Strong CP Phase and Parity in the Hamiltonian Formalism
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| Format: | Preprint |
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2025
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| _version_ | 1866915510314074112 |
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| author | Kuchimanchi, Ravi |
| author_facet | Kuchimanchi, Ravi |
| contents | We show using the Hamiltonian formalism that if parity is a good symmetry of QCD, then the strong CP phase $\barθ$ must be $0$ or $π$. We find that for $P$ to be a physical symmetry, it must leave the Hilbert space $\mathcal{H}_θ$ associated with the $θ$-vacuum invariant ($P: \mathcal{H}_θ\rightarrow \mathcal{H}_θ$), which is possible only for $θ= 0$ or $π$. We also show that forming linear combinations of states from different $θ$-sectors produces only classical statistical mixtures, consistent with superselection rules, confirming that $\mathcal{H}_θ$ is the most general Hilbert space for the quantum theory. Furthermore, we demonstrate that requiring $[P,Ω]=0$, where $Ω$ is the generator of large gauge transformations, independently enforces $\barθ=0$ (mod $π$), and that for complex quark mass matrix $M$, if a generalized parity operator $\mathcal{P}$ is a symmetry, then the value of $θ$ gets determined so that it exactly cancels $Arg Det M$, again giving $\barθ=0$ (mod $π$). These results establish the equivalence of the Hamiltonian and Lagrangian approaches to the strong CP problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_18620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong CP Phase and Parity in the Hamiltonian Formalism Kuchimanchi, Ravi High Energy Physics - Phenomenology High Energy Physics - Experiment High Energy Physics - Theory We show using the Hamiltonian formalism that if parity is a good symmetry of QCD, then the strong CP phase $\barθ$ must be $0$ or $π$. We find that for $P$ to be a physical symmetry, it must leave the Hilbert space $\mathcal{H}_θ$ associated with the $θ$-vacuum invariant ($P: \mathcal{H}_θ\rightarrow \mathcal{H}_θ$), which is possible only for $θ= 0$ or $π$. We also show that forming linear combinations of states from different $θ$-sectors produces only classical statistical mixtures, consistent with superselection rules, confirming that $\mathcal{H}_θ$ is the most general Hilbert space for the quantum theory. Furthermore, we demonstrate that requiring $[P,Ω]=0$, where $Ω$ is the generator of large gauge transformations, independently enforces $\barθ=0$ (mod $π$), and that for complex quark mass matrix $M$, if a generalized parity operator $\mathcal{P}$ is a symmetry, then the value of $θ$ gets determined so that it exactly cancels $Arg Det M$, again giving $\barθ=0$ (mod $π$). These results establish the equivalence of the Hamiltonian and Lagrangian approaches to the strong CP problem. |
| title | Strong CP Phase and Parity in the Hamiltonian Formalism |
| topic | High Energy Physics - Phenomenology High Energy Physics - Experiment High Energy Physics - Theory |
| url | https://arxiv.org/abs/2507.18620 |