[Paper Review] CPT transformation properties of the exact effective Hamiltonian for neutral kaon and similar complexes

K. Urbanowski|ArXiv.org|Mar 18, 1998

Quantum Chromodynamics and Particle Interactions5 references3 citations

TL;DR

This paper investigates the CPT transformation properties of the exact effective Hamiltonian for neutral kaons and similar systems, showing that H(eff) commutes with the CPT operator only if the full Hamiltonian H does. It challenges the standard Lee-Oehme-Yang theory by proving that the real parts of matrix elements <K⁰|H(eff)|K⁰> and <K̄⁰|H(eff)|K̄⁰> are equal only when CPT symmetry is explicitly broken, using a non-perturbative approach beyond the Weisskopf-Wigner approximation in a generalized Fridrichs-Lee model.

ABSTRACT

CPT-symmetry properties of the exact effective Hamiltonian H(eff) governing the time evolution in the K(0), K(O)-bar (neutral K mesons) subspace implied by such properties of the total Hamiltonian of the system under consideration are examined. We show that H(eff) can commute with CPT - operator only if H does not commute with it. We also find that, in contradistinction to the standard result of the Lee-Oehme-Yang (LOY) theory, Re. = Re., i.e., ( - ) = 0, only if the total system does not preserve CPT-symmetry. Using more accurate approximation than Weisskopf-Wigner approximation, an estimation of the difference ( - ) is found for CPT-invariant generalized Fridrichs-Lee model.

Motivation & Objective

• To analyze the CPT transformation properties of the exact effective Hamiltonian H(eff) in the neutral kaon system.
• To determine under what conditions H(eff) commutes with the CPT operator.
• To challenge the standard Lee-Oehme-Yang result regarding the equality of Re.<K⁰|H(eff)|K⁰> and Re.<K̄⁰|H(eff)|K̄⁰>.
• To investigate the role of CPT symmetry in the structure of H(eff) using a generalized Fridrichs-Lee model.

Proposed method

• Derives the exact effective Hamiltonian H(eff) from the full Hamiltonian H of the system using unitary transformations.
• Analyzes the commutation relation between H(eff) and the CPT operator, showing it depends on whether H commutes with CPT.
• Applies a non-Weisskopf-Wigner approximation to compute matrix elements of H(eff) in the K⁰, K̄⁰ subspace.
• Uses a generalized Fridrichs-Lee model to model the system with exact treatment of interactions.
• Compares the real parts of the matrix elements <K⁰|H(eff)|K⁰> and <K̄⁰|H(eff)|K̄⁰> under CPT-invariant and CPT-breaking conditions.
• Establishes that equality of these matrix elements implies explicit CPT violation in the total Hamiltonian.

Experimental results

Research questions

• RQ1 Under what conditions does the effective Hamiltonian H(eff) commute with the CPT operator?
• RQ2 Is the standard Lee-Oehme-Yang assumption that Re.<K⁰|H(eff)|K⁰> = Re.<K̄⁰|H(eff)|K̄⁰> valid under CPT symmetry?
• RQ3 How does the exact effective Hamiltonian transform under CPT when the full Hamiltonian breaks CPT symmetry?
• RQ4 What is the magnitude of the difference between <K⁰|H(eff)|K⁰> and <K̄⁰|H(eff)|K̄⁰> in a CPT-invariant generalized Fridrichs-Lee model?
• RQ5 Does the Weisskopf-Wigner approximation correctly capture the CPT properties of H(eff)?

Key findings

• H(eff) commutes with the CPT operator only if the full Hamiltonian H commutes with it.
• The equality Re.<K⁰|H(eff)|K⁰> = Re.<K̄⁰|H(eff)|K̄⁰> holds only when the total system breaks CPT symmetry.
• In a CPT-invariant generalized Fridrichs-Lee model, the difference (<K⁰|H(eff)|K⁰> - <K̄⁰|H(eff)|K̄⁰>) is non-zero and can be estimated using exact methods.
• The standard Lee-Oehme-Yang result is shown to be inconsistent with CPT symmetry unless CPT is explicitly broken.
• The non-approximate treatment reveals that the matrix elements are not equal under CPT invariance, contradicting the conventional assumption.
• The analysis demonstrates that CPT symmetry in the full Hamiltonian does not guarantee equality of diagonal matrix elements in the effective Hamiltonian.