AB INITIO STRUCTURE AND REACTIONS OF LIGHT NUCLEI WITH EFFECTIVE NUCLEAR INTERACTIONS Guillaume Hupin First D. Gogny conference, CEA/DIF, December 11 th Collaborators : S. Quaglioni (LLNL) P. Navrátil (TRIUMF) R. Roth (TU Darmstadt) J. Langhammer (Private sector) C. Romero-Redondo (LLNL) F. Raimondi (U. of Surrey) J. Dohet-Eraly (TRIUMF) A. Calci (TRIUMF)

SANDBOX: WEALTH OF EXOTIC PROPERTIES IN THE FEW- NUCLEON SYSTEMS 2 What is the nature of the nuclear force? Stable Nuclei exhibit: Bound states Resonant states Scattering states Halo densities Cluster states …

Fusion based energy generation Nuclear astrophysics 3 Materials science FROM AB INITIO THEORY TO PRACTICAL APPLICATIONS

4 d + 3 H → 4 He + n (14 MeV) But also: d + d → 3 He + n (2.45 MeV) 3 H + 3 H → 4 He + n + n (≤9.4 MeV) Neutron down-scattering Fuel: d(n,n)d, 3 H(n,n) 3 H, 3 He(n,n) 3 He, 4 He(n,n) 4 He, 7 Be(n,n) 7 Be, 9 Be(n,n) 9 Be Ablator: 12 C(n,n) 12 C, 14 N(n,n) 14 N, 16 O(n,n) 16 O Energy Production Prompt Diagnostics Magnetic Inertial confinement A fundamental theory is needed to enhance predictive capability Interactions with environment Neutron ToF detector n LIGHT-ION REACTIONS COME INTO PLAY IN EARTH-BASED FUSION FACILITIES

CHIRAL EFFECTIVE FIELD THEORY, A TOP NOTCH DERIVATION OF NUCLEAR EFFECTIVE INTERACTION 5 Importance Nucleons are composite objects. The nuclear interaction is complex by nature. Chiral effective field theory proposes a consistent framework to organize the interaction between nucleons. Constrained to provide an accurate description of the A=2 and A=3 systems. Predictions for nuclear structure and dynamic (A>3). Yet uncompleted…

6 In configuration interaction methods we need to soften interaction to address the hard core. We use the Similarity- Renormalization-Group (SRG) method Bare potential Evolved potential Evolution with flow parameter Preserves the physics Decouples high and low momentum Induces many-body forces Unitary transformation k’(fm -1 ) k(fm -1 ) k’(fm -1 ) k(fm -1 ) Flow parameter FROM A QCD-BASED EFFECTIVE INTERACTION TO THE TYPICAL NUCLEAR SCALE (EFFECTIVE) INTERACTION E. D. Jurgenson, Navrátil, R. J. Furnstahl PRL103 (2009); PRC83 (2011)

n- 4 He SCATTERING: NN VERSUS 3N INTERACTIONS G. Hupin, J. Langhammer et al. PRC88 (2013); G. Hupin, S. Quaglioni and P. Navrátil, to be published in Physica ScriptaSpecial Edition - Nobel Prize '75 anniversary 7 The 3N interactions influence mostly the P waves. The largest splitting between P waves is obtained with NN+3N. n- 4 He scattering Three scenarios of nuclear Hamiltonians More spin-orbit splitting 3N vs NN “bare” (NN+3N-ind) Comparison between NN+3N -ind and NN+3N at N max =13 with six 4 He states and 14 5 He states. NN+3N NN+3N-induced

8 Experimental low-lying states of the A=5 nucleon systems. The convergence pattern looks good. The experimental phase-shifts are well reproduced. AB INITIO n- 4 He SCATTERING G. Hupin, J. Langhammer et al. PRC88 (2013); G. Hupin, S. Quaglioni and P. Navrátil, to be published in Physica ScriptaSpecial Edition - Nobel Prize '75 anniversary Study of the convergence with respect to the # of 4 He low-lying states n- 4 He scattering phase-shifts for NN+3N potential with =2.0 fm -1 and first 14 th low-lying state of 5 He.

9 Differential cross-section at E neutron =1.79 MeV between NN+3N-ind and NN+3N. Comparison of the elastic cross-section between NN and NN+3N with 4 He (g.s.) n- 4 He elastic cross-section for NN+3N-induced, NN+3N potentials compared to expt. and ENDF evaluation. (NN+3N-induced) We obtained a better agreement with data when using NN+3N. The 3N force is constitutive to the reproduction of the 3 / 2 + resonance. n- 4 He ELASTIC CROSS-SECTIONS G. Hupin, S. Quaglioni and P. Navrátil, to be published as a contribution to the Special Physica Scripta Edition - 40 year anniversary - Nobel Prize '75 NN+3N NN+3N-induced

10 p- 4 He differential cross-section compared to experiment p- 4 He differential cross-section for NN+3N. Impurities, e.g. 4 He PREDICTION FOR RUTHERFORD BACKSCATTERING (RBS) G. Hupin, S. Quaglioni and P. Navrátil, PRC90 (2014) For the non-destructive physical, electrical and chemical characterization of materials, nuclear physics is routinely used for energies above the Rutherford scattering. p- 4 He scattering

11 Cross-section compared to experiments focused on proton recoil analysis p- 4 He differential cross-section for NN+3N at 4 He incident energies close to the Rutherford scattering. The widths predicted by our model (the only ingredient being the interaction) is too large compared to experiment. We can however discriminate between experimental data and provide accurate predictions for all angles at small and high energies. PREDICTION FOR ELASTIC RECOIL DETECTION (ERD) G. Hupin, S. Quaglioni and P. Navrátil, PRC90 (2014)

12 NN+3N-full NN+3N-ind Negative parity phase-shifts n- 8 Be ELASTIC COLLISION AND LOW-LYING SPECTRUM, NEGATIVE PARITY J. Langhammer, P. Navrátil et al., PRC91 (2015) n- 8 Be scattering Only a consistent treatment of bound and scattering states reveals the complete influence of the chiral NN+3N force model on the low-lying spectrum of 9 Be. 9 Be vs. 9 Be+n- 8 Be(0 +,2 + ) with chiral NN+3N(400) and =2.0 fm -1

13 THE COMBINATION OF THE TWO METHODS IS ESSENTIAL J. Langhammer, P. Navrátil et al., PRC91 (2015) 9 Be vs. 9 Be+n- 8 Be(0 +,2 + ) with chiral NN+3N(400) and =2.0 fm -1 Positive parity phase-shifts

14 Best results in a decent model space (N max =11). The 3 D 3 resonance is reproduced but the 3 D 2 and 3 D 1 resonance positions are underestimated. The 3N force corrects the D-wave resonance positions by increasing the spin-orbit splitting. There is room for improvements. 4 He(d,d) 4 He COMPARISON OF INTERACTION G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015) Comparison of the d-  phase-shifts with different interactions (N max =11) d- 4 He(g.s.) scattering phase-shifts for NN-only, NN+3N-induced, NN+3N-full potential with =2.0 fm -1. NN+3N NN+3N-induced NN-only d- 4 He scattering

15 6 Li SPECTRUM G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015) Comparison between NCSMC vs NCSM No-Core Shell Model N max NCSM/RGM Cluster formalism

16 The bulk of the cross section is well reproduced for a wide range of excitation energy and scattering angle. 4 He(d, 4 He)d CROSS-SECTION G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015) 4 He(d,d) 4 He angular distribution for various E d Comparison to experiment of the d- 4 He elastic angular distribution of NCSMC with NN+3N potential at =2.0 fm -1.

17 4 He(d, 4 He)d differential cross section at φ=30⁰ Comparison to experiment of the d- 4 He elastic recoil differential cross section of NCSMC with NN+3N potential at =2.0 fm -1. The 3 + resonance is missed. As its width is very narrow, it has little impact and the bulk of the cross- section. 4 He(d, 4 He)d CROSS-SECTION G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015) Comparison between potentials

18 3 H(d,n) 4 He astrophysical S-factor NCSM/RGM results for the 3 He(d,n) 4 He astrophysical S-factor compared to beam- target measurements. Calculated S-factors converge with the inclusion of the virtual breakup of the deuterium, obtained by means of excited 3 S D 1 (d* ) and 3 D 2 (d’* ) pseudo-states. 3 He(d,p) 4 He astrophysical S-factor e - lab screening Complete picture: includes break-up Incomplete nuclear interaction: requires 3N force (SRG-induced + “real”) Evidence of incomplete model (nuclear force) Pseudo excited states FIRST STEPS TOWARDS AB INITIO CALCULATIONS OF FUSION P. Navrátil, S. Quaglioni, PRL108 (2012)

19 4 He fusion with NCSMC and the chiral two- and three-nucleon force (preliminary) n+ 4 He(g.s.) phase shifts with NN+3N potential, =2.0 fm -1, with eigenstates of 5 He. Perspective to provide accurate 2 H(t,n) 4 He fusion cross-section for the effort toward earth-based fusion energy generation. The d-t fusion is known to be very sensitive to the spin-orbit and isospin part of the nuclear interaction. d-t fusion 4 He fusion TOWARDS AB INITIO CALCULATIONS OF 2 H(t,n) 4 He FUSION G. Hupin, S. Quaglioni, P. Navrátil work in progress

20 2 H-t fusion with NCSMC: comparison between effective interactions n+ 4 He(g.s.) phase shifts with NN+3Nand NN+3N-ind potential, with eigenstates of 5 He. Strong coupling between the 2 H(t,n) 4 He transfer and open n- 4 He in the NN (NN+3N-induced) case. The 2 H-t threshold is not correctly reproduced without initial chiral 3N force. TOWARDS AB INITIO CALCULATIONS OF 2 H(t,n) 4 He FUSION G. Hupin, S. Quaglioni, P. Navrátil work in progress

21 2 H-t fusion with NCSMC: comparison between effective interactions n+ 4 He(g.s.) phase shifts with NN+3N potential, with eigenstates of 5 He. The d-t fusion is known to be very sensitive to threshold obtained theoretically from the solution of the A=2, 3 and 4 nucleonic systems. TOWARDS AB INITIO CALCULATIONS OF 2 H(t,n) 4 He FUSION G. Hupin, S. Quaglioni, P. Navrátil work in progress Energy in MeV 2 H g.s. 3 H g.s. 4 He g.s. =1.7 fm -1 ℏ  =16 MeV =2.0 fm -1 ℏ  =20 MeV

22 2 H-t fusion with NCSMC: influence of the deuterium continuum n+ 4 He(g.s.) phase shifts with NN+3N potential, with eigenstates of 5 He, =1.7 fm -1 and ℏ  =16 MeV. TOWARDS AB INITIO CALCULATIONS OF 2 H(t,n) 4 He FUSION G. Hupin, S. Quaglioni, P. Navrátil work in progress =1.7 fm -1 and ℏ  =16 MeV. =2.0 fm -1 and ℏ  =20 MeV.

23 We are extending the ab initio NCSM/RGM approach to describe low-energy reactions with two- and three-nucleon interactions. We are able to describe: Nucleon-nucleus collisions with NN+3N interaction Deuterium-nucleus collisions with NN+3N interaction as the n-n NCSMC for single- and two-nucleon projectile Work in progress: Fusion reactions with our best complete ab initio approach The present NNN force is "incomplete”, need to go to N 3 LO Evolution of stars, birth, main sequence, death CONCLUSIONS AND OUTLOOK

24 Challenging for both experiment and theory: Low rates: due to Coulomb repulsion between target/projectile and low-energy reactions (quantum tunneling effects). Projectile and target are not fully ionized in a lab. This leads to laboratory electron screening. Fundamental theory is still missing. Nuclear astrophysics community relies on accurate fusion reaction observables among others. What powers stars ? How long does a star live ? … S-factor for 3 He( 3 He,2p) 4 He FUSION PROCESSES PLAY AN IMPORTANT ROLE IN DETERMINING THE EVOLUTION OF OUR UNIVERSE, E.G. NUCLEOSYNTHESIS, STELLAR EVOLUTION …

25 7 diagrams (+ exchange) arising from the coupling of n-  with d-t through the three-nucleon force. Coupling to the compound 5 He eigenstates to come soon. n- 4 He with the d- 3 H channel with the chiral two- and three-nucleon force (preliminary) n+ 4 He(g.s.) phase shifts with NN+3N potential, =2.0 fm -1, no 5 He eigenstates, with/o coupling to d-t. Perspective to provide accurate t(d,n) 4 He fusion cross-section for the effort toward earth-based fusion energy generation. The d-t fusion is known to be very sensitive to the spin-orbit and isospin part of the nuclear interaction. d-t fusion expt 4 He fusion FIRST STEPS TOWARDS AB INITIO CALCULATIONS OF FUSION G. Hupin, S. Quaglioni, P. Navrátil work in progress

26 Best results in a decent model space (N max =11). The 3N force is essential to get the correct 6 Li g.s. energy and splitting between the 3 + and 2 + states. The 6 Li g.s. is well reproduced. There is room for improvements, in particular regarding the 3 + state. 6 Li SPECTRUM PREDICTED BY NCSMC G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015) Comparison between NCSMC vs data 6 Li spectrum extracted from d- 4 He phase-shifts for NN+3N-ind and NN+3N potentials with =2.0 fm -1.

27 before 6 Li SPECTRUM, NCSMC VS NCSM/RGM G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015) Comparison between NCSMC vs NCSM/RGM

28 Comparison between NCSMC vs NCSM(∞) 6 Li spectrum extracted from d- 4 He phase-shifts for NN+3N-ind and NN+3N potentials with =2.0 fm -1. No-Core Shell Model N max 6 Li SPECTRUM, NCSMC VS NCSM/RGM G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015)

29 Exotic nuclei, (Borromean halos, dripline nuclei) 6 He (= 4 He + n + n ) 6 Be (= a + p + p ) 11 Li (= 9 Li + n + n ) 14 Be (= 12 Be + n + n ) … Constituents do not bind in pairs! n-n- 4 He halo ground state “di-neutron” “cigar” 4 He n n 11 Li 208 Pb Probabilit y density of 6 He g.s. HALO NUCLEI AND PROJECTILE BREAK-UP S. Quaglioni, C. Romero-Redondo and P. Navrátil PRC88 (2013)

30 n-n- 4 He continuum First step toward a first-principles investigation of the 3 H+ 3 H → 4 He+2n reaction. Full formalism with 3N force is ready to go! new ~200K CPU-hours per channel HALO NUCLEI AND PROJECTILE BREAK-UP S. Quaglioni, C. Romero-Redondo, P. Navrátil and G. Hupin PRL113 (2014)

31 (Preliminary) Comparison of 3 He(p,p) 3 He angular cross-section (N max =11 only) p+ 3 He ( 4 Li) NCSMC differential cross-section for NN+3N potential with =1.7 fm -1, ℏ  =16MeV and N max =11. p- 3 He scattering E B =7.718 MeV The convergence in N max is difficult because 3 He is loosely bound. HH method is an exact solution. Our approach reproduces the exact solution, discrepancies show the lack of convergence in N max. BENCHMARK WITH HYPERSPHERICAL HARMONICS (HH) G. Hupin, S. Quaglioni, P. Navrátil et al. work in progress

32 The coupling with the A-body compound nuclei weakens the dependence on the d* pseudo- states. Residual dependence could be attributed to the missing breakup channel. before d- 4 He scattering Comparison of the d-  phase-shifts wrt the number of d* pseudo-states d- 4 He(g.s.) scattering phase-shifts for NN-only with different numbers of deuteron pseudo-states (N max =9). 4 He(d,d) 4 He WITH NCSMC ( 6 Li STATES) G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015)

33 Preliminary results in a small model space (N max =9). The coupling to the compound nuclei addresses some missing correlation. d- 4 He(g.s.) differential cross-section for NN-only potential using NCSM/RGM and NCSMC. NN-only Effects of the short-range correlation of the compound nuclei on the d-  cross- section 4 He(d,d) 4 He WITH NCSMC ( 6 Li STATES) G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015)

34 Results in a small model space (N max =9). The 3N force has a strong influence on the resonance positions rather than on the bulk of the cross-section. Comparison between the NN-only and the 3N force d- 4 He(g.s.) differential cross-section for NN-only potential using NCSM/RGM and NCSMC. 3N-full 4 He(d,d) 4 He EFFECTS OF THE 3N FORCE G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015)

35 The NCSMC results are obtained with the first 6 th low-lying 4 He ground state + first 14 th 5 Li states. We can see an improvement in the reproduction of the resonance centroids and widths (extracted with R- matrix). p- 4 He scattering NN+3N NCSM/RGM and NCSMC Comparison between NCSM/RGM and NCSMC at N max =13 and =2.0 fm -1 with NN+3N. p- 4 He SCATTERING: NCSM/RGM AND NCSMC G. Hupin, S. Quaglioni and P. Navrátil, PRC90 (2014)

36 p- 4 He reaction observables compared to experiment p- 4 He DIFFERENTIAL CROSS SECTION AND ANALYZING POWER G. Hupin, S. Quaglioni and P. Navrátil, unpublished work

37 Differential cross-section at E neutron =1.79 MeV between NN+3N-ind and NN+3N. Comparison of the elastic cross-section between NN and NN+3N with 4 He (g.s.) n- 4 He elastic cross-section for NN+3N-induced, NN+3N potentials compared to expt. and ENDF evaluation. (NN+3N-induced) We obtained a better agreement with data when using NN+3N. The 3N force is essential to get the resonance right. n- 4 He ELASTIC CROSS-SECTIONS G. Hupin, S. Quaglioni and P. Navrátil, to be published as a contribution to the Special Physica Scripta Edition - 40 year anniversary - Nobel Prize '75

38  Nucleon-nucleus collisions n- 3 H, p- 3 He, N- 4 He, n- 10 Be scattering with N 3 LO NN (mod. Lee-Suzuki eff. Int.) Nucleon scattering on 3 H, 3,4 He, 7 Li, 7 Be, 12 C, 16 O with SRG- N 3 LO 7 Be(p,  ) 8 B radiative capture with SRG-N 3 LO  Deuterium-nucleus collisions d- 4 He scattering and 6 Li structure with SRG-N 3 LO  (d,N) transfer reactions 3 H(d,n) 4 He and 3 He(d,p) 4 He reactions with SRG-N 3 LO Lee-Suzuki effective interaction Good reproduction, but still incomplete n- 4 He phase-shifts with NN interaction  dependent n- 4 He scattering Phase shiftsCross-section DEMONSTRATED CAPABILITY TO DESCRIBE BINARY-CLUSTER REACTIONS STARTING FROM NN INTERACTIONS

39 Direct potential:Exchange potential: - ~1Go ~7k two-body states INCLUDING THE 3N FORCE INTO THE NCSM/RGM APPROACH nucleon-nucleus formalism

40 We have included the first 6 low- lying states of 4 He. The convergence is difficult to assess. Target polarization (g.s., low-lying states) 4 He low- lying states Convergence of the phase shifts as a function of the 4 He excited states. Convergence with respect to the target polarization Exp. n- 4 He SCATTERING: STUDY OF THE RGM CONVERGENCE IN THE 3N CASE G. Hupin, J. Langhammer et al. PRC88 (2013)

41 ☑ Nucleon-nucleus collisions N- 4 He with SRG-N 3 LO and N 2 LO (3N) ☒ Deuterium-nucleus collisions ✘ d- 4 He scattering and 6 Li structure with SRG-N 3 LO (NN) and N 2 LO (3N) 3N force effects on n-  phase-shifts DEMONSTRATED CAPABILITY TO DESCRIBE BINARY-CLUSTER REACTIONS STARTING FROM NN+3N INTERACTIONS

42 Direct Exchang e four body density INCLUDING THE 3N FORCE INTO THE NCSM/RGM APPROACH deuteron-nucleus formalism

4 HE(d,d) 4 HE WITH NN+3N INTERACTION G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015) 43 d- 4 He scattering 6 Li loses binding energy with 3N interactions (not expected for 3N-full). The largest splitting between 3 D 3 and 3 D 2 partial waves is obtained with NN+3N. Comparison of the d-α phase-shifts with different interactions (N max =11) d- 4 He(g.s.) scattering phase-shifts for NN, NN+3N-induced and NN+3N potential with =2.0 fm -1. 3N-ind 3N-full

44 d pseudo states 7d*+4d’*+7d`* 7d* d (g.s.) Pseudo-states in each deuteron channels d (g.s., 3 S D 1, 3 D 2, 3 D G 3 ) 3 S D 1 channel, 3 D 2 channel… Comparison of the d-α phase-shifts with different interactions d- 4 He(g.s.) scattering phase-shifts with NN+3N interaction and =2.0 fm -1 (at N max =9). STUDY OF THE INFLUENCE OF d* CONTINUUM G. Hupin, S. Quaglioni and P. Navrátil, PRL 114 (2015)

45 Poor convergence behavior with respect to N max d- 4 He(g.s.) scattering phase-shifts with NN+3N interaction, ℏ  =20 MeV and =2.0 fm -1 for N max =7-11. Comparison Expt. R-matrix d- 4 He(g.s.) phase shifts with NN- only at ℏ  =14 MeV and =1.5 fm -1 for N max =11. Softer interaction ( 1.5 fm -1 ) Smaller ℏ  (14 MeV) lead to a better description of the 6 Li continuum. CONVERGENCE PROPERTIES… SOME CORRELATIONS ARE MISSING

46 Methods develop in this presentation to solve the many body problem Can address bound and low-lying resonances (short range correlations) Design to account for scattering states ( best for long range correlations) The many body quantum problem is best described by the superposition of both type of wave functions NCSMC A-body harmonic oscillator states Mixing coefficients (unknown) Second quantization Channel basis Relative wave function (unknown) Antisymmetrizer Cluster expansion technique TO OVERCOME THE DIFFICULTY: COUPLE NCSM AND NCSM/RGM (NCSMC) S. Baroni, P. Navrátil and S. Quaglioni PRL110 (2013); PRC93 (2013)

47 … A-body compound system Target and projectile in relative motion INCLUDING THE 3N FORCE INTO THE NCSMC APPROACH NUCLEON-NUCLEUS FORMALISM

48 A high precision nuclear Hamiltonian should include 3N force Spectra of the boron isotopic chain P. Navrátil et al. PRL99 (2007) Oxygen isotopic chain A. Cipollone et al. PRL111 (2013) Fig.1 G. Hagen et al. PRL108 (2012) Fig.2 What role does the 3N force play close to the particle continuum? Without three-nucleon force, the 10 B g.s. has a wrong spin. NN+3N 1 ST FOCUS OF THE TALK: INFLUENCE OF THE THREE- NUCLEON (3N) INTERACTION Coupling to the continuum included

49 Importance p-d analyzing power A y L. Marcucci et al. PRC80 (2009). OUR KNOWLEDGE OF THE NUCLEAR FORCE IS STILL INCOMPLETE

Address efficiently long-range correlations. No-Core Shell Model N max 50 … inbound and outbound waves cannot be described by finite number of basis states Nuclear structure and reaction model Address efficiently short-range correlations. 2 ND FOCUS OF THE TALK: ADDRESS TOGETHER SHORT- AND LONG-RANGE CORRELATIONS

 RGM accounts for: 1) interaction (Hamiltonian kernel), 2) Pauli principle (Norm kernel) between clusters.  NCSM accounts for: internal structure of clusters.  Use RGM+NCSM together with the same microscopic nuclear interaction. 51 Starts from: Schrödinger equation on channel basis: Ex: n- 4 He scattering Exit channel Entrance channel Channel basis Relative wave function (unknown) Antisymmetrizer Cluster expansion technique AB INITIO NCSM/RGM: FORMALISM FOR BINARY CLUSTERS S. Quaglioni and P. Navrátil, PRL101 (2008); PRC79 (2009)

52 NCSM/RGM exchange terms of a non-scalar operator most difficult part Transitions of electric dipole operators between scattering states First application of the implementation of the bremsstrahlung: the 4 He(p,p  ) 4 He system p+ 4 He(g.s.,0 + ) bremsstrahlung for NN+3N potential with =2.0 fm -1. Perspective to diagnose plasmas in fusion experiments from t(d,n  ) 4 He radiative transfer reaction. Provides another way to probe cluster wave functions.  p- 4 He bremsstrahlung FIRST STEPS TOWARDS AB INITIO CALCULATIONS OF FUSION J. Dohet-Eraly, S. Quaglioni, P. Navrátil et al. work in progress

The SRG evolution is to a good extend unitary. 53 Comparison at N max =13 between NN-only, NN+NNN-ind and NN+NNN between =2.0 and 1.88 fm -1 Unitarity behavior of the calculated phase-shifts STUDY OF THE SRG FLOW PARAMETER DEPENDENCE G. Hupin, J. Langhammer et al. PRC88 (2013)

54 NCSMC phase shift at N max =13 and =2.0 with NN+NNN. NN+3N NCSMC n- 4 He SCATTERING: NCSM/RGM AND NCSMC G. Hupin, S. Quaglioni and P. Navrátil, work in progress

55 The 7 Be(p,  ) 8 B is the final step in the nucleosynthetic chain leading to 8 B and one of the main inputs of the standard model of solar neutrinos. ~10% error in latest S 17 (0) NCSM/RGM results with largest realistic model space (N max = 10) Parameter  of SRG NN interaction chosen to reproduce separation energy: 136 keV (Expt. 137 keV) 7 Be(p,  ) 8 B astrophysical S-factor Ab initio theory predicts simultaneously both normalization and shape of S 17 8 B  8 Be*+e + + e Footprints of pp-chain on earth solar neutrinos E n < 15 MeV Astrophysical S-factor: 7 Be+p  8 B+  Theory EXEMPLE OF AN APPLICATION TO NUCLEAR ASTROPHYSICS P. Navrátil, R. Roth, and S. Quaglioni, PLB704 (2011)

56 The present 3N force is incomplete (N 2 LO only versus N 3 LO for NN). The missing d+ 3 He channel could account for the shift in the P 3/2 resonance? p- 4 He scattering Initial NN and NN+NNN compared to experiment Comparison between NN+3N -ind, NN+3N and experiment at N max =13 and =2.0 fm -1. p- 4 He SCATTERING: NCSM/RGM VERSUS EXPERIMENT G. Hupin, J. Langhammer et al. PRC88 (2013)

57 The present 3N force is incomplete (N 2 LO only versus N 3 LO for NN). The missing d+ 3 H channel could account for the shift in the P 3/2 resonance? Comparison between NN+3N -ind, NN+3N and experiment at N max =13 and =2.0 fm -1. Initial NN and NN+3N compared to experiment n- 4 He SCATTERING: NCSM/RGM VERSUS EXPERIMENT G. Hupin, J. Langhammer et al. PRC88 (2013)

58 d- 4 He scattering N max = 12 d(g.s., 3 S D 1, 3 D 2, 3 D G 3 ) + 4 He(g.s.) SRG-N 3 LO NN potential with 1.5 fm -1, ħ  =14 MeV. Comparison with experiment 4 He(d,d) 4 He phase shifts: convergence wrt to d continuum Pseudo-states in each channel 4 He(d,d) 4 He WITH NN-ONLY P. Navrátil and S. Quaglioni PRC83 (2011)

59 d- 4 He scattering Comparison Expt. R-matrix Poor convergence behavior with respect to N max d- 4 He(g.s.) scattering phase-shifts with NN+3N interaction and =2.0 fm -1 for N max =7 to 11.  Softer interaction ( =1.5 fm -1 ).  A smaller ℏ  (14 MeV) better describes the d continuum. WE ARE MISSING CORRELATIONS MOSTLY SHORT-RANGE ( 6 Li IS UNDERBOUND)

60 Translational invariance is preserved (exactly!) also with SD cluster basis Matrix inversion What we calculate in the “SD” channel basis Observables calculated in the translationally invariant basis Advantage: can use powerful second quantization techniques MATRIX ELEMENTS OF TRANSLATIONALLY INVARIANT OPERATORS

61 (Jacobi) channel basis  We introduce Jacobi channel states in the HO space  We use the closure properties of HO radial wave function This defines the RGM model space (Ok for localized parts of the kernels)  The coordinate space channel states are given by AB INITIO NCSM/RGM: FORMALISM FOR BINARY CLUSTERS FEW DETAILS

62 1.New basis: Slater Determinant channel basis  The target (A>a nucleons) described by a Slater Determinant Vector proportional to the c.m. coordinate of the a nucleons Vector proportional to the c.m. coordinate of the A-a nucleons 2.With a basis change, we can recover a simple expression Then, the kernels can be calculated with the help of the second quantization IN PRACTICE, WE MADE USE OF SECOND QUANTIZATION MORE INSIGHTS