$nnn$ and $ppp$ correlation functions

Scattering experiments with three free nucleons in the ingoing channel are extremely challenging in terrestrial laboratories. Recently, the ALICE Collaboration successfully measured the scattering of three protons indirectly, by using the femtoscopy method in high-energy proton-proton collisions at the Large Hadron Collider. In order to establish a connection with current and future measurements of femtoscopic three-particle correlation functions, we analyze the scenarios involving $nnn$ and $ppp$ systems using the hyperspherical adiabatic basis. The correlation function is a convolution of the source function and the corresponding scattering wave function. The finite size of the source allows for the use of the free scattering wave function in most of the adiabatic channels except the lowest ones. The scattering wave function has been computed using two different potential models: ($i$) a spin-dependent Gaussian potential with parameters fixed to reproduce the scattering length and effective range and ($ii$) the Argonne $v_{18}$ nucleon-nucleon interaction. Moreover, in the case of three protons, the Coulomb interaction has been considered in its hypercentral form. The results presented here have to be considered as a first step in the description of three-particle correlation functions using the hyperspherical adiabatic basis, opening the door to the investigation of other systems, such as the $pp\mathrm{\Lambda }$ system. For completeness, the comparison with the measurement by the ALICE Collaboration is shown assuming different values of the source radius.

Figure 1

The norm of the scattering wave function, Eq. (10), for the $nn$ and $pp$ cases. For the latter, different values of $\eta$ have been considered.

Figure 2

Top panel: The $pp$ correlation function calculated using a Gaussian potential acting in the singlet $\ell =0$ channel. The source radius has been fixed to $R=1.249$ fm. The $\ell =0$ contribution is given by the black curve, the contribution of the other (free) partial waves is given by the red curve while the total result is given by the green curve. The correlation function considering only the Coulomb force is given by the red dashed curve. Bottom panel: The $pp$ correlation function measured by the ALICE Collaboration (cyan data points) compared to the calculations obtained using the AV18 potential, considering only $s$ wave (magenta band) and the prediction obtained using the Gaussian potential (orange band). The width of the bands represents the uncertainty due to the experimental determination of the source radius. The theoretical curves have been corrected to account for the experimental effects (see the text for the details).

Figure 3

Norm of the free scattering wave function for three neutrons, Eq. (36).

Figure 4

The correlation function using the free scattering wave function for three neutrons, Eq. (42), calculated using different source sizes.

Figure 5

The $J^{\pi }=1/2^{-}$ contributions to the correlation function. One or two adiabatic channels are considered with and without the inclusion of higher contributions treated as free (tail).

Figure 6

The contribution to the correlation function of the states $J^{\pi }=1/2^{-},3/2^{-}$ for three neutrons computed with one ($n=1$) or two ($n=2$) adiabatic channels calculated up to the indicated value of $K_{\max }$. The curves indicated by “Bessel” use Bessel functions as hyperradial functions in the indicated channels (see text). The total curve (black) is the correlation function computed using one adiabatic channel for the states $J^{\pi }=1/2^{-},3/2^{-}$, calculated using HH functions up to $K_{\max }=31$, with the other states starting at $K=2$ considered free.

Figure 7

The norm of the free scattering wave function for three protons with and without considering the hypercentral Coulomb force and antisymmetrization.

Figure 8

Same as Fig. 7 but for the correlation function.

Figure 9

The $ppp$ correlation function with a source size of ${\rho }_0=2\mathrm{fm}$, calculated using the first adiabatic channel for asymptotic states with $K=1,2$, considering different values of $K_{\max }$. For $K>2$ the free scattering function is considered. The full free case is also shown.

Figure 10

The $ppp$ correlation function using the AV18 potential (solid lines) and comparison to the results using a Gaussian interaction (dashed lines). The source size ${\rho }_0=2.0$ fm was used. The Coulomb free contribution is also shown.

Figure 11

Top panel: Overlap between the average $s$-wave potential and the source function for different radii. Bottom panel: The $ppp$ correlation function calculated using the AV18 potential for different source radii (solid lines). The Coulomb free contributions are shown with dashed lines.

Figure 12

The comparison of the $ppp$ correlation function measured by the ALICE Collaboration [5] (cyan full squares) and the calculated correlation functions corrected for experimental effects (bands). The vertical lines on the data points correspond to the statistical uncertainties and the boxes to the systematic ones. Different color bands correspond to the different source sizes. For details of corrected correlation functions, refer to the text.