Generalization of the quark-confining string

It is natural to generalize the quark-confining string model by introducing additional terms coming from the Riemann curvature tensor. For simplicity, only terms linear in the Riemann tensor are considered. In the absence of quark fields, such terms are trivial (as in two-dimensional gravity). In the presence of quark fields, embedding of the string in four-dimensional Minkowski space renders such terms nontrivial. We formulate this generalized quark-confining string in the first-order form, from which we derive its second-order form. The coupling between the quark spin and the string curvature via the connections induces a spin-spin interaction among the quark fields. This generalized version has a dimensionless parameter κ in addition to the quark-gluon coupling e and quark masses. The original version is recovered as a special case (i.e., κ = 0).