Short range correlations and the isospin dependence of nuclear correlation functions

Pair densities and associated correlation functions provide a critical tool for introducing many-body correlations into a wide-range of effective theories. Ab initio calculations show that two-nucleon pair-densities exhibit strong spin and isospin dependence. However, such calculations are not available for all nuclei of current interest. We therefore provide a simple model, which involves combining the short and long separation distance behavior using a single blending function, to accurately describe the two-nucleon correlations inherent in existing ab initio calculations. We show that the salient features of the correlation function arise from the features of the two-body short-range nuclear interaction, and that the suppression of the pp and nn pair-densities caused by the Pauli principle is important. Our procedure for obtaining pair-density functions and correlation functions can be applied to heavy nuclei which lack ab initio calculations.


Introduction
Correlation functions are a valuable tool for describing interacting many-body systems, providing a means of encapsulating complex many-body dynamics. In the absence of correlations, a many-body probability density, such as that from a manybody quantum mechanical wave-function, can be written as an anti-symmetrized product of single-particle probability densities. The correlation function describes important deviations from this picture.
Correlation functions are widely used in nuclear physics. The nucleus is a strongly-interacting, quantum mechanical, manybody system with high density and a complicated interaction between constituent nucleons. There is no fundamental central potential, so the correlations must exist. An early paper that modeled nuclear correlation functions [1] was used in a wide variety calculations, see the early review [2], involving the strong and weak interactions, demonstrating the impact of correlation functions on the field. More recent examples in which correlation functions are crucial ingredients include: calculations of neutrinoless double beta decay [3][4][5][6][7][8], nuclear transparency in quasielastic scattering [9][10][11][12][13][14], shadowing in deep inelastic scattering [15], and parity violation in nuclei [16,17].
Despite the wide use of correlation functions, their spin and isospin dependence has rarely been discussed. The nucleonnucleon interaction is both spin and isospin dependent, and these dependencies become very important at short-range, leading to phenomena such as the strong preference for protonneutron short-range correlated pairs [18][19][20][21][22][23][24].
The calculations in this paper use the formalism of nuclear contacts [25,26] to determine the spin and isospin decomposition of the nuclear correlation function. This formalism is based on the separation of scales inherent in the long-and short-range structure of nuclei [27]. At short distances, the aggregate effect of long-range interactions can be encapsulated into coefficients, called "contacts," which are nucleus-specific, while the underlying short-range behavior is a universal property of the two-body nuclear interaction. In the Contact formalism, the two-body density, ρ NN (r), defining the probability for finding a nucleon-nucleon pair with separation distance r, can be modeled at short distance (r 1 fm) by: for nucleus, A, where C A is the contact coefficent, NN stands for proton-proton (pp), proton-neutron (pn), or neutron-neutron (nn) pairs and the index s denotes the spin 0, 1 of the twonucleon systems. The wave functions ϕ NN,s (r) are zero-energy (S-or S-D wave) solutions to the Schrödinger equation with a modern nucleon-nucleon potential, e.g., AV18 [28]. Equation 1 assumes angle averaging, and the zero-energy nature restricts the number of contacts. The key assumption in this formalism is that these functions, ϕ NN,s (r) can be used for all nuclei. Contact coefficients can be determined for the different possible spin and isospin configurations of a nucleon-nucleon pair from experiment or from fitting ab initio calculations. Previous studies [25], show that the NN state with deuteron quantum numbers is dominant: the product C np,s=1 A |ϕ np,s=1 (r)| 2 is typically at least an order of magntiude larger than for any other combination. This dominance must be caused by the tensor force [29][30][31] As an example, the decomposition of the two-body density from contact formalism for 40 Ca is shown in figure 1.

Formalism and Results
In this work, we define the nuclear correlation function as the ratio between the two-nucleon density and that obtained from product of single-nucleon densities. Here, the one-body density distribution is denoted as ρ N ( x), which is the probability density for finding a nucleon at position x relative to the centerof-mass of the nucleus, normalized so that its integral is the number of particles of type N. The two-body density distribution ρ NN ( r), is defined as the probability density for finding a nucleon-nucleon pair separated by r, normalized so that its integral is the number of possible NN pairs. The two-body density is defined in terms of the nuclear wave function |ψ by where r i j is the separation between nucleons i and j and P s is a projection operator onto the spin of the nucleon pair. A one-body density function can be used to define an uncorrelated two-body density, where R represents the center-of-mass position of a nucleonnucleon pair, and S NN represents a symmetry factor, which equals 1 for pn pairs, equals Z(Z − 1)/2Z 2 for pp pairs-since there are only Z(Z − 1)/2 unique pp pairs in a nucleus-and equals N(N − 1)/2N 2 for nn pairs. ρ uncorr.

NN
describes the probability density that would result if the individual nucleons were drawn independently from a given one-body density and includes a sum over the nucleon spin. Thus ρ uncorr.

NN
is independent of spin s.
The correlation function, F NN,s (r), a function of the separation distance between nucleons r ≡ | r|, is given by the ratio of the fully correlated to the uncorrelated two-body densities, i.e., The notation, F NN,s (r), is meant to convey that there can be differences in correlations between different spin and isospin configurations. In cases where we refer to a generic correlation function, we will suppress the indices and use F(r). For an uncorrelated distribution, F = 1. Thus, deviations from 1 indicate correlations in a distribution.
A calculation of F(r) requires a model for both ρ N ( x) (in order to calculate ρ uncorr. NN ) and ρ NN (r). Since experimental data and parameterizations of nuclear one-body densities have been made for a wide range of nuclei, creating a model of ρ NN (r) is our main focus here.  (6), with only a single fitted parameter, can reproduce the correlation functions for both pp and pn pairs, and for both 16 O and 40 Ca, to within ±10%. The results here are shown for κ = 2.
One possible approach is to use ab initio calculations to supply ρ NN (r). In this work, we compare to ab initio calculations performed using Cluster Variational Monte Carlo (CVMC) [32] of 16 O and 40 Ca, the two heaviest nuclei studied so far using CVMC [33]. Several other calculations that include the necessary spin and isospin dependence in computing densities are those of Refs. [29,32,[34][35][36].
The points in figure 2 show correlation functions calculated using equations 3 and 4, with CVMC providing the one-and two-body densities. As can be seen, the correlation functions are similar for 16 O and 40 C, but exhibit a strong isospin dependence, which is clear from the large differences at r < 2 fm between pp-and pn-pairs (dominated by s = 1). The strong peak in the np correlation function is due to the attractive nature of the tensor force.As these CVMC calculations treat p and n symmetrically, and since 16 O and 40 Ca are both symmetric nuclei, the results for pp and nn pairs are the same.
For convenience, we provide the following parameterization with parameter values given in table 1. This single function reproduces the correlation functions of both 16 O and 40 Ca. While CVMC calculations determine the correlation functions, it is desirable to provide a better, simple understanding of the underlying mechanisms that produce the large isospin dependence. To achieve this goal we design a model in which the two-body density is formed from a combination of the correlated density coming from nuclear contact formalism and the uncorrelated density, ρ uncorr. NN (r) with the relative weighting determined by a blending function, g NN (r), and constant, κ, such that ρ NN (r) = g NN (r)ρ contact NN (r) + κ(1 − g NN (r))ρ uncorr. NN (r), (6) We can understand how the correlated and uncorrelated densities contribute to produce the specific behavior of the correlation function seen through CVMC by assessing the quality of this model and by determining the blending function. In order to parameterize g NN (r), we consider the short-and long-range constraints. At short-distance, where ρ contact NN (r) is an accurate description of the two-body density [25], g NN (r) equals 1. For large distances, ρ NN must approach ρ uncorr.
NN . Since ρ contact NN falls off approximately as 1/r 2 for r > 2 fm, g NN must approach (κ − 1)/κ in the long-range limit. We propose the following model which meets these requirements: For r < 0.9 fm, ρ NN (r) is modeled well by the contact expression Eq. (1) (see [25]). For r > 0.9 fm, the contact density and the uncorrelated densities are blended, with a characteristic length-scale, a. In principle, a would depend on the isospin of the pairs and on the specific nucleus being studied.
Varying the parameters of Eq. (7) to describe pp, nn and pn pairs in 16 O and 40 Ca shows that the same blending function g(r) can be used to describe all of the correlation functions extracted from CVMC, shown in Fig. 2. CVMC correlation functions are shown as points, while our model, described in equation 6, is shown with bands, for which the dominant contribution to the uncertainty comes from the contact coefficients, C NN . The uncorrelated uncertainty, ρ uncorr NN , used by our model is supplied by CVMC calculations of the one-body density ρ N . Our model is able to reproduce the correlation functions for both pp and pn pairs in two different nuclei. In achieving this description we find that the parameter a depends smoothly on κ. With κ = 2, a = 1.43 ± 0.02 fm. Figure 2 demonstrates that the spin-isospin dependence of the correlation function (differences between the pp and pn correlation functions) are driven by the differences in short-range correlations, i.e., those coming from the contact densities of Eq. (1), while the long range behavior is universal between different kinds of pairs and in different nuclei.

Discussion
The universal character described above enables predictions of correlation functions for nuclei that are too large for adequate ab initio calculations. Our model requires, as input, a one-body density function to form ρ uncorr. NN (r) and the nuclear contact coefficients. One-body densities have been well-measured experimentally, and simple parameterizations exist for many different nuclei, e.g., Ref. [37]. While the nuclear contacts used in this work were determined from CVMC calculations, they can also be determined from experimental data, as shown in Ref. [25]. Since the blending between the short-range contact densities and the long-range uncorrelated densities is largely isospinindependent and universal, the same scheme can be used to predict the correlation function for any heavy nucleus, for which suitable ab initio calculations do not exist. A nice ab initio treatment of light nuclei has recently appeared [38]. See also Ref. [39], which is based on nuclear matter calculations.   Figure 3: The typical contribution to a 0νββ matrix element, given by F(r)/r, is significantly different for pp pairs relative to pn pairs. The use of an average correlation function, shown here by NN, is problematic. The isospinindependent parameterizations of Miller and Spencer [1], as well as Simkovic et al. [7] are shown for comparison.
We provide an example of the utility of this approach. One of the important applications of correlation functions is the calculation of nuclear matrix elements for neutrinoless double betadecay (0νββ). If such a decay were observed, the nuclear transition matrix element would allow an extraction of the abso-lute neutrino mass from the decay rate. Typically, these matrix elements have modeled the nucleon-nucleon short ranged correlations by using a so-called Jastrow approximation without considering the significant isospin dependence of the correlation function [2]. The basis of this widely used, see e.g. [3]- [17], approximation is that the short-range correlations are the same in the initial and final states and that the same correlation function can be used for a wide range of nuclei.
To illustrate this point, let us consider a typical double betadecay transition operator, which has a 1/r radial dependence as computed in the Jastrow approximation with our correlation functions. Then the matrix element is given by A summary of the operatorsÔ s,s appears in Ref. [38]. In the case where the operator,Ô s,s is diagonal in the spin index, the contribution from the correlation function to the matrix element scales as F NN,s (r)/r. This can be used as a figure of merit. Such an approach neglects large isospin differences that enhance pn correlations. The model presented in this work provides a method for including these isospin differences in more sophisticated calculations.
In summary, our model of the correlation function reproduces CVMC calculations of 16 O and 40 Ca, the two heaviest nuclei studied so far [33]. Using the same blending function g(r) suffices to describe the correlation functions for all quantum numbers that we consider. This implies a universality in the blending of correlated and uncorrelated contributions to the two-body density, independent of the specific nucleus or of the pair isospin. Rather, the large differences between F pp (r) and F pn (r) from figure 2 are driven entirely by the isospin dependence of the two-body NN interaction at short distances. This isospin dependence is, in turn, driven by the two-nucleon tensor force.