Associate $K^0$ production in $p+p$ collisions at 3.5 GeV: The role of $\Delta {\left(1232\right)}^{++}$

An exclusive analysis of the four-body final states $\Lambda +p+{\pi }^{+}+K^0$ and ${\Sigma }^0+p+{\pi }^{+}+K^0$ measured with HADES for $p+p$ collisions at a beam kinetic energy of 3.5 GeV is presented. The analysis uses various phase space variables, such as missing mass and invariant mass distributions, in the four-particle event selection $(p$,${\pi }^{+}$,${\pi }^{+}$,${\pi }^{-})$ to find cross sections of the different production channels, contributions of the intermediate resonances ${\Delta }^{++}$ and $\Sigma {\left(1385\right)}^{+}$, and corresponding angular distributions. A dominant resonant production is seen, where the reaction $\Lambda +{\Delta }^{++}+K^0$ has a cross section about ten times higher $\left(29.45\pm 0.{08}_{-1.46}^{+1.67}\pm 2.06\mu \text{b}\right)$ than the analogous nonresonant reaction $\left(2.57\pm 0.{02}_{-1.98}^{+0.21}\pm 0.18\mu \text{b}\right)$. A similar result is obtained in the corresponding ${\Sigma }^0$ channels with $9.26\pm 0.{05}_{-0.31}^{+1.41}\pm 0.65\mu \text{b}$ in the resonant and $1.35\pm 0.{02}_{-1.35}^{+0.10}\pm 0.09\mu \text{b}$ in the nonresonant reactions.

Figure 1

${\pi }^{+}{\pi }^{-}$-invariant mass distribution after secondary vertex cuts for the selected event sample. The reconstructed values from a fit with the sum of two Gaussians, a polynomial, and a Landau function are $\langle m_{K_S^0}\rangle =494.9\text{MeV}/c^2$, $\langle {\sigma }_{K_S^0}\rangle =7.1\text{MeV}/c^2$, $N_{K_S^0}=6102$, and S/B 0.64. The dashed-dotted lines show the 3$\sigma$ region around the $K_S^0$ signal. The red and green areas indicate the chosen low and high mass sideband samples respectively.

Figure 2

Missing mass distribution with respect to the $p$, ${\pi }^{+}$, ${\pi }^{+}$, and ${\pi }^{-}$ with a cut on the $K_S^0$ mass in the ${\pi }^{+}{\pi }^{-}$-invariant mass spectrum (Fig. 1). The gray histogram corresponds to the sum of simulated contributions plus the background defined by the sideband sample. The double arrows indicate the so-called $\Lambda$ cut and ${\Sigma }^0$ cut. See text for details.

Figure 3

Missing mass distribution with respect to the $p$, ${\pi }^{+}$, and ${\pi }^{-}$ with a cut on the $K_S^0$ mass in the ${\pi }^{+}{\pi }^{-}$-invariant mass spectrum (Fig. 1). The gray histogram corresponds to the sum of simulated contributions plus the background defined by the sideband sample. The dotted and dashed lines at 1100 and 1270 MeV/$c^2$ are used as cuts in other variables.

Figure 4

Momentum distribution of the $K_S^0$ background (full squares). Applied conditions are a 3$\sigma$ cut around the $K_S^0$ peak in the ${\pi }^{+}{\pi }^{-}$-invariant mass distribution (Fig. 1), and a cut on MM$(p$,${\pi }^{+},{\pi }^{-})$ $<$ 1100 MeV/$c^2$. Relative weights of low mass (full triangles) and high mass (open triangles) sideband samples were fitted simultaneously to the spectrum. The sum of both is shown as a gray filled histogram.

Figure 5

Momentum spectra for several bins in ${\Theta }_{\mathrm{lab}}$ of the $K_S^0$ background. Applied conditions are a 3$\sigma$ cut around the $K_S^0$ peak in the ${\pi }^{+}{\pi }^{-}$-invariant mass distribution (Fig. 1), and a cut on MM(p,${\pi }^{+},{\pi }^{-}$) $<$ 1100 MeV/c${}^2$. The scaling of the low and high mass sideband is a result of the simultaneous fit of both sidebands to the momentum distribution of the $K_S^0$ background in Fig. 4. The symbols are the same as in Fig. 4.

Figure 6

Missing mass distribution with respect to the $p$, ${\pi }^{+}$, ${\pi }^{+}$ and ${\pi }^{-}$ obtained after the cut MM$(p$,${\pi }^{+},{\pi }^{-})>1270$ MeV/$c^2$ and a cut on the $K_S^0$ mass in the ${\pi }^{+}{\pi }^{-}$-invariant mass spectrum (Fig. 1). The gray histogram corresponds to the sum of simulated contributions plus the background defined by the sideband sample.

Figure 7

$p{\pi }^{+}$-invariant mass distribution after the $\Lambda$ cut on the MM$(p$,${\pi }^{+},{\pi }^{+},{\pi }^{-})$ distribution (Fig. 2) and with a cut on the $K_S^0$ mass in the ${\pi }^{+}{\pi }^{-}$-invariant mass spectrum (Fig. 1). The gray histogram corresponds to the sum of simulated contributions plus the background defined by the sideband sample. The same color code and line styles are used as in Figs. 2, 3, and 6.

Figure 8

$p{\pi }^{+}$-invariant mass distribution after the ${\Sigma }^0$ cut on the MM$(p$,${\pi }^{+},{\pi }^{+},{\pi }^{-})$ distribution (Fig. 2) and with a cut on the $K_S^0$ mass in the ${\pi }^{+}{\pi }^{-}$-invariant mass spectrum (Fig. 1). The gray histogram corresponds to the sum of simulated contributions plus the background defined by the sideband sample. The same color code and line styles are used as in Figs. 2, 3, and 6.

Figure 9

Angular distributions of $p{\pi }^{+}$ pairs in the center-of-mass reference system either in the $\Lambda$ cut [panel (a)] or in the ${\Sigma }^0$ cut [panel (b)] with a cut on the $K_S^0$ mass in the ${\pi }^{+}{\pi }^{-}$-invariant mass spectrum (Fig. 1). The gray histogram corresponds to the sum of simulated contributions plus the background defined by the sideband sample. The same color code and line styles are used as in Figs. 2, 3, and 6.

Figure 10

${\chi }^2/\mathrm{NDF}$ distribution in the $cos{\Theta }_{\text{c.m.}}^{p{\pi }^{+}}$ spectra in Fig. 9 for the variation of the second coefficients $A_2$ of the Legendre polynomials of the channels $p+p\to \Lambda +{\Delta }^{++}+K^0$ and $p+p\to {\Sigma }^0+{\Delta }^{++}+K^0$.

Figure 11

Absolute normalized angular distributions of $p{\pi }^{+}$ pairs in the center-of-mass reference system corrected for acceptance and efficiency. Panel (a) includes contributions from the reactions $p+p\to \Lambda +(p+{\pi }^{+})/{\Delta }^{++}+K^0$ and panel (b) from the reactions $p+p\to {\Sigma }^0+(p+{\pi }^{+})/{\Delta }^{++}+K^0$. The gray histogram corresponds to the sum of simulated contributions. The gray bands are the systematical uncertainties, whereas the red cups indicate the $7\%$ uncertainty from the absolute normalization to elastic scattering cross sections [23].