1 Introduction

Understanding the nucleon force is one of the main goals in nuclear physics [1, 2], which is a necessary step to understand the structure of nuclei. In addition, behaviors of antiparticles or antiparticle induced reactions are always of interest [3-5]. So far the large body of knowledge on nuclear force was derived from studies made on nucleons or nuclei, and not much is known about the nuclear force between anti-nucleons. The knowledge of interaction among two anti-protons, the simplest system of anti-nucleons (nuclei), is a fundamental ingredient for understanding the structure of more sophisticated anti-nuclei and their properties. The important parameters to describe the strong interaction between two particles are the scattering length (f₀) and the effective range of the interaction (d₀). The parameter f₀ is related to the scattering cross section. At low energy limit, the scattering cross section is given by $\sigma =4\pi f_0^2$. The parameter d₀ is related to the range of the potential. In the case of square well potential, d₀ is the range (radius) of the potential. For a short range potential, f₀ and d₀ are related to the s-wave scattering phase shift, δ₀, and the collision momentum, k, by $k_0\approx 1/f_0+\frac{1}{2}{d}_0{k}^2$. To illustrate the scattering length (f₀) and the effective range of the interaction (d₀) cleanly, we use a square potential as an example. Figure 1 shows the square potential and the wave functions corresponding to different setups. V₀ represents the potential, and r₀ is for the radius part of the wave function. Here r₀ = d₀, i.e. the effective range. In case a) we have V₀ > 0 for a repulsive potential and the other three are attractive, with different depths. For square well potential, the effective range is the range (radius) of the potential, and a (or -f₀) is the intercept on the r axis when one does a linear extrapolation of the wave function at the barrier. If a bound state can be formed, one needs an attractive potential and the maximum of the wave function inside the potential range, like case b). However, in case c) when a < 0 (f₀ > 0, no bound state will be formed even though the potential is still attractive. In case d), the a is divergent at infinite. This is a special case usually called "zero energy resonance". Put it in short, for a bound state to be formed, one needs an attractive potential and a > 0 (or f₀ < 0).

Fig. 1.Full-screen View

(Color online) The square potential versus the wave functions corresponding to different setups. a) repulsive potential; b) attractive potential which can form the bound state; c) attractive potential but no bound state; and d) attractive potential with "zero energy resonance".

Although the existence and production rates of antimatter nuclei [6, 7] include antideuterons, antitritons, antihelium-3, antihypertriton [8-10], and antihelium-4 [8, 11, 12], offer indirect information about interactions between antinucleons. The interaction between two antinucleons, which is the basic interaction that binds the antinucleons into antinuclei, has not been directly measured in previous experiments. On the other hand, the measurement between antiprotons offers us a test of matter-antimatter symmetry, which is known as charge conjugation-parity transformation-time reversal (CPT) symmetry. Different CPT tests were conducted in other experiments. For example, one of which is the precious mass difference measurement for light nuclei and light anti-nuclei, such as deuteron and anti-deuteron, there is ³He and anti-³He by the ALICE collaboration [13]. Another example is a high-precision comparison of the antiproton-to-proton charge-to-mass ratio carried out in a Penning trap system [14]. In this work we would test the CPT from strong interaction aspect which is the nuclear force [15].

Relativistic heavy-ion collision provides a unique environment for not only the formation of quark-gluon plasma (QGP) [16-18], but also the production of antimatter nuclei [8]. With abundantly produced antiprotons, we can for the first time measure the scattering length (f₀) and the effective range (d₀) of the strong interaction between antinucleons. To do that, we use the technique that involves momentum correlation for probing the antiproton-antiproton interaction. It resembles the space-time correlation technique used in Hanbury-Brown and Twiss (HBT) intensity interferometry. Since it was first used in astronomy by Hanbury-Brown and Twiss in the 1950’s [19], this technique [20-22] has been used in many areas of physics, including the study of the quantum state of Bose-Einstein condensates [23], the correlation among electrons [24], and among atoms in cold Fermi gases [25]. In the late 1950’s, the Bose-Einstein enhancement, which is an enhanced number of pairs of identical pions produced with small opening angles, was first observed (GGLP effect) [26]. Later on, Kopylov and Podgoretsky devised the basics of the momentum correlation interferometry technique. In this technique, they defined the correlation functions (CFs) as ratios of the momentum distributions of correlated and uncorrelated particles,

with C=1 for no correlations. One can apply the mixing technique to construct the uncorrelated distribution by using particles from different collisions and extracting the space-time structure of the particle emission from the correlation function. So far the method of momentum correlation has been widely used by the nuclear physics community [27-33]. In particular, the same technique can even be applied for the complete kinematically three body decay of the nuclei very recently, which is very powerful to investigate the proton emission mechanism [34, 35].

2 STAR experiment

The process of two-particle correlation can be illustrated in Fig. 2 at RHIC-STAR detector. Colorful tracks display the different charged particles which are recorded by the Time Projection Chamber (TPC) at STAR. Assuming both correlated two (anti-)protons (red and yellow curves) are emitted from the source, they will interfere each other and finally produce intensity interferometry spectra, i.e. momentum correlation function. For each correlation function, in addition to quantum statistics effects, final state interactions (FSI) play an important role in the formation of correlations between particles. FSI includes the formation of resonances, the Coulomb repulsion, and the nuclear interactions between two particles, etc [27, 28, 36, 37]. They allows for (see Ref. [29, 38] and references therein) coalescence femtoscopy, correlation femtoscopy with non-identical particles, including access to the relative space-time production asymmetries, and a measurement of the strong interaction between specific particles. The latter measurement is often difficult to access by other means and is the focus of this proceeding paper (for recent studies see Refs. [39, 40]).

Fig. 2.Full-screen View

(Color online) Demonstration of two-particle correlation in heavy-ion collisions from the TPC in the STAR detector. The curves show particle trajectories where the track momenta can be determined. The 3-vector momenta can be measured by STAR detector in a wide range from about 0.1 GeV/c. Two particles were emitted from two separated points, with four-coordinates, x₁ and x₂. Their four-momenta, p₁ and p₂, were detected.

The data used in this analysis are Au + Au minimum bias events taken by the STAR experiment at RHIC during year 2011 at center of mass energies $\sqrt{s_{NN}}$ = 200 GeV . All data used here were taken by STAR Time Projection Chamber (TPC) [41] and Time of Flight detector (TOF) [42]. The TPC provides kinematic parameters of a track as well as the particle identification information based on the ionization energy loss. The TOF provides the mass information of the track. A Zero Degree Calorimeter (ZDC) [43] and a Vertex Position Detector (VPD) are combined to implement a minimum bias trigger. For event level, we use |Vz| < 30 cm, |Vr| < 2 cm and |TPCVz - VPDVz|< 3 cm. In our analysis, we select 30-80% centrality. A C++ class (StRefMultCorr) has been provided in STAR as a standard class to handle the centrality determination and generate event weight according to the centrality and luminosity at the same time. All centrality determination in this analysis is from the StRefMultCorr class. A set of basic track quality cuts used are listed in Table 1. A TPC track can have up to 45 hits, and a minimum of 15 TPC hits used in track fitting is required for every track participating the analysis. A single, long TPC track may be reconstructed as two shorter tracks if track splitting happens. These two short tracks have almost identical track parameters and only one of them should be used. Requiring the ratio of number of fitting hits over the number of maximum possible hits larger than 0.52 can effectively suppress the track splitting. A minimum transverse momentum of 0.4 GeV is required. The distance of closest approach (DCA) to primary vertex of global tracks is required to be smaller than 2 cm to select primary tracks. In order to select proton samples from the track, we use the nSigma cut and mass square cut. The information of the mass square can be derived from the TOF detector. Figure 3 shows a mass squared (m²) distribution versus $n_{{\sigma }_z}$ for antiprotons. By using both TPC and TOF, the purity for (anti)protons can be over >99% with transverse momentum (pT) less than 2 GeV/c.

Fig. 3.Full-screen View

(Color online) m² versus $n_{{\sigma }_z}$ for negatively charged particles. Here m² = (p²/c²)(t²c²/L²-1), where t and L are the time of flight and path length, respectively. c is the light velocity.

3 Method

In this experiment, two particle correlation function is defined as $\frac{A(k^{*})}{B(k^{*})}$, where A(k^*) is the distribution of half of the relative momentum (k^*) measured for the correlated pairs from the same event. And B(k^*) is the non-correlated pairs from two different (mixed) events. Impurities in (anti)protons will reduce the observed correlation strength. We use the following equation for the pair purity correction:

where PairPurity(k^*) is the product of the purities for the two particles and CF_measured(k^*) and CF_corrected(k^*) are respectively the corrected and measured correlation functions.

Inside our (anti)proton sample, there are secondary (anti)protons that are indistinguishable from primordial ones. When considering resonances, one should distinct the short-lived ones (like Delta) from the long-lived ones (decaying electromagnetically or weakly, like Lambda). The latter give rise to the effect of residual correlations (due to small decay momenta) as discussed in the paper. Their decay lengths are huge and do not affect the measured invariant radius Rpp; they reveal themselves only through the suppression parameter, xpp, and through the residual correlation contributions. The short-lived resonances do not contribute to residual correlations (not only due to substantial decay momenta, but also due to short decay times comparable with the production time, thus not allowing for the FSI to be developed). The residual correlations are mainly from the p-Λ and Λ-Λ correlations or their antiparticle pairs. We need to consider the corresponding contributions when we fit our CF. Taking the two-proton correlation function as an example [44],

where C_inclusive(k^*) is the inclusive CF, and C_pp(k^*;R_pp) is the true proton-proton CF, which can be described by the Lednický and Lyuboshitz analytical model [37]. In this model, for given s-wave scattering parameters, the correlation function with FSI is calculated as the square of the properly symmetrized wave function averaged over the total pair spin and the distribution of relative distances of particle emission points in the pair rest frame. C_pΛ(k^*;R_pΛ) is the p-Λ CF from a theoretical calculation [37] which has explained experimental data well [39]. C_ΛΛ(k^*) is from an experimental measurement corrected for misidentified Λ’s [40]. Rpp and R_pΛ are the invariant Gaussian radii [39] from the proton-proton correlation and the proton-Λ correlation, respectively. Here they are assumed numerically to be the same. xpp, x_pΛ, and x_ΛΛ represent the relative contributions from pairs with both daughters from the primary collision, pairs with one daughter from the primary collision and the other one from a Λ decay, and pairs with both daughters from a Λ decay, respectively. THERMINATOR2 model can give such parameters [45]. $C_{p\Lambda }(k_{pp}^{*})={\displaystyle \int }{C}_{p\Lambda }(k_{p\Lambda }^{*})T(k_{p\Lambda }^{*}\mathrm{,}{k}_{pp}^{*})d{k}_{p\Lambda }^{*}$, where $T(k_{p\Lambda }^{*}\mathrm{,}{k}_{pp}^{*})$ is a matrix that transforms $k_{p\Lambda }^{*}$ to $k_{pp}^{*}$ [44].

The proton-proton correlation function, C_pp(k^*;R_pp) in Eq. 3, can be described by the Lednický and Lyuboshitz analytical model [37]. In this model, the correlation function is calculated as the square of the properly symmetrized wave function averaged over the total pair spin, S, and the distribution of relative distances (r^*) of particle emission points in the pair rest frame, assuming 1/4 of the singlet and 3/4 of triplet states and a simple Gaussian distribution $dN/d^3{r}^{*}\sim \exp (-r^{*2}/4R_{pp}^2$). Starting with the FSI weight of nucleons emitted with the separation r^* and detected with the relative momentum k^*,

where ${\psi }_{-k^{*}}^{S(+)}(r^{*})$ is the equal-time (t^*=0) reduced Bethe-Salpeter amplitude which can be approximated by the outer solution of the scattering problem [46]. This is

where η=(k^* ac)^-1, ac= (57.5 fm) is the Bohr radius for two protons, ρ= k^* r^*, ξ= k^*r^*+ρ, Ac(η) is the Coulomb penetration factor given by Ac(η)=2πη[exp(2πη)-1]^-1, F is the confluent hypergeometric function, $\tilde{G}(\rho \mathrm{,}\eta )=\sqrt{A_c(\eta )}[G_0(\rho \mathrm{,}\eta )+i{F}_0(\rho \mathrm{,}\eta )]$ is a combination of the regular (F₀) and singular (G₀) s-wave Coulomb functions,

is the s-wave scattering amplitude renormalized by the Coulomb interaction, and $h(\eta )={\eta }^2{\displaystyle \sum _{n=1}^{\infty }}{[n(n^2+{\eta }^2)]}^{-1}-C-\ln \left|\eta \right|$ (here C 0.5772 is the Euler constant). The dependence of the scattering parameters on the total pair spin, S, is omitted since only the singlet (S=0) s-wave FSI contributes in the case of identical nucleons. The theoretical CF at a given k^* can be calculated as the average FSI weight w(k^*, r^*) obtained from the separation r^*, simulated according to the Gaussian law, and the angle between the vectors k^* and r^*, simulated according to a uniform cosine distribution. This CF is subject to the integral correction [37] $-A_c(\eta )|f_c(k^{*}{)|}^2{d}_0/(8\sqrt{\pi }{R}_{pp}^3)$ due to the deviation of the outer solution from the true wave function in the inner potential region. Considering that the emitting source has a net positive charge in Au+Au collisions, which influences the CF differently for proton and antiproton pairs, this effect is included in the consideration according to Ref. [47, 48]

4 Results and discussion

When we fit the proton-proton correlation function, only the radius is the free parameter, and we fixed f₀ = 7.82 fm and d₀ = 2.78 fm as they are well determined (from proton-proton elastic-scattering experiments [49]). While when we fit the antiproton-antiproton correlation function, the radius R, f₀, and d₀ are treated as free parameters. In Fig. 4, we present the PID purity corrected CF for proton-proton pairs (Fig. 4a) and antiproton-antiproton pairs (Fig. 4b), for 30-80% centrality of Au + Au collisions at centre-of-mass energy of 200 GeV. The fit to the data is plotted as red solid lines, shown for C_inclusive, while the CF of pp contribution is shown as dashed lines. For proton-proton CF, R_pp = 2.75 ± 0.01 fm; χ²/NDF = 1.66. For antiproton-antiproton CF, $R_{\overline{p}\overline{p}}=2.80\pm 0.02$ fm; f₀=7.41 ± 0.19 (stat) ± 0.36(sys) fm; d₀=2.14 ± 0.27 (stat) ± 1.34 (sys) fm; χ²/NDF = 1.61. The proton-proton CF exhibits a maximum at k^* around 0.02 GeV, which is caused by the attractive S-wave interaction between the two protons and is consistent with previous measurements. The antiproton-antiproton CF shows an exact similar structure, indicating that the interaction between two antiprotons is also attractive. Fig. 4c shows the ratio of the inclusive CF for proton-proton pairs to that of antiproton-antiproton pairs, and it is unity except for a very small k^* region where the error becomes large. This indicates that both proton-proton and antiproton-antiproton have the same strong interaction.

Fig. 4.Full-screen View

(Color online) Proton-proton correlation (a), antiproton-antiproton correlation (b), and the ratio of the former to the latter (c). Errors are statistical only. The fits to the data with equation 3, C_inclusive(k^*), are plotted as solid lines, and the term 1 + x_pp[C_pp(k^*;R_pp)-1] is shown as dashed lines. The χ²/NDF of the fit is 1.66 and 1.61 for (a) and (b), respectively.

Table 2 presents strong interaction parameters f₀ and d₀ of the antiproton-antiproton interaction as well as prior measurements for nucleon-nucleon interactions [49, 50]. It is found that the f₀ and d₀ for the antiproton-antiproton interaction are consistent with the proton-proton interaction within errors.

5 Summary and Outlook

Our STAR measurements on momentum correlation function of antiproton-antiproton pair from the RHIC-STAR detector provide quantitative parameters of scattering length and effective range for antiproton-antiproton interaction [51]. These parameters illustrate the strong interaction between antiprotons is attractive, which is the basis of the formation of complex antinuclei. The parameters also provide input for describing the interaction among cold-trapped gases of antimatter ions, as in an ultra-cold environment, where s-wave scattering dominates and effective-range theory shows that the scattering length and effective range are parameters that suffice to describe elastic collisions [52]. The result provides a new quantitative verification of matter-antimatter symmetry in the important and ubiquitous context of the forces responsible for the binding of (anti)nuclei.

Finally, considering the relatively large error of the current data, possible future improvement on the measurement can be made by reducing the uncertainty from the Λ-Λ CF (C_Λ-Λ(k^*)), which dominates our systematical error, with further accumulation of data. In addition, a similar effort of extracting f₀ and d₀ could also be repeated with (anti)proton CF [53] measured at the Large Hadron Collider, where the yield ratio of antiproton to proton is close to unity. In an additional way, the interaction between antiprotons could be also measured by antiproton-antiproton scattering, which could be realised in the future, by which f₀ and d₀ can be extracted by studying the s-wave scattering phase shift versus energy.