Chiral forces and quantum Monte Carlo: from nuclei to neutron star matter

Transcription

1 Chiral forces and quantum Monte Carlo: from nuclei to neutron star matter Diego Lonardoni FRIB Theory Fellow In collaboration with: J. Carlson, LANL S. Gandolfi, LANL X. Wang, Huzhou University, China A. Lovato, ANL S. C. Pieper, ANL R. B. Wiringa, ANL Transport 017, March 9, 017

2 Introduction Neutron Star (NS): compact objects made mostly by neutrons R 1 km M 1.4 M atmosphere: atomic and plasma physics outer crust: physics of superfluids (neutrons, vortex), solid state physics (nuclei) inner crust: deformed nuclei, pasta phase outer core: nuclear matter inner core: hyperons?? quark matter?? condensates (, K)??? curtesy of Dany Page

3 Introduction 3 The hydrostatic equilibrium of a static spherically symmetric star is given by the solution of Tolman-Oppenheimer-Volkoff equations (TOV): derivation of static properties of the stars like mass and radius E M M max TOV 1:1 0 b 1 km R Note: for a given EOS the solution of the TOV is unique!! unique M(R) relation!! if you know the EOS you can predict the maximum mass and compare with observations Problem: do we know the EOS?? not really but: EOS for NS: energy/pressure of a nuclear medium nuclear interactions!!

4 Introduction 4 nuclei neutron stars same underlying physics!! nuclear interactions R fm m M 10 7 kg look at experiments & observations R 10 km 10 4 m M 1.4 M kg Introduction Quantum Monte Carlo methods & Nuclear Hamiltonians Moving towards medium-mass nuclei AFDMC & chiral forces Preliminary results: binding energies & radii Preliminary results: single- & two-nucleon momentum distributions Future directions and conclusions

5 Quantum Monte Carlo methods 5 { { Goal: solve the many-body problem for correlated systems in a non-perturbative fashion VMC, GFMC: sampling in coordinate space reduced number of nucleons: A=1 w.f. with -body & 3-body correlations CVMC: sampling in coordinate space + cluster expansion closed shell nuclei (+/-1): A=40 real-space (but not only) AFDMC: sampling in coordinate & spin-isospin space large number of nucleons local forces (possibly) VMC CVMC GFMC AFDMC 1 GFMC: light systems A apple 1 40 AFDMC: light to mediumheavy nuclei A 50 A AFDMC: nuclear matter A = 1

6 Nuclear Hamiltonians 6 Model: non-relativistic nucleons interacting with an effective nucleon-nucleon (NN) force and 3-nucleon interaction (NNN) H = ~ m X i r i + X i<j v ij + X i<j<k v ijk v ij v ijk fitted on NN scattering data & deuteron fitted on properties of (light) nuclei (+constraints) Focus on two families of nuclear interactions: Real-space, local (mostly), phenomenological: Argonne (NN) + Urbana-Illinois (NNN) Momentum-space, non-local, chiral effective field theory: X-EFT (NN+NNN+ ) Note: local vs non-local ( p =(p1 p )/ ( q = p 0 p local r p 0 =(p 0 1 p 0 )/ k =(p 0 + p)/ non-local r r

7 Nuclear Hamiltonians: phenomenological potentials NN: Argonne AV18 (AV8 ) X X X 7 v ij = X p X O p ij vp (r ij ) O p=1,8 ij = {, i j,s ij, L ij S ij } {, i j } NNN: Urbana-Illinois π π π π π π π π π π + short-range term Pros: Argonne interactions fit phase shifts up to high energies. Accurate up to (at least) -3 saturation density. Suitable for QMC calculations. Very good description of several observables in light nuclei (GFMC ground-state: uncertainties within 1-%). Cons: Phenomenological interactions are phenomenological, not clear how to improve their quality. Theoretical uncertainties hard to quantify. 3-body forces?

8 Nuclear Hamiltonians: phenomenological potentials 8 Energy (MeV) He He Li 7/ 5/ 5/ 7/ 1/ 3/ 7 Li Argonne v 18 with UIX or Illinois-7 GFMC Calculations 1 June He + 8 Li Be / + 5/ + 7/ 7/ 3/ 3/ + 5/ + 1/ 5/ 1/ + 3/ Be B E (MeV) AV8' AV8' + UIX AV8' + IL7 PNM.45(1)M 1.74(1)M obs : M AV18 AV18 +UIX AV18 +IL7 Expt. 1 C ρ b (fm -3 ) Note: AV8 +UIX and (almost) AV8 are sti enough to support observed neutron stars, light nuclei but AV8 +IL7 too how soft. to! How to reconcile infinite with matter nuclei??? terrestrial experiments reconcile? P. Maris et al., Phys. Rev. C 87, (013) astrophysical observations (M, R) medium-mass region, neutron-rich system AFDMC Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 7 / other framework local chiral forces

9 Nuclear Hamiltonians: chiral EFT potentials 9 LO NN NNN 1 0 O Q b X-EFT is an expansion in powers of Q m 100 MeV b m 800 MeV soft scale hard scale Q/ b NLO 1 O Q b Long-range physics: given explicitly (no parameters to fit) by pion-exchanges 1 N 3 LO O Q b + Short-range physics: parametrized through contact interactions with low-energy constants (LECs) fit to low-energy data 1 N 3 4 LO O Q b + + Many-body forces enter systematically and are related via the same LECs

10 Nuclear Hamiltonians: chiral EFT potentials 10 Pros: Chiral interactions have a theoretical derivation and they can be systematically improved. They are typically softer than the phenomenological forces, making most of the calculations easier to converge. Many-body forces are naturally accounted for. Cons: Chiral interactions describe low-energy (momentum) physics. How do they work at large momenta? In the standard formulation they are non-local and they are written in momentum-space. Not suitable for AFDMC calculations. local chiral N LO potentials -body NN 3-body NNN A. Gezerlis et al., Phys. Rev. Lett. 111, (013) A. Gezerlis et al., Phys. Rev. C 90, (014) I. Tews et al., Phys. Rev. C 93, (016) J. E. Lynn et al., Phys. Rev. Lett. 116, (016) J. E. Lynn et al., Phys. Rev. Lett. 113, (014)

11 Nuclear Hamiltonians: local chiral EFT potentials 11 -body N LO pion exchanges up to N LO depend only on p, p 0, q contact terms: LO 7 NLO - N LO no momentum dependence depend on q, q k local regulators in real space for both long and short range physics e (r/r 0) 4 cutoff: R 0 = fm momentum cutoff MeV v ij = X p O p ij vp (r ij ) O p=1,7 ij = {, i j,s ij } {, i j } + L ij S ij { local non-local { included in both GFMC and AFDMC propagators

12 Nuclear Hamiltonians: local chiral EFT potentials 1 3-body N LO π π π c 1,c 3,c 4 c D c E same as NN need to be fit fit on: 4 He binding energy low-energy phase shifts n- scattering p-wave Note: same regulator functions and cutoff as -body NN Note: finite cutoff different possible operator structures: V D local!! D1, D V E E,E,E P,... appealing for QMC not only TABLE I. Fit values for the couplings c D and c E for different choices of 3N forces and cutoffs. V 3N R 0 (fm) c E c D N LO ðd1;eτþ N LO ðd;eτþ N LO ðd;e1þ N LO ðd;epþ J. E. Lynn et al., Phys. Rev. Lett. 116, (016) 06501

13 Nuclear Hamiltonian: local chiral EFT potentials (a) K. M. Nollett et al., Phys. Rev. Lett. 99, 050 (007) (b) 1+ NLO AV18 AV18+UIX AV18+IL R-Matrix (a) NLO (D, E ) R0 = 1. fm NLO (D, E 1) NLO (D, EP) ce 1 NLO (D1, E ) R0 = 1.0 fm R0 = 1.0 fm ce 3 (deg.) δjl (degrees) NLO (D, E ) R0 (D, = 1.EP) fm NLO R0 = 1.0 fm R matrix (a) ce 150 R0 = 1.0 fm 0 R = 1. fm cd 1 R0 = 1.0 fm NLO (D1, E ) NLO (D, E ) NLO (D, E 1) Ec.m. (MeV) phenomen Ecm 1.0(MeV) 1.0 R0 = 31.0 fm cd 4 NLON(D, EP)E ) LO (D1, N3.0 LO (D, E 4.0).0 chiral NLO (D, E 1) 18 AV8' 16 AV8' + UIX AV8' + IL cd.0 1 E/A (MeV) E (MeV) NLO (D, E 1) 1.5 NLO (D, EP) NLO (D, E ) N LO (D, EP) ρb (fm ) P. Maris et al., Phys. Rev. C 87, (013) n (fm 3) J. E. Lynn et al., Phys. Rev. Lett. 116, (016)

14 AFDMC & local chiral EFT potentials 14 Status: -body and 3-body local chiral N LO potentials have been implemented in GFMC but so far they have been tested on s-shell nuclei only (A=3,4) and neutron matter -body potentials have been implemented in AFDMC: need to be tested in nuclei 3-body potentials have been implemented in AFDMC: need to be tested Problem: commutators from TPE in NNN cannot be included in the AFDMC propagator Idea: use an approximate propagator for NNN 1. enhance the other NNN components in the propagator to compensate for the missing operators. minimize the expectation value of the difference between enhanced and real NNN force: hv diff 3b i Note: similar idea used in GFMC for AV18: propagation done with AV8, where is chosen so as to minimize the difference <AV18 - AV8 >

15 AFDMC & local chiral EFT potentials: A = H( 1 +, 1 ) E b = 8.48 MeV q hrch i =1.759(36) fm Method cut-o b b+3b (D,E ) GFMC AFDMC R 0 (fm) E b (MeV) q r ch (fm) Eb (MeV) q r ch (fm) (1) 1.79() 8.34(1) 1.74(3) (1) 1.76() 8.35(4) 1.74(4) (4) 1.76() 8.35(7) 1.73() (3) 1.75() 8.7(6) 1.73() 3 He ( 1 +, 1 ) E b = MeV q hrch i =1.966(3) fm Method cut-o b b+3b (D,E ) GFMC AFDMC R 0 (fm) E b (MeV) q r ch (fm) Eb (MeV) q r ch (fm) (1).07() 7.65() 1.98() (1).0(1) 7.63(4) 1.98(1) (5).0() 7.55(8) 1.97() (3) 1.99(1) 7.60(6) 1.90() GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (014) & Phys. Rev. Lett. 116, (016)

16 AFDMC & local chiral EFT potentials: A = He (0 +, 0) E b = 8.96 MeV q hrch i =1.676(3) fm Method cut-o b b+3b (D,E ) GFMC AFDMC R 0 (fm) E b (MeV) q r ch (fm) Eb (MeV) q r ch (fm) (1) 1.74(1) 8.30(1) 1.66() (1) 1.69(1) 8.30(1) 1.66(1) (6) 1.73(1) 7.97(1) 1.69(1) (6) 1.70(1) 8.33(10) 1.67(1) GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (014) & Phys. Rev. Lett. 116, (016) good agreement AFDMC-GFMC both at -body and 3-body level Note: same coefficients in the NNN propagator for A = 3, 4 iapple1% compared to the total binding energy hv diff 3b variations of the coefficients do no affect the final result AFDMC open-shell systems & larger systems

17 AFDMC & local chiral EFT potentials: A = He (0 +, 1) E b = 9.71 MeV q hrch i =.066(11) fm Method cut-o b b+3b (D,E ) AFDMC R 0 (fm) E b (MeV) q r ch (fm) Eb (MeV) q r ch (fm) 1.0.1(5).11() 8.0(5).06() 1. 4.().07() 9.4(8) 1.98() 6 Li (1 +, 0) E b = MeV q hrch i =.589(34) fm Method cut-o b b+3b (D,E ) AFDMC R 0 (fm) E b (MeV) q r ch (fm) Eb (MeV) q r ch (fm) (5).6(3) 9.6(8).43(3) (5).44(5) 31.0(8).6(3) Note: same coefficients in the NNN propagator for A = 4: w.f. built with s-p-d single particle states hv diff 3b i < 3%

18 AFDMC & local chiral EFT potentials: A = O(0 +, 0) E b = MeV q hrch i =.699(5) fm Method cut-o b b+3b (D,E ) AFDMC R 0 (fm) E b (MeV) q r ch (fm) Eb (MeV) q r ch (fm) (3).76(3) 97(6).78(5) (5).48() 150(5).18(5) 40 Ca (0 +, 0) E b = MeV q hrch i =3.478(1) fm Method cut-o b b+3b (D,E ) AFDMC R 0 (fm) E b (MeV) q r ch (fm) Eb (MeV) q r ch (fm) (10) (10) Note: need to change the coefficients in the propagation still possible to keep hv diff 3b i small Question: how is this interaction working?

19 AFDMC & local chiral EFT potentials: n(k) 1 single-nucleon momentum distribution 10 0 VMC: AV18+UX n p (k)/z (fm 3 ) CVMC: AV18+UIX AFDMC: AV6' AFDMC: AV7' AFDMC: N LO R 0 =1.0 fm AFDMC: N LO R 0 =1. fm He k (fm -1 ) CVMC: D.L., A. Lovato, S. C. Pieper, R. B. Wiringa, in preparation; AFDMC: S. Gandolfi, J. Carlson, D.L., X. Wang, in preparation

20 AFDMC & local chiral EFT potentials: n(k) single-nucleon momentum distribution 1.0 n p (k) integrated strenght % 9.8% 4 He AV6' 0. N LO R 0 =1. fm k (fm -1 )

21 AFDMC & local chiral EFT potentials: n(k) 4 single-nucleon momentum distribution n p (k)/z (fm 3 ) CVMC: AV18 AFDMC: AV6' AFDMC: N LO R 0 =1.0 fm AFDMC: N LO R 0 =1. fm O k (fm -1 ) CVMC: D.L., A. Lovato, S. C. Pieper, R. B. Wiringa, in preparation; AFDMC: S. Gandolfi, J. Carlson, D.L., X. Wang, in preparation

22 AFDMC & local chiral EFT potentials: n(k) 5 single-nucleon momentum distribution 1.0 n p (k) integrated strenght % 94.5% 16 O AV6' 0. N LO R 0 =1.0 fm k (fm -1 )

23 AFDMC & local chiral EFT potentials: n(k) 6 single-nucleon momentum distribution He: AV6' 16 O: AV6' 4 He: N LO R0 =1.0 fm 16 O: N LO R0 =1.0 fm n p (k)/z (fm 3 ) k (fm -1 ) S. Gandolfi, J. Carlson, D.L., X. Wang, in preparation

24 AFDMC & local chiral EFT potentials: n (q) 1 9 two-nucleon momentum distribution (integrated Q) 10 n 1 (q) (fm -1 ) np, AV18 (VMC) pp, AV18 (VMC) np, AV6' pp, AV6' np, N LO R 0 =1. fm pp, N LO R 0 =1. fm He 5.15(3) MeV/nuc 5.30(6) MeV/nuc q (fm -1 ) VMC: R. B. Wiringa et al., Phys. Rev. C 89, (014); AFDMC: S. Gandolfi, J. Carlson, D.L., X. Wang, in preparation

25 AFDMC & local chiral EFT potentials: n (q) 1 30 two-nucleon momentum distribution (integrated Q) 1.0 n 1 (q) integrated strenght % 95.4% 4 He np, AV6' (3) MeV/nuc 5.30(6) MeV/nuc np, N LO R 0 =1. fm q (fm -1 )

26 AFDMC & local chiral EFT potentials: n (q) 1 31 two-nucleon momentum distribution (integrated Q) 1.0 n 1 (q) integrated strenght % 97.0% 4 He pp, AV6' (3) MeV/nuc pp, N LO R 0 =1. fm 5.30(6) MeV/nuc q (fm -1 )

27 AFDMC & local chiral EFT potentials: n (q) 1 33 two-nucleon momentum distribution (integrated Q) 10 3 n 1 (q) (fm -1 ) np, AV6' pp, AV6' np, N LO R 0 =1.0 fm pp, N LO R 0 =1.0 fm O 5.15(3) MeV/nuc 5.30(6) MeV/nuc q (fm -1 ) S. Gandolfi, J. Carlson, D.L., X. Wang, in preparation

28 AFDMC & local chiral EFT potentials: n (q) 1 34 two-nucleon momentum distribution (integrated Q) 1.0 n 1 (q) integrated strenght % 97.% 16 O np, AV6' (3) MeV/nuc 5.30(6) MeV/nuc np, N LO R 0 =1.0 fm q (fm -1 )

29 AFDMC & local chiral EFT potentials: n (q) 1 35 two-nucleon momentum distribution (integrated Q) 1.0 n 1 (q) integrated strenght % 97.4% 16 O pp, AV6' (3) MeV/nuc 5.30(6) MeV/nuc pp, N LO R 0 =1.0 fm q (fm -1 )

30 SRC Pair Fraction (%) was 11%. Taking this into account, the corrected experimental ratio for 1 C(e,e'pn)/ 1 C(e,e'pp) was 9.0 ±.5. To deduce the ratio of p-n to p-p SRC pairs in the ground state of 1 C, we used the measured 1 C(e,e'pn)/ 1 1 C(e,e'pp) ratio. Because we used (e,e'p) events to search for SRC nucleon pairs, the probability of detecting p-p pairs was twice that of p-n pairs; thus, we conclude that the ratio of p-n/p-p pairs in the 1 Cgroundstateis18±5 (Fig. ). To get a comprehensive picture of the structure of 1 C, we combined the pair faction results with the inclusive 1 C(e,e') measurements (4, 5, 14) andfoundthatapproximately0%of the nucleons in 1 CformSRCpairs,consistent In the 1980s and several femtometers, known as long-range correlations, has been established (3), but these accounted scattered electrons at a central scattering events for 1 C(e,e'p). The spectrometers, left HRS detected we read out the BigBite and neutron-detector electronics; thus, we could determine the 1 angle riments using elecf several C(e,e'pp)/ 1 C(e,e'p) and the 1 C(e,e'pn)/ AFDMC hundred for less than half of & the predicted local correlated nucleon pairs. Recent high-momentum transfer meaues correspond to the quasi-free chiral (momentum) of 19.5 EFT ( GeV/c). These C(e,e'p) ratios. potentials: val- For the 1 C(e,e'pp)/ 1 C(e,e'p) n (q) ratio, we knockout found that of 9.5 a ± % of the (e,e'p) events OH 444, USA. Tel surements (4 1) haveshownthat nucleonsin single proton with transferred had anthree-momentum associated recoiling proton, as reported in el. 3 Massachusetts Insti- nuclear ground states can form pairs with large q =1.65GeV/c, transferredenergyw (1). Taking into =0.865 account the finite acceptance of 0139, USA. 4 Califoros Angeles, CA 9003, relative momentum and small center-of-mass GeV, Q = q (w/c) =(GeV/c) the neutron (where detector Q [using is the same procedure (CM) momentum due to the short-range (scalar the four-momentum, squared), as with the and proton Bjorken G1 8QQ, Scotland, UK. detector (1)] and the neutron onne, IL 60439, USA. and tensor) components of the nucleon-nucleon scaling parameter x B = Q detection /mw =1.,wherem efficency, we found that 96 ± % of 1344, USA. 8 Prince interaction. These pairs are referred to as shortrange correlated (SRC) pairs. The study of these knocked-out protons at three 300 different MeV/c had values a recoiling for neutron. This result is the mass of the proton. The the (e,e'p) right HRS eventsdetected with a missing momentum above 95, Saudi Arabia. 9 Uni- 074, USA. 10 Florida SRC pairs allows access to cold dense nuclear the central angle (momentum): agrees 40.1 with (1.45 a GeV/c), hadron beam measurement of 33199, USA. 11 Thomas matter, such as that found in a neutron star (1.4 GeV/c), and(p,pn)/(p,p), 3.0 (1.36 in GeV/c). which 9 ± 18% of the (p,p) lity, Newport News, VA versity, Seoul , events with a missing momentum above the Fermi cleare (INFN), Sezione di ity of Virginia, Charlottes- Fig. 1. Illustration of the 1 C(e,e'pN) State University of New reaction. The incident electron beam. 16 Kharkov Institute of 10 couples to a nucleon-nucleon pair via 108, Ukraine. 17 College avirtualphoton.inthefinalstate, VA 3187, USA. 18 Old 08, USA. 19 University of the scattered electron is detected wan, S7N 5E Canada. along with the two nucleons that aire Fysica, Amsterdam, are ejected from the nucleus. Typical nuclear density is about pp/np from [ C(e,e pp) / C(e,e pn) ] / usetts Amherst, Amherst, 1 1 entucky, Lexington, KY nucleons/fm 3,whereasforpairsthepp/N from [ C(e,e pp) / C(e,e p) ] / and INFN sezione Bari, local density is approximately five 1 1 tomiqueetdecosmologie, np/n from C(e,e pn) / C(e,e p) te Jožef Stefan, 1000 times larger. 1 1 np/n from C(p,pn) / C(p,p) University, Norfolk, VA Glasgow G1 8QQ, Scotelphia, PA 191, USA , Armenia. 30 Deubljana, 1000 Ljubljana, na Champaign, Urbana, be addressed. Downloaded from two-nucleon momentum distribution (q & Q): on the way!! short-range interaction reported here suggests that momentum distribution for the protons and neutrons in neutron stars will be substantially different from that characteristic of an ideal Fermi gas. A theoretical calculation that takes into account the p-n correlation effect at relevant neutron star densities and realistic proton concentration shows the correlation effect on the momentum distribution of the protons and the neutrons (3). We therefore speculate that the small concentration of protons inside neutron stars might have a disproportionately large effect that needs to be addressed in realistic descriptions of neutron stars. References and Notes 1. L. Lapikas, Nucl. Phys. A. 553, 97 (1993).. J. Kelly, Adv. Nucl. Phys. 3, 75 (1996). 3. W. H. Dickhoff, C. Barbieri, Prog. Part. Nucl. Phys. 5, 377 (004). 4. K. S. Egiyan et al., Phys. Rev. C Nucl. Phys. 68, (003). on Downloaded from JUNE 008 VOL 30 SCIENCE R. Subedi et al., Science 30, 1476 (008) Missing Momentum [GeV/c] Fig.. The fractions of correlated pair combinations in carbon as obtained from the (e,e'pp) and (e,e'pn) reactions, as well as from previous (p,pn) data. The results and references are listed in table S1. Fig. 3. The average fraction of nucleons in the various initial-state configurations of 1 C. short-range correlated pairs induced by the nuclear force (tensor force) ρ pn (q,q=0) (fm 3 ) 1 C B 5 np 8 Be pp 6 Li He q (fm -1 ) SCIENCE VOL JUNE behavior expected to be universal across a wide range of nuclei true?? model dependent?? R. B. Wiringa et al., Phys. Rev. C 89, (014)

31 AFDMC & local chiral EFT potentials: n (q) 1 37 two-nucleon momentum distribution (q & Q): on the way!! 10 References and Notes 1. L. Lapikas, Nucl. Phys. A. 553, 97 (1993).. J. Kelly, Adv. Nucl. Phys. 3, 75 (1996). 3. W. H. Dickhoff, C. Barbieri, Prog. Part. Nucl. Phys. 5, 377 (004). 4. K. S. Egiyan et al., Phys. Rev. C Nucl. Phys. 68, (003). SRC Pair Fraction (%) pp/np from [ C(e,e pp) / C(e,e pn) ] / 1 1 pp/n from [ C(e,e pp) / C(e,e p) ] / 1 1 np/n from C(e,e pn) / C(e,e p) 1 1 np/n from C(p,pn) / C(p,p) R. Subedi et al., Science 30, 1476 (008) Missing Momentum [GeV/c] Fig.. The fractions of correlated pair combinations in carbon as obtained from the (e,e'pp) and (e,e'pn) reactions, as well as from previous (p,pn) data. The results and references are listed in table S1. Fig. 3. The average fraction of nucleons in the various initial-state configurations of 1 C. neutron stars: 5-10% protons realistic calculations of NS need to take into account these correlation effect employed interaction (and method)

32 Future directions 38 progresses in AFDMC calculations moving towards an ab-initio description of the medium region of the nuclear chart neutron-rich nuclei better understanding of the 3n force: fundamental for NS description A O Estimated by theory Produced at FRIB A Ca moving towards the neutron drip line: appearance of new magic number & last stable isotopes nuclear correlations A. B. Balantekin et al., Mod. Phys. Lett. A 9, (014) neutron skin JLab ( 48 Ca) constraints for the symmetry energy (and its slope) NS observables: the mass-radius relationship (cooling rates, the thickness of the crust) constraints on NNN force

33 Conclusions 39 Substantial progresses in AFDMC calculations inclusion of b & 3b local N LO chiral forces good agreement with GFMC for A = 3, 4 first QMC calculations for A = 6, 16, 40 (binding energies & radii) single- and two-nucleon momentum distributions for different forces Develop the technology to study the medium-mass region of the nuclear chart in a non perturbative fashion Access the physics of neutron-rich systems: better understanding of nuclear forces & the connection to the physics of neutron stars Thank you!!