Short range correlations in the extended quantum molecular dynamics model

Momentum distribution of nucleons of ¹²C with SRC can be also obtained after cooling process as shown in Fig. 2 where the red, blue and green lines indicate the momentum distribution with different parameters for repulsive potential. They show that the high momentum part of distribution increases with the addition of stronger repulsive potential, which is of course consistent with the SRC effect.

It should be noted that the SRC effect may be also from other contributions such as tensor force Tensor1 ; Tensor2 ; Bai ; Tensor3 ; Bai2 , which is not presented in the EQMD model. The tensor force mainly acts on spin-triplet, isospin-singlet neutron-proton pairs, and it significantly reduces the kinetic symmetry energy to even negative values at saturation density Tensor4 ; Tensor5 ; Tensor6 ; Tensor7 . The tensor force is spin and isospin dependent. The current EQMD model does not contain spin quantum numbers. Adding a new quantum number to EQMD model could be another study, which is not included in this work. Thus, the present modified EQMD model has limitation and thus room is open for further improvement in the future. In the present framework, we calculate the process of proton-target reaction. Here we only focus on the emission of two protons. Though the momentum distribution seems reasonable as mentioned above, we should also have reliability check due to the addition of repulsive potential in the present work. In order to demonstrate a reasonable result in our calculation intuitively, the process of ¹²C proton-pair knock-out at 250 MeV incident energy has been setup in our calculation and used for comparison with the experimental data. Here the target nucleus with two different initial structures, namely, triangular and spherical configurations, are respectively simulated. The separation energy of the process is calculated, which is compared with the experimental results of Kobayashi et  al. in 2008 Kobayashi , as shown in Fig. 3.

In the figure, the orange line indicates the target nucleus ¹²C with three-$\alpha$ triangular structure while the dark one with spherical nucleon distribution. Here we use the area normalization, i.e. 1 for the spherical ¹²C and 1/5 for the triangular ¹²C. It shows that there is one peak in the separation energy spectrum for triangular three-$\alpha$ structure, and two peaks in the separation energy spectrum of ¹²C nucleus for the spherical nucleon distribution. The second peak is consistent with the peak of ¹²C nucleus with triangular structure. It is considered that both structures are not completely separated when screening the cluster structures of the initial nucleus after the cooling process, which leads to the possible existence of triangular structure in spherical nucleus. So we think that only the first peak is the separation energy spectrum of ¹²C nucleus for the Woods Saxon distribution. According to experiment by Bhowmik et  al. in 1976 Ranjan , different peaks will be formed in the separation energy spectrum according to the state of the nucleus, and the largest peak should correspond to the ground state of the nucleus. The EQMD model is a phenomenological transport model, which is not capable to simulate the excited states of nucleus very well. Therefore, the highest point for the experimental data in the figure should correspond to the first peak of the separation energy spectrum of the Woods Saxon distribution. The peak value of the separation energy spectrum of ¹²C nucleus in the Woods Saxon distribution is slightly higher than the experimental data. Here we think that the main reason is that the EQMD model does not take into account the effect of pion production. The incident energy of 250 MeV is higher than the threshold of pion production, and the pion generation will take away part of the energy. The separation energy calculation of the EQMD model without considering pion production effect includes this part of energy, which leads to a slightly larger separation energy than the experimental value. The separation energy of ¹²C nucleus for the triangular three-$\alpha$ clustering structure is higher than that of ¹²C nucleus for the spherical nucleon distribution in Fig. 3, this is because that spherical distribution is more uniform, while the triangular cluster structure has three $\alpha$-clusters, and the $\alpha$ cluster binding energy is relatively higher than spherical distribution. The pair of protons knock-out reaction for triangular configuration is similar to the proton pair knock-out reaction from one $\alpha$ cluster, so the calculated separation energy of the triangular clustering ¹²C nucleus is higher than that of the spherical distribution ¹²C nucleus. However, we notice that a small peak seems visible in the data at the same separation energy (around 20 MeV) as the triangle three-$\alpha$ configuration, which indicates that triangle three-$\alpha$ configuration is reasonable for explaining the separation energy in the region around 20 MeV. In one word, the separation energy distribution of the ¹²C(p, 2p)¹¹B reaction simulated by the present modified EQMD model is approximately agreement with the experimental data, which indicates that the present model in which the short-range repulsive potential was taken into account is a suitable tool for studying the proton pair knockout reaction.

Based on the above check of separation energy spectrum, we can simulate the proton pair knock-out reaction to study the SRC effect. In this calculation, we only consider the channel of two-proton emission, and obtain the information of phase space and emission time when the two protons eject.

Fig. 4 shows the difference of emission time distribution of the two emitted protons in different situations. The results show that most protons are emitted before 100 fm/c, which can be considered that they are caused by the knockout reaction. The other emitted protons are created in a uniform platform over 100 fm/c, which can be considered as a result of sequential decay. The higher the platform, the less stable the nucleus is. Thus, Fig. 4 shows that the spherical distribution ¹²C nucleus is more stable than the triangular clustering ¹²C nucleus. Fig. 4(b) and (c) show that for both structures, the more SRC is added, the less stable the nucleus is. In addition, for triangle ¹²C case, the stronger SRC effect (red line, $r_{0}=1.1$) induces earlier emission of protons.

Fig. 5 shows the difference of momentum distributions of the two emitted protons in different situations. Fig. 5(a) shows that the momentum distribution of the two emitted protons from ¹²C nucleus with different structures are almost the same. Fig. 5(b) and (c) show that the high momentum parts increase with the addition of stronger SRC, especially for the triangle case.

Fig. 4(b) and (c) and Fig. 5(b) and (c) show that both of emission time distribution and momentum distribution of the triangular clustering ¹²C nucleus change more obviously with the addition of SRC. It is generally considered that the closer the nucleons are combined, the greater the influence of SRC. Table 1 indicates that the RMS radius of the spherical distribution ¹²C nucleus is larger than the triangular clustering ¹²C nucleus, which means that globally the spherical distribution ¹²C nucleus combine closer than the triangular clustering ¹²C nucleus. We think that in each single $\alpha$ cluster the nucleons combine closer than the spherical structure, so the triangular clustering ¹²C nucleus is more sensitive to SRC. For triangle ¹²C case, stronger SRC potential (red line, $r_{0}=1.1$) induces obvious higher momentum component. In other word, initial high momentum tail can be somehow inherited by the higher momentum component of ejected protons.

Finally, the momentum correlation function of the emitted proton pair can be calculated by taking this phase space and emission time information as the input of LL model which is described in section C. The calculation result of the proton pair knock-out reaction of ¹²C with different initial ¹²C configuration at 250 MeV is shown in Fig. 6. It shows that the momentum correlation as a function of relative momentum of the two emitted protons. There is a peak at $q$ = 20 MeV/$c$, which is due to the contribution of strong interaction as well as Coulomb interaction. The function then tends toward unity at larger relative momentum ($q$) because of the vanishing correlation. In Fig. 6(a), we can see that the black line which refers to target nucleus with spherical nucleon distribution is significantly higher than the red line with the triangular cluster structure. The reason can be explained by the effective emission source size theory as Ref. HuangBS-2019 , which provides a similar result.

The momentum correlation function of the proton knock-out reaction with the SRC effect is calculated by the same method, and the result is shown in Fig. 6(b) and (c) for ¹²C nucleus with the spherical distribution and the triangular cluster structure, where the momentum correlation function calculated with different parameters $r_{0}$ for short range repulsive potential are displayed with green short dash line, blue dash line and red solid line, respectively, for $r_{0}$ = 1.3, 1.2 and 1.1, as well as the result without the SRC effect which is depicted in black solid line. It is seen from Fig. 6(a) that with the increase of the added repulsive potential, the peak gradually decreases, while the calculation without the short range repulsive potential gives the strongest correlation. Based on the explanation in Ref. HuangBS-2019 , the SRC leads to a larger size of the effective emission source, which leads to a lower peak. Besides, the increase of the natural decay part as shown in Fig. 4(b) and (c) can reduce the peak in the momentum correlation function, because there is no stable momentum or emission time correlation between randomly emitted natural decaying particles, which is also one of the explanations for the changing trend of the function. The result is also similar to the BUU result obtained in Ref. Wei ; Wei2 .