$p\Xi^- $ Correlation in Relativistic Heavy Ion Collisions with Nucleon-Hyperon Interaction from Lattice QCD

TL;DR

This paper uses lattice QCD to analyze the proton-Xi minus interaction and predicts a measurable correlation enhancement in heavy ion collisions, aiding understanding of hyperon-nucleon forces.

Contribution

It provides the first lattice QCD-based prediction of $p ext{-}\Xi^-$ correlations in heavy ion collisions, linking fundamental interactions to experimental observables.

Findings

Enhanced low-momentum correlation due to strong attraction.

Potential to constrain hyperon-nucleon interactions through experiments.

Provides a theoretical framework connecting lattice QCD and heavy ion collision data.

Abstract

On the basis of the $p\Xi^-$ interaction extracted from (2+1)-flavor lattice QCD simulations at the physical point, the momentum correlation of $p$ and $\Xi^-$ produced in relativistic heavy ion collisions is evaluated. $C_{\rm SL}(Q)$ defined by a ratio of the momentum correlations between the systems with different source sizes is shown to be largely enhanced at low momentum due to the strong attraction between $p$ and $\Xi^-$ in the $I=J=0$ channel. Thus, measuring this ratio at RHIC and LHC and its comparison to the theoretical analysis will give a useful constraint on the $p\Xi^-$ interaction.

Equations

$$R^{\alpha }(\vec{r},t)$$

$$\left(D_t^{\alpha }-H_0^{\alpha }\right)R^{\alpha }(\vec{r},t)$$

$$U^{\alpha \beta }(\vec{r},{\vec{r}}^{\prime })=(V_{\mathrm{LO}}^{\alpha \beta }(\vec{r})+V_{\mathrm{NLO}}^{\alpha \beta }(\vec{r})+\cdots )\delta (\vec{r}-{\vec{r}}^{\prime }).$$

$$C(\mathbf{Q},\mathbf{K})\equiv \frac{N_{p\Xi }({\mathbf{k}}_p,{\mathbf{k}}_{\Xi })}{N_p({\mathbf{k}}_p)N_{\Xi }({\mathbf{k}}_{\Xi })}\simeq \frac{\int d^4{x}_p\int d^4{x}_{\Xi }{S}_p(x_p,{\mathbf{k}}_p)S_{\Xi }(x_{\Xi },{\mathbf{k}}_{\Xi }){\left|{\Psi }_{p\Xi }({\mathbf{r}}^{\prime })\right|}^2}{\int d^4{x}_p{S}_p(x_p,{\mathbf{k}}_p)\int d^4{x}_{\Xi }{S}_{\Xi }(x_{\Xi },{\mathbf{k}}_{\Xi })},$$

$$S_i(x_i,{\mathbf{k}}_i)\propto E_i{e}^{-\frac{{\mathbf{x}}_i^2}{2R_i^2}}\delta (t-t_i),(i=p,{\Xi }^{-}),$$

$$C(Q)$$

$$C_{\mathrm{SL}}(Q)\equiv C_{R_{p,\Xi }=2.5\mathrm{fm}}(Q)/C_{R_{p,\Xi }=5\mathrm{fm}}(Q),$$

Full text

1 Introduction

The coupled-channel Nambu-Bethe-Salpeter (NBS) wave function measured in lattice QCD [1, 2] can now provide “theoretical” information of hyperon-nucleon and hyperon-hyperon interactions through the HAL QCD method [3, 4, 5, 6]. The energy-independent non-local potentials $U(r,r^{\prime})$ obtained by the method allow us to calculate the scattering phase shifts and binding energies of two baryons.

These potentials are also useful for analyzing the two-particle momentum correlations in relativistic heavy ion collisions [7]. It was recently studied in [8] that the possible spin-2 $p\Omega^{-}$ dibaryon state suggested by lattice QCD [9] can be probed by the $p\Omega^{-}$ momentum correlation at RHIC and LHC. In particular, the ratio of correlation functions between small and large collision systems, $C_{\rm SL}(Q)$, is shown to be a good measure to extract the strong interaction effect without much contamination from the Coulomb effect [8]. In the present paper, we extend the analysis to the $p\Xi^{-}$ system in $I=J=0$ channel which was recently predicted to have large attraction by the lattice QCD simulations at physical quark masses [4].

2 Lattice QCD formulation

We start with the normalized four-point function $R$ in channel $\alpha$ defined by

[TABLE]

where $B_{\alpha_{1}}(\vec{x},t)$ and $B_{\alpha_{2}}(\vec{x},t)$ are the sink operators for octet baryons. $\sqrt{Z_{\alpha_{1}}}$ $\sqrt{Z_{\alpha_{2}}}$ are the corresponding wave-function renormalization factors, and ${\mathcal{J}}(0)$ is a source operator at zero initial-time to create two baryons. The coupled channel potential is obtained through the linear partial differential equation  [2];

[TABLE]

with ${H_{0}}^{\alpha}=-\frac{\nabla^{2}}{2\mu^{\alpha}}$ and $\Delta^{\alpha\beta}=\exp[-(m_{\beta_{1}}+m_{\beta_{2}})t]/\exp[-(m_{\alpha_{1}}+m_{\alpha_{2}})t]$. $D_{t}^{\alpha}$ is a time-derivative operator whose leading-order term reads $-\partial/\partial t$. We introduce a derivative expansion to treat the non-local potential as

[TABLE]

In the following, we truncate the expansion at the leading order.

We employ $(2+1)$-flavor QCD configurations on the $L^{4}=96^{4}$ lattice with the lattice spacing $a\simeq 0.085$fm. This corresponds to the physical size, $La=8.1{\rm{fm}}$, which guarantees that the finite volume effect on $U^{\alpha\beta}(\vec{r},\vec{r}^{\prime})$ is negligible. The quark masses are chosen for the system to be almost at the physical point; $m_{\pi}\simeq 146$ MeV and $m_{K}\simeq 525$ MeV [4]. The total number of configurations is $414$ $\times$ $4$ space-time rotations $\times$ $48$ wall sources. The baryon masses measured in this setup are listed below.

[TABLE]

3 $p\Xi^{-}$ potential in $I=0$ channel

The $S=-2$ baryon-baryon interactions including the I=0 $\Lambda\Lambda-N\Xi-\Sigma\Sigma$ coupled-channel system have been recently reported in [4]. In particular, one of the diagonal components $V_{N\Xi,N\Xi}(r)$ in the $(I,J)=(0,0)$ channel (${}^{1}S_{0}$) was shown to have large attractive well at intermediate distance and relatively weak repulsive core at short distance, while $V_{N\Xi,N\Xi}(r)$ in the $(I,J)=(0,1)$ channel (${}^{3}S_{1}$) has weaker attractive well and stronger repulsive core. Also, $V_{N\Xi,N\Xi}(r)$ in the $I=1$ channels do not have appreciable attraction. Motivated by these observations, we parametrize the lattice results of $V_{N\Xi,N\Xi}(r)$ in the $I=0$ channels by a combination of the Gauss and Yukawa functions as shown in Fig.1. Curves with different $t$ correspond to the potentials obtained from $R({\vec{x}},t)$ for different $t$, so that the $t$ dependence of $V(r)$ reflects typical magnitude of the systematic error of the lattice data. We found that the strong QCD attraction in Fig.1(Left) together with the Coulomb attraction leads to the ${}^{1}S_{0}$ system close to the unitary region where the inverse of the scattering length is close to zero. On the other hand, the ${}^{3}S_{1}$ system described by Fig.1(Right) has strong repulsion even with the Coulomb attraction.

4 $p\Xi^{-}$ momentum correlation

The correlation function of non-identical pair such as $p\Xi^{-}$ is given in terms of the two-particle distribution $N_{p\Xi}({\bf k}_{p},{\bf k}_{\Xi})$ normalized by a product of the single particle distributions, $N_{\Xi}({\bf k}_{\Xi})N_{p}({\bf k}_{p})$,

[TABLE]

where relative and total momenta are defined as ${\bf Q}=(m_{p}{\bf k}_{\Xi}-m_{\Xi}{\bf k}_{p})/M$ and ${\bf K}={\bf k}_{p}+{\bf k}_{\Xi}$, respectively, with $M\equiv m_{p}+m_{\Xi}$. The source functions $S_{i}(x_{i},{\bf k}_{i})\equiv E_{i}\frac{dN_{i}}{d^{3}{\bf k}_{i}d^{4}x_{i}}$ (with $i=p,\Xi$ and $E_{i}=\sqrt{{\bf k}_{i}^{2}+m_{i}^{2}}$) correspond to the phase space distributions of $p$ and $\Xi$ at freeze-out. The final state interaction after the freeze-out is described by the two-particle wave function $\Psi_{p\Xi}$ with a shifted relative coordinate ${\bf r}^{\prime}={\bf x}_{\Xi}-{\bf x}_{p}-{\bf K}(t_{p}-t_{\Xi})/M$.

Here we consider the static source function with spherical symmetry to extract the essential part of physics;

[TABLE]

where $R_{i}$ is a source size parameter. Assuming the equal-time emission $t_{p}=t_{\Xi}$, we obtain

[TABLE]

where $[dr]=\frac{1}{2\sqrt{\pi}R^{3}}\!dr\,r^{2}e^{-\frac{r^{2}}{4R^{2}}}$ with $R=\sqrt{(R_{p}^{2}+R_{\Xi}^{2})/2}$ being the effective size parameter. $\int d\Omega$ is the integration over the solid angle between ${\bf Q}$ and ${\bf r}$. Note that $\psi^{C}({\bf r})$ is the Coulomb wave function characterized by the reduced mass and the Bohr radius of the $p\Xi^{-}$ system. Its S-wave component is denoted by $\psi_{0}^{C}(r)$. The scattering wave functions obtained by solving the Schrödinger equation with both strong interaction and Coulomb interaction are denoted by $\chi_{\rm sc}^{J=0}(r)$ and $\chi_{\rm sc}^{J=1}(r)$ for the ${}^{1}S_{0}$ channel and ${}^{3}S_{1}$ channel, respectively. We assume that the $I=1$ sector does not contribute substantially to $C(Q)$, which is supported by the fact that the $I=1$ $p\Xi^{-}$ potential has only short-range repulsion [4]. The factors $1/8=1/2\times 1/4$ and $3/8=1/2\times 3/4$ originate from the isospin and spin multiplicities. Also, we assume that the absorptive contribution by the coupling to the $\Lambda\Lambda$ channel is negligible since it is reported to be weak due to its short range nature [4].

In [8], the “SL (small-to-large) ratio” was introduced: It is defined as a ratio of $C(Q)$ between the systems with different source sizes,

[TABLE]

which has good sensitivity to the strong interaction without much contamination from the Coulomb interaction [8]. Shown in Fig.2 is $C_{\rm SL}(Q)$ of the $p\Xi^{-}$ system with the Coulomb interaction under the assumption of the static source given in Eq.(4).

The large enhancement of this ratio at small $Q$ originates from the fact that the $p\Xi^{-}$ system in the ${}^{1}S_{0}$ channel is close to the unitary region. The result has rather weak dependence on $t$, which indicates that the systematic errors of the lattice data do no aftect the final results significantly. We have also checked that taking the expanding source as discussed in [8] does not change the present result.

5 Summary

The momentum correlation of the $p\Xi^{-}$ system was presented by employing the $p\Xi^{-}$ potential extracted from the coupled channel analysis of the (2+1)-flavor lattice QCD data at the physical point. So-called the SL-ratio of the momentum correlation ($C_{\rm SL}(Q)$) was calculated and was shown to have large enhancement at small $Q$ due to the strong attraction between $p$ and $\Xi^{-}$ in the ${}^{1}S_{0}$ channel. Measuring this ratio at RHIC and LHC and its comparison to the present theoretical analysis will give useful constraint on the $p\Xi^{-}$ interaction. Such information is particularly important not only for the nature of the possible $H$-dibaryon coupled to $p\Xi^{-}$ [4] but also for the properties of $\Xi$-hypernuclei [10] and for $\Xi^{-}$ in the central core of the neutron star [11].

Acknowledgments

This work is supported in part by MEXT Grant-in-Aid for Scientific Research (JP15K17667, 24105008), SPIRE (Strategic Program for Innovative REsearch) Field 5 project and ”Priority Issue on Post-K computer” (Elucidation of the Fundamental Laws and Evolution of the Universe). K.M. was supported by JSPS Grant 16K05349, and National Science Center, Poland under grants:, Maestro DEC-2013/10/A/ST2/00106. T.H. was partially supported by RIKEN iTHES Project and iTHEMS Program. The authors thank HAL QCD Collaboration for proving us with the data for $p\Xi^{-}$ interactions and for stimulating discussions.