Coupling to SU(2)-gauge fields

3.10 Coupling to SU(2)-gauge fields

Berg and collaborators [42

, 43

, 44

, 27

, 28

, 41

] coupled the pure-curvature action geometrically to the Wilson action for $SU (2)$ -gauge fields via dimensionless weight factors $Wb$ ,

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 $k = 0.01$ [42 , 44 ], 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. [27 , 28 ] 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.