Phase-space distributions of nuclear short-range correlations

Nuclear short-range correlations (SRCs) induce high-momentum/high-energy fluctuations in the nuclear medium. In order to assess their impact on nuclear bulk properties, like nuclear radii and kinetic energies, it is instrumental to determine how SRCs are distributed in phase space as this sheds light on the connection between their appearance in coordinate and momentum space. Using the lowest-order correlation operator approximation (LCA) to include SRC, we compute two-dimensional nuclear Wigner quasiprobability distributions $w(r, k)$ to locate those $({r}, {k})$ phase-space regions that are most heavily impacted by SRCs. The SRC-induced high-momentum components find their origin in a radial range that is confined to the nuclear interior. Significant SRCs strength is generated in the full momentum range $0 \leq k \lesssim 5 ~\text{fm}^{-1} $ covered in this work, but below the Fermi momentum those are dwarfed by the mean-field contributions. As an application of $w(r, k)$, we focus on the radial dependence of the kinetic energy $T$ and the momentum dependence of the radius $r_{\text{rms}}$ for the symmetric nuclei $^{12}$C, $^{40}$Ca and the asymmetric nucleus $^{48}$Ca. The kinetic energy almost doubles after including SRCs, with the largest increase occurring in the nuclear interior $r \lesssim 2$ fm. The momentum dependence of the $r_{\text{rms}}$ teaches that the largest contributions stem from $k \lesssim 2 $ fm$^{-1}$, where the SRCs induce a slight reduction of the order of a few percent. The SRCs systematically reduce the $^{48}$Ca neutron skin by an amount that can be 10\%.


Introduction
The size of an atomic nucleus [1,2] and how protons and neutrons are spatially arranged for various proton-to-neutron ratios [3][4][5][6] are topics of continued great interest in the precision era of nuclear physics. Detailed nuclear-structure studies have shown that long-range correlations connected with core-breaking effects have a substantial impact on computed proton and neutron nuclear radii and are sources of uncertainties in advanced nuclear-structure calculations. For example, a systematic study for 48 Ca [1] indicated that in ab initio theory with a family of modern chiral effective forces, the variations in the computed proton and neutron radii can be of the order of 10%. The impact of short-range correlations (SRCs) on bulk nuclear properties like radii is not that well known and has recently been addressed in [7]. In that paper, qualitative arguments are developed as to why the omission of SRCs may have a non-negligible impact on the computed radii of neutron-rich nuclei and it was suggested that more quantitative calculations are in order.
Wigner distributions [8,9] provide a distinct view on the spatial and momentum structure of quantum systems and are widely applicable including in subatomic physics [10]. It is a subject of great interest in non-perturbative quantum chromodynamics (QCD) [11][12][13][14][15][16][17][18]. The QCD Wigner distribution provides information about the joint positionmomentum distributions of partons in the nucleon and as such can be considered as the mother partonic distribution from which all others can be derived. Selected results of Wigner distributions for finite nuclei have been presented in Refs. [19][20][21][22]. A recent calculation using realistic potentials highlighted the influence of SRCs on the deuteron's Wigner distribution [23].
In this work we aim at providing a study of the positionmomentum structure of SRCs by presenting calculations of Wigner distributions for finite nuclei. The SRCs have been connected with fat momentum tails in the nuclear momentum distributions [24][25][26][27]. The spatial structure of the SRCs in finite nuclei has received less attention and will be one of the topics of discussion here. To quantify the impact of SRCs, we use the lowest-order correlation operator approximation (LCA) [28][29][30] that is based on a number of assumptions: (i) the scale separation between the long-range and short-range nuclear correlations; (ii) the universal local character that make SRCs a property that can be imposed on the mean-field behavior through the operation of universal operators [31][32][33]. The LCA shares these assumptions with alternate theoretical approaches to quantify the impact of SRCs, including the generalized contact formalism (GCF) [34,35]. As LCA shifts the complexity from the wave functions to the operators, it can be used for SRCs estimation in nuclear structure and nuclear reaction applications. Since the LCA formalism does not account for long-range correlations, our focus is on the relative contribution of SRCs to nuclear bulk properties like point-nucleon radii and kinetic energies. We deem that our calculations serve as an important comparative benchmark for highlighting the impact of SRCs on nuclear bulk properties.
The kinetic energy is connected with the momentum structure whereas the radii are connected with the spatial structure. With the Wigner distributions one gains access to the momentum structure of radii and the spatial structure of the kinetic energy, and how those are impacted by SRCs. Alternate approaches [35] have addressed the SRCs in both coordinate and momentum space. Wigner distributions provide a unique window of insight into the phase-space distributions as they provide information in both variables simultaneously.
In what follows, we first develop a formalism to compute Wigner distributions that include the effect of SRCs. We then proceed with the presentation of the results of numerical calculations for the separated proton and neutron Wigner distribution for the nuclei 12 C, 40 Ca and 48 Ca. Those distributions form the basis to elucidate the phase-space dependence of SRCs in the proton and neutron radii and kinetic energies.

Formalism
The Wigner distribution is the central quantity of interest in this work. The quantum Wigner operator has the following spectral decomposition in three-dimensional coordinate (≡ ) or momentum (≡ ) space: An outline of the derivation of the Wigner distribution in the LCA is given in App. A. Through the introduction of SRCs operators, many-body operators between Slaterdeterminant states are generated. In LCA, the many-body operators generated from one-body operators are truncated at the level of two-body operators and the ( , ) can be separated in a proton and a neutron part by considering the four isospin pair combinations (see Eq. (16)) Hereby, the has two categories of contributions. The first category stems from an uncorrelated proton and neutron that are both described as quasi-particles in the mean field. The second category is the SRCs one whereby the tagged proton and neutron are correlated through one or a product of two correlation operators. Similar discussions hold for the other three pair combinations.
To obtain the nuclear rms radius and kinetic energy, we consider scalar operatorŝ 2 ,̂ 2 which have Wigner transforms 2 , 2 respectively. We can use Eq. (5) to extract the radial dependence of the kinetic energy operator in coordinate space, and that of the rms radius in momentum space. The first method permits to calculate the quasi-expectation value of̂ =̂ 2 ∕2 (̂ 2 ) at a given position (momentum) Due to quantum effects in the Wigner distribution, these quasi-expectation values of positive definite operators can be negative. These variables allow to infer the magnitude of the nucleon kinetic energy at a certain , or the size of the nuclear rms radius with a certain . Because of the -( -) dependence in both numerator and denominator, the ( ) ( 2 rms ( )) do not integrate to the full ( 2 rms ). A distribution that is both reminiscent of the spatial structure of the kinetic energy (momentum structure of the nuclear radius) and provides the proper scalar quantity after integration, can be obtained in the second method that is based on the computation of the densities ( ) and 2 ( ): The ( ) encodes the contribution to the kinetic energy at given nucleon radial coordinate and was also considered in Ref. [20]. Analogously, 2 ( ) encodes the contribution to the nuclear radius squared at given momentum . In this work we assess the impact of SRCs on the ( ) ( rms ( )) and ( ) ( 2 ( )), each offering complementary insight in what actually happens to bulk nuclear properties after including high-momentum/high-energy fluctuations in a model.

Results
The LCA has the following inputs: (i) a set of universal strength correlation functions ( 12 ) entering the matrix elements of Eq. (17); (ii) the HO frequency for which we adopt a global parameterization of the form As in previous publications [28][29][30], we use the Argonne VMC correlation functions [36] for the tensor and spinisospin correlation functions, and use two options for the central correlation function ( 12 ): a hard one that is computed with the aid of the Reid potential (denoted [ ]) [37], and the softer Argonne correlation function ( [ ]) [36]. We stress that with the [ ] we obtain momentum distributions that are very similar to those obtained in ab initio calculations [29]. Furthermore, the choice for [ ] was a data driven one, as 12 C( , ′ ) data could be described with this choice [38]. For the HO frequency of Eq. (14), we have in previous LCA works systematically used the "default" values 1 = 45 MeV, 2 = 25 MeV, a choice that we refer to as ℏ [ ]. As we quantify the effect of SRCs on nuclear radii in this work and wish to quantify uncertainties stemming from the model parameters, we also explore other values of 1 , 2 by fitting them to the measured nuclear rms charge radii of 4 He, 9 Be, 12 C, 16 O, 27 Al, 40 Ca, 48 Ca, 56 Fe, 108 Ag, 197 Au and 208 Pb [39]. This was done using a standard minimum 2 fit. These inputs carry the label ℏ [ ]. In this work, the IPM corresponds with the HO model that can be formally reached after setting all correlation operators equal to zero in LCA. We consider this HO model as the benchmark against which to measure the impact of SRCs. As a robustness check and sensitivity analysis, we consider two choices for the two inputs to LCA. Together with the two HO frequency variants of the IPM we are left with six models that are listed in Table 1. In Fig. 1 we compare the computed charge radii to data for these models. As we compute point-nucleon radii, corrections (see Ref. [1]) were applied before comparing to data. Note that the fits produce 1 that are somewhat smaller than the default value and a small 2 . We stress that the variation of ℏ across the various model variants does not exceed 13%. In Fig. 1 IPM LCA LCA LCA LCA  [1]) and data [39]. Bottom: One-body momentum distribution ( ) for 48 Ca as computed in LCA with the four model variants detailed in Table 1. the measured rms radii. The LCA calculations with those HO parameters produce rms radii that are a few percent smaller whereby the strongest impact of the SRCs is observed for the LCA model that uses a hard central correlation function. For medium and heavy nuclei, the fitted IPM and LCA ℏ [ ] parameterizations yield almost identical charge radii, in overall good agreement with the data. For the light nuclei there are larger deviations between the predicted radii obtained with the ℏ [ ] and ℏ [ ] parameters. With the ℏ [ ] HO frequency the measured radius for 12 C can be reproduced. The larger deviations between computed and measured radii for 4 He and 9 Be can be attributed to the absence of long-range effects in the LCA. For these light nuclei, also the center-of-mass (c.o.m.) corrections can be sizable [40].
The numerical cost of LCA scales polynomially with mass number which makes it applicable throughout the mass table. In this work, our focus is on 12 C, 40 Ca, and 48 Ca. The latter nucleus allows us to assess the SRC effects in asymmetric nuclei. For 48 Ca, the bottom panel in Fig. 1 shows the LCA one-body momentum distribution for the four model variants. Contrary to the radii, the choice of HO frequency has an almost negligible effect on the momentum distribution. The choice for the central correlation function, on the other hand, mainly affects the momentum distribution for momenta larger than half the nucleon mass ≳ 3 fm −1 . As reported in Refs. [29,30], as relatively little strength is present at those momenta this difference results in a variation of the order of a few percent in comparisons with quantities extracted from electron scattering data. For the LCA momentum distributions for 12 C and 40 Ca (shown in Ref. [29]), similar remarks hold for the sensitivity to the two input sources as for 48 Ca.
Over the last decade, constraints of the SRCs models have been improved through the increased availability of two-nucleon knockout data from proton-and electronnucleus experiments in selected kinematics [41][42][43][44][45][46][47]. Thanks to its flexibility, the LCA model could be well tested against results for the isospin [29] and mass [30,48] dependence of SRCs. The LCA provides a good basis for accurate SRCs modeling across the nuclear mass table. As an illustration of this we mention the so-called 2 scaling factors that are extracted from inclusive electron-nucleus data and can be connected to the aggregated effect of SRCs in nucleus relative to the deuteron. A recent 48 Ca( , ′ ) / 40 Ca( , ′ ) measurement has addressed the isospin structure of SRCs [49] and provided the result 0.971 ± 0.012 for the 2 ( 48 Ca)∕ 2 ( 40 Ca) ratio of measured cross sections per nucleon. Loosely speaking, this implies that per nucleon there is about 3% less impact from SRCs in 48 Ca as compared to 40 Ca. The LCA predictions for this quantity (that were published before the data [30]) can be extracted from the high-momentum tails of the LCA momentum distribution for 40 Ca and 48 Ca. The computed numbers for the 48 Ca / 40 Ca ratio are 4.89/4.99=0.98 (see Table I of Ref. [30]) which is in close agreement with the data. Figure 2 shows the results for the two-dimensional Wigner distribution ( , ). These LCA numerical results were subjected to several checks. One-body momentum distributions ( ) obtained using Eq. (3) were confronted with those from direct computation [30]. Similarly, the rms radii obtained with Eq. (13) that involves the ( , ) were compared with those obtained through direct calculations with the rms radius operator. Finally, the normalization of the four isospin pair combinations ( , , , ) of the LCA ( , ) of Eq. (9) are constrained by and . 1 We find sub-percent deviations that can be attributed to the truncation of the summation over the quantum numbers , in Eq. (19) and the introduction of finite grid sizes in ( , ). Note that in computing matrix elements with ( , ) one multiplies it with 2 2 , see for example Eqs. (12) and (13).
Inspecting the results of Fig. 2, one observes that in IPM the ( , ) extends over the entire radial range and over a well-constrained -range that is almost identical for all three nuclei considered. The SRCs generate a fat momentum tail in ( , ) that is mainly confined to the interior of the nucleus. Correspondingly, the high-momentum SRC contributions are distributed in a narrower range than the mean-field contributions. Indeed, the weight of the fat tails diminishes with increasing , with the largest weight at < rms , and hardly any high-momentum components for ≳ 2 rms . This is a reflection of the fact that in LCA the SRCs are chiefly generated from correlation operators acting on IPM nodeless relative -pairs [50]. Obviously, the wave functions for these -pairs have a finite density at relative 12 = 0 and are very much confined to the nuclear interior. The panels in the bottom row of Fig. 2 consistently show that the strength of the SRCs contribution to the LCA ( , ) shifts to smaller compared to the IPM. Note that the use of the term SRC does not imply a momentum cut here. There are significant contributions to the SRCs part from < as the bottom row in Fig. 2 shows. The normalization of the SRC part (1.8/6 nucleons for 12 C) cannot be directly connected with experimental measures for the number of SRC pairs as those exclusively refer to high-momentum SRCs. The maximum of the Wigner distribution (and the rms radius) for the SRCs contribution sits at smaller values of in comparison to the IPM and LCA ones. We verified that SRCs make up almost 100% of the LCA result for > 2 fm −1 at any . As mentioned earlier the absolute SRC contribution diminishes strongly for large at these higher momenta. This might raise the question to what extent the SRCs in the interior can be probed in scattering experiments, where the ejected particles are subject to strong final-state interactions (FSI) while traversing the nuclear medium. A previous study [51] has shown, however, that reactions probing correlated nucleon pairs are still sensitive to the nuclear interior, even after correcting for FSI effects. This is in contrast to singlenucleon knockout reactions which become much more surface dominated after FSI corrections.
For 12 C, the two-dimensional Wigner distributions ( , ) in Fig. 2 has a distinctive negative region for small and , illustrative of quantum effects. For the two calcium isotopes, the area with ( , ) < 0 is also located at small and . The proton ( , ) are almost identical in 40 Ca and 48 Ca (note the log scale), whereas for neutrons in 48 Ca the emergence of a surface-located neutron skin is clearly visible. In the forthcoming, it will be shown that the relative impact of SRCs on the proton and neutron kinetic energies differs in an asymmetric nucleus [29,45,52,53]. Compared to the deuteron results for ( , ) of Ref. [23], the fat momentum tails in LCA for finite nuclei extend over a larger momentum range and do not display the oscillations. These differences are likely attributed to the smoothing effect from pair c.o.m. motion and the fact that many pairs are contributing. We now discuss the numerical results for the Wignerbased functions that are introduced to investigate the spatial dependence of the kinetic energy ( ( ) and ( ) of Eqs. (10) and (12)) and the momentum dependence of the nuclearradii ( rms ( ) and 2 ( ) of Eqs. (11) and (13)). In what follows, we systematically compare IPM with LCA results and we stress that both are obtained for identical normalizations of the ( , ). The 12 C results for the ℏ [ ] model variants of Table 1  For < rms the SRCs account for roughly 60-70% of the kinetic energy. The ( ) demonstrates that the kinetic energy receives significant contributions from > rms . For the rms radius, the overall impact of the SRCs is far more modest than what is observed for the kinetic energies. The LCA predictions for 2 ( ) are below the IPM one except for > 4 fm −1 . Overall this implies smaller radii when including SRCs using the same HO frequency, as was shown in Fig 1. From the 2 ( ) one infers that both the IPM and LCA contributions to rms from > 2 fm −1 are very small despite the fact that the LCA 2 ( ) is much larger than the IPM one for high momenta. We look at these highmomentum contributions in more detail in Fig. 6. In Figs Fig. 4, we can make the following observations. In the IPM neutrons possess more kinetic energy than protons for all . For the SRC contribution to the LCA result, the situation is reversed. As far as the SRCs contributions are concerned, protons are more kinetically energetic for all . The weighted sum of the IPM and SRCs contributions results in the radial dependence of the LCA ( ). In the deep interior ≲ 1.8 fm of the nucleus, where the SRC effects are in full swing, protons have more kinetic energy, while for ≳ 1.8 fm neutrons have more. The difference is very small, and in the deep surface region > 6 fm there is no effect from the SRC. The  Table 2.  Fig. 3. Legends (linestyle/color) apply to all panels. The vertical bands cover the range of point rms radii for the depicted models, see Table 2. 48 Ca ( ) shows that counted per nucleon the IPM protons in the nuclear interior contribute more to the bulk kinetic energy than the neutrons, while the situation is reversed in the exterior. In the LCA, the first effect is enhanced while the second is reduced. The combination of these effects results in more kinetically energetic protons than neutrons after including SRCs (see Table 2).
In line with the carbon results of Fig. 3, the radius rms ( ) of the calcium isotopes in Fig. 5 reaches a plateau at ≳ 2 fm −1 . The 48 Ca neutron radii are larger than the proton ones for momenta below the Fermi one, but in the momentum region dominated by the tensor correlations (1.5 fm −1 ≲ ≲ 2.5 fm −1 ) the LCA proton and neutron radii almost coincide. As with 12 C, the 2 ( ) indicate that high-momentum nucleons only contribute marginally to the rms radius, even in the LCA.
The resulting kinetic energies and rms radii rms are summarized in Table 2 for the different model variants.
The mentioned values are directly obtained through phasespace integration of the ( , ) (see Eqs. (12) and (13)). The 40 Ca results can be compared to those of recent ab initio calculations (Tables V and VI of [27]). The calculations with the AV18 two-body force lead to = 32.29 MeV and rms = 3.41 fm. In LCA we find values of and rms that are slightly smaller but the deviation is at most 10%. We stress that in LCA we do not have long-range correlations. In that light it is important to note that the results of [27] indicate that three-body forces decrease the by about 1 MeV and increase the rms by about 0.1 fm.
For 48 Ca, one observes that all LCA model variants systematically predict that the proton kinetic energy is about 1 MeV larger than the neutron one, a phenomenon known as "kinetic energy inversion" [53]. For the rms radii, the LCA result shows a slight reduction of the different radii for the ℏ [ ] inputs, while SRC connected changes in the size of the 48 Ca neutron skin are of the order 5-10%. Note that the kinetic energy inversion for protons and neutrons in 48 Ca cannot be directly inferred from the shapes of the ( , ) of Fig. 2. Indeed, the difference between 48 Ca proton and neutron values for rms (which is related to ⟨ ⟩ as ⟨ ⟩ = 2 rms ∕2 ) is hardly visible in Fig. 2.
Overall, the impact of the SRCs relative to the IPM result for the radii and kinetic energies is rather insensitive to the input choices of the calculations. For the radii the largest uncertainties stem from the choice of the HO parameter. As illustrated by the ℏ [ ] results that use a fixed HO frequency, the impact of SRCs is a reduction of the order 3-6%. The SRCs systematically reduce the neutron skin. For the kinetic energies, the impact of the SRCs is large and the uncertainty stemming from the choice for the central Table 2 Computed kinetic energies per nucleon (in MeV) and point-nucleon rms radii (in fm). The comparison between the computed and measured charge radii for the different model variants is displayed in Fig. 1  So far, no momentum selection was imposed when investigating the influence of SRCs on bulk nuclear properties. In order to keep the dimensionality of the numerical ab-initio calculations in check one often relies on softened nucleonnucleon interactions that involve a momentum cut-off. In light of this, we wish to quantify the influence of SRCs on the nuclear radii as a function of a momentum cut-off. To this end, we introduce a momentum cutoff Λ in the momentum integrals of Eq. (13), both in the numerator and denominator. We show the relative change in the rms radii as a function of Λ in Fig. 6. The IPM (LCA) radii converge at Λ ≈ 2 fm −1 (Λ ≈ 4 − 5 fm −1 ). The momentum range 2 ≲ ≲ 4 fm −1 shrinks the LCA rms radii by about 1%. This might seem to be at odds with the obvious fact that at large momentum the LCA rms ( ) of Fig. 5 overshoots the IPM one. These rms ( > 2 fm −1 ) values, however, are smaller than those for rms ( < 2 fm −1 ). This means that excluding the > 2 fm −1 (see 2 ( ) in Figs. 3 and 5), increases the LCA rms radii. For 48 Ca the per nucleon influence of SRCs is different for protons than for neutrons, resulting in a slight decrease of the neutron skin on the order of 3% when including highmomentum SRCs with > 2 fm −1 , independent of the chosen LCA model. This agrees with the naive picture of the dominant tensor correlations pulling the high-momentum protons close to the neutrons, see rms ( ) in Fig. 5.

Conclusion and Outlook
Most often nuclear short-range correlations in finite nuclei are addressed from the perspective of distributions in momentum space and the operational definition of SRCs is one whereby one refers to nucleons with a momentum well above the Fermi momentum ( ≳ = 1.2 fm −1 ). This implies that the SRC terminology is exclusively used for nucleons that have a momentum larger than . In this work, we have added the spatial perspective by presenting calculations of nuclear Wigner distributions ( , ) that include the effect of SRCs. The calculations are conducted within the lowest-order correlation operator approximation (LCA), a framework that has shown its interpretative potential in dealing with observations. Common features emerged for the ( , ) of the three nuclei 12 C, 40 Ca and 48 Ca considered in this work. The SRCs impact the ( , ) over the full momentum range considered (from 0 up to the nucleon mass) and a radial range that is confined to the deep nuclear interior. The SRC contribution to the Wigner distribution has a rms of about 1.6-1.7 fm −1 , whereas the IPM ( , ) has rms ≈ 0.8 fm −1 . The resulting LCA ( , ) that has both mean-field and SRC contributions has rms ≈ 1.2 fm −1 . As the SRCs are confined to the deep interior of the nucleus, the LCA ( , ) has an rms that is approximately 0.2 fm smaller than the IPM one when using the same HO frequency.
From the ( , ) we computed a prototypical bulk nuclear momentum-space and coordinate-space feature, namely the non-relativistic kinetic energy and the point-nucleon radius. The SRCs almost double the proton and neutron kinetic energies. Furthermore, the proton-neutron dominance of the SRCs gives rise to peculiar effects in asymmetric nuclei with a neutron abundance whereby the per-nucleon kinetic energy is larger for protons than for neutrons. The impact of SRCs on proton and neutron radii is at the percent level. In asymmetric nuclei the proton and neutron radii are impacted differently. In the asymmetric nucleus 48 Ca studied here, the SRCs reduce the neutron skin by an amount that is of the order of 5-10%, with high-momentum SRC components accounting for 3%.
The Wigner distributions discussed here could provide valuable input in semi-classical transport calculations for nuclear reaction cross sections [54][55][56] that wish to account for the effects of SRCs. It would be interesting to see whether a model with a realistic description of both long-range and short-range correlations leads to sizable changes in the nuclear radii.
The computational resources (Stevin Supercomputer Infrastructure) and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by Ghent University, FWO and the Flemish Government -department EWI. We thank M. Sargsian for comments on an earlier draft.
In the numerical evaluation of Eq. (19), the summation over is truncated after > 10, while for the other variables all combinations with non-vanishing 3 -symbols are retained. This truncation in the variable results in normalization errors on the sub-percent level for heavy nuclei.