Femtoscopic study of coupled-channels $N\mathrm{\Xi }$ and $\mathrm{\Lambda }\mathrm{\Lambda }$ interactions

The momentum correlation functions of $S=-2$ baryon pairs ($p{\mathrm{\Xi }}^{-}$ and $\mathrm{\Lambda }\mathrm{\Lambda }$) produced in high-energy $pp$ and $pA$ collisions are investigated on the basis of the coupled-channels formalism. The strong interaction is described by the coupled-channels HAL QCD potential obtained by lattice QCD simulations near physical quark masses, while the hadronic source function is taken to be a static Gaussian form. The coupled-channels effect, the threshold difference, the realistic strong interaction, and the Coulomb interaction are fully taken into account for the first time in the femtoscopic analysis of baryon-baryon correlations. The characteristic features of the experimental data for the $p{\mathrm{\Xi }}^{-}$ and $\mathrm{\Lambda }\mathrm{\Lambda }$ pairs at the Large Hadron Collider are reproduced quantitatively with a suitable choice of nonfemtoscopic parameters and the source size. The agreement between theory and experiment indicates that the $N\mathrm{\Xi }$ ($\mathrm{\Lambda }\mathrm{\Lambda }$) interaction is moderately (weakly) attractive without having a quasibound (bound) state.

Figure 1

Categories of the $H$ dibaryon mass region. The bound $H$ region is defined as $M_H<2M_{\mathrm{\Lambda }}$. The “quasibound” state is defined as a physical state between the $\mathrm{\Lambda }\mathrm{\Lambda }$ and $N\mathrm{\Xi }$ thresholds, and it is a bound state of $N\mathrm{\Xi }$ and a resonance state of $\mathrm{\Lambda }\mathrm{\Lambda }$. For a quasibound state, the eigenmomenta $q$ in $N\mathrm{\Xi }$ channels need to have positive imaginary parts, so that the asymptotic wave function $\propto exp\left(iqr\right)/r$ converges to zero. The imaginary part of the eigenmomentum in the $\mathrm{\Lambda }\mathrm{\Lambda }$ channel needs to be negative, and the resonance wave function diverges asymptotically. If these eigenmomentum conditions are not satisfied for a pole, that pole does not represent a physical state and is called a virtual pole. See Appendix pp2 for details.

Figure 2

The $s$-wave coupled-channels HAL QCD potential for three temporal distances, $t=11,12$, and 13 at almost physical quark masses [12]. The colored shadow denotes the statistical error of each potential.

Figure 3

The $p{\mathrm{\Xi }}^{-}$ correlation function with the HAL QCD potential. The solid (dashed) line corresponds to the case with (without) the Coulomb interaction. The statistical error from the lattice QCD data is shown by the shaded area. The correlation function only with the Coulomb interaction is shown by the dotted line.

Figure 4

The breakdown of the $p{\mathrm{\Xi }}^{-}$ correlation function. The left panel shows the spin-averaged correlation function given by Eq. (6). The middle and right panels show the correlation function of spin single and triplet channels, respectively. The dashed lines denote the correlation function calculated only with the $p{\mathrm{\Xi }}^{-}$ wave function. The dash-dotted line and solid line denote the results with the contributions of $p{\mathrm{\Xi }}^{-}+\mathrm{\Lambda }\mathrm{\Lambda }$ and $p{\mathrm{\Xi }}^{-}+n{\mathrm{\Xi }}^0+\mathrm{\Lambda }\mathrm{\Lambda }$ channels, respectively.

Figure 5

The $\mathrm{\Lambda }\mathrm{\Lambda }$ correlation function. The statistical error of the lattice QCD data is shown by the shaded area. The result of pure quantum statistics without strong interaction is shown by dotted line.

Figure 6

The breakdown of the $\mathrm{\Lambda }\mathrm{\Lambda }$ correlation function. The dashed line denotes the correlation function calculated only with the $\mathrm{\Lambda }\mathrm{\Lambda }$ wave function component. The dash-dotted (the solid line) denote the results in which the contribution from the $n{\mathrm{\Xi }}^0$ (all the coupled channels) are added. The dotted line denotes the pure quantum statics case, where all the final state interactions are switched off.

Figure 7

Experimental and theoretical correlation functions of the $p{\mathrm{\Xi }}^{-}$ pairs (the upper panels) and the $\mathrm{\Lambda }\mathrm{\Lambda }$ pairs (the lower panels). The blank squares are the ALICE data taken from Refs. [11, 16, 17]; the statistical error and systematic error are denoted by the vertical line and the shaded bar, respectively. Solid lines are the theoretical results with statistical and systematic uncertainties represented by the shaded region. The left (right) panels correspond to the results in $pp$ collisions at 13 TeV ($p\mathrm{Pb}$ collisions ar 5.02 TeV). The dotted lines show the results with only Coulomb interaction (only quantum statistics) for the $p{\mathrm{\Xi }}^{-}$ ($\mathrm{\Lambda }\mathrm{\Lambda }$) correlation functions. The dash-dotted lines show the correlation function calculated with the LL formula.

Figure 8

The contour plot of the correlation function $C\left(q\right)$ in the LL analytic model at $r_{\mathrm{eff}}=0$ as a function of $x=qR$ and $y=R/a_0$.

Figure 9

Source size dependence of the $p{\mathrm{\Xi }}^{-}$ and $\mathrm{\Lambda }\mathrm{\Lambda }$ correlation functions. The thick lines denote the results with full coupled-channels calculation. For comparison, the calculations with the pure Coulomb cases (pure quantum statics cases) are shown for $p{\mathrm{\Xi }}^{-}$ ($\mathrm{\Lambda }\mathrm{\Lambda }$) correlation function by thin lines.

Figure 10

A schematic picture of the $s$-wave pole position generated by the strong interaction in the complex energy plane. The upper left (bottom left) figure shows the pole in the ($+$) [($-$)] sheet of the single channel case and the upper right [bottom right] right figure shows that in the ($-,+$) [($+,-$)] sheet of the coupled-channels case (see the text for the notation of the Riemann sheets). Energy region where the pole corresponds to physical eigenstate is denoted by the blue lines. As the attractive interaction becomes weaker from the bound (quasibound) region, the bound (quasibound) pole becomes the virtual pole.

Figure 11

Same as Fig 8 but with Gamow factor. For given $(x,y)=(qR,R/a_0)$, $\eta =-\mu \alpha /q$ is calculated as $\eta (x,y)=-\alpha |\mu y{a}_0|/x$ where we adopt $\mu a_0={\mu }_{p{\mathrm{\Xi }}^{-}}{a}_0^{N\mathrm{\Xi }(J=0)}=-3.32$.