Many-body factorization and position–momentum equivalence of nuclear short-range correlations


While mean-field approximations, such as the nuclear shell model, provide a good description of many bulk nuclear properties, they fail to capture the important effects of nucleon–nucleon correlations such as the short-distance and high-momentum components of the nuclear many-body wave function1. Here, we study these components using the effective pair-based generalized contact formalism2,3 and ab initio quantum Monte Carlo calculations of nuclei from deuteron to 40Ca (refs. 4,5,6). We observe a universal factorization of the many-body nuclear wave function at short distance into a strongly interacting pair and a weakly interacting residual system. The residual system distribution is consistent with that of an uncorrelated system, showing that short-distance correlation effects are predominantly embedded in two-body correlations. Spin- and isospin-dependent ‘nuclear contact terms’ are extracted in both coordinate and momentum space for different realistic nuclear potentials. The contact coefficient ratio between two different nuclei shows very little dependence on the nuclear interaction model. These findings thus allow extending the application of mean-field approximations to short-range correlated pair formation by showing that the relative abundance of short-range pairs in the nucleus is a long-range (that is, mean field) quantity that is insensitive to the short-distance nature of the nuclear force.

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Fig. 1: Short-distance universality of two-nucleon density in nuclei.
Fig. 2: High-momentum universality of two-nucleon density in nuclei.
Fig. 3: Nuclear contact density distribution.
Fig. 4: Spin-dependent nuclear contact term ratios.
Fig. 5: Nuclear contact terms.

Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request


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We thank J. E. Lynn for providing some of the deuteron momentum distributions. We also thank J. Carlson, C. Ciofi degli Atti, W. Cosyn, S. Gandolfi, A. Lovato, G. A. Miller, J. Ryckebusch, M. Sargsian and M. Strikman for discussions. This work was supported by the US Department of Energy, Office of Science, Office of Nuclear Physics under award nos. DE-FG02-94ER40818, DE-FG02-96ER-40960, DE-AC02-06CH11357, DE-AC05-06OR23177 and DE-SC0013617 (FRIB Theory Alliance Award), the Pazy foundation, and the Israeli Science Foundation (Israel) under grant nos. 136/12 and 1334/16, the NUCLEI SciDAC program, the INCITE program and the Clore Foundation. Computational resources have been provided by the Los Alamos National Laboratory Institutional Computing Program, which is supported by the US Department of Energy National Nuclear Security Administration under contract no. 89233218CNA000001, by the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the US Department of Energy, Office of Science, under contract no. DE-AC02- 06CH11357, and by the National Energy Research Scientific Computing Center (NERSC), which is supported by the US Department of Energy, Office of Science, under contract no. DE-AC02-05CH11231.

Author information




D.L., M.P. and R.B.W. performed the many-body QMC calculations. N.B. and R.W. calculated the two-body universal functions. R.C.-T and R.W. analysed the QMC calculated densities and extracted the nuclear contacts. N.B., D.W.H., E.P., A.S., L.B.W. and O.H. initiated and guided the work. All authors contributed to writing the paper and reviewing the results.

Corresponding author

Correspondence to O. Hen.

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Extended data

Extended Data Fig. 1 Scale and scheme dependence of two-nucleon densities.

Two-nucleon ρpNA (r) coordinate (a-d) and (npNA(q) momentum (e-h) space distributions calculated for different nuclei using phenomenological (AV18+UX and AV4’+UIXc) and chiral (N2LO(1.0fm), N2LO(1.2fm), and NV2+3-Ia*) potentials. The top (a, b, e, f) and bottom (c, d, g, h) rows show distributions for 4He and 16O respectively. Source data

Extended Data Fig. 2 Universal two-body functions.

NN interaction model dependence of the universal two-body functions φNNa2 for spin-1 pn (a, b) and spin-0 pp (c, d) pairs calculated in both coordinate (a, c) and momentum (b, d) space. Distributions are shown for both phenomenological (AV18+UX and AV4’+UIXc) and chiral (N2LO(1.0fm), N2LO(1.2fm), and NV2+3-Ia*) interaction models. Source data

Extended Data Fig. 3 Nuclei and models included in this study.

QMC-calculated two-nucleon distributions for different nuclei and NN+3N potentials. Checkmarks indicate calculations used in the current study. All calculations are available for both coordinate and momentum space, except for 16O and 40Ca with AV18 (labeled with an * below), for which the UIX potential is used and results are only available in coordinate space5. Calculations with the N2LO(1.2fm) potential for heavier systems are not considered in this work due to the large regulator artifacts found for A ≥ 12 (see ref. 20).

Extended Data Fig. 4 pp and nn two-nucleon distributions.

Same as Fig. 2 but for pp (a-d, i-k) and nn (e-h, m-p) distributions. Source data

Extended Data Fig. 5 pp-to-pn pairs ratio from experiment and theory.

Ratio of pp-to-pn back-to-back pairs in 4He as a function of pair relative momentum q, npp(Q = 0, q)/npn(Q = 0, q), for different NN+3N potentials, compared with the experimental extractions of ref. 8 using (e,e’pp) and (e,e’pn) data. Data error bars show the combined statistical and systematical uncertainties at the 1s or 68% confidence level. Source data

Extended Data Fig. 6 Nuclear contacts values.

The extracted contact values have been divided by A/2 and multiplied by 100 to give the percent of nucleons above kF. For symmetric nuclei, Cnns=0 = Cpps=0. In the case of 3He, Cnns=0, as there is only one neutron in this nucleus. In the case of 3H, Cpps=0, as there is only one proton in this nucleus, and the values shown under Cpps=0 correspond to Cnns=0.

Extended Data Fig. 7 Impact of three body forces and correlations on N2LO k-r consistency.

Ratios of coordinate-space to momentum-space pn (a) and pp (b) contact terms in 3H and 3He for the N2LO(1.0fm) interaction. The contact values for 3H in the spin-0 pp panel corresponds to the spin-0 nn values, as there are no pp pairs in this nucleus. V2+V3 refers to calculations employing the full 2+3-body Hamiltonian, while results labelled as V2 are obtained turning off the 3-body potential. The second part of the labels refers instead to different wave functions, obtained by optimizing different correlations for the given V2+V3 or V2 Hamiltonian: 2b+3b for the full 2+3-body correlations; 2b2+3b for the full 2+3-body correlations, where more sophisticated 2b correlations are used (see ref. 20); 2b for 2-body correlations only; 2b* for 2-body correlations only, obtained by turning off the 3-body correlations of the 2b+3b wave function (no re-optimization). Source data

Extended Data Fig. 8 Calculation accuracy estimation.

Ratios of contacts extracted from VMC densities to contacts extracted from DMC and extrapolated (EXT = 2 × DMC - VMC20) distributions for pn (a) and pp (b) contacts. Error bars show the combined statistical and extraction systematical uncertainties at the 1σ or 68% confidence level. Source data

Extended Data Fig. 9 Scale dependence of N2LO-based QMC calculations.

a-d, Relative-distance densities for pn (a, b) and pp (c, d) pairs for several nuclei (colored lines) integrated over R and compared with the two-body universal functions (black lines). For each interaction, all calculations are scaled to have the same value at ~ 1 fm and show the same short-distance behavior for all nuclei. e-h, same as the left panel but for the two-nucleon momentum-space distribution ratios, \({n}_{p}{n}^{A}(q)/| {\tilde{\varphi }}_{p}{n}^{s = 1}(q){| }^{2}\) (e, f) and \({n}_{p}{p}^{A}(q)/| {\tilde{\varphi }}_{p}{p}^{s = 0}(q){| }^{2}\) (g, h) normalized to unity at q = 3.5 fm−1. Scaling is clearly observed at high momenta. The N2LO 1.0fm and 1.2fm distributions are only shown up to 4.4 and 3.8fm−1 respectively, above which statistics is poor and regulator/cutoff artifacts dominate. Source data

Extended Data Fig. 10 4He two-nucleon relative-momentum distribution ratios, \({n}_{N}N(q)/| {\tilde{\varphi }}_{N}N(q){| }^{2}\).

Spin-1 pn (a) and spin-0 pp (b) distributions. All curves are divided by A/2 = 2 and multiplied by 100. Results for five potentials are shown: AV18+UX, AV4’+UIXc, NV2+3-Ia*, N2LO R0 = 1.0fm, and N2LO R0 = 1.2fm. Horizontal lines with error bands correspond to the extracted contacts (see Methods and Extended Data Fig. 6). For N2LO potentials, results for one choice of three-body contact operators is shown here. Error bars show the monte-carlo statistical uncertainties of the QMC calculation at the 1s or 68% confidence level. Source data

Supplementary information

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Cruz-Torres, R., Lonardoni, D., Weiss, R. et al. Many-body factorization and position–momentum equivalence of nuclear short-range correlations. Nat. Phys. (2020).

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