hoppingparameter

The hopping parameter, usually denoted by kappa (κ), is a parameter used in lattice gauge theory, particularly in simulations with Wilson-type fermions, to control the strength of fermion hopping between neighboring lattice sites in the discretized Dirac operator. It plays a central role in linking the lattice action to the quark mass.

In Wilson fermion formulations, κ is related to the bare quark mass m0 in lattice units by am0 = (1/(2κ)) − 4, or equivalently κ = 1/(2am0 + 8). The massless (chiral) limit corresponds to a critical value κc where am0 = 0. For κ < κc, the effective quark mass is positive; as κ increases toward κc, the quark mass decreases.

Practically, κ is varied in simulations to tune the quark masses to their physical values. Since larger κ corresponds to smaller mass, κ is chosen below κc and then adjusted to reproduce observed hadron masses through extrapolation or interpolation. The exact κc and the mapping from κ to physical masses depend on the gauge coupling and the lattice spacing, so κ tuning is typically part of a broader calibration procedure.

Conceptually, the hopping parameter appears in the Wilson-Dirac operator as the coefficient of the nearest-neighbor hopping terms, governing how readily a fermion field can move from one lattice site to an adjacent one. It is specific to Wilson-type discretizations (including Clover) and differs from gauge-field parameters such as the coupling or the lattice spacing.

In summary, κ is a tunable input that controls the quark mass in Wilson-type lattice simulations, with the massless limit occurring at κc and physical masses obtained by calibrating κ against measured hadron observables.