Ab initio No-Core Shell Model with the continuum of 3 H(d,n) 4 He fusion reaction Guillaume Hupin FUSTIPEN, March 19 th Collaborators: S. Quaglioni (LLNL) P. Navrátil (TRIUMF) R. Roth (TU Darmstadt) J. Langhammer (TU Darmstadt) C. Romero-Redondo (LLNL/TRIUMF) F. Raimondi F. Raimondi (TRIUMF) J. Dohet-Eraly (TRIUMF)

2 Exact Ab initio Configuration interaction Nuclear energy density functional One small step

3 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). Input: Interaction between nucleon from QCD

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 Light-ion reactions come into play in earth-based fusion facilities Interactions with environment Neutron ToF detector n

5 No-Core Shell Model N max Ab initio NCSM/RGM: formalism S. Quaglioni and P. Navrátil, PRL101 (2008); PRC79 (2009) NCSM NCSM/RGM: Binary cluster NCSM/RGM: Binary cluster NCSM/RGM: Three-body cluster NCSM/RGM: Three-body cluster Large expansions in A-body harmonic oscillator (HO) basis Preserves translational invariance Slater-Determinant basis are simple Can include NN+3N interactions Uses effective interaction to accelerate convergence to exact solution with N max

6 No-Core Shell Model N max Ab initio NCSM/RGM: formalism S. Quaglioni and P. Navrátil, PRL101 (2008); PRC79 (2009) Large expansions in A-body harmonic oscillator (HO) basis Preserves translational invariance Slater-Determinant basis are simple Can include NN+3N interactions Uses effective interaction to accelerate convergence to exact solution with N max  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.

7 Low-lying resonances above reaction threshold: a simple example with the tightly bound 4 He Exit channel Entrance channel Exit channel Entrance channel e.g. 3 H(d,n) 3 He Exit channel Entrance channel

8 Ab initio NCSMC: formalism S. Baroni, P. Navrátil and S. Quaglioni PRL110 (2013); PRC93 (2013) A-body harmonic oscillator states Mixing coefficients (unknown) Channel basis Relative wave function (unknown) Antisymmetrizer Can address bound and low- lying resonances (short range correlations) Design to account for scattering states ( best for long range correlations)

Effects of closed channels: NCSM/RGM vs NCSMC G. Hupin, J. Langhammer et al. PRC88 (2013),, PRC90 (2014) Convergence with respect to the # of 4 He low-lying states 9 Same plot once the first 14 th low-lying state of 5 He are accounted for. : closed channels The coupling with the A-body compound nuclei accounts partially for the effect of closed channels !

And for the n- 4 He elastic cross-sections 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-ind The 3 H(d,n) 3 He reaction channel is emulated although shifted since the reaction threshold ( 3 H and d binding energies) is not included> We obtained a better agreement with data when using NN+3N. The 3N force is essential to get the resonance right. 10 Exit channel Entrance channel expt. threshold

4 He(d,d) 4 He with NCSMC ( 6 Li states) G. Hupin, S. Quaglioni and P. Navrátil, submitted to PRL 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 11 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).

6 Li spectrum predicted by NCSMC G. Hupin, S. Quaglioni and P. Navrátil, submitted to PRL 12 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 still room for improvements, in particular regarding the 3 + state (known). 6 Li spectrum extracted from d- 4 He phase-shifts for NN+3N-ind and NN+3N potentials with =2.0 fm -1.

6 Li spectrum: NCSMC vs NCSM G. Hupin, S. Quaglioni and P. Navrátil, submitted to PRL 13 No-Core Shell Model N max 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. S. Coon et al. arXiv:

6 Li spectrum: NCSMC vs NCSM G. Hupin, S. Quaglioni and P. Navrátil, submitted to PRL 14 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. Calculation with two-nucleon projectile requires a huge three- nucleon force model-space which is, in practice, truncated up to E 3max =17. Jacobi JT coupled M scheme R. Roth et al. PRC 90 (2014)

Another example of an application to nuclear astrophysics from the work of Francesco Raimondi Footprints of pp-chain on earth 8 B → 8 Be*+e + + e solar neutrinos E n < 15 MeV 7 Be+p → 8 B+  15 Indirect measurement of 8 B: measuring the 7 Li(d,p) 8 Li yield on the resonance (Γ≈ 0.2 MeV) at E d = 0.78 MeV) Kavanagh 1960, Parker 1966, Filippone 1982, Weissman 1998

16 Analysis of the low-lying resonance from the work of Francesco Raimondi Eigenphase shifts Effect of the off-diagonal coupling part of the collision matrix Elastic collisions

17 Reaction mechanism from Ab initio theory from the work of Francesco Raimondi

18 Preliminary results versus experiment from the work of Francesco Raimondi

First step towards Ab initio many-body calculations of fusion P. Navrátil, S. Quaglioni, PRL108 (2012) 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 19

First steps towards Ab initio calculations of fusion G. Hupin, S. Quaglioni, P. Navrátil work in progress 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. 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. n- 4 He including the d- 3 H channel with the chiral two- and three-nucleon force (preliminary) phase shifts with NN+3N potential, =2.0 fm -1, no 5 He eigenstates, with/o coupling to d-t. d-t fusion expt 4 He fusion 20

First steps towards Ab initio calculations of fusion G. Hupin, S. Quaglioni, P. Navrátil work in progress 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. 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. n- 4 He including the d- 3 H channel: effect of the deuteron pseudostates phase shifts with NN+3N potential, =2.0 fm -1, no 5 He eigenstates, with/o coupling to d-t. 4 He fusion 21

First steps towards Ab initio calculations of fusion G. Hupin, S. Quaglioni, P. Navrátil work in progress Preliminary but already decent N max (=9) 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. 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 5He. d-t fusion expt 4 He fusion 22

Conclusions and Outlook We have extending the ab initio NCSM/RGM approach to describe low-energy reactions with two- and three-nucleon interactions. We have developed a new technique that treats bound and continuum states alike. We are able to describe: Nucleon-nucleus collisions with NN+3N interaction Deuterium-nucleus collisions with NN+3N interaction Transfer reactions Work in progress Fusion reactions calculations to be completed and analyzed We would like to test different nuclear interaction Three-body cluster with NN+3N and NCSMC coming soon Radiative capture and bremsstrahlung under development 23

4 He(d,d) 4 He differential cross- section G. Hupin, S. Quaglioni and P. Navrátil, submitted to PRL Final results in a N max =11 model space. All relevant negative and positive partial waves are included as well as up to fifteen excited states of the compound nuclei. 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 24