Nucleon-nucleon correlations, short-lived excitations, and the quarks within

This article reviews our current understanding of how the internal quark structure of a nucleon bound in nuclei differs from that of a free nucleon. The interpretation of measurements of the European Muon Collaboration (EMC) effect for valence quarks, a reduction in the deep inelastic scattering cross-section ratios for nuclei relative to deuterium, and its possible connection to nucleon-nucleon short-range correlations (SRCs) in nuclei are focused on. This review and new analysis (involving the amplitudes of non-nucleonic configurations in the nucleus) of the available experimental and theoretical evidence shows that there is a phenomenological relation between the EMC effect and the effects of SRCs that is not an accident. The influence of strongly correlated neutron-proton pairs involving highly virtual nucleons is responsible for both effects. These correlated pairs are temporary high-density fluctuations in the nucleus in which the internal structure of the nucleons is briefly modified. This conclusion needs to be solidified by the future experiments and improved theoretical analyses that are discussed herein.

Figure 1

The $A(e,e^{\prime }p)$ reaction in the plane wave impulse approximation. A nucleus of four-momentum $P_A$ emits a nucleon of four-momentum $P_{\text{miss}}$ that absorbs a virtual photon of four-momentum $q$ to make a nucleon of four-momentum $P_{\text{miss}}+q$, with $(P_{\text{miss}}+q{)}^2=M^2$, where $M$ is the nucleon mass. The blob represents the in-medium electromagnetic form factors.

Figure 2

(Upper) The $\mathrm{O}(e,e^{\prime }p)$ cross section plotted vs missing energy at $Q^2=0.8{\mathrm{GeV}}^2$ and $\nu =0.439\mathrm{GeV}$ for different angles ${\theta }_{pq}$ between the proton spectrometer and $\mathbf{q}$. The curve labeled DWIA is a distorted wave impulse approximation calculation of $s$-shell knockout; the other curves are calculations of two-nucleon knockout including meson exchange currents (MEC), delta production (IC), and central and/or tensor correlations. From ([182]. (Lower) The cross section plotted vs missing momentum for the $1p_{1/2}$ and $1p_{3/2}$ states. The curves show DWIA calculations. See [119], [182], and [103] for details. From [119].

Figure 3

The fractional spectroscopic factors (the ratio of measured cross sections to those calculated with the independent particle shell model) for valence nucleon knockout $(e,e^{\prime }p)$. Adapted from [177].

Figure 4

The nucleon momentum distributions $n_0(k)$ (dashed lines) and $n(k)$ (solid lines) plotted vs momentum in ${\mathrm{fm}}^{-1}$ for the deuteron, ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, and ${}^{56}\mathrm{Fe}$. Adapted from [62].

Figure 5

Scaled two-body distribution function ${\rho }_{2,1}^A(r)/A$ [see Eq. (a11)] for nuclei with $A=2$, 3, and 4. A correlation hole is seen for all of these nuclei. The two sets of curves are obtained with the (left) $\mathrm{AV}18+\mathrm{UIX}$ and (right) $N^2\mathrm{LO}$ potentials. The meaning of $R_0$ is discussed in the text. Adapted from [56].

Figure 6

Distributions of the relative angle ($\gamma$) between the reconstructed initial momentum of the knockout proton and the recoil neutron. Results for ${}^{12}\mathrm{C}(p,2pn)$ events from BNL, shown as a function of the momentum of the recoil neutron (a) and for events with recoiling neutron momentum greater than (b) and less than (c) $k_F=225\mathrm{MeV}/c$. Note the transition from an isotropic distribution to a correlated one at about $k_F=225\mathrm{MeV}/c$. Adapted from [267] and [231].

Figure 7

Distributions of the relative angle ($\gamma$) between the reconstructed initial momentum of the knockout proton and the recoil nucleon. Results for ${}^{12}\mathrm{C}(e,e^{\prime }pp)$ events from JLab at kinematics corresponding to scattering off $\sim 500\mathrm{MeV}/c$ initial momentum protons. From [253].

Figure 8

The ratio of ${}^{12}\mathrm{C}(e,e^{\prime }pN)$ double knockout events to ${}^{12}\mathrm{C}(e,e^{\prime }p)$ single knockout events, shown as a function of the reconstructed initial (missing) momentum of the knocked-out proton from the ${}^{12}\mathrm{C}(e,e^{\prime }p)$ reaction. Triangles and circles mark ${}^{12}\mathrm{C}(e,e^{\prime }pN)$ and ${}^{12}\mathrm{C}(e,e^{\prime }pp)$ events, respectively. The square shows the ${}^{12}\mathrm{C}(e,e^{\prime }pp)/{}^{12}\mathrm{C}(e,e^{\prime }pn)$ ratio. A clear dominance of ${}^{12}\mathrm{C}(e,e^{\prime }pn)$ events is observed, evidence of the tensor nature of the nucleon-nucleon interaction in the measured momentum range. The pie chart on the right illustrates our understanding of the structure of ${}^{12}\mathrm{C}$, composed of 80% mean-field nucleons and 20% SRC pairs, where the latter is composed of $\sim 90\%$ $np$-SRC pairs and 5% $pp$ and $nn$ SRC pairs each. Adapted from [266].

Figure 9

The distribution of the cosine of the opening angle $\gamma$ between ${\mathbf{p}}_{\text{miss}}$ and ${\mathbf{p}}_{\text{recoil}}$ for the ${}^4\mathrm{He}(e,e^{\prime }pn)$ reaction. The solid curve is a simulation of scattering off a moving pair with a c.m. momentum distribution having a width of $100\mathrm{MeV}/c$. The insets show the missing mass distributions. Adapted from [170].

Figure 10

The measured $pp$ to $pn$ ratio as function of the proton missing momentum ( labeled $q$) ([170]) compared to calculations of the two-nucleon momentum distribution ([283]) integrated over various ranges of the c.m. momentum from zero (bottom curve) to infinity (top curve) ([281]). The data are shown as a function of the nucleon momentum and the calculations are shown as a function of the pair relative momentum. The two are equivalent for low c.m. momentum of the pair but differ at large c.m. momentum. Adapted from [281].

Figure 11

The ratio of $pp$ to $pn$ pairs in ${}^3\mathrm{He}(e,e^{\prime }pp)n$. The solid and star points show the ratios for $300\le p_{\mathrm{rel}}\le 500$ and $400\le p_{\mathrm{rel}}\le 600\mathrm{MeV}/c$, respectively, as a function of the total (e.g., center-of-mass) momentum of the pair. The $pp/pn$ ratio is much less than 1 for small $p_{\mathrm{tot}}$, increasing to the pair-counting ratio of 0.5 at large $p_{\mathrm{tot}}$. The ratio at small $p_{\mathrm{tot}}$ is about 0.1 for $300\le p_{\mathrm{rel}}\le 500\mathrm{MeV}/c$, and about 0.25 for $400\le p_{\mathrm{rel}}\le 600\mathrm{MeV}/c$. The solid line shows a calculation by Golak for $300\le p_{\mathrm{rel}}\le 500\mathrm{MeV}/c$ which neglects rescattering of the struck nucleon but includes the reinteraction of the two nucleons in the SRC pair. The dashed line (blue) shows the ${}^3\mathrm{He}$ momentum distribution integrated over the experimental acceptances. From [26].

Figure 12

The relative fraction of $np$ and $pp$ SRC pairs (excluding $nn$ pairs) derived from $A(e,e^{\prime }p)$ and $A(e,e^{\prime }pp)$ measurements on a range of nuclei. From [145].

Figure 13

The minimum momentum of the struck nucleon in inclusive $(e,e^{\prime })$ scattering as a function of $x_B$. (a) The minimum momentum for deuterium for a variety of momentum transfers. (b) The minimum momentum for a variety of nuclei at $Q^2=2{\mathrm{GeV}}^2$. The residual $A-1$ system is assumed to be in its ground state. From [92].

Figure 14

Momentum distribution of the deuteron. Points show the results extracted from the experimental scaling function $F(y)$ at four different momentum transfers. Curves show the calculated momentum distributions using three different $NN$ potentials Paris ([176]), Nijmegen ([262]), and Argonne V14 ([284]). The figure uses $k$ for momentum instead of $p$. From [104].

Figure 15

Electron quasielastic scattering from a nucleon in deuterium (left) and from a nucleon in a SRC pair in a heavier nucleus (right). The labels ${\mathrm{\Gamma }}_{NN}$ and ${\mathrm{\Gamma }}_A$ refer to the deuteron and nuclear vertex functions, respectively.

Figure 16

Inclusive per nucleon cross-section ratios of (top) nuclei to ${}^3\mathrm{He}$ at $1.4<Q^2<2.6{\mathrm{GeV}}^2$ and (bottom) nuclei to deuterium at $Q^2=2.7{\mathrm{GeV}}^2$. Adapted from (top) [93], and (bottom) [104].

Figure 17

A qualitative sketch of the dominant features of the nucleon momentum distribution in nuclei. At $k<k_F$, the nucleon momentum is balanced by that of the other $A-1$ nucleons and can be described by mean-field models. At $k>k_F$, the nucleon belongs to a $pn$-SRC pair and its momentum is balanced by that of one other nucleon.

Figure 18

Deep inelastic scattering at large values of $Q^2$. A lepton ( labeled $e$) scatters from a nucleon by emitting a spacelike virtual photon with four-momentum $q$, which is absorbed on a single quark with momentum fraction $x_B{P}^{+}$. Only the outgoing lepton is subsequently detected.

Figure 19

The bands show $x_B$ times the unpolarized parton distributions for the different parton flavors $\{u_v,d_v,u,d,s=\overline{s},c=\overline{c},b=\overline{b}$, and $g\}$ obtained in NNLO NNPDF2.3 global analysis ([27]), at $Q^2=10$ and $10^4(\mathrm{GeV}/c{)}^2$, with ${\alpha }_s(M_Z^2)=0.118$. Here $x=x_B$. From the PDG, [224].

Figure 20

Image of the EMC data as it appeared in the November 1982 issue of the CERN Courier. This image nearly derailed the refereed publication, as the editor argued that the data had already been published. From [24].

Figure 21

The per nucleon cross-section ratio of various nuclei to deuterium as measured at (top) SLAC ([125]) and (bottom) Jefferson Lab ([252]). The solid curves show the $A$-dependent fit to the SLAC data ([125]), while the dashed curve is the SLAC fit to ${}^{12}\mathrm{C}$. From (top) [125], and (bottom) [252].

Figure 22

The $Q^2$ dependence of the EMC ratio for iron at various values of $x_B$. From [125].

Figure 23

The slope of the EMC effect for $0.35\le x_B\le 0.7$ plotted vs the average nuclear density for various light nuclei as measured at Jefferson Lab. From [252].

Figure 24

Deep inelastic scattering diagram. A virtual photon ${\gamma }^{*}$ of momentum $q$ is absorbed on a quark of momentum $k$ contained in a nucleon of momentum $p$ in a nucleus of momentum $P$. The imaginary part of this diagram corresponds to the hadronic tensor $W^{\mu \nu }$. Adapted from [213].

Figure 25

The measured EMC effect in gold ([125]) compared to a nucleons-only calculation of the EMC effect in lead. From [256].

Figure 26

The Drell-Yan process. A quark with momentum fraction $x_1$ from the incident proton annihilates with an antiquark from the nuclear target with momentum fraction $x_2$ to form a timelike virtual photon which decays to a ${\mu }^{+}{\mu }^{-}$ pair. Adapted from [40].

Figure 27

Drell-Yan experimental results. Ratio of Drell-Yan cross sections as a function of the momentum fraction $x_2$ of a quark in the nucleus. The version of the rescaling model does not reproduce the nuclear deep inelastic scattering data ([39], [40]). Adapted from [8].

Figure 28

The $A(e,e^{\prime }p)$ reaction in the plane wave impulse approximation. A nucleus of four-momentum $P$ emits a nucleon of four-momentum $p$ that absorbs a virtual photon of four-momentum $q$ to make a nucleon of four-momentum $p+q$, with $(p+q{)}^2=M^2$, where $M$ is the nucleon mass. The blob represents the in-medium electromagnetic form factors.

Figure 29

Evolution of nuclear physics from structureless nucleons in the 1940s to independent three-quark nucleons in the 1970s to the modified nucleons of today, either modified single nucleons (left) or modified two-nucleon configurations (right).

Figure 30

The measured EMC effect for ${}^{27}\mathrm{Al}$ ([125]) compared to QMC calculations of both the regular and the polarized EMC effects. From [69].

Figure 31

Two-component nucleon model: normal-sized component plus pointlike configuration component.

Figure 32

The ratio of $F_2$ in the nucleus to the free $F_2$ (the EMC ratio) in the pointlike configuration suppression model. From [105].

Figure 33

Medium modification of form factors. Figure adapted with slight modifications. Adapted from [105].

Figure 34

The slope of the EMC effect ($R_{\mathrm{EMC}}$, ratio of nuclear to deuteron cross section) for $0.35\le x_A\le 0.7$ plotted vs $a_2(A)$, the SRC scale factor (the relative probability that a nucleon belongs to an SRC $NN$ pair) for a variety of nuclei. The fit parameter $a=-0.070\pm 0.004$ is the intercept of the line constrained to pass through the deuteron (and is therefore also the negative of the slope of that line). From [146].

Figure 35

$\alpha$ (solid line), $|U|$ (dashed line) for $Q^2=2.7{\mathrm{GeV}}^2$ and $p_{\perp }=0$. The quantity $|U|$ is presented in units of the nucleon mass $M$ and is proportional to the virtuality $v(\mathbf{p},E)$ via Eq. (45)

Figure 36

The deuteron IMC ratio $R_{\mathrm{EMC}}^d=F_2^d/(F_2^n+F_2^p)$ as extracted from the BONuS data. Total systematic uncertainties are shown as a band arbitrarily positioned at 0.91 (blue). The yellow band shows the CJ12 ([225]) limits expected from their nuclear models. The black points are the combined 4- and 5-GeV data, whereas the red points are the 4-GeV data alone. The dashed blue line shows the calculations of [173]. The solid line (black) is the fit to the black points for $0.35<x_B<0.7$. From [127].

Figure 37

The ratio of neutron to proton structure functions $F_2^n(x_B,Q^2)/F_2^p(x_B,Q^2)$ as extracted from the measured deuteron and proton structure functions $F_2^d$ and $F_2^p$. The solid symbols show $F_2^n/F_2^p$ extracted by [280] from the deuteron IMC ratio and the world data for $F_2^d/F_2^p$ at $Q^2=12{\mathrm{GeV}}^2$ ([17]). The open symbols show $F_2^n/F_2^p$ extracted from the same data correcting only for nucleon motion in deuterium using a relativistic deuteron momentum density ([17]). From [280].

Figure 38

$F_2^n/F_2^p$ as a function of $x_B$. Results from the IMC and other phenomenological extractions are compared to selected theoretical predictions. See [150] and [240] for details. From [150].

Figure 39

$d/u$ ratio at $Q^2=12{\mathrm{GeV}}^2$ with the full theoretical uncertainty from [5] (black) and with the IMC constraint at the 90% C.L. (red) from [137]. Also shown for comparison are recent extractions that do not include nuclear correactions from the CJ15 ([4]) and CT14 ([90]) PDF extractions.

Figure 40

The measured ${}^4\mathrm{He}(\overrightarrow{e},e^{\prime }\overrightarrow{p}){}^3\mathrm{H}$ polarization-transfer double ratio $R$ (upper panel) and induced polarization $P_y$ (lower panel) as a function of $Q^2$. Open symbols: [88] and [264]; solid circles: [227] and [192]. The data are compared to DWIA calculations from [248] (updated in 2010) using unmodified form factors and from the Madrid group ([51]; [270]; [271]) using the cc1 (lower set of curves) and cc2 (upper set of curves) off-shell current operators in combination with unmodified (black dashed lines), QMC modified (red solid lines), and CQS modified (red dashed line) in medium form factors. See text for details. From [265].

Figure 41

The expected results from future Jefferson Lab tagged DIS measurements. The dashed lines are obtained from the color screening models ([111]), the dotted lines are from the color delocalization models ([74]), and the dot-dashed lines are from the off-shell models ([198]). From [144] and [143].

Figure 42

Phase space of nuclei considered for future EMC and SRC measurements at Jefferson Lab as a function of their proton ($Z$) and neutron ($N$) numbers. The two insets focus on the light and medium mass nuclei regimes. For light nuclei one can systematically study a series of symmetric nuclei and the detailed effect of the addition of one neutron (proton). For medium mass nuclei the horizontal and vertical bands mark nuclei with a similar number of protons (neutrons) and a varying number of neutrons (protons) allowing one to study the effect of nuclear asymmetry.

Figure 43

The expected results of the polarized EMC effect measurement at Jefferson Lab. The ratio of the parallel double-spin asymmetry $A_{\parallel }$ for $\overrightarrow{{}^7\mathrm{Li}}(\overrightarrow{e},e^{\prime })$ to $\overrightarrow{p}(\overrightarrow{e},e^{\prime })$, normalized by multiplying it with the “naive” unpolarized structure function ratio for ${}^7\mathrm{Li}$ over hydrogen, plotted vs $x_B$. The models are NNM (naive nuclear model with no Fermi motion, dashed line), SNM (standard nuclear model with Fermi motion and kinematical binding energy effects, upper solid line at low $x$), QMC (quark-meson coupling model, bottom solid line at low $x$), and MSS (modified sea scheme) [$x$ rescaling ([97]), mid solid line at low $x$]. Adapted from [172].