A sign of three-nucleon short-range correlation from an analysis of nuclear mass and short-range correlation probability

A breakthrough in high-energy electron/proton-scattering experiments off the nuclear targets initiated the study of the nucleon-nucleon short-range correlation Frankfurt et al. (1993); Aclander et al. (1999); Tang et al. (2003); Piasetzky et al. (2006); Egiyan et al. (2006); Subedi et al. (2008); Hen et al. (2014); Duer et al. (2018); Schmookler et al. (2019), which is a novel phenomenon driven by nuclear dynamics range from medium distance to short distance Frankfurt and Strikman (1988); Sargsian et al. (2005); Vanhalst et al. (2011); Weiss et al. (2018); Schiavilla et al. (2007); Alvioli et al. (2008); Neff et al. (2015). A surge of interests in the SRC research, with a significant amount of experiments and theoretical progresses, provided the criteria for the identification of SRC, and gradually revealed the underlying physical mechanism of SRC and the significance of SRC in the frontiers of nuclear physics and astrophysics. Measurements from both electron-nucleus ($eA$) and proton-nucleus ($pA$) experiments finally lead to some consensuses regarding nucleon-nucleon SRC: the nucleon inside SRC pair has a large relative momentum compared with the Fermi momentum ($k_{F}$) of the system while the SRC pair as a whole has a small center-of-mass momentum, and under the interplay of tensor force and short-range repulsive force, the SRC pair has two dominant features of locality and high momentum Hen et al. (2017); Arrington et al. (2012a); Fomin et al. (2017); Arrington et al. (2022).

Multi-nucleon short-range correlations represent the most intriguing part of the nuclear dynamic beyond the shell model and they are closely related to the strong interaction physics at the quark level as well as the relevant high-density dynamics in neutron star Subedi et al. (2008); Frankfurt et al. (2008). (i) SRC occurs in short-distance region, hence it is closely related to the phenomena governed by quantum chromodynamics (QCD), including chiral symmetry, quark interchanges between nucleons, and quark-gluon degrees of freedom, to name just a few examples. (ii) An ingenious analogy between the waltz and the nucleon-nucleon SRC Hen et al. (2014) vividly explains a universal nature of the dominant $np$ SRC pairs in both symmetric and asymmetric nuclei. This is vital for understanding the dynamics in the interior of neutron stars with a small fraction of protons. (iii) A remarkable linear correlation between the strength of the European Muon Collaboration (EMC) effect and the nucleon-nucleon SRC probability links these two seemingly disconnected phenomena, and implies that the nucleon-nucleon SRC induces a substantial change of the structure of bound nucleon thus yields the nuclear EMC effect Schmookler et al. (2019); Weinstein et al. (2011); Hen et al. (2012); Arrington et al. (2012b); Chen et al. (2017); Wang et al. (2023); Ma et al. (2023). In view of these aspects, it has been an area of continuing interests to investigate the details of multi-nucleon short-range correlations.

Up to date, the properties and details of 2N SRC have been extensively investigated, while the existence of multi-nucleon SRC of more two nucleons is still not clear Ye et al. (2018); Sargsian et al. (2019); Day et al. (2023). Beyond the 2N SRC, the main subject of this study is 3N SRC. First of all, it is an interesting topic whether there is a relationship between the 3N SRC and the three-body nuclear force Sargsian et al. (2005, 2019); Day et al. (2023). Second, from our previous studies, we found that the 2N SRC only is not enough to reproduce the observed nuclear EMC effect within both a $x$-rescaling model and a nucleon-swelling model Wang et al. (2023); Ma et al. (2023). Besides the dominant 2N SRC, 3N SRC could be another origin of the EMC effect. Third, 3N SRC is an important intermediate state in forming the four-nucleon SRC and other types of nucleon clusters. Therefore, it is imperative to study the $3N$ SRC, so as to unveil the diverse and new micro structures inside nucleus.

The $3N$ SRC is a type of short-distance configuration of nucleons with small inter-distances ($\leq 1.2$ fm), which has the very similar characteristics of the two-nucleon SRC Hen et al. (2017); Fomin et al. (2017). The nucleon momentum in $3N$ SRC significantly exceeds $k_{F}$, but in this case of $3N$ SRC, the high-momentum nucleon is balanced by the other two correlated nucleons with momenta around $k_{F}$. The c.m. momentum of the 3N SRC pair $p_{cm}$ is argued to be less than or equal to $k_{F}$ Sargsian et al. (2019); Day et al. (2023). On the experimental side, the appearance of a plateau of cross-section ratio in the region of the Bjorken variable $x_{B}\geq 2$ is the most direct signal for the existence of 3N SRC Ye et al. (2018). The pioneering experimental result by CLAS collaboration more than one decade ago displayed the second plateau after the $2N$ SRC plateau with large uncertainties as well, which hints the existence of $3N$ SRC Egiyan et al. (2006). However the subsequent experiment by JLab Hall C collaboration reported that the $3N$ SRC plateau disappears confronting the more precise experimental data. The recent analysis of the CLAS data found that this previous observed $3N$ SRC plateau is an effect from the bin migrations Higinbotham and Hen (2015). It is worth mentioning that the E02-019 experiment at JLab Hall C display no $3N$ SRC plateau at all, with the assistance of all possible experimental technologies. At present, there are more experiments planed to intensively verify the existence of $3N$ SRC, and parallel with the future experiments, exploring more new methods to pin down the $3N$ short-range correlations is also a feasible direction in the field.

In this paper, we try to find an evidence or a hint of $3N$ SRC from an in-depth analysis of the nuclear mass in terms of the 2N-SRC probability. In Sec. II, we introduce the definitions of 2N-SRC and 3N-SRC probabilities that will used in this analysis, and the relation between the 2N-SRC probability and the 3N-SRC probability. The decomposition of nuclear mass in terms of mean-field nucleons and multi-nucleon short-range correlations is given in Sec. III. The correlation between the nuclear mass and the SRC probability is analyzed with and without the 3N SRC, which is shown in Sec. IV. Finally, some discussions and a summary of our analysis are present in Sec. V.

In this study, the 2N SRC probability is defined as the probability of a nucleon being in a 2N SRC pair, which is written as,

where $n_{\rm 2NSRC}$ is the number 2N SRC pairs and $A$ is the total number of nucleons. Similarly, the 3N SRC probability is defined as,

in which $n_{\rm 3NSRC}$ is the number of 3N SRC clusters. With decades of efforts by experimentalists and theorists, the number of 2N SRC pairs in nucleus is already quantified Frankfurt et al. (1993); Egiyan et al. (2006); Fomin et al. (2012); Schmookler et al. (2019). Hence, the 2N SRC probability also can be estimated. However, there is few information about the 3N SRC in experiments, and its existence also need to be confirmed. It is not easy to have a reliable estimation of the 3N SRC probability.

In this work, we make an estimation of 3N SRC probability based on a simple assumption that the 3N SRC cluster is formed with sequential processes, and it parameterized as $P_{\rm 3N}=\kappa P^{2}_{\rm 2N}$ in which $\kappa$ is a free parameter. Fig. 1 schematically shows how the 3N SRC configuration is produced with two sequential processes of cohesion. The first process is that two nucleons attract each other and form into the close-proximity configuration as a 2N SRC pair. If luckily, within the short lifetime of the 2N SRC state, another nucleon is attached to the 2N SRC and generate a 3N SRC cluster. This is the second process for the 3N SRC formation. The probabilities of the first and the second processes are similar, for the similar intermediate attraction force is needed for both the first and the second processes. Therefore, the probability for the second process to happen is $\kappa P^{2}_{\rm 2N}$, with the parameter $\kappa$ is probably near 1. Then the probability of a nucleon in a 3N SRC is given by,

Combing Eqs. (1) and (3), finally we have an estimate of $P_{3N}$, which is written as,

in which $\kappa$ is a free parameter probably around 1.

Note that Fig. 1 just shows a simple picture of cluster formation processes. As the nucleus is complex many-body system, rigorous theoretical calculation on the probability of the cohesion process is very challenging, yet looking forward in the future. For multi-nucleon short-range correlations, the quark-exchange and the gluon-exchange processes also should be considered for the short-distance interactions. Nevertheless, according to the simple physical mechanism shown in Fig. 1, the probability of short-range correlations of more nucleons goes down quickly.

A simple global property of a nucleus is the mass, which is closely related to the nuclear binding or the mass deficit. Although the nucleus is a complex quantum system of complicated nucleon motions and diversified micro-structures, the sophisticated nuclear structure should be reflected in the nuclear mass. More importantly, the masses of most nuclei are already precisely measured in experiments. In this work, we try to analyze the nuclear mass at length in terms of the micro-structures.

Suppose in a nucleus, the nucleons can be classified into three categorises: the mean-field nucleon, the nucleon in 2N SRC, and the nucleon in 3N SRC. Then the nuclear mass can be decomposed as,

where $m_{\rm\bar{N}}$, $m_{\rm\bar{N}}^{\rm 2N}$, and $m_{\rm\bar{N}}^{\rm 3N}$ denote respectively the average mean-field nucleon mass, the average nucleon mass in 2N SRC, and the average nucleon mass in 3N SRC. If we remove the third term on the right side in Eq. (5), then we just a nuclear mass decomposition without 3N SRC clusters. By using this mass decomposition formula, we assume that the properties of 2N SRC or 3N SRC are universal among different nuclei. Actually, the approximate universality of 2N SRC has been tested and predicted in experiments Egiyan et al. (2006); Fomin et al. (2012); Schmookler et al. (2019) and theoretical calculations Feldmeier et al. (2011); Alvioli et al. (2016). Therefore the proposed nuclear mass decomposition is a good approximation of the nuclear mass in terms of some special micro-structures. For the nuclear mass per nucleon, rearranging Eq. (5), thus we have,

Combing Eqs. (3) and (6), finally we have the per-nucleon nuclear mass in terms of the probability of 2N SRC, which is written as,

From Eq. (7), one sees that the per-nucleon nuclear mass is a quadratic function of the 2N SRC probability, $aP_{\rm 2N}^{2}+bP_{\rm 2N}+c$, so long as there are 3N SRCs exist in the nucleus and the 3N SRC is generated from the combination of a nucleon and a 2N SRC pair. At the same time, if there is no short-range correlation of more than two nucleons, then the nuclear mass is just a linear function of the 2N SRC probability, as $bP_{\rm 2N}+c$. Therefore, by analyzing the correlation between the nuclear mass and the probability of 2N SRC, we could have some indications of multi-nucleon SRCs beyond the 2N SRC. If the nuclear mass is not linearly correlated with the 2N SRC probability, then there probably are some short-range correlations of more than two nucleons.

The 2N SRC probability in a nucleus is closely related to the number of 2N SRC pairs in the nucleus, according to the definition in Eq. (1). And the number of 2N SRC pairs in a nucleus is proportional to the SRC scaling factor $a_{2}$ ($a_{2}$ is defined as the quasielastic cross-section ratio between a heavy nucleus and the deuteron, in the 2N SRC kinematical region). Thanks to the developments of high-intensity and high-energy scattering experiments off the nuclear targets, the SRC scaling factor $a_{2}$ of many nuclei have been measured by SLAC Frankfurt et al. (1993), CLAS Egiyan et al. (2006); Schmookler et al. (2019) and JLab Hall C collaborations Fomin et al. (2012), which are listed in Table 1. The combined mean values are provided in the table as well. With the measured value of $a_{2}$, the number of 2N SRC pairs in the nucleus (mass number $A$) can be deduced from the following formula Wang et al. (2023); Ma et al. (2023):

where $n_{\rm SRC}^{\rm d}$ is the number of SRC pairs in deuteron. If $n_{\rm SRC}^{\rm d}$ is determined, then the absolute number of 2N SRC pairs in the nucleus is obtained. Once the absolute number of 2N SRC pairs is obtained, the 2N SRC probability in a nucleus is easily computed using Eq. (1).

Currently, there are some estimations on the number of SRC pairs in the deuteron. In our previous analysis, $n_{\rm SRC}^{\rm d}$ is determined to be $0.021\pm 0.005$ from a correlation analysis of the nuclear mass and $a_{2}$ Wang et al. (2021a). By counting the high-momentum nucleons of momentum above 275 MeV/c, $n_{\rm SRC}^{\rm d}$ is estimated to be $0.041\pm 0.008$ by CLAS collaboration Egiyan et al. (2006). Regardless of the obvious inconsistence between the two estimations, both values of $n_{\rm SRC}^{\rm d}$ from the models are used in this analysis. The resulting 2N SRC probabilities based on the values of $n_{\rm SRC}^{\rm d}$ from the two models are denoted as $P_{\rm 2N}^{\rm I}$ and $P_{\rm 2N}^{\rm II}$, respectively. The computed 2N SRC probabilities of various nuclei are listed in Table 2.

For the nuclear masses of the studied nuclei, they are very precisely measured in experiments and the combined average values are provided by atomic mass evaluation group. We take the nuclear mass data from the up-to-date evaluation in Ref. Huang et al. (2021); Wang et al. (2021b). The resulting per-nucleon nuclear masses of the studied nuclei are listed in Table 2.

Finally, the per-nucleon nuclear mass as a function of the 2N SRC probability $P_{\rm 2N}^{\rm I}$ is shown in Fig. 2. Two fits to the correlation between the nuclear mass and $P_{\rm 2N}^{\rm I}$ are performed based on the models with and without the 3N SRC clusters. The solid and dashed curves respectively show the fits of the two models with and without 3N SRCs. One sees that the model with 3N SRC fits much better the experimental data regarding the nuclear mass and the 2N SRC probability. The quality of the fit $\chi^{2}/N$ are 58.37/10 and 160.5/10 for the fits with and without 3N SRC, respectively. The sum of residual error squares is 2.35 for the linear fit without 3N SRC configurations, while it is 1.90 for the quadratic fit with 3N SRC configurations. These analyses indicate that there is a positive sign of the 3N SRC clusters in the nucleus. From the fit with 3N SRC configuration, we also determined the free parameters: $(m_{\rm\bar{N}}^{\rm 2N}-m_{\rm\bar{N}})=-48.0\pm 5.0$ MeV and $\kappa(m_{\rm\bar{N}}^{\rm 3N}-m_{\rm\bar{N}})=-367\pm 68$ MeV. The mass deficit of 2N short-range correlated nucleon is smaller than our previous estimation Wang et al. (2021a). The mass of 3N SRC nucleon is $572\pm 68$ MeV if $\kappa=1$, and it is $755\pm 39$ MeV if $\kappa=2$. These obtained masses of 2N and 3N SRC nucleons would provide some information on the modified inner structures of SRC nucleons. As the SRC nucleon mass is much smaller than the mean-field nucleon mass, the mean-field nucleon mass is taken as the average of free proton and free neutron masses, $m_{\rm\bar{N}}=938.92$ MeV. In this way, the mass decomposition formula has a fixed point of the free nucleon ($m(P_{\rm 2N}=0)=m_{\rm\bar{N}}$).

Fig. 3 shows the per-nucleon nuclear mass as a function of the 2N SRC probability $P_{\rm 2N}^{\rm II}$ from CLAS’ estimation Egiyan et al. (2006). We also performed two fits to the correlation between the nuclear mass and $P_{\rm 2N}^{\rm II}$ based on the models with and without the 3N SRC clusters. With no surprise, the model with 3N SRC fits much better the experimental data regarding the nuclear mass and $P_{\rm 2N}^{\rm II}$. The quality of the fit $\chi^{2}/N$ are 58.39/10 and 161.3/10 for the fits with and without 3N SRC, respectively. Judged by the analysis of the correlation between nuclear mass and $P_{\rm 2N}^{\rm II}$, there is a strong sign of 3N SRC clusters in the nucleus. Although the obtained free parameters are different from those obtained from the fit to $P_{\rm 2N}^{\rm I}$ data, the conclusion on the existence of 3N SRC is the same.

The fitting results of the fits discussed above are all summarized in Table 3, including the quality of the fit, the extracted values of the free parameters.

In this work, we find a positive sign of the existence of 3N SRC cluster in the nucleus, by the analysis of the correlation between the nuclear mass and the 2N SRC probability. A quadratic function is much better in describing the correlation between the mass and the 2N SRC probability, compared to the linear function. The quadratic function is explained with a nuclear mass decomposition formula considering there are three types of nucleon in the nucleus: the mean-field nucleon, the 2N SRC nucleon and the 3N SRC nucleon. It is worth noting that the absolute 2N-SRC probability is derived from the experimental measurements of $a_{2}$ and the number of 2N SRC pairs in the deuteron. Hence, the absolute 2N-SRC probability is more or less the experimental determined quantity. Actually, no matter how large the number of 2N SRC pairs in the deuteron is, the correlation between the nuclear mass and the 2N SRC probability is better explained with the model considering the existence of 3N SRC clusters. Therefore, we conclude that there is a sign of 3N SRC, showing in the the experimental data on the 2N SRC probabilities of the studied nuclei: ²H, ³He, ⁴He, ⁹Be, ¹²C, ²⁷Al, ⁵⁶Fe, Cu, ¹⁹⁷Au and ²⁰⁸Pb.

The analysis demonstrated in this work should be treated as a preliminary study. The nuclear mass decomposition formula used in the correlation analysis of the mass and the 2N SRC probability is just an approximate equation. The formula is based on three main assumptions: (i) there are 2N and 3N SRCs in the nucleus in addition to the independent mean-field nucleons; (ii) the masses of 2N SRC and 3N SRC nucleons are universal quantities in different nuclei; (iii) the 3N SRC is formed from the cohesion of a nucleon and a 2N SRC pair. Therefore, the analysis result present in this work is model-dependent.

From the analysis, the masses of 2N SRC and 3N SRC nucleons are also extracted from the fits. We find that the extracted SRC nucleon mass is very sensitive to the absolute probability of 2N SRC applied in the analysis. Nevertheless, the SRC nucleon mass extracted in this work would provide some valuable references on the nuclear modification of the inner structure of SRC nucleon.

At the end, we want to point out that the nonlinear correlation between the nuclear mass and the 2N SRC probability may also arise from 4N SRC, or $\alpha$ cluster, or even much bigger cluster of more than four nucleons. If the multi-nucleon cluster is formed from the sequential processes of nucleon cohesion, then the bigger cluster has the smaller formation probability. Hence, the the nonlinear correlation between the mass and the 2N SRC probability is mainly attributed to the 3N SRC cluster. There is still some rooms for an improvement of the current analysis. We can imagine that the nucleus is much complex quantum system of many nucleons, and there are much diversified and unknown micro-structures inside it. We still need more theoretical developments in revealing the underlying mechanism of 3N SRC, and the novel experimental techniques to confirm the existence of 3N SRC, and even to study the properties of these minorities in the nucleus.