitions were found in both of these theories, it seems reasonable that these theories should be studied together so that they can be compared and contrasted. To make a more detailed exploration of the phase transitions in these theories, we study the string tensions, via the Wilson loops, for these gauge groups. The decoupling of the SU(4) and U(1) components of the U(4) gauge group, in the low-temperature region, should be observed so that U(4) and SU(4) gauge theories should merely be shifted in the inverse temperature. Both U(4) and SU(4) color should be confined while the U(1) charges of U(4) gauge theory should be deconfined. Similar behavior has recently been observed for the gauge groups U(2) [3], SU(2) [4], U(3) [5] and SU(3) [6]. The partition function for the U(N) and SU(N) non-abelian gauge theories on a four-dimensional euclidean lattice is defined by