Larin (EPL 129, 21003, 2020) proposed that the Standard Model could violate CPT symmetry without contradicting the CPT theorem if the d and s quarks carried opposite CPT phases under the physical CPT operator, a pattern realizable in a composite-quark model, and derived from a two-state proton picture a fractional proton–antiproton mass difference of order 6×10−116\times10^{-11}. The proposal leaves open whether the composite construction has any consistent nonperturbative realization, and whether the mass-splitting estimate survives a realistic hadronic treatment and remains compatible with proton–antiproton precision constraints. These questions remain unaddressed in the literature, while the BASE 16-parts-per-trillion charge-to-mass ratio comparison (2022) already probes the predicted magnitude with a null result under the equal-charge assumption. The open problem is to establish or refute the theoretical consistency, quantitative prediction, and phenomenological compatibility of this CPT-violation mechanism.

Background

CPT invariance — invariance under the combined antiunitary operation of charge conjugation C (particle–antiparticle interchange), parity P (spatial reflection), and time reversal T — is an exact symmetry of any local, Lorentz-invariant quantum field theory with the usual connection between spin and statistics; this is the content of the CPT theorem. Its most testable consequence is that a particle and its antiparticle have exactly equal masses and lifetimes and equal-and-opposite charges. In every field transformation law the CPT operator Θ\Theta appears only up to a phase, e.g. for a spinor field Θψ(x)Θ−1=ηΘ(ψ)γ5ψT+(−x)\Theta\psi(x)\Theta^{-1}=\eta_\Theta(\psi)\gamma^5\psi^+_T(-x), where γ5=iγ0γ1γ2γ3\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3, ψT+\psi^+_T denotes the complex-conjugate spinor (the transpose of the Hermitian conjugate), and ηΘ\eta_\Theta is a CPT phase; for free fields these phases are pure convention, but with interactions the relative phases between different fields are physically meaningful because they fix the signs with which the various terms of an interaction Lagrangian transform. S. A. Larin (EPL 129, 21003 (2020)) observed that if the d and s quarks are assigned opposite CPT phases, ηΘ(s)=−ηΘ(d)\eta_\Theta(s)=-\eta_\Theta(d), under a physical CPT operator Θph\Theta_{\rm ph} that maps physical states to physical states, then the Standard Model weak charged-current term LCC=(g/(22))Wμ+uˉγμ(1−γ5)(dcos⁡θc+ssin⁡θc)+h.c.L_{CC}=(g/(2\sqrt{2}))W^+_\mu\bar u\gamma^\mu(1-\gamma^5)(d\cos\theta_c+s\sin\theta_c)+{\rm h.c.} (with gg the weak coupling and θc≈13∘\theta_c\approx13^\circ the Cabibbo angle mixing d and s in the weak charged current) is no longer invariant: the pieces linear in d and s flip with opposite signs, so the Hamiltonian splits into a CPT-even part H+H_+ and a CPT-odd part H−H_-. Larin argues this does not contradict the Jost theorem (the Wightman-function formulation of the CPT theorem) because Θph\Theta_{\rm ph} does not commute with the Hamiltonian and is therefore time dependent in the Heisenberg representation, outside that theorem's assumptions. He shows the required phase pattern can be arranged in a toy model of composite quarks, d=(ψˉγ0ψ)ψd=(\bar\psi\gamma^0\psi)\psi and s=(ψˉγ5ψ)ψs=(\bar\psi\gamma^5\psi)\psi (with ψˉ=ψ+γ0\bar\psi=\psi^+\gamma^0 the Dirac adjoint and normal ordering — rearrangement of the operator product so annihilation operators stand to the right — understood), built from a single constituent spinor field ψ\psi that is not part of the Standard Model. Modeling the proton as ∣p⟩=∣uud⟩+ξ∣uus⟩|p\rangle=|uud\rangle+\xi|uus\rangle, with weak-interaction mixing amplitude ξ≈⟨uud∣Hw∣uus⟩/(mΣ−mp)≈GFsin⁡θc/(mΣ−mp)≈0.8×10−5\xi\approx\langle uud|H_w|uus\rangle/(m_\Sigma-m_p)\approx G_F\sin\theta_c/(m_\Sigma-m_p)\approx0.8\times10^{-5} (where HwH_w is the weak-interaction Hamiltonian, whose hadronic matrix element ⟨uud∣Hw∣uus⟩\langle uud|H_w|uus\rangle supplies the mass dimension in this schematic estimate; GF≈10−5mp−2G_F\approx10^{-5}m_p^{-2} is the Fermi constant and mΣ≈1189m_\Sigma\approx1189 MeV the Σ+\Sigma^+ hyperon mass), Larin obtains mp−mpˉ=2⟨p∣H−∣p⟩≈4ξ⟨uud∣H−∣uus⟩m_p-m_{\bar p}=2\langle p|H_-|p\rangle\approx4\xi\langle uud|H_-|uus\rangle, which he estimates schematically as the fractional proton–antiproton mass difference ∣mp−mpˉ∣/mp≈4ξsin⁡θcGF≈6×10−11|m_p-m_{\bar p}|/m_p\approx4\xi\sin\theta_c G_F\approx6\times10^{-11}. The relevant experimental context is: the direct bound ∣mp−mpˉ∣/mp<7×10−10|m_p-m_{\bar p}|/m_p<7\times10^{-10} (as quoted by Larin from the Particle Data Group review), the antiproton-to-proton charge-to-mass ratio comparison at 69 parts per trillion (ppt; 1 ppt =10−12=10^{-12} fractional) from Penning-trap measurements by the BASE collaboration (a Penning trap confines a single charged particle with static electric and magnetic fields, and the cyclotron frequency νc=(q/m)B/2π\nu_c=(q/m)B/2\pi determines the charge-to-mass ratio), and the neutral-kaon bound ∣mK0−mKˉ0∣/mK0≤0.8×10−18|m_{K^0}-m_{\bar K^0}|/m_{K^0}\le0.8\times10^{-18}, which however rests on the Wigner–Weisskopf approximation — the standard effective two-level treatment of the K0K^0–Kˉ0\bar K^0 oscillation — a sector Larin explicitly excludes from his construction, asserting that his considerations do not contradict that restriction. The paper's own statement of what remains open is experimental: checking the mechanism requires improving the precision of the direct proton–antiproton mass comparison by roughly an order of magnitude, or the charge-to-mass ratio precision by roughly a factor of two. Larin emphasizes that the composite-quark realization uses a constituent field outside the Standard Model and that the mass-splitting estimate rests on a simplified two-state quantum-mechanical picture of the proton; he also notes the equality of scattering cross-sections between particles and reversed antiparticle reactions as another well-known experimentally checkable consequence of CPT invariance, while electric dipole moments and anomalous magnetic moments are not considered in the paper.

Problem Statement

Determine whether Larin's opposite-CPT-phase mechanism for CPT violation in the Standard Model is physically viable and quantitatively robust. Concretely: (1) Can opposite CPT phases for the d and s quarks, ηΘ(s)=−ηΘ(d)\eta_\Theta(s)=-\eta_\Theta(d), be realized in a dynamics consistent with the Standard Model — in particular, can the composite-quark ansatz d=(ψˉγ0ψ)ψd=(\bar\psi\gamma^0\psi)\psi, s=(ψˉγ5ψ)ψs=(\bar\psi\gamma^5\psi)\psi, or an alternative construction, be given a consistent nonperturbative origin, and does the resulting time-dependent physical CPT operator Θph\Theta_{\rm ph} genuinely evade the Jost theorem rather than contradicting it? (2) What is the robust prediction for the proton–antiproton mass splitting once the two-state ∣uud⟩+ξ∣uus⟩|uud\rangle+\xi|uus\rangle model is refined to include sea quarks, gluons, and higher-order weak contributions — in particular, does the estimate ∣mp−mpˉ∣/mp∼6×10−11|m_p-m_{\bar p}|/m_p\sim6\times10^{-11} survive, and is it compatible with the 16-ppt null antiproton-to-proton charge-to-mass ratio comparison of Borchert et al. (2022) under the equal-charge assumption, or does the mechanism require a compensating CPT-odd charge contribution? The mechanism counts as viable if a consistent realization can be given with a defensible quantitative prediction compatible with the existing proton–antiproton bounds, and as refuted if an inconsistency with the CPT theorem's assumptions, an unresolvable conflict with existing data, or the impossibility of realizing the opposite-phase pattern ηΘ(s)=−ηΘ(d)\eta_\Theta(s)=-\eta_\Theta(d) in any dynamics consistent with the Standard Model, of which the composite-quark ansatz is only one candidate realization — can be demonstrated.

References

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