Lorentz and CPT Tests in Neutron and Storage-Ring EDM Experiments

Technical Summary: Lorentz and CPT Tests in Neutron and Storage-Ring EDM Experiments

Problem Statement
Electric dipole moment (EDM) experiments serve as highly sensitive probes for new sources of parity (P) and charge-parity (CP) violation. Beyond these standard symmetry tests, the extreme precision of current EDM measurements (e.g., neutron EDM limits at $|d_n| < 1.8 \times 10^{-26} \, e \, \text{cm}$ and muon EDM limits at $\sim 10^{-19} \, e \, \text{cm}$) renders them powerful tools for testing fundamental Lorentz and CPT symmetries. Within the framework of the Standard-Model Extension (SME), which incorporates all possible Lorentz-violating terms into the action of General Relativity and the Standard Model, CPT violation is necessarily accompanied by Lorentz violation. While extensive limits exist on many SME coefficients, specific coefficients relevant to spin precession in confined neutron and storage-ring environments remain unconstrained. The problem addressed is the lack of explicit theoretical correspondences between measured EDMs in these specific experimental setups and the specific SME coefficients governing Lorentz and CPT violation.

Methodology
The paper analyzes Lorentz- and CPT-violating contributions to spin precession frequencies within the SME framework for two distinct experimental configurations:

1. Confined Neutrons: For neutron EDM experiments, the authors employ perturbation theory to derive leading-order effects. The analysis assumes an apparatus frame with magnetic ($\vec{B}$) and electric ($\vec{E}$) fields aligned along the $x_3$ axis. The energy shift ($\delta E_s$) is calculated as the expectation value of the perturbative Lorentz-violating Hamiltonian ($\delta H$) over unperturbed stationary neutron eigenstates ($\chi_s$). By reversing the electric field direction while holding the magnetic field fixed, the authors isolate a Lorentz-violating frequency difference ($\Delta \omega_{n, LV}$).

2. Storage-Ring Experiments: For charged particles (protons, muons, and antiparticles) in storage rings, the study utilizes a generalized Bargmann-Michel-Telegdi (BMT) equation that incorporates Lorentz and CPT violation. The analysis considers particles confined by radial electric and uniform upward magnetic fields. The authors derive the modification to the radial component of the spin-precession frequency ($\delta \omega_{s, LV}^{rad}$) arising from Lorentz violation, expressing it in terms of specific "hacheck" coefficients ($\check{H}, \check{b}, \check{g}$) which are functions of the fundamental SME coefficients, particle mass ($m_w$), velocity ($\vec{\beta}$), and Lorentz factor ($\gamma$).

Key Contributions and Results
The primary contribution of this work is the establishment of explicit mathematical correspondences between the standard EDM observables and specific SME coefficients:

• Neutron EDM: The analysis reveals that a Lorentz-violating frequency shift mimics an EDM signal. By comparing the standard EDM frequency shift ($\Delta \omega_{n, EDM} = 4d_n E$) with the derived Lorentz-violating shift, the paper establishes the correspondence:
  $$|d_n| \longleftrightarrow |\tilde{b}^{303}_{F,n}|$$
  where $\tilde{b}^{303}_{F,n}$ is a linear combination of SME coefficients ($b^{(5)}, H^{(5)}, d^{(6)}, g^{(6)}$) specific to the neutron.

• Storage-Ring EDM: Similarly, by comparing the standard BMT EDM contribution with the generalized BMT Lorentz-violating term, the paper derives the correspondence:
  $$|d_w| \longleftrightarrow \frac{|\delta \omega_{s, LV}^{rad}|}{2|\vec{E} + \vec{\beta} \times \vec{B}|}$$
  This relation links the measured EDM of a particle species $w$ to a specific combination of SME coefficients ($\check{H}^{03}_w, \check{b}^{11}_{F,w}, \check{b}^{22}_{F,w}, \check{g}^{33}_{F,w}, \check{H}^{33}_{F,w}$) dependent on the particle's kinematics.

Significance and Claims
The paper claims that these derived relations provide the necessary theoretical basis for setting the first limits on several previously unconstrained SME coefficients. By translating existing experimental limits on neutron, muon, and antimuon EDMs into constraints on the relevant SME coefficients, future studies can utilize current data to probe Lorentz and CPT symmetry in new sectors. The authors note that while the relations are established in the apparatus frame, transforming these bounds to the Sun-centered frame (required for standard reporting) involves coordinate transformations detailed in prior literature. The work does not propose new experiments but rather provides the analytical framework to reinterpret existing high-precision EDM data as tests of Lorentz and CPT symmetry.