A Dalitz plot analysis of the $D_s^+ \to \pi^- \pi^+ \pi^+$ decay is presented. The analysis is based on proton-proton collision data recorded by the LHCb experiment at a centre-of-mass energy of $8$ TeV and corresponding to an integrated luminosity of $1.5 fb^{-1}$. The resonant structure of the decay is obtained using a quasi-model-independent partial-wave analysis, in which the $\pi^+\pi^-$ S-wave amplitude is parameterised as a generic complex function determined by a fit to the data. The S-wave component is found to be dominant, followed by the contribution from spin-2 resonances and a small contribution from spin-1 resonances. The latter includes the first observation of the $D_s^+ \to \omega(782) \pi^+$ channel in the $D_s^+ \to \pi^- \pi^+ \pi^+$ decay. The resonant structures of the $D_s^+ \to \pi^- \pi^+ \pi^+$ and $D^+ \to \pi^- \pi^+ \pi^+$ decays are compared, providing information about the mechanisms for the hadron formation in these decays.
Dominant tree-level diagrams leading to the (left) $D^+_s\rightarrow \pi^-\pi^+\pi^+$ and (right) $ D ^+ \rightarrow \pi ^- \pi ^+ \pi ^+ $ decays. The resonances are produced from an $s\bar s$ pair in the $D^+_s\rightarrow \pi^-\pi^+\pi^+$ decay and from a $d\bar d$ pair in the $ D ^+ \rightarrow \pi ^- \pi ^+ \pi ^+ $ decay. For both decays, the annihilation diagram is suppressed. |
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Invariant-mass distribution of $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ candidates after the final selection, with the fit result superimposed. |
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(Top) Dalitz plot from selected $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ candidates in the signal region. The colour scale indicates the density of candidates. (Bottom) Perspective view of the Dalitz plot. |
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(Left) efficiency variation over the Dalitz plot and (right) background model for $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ decays. |
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{The folded Dalitz plot projections (left) $s_{high}$ and (right) $s_{low}$}. |
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(Left) The unfolded Dalitz plot projection $s_{13}$; (right) the folded distribution of the normalised residuals across the Dalitz plot. |
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(Left) Magnitude and (right) phase of the S-wave amplitude as a function of $m(\pi^+\pi^-)$. The uncertainties are statistical. |
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Argand plot of the $\pi^+\pi^-$ S-wave amplitude. The values of $m(\pi^+\pi^-)$ at the edge of each interval are indicated next to the corresponding experimental point. The amplitude starts at the point circled in red and undergoes two counterclockwise circles. |
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(Left) Dalitz plot of the $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ and (right) $ D ^+ \rightarrow \pi ^- \pi ^+ \pi ^+ $ decays [4]. The colour scale indicates the density of candidates. |
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Dominant amplitudes for the $ D ^+ \rightarrow \pi ^- \pi ^+ \pi ^+ $ and $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ decays. The scalar resonances are produced from rescattering of the two pseudoscalar mesons $ab$ ($a'b'$) formed by a $q\bar q$ pair from the vacuum and the $d\bar d$ ($s\bar s$) pair from the decay of the $c$ quark. |
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(Left) Magnitude and (right) phase of the $\pi ^-$ $\pi ^+$ S-wave amplitude for the $ D ^+ \rightarrow \pi ^- \pi ^+ \pi ^+ $ (black line) and $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ decays (blue dot). |
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(Left) Comparison of the $\pi^+\pi^-$ S-wave phase from $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ decays and the phase from $\pi^+\pi^-\rightarrow \pi^+\pi^-$ scattering. Data on $\pi^+\pi^-\rightarrow \pi^+\pi^-$ scattering above 0.6 $\text{ Ge V}$ are from a re-analysis [5] of original data from [34], and below 0.4 $\text{ Ge V}$ are from $K_{e4}$ decays [35]. (Right) The S-wave phase from the $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ decay is shifted by 210$^{\circ}$. |
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Comparison of $\pi ^+$ $\pi ^-$ S-wave amplitude from the $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ decay with previous results from BaBar [6] and BESIII [7]. |
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Animated gif made out of all figures. |
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Masses and widths of the resonances used in the fit [2]. Quoted uncertainties are used to estimate systematic uncertainties. |
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Spin-dependent Blatt--Weisskopf barrier factors. The normalisation ensures that the barrier factors are equal to unity at the resonance mass. |
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Results from the default fit. The row "combined" has the combined fit fractions of the $\rho(1450)^0$ and $\rho(1700)^0$ contributions, including the interference between them. The uncertainties are statistical. |
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Results from the default fit. Interference fit fractions (%) between the resonant amplitudes. The uncertainties are statistical. |
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Summary of the systematic uncertainties on the magnitudes and phases of the complex coefficients from the P- and D-wave amplitudes. The column "total exp." is the sum in quadrature of the first four columns. For comparison, the statistical uncertainties are included in the last column. |
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Systematic uncertainties on the fit fractions (%). The combined fit fractions of the $\rho(1450)^{0}\pi^+$ and $\rho(1700)^{0}\pi^+$ are quoted in the row "combined". |
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Final results of the $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ Dalitz plot fit. The uncertainties are statistical, experimental systematic and associated to the decay amplitude model, respectively. |
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Resonant structure of the $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ decay from this analysis compared to previous determinations from BaBar [6] and BESIII [7]. The fit fractions are given in per cent. The statistical and systematic uncertainties are added in quadrature. |
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Magnitude and phase of the S-wave amplitude as a function of $\pi ^-$ $\pi ^+$ invariant mass. The uncertainties are statistical, experimental and model, respectively. The $\pi^+\pi^-$ invariant mass is expressed in $\text{ Ge V}$ . |
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Resonant structures of the $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ $ and $ D ^+ \rightarrow \pi ^- \pi ^+ \pi ^+ $ [4] decays, expressed fit fractions (%). Uncertainties are only statistical. |
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Created on 27 April 2024.