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Bargmann Invariants and Correlated Geometric CP-Violating Structures in Neutral Meson Systems

Swarup Sangiri

Abstract

Bargmann invariants provide a rephasing-invariant description of phase relations among quantum states and offer a geometric perspective on interference phenomena. In this work, we investigate their role in neutral meson systems by constructing cyclic products involving the heavy and light mass eigenstates together with decay-projected states arising from correlated meson decays. Explicit expressions for third-order and fourth-order invariants are obtained in terms of mixing parameters and decay amplitudes. The analysis shows that the associated geometric phases encode CP-sensitive interference effects between meson-antimeson mixing and decay amplitudes and become trivial in the CP-conserving limit. Expressing the decay amplitudes in terms of CKM matrix elements reveals quartic combinations with analogous rephasing-invariant weak-phase structure to that of the Jarlskog invariant. We further introduce a rephasing-invariant ratio constructed from third- and fourth-order Bargmann invariants, which isolates correlated CP-violating structures that cannot, in general, be factorized into independent decay-channel contributions and can enhance sensitivity to small deviations from CP symmetry. The invariants can also be related to parameters governing time-dependent CP asymmetries in neutral meson decays, thereby providing a geometric interpretation of observable CP-violating interference effects.

Bargmann Invariants and Correlated Geometric CP-Violating Structures in Neutral Meson Systems

Abstract

Bargmann invariants provide a rephasing-invariant description of phase relations among quantum states and offer a geometric perspective on interference phenomena. In this work, we investigate their role in neutral meson systems by constructing cyclic products involving the heavy and light mass eigenstates together with decay-projected states arising from correlated meson decays. Explicit expressions for third-order and fourth-order invariants are obtained in terms of mixing parameters and decay amplitudes. The analysis shows that the associated geometric phases encode CP-sensitive interference effects between meson-antimeson mixing and decay amplitudes and become trivial in the CP-conserving limit. Expressing the decay amplitudes in terms of CKM matrix elements reveals quartic combinations with analogous rephasing-invariant weak-phase structure to that of the Jarlskog invariant. We further introduce a rephasing-invariant ratio constructed from third- and fourth-order Bargmann invariants, which isolates correlated CP-violating structures that cannot, in general, be factorized into independent decay-channel contributions and can enhance sensitivity to small deviations from CP symmetry. The invariants can also be related to parameters governing time-dependent CP asymmetries in neutral meson decays, thereby providing a geometric interpretation of observable CP-violating interference effects.

Paper Structure

This paper contains 11 sections, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formalism for Neutral Meson Mixing and Correlated Pairs
  3. Third-Order Bargmann Invariant
  4. Derivation in Terms of Mixing and Decay Amplitudes
  5. Physical Interpretation and $\cal CP~$ Limit
  6. Fourth-Order Bargmann Invariant
  7. Derivation in Terms of Mixing and Decay Amplitudes
  8. Physical Interpretation and $\cal CP~$ Limit
  9. Connection to Quark-Level $\cal CP~$ Invariant
  10. Geometric $\cal CP~$ Observables and Correlated Interference Structures in Neutral Mesons
  11. Summary

Figures (2)

  • Figure 1: Geometric representation of the third-order Bargmann invariant $\Delta_3$. The closed triangular loop in projective Hilbert space connects the states $P_H \rightarrow \psi_f \rightarrow P_L \rightarrow P_H$. The phase associated with this loop corresponds to the geometric phase $\gamma_{\Delta_3}$.
  • Figure 2: Geometric representation of the fourth-order Bargmann invariant $\Delta_4$. The closed quadrilateral loop involves two decay channels and connects the sequence $P_H \rightarrow \psi_f \rightarrow P_L \rightarrow \psi_g \rightarrow P_H$. This structure encodes interference between mixing and decay processes across multiple channels.