Measurement of the $\Sigma \pi$ photoproduction line shapes near the $\Lambda \left(1405\right)$

The reaction $\gamma +p\to K^{+}+\Sigma +\pi$ was used to determine the invariant mass distributions or “line shapes” of the ${\Sigma }^{+}{\pi }^{-}$, ${\Sigma }^{-}{\pi }^{+}$, and ${\Sigma }^0{\pi }^0$ final states, from threshold at 1328 $\mathrm{MeV}/c^2$ through the mass range of the $\Lambda \left(1405\right)$ and the $\Lambda \left(1520\right)$. The measurements were made with the CLAS system at Jefferson Lab using tagged real photons, for center-of-mass energies $1.95<W<2.85$ GeV. The three mass distributions differ strongly in the vicinity of the $I=0$ $\Lambda \left(1405\right)$, indicating the presence of substantial $I=1$ strength in the reaction. Background contributions to the data from the ${\Sigma }^0\left(1385\right)$ and from $K^{*}\Sigma$ production were studied and shown to have negligible influence. To separate the isospin amplitudes, Breit-Wigner model fits were made that included channel-coupling distortions due to the $N\overline{K}$ threshold. A best fit to all the data was obtained after including a phenomenological $I=1$, $J^P=1/2^{-}$ amplitude with a centroid at $1394\pm 20$ $\mathrm{MeV}/c^2$ and a second $I=1$ amplitude at $1413\pm 10$ $\mathrm{MeV}/c^2$. The centroid of the $I=0$ $\Lambda \left(1405\right)$ strength was found at the $\Sigma \pi$ threshold, with the observed shape determined largely by channel coupling, leading to an apparent overall peak near 1405 $\mathrm{MeV}/c^2$.

Figure 1

Creation of the three-body $K^{+}$ $\Sigma$$\pi$ final state via an intermediate hyperon in the reaction $\gamma +p\to K^{+}+\Sigma +\pi$. In this particular example, a $t$-channel exchange enables an off-shell kaon to create a $\Lambda \left(1405\right)$ that is subthreshold for on-shell $N\overline{K}$ reactions.

Figure 2

Representation of reaction and decay channels used in this analysis. Particles $X$ given in the columns were reconstructed via kinematic fitting or the missing mass determination. Branching fractions are indicated as percentages. The rightmost column lists the figures related to the given channel.

Figure 3

Difference in particle time of flight, $\Delta \mathrm{TOF}$, versus the measured magnitude of momentum for the ${\pi }^{+}$ (top) and ${\pi }^{-}$ (bottom) for a given energy bin of $2.35<W<2.45$ $\mathrm{GeV}$. The different horizontal bands correspond to the 2-ns time structure of the CEBAF beam. The magenta lines show where the cuts were applied to select each particle.

Figure 4

$\Delta \mathrm{TOF}$ versus the measured magnitude of momentum for the $K^{+}$ candidates at several analysis stages for a given energy bin of $2.35<W<2.45$ $\mathrm{GeV}$. From top to bottom, the plots correspond to the following: after the $K^{+}$ misidentification rejection cut; after selecting the detected in-time ${\pi }^{-}$ as shown in Fig. 3; and selection on both the detected ${\pi }^{-}$ and ${\pi }^{+}$. The magenta lines represent the selection cut on the $K^{+}$. The last spectrum is seen to be much cleaner after the $\Delta \mathrm{TOF}$ selection cuts on the pions.

Figure 5

${\mathrm{M}}^2\left(p{\pi }^0\right)$ versus ${\mathrm{M}}^2\left(p{\pi }^{-}\right)$ for a given energy bin, with projections. The bands corresponding to the $\Lambda$ (vertical) and ${\Sigma }^{+}$ (horizontal) are clearly seen.

Figure 6

Fits to the ground-state hyperons (a) $\Lambda$ and (b) ${\Sigma }^{+}$for a single bin in energy and angle. The data of each invariant mass squared are shown as the histograms, and the fits are shown as the solid curve (total), dashed curve (Gaussian), and dot-dashed curve (background). The outer dotted vertical lines show the range of the fits; the inner lines show $\pm 3\sigma$ around the peaks, which is used to define the signal events.

Figure 7

$\mathrm{M}\left(\Lambda {\pi }^0\right)$ versus $\mathrm{M}\left(K^{+}{\pi }^0\right)$ for four energy bins increasing from (a) to (d). A clear horizontal band corresponding to the $\Sigma \left(1385\right)$ is seen, as well as a vertical band corresponding to the $K^{*+}$. The contours, as well as the dashed lines, show the kinematic boundaries allowed in that energy bin. The blue dashed lines show the masses of each resonance [$\Sigma \left(1385\right)$, $K^{*+}$] as given by the PDG, while the vertical dashed lines show where the $K^{+}{\pi }^0$ invariant mass is $M_0\pm \Gamma$, and $M_0$ and $\Gamma$ are the mass and width of the $K^{*+}$ as given by the PDG.

Figure 8

$\mathrm{M}\left({\Sigma }^{+}{\pi }^{-}\right)$ versus $\mathrm{M}\left(K^{+}{\pi }^{-}\right)$ for four bins of increasing energy from (a) to (d). Clear horizontal bands corresponding to the $\Lambda \left(1405\right)$, $\Lambda \left(1520\right)$, and higher $Y^{*}$ resonances are seen, as well as a vertical band corresponding to the $K^{*0}$. The contours as well as the dashed lines show the kinematic boundaries allowed in that energy bin. The blue dashed lines show the masses of each resonance [$\Lambda \left(1405\right)$, $\Lambda \left(1520\right)$, $K^{*}$] as given by the PDG, while the vertical dashed lines show where the $K^{+}{\pi }^{-}$ invariant mass is $M_0\pm \Gamma$, and $M_0$ and $\Gamma$ are the mass and width of the $K^{*0}$ as given by the PDG.

Figure 9

${\mathrm{M}}^2\left(n{\pi }^{+}\right)$ against ${\mathrm{M}}^2\left(n{\pi }^{-}\right)$ for a single energy bin, with projections. The bands corresponding to the ${\Sigma }^{-}$ (vertical) and ${\Sigma }^{+}$ (horizontal) are clearly seen. The faint diagonal band corresponds to the events of $n{K}_S^0\to n{\pi }^{+}{\pi }^{-}$.

Figure 10

Fits to the ground-state hyperons (a) ${\Sigma }^{+}$ and (b) ${\Sigma }^{-}$ for a single bin in energy and angle. The data of each invariant mass squared are shown as the histograms, and the fits are shown as the solid curve (total), dashed curve (Gaussian), and dot-dashed curve (background). The outer dotted vertical lines show the range of the fit, and the inner lines show $\pm 2\sigma$ around the peaks used to select events. The opposing signal region was excluded, as well as the $\pm 2\sigma$ peak around the $K_S^0$.

Figure 11

$\mathrm{M}\left({\Sigma }^{-}{\pi }^{+}\right)$ versus $\mathrm{M}\left(K^{+}{\pi }^{+}\right)$ for four energy bins increasing from (a) to (d). Clear horizontal bands corresponding to the $\Lambda \left(1405\right)$, $\Lambda \left(1520\right)$, and higher $Y^{*}$ resonances are seen. The blue dashed lines show the nominal masses of the $\Lambda \left(1405\right)$ and $\Lambda \left(1520\right)$ from the PDG. Note that in this channel the combination of $K^{+}{\pi }^{+}$ shows no resonant structure. The contours as well as the dashed lines show the kinematic boundaries allowed in that energy bin.

Figure 12

Examples of the $\gamma p\to K^{+}p{\pi }^{-}\left(X\right)$ missing mass squared (${\mathrm{MM}}^2$) spectrum for selected kinematic bins. The vertical dashed lines show the selection range for the ${\Sigma }^0{\pi }^0$ channel between the ${\pi }^0$ and $2{\pi }^0$ limits.

Figure 13

Sample fit results of the strong final state of $K^{+}\Lambda {\pi }^0$. The events are plotted versus the missing mass from the $K^{+}$, which is equivalent to the invariant mass of the $\Lambda {\pi }^0$ system. The data are shown by the black crosses, while the ${\Sigma }^0\left(1385\right)$ signal MC and the $K^{*+}\Lambda$ background are shown by the red crosses and green circles, respectively. The sum of the MC templates are shown by the solid magenta line.

Figure 14

Sample invariant mass spectra for $W=2.6$ GeV and $\cos {\theta }_{K^{+}}^{\mathrm{c}.\mathrm{m}.}=0.65$ showing the ${\Sigma }^0\left(1385\right)$ peak using a MC template based on (a) relativistic Breit-Wigner (mass-dependent width) and (b) nonrelativistic Breit-Wigner (mass-independent width). The fit with the relativistic Breit-Wigner form clearly does not fit the data well.

Figure 15

Fit result to the strong final state of ${\Sigma }^{+}{\pi }^{-}$ before and after MC iteration, for $W=2.5$ GeV, $\cos {\theta }_{K^{+}}^{\mathrm{c}.\mathrm{m}.}=0.35$, as a function of the ${\Sigma }^{+}{\pi }^{-}$ invariant mass. The data are shown with black crosses. (a) Before iteration. Each MC template and the Breit-Wigner function for the $Y^{*}\left(1670\right)$ is shown by a separate color. The total simulation is shown in cyan. (b) After iterations of the $\Lambda \left(1405\right)$ template.

Figure 16

Line shape results for the two ${\Sigma }^{+}{\pi }^{-}$ channels. The ${\Sigma }_p^{+}{\pi }^{-}$ channel is shown with light magenta open circles, while the ${\Sigma }_n^{+}{\pi }^{-}$ channel is shown with light blue triangles. The weighted average of the two line shapes is taken as the final ${\Sigma }^{+}{\pi }^{-}$ line shape, and is shown as red downward triangles. The dashed line represents a relativistic Breit-Wigner function with a mass-dependent width, with the mass and width taken from the PDG. The blue-hatched histogram at the bottom shows the averaged estimated systematic discrepancy between the two reconstructed decay modes.

Figure 17

Mass distribution results for all three $\Sigma \pi$ channels. The weighted average of the two ${\Sigma }^{+}{\pi }^{-}$ channels is shown in the red circles, the ${\Sigma }^0{\pi }^0$ channel is shown as the blue squares, and the ${\Sigma }^{-}{\pi }^{+}$ channel is shown as the green triangles. The ${\Sigma }^0{\pi }^0$ and ${\Sigma }^{-}{\pi }^{+}$ channels have inner error bars representing the statistical errors and outer error bars that have the estimated residual discrepancy of the data and fit results added in quadrature. The dashed line represents a relativistic Breit-Wigner function with a mass-dependent width, with the mass and width taken from the PDG and with arbitrary normalization. The vertical dashed lines show the opening of each $\Sigma \pi$ threshold.

Figure 18

Mass distributions at $W=2.10$ GeV and $E_{\gamma }=1.88$ GeV in comparison to the model of Nacher et al. [7] scaled down by a factor of 2.0. The ${\Sigma }^{+}{\pi }^{-}$ channel is shown as red circles and the red dot-dashed line; the ${\Sigma }^0{\pi }^0$ channel is shown as the blue squares and the blue dashed line; the ${\Sigma }^{-}{\pi }^{+}$ channel is shown as the green triangles and the green solid line. The dashed vertical colored lines at the left side show the reaction thresholds, and the vertical dashed lines at 1.405- and 1.437-GeV mark the nominal centroid and the $N\overline{K}$ thresholds, respectively. The error bars on the data points are combined statistical and point-to-point systematic uncertainties.

Figure 19

Final results for the line shape in the ${\Sigma }_p^{+}{\pi }^{-}$ channel when the $K^{*}$ is removed successively in steps of $\pm \frac{1}{2}\Gamma$, where $\Gamma$ is the width of the $K^{*}$ quoted in the PDG. The black circles represent our final results without a cut on the $K^{*}$, while the red squares, blue triangles, and green downward triangles represent cuts of $\pm \Gamma /2$, $\pm \Gamma$, $\pm \frac{3}{2}\Gamma$ centered around the $K^{*}$, respectively.

Figure 20

Data and fits for ${\Sigma }^{+}{\pi }^{-}$, with each panel showing a different value of $W$. Data and fitted shapes are in red. The isospin contributions are $I=0$ (solid black), narrow $I=1$ (dotted black), and wide $I=1$ (dashed black). The black curves are the same in all panels except for normalization. The vertical dashed lines show the $\Sigma \pi$ thresholds on the left, the nominal 1.405-GeV location, and the $N\overline{K}$ threshold. The incoherent background is shown as a thin dashed line (red).

Figure 21

Data and fits for ${\Sigma }^0{\pi }^0$, with each panel showing a different value of $W$. Data and fitted shapes are in blue. All the other lines and curves are exactly the same as in Fig. 20. The $I=1$ curves are not included here. The incoherent background is shown as a thin dashed line (blue).

Figure 22

Data and fits for ${\Sigma }^{-}{\pi }^{+}$, with each panel showing a different value of $W$. Data and fitted shapes are in green. All the other lines and curves are exactly the same as in Fig. 20. The incoherent background is shown as a thin dashed line (green).

Figure 23

Strength of each of the isospin amplitudes as a function of $W$. These are the real coefficients of the amplitudes, of which the magnitudes give the contributions of each isospin component.