# Energy and momentum dependence of nuclear short-range correlations - Spectral function, exclusive scattering experiments and the contact formalism

## Abstract

Results of electron-induced one- and two-nucleon hard knockout reactions, and , in kinematics sensitive to nuclear short-range correlations, are studied using the nuclear contact formalism. A relation between the spectral function and the nuclear contacts is derived and used to analyze the dependence of the data on the initial energy and momentum of the knocked-out proton. The ratio between the number of emitted proton-proton pairs and proton-neutron pairs is shown to depend predominantly on a single ratio of contacts. This ratio is expected to present a deep minima in the initial energy and momentum plane, associated with the node in the proton-proton wave function.

The formalism is applied to analyze data from recent He and C electron-scattering experiments performed at Jefferson laboratory. Two different nucleon-nucleon potentials were used to asses the model-dependence of the results. For the ratio of proton-proton to proton-neutron pairs in He, a fair agreement with the experimental data is obtained using the two potentials, whereas for the ratio of proton-proton pairs to the total knocked-out protons in C, some of the features of the theory are not seen in the experimental data. Several possible explanations for this disagreement are discussed. It is also observed that the spectral function at specific domains of the momentum-energy plane is sensitive to the nucleon-nucleon interaction. Based on this sensitivity, it might be possible to constrain the short range part of the nuclear potential using such experimental data.

###### pacs:

67.85.-d, 05.30.Fk, 25.20.-xIn order to fully describe nuclear systems, it is necessary to understand the short-range behavior of interacting nucleons, i.e. the implications of few nucleons being close to each other inside the nucleus. These nuclear short-range correlations (SRCs) have been studied intensively in the last decades. High-energy and large momentum-transfer electron and proton-scattering experiments show that almost all of the nucleons with momentum larger than the Fermi momentum are part of an SRC pair, which amount to about 20% of the nucleons in medium-size and heavy nuclei FraSar93 (); Egiyan03 (); Egiyan06 (); Fomin12 (); Tang2003 (); Piasetzky06 (); Subedi08 (); Korover14 (); HenSci14 (). A dominance of neutron-proton pairs was observed among the different possible pairs Tang2003 (); HenSci14 (); Piasetzky06 (); Subedi08 (); Korover14 (); Baghdasaryan10 (). These conclusions are also supported by theoretical works, in which ab-initio calculations of momentum distributions in nuclei show a universal high-momentum tail, similar in shape to the deuteron high-momentum tail Schiavilla07 (); AlvCioMor08 (); FeldNeff11 (); AlvCio13 (); WirSchPie14 (); Sargsian2005 (). For more details, see recent reviews Hen_review (); Cio15_review ().

Recently, the nuclear contact formalism, a new approach for analyzing nuclear SRCs, was presented WeiBazBar15 (); WeiBazBar15a (); WeiBar17 (); WeiHen17 (). In this theory, new parameters, called the nuclear contacts, describe the probability of finding two nucleons close to each other inside the nucleus. The values of these contacts depend on the specific nucleus discussed. Another important ingredients of this theory are the universal two-body functions that describe the motion of the SRC pairs. These functions can be model-dependent, i.e. depend on the nucleon-nucleon interaction, however they are identical for all nuclei. This theory was used previously to derive the nuclear contact relations, which are relations between the nuclear contacts and different nuclear quantities, such as the one-body and two-body momentum and coordinate space distributions WeiBazBar15a (); WeiHen17 (), the photo-absorption cross section WeiBazBar15 (); WeiBazBar16 (), the Coulomb sum rule WeiPazBar (), and the correlation function correlation_func ()

The purpose of this paper is to study and analyze electron-scattering experimental data using the contact theory. We will focus on hard semi-exclusive and exclusive scattering experiments, in which one or two emitted nucleons are measured in addition to the scattered electron Subedi08 (); Korover14 (); HenSci14 (); Shneor2007 (); Erez18 (); Duer18 (). These measurements, in appropriate kinematics, are one of the main experimental methods for studying nuclear SRCs, and thus it is important to have a good theoretical description of their results.

In electron-scattering experiments, under the one-photon exchange approximation, momentum and energy are transferred to the nucleus by a virtual photon. If is large enough ( GeV), the photon is predominantly absorbed by a single nucleon. This nucleon is knocked out from the nucleus and its momentum and energy are measured. Neglecting final-state interaction (FSI), the initial momentum and (off-shell) energy of the nucleon in the nucleus ground state, before it was knocked out, can be reconstructed

(1) |

If the initial momentum is larger than the typical Fermi momentum MeV/c fm, then it is most likely that the knocked-out nucleon was part of an SRC pair. In this case, an emission of a second nucleon is to be expected. This nucleon is the correlated partner. Its final momentum equals its initial-state momentum inside the nucleus .

This description indicates that the semi-exclusive and exclusive cross sections should be proportional to the probability of finding a nucleon with momentum and energy in the initial state, which is just the definition of the spectral function . Indeed, it was shown in DeForest83 () that within the plane-wave impulse approximation (PWIA), the cross section is given by

(2) |

where, is the final electron four-momentum, denotes a knocked-out neutron or a proton, and is the off-shell electron-nucleon cross section.

Under few simple assumptions, which will be presented below, the asymptotic high-momentum proton spectral function can be written as

(3) |

Here, are the nuclear contacts, that measure the probability to find a proton-proton () pair or a proton-neutron () pair close together, with quantum numbers denoted by , while the functions are the contributions of these pairs to the spectral function. corresponds to the spin-one deuteron quantum numbers, and corresponds to the spin-zero s-wave quantum numbers. These are the main two-body channels of nuclear SRC pairs WeiHen17 (). Based on the experience with the one-body momentum distribution WeiHen17 (), Eq. (Energy and momentum dependence of nuclear short-range correlations - Spectral function, exclusive scattering experiments and the contact formalism) is expected to be valid for . The probability to find a proton with energy and large momentum , has contribution from both and pairs. The equivalent neutron spectral function is obtained by changing between and .

The derivation of Eq. (Energy and momentum dependence of nuclear short-range correlations - Spectral function, exclusive scattering experiments and the contact formalism) starts with the definition of the spectral function

(4) |

where is the ground state wave function, is the ground state energy and is its binding energy, is an -body eigenstate of the nuclear Hamiltonian with energy , and is an average over the magnetic projections of the ground state. is the nucleon mass and is the annihilation operator of a nucleon with momentum and spin .

For , neglecting three-body or higher correlations, the ground state wave function is dominated by an SRC pair with very large relative momentum and can be written as

(5) |

This is the basic assumption of the contact theory, and it was validated using ab-initio calculations WeiHen17 (); AlvCio16 (). are universal two-body functions, while describe the motion of the rest of the particles, and the pair’s center of mass (CM) motion, . In this picture, once particle is removed, particle is left with high momentum and can be treated as a spectator. Consequently, we may write

(6) |

Here, is an eigenstate of the -body nuclear Hamiltonian with energy , is the spin of particle , is the anti-symmetrizing operator, and normalization factor. It follows that

(7) |

where is the energy of the second correlated nucleon, is the binding energy of the -nucleon system, and the last term is the contribution of the CM motion of the -nucleon system.

Substituting Eqs. (5) and (6) into Eq. (Energy and momentum dependence of nuclear short-range correlations - Spectral function, exclusive scattering experiments and the contact formalism), and assuming that the -nucleon binding energy is narrowly distributed around a central value , we arrive at Eq. (Energy and momentum dependence of nuclear short-range correlations - Spectral function, exclusive scattering experiments and the contact formalism). For a pairs of nucleons , the SRC functions are given by

(8) |

and

(9) |

where , the CM momentum distribution of the SRC pair, is given by . In practice, it can be assumed that all SRC pairs have similar CM distribution , which we shall take as a three-dimensional Gaussian with a width CioSim96 (); Erez18 (); Colle14 (). The spectral functions are expected to be almost identical across the table of nuclides, as the CM and binding energy corrections are relatively small for nuclei heavier than C Erez18 ().

The delta function in Eq. (8) can be used to eliminate the integration over the angles, and can be obtained through numerical integration over , without further approximations. In this integration we also require that . Alternatively, we can continue analytically if we replace the CM term of Eq. (9) by its mean value . This should be a good approximation for small values of or large values of . Then, the delta function can be used to fix the magnitude of , given by

(10) |

We can also see that if the CM momentum distribution has a zero width, i.e. is a delta function which dictates , the spectral function becomes simply a delta function, centered around

(11) |

According to Eq. (10), the momentum magnitude of the second-emitted nucleon in experiments depends only on the initial energy but not on the initial momentum of the knocked-out proton. This might not seem reasonable at first glance, since we expect that Tang2003 (); Piasetzky06 (). But, if one substitutes the value of of Eq. (11) together with , into Eq. (10), we obtain , as expected. For a given , the value of of Eq. (11) should be close to a maximum point in the spectral function, and thus most experimental data is centered around such values of and , leading to the observation of . If sufficient experimental data of exclusive experiments in other domains of the momentum-energy plane will be available, it might be possible to see the energy dependence of and compare it to Eq. (10). We expect for corrections to this relation due finite , the distribution of the around the mean value , and FSI effects.

To calculate the spectral function we must first calculate the universal functions . These are the zero-energy solutions of the two-body Schrodinger equation for the spin-zero channel, and the deuteron wave-function for . In Fig. 1 we present the resulting functions using the AV18 nucleon-nucleon (NN) potential av18 () and the chiral EFT NN force N3LO(600) N3LO () for the spin-zero channel and the deuteron channel. It can be seen that the two potentials produce similar functions up to the cutoff value of the N3LO potential ( fm). Some differences in the functions, like the location of the node, are observed.

Before presenting our calculations for the spectral function, we note that is expected to have a narrow distribution around zero, in each axis, with . Therefore, the main contribution to the spectral function comes from being anti-parallel to . As can be seen in Fig. 1, the function has a node around , and thus we expect to have a minimum for

(12) |

The calculations of and , based on Eqs. (8) and (9), are presented in Figs. 2 and 3, using the AV18 NN interaction. In Fig. 2, they are presented as a function of , at GeV/c and different values of . In Fig. 3, the calculations are a function of at MeV/c. The calculations were done for He, taking to be its binding energy and the binding energy of the deuteron for the case and zero for the case. Calculations for heavier nuclei are similar, with shifted due to the different values of and . Similar calculations using the N3LO(600) potential are presented in the supplemental materials. It can be seen that for small values of , the spectral function is very close to the zero-CM prediction of Eq. (11), corresponding to back-to-back SRC pairs. As the CM width is increased, deviates from this back-to-back picture. In addition, we can see that the spectral function has an interesting structure as it develops two maxima for MeV/c. This structure reflects the node in the function, as predicted in Eq. (12).

To compare between the results of the AV18 and N3LO(600) potentials, we present in Fig. 4 the He calculations of and , as a function of for fixed MeV/c and MeV/c. Here, the results are normalized to at GeV. The bands around the results show the effect of changing the value of between MeV. It is clear that the results are very similar for the two potentials, while the results for show significant differences. This is due to the differences seen in the functions presented in Fig. 1 around their node. Based on this sensitivity of to the potential, it might be possible to constrain the short-range part of the potential using SRCs experimental data, as we will further discuss below. We note that becomes less sensitive to the potential for higher or lower values of .

It should be noted that our expressions for the spectral function derived from the contact formalism are similar to the convolution model presented by Ciofi degli Atti et al. in CioStr91 (); CioSim96 (), and revisited recently CioMezMor17a (); CioMor17b (). The convolution model was shown to agree with ab initio calculations of the spectral function of He CioMezMor17a (), which strengthen the validity of the current derivation as well. Nevertheless, our model differs slightly from the convolution model. In our expressions, we directly use the contact parameters, adding the spectral function to the list of the contact relations. The contact formalism also include different channels, e.g. two different two-body functions, instead of a single deuteron function used in the convolution model.

Equipped with our contact relation for the spectral function, we can go back to the exclusive electron-scattering experiments. One of the main results of these experiments is the ratio between the number of emitted pairs and pairs, extracted from the and cross sections. Based on Eq. (Energy and momentum dependence of nuclear short-range correlations - Spectral function, exclusive scattering experiments and the contact formalism), we can see that if there is a proton in some nucleus with off-shell energy and momentum , then it is part of an SRC pair, which is either a pair or a pair. The ratio of the number of such to pairs is given by

(13) |

For symmetric nuclei () we expect that WeiHen17 (), and thus this ratio depends only on a single parameter . We can see that this ratio generally depends on both the initial momentum of the proton and its energy . Within the PWIA, and based on Eq. (2), this ratio can be extracted from the exclusive-scattering experiments and is given by .

The relation of the measured nucleon knockout cross-section ratios to PWIA calculations and ground-state energy-momentum densities relies on the fact that for the high- kinematics used in the measurement, according to calculations, reaction mechanisms other than the hard breakup of SRC pairs are suppressed and any residual effects are significantly reduced when considering cross-section ratios as oppose to absolute cross-sections Hen_review (); Arrington2011xs (); ColleHen15 (); Colle2015ena (); Boeglin2011mt (). The cancellation of reaction mechanisms in the cross-section ratio steams from the approximate factorization of the experimental cross-section at high-, which also allows correcting the data for any remaining effects of FSI and Single-Charge Exchange (SCX) of the outgoing nucleons using an Eikonal approximation in a Glauber framework Cio15_review (); Frankfurt1996xx (); ColleCosyn2014 (); Dutta2013 (). The experimental data discussed in this work is already corrected for such effects Subedi08 (); Korover14 (); HenSci14 (); Shneor2007 (). It should be noted that these corrections were verified experimentally, see discussion in Colle2015ena (); Hen:2012yva (); Hen_review (); Arrington2011xs (); Dutta2013 (); Frankfurt2001 (); Pieper1992 ().

The ratio was extracted from exclusive-scattering experimental data for He Korover14 () and C Subedi08 (); Shneor2007 (). In these experiments, the main focus was the dependence of these ratios on the initial momentum , and not the dependence on . In both experiments, the ratios were measured in several kinematical settings, each corresponding to specific central values of and . The momentum-dependence of the ratio was highlighted, but the effects of the initial energy were not discussed. This discussion is also missing in previous theoretical works that used the momentum distribution as a starting point to predict the ratio AlvCio16 (); WeiHen17 (); Ryc15 (); NeffFeldHor15 (). The study of this ratio, and SRC pairs in general, should be extended to include the full energy and momentum dependence.

Using Eq. (13) we can predict the value of the ratio as a function of both and , for any nucleus, if the values of the contacts and for this nucleus are known. The values of the contacts for several nuclei with mass number up to were extracted recently WeiHen17 () using variational Monte Carlo (VMC) two-body densities in momentum and coordinate space WirSchPie14 (); Wiringa_CVMC (), calculated using the AV18 NN potential and the Urbana X (UX) three-nucleon force ubx (). We will focus here on He and C, for which the experimental data is also available. As mentioned before, for symmetric nuclei as these, the ratio depends only on one contact ratio. We use the available experimental data of Refs. Korover14 () to fit this ratio of contacts for He, utilizing Eq. (13). For C, we fit the ratio of contacts to the ratio of Ref. Shneor2007 (), which will be discussed below. The fitted values for He and C are given in Table 1, using the AV18 and the N3LO(600) potentials for the calculation of the spectral function, the experimental estimate Korover14 (), and Tang2003 (); Shneor2007 (); Erez18 (), and the relevant bound-state energies for and . Previously extracted contact values, using the AV18 NN potential and the UX three-body force, are also given in the table, and agree with the AV18 ratio extracted here. This ratio of contacts gives us the ratio between the total number of SRC pairs in the deuteron channel and the number of SRC pairs.

The extracted contact ratio using N3LO(600), also shown in table 1, is larger than the one obtained using AV18, which shows that this ratio is model dependent. The main source for this model dependence is the sharp fall of the N3LO(600) functions for fm (Fig. 1). This reduces significantly the number of SRC pairs, i.e. the value of , because the contribution of fm is small, while the AV18 function has significant contribution to SRC pairs for fm. We can look on the total number of deuteron pairs over pairs with relative momentum restricted to fm, given by

(14) |

For AV18 we get a ratio of for He, which is much larger than the ratio of all pairs of table 1. For N3LO(600) we get a ratio of for He, similar to the original ratio shown in the table. We can see that the two potentials give consistent values when restricting the momentum range to fm, and the model dependence disappears. Similar result is obtained also for C. In this discussion, it is important to distinguish between two SRC ratios. One is measured in exclusive scattering, given by Eq. (13), and depends on both the initial momentum and the initial energy of the knocked out proton. The second, describes the number of and (deuteron) pairs with relative momentum , and is given by .

A | potential | (e,e’pN) | k-VMC | r-VMC |
---|---|---|---|---|

He | AV18 | |||

N3LO(600) | - | - | ||

C | AV18 | |||

N3LO(600) | - | - |

Using the fitted contact ratio for He, we can now predict the full dependence of the ratio. The results are presented in Fig. 5 using the AV18 and N3LO(600) potential. We can see that the surface describes well the exclusive-scattering experimental data of Ref. Korover14 () (the black points) using both potentials. We also include our analytic prediction for the points for which the ratio is minimal (red line), based on Eq. (12). There is a good agreement with the full numerical calculations. One can see that the available experimental data sits on a diagonal line in the plane, while there is no experimental data for substantial parts of this plane. Thus, additional experimental data, covering the plane, is needed to fully investigate the theoretical predictions presented in Fig. 5.

Based on Fig. 5, it seems that AV18 and N3LO(600) predict a similar structure for . This takes us back to Fig. 4, which showed that is sensitive to the NN potential around MeV. Thus, if the number of SRC pairs will be measured in future exclusive experiments as a function of with fixed MeV, it might be possible to use it to constrain the NN potential. Since we are discussing pairs with high relative momentum, it should be sensitive to the short distance part of the potential. Based on the bands presented in Fig. 5, we note that the experimental uncertainty of the value of should not be larger than MeV, in order to differentiate between AV18 and N3LO(600).

One can also consider the ratio, i.e. the number of correlated pairs consisting of a proton with off-shell momentum-energy , divided by the total number of such protons. For , this ratio should be given by

(15) |

This ratio was extracted from exclusive scattering experiments for He Korover14 () and C Shneor2007 (). We note that similar corrections to those discussed above (for FSI and SCX) were already applied to the cross sections to obtain the experimental ratio. These corrections are much more significant here, comparing to the corrections, and include transparency effects and significant model-dependent acceptance corrections (of the order of a factor of for the experimental data analyzed here).

Fig. 6 depicts the ratio for C using the AV18 potential, based on Eq. (15) and the contact ratio fitted in this work (table 1), compared to the experimental data of Ref. Shneor2007 (). Here, one can see that while the theory predicts a deep minima in the ratio, the experimental data seems to show a constant ratio of about 5%. Similar figure is presented in the supplemental materials using the N3LO(600) potential. There are few possible explanations for this disagreement between our theory and the data. As mentioned above, the corrections applied to the data in order to obtain the ratio are quite significant. The disagreement shown in Fig. 6 might indicate that these corrections should be re-examined. Experimental data which requires smaller corrections can be useful here, for example using large-acceptance detectors (see e.g. Ref. HenSci14 ()). It is also possible that the limited statistics and the large bins of the data presented in Fig. 6 smears the finer details of the ratio, yielding approximately a constant ratio. If this is the case, to verify the theoretical predictions of this work, better data is needed. Finally, corrections to the theory should also be studied, such as the effects of the energy distribution of the system () around its mean value.

In the supplemental materials, we present the ratio also for He and the ratio for , using the same values of the contacts (table 1). Similar to C, the experimental data for the ratio of He Korover14 () seems to indicate a constant value for the ratio, while the theory shows a different picture. The single experimental point for the C ratio is in agreement with the theoretical predictions. The analysis of the ratio is also presented in the supplemental materials for He and C. The experimental data for this ratio Korover14 (); Subedi08 () includes quite large errorbars and better data is needed to investigate the theoretical predictions.

To summarize, the nuclear contact formalism was used to derive a relation between the nuclear contacts, describing the probability to find SRC pairs in the nucleus, and the spectral function. This relation was utilized to analyze the , and ratios for He and C, emphasizing the full dependence in the plane and revealing a richer structure than was assumed so far, using two different nuclear potentials. For there is a good agreement with the available experimental data, extracted from exclusive electron-scattering experiments, while for there seems to be a disagreement. Possible explanations for this disagreement were discussed. Better experimental data is needed for in order to compare with the theoretical predictions. The contact ratio for He and C extracted using the AV18 potential agrees with previous values, extracted using the same potential. The contact values seem to depend on the NN interaction, but this model dependence is resolved if one is looking on a limited high-momentum range. It was also shown that the contribution of SRC pairs to the spectral function is sensitive to the NN potential, which can be used to constrain the short-range part of the potential, if appropriate experimental data is available.

A main conclusion of this work is that the full energy and momentum dependence of exclusive electron-scattering experiments should be studied, experimentally and theoretically, in order to obtain a full picture regarding nuclear SRCs. Further experimental data for the , and ratios and other observables, for different nuclei, covering the energy-momentum plane, is required for investigating the predictions presented in this work.

###### Acknowledgements.

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