Long-distance heavy-quark potentials in the effective string theory

We use nonlinear transformations for non-relativistic fields to implement Poincare symmetry in Non-Relativistic QCD (NRQCD) and
potential Non-Relativistic QCD (pNRQCD), and show how the Poincare algebra is satisfied for these transformations, which thereby gives complementary information on the Wilson coefficients. Upon the established constraints on the Wilson coefficients, which are
potential-terms especially in pNRQCD, we explore their analytic behaviors at large separations between the static heavy quark and heavy antiquark, using the framework of the effective string theory (EST). Finally, we discuss how the (dimensionful) parameters that arise from the EST can be reduced/simplified by the realization of Poincare symmetry in QCD.