A mathematical review on the multiple-solution problem

Abstract

The recent multiple-solution problem in extracting physics information from a fit to the experimental data in high energy physics is reviewed in a mathematical viewpoint. All these multiple solutions were found via a fit process previously, while in this letter we prove that if the sum of two coherent Breit-Wigner functions is used to fit the measured distribution, there should be two and only two non-trivial solutions, and they are related to each other by analytical formulae. For real experimental measurements in more complicated situations, we also provide a numerical method to derive the other solution from the already obtained one. The excellent consistency between the exact solution obtained this way and the fit process justifies the method. From our results it is clear that the physics interpretation should be very different depending on which solution is selected. So we suggest that all the experimental measurements with potential multiple solutions be re-analyzed to find the other solution because the result is not complete if only one solution is reported.

arXiv comment10 pages, no figure; IJMPA's Volume No.26, Issue No. 25, Year 2011

References in corpus (8)

• Observation of Two Resonant Structures in e+e- to pi+ pi- psi(2S) via Initial State Radiation at Belle
• Measurement of e^+e^- to pi^+pi^-J/psi Cross Section via Initial State Radiation at Belle
• High-statistics measurement of the pion form factor in the rho-meson energy range with the CMD-2 detector
• Study of the process e+e- -> omega pi0 in the phi-meson mass region with the KLOE detector
• Combined fit to BaBar and Belle data on e+e- to pi+pi- psi(2S)
• On the leptonic partial widths of the excited psi states
• Multiple solutions in extracting physics information from experimental data
• Branching fraction of the isospin violating process phi to omega pi0

Cited by in corpus (28)

• Precise measurement of the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}J/\psi$ cross section at center-of-mass energies from 3.77 to 4.60 GeV
• Evidence of Two Resonant Structures in $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{h}_c$
• Evidence of a resonant structure in the $e^{+}{e}^{-}\to {\pi }^{+}{D}^0{D}^{\ast -}$ cross section between 4.05 and 4.60 GeV
• Observation of the $Y(4230)$ and a new structure in $e^{+}{e}^{-}\to K^{+}{K}^{-}J/\psi$
• Observation of an anomalous line shape of the ${\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ mass spectrum near the $p\bar{p}$ mass threshold in $J/\psi \to \gamma {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$
• Evidence for resonant structures in e+e- to pi+pi-h_c
• Measurement of cross-section for $e^{+}{e}^{-}\to {\Xi }^{-}{\bar{\Xi }}^{+}$ near threshold at BESIII
• Measurement of the cross section for $e^{+}{e}^{-}\to \Lambda \bar{\Lambda }$ and evidence of the decay $\psi (3770)\to \Lambda \bar{\Lambda }$
• Hadronic cross section of $e^{+}{e}^{-}$ annihilation at bottomonium energy region
• Observation of a state $X(2600)$ in the ${\pi }^{+}{\pi }^{-}{\eta }^{\prime }$ system in the process $J/\psi \to \gamma {\pi }^{+}{\pi }^{-}{\eta }^{\prime }$
• Study of $e^{+}{e}^{-}\to p\bar{p}$ in the vicinity of $\psi (3770)$
• General mathematical analysis on multiple solutions of interfering resonances combinations
• Recent progress on the study of the charmoniumlike states
• Study of e${}^{+}$e${}^{-}$ $\to \text{p}\overline{\text{p}}{\pi }^0$ in the Vicinity of the $\psi \text{(3770)}$
• Study of $J/\psi$ and $\psi (3686)$ decays to ${\pi }^{+}{\pi }^{-}{\eta }^{\prime }$
• Observation of the Anomalous Shape of $X(1840)$ in $J/\psi \to \gamma 3({\pi }^{+}{\pi }^{-})$ Indicating a Second Resonance Near $p\bar{p}$ Threshold
• Observation of $J/\psi$ Electromagnetic Dalitz Decays to $X(1835)$, $X(2120)$ and $X(2370)$
• Measurement of $e^{+}{e}^{-}\to \gamma {\chi }_{c0,c1,c2}$ cross sections at center-of-mass energies between 3.77 and 4.60 GeV
• Observation of di-structures in $e^{+}{e}^{-}\to J/\psi \mathrm{X}$ at center-of-mass energies around 3.773 GeV
• Electronic width of the $\psi (3770)$ resonance interfering with the background
• Symmetry analysis of charmonium two-body decay
• Selecting the physical solution via $\eta$-${\eta }^{\prime }$ mixing
• Cross section measurements of $e^{+}{e}^{-}\to \Phi K^{+}{K}^{-}$ and $e^{+}{e}^{-}\to \Phi K_S^0{K}_S^0$ at center-of-mass energies between 3.7730 GeV and 4.7008 GeV
• Theory of second optimization for scan experiment
• Measurement of $e^{+}{e}^{-}\to \varphi {\eta }^{\prime }$ cross sections at center-of-mass energies between 3.508 and 4.600 GeV
• $\varphi (2170)$ decaying to $\varphi \pi \pi$ and $\varphi K\bar{K}$
• Improved description of e^+e^- \to p\bar{p}π^0 cross section line shape and more stringent constraints on ψ(3770) and ψ(4230) \to p\bar{p}π^0
• Pinpointing Physical Solutions in Y(4230) Decays