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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.28756 |
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| _version_ | 1866917540715823104 |
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| author | Mannheim, Philip D. |
| author_facet | Mannheim, Philip D. |
| contents | In the standard Breit-Wigner approach to scattering the phase shift is to have a form $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ at a real energy resonance. This leads to complex energy poles in the scattering amplitude at $E_{\rm BW}=E_1-iΓ_1$, poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ is not always a good description of the real energy scattering amplitude, that $Γ_1$ can be negative, that $E_{\rm BW}$ is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear $PT$ symmetry solutions to the square well Schrödinger equation appear in complex conjugate energy pairs $E_{\mp}=E_2\mp i Γ_2$ with $E_- \neq E_{\rm BW}$, doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28756 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Critique of Breit-Wigner resonance scattering Mannheim, Philip D. High Energy Physics - Phenomenology In the standard Breit-Wigner approach to scattering the phase shift is to have a form $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ at a real energy resonance. This leads to complex energy poles in the scattering amplitude at $E_{\rm BW}=E_1-iΓ_1$, poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ is not always a good description of the real energy scattering amplitude, that $Γ_1$ can be negative, that $E_{\rm BW}$ is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear $PT$ symmetry solutions to the square well Schrödinger equation appear in complex conjugate energy pairs $E_{\mp}=E_2\mp i Γ_2$ with $E_- \neq E_{\rm BW}$, doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two. |
| title | Critique of Breit-Wigner resonance scattering |
| topic | High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2605.28756 |