Constructive Heavy Particle Effective Theory with Nonlinear Poincaré Symmetry

TL;DR

The paper develops a constructive Heavy Particle Effective Theory (HPET) from the nonlinear realization of spontaneously broken Poincaré symmetry, linking heavy-particle boosts to reparameterization invariance via a CCWZ coset framework. It introduces a general, gauge-covariant function $f(\frac{D_{\mu}}{m})$ that encodes anti-particle information and determines the nonlinear boost generator $\vec{\mathcal{K}}_x=m\vec{x}+\vec{\mathcal{K}}$, ensuring Lorentz covariance order-by-order in $1/m$ and constraining HPET Wilson coefficients. The work presents both bottom-up (invariance-based) and top-down (matching from a general relativistic Lagrangian) approaches, showing they yield compatible relations among NR Wilson coefficients up to $\mathcal{O}(1/m^3)$, with gauge-field dependent terms demanding a covariant $f(D)$ beyond tree level. This framework unifies traditional FW and NR reductions with a symmetry-guided method, improving UV-insensitive operator construction for HPET and potentially enabling systematic extensions to HQET, NRQED, and related theories with long-range interactions.

Abstract

We develop a constructive heavy particle effective theory (HPET) through the nonlinear realization of the spontaneously broken Poincaré 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 $δ\mathcal{L}$ at the $1/m^3$.

Figures (3)

• Figure 1: This diagram illustrates the boost transformation of the residual momentum from $k^{\mu}$ to $k^{\prime\mu}$. The red curve denotes the mass-shell $p^2 = m^2$. The black dot marks the reference momentum $m v^{\mu}$, while the blue box centered around it represents the validity region of the Heavy Particle Effective Theory for this specific velocity $v^{\mu}$. Each small green dot represent a residual momentum. Under a boost, the residual momentum transforms as $k^{\mu} \to k^{\prime\mu}$, whereas the reference velocity $v^{\mu}$ remains unchanged.
• Figure 2: This figure illustrates the reparameterization of the velocity from $v^{\mu}$ to $v^{\prime\mu}$. The blue box, centered on the original velocity $v^{\mu}$, and the dashed green box, centered on the new velocity $v^{\prime\mu}$, represent the validity regions of the effective theory in their respective frames. Under the reparameterization, the total physical momentum $p^{\mu}$ remains invariant.
• Figure 3: The workflow of the constructive HPET.