3.10 Coupling to SU(2)-gauge fields

Berg and collaborators [42Jump To The Next Citation Point43Jump To The Next Citation Point44Jump To The Next Citation Point27Jump To The Next Citation Point28Jump To The Next Citation Point41Jump To The Next Citation Point] coupled the pure-curvature action geometrically to the Wilson action for SU (2)-gauge fields via dimensionless weight factors Wb,
Regge β-∑ --Vb-- S = S − 2 Wb Re [Tr(1 − Ub)], Wb = const (Ab )2, (19 ) b
where Ub denotes the SU (2)-holonomy around b and β is proportional to the inverse square coupling constant, β = -42 g. One motivation was to understand whether in the well-defined pure-gravity region, one can choose the elementary particle masses to be ≪ m Pl as β → ∞, as one might expect for a realistic gravity+matter system. This seems a rather distant hope, since in the simulations performed so far, the ratio mPl ∕mhadron is of order unity.

Initial computations were performed on a 2 × 43-lattice with the scale-invariant measure, and at k = 0.01 [4244], and extended to larger k-values in [43]. For k ≤ 0.04, one finds some evidence for a (first-order?) transition; the region of β where the transition occurs does not change much with k. Beirl et al. [2728] extended this analysis by measuring the static potential V of a quark–antiquark pair on lattices of size 4 × 63 and 4 × 83. With and without gravity, one finds both a confined and a deconfined phase; in the presence of gravity, the transition occurs at a smaller β-value. More recently, Berg et al. [41] have gathered further data on the location and stability of the well-defined phase in the (k, β)-plane, and extracted a string tension for various β-values.

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