Constructive Heavy Particle Effective Theory with Nonlinear Poincar\'{e} Symmetry

We develop a constructive heavy particle effective theory (HPET) through the nonlinear realization of the spontaneously broken Poincar\'{e} symmetry $R^{3,1} \rtimes SO(3,1) \rightarrow R^{3,1} \rtimes SO(3)$. Starting from the heavy one-particle state, we find the nonlinear boost transformation indicates the shift symmetry in the coset construction, corresponding to the reparameterization invariance. Using the little group Wigner rotation, we obtain the nonlinear boost transformation for corresponding heavy field, recovering the Foldy-Wouthuysen transformation. At the operator level, since interaction terms would modify the nonlinear transformation, we propose a most general parametrization on the boost transformation only based on symmetry. The nonlinear boost transformation relates different Wilson coefficients of the HPET operators, providing a bottom-up approach of constructing the independent HPET operators, and generalizing the top-down HPET operators beyond the tree-level integrating out. Utilizing the HPET as example, we obtain additional constraints for the boost transformation as well as the additional variation $\delta \mathcal{L}$ at the $1/m^3$.