The center of mass motion of short-range correlated nucleon pairs studied via the A(e,e'pp) reaction
Short-Range Correlated (SRC) nucleon pairs are a vital part of the nucleus, accounting for almost all nucleons with momentum greater than the Fermi momentum (). A fundamental characteristic of SRC pairs is having large relative momenta as compared to , and smaller center-of-mass (c.m.) which indicates a small separation distance between the nucleons in the pair. Determining the c.m. momentum distribution of SRC pairs is essential for understanding their formation process. We report here on the extraction of the c.m. motion of proton-proton () SRC pairs in Carbon and, for the first time in heavier and ansymetric nuclei: aluminum, iron, and lead, from measurements of the () reaction. We find that the pair c.m. motion for these nuclei can be described by a three-dimensional Gaussian with a narrow width ranging from to MeV/, approximately consistent with the sum of two mean-field nucleon momenta. Comparison with calculations appears to show that the SRC pairs are formed from mean-field nucleons in specific quantum states.
On sabbatical leave from ]Nuclear Research Centre Negev, Beer-Sheva, Israel Current address:]Idaho State University, Pocatello, Idaho 83209 On sabbatical leave from ]Nuclear Research Centre Negev, Beer-Sheva, Israel
The CLAS Collaboration
The atomic nucleus is a complex, strongly interacting, many body system. Effective theories can successfully describe the long-range part of the nuclear many-body wave function. However, the exact description of its short-range part is challenging. This difficulty is due to the complexity of the nucleon-nucleon () interaction and the large nuclear density, which make it difficult to simplify the problem using scale-separated approaches when describing the short-range part of the nuclear wave-function.
Recent experimental studies have shown that approximately 20% of the nucleons in the nucleus belong to strongly interacting, momentary, short-range correlated (SRC) nucleon pairs Frankfurt et al. (1993); Egiyan et al. (2003, 2006); Fomin et al. (2012). These pairs are predominantly proton-neutron pairs with a center-of-mass (c.m.) momentum that is comparable to any two nucleons in the nuclear ground state and a much higher relative momentum between the nucleons in the pair (, the nuclear Fermi momentum) Tang et al. (2003); Piasetzky et al. (2006); Shneor et al. (2007); Subedi et al. (2008); Korover et al. (2014); Hen et al. (2014). They account for almost all of the nucleons in the nucleus with momentum greater than and for to of the kinetic energy carried by nucleons in the nucleus Ryckebusch et al. (2015); Wiringa et al. (2014); Lonardoni et al. (2017); Ciofi degli Atti and Simula (1996); Hen et al. (2014). See Refs. Hen et al. (2017); Ciofi degli Atti (2015); Frankfurt et al. (2008) for recent reviews. SRC pairs are thus a vital part of nuclei with implications for many important topics including the possible modification of bound nucleon structure and the extraction of the free neutron structure function Hen et al. (2017); Weinstein et al. (2011); Hen et al. (2012, 2013a, 2011); Chen et al. (2017), neutrino-nucleus interactions and neutrino oscillation experiments Gallagher et al. (2011); Fields et al. (2013); Fiorentini et al. (2013); Acciarri et al. (2014); Weinstein et al. (2016); Van Cuyck et al. (2016), neutrino-less double beta decay searches Kortelainen et al. (2007); Song et al. (2017), as well as neutron star structure and the nuclear symmetry energy Hen et al. (2015); Cai and Li (2016); Hen et al. (2016).
The smaller c.m. momentum as compared to the large relative momentum of SRC pairs is a fundamental characteristic of such pairs, and is an essential indications that the nucleons in the pair are in close proximity with limited interaction with the surrounding nuclear environment Weiss et al. (2018).
Modern calculations Colle et al. (2014) indicate that SRC pairs are temporary fluctuations due to the short-range part of the interaction acting on two nucleons occupying shell-model (“mean-field”) states. The exact parentage and formation process of SRC pairs is not well understood. While state-of-the-art many-body calculations of one- and two-body momentum densities in nuclei Wiringa et al. (2014); Carlson et al. (2015); Neff et al. (2015) seem to produce SRC features that are generally consistent with measurements, they do not offer direct insight into the effective mechanisms of SRC pair formation.
Effective calculations using scale-separated approaches agree with many-body calculations Weiss et al. (2018); Ryckebusch et al. (2015); Alvioli et al. (2016); Ciofi degli Atti et al. (2017), suggesting that, at high-momenta, the momentum distribution of SRC pairs can be factorized into the c.m. and relative momentum distributions ,
where is greater than and Weiss et al. (2018); Ciofi degli Atti and Morita (2017); Ciofi degli Atti et al. (2017). This implies that the relative momentum distribution of SRC pairs, , is a universal function of the short-range part of the () interaction, such that the many-body nuclear dynamics affect only the c.m. momentum distribution, . Therefore, extracting the c.m. momentum distribution of SRC pairs can provide valuable insight into their formation process.
The c.m. momentum distributions of SRC pairs in He and C have been extracted previously from and measurements Tang et al. (2003); Shneor et al. (2007); Korover et al. (2014). Here we present the first study of the c.m. momentum distribution of SRC pairs in nuclei heavier than C using the () reaction. The cross-section for this () two-nucleon knockout reaction in some kinematics approximately factorizes as a kinematic term times the elementary electron-proton cross section times the nuclear decay function, which defines the combined probability of finding the knocked-out nucleon pair with given energies and momenta Frankfurt and Strikman (1981, 1988); Piasetzky et al. (2006); Ryckebusch (1996); Colle et al. (2014). The decay function also factorizes into relative and c.m. parts, just like Eq. 1 Piasetzky et al. (2006). Therefore, the () cross section is approximately proportional to the c.m. momentum distributions of SRC pairs Ciofi degli Atti et al. (1991); Piasetzky et al. (2006); Sargsian et al. (2005); Ryckebusch (1996); Colle et al. (2014):
To increase sensitivity to the initial state properties of -SRC pairs, the measurement was done using high energy electrons scattering at large momentum transfer (hard scattering), in kinematics dominated by the hard breakup of SRC pairs, as discussed in detail in Hen et al. (2017). In this kinematics, Eq. 2 is a good approximation since rescattering of the two outgoing nucleons does not distort the width of the momentum distribution (see discussion below).
The data presented here were collected as part of the EG2 run period that took place in 2004 in Hall B of the Thomas Jefferson National Accelerator Facility (Jefferson Lab). The experiment used a 5.01 GeV electron beam, impinging on H and natural C, Al, Fe, and Pb targets at the CEBAF Large Acceptance Spectrometer (CLAS) Mecking et al. (2003). The analysis was carried out as part of the Jefferson Lab Hall B Data-Mining project.
CLAS used a toroidal magnetic field and six independent sets of drift chambers for charged particle tracking, time-of-flight scintillation counters for hadron identification, and Čerenkov counters and electro-magnetic calorimeters for electron/pion separation. The polar angular acceptance was and the azimuthal angular acceptance ranged from 50% at small polar angles to 80% at larger polar angles. See Refs. Hen et al. (2014, 2013b) for details on the electron and proton identification and momentum reconstruction procedures.
The EG2 run period used a specially designed target setup, consisting of an approximately 2-cm LD cryotarget followed by one of six independently-insertable solid targets (thin Al, thick Al, Sn, C, Fe, and Pb, all with natural isotopic abundance, ranging between 0.16 and 0.38 g/cm), see Ref. Hakobyan et al. (2008) for details. The LD target cell and the inserted solid target were separated by about 4 cm. The few-mm vertex reconstruction resolution of CLAS for both electrons and protons was sufficient to unambiguously separate particles originating in the cryotarget and the solid target.
The kinematics of the () reaction is shown schematically in Fig. 1. Identification of exclusive () events, dominated by scattering off -SRC pairs, was done in two stages: (1) selection of () events in which the electron predominantly interacts with a single proton belonging to an SRC pair in the nucleus Subedi et al. (2008); Hen et al. (2013b, 2014), and (2) selection of () events by requiring the detection of a second, recoil, proton in coincidence with the () reaction.
We selected () events in which the knocked-out proton predominantly belonged to an SRC pair by requiring a large Bjorken scaling parameter (where , and are the three-momentum and energy, respectively, transferred to the nucleus, and is the proton mass). This requirement also suppressed the effect of inelastic reaction mechanisms (e.g., pion and resonance production) and resulted in GeV Frankfurt et al. (1997); Shneor et al. (2007). We also required large missing momentum MeV/, where with the measured proton momentum. We further suppressed contributions from inelastic excitations of the struck nucleon by limiting the reconstructed missing mass of the two-nucleon system GeV/ (where is the total energy of the leading proton). We identified events where the leading proton absorbed the transferred momentum by requiring that its momentum was within 25 of and that Hen et al. (2013b, 2014). As shown by previous experimental and theoretical studies, these conditions enhance the contribution of scattering off nucleons in SRC pairs and suppress contribution from competing effects Groep et al. (2000); Blomqvist et al. (1998); Kester et al. (1995); Arnold et al. (1990); Laget (1987); Frankfurt et al. (1997); Colle et al. (2016); Sargsian (2001).
() events were selected by requiring that the () event had a second, recoil proton with momentum MeV/. There were no events in which the recoil proton passed the leading proton selection cuts described above. The recoil proton was emitted opposite to Hen et al. (2014), consistent with the measured pairs having large relative momentum and smaller c.m. momentum.
In the Plane Wave Impulse Approximation (PWIA), where the nucleons do not rescatter as they leave the nucleus, and are equal to the initial momenta of the two protons in the nucleus before the interaction. In that case we can write
We use a coordinate system where is parallel to , and and are transverse to it and defined by: and .
Figure 2 shows the number of () events plotted versus the - and - components of (see Eq. 3). The data shown are not corrected for the CLAS acceptance and resolution effects. As the () cross section is proportional to , we can extract the width of from the widths of the measured distributions. Both and are observed to be normally distributed around zero for all nuclei. Thus, as expected, can be approximated by a three-dimensional Gaussian Tang et al. (2003); Shneor et al. (2007); Korover et al. (2014); Ciofi degli Atti and Simula (1996); Colle et al. (2014), and we characterize its width using and , the standard deviation of the Gaussian fits in the two directions transverse to . We average and for each nucleus to get , the Gaussian width of one dimension of . These widths are independent of the magnitude of , supporting the factorization of Eq. 3.
There are three main effects that complicate the interpretation of the raw (directly extracted) c.m. momentum distribution parameters (i.e., ): (1) kinematical offsets of the c.m. momentum in the direction, (2) reaction mechanism effects, and (3) detector acceptance and resolution effects. We next explain how each effect is accounted for in the data analysis.
(1) Kinematical offsets in the c.m. momentum direction: Since the relative momentum distribution of pairs falls rapidly for increasing , it is more likely for an event with a large nucleon momentum () to be the result of a pair with smaller and a oriented in the direction of the nucleon momentum. This kinematical effect will manifest as a shift in the mean of the c.m. momentum distribution in the (nucleon initial momentum) direction. To isolate this effect, we worked in a reference frame in which and and are perpendicular to . The extracted c.m. momentum distributions in the and directions were observed to be independent of , as expected.
(2) Reaction mechanism effects: These include mainly contributions from meson-exchange currents (MECs), isobar configurations (ICs), and rescattering of the outgoing nucleons (final-state interactions or FSI) that can mimic the signature of SRC pair breakup and/or distort the measured distributions Groep et al. (2000); Blomqvist et al. (1998); Kester et al. (1995).
This measurement was performed at an average of about 2.1 GeV and to minimize the contribution of MEC and IC relative to SRC breakup Arnold et al. (1990); Laget (1987); Frankfurt et al. (1997); Colle et al. (2016). Nucleons leaving the nucleus can be effectively “absorbed”, where they scatter inelastically or out of the phase-space of accepted events. The probability of absorption ranges from about 0.5 for C to 0.8 for Pb Hen et al. (2013b); Garino et al. (1992); O’Neill et al. (1995); Abbott et al. (1998); Garrow et al. (2002). Nucleons that rescatter by smaller amounts (i.e. do not scatter out of the phase-space of accepted events) are still detected, but have their momenta changed. This rescattering includes both rescattering of the struck nucleon from its correlated partner and from the other nucleons. Elastic rescattering of the struck nucleon from its correlated partner will change each of their momenta by equal and opposite amounts, but will not change (see Eq. 3) Frankfurt et al. (1997); Colle et al. (2016). To minimize the effects of rescattering from the other nucleons, not leading to absorption, we selected largely anti-parallel kinematics, where has a large component antiparallel to Frankfurt et al. (1997). Relativistic Glauber calculations show that, under these conditions, FSI are largely confined to within the nucleons of the pair Frankfurt et al. (1997); Sargsian (2001); Colle et al. (2016); Frankfurt et al. (2008); Arrington et al. (2012).
The probability of the struck nucleon rescattering from the nucleons is expected to increase with . Such rescattering, when not leading to reduction of the measured flux (i.e., absorption), should broaden the extracted c.m. momentum distribution. The measured widths do not increase strongly with . This provides evidence that, in the kinematics of this measurement, FSI with the other nucleons do not distort the shape of the measured c.m. momentum distribution, in agreement with theoretical calculations Frankfurt et al. (1997); Sargsian (2001); Colle et al. (2016).
In addition, Single Charge Exchange processes can lead to the detection of an () event that originate from the hard breakup of an -SRC pair. While such SCX processes have relatively low cross-sections, the predominance of SRC pairs by pairs enhances its impact in measurements of the () reaction. Using the formalism of Ref. Colle et al. (2016), assuming the abundance of -SRC pairs is times higher than that of -SRC pairs, we estimate that such SCX processes account for approximately 40% of the measured () events. This is a large fraction that could impact the interpertation of the data. However, as - and -SRC pairs are expected to have very similar c.m. momentum densities Colle et al. (2014, 2016), this effect should not have a significant impact on the width of the c.m. momentum density.
(3) Detector acceptance and resolution effects: While CLAS has a large acceptance, it is not complete, and the measured c.m. momentum distributions need to be corrected for any detector related distortions. Following previous analyses Shneor et al. (2007); Subedi et al. (2008); Korover et al. (2014), we corrected for the CLAS acceptance in a 6-stage process: (1) We modeled the c.m. momentum distribution as a three-dimensional Gaussian, parametrized by a width and a mean in each direction. In the directions transverse to the widths were assumed to be constant and equal to each other () and the means were fixed at zero. In the direction parallel to , both the mean and the width were varied over a wide range. (2) For a given set of parameters characterizing the c.m. momentum distribution in step (1), we generated a synthetic sample of () events by performing multiple selections of a random event from the measured events and a random from the 3D Gaussian. The combination of the two produced a sample of recoil protons with momentum (). (3) We determined the probability of detecting each recoil proton using GSIM, the GEANT3-based CLAS simulation Holtrop (). (4) We analyzed the Monte Carlo events in the same way as the data to extract the c.m. momentum distributions and fit those distributions in the directions transverse to with a Gaussian to determine their reconstructed width. (5) We repeated steps (1) to (4) using different input parameters for the 3D Gaussian model used in step (1) and obtained a ‘reconstructed’ for each set of input parameters. was varied between 0 and 300 MeV/. The mean and width in the direction were sampled for each nucleus from a Gaussian distribution centered around the experimentally measured values with a nucleus dependent width (1) ranging from 45 to 125 MeV/ for the mean and 30 to 90 MeV/ for the width. The exact value of the width of the distribution is a function of the measurement uncertainty for each nucleus. It extends far beyond the expected effect of the CLAS acceptance. (6) We examined the distribution of the generated vs. reconstructed widths in the directions transverse to to determine the impact of the CLAS acceptance on the measured values.
The net effect of the acceptance corrections was to reduce the widths of the c.m. momentum distributions by 15–20 MeV/ for each nucleus and to increase the uncertainties.
As a sensitivity study for the acceptance correction procedure, we examined two additional variations to the event generator in the direction: (A) a constant width of 70 MeV/ and (B) a width and mean that varied as a linear function of . The variation among the results obtained using each method was significantly smaller than the measurement uncertainties and gives a systematic uncertainty of 7%. We also performed a ‘closure’ test where we input pseudo-data with known width and statistics that matched the measurements, passed it through the CLAS acceptance to see the variation in the ‘measured’ width and then applied the acceptance correction to successfully retrieve the generated value.
The CLAS reconstruction resolution, , for the c.m. momentum of pairs was measured using the exclusive reaction and was found to equal 20 MeV/. We subtracted this in quadrature from the measured c.m. width: , which amounts to a small, 2–3 MeV/, correction.
Figure 3 shows the extracted , in the directions transverse to , including acceptance corrections and subtraction of the CLAS resolution. The uncertainty includes both statistical uncertainties as well as systematical uncertainties due to the acceptance correction procedure.
The extracted value of for C is consistent with previous C measurements of Shneor et al. (2007) and C measurements of Tang et al. (2003), with significantly reduced uncertainty. The extracted width grows very little from C to Pb, and is consistent with a constant value within uncertainties (i.e., it saturates). The saturation of with supports the claim that, in the chosen kinematics, FSI with the nucleons primarily reduces the measured flux, while not significantly distorting the shape of the extracted c.m. momenutm distribution.
Figure 3 also compares the data to several theoretical predictions for the c.m. momentum of the nucleons which couple to create the SRC pairs. Ref. Ciofi degli Atti and Simula (1996) considers all possible pairs from shell-model orbits, while Ref. Colle et al. (2014) considers both all pairs, and nucleons in a relative state (i.e., nodeless -wave with spin 0) Vanhalst et al. (2011, 2012). The simplistic Fermi-Gas prediction samples two random nucleons from a Fermi sea with from Moniz et al. (1971).
The agreement of the data with calculations supports the theoretical picture of SRC pair formation from temporal fluctuations of mean-field nucleons Hen et al. (2017). The experimentally extracted widths are consistent with the Fermi-Gas prediction and are higher than the full mean-field calculations that consider formation from all possible pairs. The data are lower than the calculation that assumes restrictive conditions on the mean-field nucleons that form SRC pairs Colle et al. (2014).
We note that the SRC-pair c.m. momentum distributions extracted from experiment differ from those extracted directly from ab-initio calculations of the two-nucleon momentum distribution. The latter are formed by summing over all two-nucleon combinations in the nucleus and therefore include contributions from non-SRC pairs. See discussion in Ref. Weiss et al. (2018).
In conclusion, we report the extraction of the width of the c.m. momentum distribution, , for -SRC pairs from () measurements in C, Al, Fe, and Pb. The new data is consistent with previous measurements of the width of the c.m. momentum distribution for both and pairs in C. increases very slowly and might even saturate from C to Pb, supporting the claim that final state interactions are negligible between the two outgoing nucleons and the residual nucleus. The comparison with theoretical models supports the claim that SRC pairs are formed from mean-field pairs in specific quantum states. However, improved measurements and calculations are required to determine the exact states.
Acknowledgements.We acknowledge the efforts of the staff of the Accelerator and Physics Divisions at Jefferson Lab that made this experiment possible. We are also grateful for many fruitful discussions with L.L. Frankfurt, M. Strikman, J. Ryckebusch, W. Cosyn, M. Sargsyan, and C. Ciofi degli Atti. The analysis presented here was carried out as part of the Jefferson Lab Hall B Data-Mining project supported by the U.S. Department of Energy (DOE). The research was supported also by the National Science Foundation, the Israel Science Foundation, the Chilean ComisiÃ³n Nacional de InvestigaciÃ³n CientÃfica y TecnolÃ³gica, the French Centre National de la Recherche Scientifique and Commissariat a lâEnergie Atomique the French-American Cultural Exchange, the Italian Istituto Nazionale di Fisica Nucleare, the National Research Foundation of Korea, and the UKâs Science and Technology Facilities Council. Jefferson Science Associates operates the Thomas Jefferson National Accelerator Facility for the DOE, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177. The raw data from this experiment are archived in Jefferson Labâs mass storage silo. E. O. Cohen would like to acknowledge the Azrieli Foundation.
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