ReviewHard probes of short-range nucleon–nucleon correlations
Introduction
Despite a fairly detailed understanding of the rich structure of the nucleon–nucleon strong interaction, its implications for the dynamics of atomic nuclei are not yet fully understood. Bulk properties of medium and heavy nuclei follow from the rather general characteristics of nuclear forces such as nucleons being fermions and nuclear forces being short-range and having a repulsive core. Because of these characteristics, nuclei demonstrate degeneracy of the Fermi system with clear identification of the Fermi momentum, , for quantities such as momentum or kinetic energy distributions of bound nucleons [1]. Refinements of such approximations within the framework of Bethe–Goldstone [2] or Nuclear Shell [3] models require, but are not very sensitive to, the short-range properties of the nucleon–nucleon (NN) interaction. Therefore, these models provide limited ability to investigate the structure of nuclear matter beyond saturation density, where one approaches the expected transition from nucleonic to quark–gluon degrees of freedom. As such, the experimental evidence for short-range correlations within the context of these models is rather indirect.
Simple shell model calculations make predictions for the momentum distribution and occupation number (or spectroscopic factor) for each nuclear shell. Early measurements found that while such mean-field calculations provide extremely successful descriptions of the energy and momentum distribution of nucleons in these shells, they do not correctly predict the occupancy of the shells. In the early 70s, with the advent of high-quality, medium-energy experiments on various nuclei, several observations were made which suggested significant strength of the nuclear spectral function beyond the single shell excitations. For example, the apparent violation of the Koltun sum rule [4] in proton knock-out experiments [5] was clearly attributed to the high energy excitation part of the nuclear spectral function [6] while no correlation effects were found in the momentum distributions corresponding to the fixed nuclear shells [7]. More comprehensive measurements of the proton knock-out reactions [8], [9] show that the spectroscopic factor, basically the ratio of the observed strength within a given shell to the expected strength, is less than one. The observed strength is typically ∼30%–40% below the shell-model expectation for measurements in many nuclei and looking at several nuclear shells. Even the most advanced Hartree–Fock calculations, which include long-range correlations, significantly overestimate the strength observed in the nuclear shells. A very plausible explanation for this discrepancy is the presence of strong short-range NN interactions. The repulsive core and tensor components of the NN force yield hard interactions that can excite nucleons outside of the low-lying shells. Thus, the nucleons are removed from the relatively low excitations associated with the nuclear shells and moved to higher energies and momenta, where there is little or no strength in the mean-field calculations.
To understand why the independent particle shell model can yield detailed and precise predictions for the nucleon distributions yet fail to predict the absolute strengths, one needs only to consider modern nucleon momentum distributions. At lower nucleon momenta, the distributions shown in Fig. 1 clearly show the characteristics of a degenerate Fermi system with broad momentum distribution, falling off rapidly at momenta approaching . In stark contrast to this behavior, the high-momentum tail has a much less rapid falloff which is similar for all nuclei from deuterium to nuclear matter. This universality of the high momentum tails strongly argues against the role of the collective or mean-field effects. Indeed, the mean-field calculations dramatically underestimate the strength at , falling short by several orders of magnitude at high momenta, as shown in Fig. 1. This universality of the high momentum tails can be easily understood if they are generated via the short-range part of the two-nucleon potential, and are thus independent of the shell model component of the momentum distribution.
Note that we often describe these two-body short-range correlations as excitations in the nucleus where two nucleons undergo a hard interaction and end up in a configuration with large relative momenta but a small total momentum. However, these are excitations relative to a theoretical mean-field ground state of the nucleus are not related to any real excited states of the nucleus; they are a contribution to the true nuclear ground state. When trying to probe these configurations in high-energy reactions, the goal is to take a “snapshot” of the configuration of the nucleons, and so the SRCs are often described as static configurations, treating the nucleons as though they were simply an isolated pair of high-momentum nucleons, analogous to the high-momentum part of the deuteron momentum distribution. However, such configurations are components of the nuclear ground wave function, with these virtual excitations responsible for generating most of the high-momentum nucleons.
The above scenario naturally suggests that if the two-body NN interaction is responsible for the high-momentum tail of the nuclear momentum distribution, then one expects the shape of the distribution beyond the Fermi momentum to be essentially identical for all nuclei, neglecting the small center-of-mass momentum of the SRC. In this case, represents the momentum distribution generated by the two-body interaction and yields the relative strength in the high-momentum tails which is related to the probability of finding these high-momentum two-nucleon configurations, or two-nucleon short-range correlations (2N SRCs) in the nucleus relative to the two-nucleon system.
In this scenario, the NN SRC represents a pair of nucleons where each nucleon has a large momentum (exceeding ) but the total momentum of the pair is very small, i.e. a pair of nucleons with large, back-to-back momenta. In the case of an iso-singlet SRC, one expects the high-momentum part of the distribution to look much like the high-momentum tails in the deuteron, which is an iso-singlet nucleon pair with zero total momentum.
The short-range NN attraction is dominated by the tensor interaction, which yields high momentum iso-singlet pairs but does not contribute to the iso-triplet channel . Therefore, one expects the two-body distribution to be identical to the deuteron distribution, , and the ratio of scattering cross sections between a heavy nucleus and the deuteron to yield . The value of can then be interpreted as the relative probability of finding NN SRCs in the nucleus compared to the deuteron.
This simple picture should break down at momenta where the central repulsive core of the NN potential dominates, as the SRCs will no longer be dominated by deuteron-like configurations. Additionally, as one goes to extremely high momenta, inclusion of three-nucleon configurations may become important. At even higher momenta, where the nucleon kinetic energy is comparable with the excitation energies of nucleon, non-nucleonic degrees of freedom may also need to be taken into account.
Extensive theoretical investigations of the impact of nucleon correlations in the structure of nuclei have been performed using a variety of methods. See, for example the review of Ref. [28] as well as several more recent works [29], [30], [31], [32], [33], [34], [35]. Detailed calculations employing the Green’s function Monte Carlo method using various 2N and 3N interactions have been used to study the structure and impact of correlations in light nuclei [36], [37], [38].
To probe the high momentum tail experimentally one needs to deal with several issues which follow mainly from the fact that the momentum distribution of the nucleon is not an experimental observable. As a result, one needs to identify the appropriate observables which are most relevant for characterization of bound nucleons in the high momentum tail.
The second problem one faces is finding a probe that is able to distinguish genuine -Nucleon () SRCs from -body processes involving long range interactions. The most efficient solution of this problem is to study the SRC using a probe with large momentum transfer, , and energy transfer, . Most of these measurements utilize single nucleon knock-out reactions, so one must choose kinematics which minimize contributions from more inelastic scattering processes. The optimal approach depends on the reaction mechanism, and the kinematic requirements are detailed in Section 4.
Prior to the high-energy experiments reviewed herein, the effects of short-range and tensor correlations were seen in valence knock-out experiments where, as mentioned above, the strength of the cross sections is over-predicted by independent particle models by 30%–40% [8]. Also, experiments which probed the continuum of the reaction found peaks in the missing momentum spectra [24], [26]. The most straightforward interpretation of those peaks is that a large fraction of the continuum strength arises from the breaking of an initial-state correlated pair with large relative and small center-of-mass momenta [8]. Such an interpretation is consistent with calculations that include nucleon–nucleon short-range correlations such as Muther and Dickhoff [39].
These initial indications of the importance of short-range correlations, missing strength relative to mean-field expectations and peaks in the missing momentum distribution, did not cleanly isolate SRCs or provide direct information that could elucidate the structure of SRCs. High energy scattering experiments, aimed at isolating scattering from high-momentum nucleons, provide a more direct way to probe SRCs in nuclei. Over the last decade, several such high energy nucleon knock-out experiments were performed which provided significant advances in our understanding of short-range nucleon correlations in nuclei. In this work we review these experiments, emphasizing their impact on our understanding the dynamics of short-range correlations.
Section snippets
The origin and the features of short-range correlations in nuclei
Already in the 1950s, it was observed the nucleons in nuclei exhibit collective behavior in response to the absorption of specific probes such as real photons or pions. These effects were understood on the basis of the simple observation that a free nucleon cannot absorb a real photon below the pion production threshold and must couple to other nuclear constituents for photon absorption to take place. Based on the experimental fact that the real photon has a large absorption cross section on
Main approaches of probing short-range correlations in nuclei
This section details the desired requirements for a complete and detailed investigation of SRCs in both light and heavy nuclei. Some of these are universal, applying to all hard probes, while others are reaction dependent.
- I.
Instantaneous interaction involving a nucleon from the SRC: One should have a clear way of identifying events in which the high energy probe interacts with the correlated nucleon and instantaneously removes it from the SRC. The requirement that the process is instantaneous is
General kinematical considerations for SRC studies
The above conditions will drive the restrictions on momentum and energy transfers in reactions aimed at probing SRCs. While the exact kinematic requirements depend on the reaction, one can make some general estimates for the different classes of reactions. Condition (I), the desire to have a nucleon from the SRC removed instantaneously, can be achieved if the energy and momentum transfer scales are much larger than the excitation energy scale characteristic to the nuclei in general and the SRC
Studies of the deuteron
The deuteron plays a special role in SRC studies. The measurement of the momentum distribution is necessary for identification of the iso-singlet component of 2N SRCs in nuclei, which is expected to be a universal feature in the high-momentum tail of all nuclei (Eq. (1)). As the simplest nucleus, the deuteron also provides an ideal testing ground for many issues related to details of the reaction mechanism: meson exchange contributions (MECs), isobar contributions (ICs), final state
Inclusive reactions beyond the deuteron
The first high- experiments that probed the high-momentum component of the ground state wave functions of more complex nuclei were inclusive measurements performed at SLAC in the 80s [119], [120] and 90s [121], [122], [123], [77], [124], followed by series of measurements at Jefferson Lab [91], [63], [125], [126], [93], [94]. A compilation of the data from the many of these experiments as well as a detailed discussion of quasi-elastic scattering (with a focus on probing the mean-field
Semi-inclusive knock-out reactions beyond the deuteron
Semi-inclusive reactions on nuclei, in which the struck nucleon is detected in coincidence with the scattered electron, provides the next highest level of complexity along with new details of the SRC structure. The conditions required for isolation of the structure of SRCs are discussed in Section 4.4. While inclusive reactions within the PWIA framework can provide some information about the momentum distribution of the nucleon, semi-inclusive reactions can, in principle, fully probe
Triple-coincidence reactions
A newer approach to resolving the detailed structure of short-range correlations involves making nucleon-knockout measurements in which two outgoing nucleons are detected, either in photo-nuclear reactions [150] or in coincidence with the scattered electron in electroproduction reactions [151], [152], [153], [154]. For most of these initial measurements, two protons were detected so as to minimize meson exchange effects, since only neutral mesons can be exchanged between two protons. However,
Summary and outlook
We have reviewed the physics importance of studying short-range, high-momentum nucleon configurations in nuclei. We then presented in some detail the necessary experimental requirements for measurements that can provide reliable information on these SRCs, and discussed the main results of a new generation of high energy and high momentum transfer measurements aimed at verifying the presence and probing the nature of these correlations.
We emphasized the special role of high studies of the
Acknowledgments
This work was supported by the US Department of Energy, Office of Nuclear Physics, under contracts DE-AC02-06CH11357, DE-AC05-06OR23177, and DE-FG02-01ER-41172. We thank our colleagues who assisted in the preparation of this work, including Werner Boeglin, Donal Day, Nadia Fomin, Roy Holt, Ushma Kriplani, Eli Piasetzky, Mark Strikman, and Larry Weinstein for useful comments and discussions, including preliminary results from recent measurements, and providing figures.
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