1. Francesca Sammarruca University of Idaho Research supported in part by the US Department of Energy. Systematic Uncertainty Quantification of Nucleonic Matter Predictions CUSTIPEN Workshop on Advances in the Computations of Nuclear Structure and Nucleon-Nucleon Force August 1 – August 6, 2015 PKU

2. Collaborators: L. Coraggio (INFN, Naples) J.W. Holt (University of Washington) N. Itaco (INFN and University of Naples) R. Machleidt (University of Idaho) L.E. Marcucci (INFN, Pisa) 2

3. OUTLINE: Nuclear/neutron matter as a testing ground for many-body theories. Different approaches to predicting nuclear/neutron matter properties: advantages/disadvantages. Chiral EFT and error quantification. Chiral forces: Overview. Conclusions, outlook. Order-by-order nuclear/neutron matter calculations. 3

4. After more than 8 decades of nuclear physics, still a lot is unknown about the nuclear chart. Particularly true for systems with large isospin asymmetry 4

5. Experimental programs at RIBF (and the upcoming FRIB) will have widespread impact, filling some of the gaps in our incomplete knowledge of the nuclear chart. Studies of IANM (particularly the symmetry energy contribution to the EoS), are now especially timely, as they support rich on-going and future experimental effort. Isospin-Asymmetric Nuclear Matter (IANM) is closely related to neutron-rich nuclei and is a convenient theoretical laboratory. 5 The EoS of highly neutron-rich matter is crucial to understand wide-ranging questions in nuclear structure.

6. To derive the properties of nuclear systems from the basic nuclear interactions (AB INITIO) The goal of microscopic nuclear physics: 6

7. Meson-theoretic interactions have been used extensively in nuclear matter calculations (and still are). Problem: The 3NFs employed in these calculations have only a lose connection to the associated 2NFs. 7 How to select the contributions to be retained?

8. Relativistic approaches have focused on the Dirac-Brueckner-Hartree-Fock (DBHF) framework together with relativistic OBEPs, and are suitable to address a broad range of momenta/densities. DBHF: An efficient way to include a particular class of 3NF. (virtual nucleon-antinucleon excitation) 8

9. The philosophy of EFT and power counting: When correctly implemented, EFT provides a well-defined path to calculate observables whose truncation error should decrease systematically as higher orders are included. Problem shared by all non-EFT-based approaches: It is essentially impossible to estimate reliably the actual uncertainty associated with a particular prediction. 9

10. EFT: A framework in which the properties governed by low-energy physics are specified by the choice of degrees of freedom and symmetries, and can be computed systematically. 10 Short-range physics is included through the processes of regularization and renormalization Power counting: an organizational scheme to rank-order the various diagrams. Nuclear two- and few-body forces emerge on equal footing in a controlled hierarchy.

11. 11

12. Sources of uncertainty: 1) Many-body method (not inherent to EFT) 2) ERRORS IN THE LECs AND THEIR PROPAGATION IN THE HAMILTONIAN: 12 Short-range (NN) LECs Long-range (pi-N) LECs QUANTIFICATION OF THE UNCERTAINTY IN NUCLEONIC MATTER 3) Regulator dependence 4) Truncation error

13. 13 Nuclear matter has been approached with a broad spectrum of many-body methods: Coupled-cluster Many-body perturbation Monte Carlo: Variational MC Green’s function MC

14. 14 Uncertainty from the many-body method: We adopt infinite summation of particle-particle (pp) ladder diagrams. From recent coupled-cluster calculations extended beyond pp and hh ladders [Hagen et al., PRC89, 014319 (2014)], we conclude: Negligible effect in neutron matter About +/-1 MeV in symmetric nuclear matter We determine that the pp ladder approximation provides sufficient accuracy for our purposes.

15. Thus: For the NN LECs, we believe that the uncertainty arising from errors in the NN data has only negligible impact in the many-body system. 15 Findings from the Granada group (arXiv:1407.7784 (2014)): Applied 205 MC samples of smooth local potentials (all with chi-squared/datum of about 1) and found an uncertainty of +/- 15 keV for the triton B.E. In a standard BHF calculation of nuclear matter with local high-precision potentials from the Nijmegen group, we find an uncertainty of +/- 0.6 MeV in the E/A at normal density. (Consistent with a similar study of the Granada group using Skyrme forces.) Uncertainty from the NN LECs:

16. 16 Uncertainty in Pi-N LECs: They impact mostly high partial waves (those where no contacts are available). At NLO: D waves and higher At N3LO: F waves and higher Therefore, we expect this to cause only small uncertainty in nuclear matter. However, a systematic investigation with consideration of pi-N LECs uncertainty in both 2NF and (consistent) 3NF has not yet been done.

17. REGULARIZATION and CUTOFF DEPENDENCE: Chiral NN potentials are multiplied by a regulator function: which sets the “UV scale”. Potentials with different cutoffs constitute a family with the same long-range properties. Cutoff dependence is expected to decrease with increasing order. 17

18. 18 NLO Cutoff = 450-800 MeV

19. 19R. MachleidtChiral NFs NTSE2014, June 23-27, 2014 19 R. Machleidt Chiral NFs NTSE2014, June 23-27, 2014 19 NLO NNLO Cutoff = 450-800 MeV

20. 20R. MachleidtChiral NFs NTSE2014, June 23-27, 2014 20 R. Machleidt Chiral NFs NTSE2014, June 23-27, 2014 20 R. Machleidt Chiral NFs NTSE2014, June 23-27, 2014 20 NLO NNLO Cutoff = 450-800 MeV NLONNLO Cutoff = 450-600 MeV N3LO

21. 21 At the 2N level: Excellent order-by-order improvement. From NLO to N2LO: No new contacts are generated. At N2LO, subleading 2PE contributions allow for a better description of the intermediate-range attraction (they encode the important physics of correlated 2PE). Only at N3LO high-precision quality description of the NN data becomes possible (additional contacts).

22. 22 Moving on to nuclear matter: We proceed with the particle-particle ladder approximation (see above), with NN forces as described above, and 3NF…..

23. 3NFs make their first appearance at N2LO: The 3NF LECs are fitted to the binding energies of 3H and 3He and the Gamow-Teller matrix element for triton beta-decay. (Marcucci et al.). 23

24. Predictions at NLO (yellow), N2LO (red), N3LO (blue), varying in each case the cutoff in the regulator function applied to the chiral NN potential between 450 and 600 MeV. 24

25. Observations from the nuclear matter (SNM) calculations: Large spread at NLO and N2LO; Bands do not overlap. Thus, their width is not a suitable representation of the uncertainty in the respective orders. Cutoff dependence is reduced in our N3LO calculation. But, still large uncertainty. Missing contributions (3NF at N3LO) could be a source of uncertainty. 25 Better convergence pattern with the lower (450-500 MeV) cutoffs.

26. Same study, for neutron matter.. Yellow: NLO Red: N2LO Blue: N3LO 26

27. For neutron matter (NM): Similar observations apply as in SNM with regard to the bands not overlapping. Results are not well converged. Generally weaker cutoff dependence than in SNM. Overall: Although cutoff sensitivity decreases with increasing order, cutoff variations generally underestimate the uncertainty. 27

28. …and, similar study for the symmetry energy Yellow: NLO Red: N2LO Blue: N3LO 28

29. 29 Our goal is to lay the foundation for such calculations of nuclear many-body systems. Order-by-order convergence studies with chiral forces are very important. At this time, we conclude: At low orders in the chiral expansion, no indication of good convergence. Our N3LO calculations indicate a pattern of slow convergence for the lower values of the cutoff. Summary and Conclusions:

30. 30 OUTLOOK Program to make the analysis broader and more systematic: Start series of calculations at N4LO! Vary the pi-N LECs. For each set of c i, construct NN potentials and use them consistently in the 2NF and the 3NF. For each set, refit the c D and c E LECs to the A=3 system. N4LO is very promising…..

31. 31 In progress/planned: Predictions of neutron skins, order by order Mass/radius of compact stars

32. End 32