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

## Abstract

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.

## Access options

from\$8.99

All prices are NET prices.

## 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

## References

1. 1.

Hen, O., Miller, G. A., Piasetzky, E. & Weinstein, L. B. Nucleon-nucleon correlations, short-lived excitations, and the quarks within. Rev. Mod. Phys. 89, 045002 (2017).

2. 2.

Weiss, R., Bazak, B. & Barnea, N. Generalized nuclear contacts and momentum distributions. Phys. Rev. C 92, 054311 (2015).

3. 3.

Weiss, R., Cruz-Torres, R., Barnea, N., Piasetzky, E. & Hen, O. The nuclear contacts and short range correlations in nuclei. Phys. Lett. B 780, 211–215 (2018).

4. 4.

Wiringa, R. B., Schiavilla, R., Pieper, S. C. & Carlson, J. Nucleon and nucleon-pair momentum distributions in A ≤ 12 nuclei. Phys. Rev. C 89, 024305 (2014).

5. 5.

Lonardoni, D., Lovato, A., Pieper, S. C. & Wiringa., R. B. Variational calculation of the ground state of closed-shell nuclei up to A = 40. Phys. Rev. C 96, 024326 (2017).

6. 6.

Lonardoni, D. et al. Properties of nuclei up to A = 16 using local chiral interactions. Phys. Rev. Lett. 120, 122502 (2018).

7. 7.

Subedi, R. et al. Probing cold dense nuclear matter. Science 320, 1476–1478 (2008).

8. 8.

Korover, I. et al. Probing the repulsive core of the nucleon-nucleon interaction via the 4He(e,epN) triple-coincidence reaction. Phys. Rev. Lett. 113, 022501 (2014).

9. 9.

Hen, O. et al. Momentum sharing in imbalanced Fermi systems. Science 346, 614–617 (2014).

10. 10.

Duer, M. et al. Probing high-momentum protons and neutrons in neutron-rich nuclei. Nature 560, 617–621 (2018).

11. 11.

Schmidt, A. et al. Probing the core of the strong nuclear interaction. Nature 578, 540–544 (2020).

12. 12.

Korover, I. et al. Tensor-to-scalar transition in the nucleon-nucleon interaction mapped by 12 C(e,e′pn) measurements. Preprint at https://arxiv.org/abs/2004.07304 (2020).

13. 13.

Schmookler, B. et al. Modified structure of protons and neutrons in correlated pairs. Nature 566, 354–358 (2019).

14. 14.

Segarra, E. P. et al. Neutron valence structure from nuclear deep inelastic scattering. Phys. Rev. Lett. 124, 092002 (2020).

15. 15.

Simkovic, F., Faessler, A., Muther, H., Rodin, V. & Stauf, M. The 0νββ -decay nuclear matrix elements with self-consistent short-range correlations. Phys. Rev. C 79, 055501 (2009).

16. 16.

Cirigliano, V. et al. New leading contribution to neutrinoless double-β decay. Phys. Rev. Lett. 120, 202001 (2018).

17. 17.

Miller, G. A. et al. Can long-range nuclear properties be influenced by short range interactions? A chiral dynamics estimate. Phys. Lett. B 793, 360–364 (2019).

18. 18.

Li, B. A., Cai, B.-J., Chen, L. W. & Xu, J. Nucleon effective masses in neutron-rich matter. Prog. Part. Nucl. Phys. 99, 29–119 (2018).

19. 19.

Lonardoni, D., Gandolfi, S., Wang, X. B. & Carlson, J. Single- and two-nucleon momentum distributions for local chiral interactions. Phys. Rev. C 98, 014322 (2018).

20. 20.

Lonardoni, D. et al. Auxiliary field diffusion Monte Carlo calculations of light and medium-mass nuclei with local chiral interactions. Phys. Rev. C 97, 044318 (2018).

21. 21.

Lynn, J. et al. Ab initio short-range-correlation scaling factors from light to medium-mass nuclei. J. Phys. G 47, 045109 (2020).

22. 22.

Cohen, E. O. et al. Center of mass motion of short-range correlated nucleon pairs studied via the A(e,e'pp) reaction. Phys. Rev. Lett. 121, 092501 (2018).

23. 23.

Wiringa, R. B., Stoks, V. G. J. & Schiavilla, R. Accurate nucleon-nucleon potential with charge-independence breaking. Phys. Rev. C 51, 38–51 (1995).

24. 24.

Chen, J. W., Detmold, W., Lynn, J. E. & Schwenk, A. Short range correlations and the EMC effect in effective field theory. Phys. Rev. Lett. 119, 262502 (2017).

25. 25.

Egiyan, K. S. et al. Observation of nuclear scaling in the A(e,e') reaction at xB ≥ 1. Phys. Rev. C 68, 014313 (2003).

26. 26.

Weiss, R. et al. Study of inclusive electron scattering scaling using the generalized contact formalism. Preprint at https://arxiv.org/abs/2005.01621 (2020).

27. 27.

Weiss, R., Korover, I., Piasetzky, E., Hen, O. & Barnea, N. Energy and momentum dependence of nuclear short-range correlations—spectral function, exclusive scattering experiments and the contact formalism. Phys. Lett. B 791, 242–248 (2019).

28. 28.

Pybus, J. R. et al. Generalized contact formalism analysis of the 4He(e,e'pN) reaction. Phys. Lett. B 805, 135429 (2020).

29. 29.

Arrington, J. & Fomin, N. Searching for flavor dependence in nuclear quark behavior. Phys. Rev. Lett. 123, 042501 (2019).

30. 30.

Hen, O. et al. Comment on ‘Searching for flavor dependence in nuclear quark behavior’. Preprint at https://arxiv.org/abs/1905.02172 (2019).

31. 31.

Wiringa, R. B. & Pieper, S. C. Evolution of nuclear spectra with nuclear forces. Phys. Rev. Lett. 89, 182501 (2002).

32. 32.

Gezerlis, A. et al. Local chiral effective field theory interactions and quantum Monte Carlo applications. Phys. Rev. C 90, 054323 (2014).

33. 33.

Lynn, J. E. et al. Chiral three-nucleon interactions in light nuclei, neutron-α scattering, and neutron matter. Phys. Rev. Lett. 116, 062501 (2016).

34. 34.

Piarulli, M. et al. Local chiral potentials with Δ-intermediate states and the structure of light nuclei. Phys. Rev. C 94, 054007 (2016).

35. 35.

Piarulli, M. et al. Light-nuclei spectra from chiral dynamics. Phys. Rev. Lett. 120, 052503 (2018).

36. 36.

Baroni, A. et al. Local chiral interactions, the tritium Gamow-Teller matrix element, and the three-nucleon contact term. Phys. Rev. C 98, 044003 (2018).

37. 37.

Cruz-Torres, R. et al. Short range correlations and the isospin dependence of nuclear correlation functions. Phys. Lett. B 785, 304–308 (2018).

## Acknowledgements

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

Authors

### Contributions

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.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## 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

### Supplementary Information

Supplementary analysis information.

## Source data

### Source Data Fig. 1

Source data for reproducing Fig. 1.

### Source Data Fig. 2

Source data for reproducing Fig. 2.

### Source Data Fig. 3

Source data for reproducing Fig. 3.

### Source Data Fig. 4

Source data for reproducing Fig. 4.

### Source Data Fig. 5

Source data for reproducing Fig. 5.

### Source Data Extended Data Fig. 1

Source data for reproducing Extended Data Fig. 1.

### Source Data Extended Data Fig. 2

Source data for reproducing Extended Data Fig. 2.

### Source Data Extended Data Fig. 4

Source data for reproducing Extended Data Fig. 4.

### Source Data Extended Data Fig. 5

Source data for reproducing Extended Data Fig. 5.

### Source Data Extended Data Fig. 7

Source data for reproducing Extended Data Fig. 7.

### Source Data Extended Data Fig. 8

Source data for reproducing Extended Data Fig. 8.

### Source Data Extended Data Fig. 9

Source data for reproducing Extended Data Fig. 9.

### Source Data Extended Data Fig. 10

Source data for reproducing Extended Data Fig. 10.

## Rights and permissions

Reprints and Permissions

Cruz-Torres, R., Lonardoni, D., Weiss, R. et al. Many-body factorization and position–momentum equivalence of nuclear short-range correlations. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-01053-7

• Accepted:

• Published:

• ### Spin Susceptibility in Neutron Matter from Quantum Monte Carlo Calculations

• Luca Riz
• , Francesco Pederiva
• , Diego Lonardoni
•  & Stefano Gandolfi

Particles (2020)

• ### Contact between nucleons

• Michael Urban

Nature Physics (2020)