Qualities of nucleons, such as the fundamental parameter mass, might be modified in extreme conditions relative to those of isolated nucleons. We show the ratio of the EMC-effect tagged nucleon mass to that of the free one (m∗/m); these values are derived from the nuclear structure function ratio between heavy nuclei and deuterium measured in the electron Deep Inelastic Scattering (DIS) reaction in 0.3 ⩽x⩽0.7. The increase in m∗/m with A−1/3 is phenomenologically interpreted via the release of a color-singlet cluster formed by sea quarks and gluons in bound nucleons holding high momentum in the nucleus, from which the mass and fraction of non-nucleonic components in nuclei can be deduced. The mass of color-singlet clusters released per short range correlated (SRC) proton in the high momentum region (k> 2 fm−1) is extracted to be 16.890±0.016 MeV/c2, which evidences the possibility of a light neutral boson and quantized mass of matter.
In recent years, nuclear processes involving weakly bound (WB) projectiles, such as 6Li and 7Li nuclei, have been extensively investigated [1–5]. The primary reason for this interest is the opportunity to investigate how distinct reaction mechanisms affect each other when the binding energy of the projectile is relatively low. The 6,7Li WB projectiles are characterized by a high clusterization probability that emerges at a relatively low excitation energy (Ex), 6Li→α + d at Ex ~ 1.474 MeV and 7Li→α + t at Ex ~ 2.468 MeV, making them good candidates for this.
The scattering processes between heavy nuclei at energies approaching the Coulomb barrier (VB) are characterized by a peculiar behavior of the optical potential (OP), which is referred to as the threshold anomaly (TA) [6]. The term TA was introduced to define the manner in which the real and imaginary potentials vary with energy around VB. For systems with tightly bound projectiles, the TA can be identified as a bell-shaped peak in the real potential and a significant reduction in the strength of the imaginary potential as the energy reduces below VB. However, for systems induced by WB projectiles, this is considerably different. As the WB projectiles are characterized by their weak nature and high dissociation probability that is not suppressed below and close to VB, the coupling to the continuum creates a dynamic polarization potential (DPP) that causes the typical TA to disappear [7]. Owing to the high breakup probability that is present strongly below and close to VB, the imaginary potential is expected to increase instead of falling to zero to account for the non-vanishing breakup cross sections below VB. These characteristics have been proposed to explain why the usual TA does not exist in systems including the WB 6,7Li projectiles. The absence of TA at energies around VB was observed in many systems induced by the WB 6,7Li projectiles, and this was introduced as a new anomaly called the breakup threshold anomaly (BTA) [8]. Although the BTA is well presented in various systems induced by 7Li projectiles, among them 7Li + 27Al [9], 28Si [10], 138Ba [11], and other systems such as 7Li + 59Co [12], 80Se [13], 134Ba [7], and 208Pb [14], revealed the occurrence of the usual TA. These controversial results related to the occurrence of the BTA in systems induced by 7Li make it an active research area [15, 16].
In addition to the available experimental measurements for the 7Li + 144Sm system [17−21], many theoretical studies have focused on investigating this system [22−26]. In Ref. [17], the 6,7Li + 144Sm elastic scattering angular distributions (ADs) were measured experimentally at energies around VB for the two systems. The ADs were investigated using the optical model (OM) with a nuclear potential of the usual Woods-Saxon (WS) shape. None of the weakly bound systems studied (6,7Li + 144Sm) exhibited the TA, which is well presented in systems with tightly bound projectiles such as (12C, 16O + 144Sm). In addition to the various studies that determined a systematic global potential for 7Li [22−24], the 7Li + 144Sm system was investigated microscopically using the continuum discretized coupled channel (CDCC) method [25, 27−30]. In Ref. [25], the ADs for 7Li projectiles scattered by 59Co, 144Sm, and 208Pb targets at energies around VB were investigated using the CDCC method to determine the significance of the Coulomb-nuclear interference. The differences and similarities between the results for the 6Li and 7Li were also explored. The analysis showed that the 6Li total breakup cross sections are greater than those for 7Li.
The current study complements our ongoing research plan that investigates the peculiarities and interaction mechanisms of different nuclear systems induced by the WB projectiles [31−42]. In the current study, we examine the ADs and reaction cross sections (σR) for the 7Li + 144Sm system within various interaction potentials and calculation approaches. Finally, information on the absence of the typical TA is presented. In Section II, the potentials that were implemented in the theoretical calculations are discussed. The obtained results are presented in Section III, along with their discussion. The findings and interpretations of this work are discussed in Section IV.
The elastic scattering 7Li + 144Sm ADs at energies between 21.6 and 52 MeV [17, 18] are investigated within various potentials and approaches, starting with the fundamental phenomenological OM and the semi-microscopic São Paulo potential (SPP). The cluster folding model (CFM) was used to account for the 7Li→α + t cluster structure. The CFM computations with and without the inclusion of a surface DPP were performed to reproduce the considered data. Finally, the CDCC computations were carried out by considering the coupling to resonant states only, as well as that to resonant and non-resonant continuum. As a result, we believe that the analyses using different potentials could aid in clarifying the various characterizations of the 7Li + 144Sm system and in determining the optimal potentials that accurately reproduce the data.
The applicability of the OM model is predicated on the existence of a smooth-varying average potential and the premise that the influence of the various reaction channels can be reflected by an imaginary potential. As a result, the OM is applied primarily to the data that varied smoothly with the energy. If one of these premises does not hold and the behavior of cross sections deviates from being reasonably smooth, the OM will be a non-useful concept. The considered 7Li + 144Sm elastic scattering ADs are analyzed within the OM, which has the following central potential:
Here, VC(r) is the Coulomb potential, characterized by the radius RC=rCA1/3t,rC=1.3 fm. The real and imaginary parts of the nuclear potential appear as the second and third terms in Eq. (1), have the standard volume WS shape, and are identified by the parameters: V0 and W0 (potential depth), rV and rW (reduced radius parameter), aV and aW (diffuseness), respectively.
The 7Li + 144Sm interaction potential was constructed microscopically to address the various ambiguities inherited from the OM potentials. The SPP is conceptually comparable to the conventional double folding potential since it is derived by folding the densities of the projectile and target nuclei with an effective interaction potential vSPP2NN [43–47], and it is expressed as:
VF(R)=∬
(2)
The density distribution \rho_{P}(r_{P})of 7Li was taken from Ref. [48] and has the following form:
where \xi =0.1387, \gamma =0.0232, and \beta =0.3341. The density distribution \rho_{T}(r_{T}) of 144Sm nucleus was calculated within the Dirac-Hartree-Bogoliubov model [49] using the REGINA code. The new version SPP2 [47] was employed, and the effective interaction v_{NN}^{{\text{SPP2}}} , which is energy- dependent, is expressed as
The CFM was implemented to describe the 7Li + 144Sm elastic scattering ADs by considering the 7Li→α + t cluster nature that emerges at a low excitation energy ~ 2.468 MeV. The real ( {V^{CF}} ) and imaginary ( {W^{CF}} ) cluster folding potentials (CFPs) for the 7Li + 144Sm system were prepared as follows:
where {V_{{\text{ }}\alpha - {{\text{ }}^{144}}{\text{ Sm}}}} , {W_{{\text{ }}\alpha - {{\text{ }}^{144}}{\text{ Sm}}}} , {V_{t{ - ^{144}}{\text{ Sm}}}} , and {W_{t{ - ^{144}}{\text{ Sm}}}} are the real and imaginary potentials for the t + 144Sm and α + 144Sm channels at appropriate energies Et≈ 3/7ELi and Eα≈ 4/7ELi [50,51]. The intercluster wave function {{{\chi }}_{\alpha - t}}(r) represents a 2P3/2 state in a real WS potential, and it is characterized by the parameters RV = 1.83 fm, aV= 0.65 fm, and the depth (V0) was varied till the binding energy for the cluster was reproduced. As the maximum considered energy was 52 MeV, the suitable t + 144Sm and α + 144Sm potentials for constructing the 7Li + 144Sm CFPs were Ut + 144Sm at Elab=3/7 × 52 = 22.29 MeV and Uα+144Sm at Elab = 4/7 × 52 = 29.71 MeV, (U = V + W). The potentials Ut + 144Sm at Elab = 22.29 MeV “taken from the global potential for tritons” [50] and Uα + 144Sm at Elab = 50 MeV [51] were adopted to generate the CFPs for the 7Li + 144Sm. The three potentials used − WS, SPP, and CFP − to study the 7Li + 144Sm system at E = 52 MeV close to the strong absorption radius (RSA) are shown in Fig. 1. Despite the observed differences between the considered potentials in the low radial region, all fairly reproduced the data. This is consistent with the fact that most of the available heavy ion scattering data is sensitive only to the tail of the nucleus-nucleus potential, in the vicinity of the strong absorption radius (RSA), which is typically ~ 1.5{\text{ }}(A_P^{1/3} + A_T^{1/3}) . The employed OM and SPP are energy-dependent, while the employed CFP is energy-independent.
Figure 1
Figure 1.
(color online) Implemented real WS, SPP, and CFP potentials utilized to reproduce the 7Li + 144Sm AD at Elab= 52 MeV in (a) whole radial range and (b) in the range 8–12 fm with a linear and logarithmic scale.
The 7Li + 144Sm elastic scattering ADs were reanalyzed phenomenologically at energies between 21.6 and 52 MeV [17, 18] using OMP with a Coulomb and nuclear part, as shown in Eq. (1). The OM calculations were performed using the geometrical parameters rV, rW, aV, and aWfixed to 1.286, 1.739, 0.853, and 0.809 fm, respectively, in accordance with Cook’s study [22]. This allowed us to explore how the potential depths evolve with energy. The OM computations employed two varying parameters, V0 and W0, which were adjusted to describe the experimental data. The FRESCO code [52] upgraded with the χ2 minimization SFRESCO code was implemented to describe the data and obtain the optimal potential parameters by minimizing the χ2 value, which gives the deviation between the experimental data and the calculations.
The experimental 7Li + 144Sm ADs at all the considered energies below and above the barrier are fairly reproduced utilizing the OM approach as depicted in Fig. 2, and the extracted optimal parameters are listed in Table I. To examine the applicability of the dispersion relation on the real (JV) and imaginary (JW) volume integrals, these quantities were calculated, and their values are displayed in Table 1. The JV and JW values do not follow the typical dispersion relation, where the imaginary potential depth does not drop to zero as the energy decreases below the barrier. The OM analysis revealed that the TA, which is well presented in systems with tightly bound projectiles [53, 54], is not present in the 7Li + 144Sm system under consideration.
Figure 2
Figure 2.
(color online) 144Sm(7Li,7Li)144Sm experimental ADs versus theoretical calculations within WS potential (solid curves) and SPP (dashed curves) at Elab = 21.6, 25, 30, 35, 40.8, and 52 MeV. Data is displaced by 0.1.
The extracted V0 and W0 values from the present OMP analysis are far from those reported in Ref. [17] owing to the different geometrical parameters considered. Hence, we compared the behavior of the obtained OM-WS potential from this work with that potential in Ref. [17]. It was found that the two potential families at the sensitive radius (~ 10.7 fm) have a minimum uncertainty and are nearly independent of the geometrical shape.
Table 1
Table 1.
Optimal parameters extracted from the analysis of the 7Li + 144Sm system within both WS and SPP. The JV, JW and reaction cross sections (σR) values are given.
It is always desirable to derive the interaction potential within microscopic approaches, such as the SPP, for minimizing the various parameter ambiguities that may arise during the phenomenological OM analysis. Therefore, the new SPP2 [47] version was implemented to investigate the 7Li + 144Sm ADs. The real portion of SPP was computed using Eq. (2), and the imaginary portion was determined by multiplying the real portion by a factor. The shape of the central potential is as follows:
The second and third terms denote the real and imaginary SPP, with their corresponding normalization factors, NRSPP and NISPP, respectively. Consequently, two changeable parameters (NRSPP and NISPP) were used to fit the data. The small number of fitting parameters and the non-biased search (NRSPP and NISPP were permitted to freely vary between 0.1 and 2.0), and they result in an obvious energy dependence on the potentials. The computations within the SPP, as shown in Fig. 2, reasonably agree with the experimental 7Li + 144Sm ADs using the NRSPP and NISPP values displayed in Table 1. With an average value of 0.69 ± 0.09, the extracted NRSPP values in the energy range of 21.6–52 MeV (close to and above the barrier VB) are close. The NRSPP values moderately change as the energy falls below the barrier VB, whereas the NISPP values demonstrate an increasing trend. This observed behavior deviates strongly from the typical TA, in which the imaginary potential vanishes as the energy falls below the barrier. In other words, the analysis performed within the SPP confirms the existence of the BTA, which agrees with the findings of Ref. [17]. The analysis reveals the necessity of decreasing the real SPP strength by ~ 31% in order to fairly fit the 7Li + 144Sm ADs. This decrease is primarily owing to the 7Li breakup effects.
Motivated by the appreciable clusterization probability of 7Li into t + α cluters, we investigate the considered 7Li + 144Sm ADs using CFPs defined in Eqs. (5) and (6). The form of the central potential is as follows:
The second and third terms denote the real and imaginary CFP, with their corresponding normalization factors, NRCF and NICF, respectively. Consequently, two changeable parameters (NRCF and NICF) were used to fit the data. The obvious advantage of the analysis performed within the CFP is that both the real and imaginary potentials are prepared semi-microscopically. Second, the limited number of fitting parameters (NRCF and NICF) can indicate a clear energy dependence on the real and imaginary potential strengths, which provides a better understanding of the presence or absence of the BTA in the studied system. Some systems induced by the WB 7Li projectile [9–11, 17] showed the presence of BTA at energies around VB, where the imaginary potential strength increaed instead of dropping to zero. Other systems induced by the 7Li projectiles [12–14] showed the absence of the BTA.
The theoretical analysis within the CFP successfully reproduced the 7Li + 144Sm ADs at all considered energies and throughout the whole angular range, as depicted in Fig. 3. The extracted NRCF values in the energy range of 25–52 MeV (above the barrier VB) are quite consistent, with an average value of 0.62 ± 0.20. Furthermore, by reducing the energy below VB, the extracted NRCF exhibits an increasing trend, as listed in Table 2. The non-vanishing trend observed for the imaginary CFP part when the energy decreases below the barrier indicates the presence of the BTA. In addition, the analysis revealed that the strength of the real CFP must be decreased by ~ 38% to adequately describe the 7Li + 144Sm ADs in the energy range 25–52 MeV (above VB).
Figure 3
Figure 3.
(color online) 144Sm(7Li,7Li)144Sm experimental ADs versus the calculations within CFP at Elab = 21.6, 25, 30, 35, 40.8, and 52 MeV. Data is displaced by 0.1.
The considerable coupling impact on the breakup channel is the main source for the evident reduction in the strength of the SPP and CFP. This impact can be modelled by including an additional DPP to simulate the effects that result from coupling to all the other states (continuum) or by carrying out the microscopic CDCC computations. Both methods were considered to reproduce the 7Li + 144Sm ADs. In the present work, we have suggested a simple approach to simulate the DPP. The following form (Eq. 9) was employed to calculate the DPP based on the previously generated CFPs.
As previously demonstrated, the 7Li + 144Sm ADs in the energy range of 25–52 MeV were reasonably described within the CFPs when the real and imaginary CFP strengths were decreased by ~ 38% and 39%, respectively. We describe the considered ADs using non-normalized CFPs such that “both NRCF and NICF are fixed to unity” by incorporating an additional DPP. As a simple approach, the DPP was taken as a factor multiplied by the derivative of the real and imaginary CFPs. Thus, two parameters, NRDPP and NIDPP, which denote the normalization factors for the DPP, were employed to fit the data. The same methodology was followed in our previous works [5, 55, 56]. As shown in Fig. 4, the experimental 7Li + 144Sm ADs agree well with the (non-normalized CFPs + DPP) calculations, and the optimally obtained NRDPP and NIDPP values are listed in Table 3. The nature of the implemented DPP changed from an attractive one (for the data above the VB) to a repulsive one (for the data below the VB), as shown from the extracted values listed in Table 3. Consequently, the nature of the DPP highly depends on the interaction energy.
Figure 4
Figure 4.
(color online) 144Sm(7Li,7Li)144Sm experimental ADs at Elab = 21.6, 25, 30, 35, 40.8, and 52 MeV versus the (non-normalized CFPs + DPP) calculations (curves), 4-channels CDCC computations (dashed curves), and 14-channels CDCC computations (dashed dot curves).
Then, we employed the microscopic CDCC approach [57–59] to investigate the considered 7Li + 144Sm ADs. The inclusion of couplings to the continuum states, which makes it possible to analyze the impacts of projectile breakup, is an obvious advantage of the CDCC approach. The CDCC computations were carried out using the FRESCO code, considering the coupling to resonant (7/2– and 5/2–) states of widths 0.2 and 2.0 MeV, the bound non-resonant (1/2–) state, and the non-resonant continuum (discretized into 10 bins). These computations were referred to as the "14-channels CDCC". In addition, the CDCC computations, named "4-channels CDCC", were performed by ignoring the coupling to the non-resonant continuum; these computations are illustrated in Fig. 4. Our test computations revealed that the coupling to resonant states has the most important contributions, while the contributions from other continuum states are minimal. By using Eqs. (5) and (6), the diagonal and coupling potentials were determined. The continuum above the α + t breakup threshold energy was divided into momentum bins of width Δk = 0.25 fm-1, k is extended from 0.0 to 0.75 fm-1. The implemented model space for truncation and discretization of 7Li was taken from Ref. [60].
It is important to clarify that the calculations performed within the CFP approach could be considered a particular case of the CDCC calculations with only a single channel. Within the CFP approach, we reproduced the 7Li + 144Sm ADs using normalized real and imaginary CFP parts, as illustrated in Table 2. However, in the CDCC calculations (4 and 14 channels), the calculations are free of any adjusting parameters; therefore, we could study the effects arising from the couplings to resonant and non-resonant continuums on the elastic scattering channel. Therefore, it is not surprising that the CFP calculations (1-channel CDCC) using normalized CFPs work very well (Fig. 3), whereas the more physical CDCC calculations (4 and 14 channels, Fig. 4) failed to reproduce the data.
Additionally, the elastic scattering ADs for the 7Li + 144Sm system were analyzed using an effective potential Ueff extracted from the CDCC computations. The Ueff was taken as the sum of the cluster folding potential (UCF) and the dynamic polarization potential (UTELP). The latter arises from the coupling to the continuum states and is generated from the CDCC computations utilizing the so-called trivially equivalent local potential (TELP) approach [61]. The implemented Ueff in the theoretical calculations is expressed in Eq. (10). The generated UCF, UTELP, and Ueff for the 7Li + 144Sm system at Elab= 40.8 and 52 MeV in the radial region 8–12 fm, near the RSA, are shown in Fig. 5.
Figure 5
Figure 5.
(color online) UCF, UTELP, and Ueff potentials generated from the CDCC computations for the 7Li + 144Sm system at E = 40.8 and 52 MeV.
{U_{\rm eff}}(r) = {U_{\rm CF}}(r) + {U_{\rm TELP}}(r), U = V + {\rm i} W.
(10)
The considered ADs are reproduced utilizing normalized Ueff, which is characterized by two normalization factors, NReff and NIeff, for the real and imaginary parts of the Ueff, respectively. The 7Li + 144Sm elastic ADs and the calculations within the normalized Ueff agreed well, as shown in Fig. 6 using the NReff and NIeff values listed in Table 4.
Figure 6
Figure 6.
(color online) 144Sm(7Li,7Li)144Sm experimental ADs versus the calculations within the normalized Ueff at Elab = 21.6, 25, 30, 35, 40.8, and 52 MeV.
The energy dependence on the JV and JW values extracted from the various implemented potentials showed a nearly consistent pattern despite the different numerical values, as shown in Figs. 7 and 8. The studied dependence could be useful for identifying the presence or absence of the usual TA, which is identified as a bell-shaped peak in the real potential and a dramatic reduction in the strength of the imaginary potential as the energy reduces below the barrier. As shown in Fig. 8, there was no dramatic reduction in the JW values as the energy decreased below the barrier; the JW values demonstrated an increasing trend rather than dropping to zero as it approached the barrier, especially for those extracted from the calculations within SPP. This analysis demonstrates the absence of the typical TA, and that the BTA is well presented in the 7Li + 144Sm system.
Figure 7
Figure 7.
(color online) Energy dependence on the JV values extracted from the various applied potentials.
Within the framework of the various employed approaches, we investigated the energy dependence on the extracted σR values, as shown in Fig. 9. A commonly utilized reduction approach [62–64] was employed to clarify the comparison of the total reaction cross sections for the system under consideration. The reduced cross section is scaled as {\sigma _{\rm red}} = \dfrac{{{\sigma _R}}}{{\left( {A_P^{1/3} + A_T^{1/3}} \right)}} and the reduced energy is scaled as {E_{\rm red}} = {E_{\rm c.m.}}\dfrac{{A_P^{1/3} + A_T^{1/3}}}{{{Z_P}{Z_T}}}. The symbols P and T denote the projectile and target, respectively; σRdenotes the total reaction cross section; Z denotes the charge; and A denotes the masses of the involved nuclei. Figure 9 illustrates the reduced energy dependence on the extracted reduced reaction cross sections for the 7Li + 144Sm system as well as for the 7Li + 28Si [56] and 7Li + 58Ni [42] systems. The black dashed curve in the figure represents the energy dependency based on the Wong formula [64], which has the following form:
Figure 9
Figure 9.
(color online) Reduced energy dependence on the extracted reduced cross section for the 7Li + 144Sm system within the different implemented approaches: OM, SPP, CFP, CFP + DPP, and CDCC. Data for 7Li + 28Si [56] and 7Li + 58Ni [42] systems are also presented.
where {\varepsilon _0} = \hbar {\omega _0}\dfrac{{A_P^{1/3} + A_T^{1/3}}}{{{Z_P}{Z_T}}} , {V_{\rm red}} = {V_0}\dfrac{{A_P^{1/3} + A_T^{1/3}}}{{{Z_P}{Z_T}}}, and {E_{\rm red}} = {E_{\rm c.m.}}\dfrac{{A_P^{1/3} + A_T^{1/3}}}{{{Z_P}{Z_T}}}. The parameters of the Wong model were used with the fitting parameters {\varepsilon _0} = 0.178, r0 = 0.13 fm, and {V_{\rm red}} = 0.83 MeV. However, the reliability of these fit parameters may be limited owing to the scarcity of data. It is clearly shown that the extracted σR values exhibit an increasing trend as the energy increased, and they are nearly consistent except for the extracted σR values from the CDCC computations, which showed a remarked increase, especially at the low energies studied, mainly owing to the breakup effects of 7Li.
To observe the effect of channel coupling on the 7Li + 144Sm AD at Elab = 40.8 MeV, we compared the 1 channel (1 Ch) CDCC calculation to those of the 4 and 14 channels, as shown in Fig. 10. As expected, the more channels were coupled, the better fitting the achieved. Further, one of the unsettled issues observed in nuclear reactions induced by the weakly bound clusterized nuclei such as 7Li projectiles is the origin of the incomplete fusion and large α particle production. A few years ago, the most accepted explanation for the aforementioned behavior was the breakup into α + t structure followed by the capture of triton (t). In recent studies [65, 66], this issue was investigated through pioneering experimental measurements for the 7Li + 93Nb and 7Li + 209Bi systems, respectively, allowing, for the first time, a significant population of the region to be accessible only to the direct triton stripping process and not to breakup followed by the capture of triton. The main conclusion drawn from these studies was the dominance of the direct cluster-stripping mechanism in the large alpha production. To further study this effect on the considered 7Li + 144Sm AD at Elab = 40.8 MeV, we studied the breakup effect and the combined effects of breakup and triton transfer on the elastic scattering channel, as depicted in Fig. 10.
Figure 10
Figure 10.
(color online) 144Sm(7Li,7Li)144Sm experimental AD at Elab = 40.8 MeV versus the 1-channel (dotted curve), 4-channels (dashed curve), and 14-channels (dashed dotted curve) CDCC computations. The figure also demonstrates the effect of breakup (solid curve) in comparison with the combined effects of breakup and triton transfer (short dashed curve).
The triton was assumed to be transferred as a single entity from the 7Li ground state to a t + 144Sm state f 147Eu. The potential for the entrance channel (7Li + 144Sm) is the normalized Ueff at Elab = 40.8 MeV (with normalizations, NReff = 1.029 and NReff = 0.794), which considers the 7Li breakup effect. For the exit channel (t + 147Eu), the SPP was employed with the standard form (NRSPP = 1.0 and NISPP = 0.78). The bound state potential for the configurations 7Li→ α + t and 147Eu→ 144Sm + t were taken in the standard WS form with RV = 1.25, aV = 0.65 fm and the depth was adjusted to reproduce the binding energies for the considered clusters, 2.467 and 10.529 MeV, respectively. The spectroscopic amplitudes for the \left\langle {{{}^{\text{7}}{\text{Li}}}} \mathrel{\left | {\vphantom {{{}^{\text{7}}{\text{Li}}} {\alpha + t}}} \right. } {{\alpha + t}} \right\rangle and \left\langle {{{}^{{\text{147}}}{\text{Eu}}}} \mathrel{\left | {\vphantom {{{}^{{\text{147}}}{\text{Eu}}} {^{144}{\text{Sm}} + t}}} \right. } {{^{144}{\text{Sm}} + t}} \right\rangle overlaps were taken as unity. The calculation, considering the combined effects of breakup and triton transfer, demonstrated a substantial increase in the cross-section values, indicating that there is a significant impact from the triton transfer mechanism.
The study provides a thorough analysis of the existing 7Li + 144Sm ADs at energies comparable to the barrier energy VB, utilizing various potentials and approaches starting from the simplest WS volume-shaped phenomenological potential of two changeable parameters (V0 and W0) and fixed geometrical parameters. As a result, either the presence or absence of TA may be indicated by the energy dependence on both the V0 and W0 values. The analysis highlighted the presence of the BTA behavior; the imaginary potential depth exhibits a non-vanishing trend as the energy reduces below the VB, which agrees with the findings of Refs. [9−11, 17] and contradicts those of Refs. [12-14].
The analysis within the SPP revealed the following features. The imaginary SPP strength showed a remarked increase as the energy reduced below VB. Therefore, the analysis using the SPP approach confirmed the presence of the BTA. In addition, the analysis revealed the need to reduce the real SPP strength by ~ 31% in order to fairly reproduce the 7Li + 144Sm ADs.
The considered data were reanalyzed later within the CFM by considering the 7Li → t + α cluster nature. The CFM analysis revealed the need to reduce the real CFP strength by ~ 38% to adequately reproduce the 7Li + 144Sm ADs, while confirming the presence of the BTA phenomenon. The real CFP strength showed an increasing trend as the energy decreased below the barrier. This rise in the real CFP strength was accompanied by a non-vanishing trend in the imaginary CFP strength.
Generally, the analyses conducted within the SPP and CFP revealed the necessity of reducing the real potential strength to accurately reproduce the data under consideration. This observed behavior originates from the strong coupling to the 7Li breakup continuum states, which generates a repulsive DPP. Two different strategies were followed for modeling the coupling effects: by introducing an additional DPP and by carrying out the full microscopic CDCC method. The CDCC computations revealed that the coupling to the resonant states has the most significant contribution. The analyses also showed that both the 7Li breakup and the triton transfer mechanism have significant impacts on the elastic channel data.
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Tao-Feng Wang, Zi-Ming Li and Xiao-Ting Yang. Possible Light Neutral Boson and Particle Mass Quantization[J]. Chinese Physics C. doi: 10.1088/1674-1137/abc536
Tao-Feng Wang, Zi-Ming Li and Xiao-Ting Yang. Possible Light Neutral Boson and Particle Mass Quantization[J]. Chinese Physics C.
doi: 10.1088/1674-1137/abc536
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School of Physics, Beihang University, Beijing 100191, China
Received Date:
2020-04-10
Available Online:
2021-01-15
Abstract: Qualities of nucleons, such as the fundamental parameter mass, might be modified in extreme conditions relative to those of isolated nucleons. We show the ratio of the EMC-effect tagged nucleon mass to that of the free one (m^{\ast}/m); these values are derived from the nuclear structure function ratio between heavy nuclei and deuterium measured in the electron Deep Inelastic Scattering (DIS) reaction in 0.3 \leqslant x\leqslant 0.7. The increase in m^{\ast}/m with A^{-1/3} is phenomenologically interpreted via the release of a color-singlet cluster formed by sea quarks and gluons in bound nucleons holding high momentum in the nucleus, from which the mass and fraction of non-nucleonic components in nuclei can be deduced. The mass of color-singlet clusters released per short range correlated (SRC) proton in the high momentum region (k> 2 fm^{-1}) is extracted to be 16.890\pm0.016 MeV/c^{2}, which evidences the possibility of a light neutral boson and quantized mass of matter.
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I.
INTRODUCTION
The widely accepted physical framework, the Standard Model (SM), can explain the fundamental properties of and force between leptons and baryons. It was substantially confirmed after the discovery of the Higgs boson, which breaks the weak isospin symmetry of the electroweak interaction [1]. A theoretical complement of SM extensions [2] has been used to explain the astrophysical observations of evidence for dark matter [3-6], thought to account for a quarter of the universe's total energy density and composed of subatomic particles. A new proposed gauge vector boson is the force mediator of Weakly Interacting Massive Particle (WIMP) annihilations [6]. This boson is expected to be a light neutral spin-1 boson with a small gauge coupling in the MeV/c ^{2} to hundred MeV/c ^{2} mass range via allowed decays into leptons and hadrons [7].
The differential cross section for electron Deep Inelastic Scattering (DIS) can be expressed as
where \alpha is the fine-structure constant, E and \theta are the incident electron energy and the scattering angle, respectively, and W_{1} and W_{2} are form factors. Under the limit of high energy and momentum transfer, the structure function has been found to scale with the form factors of F_{1}(x) = mW_{1}(x) and F_{2}(x) = \nu W_{2}(x) , where m and \nu are the free nucleon mass and the transfer energy in the electron DIS reaction, respectively. The two structure functions are connected to each other by the CallanGross relation [8], F_{2}(x) = 2xF_{1}(x) . Therefore, the ratio of differential cross-sections depending on x is proportional to that of the structure function F_{2}(x) . The ratio of the per-nucleon lepton DIS cross section of heavier nuclei to that of deuterium is less than one, known as the EMC effect [9-13], indicates a quenching of nuclear structure functions with respect to those of the isolated nucleon. Structure functions are interpreted as a linear combination of quark distribution functions. The quark distribution corresponding to the EMC effect is essentially altered accordingly. Do the properties of the EMC-effect tagged nucleon, such as its mass, also change remarkably? This is an attractive topic for both the experimental and theoretical sides of nuclear and particle physics. Only a few experimental studies have been carried out because of the extreme difficulty of producing the proper nuclear in-medium state and using an indirect method to deduce the dependent nucleon mass information. It is thus quite interesting to probe the mass of the nucleon and its dependence on the EMC effect in the nucleus via a phenomenological framework based on the quark distribution functions.
We mine the world-wide experimental EMC effect data to extract the nucleon mass in light and heavy nuclei using the Bjorken re-scaling method. A reducing trend of the mass of the EMC-effect tagged nucleon with A is clearly exhibited; this can be understood using the model of the release of non-nucleonic components from the nucleon under a configuration with a strong nucleon-nucleon interaction. The detailed description of the analysis process and results are shown in the following sections.
II.
X-RESCALING
The momentum fraction of the quark in the free nucleon is x = Q^{2}/2m\nu , where Q^{2} = \overrightarrow{q}^{2}-\nu^{2} ; \overrightarrow{q} is the 3-vector momentum transfer in the ( e, e' ) reaction. The nucleon structure functions ( F_{2}(x) ) in the nucleus detected by the charged lepton DIS reaction differ significantly from those measured for the free nucleon, i.e., the EMC effect, which probably indicates the essential modification of the quark distribution in bound nucleons. Therefore, the momentum fraction of quarks in the nucleon in the nucleus is formed as x' = Q^{2}/2m^{\ast}\nu , resulting from the altered quantity of the nucleon mass in the nucleus, especially the EMC-effect tagged nucleon. An x-rescaling was modeled as x' = xm/m^{\ast} = x\eta , reflecting the modification of the bound nucleon mass inside the nuclei [14], even though experiments have detected only a momentum-averaged m^{\ast} for the EMC-effect. The ratio of differential cross sections of the DIS reaction for nuclei and deuterium is approximately equal to their nuclear structure function ratio. The x dependence of R(A) = 2F_{2}^{(A)}(x)/AF_{2}^{(D)}(x) for different targets was measured previously using the electron DIS reaction [15]. Although m^{\ast}/m was extracted based on the electron DIS measurements using x-rescaling with the parton model [14], F_{2}(x) of nuclei were roughly expressed by x parameterizations for the valence quark distribution without gluon and anti-quark considerations [14]. We utilized the modern parton distribution function to analyze the new measured EMC-effect data and obtained a different function for m^{\ast}/m that depends on A^{-1/3} . Remarkably, a quantized-mass relationship for the reduced mass of an EMC-effect tagged nucleon and the mass of well-known mesons are observed.
To extract m^{\ast}/m , the ratio of the nuclear structure function is expressed as [16]
where u, \;\overline{u},\;d,\;\overline{d} and s are the parton distributions of the free proton [17], and u^{A},\;\overline{u}^{A},\;d^{A},\;\overline{d}^{A} and s^{A} are the nuclear parton distributions [18],
where w_{i} is a weight function that indicates nuclear modification for type-i parton distribution. Because w_{i} are x-dependent functions, we let w_{i} = 1 and utilize \eta x to replace x' to describe the nuclear modification.
Recently, R(A) of several nuclei were precisely re-measured in the J-Lab [13], and the data were corrected for isoscalar, nuclear Coulomb field, inelastic scattering, and Short Range Correlation (SRC) center-of-mass motion. We carried out the x-rescaling for the new measured R(A) as 2F_{2}^{(A)}(\eta x)/AF_{2}^{(D)}(\eta x) and adopted the new parameterization of nuclear parton distributions [18], including valence-, anti-, and sea-quark descriptions for nuclear structure functions [16, 17]; we then performed a fit (Fig. 1) to deduce the only parameter \eta , i.e., m/m^{\ast} . The new m^{\ast}/m depending on A^{-1/3} are plotted and fitted with a function of m^{\ast}/m = 0.994-0.069{\rm e}^{-4.488A^{-1/3}} , as shown in Fig. 2, which differs from the linear function given in Ref. [14]. The nucleon mass reduction is related to the density distributions of the nucleus, which are usually described by a Woods-Saxon type distribution as
Figure 1.
(color online) Fitting for EMC data [13, 15] in the range of 0.35 < x < 0.7, with x-rescaling of the nuclear structure function via the parton model [14, 16, 18].
Figure 2.
(color online) Ratio of m^{\ast}/m depending on A^{-1/3} : triangles denote the ratio of average per nucleon mass calculated from mean field theory [19] to the free nucleon mass, and solid circles indicate m^{\ast}/m extracted from 2\sigma_{A}/A\sigma_{2} experimental data in the range of 0.35\leq x \leq0.7 to the free nucleon mass [13, 15]. Squares indicate the previous m^{\ast}/m value [14]. The solid curve is the fitting by the function m^{\ast}/m = 0.994- 0.069{\rm e}^{-4.488A^{-1/3}} .
with diffuseness parameter a\sim 0.53 fm, where \rho_{0}\sim 0.17 fm ^{-1} is the density at the center of the nucleus. The radius parameter R is formed by mass number A as R\sim1.10A^{1/3} (fm). Therefore, an exponential function was utilized for fitting m^{\ast}/m depending on A^{-1/3} .
The R(A) data after the isospin correction are shown in Fig. 1. The directly measured EMC effect of R(A) of ^{3} He is larger than one, and those of the isospin symmetry and neutron rich nuclei are less than one, which indicates an enhancement of the structure function for the proton-excess nucleus and a shrinkage of the structure function for the isospin symmetry and neutron rich nuclei compared with that of the free nucleon. R(A) is the ratio of the structure function averaged by atomic number A; it reflects the characteristics of the major nucleons. Conversely, the main mass of the nucleon comes from the dynamic motion of quarks and gluon. The variation of the structure function corresponds to the changed mass of the nucleon in the EMC-effect region. Therefore, it is expected that the isospin splitting of the nucleon is m^{\ast}_{p}>m^{\ast}_{n} for the proton-excess nuclei and the isospin symmetry and neutron rich nuclei in the EMC-effect range of 0.35< x < 0.7 .
III.
COLOR SINGLET CLUSTER MODEL
The existence of a light boson with mass m_{B}c^{2}<20 MeV, coupling to baryons of the Higgs boson or the proposed scalar partners of the graviton, would give rise to a force with relatively long-range \hbar c/m_{B}c^{2}>10 fm [20]. This light boson is possibly produced from a color-singlet cluster formed by sea quarks and gluons in the bound nucleon. The color-singlet cluster might be released from the bound nucleons, rather than the valence quarks that represent the quantum number of nucleons and are confined by the QCD potential. The released color-singlet clusters can exchange momentum within the nucleus. Those clusters are proposed to be the origin of the non-nucleonic components in the nucleus and significantly influence the nuclear effect on the structure function [21]. If the non-nucleonic degree of freedom is combined with the x-rescaling model, a perfect description of the structure function ratio for the isospin scalar nucleus in the region x < 0.7 can be realized [21].
The formation of the color-singlet clusters is only connected with the color distribution and should not be related to the momentum distribution [21]. The parton momentum distribution in the color-singlet clusters can be expressed as P_{i}'(x,Q^{2}) = c_{i}(Q^{2})P_{i}(x,Q^{2}) , where i = s,g indicate sea quarks or gluons, and P_{i}(x,Q^{2}) are their momentum distribution in the nucleon. c_{i} (Q^{2}) is the fraction of momentum distribution P_{i}(x,Q^{2}) in the released cluster. Assuming \beta = c_{s}(Q^{2})S+c_{g}(Q^{2})G , where G = \int_{0}^{\infty} P_{G}(x){\rm d}x and S = \int_{0}^{\infty}P_{S}(x){\rm d}x , \beta is Q^{2} independent. It is easier for the nucleons at the center of the nucleus to produce color-singlet clusters because there are more surrounding nucleons than those on the nuclear surface to supply the nuclear medium environment. Hence, c_{i}(Q^{2}) is an averaged value, and \beta becomes larger with increasing A. According to the uncertainty principle, the longitudinal size of a parton would extend over more than one nucleon and participate in the reaction with the neighbor nucleon if x is very small, which results in the shadowing region of R(A) . The momentum of the nucleus is therefore P_{A} = A(1+\beta-\alpha)P_{N} , where \alpha is the fraction of the common parton in the nucleus and P_{N} is the nucleon momentum distribution. The fraction of parton momentum in the nucleon is
\eta = (1+\beta-\alpha)Am/M_{A} , where M_{A} is the mass of a nucleus. Hence, the total fraction of the non-nucleonic color-singlet cluster (\beta-\alpha) in nuclei can be obtained, and its A^{-1/3} dependence is shown in Fig. 3. The corresponding fitting function is 0.0032+0.0901{\rm e}^{-7.1428A^{-1/3}} .
Figure 3.
(color online) Fraction of non-nucleonic color-singlet cluster in nuclei; the corresponding fitting function is 0.0032+ 0.0901(\rm e)^{-7.1428A^{-1/3}} .
Tao-Feng Wang, Zi-Ming Li and Xiao-Ting Yang. Possible Light Neutral Boson and Particle Mass Quantization[J]. Chinese Physics C.
doi: 10.1088/1674-1137/abc536
Table 1.
Optimal parameters extracted from the analysis of the 7Li + 144Sm system within both WS and SPP. The JV, JW and reaction cross sections (σR) values are given.
Figure 1. (color online) Fitting for EMC data [13, 15] in the range of 0.35 < x < 0.7, with x-rescaling of the nuclear structure function via the parton model [14, 16, 18].
Figure 2. (color online) Ratio of m^{\ast}/m depending on A^{-1/3} : triangles denote the ratio of average per nucleon mass calculated from mean field theory [19] to the free nucleon mass, and solid circles indicate m^{\ast}/m extracted from 2\sigma_{A}/A\sigma_{2} experimental data in the range of 0.35\leq x \leq0.7 to the free nucleon mass [13, 15]. Squares indicate the previous m^{\ast}/m value [14]. The solid curve is the fitting by the function m^{\ast}/m = 0.994- 0.069{\rm e}^{-4.488A^{-1/3}} .
Figure 3. (color online) Fraction of non-nucleonic color-singlet cluster in nuclei; the corresponding fitting function is 0.0032+ 0.0901(\rm e)^{-7.1428A^{-1/3}} .
Figure 4. (color online) Mass ( \delta_{m} ) of the color-singlet cluster depending on the short-range-correlated proton number N_{p} .
Figure 5. (color online) Particle masses M (MeV/c ^{2} ) depending on quantum number n: star indicates the present deduced mass of a color-singlet cluster released per proton in the momentum region greater than 2 fm ^{-1} ; solid circle denotes a particle mass less than 1 GeV/c ^{2} depending on the quantum number n; the line is the fitting with a linear function for M and n.