1) Dept. of Phys. & Astro. “G. Galilei”, University of Padova “Two-nucleon emission with pairing interaction in three-body systems” Tomohiro Oishi 1) Dept. of Phys. & Astro. “G. Galilei”, University of Padova 2) I.N.F.N.- Sez. Padova Collaborators: Lorenzo Fortunato (Univ. of Padova) Markus Kortelainen (Univ. of Jyväskylä, Finland) Alessandro Pastore (Univ. of York, UK) Kouichi Hagino (Tohoku Univ., Japan) Hiroyuki Sagawa (RIKEN Nishina Center & Univ. of Aizu, Japan) Further Information in, L. Fortunato and T. Oishi, arXiv: 1701.04684. T.O., M. Kortelainen and A. Pastore, arXiv: 1606.03111. T.O., K. Hagino and H. Sagawa, Phys. Rev. C 90, 034303 (2014). I would like to thank all the organizers and participants to have this workshop. It is my great honor to have this chance to talk about our works. “Probing fundamental interactions by low energy excitations - Advances in theoretical nuclear physics -” Royal Institute of Technology, Stockholm, 07/June/2017.

Two-proton (2p) radioactivity K. Miernik et al., PRL99, 192501(2007). Experiment with time-projection chamber. Novel decay-mode found in 2001. A pair of protons is emitted from the parent nucleus. A typical few-body, quantum resonance problem with two interacting fermions. → Connections to the quantum entanglement, BCS-BEC crossover, and open quantum systems can be found. Nuclear force between 2p (pairing force) plays an essential role. 45Fe まんま。

System = 6Be Nucleus (Core=Alpha) 単純な模型が妥当そうで、かつペアリングの影響が如実に現れてくれそうな系である。 The lightest & simplest 2p-emitter. The 1p-emission is suppressed (the true 2p-emitter).

TDM (ours) v.s. Non-Hermitian Method Time-Dependent Non-Hermitian Initial State: Resonance State: Exponential Decay-rule: Krylov-Fockの定理は、離散化＆cutoff がある場合でも問題無く成り立つ。 生き残り率がexponentialに近ければ、用意したICが適切な近似である。 利点； （１）FSIsとICとが明瞭かつ厳密に区別して取り扱える（ただし、今回の計算では、この長所はあまり活かせていない）。 （２）崩壊過程を可視化できるので、どんな物理が潜んでいるのかが分かり易い。 In numerical calculations, Why Hamiltonian can be non-Hermitian ?  Because its model-space is defined so as.

Time-dependent 2p-emission

Problem: is our pairing model adequate ? w0  Vacuum scattering length Other parameter(s)  Q2p=1.4 MeV With the density-dependent delta pairing model, Only w0= w(r→∞) controls Γ2p. Impossible trinity of w0, Q2p, and Γ2p.  Possible improvements include (i) non-trivial density-dependence, (ii) three-body force, etc..

Recent project: L. Fortunato and T. Oishi, arXiv: 1701.04684.

Motivation Time-dependent method for multi-particles: Conceptually, it’s complicated with, e.g. multi-boundary conditions, subsystem-energy rules, etc. Numerically, it’s also demanding.  General & efficient benchmark study to solve N-body (resonant) problem. Generality = that can be feasible for any N bosons or/and fermions, in any scale. Efficiency = that asks less cost to obtain, e.g. the Hamiltonian matrix.

Benchmark in 1-dimensional 3-body problem Simplest testing field with (N>2) particles. HO-basis;

Matrix Elements of 2B-Interactions 1 2 3 𝜉 𝑖=1~3 (12) O 𝑉( 𝑥 1 − 𝑥 2 ) +𝑉( 𝑥 1 − 𝑥 3 ) +𝑉( 𝑥 2 − 𝑥 3 ) 1 2 3 𝜉 𝑖=1~3 (13) O 1 2 3 𝜉 𝑖=1~3 (23) O

Analytic !! (for HO-basis) Overlap coefficients Kinematic rotation = O(N-1) group with W(12,ij): Analytic !! (for HO-basis)

Benchmark 𝐸 𝑛, 𝑙 = 3 ℏ𝜔(2𝑛+𝑙) Ref: F. Calogero, Jour. of Math. Phys. 10, Num. 12, 2191 (1969) Calogero Hamiltonian (3-bosons, 1D, HO-Interactions): 𝐸 𝑛, 𝑙 = 3 ℏ𝜔(2𝑛+𝑙) Analytic solution: Our results: <><><><><> NOW Hamiltonian has been DIAGONALIZED !! <><><><><> e_ex 0= 1.7320508086 (MeV), 1.7320508174 e_ex 1= 3.4641016259 (MeV), 1.7320512630 e_ex 2= 5.1961528889 (MeV), 0.0000004537 e_ex 3= 5.1961533426 (MeV), 1.7320511187 e_ex 4= 6.9282044613 (MeV), 0.0000032783 e_ex 5= 6.9282077396 (MeV), 1.7320733024 e_ex 6= 8.6602810420 (MeV), 0.0000680149 e_ex 7= 8.6603490569 (MeV), 0.0000547213 e_ex 8= 8.6604037782 (MeV), 1.7319823331 e_ex 9= 10.3923861114 (MeV), 0.0001106872 e_ex10= 10.3924967986 (MeV), 0.0002580581 e_ex11= 10.3927548567 (MeV), 1.7317226328

x3-X12 Benchmark (2) x2-x1 Present achievement = 1D 3-body Hamiltonian is solved. Future Plans; Bound  Resonant (time-development, complex-gauge, etc.) 1D  3D (e.g. 12C as triple alpha, Be isotopes and many more!) 3-body  N-body 散乱や共鳴の問題は、本来、時間発展で解かなければいけない（酒井さん）。ところが、多くの理論家はあまり気を遣わずに、（摂動論的な展開の仮定とともに）時間発展の演算自体からは時間依存性を抜いてしまっている。 このモデルを共鳴に拡張する場合、初期状態から、ハミルトニアンの相互作用をあらわに時間依存させて解けるか？言い方を変えると、断熱近似無しで解けるか？単純なモデル計算ですら、今のところはなされていない（←ココは本当かどうか分からん）。

Recent topic: pn-correlation “Does pn-pair in finite nuclei behave like a deuteron ?” C n +8.67 +4.44 pn-binding energy pn-kinetic energy pn-pairing energy for pn-subsystem. +1.78 MeV -3.21 MeV -6.89 -7.65 E3B= - 3.70 MeV E3B= - 9.74 MeV

Preliminary result of pn-decay Decay-width: pn-emission  information on S=1 (T=0) pairing ( similarly as 2p-emission  information on S=0 (T=1) pairing )

We look forward for suggestions and collaborations! Summary Two-proton (and also 2n, pn) emission can provide information on the nuclear pairing correlation and dynamics. Improvements of pairing-interaction and quantum-resonance models are necessary, utilizing the existing and forthcoming experimental data. Future Work General & efficient computation for the multi-particle problem, including resonance systems with time-dependent or/and non-Hermitian procedure. Applications to 2p-decay, 2n-decay, pn-decay, alpha-cluster resonances, tetra-neutrons, cold-atomic resonances, etc. We look forward for suggestions and collaborations!

Appendix

Benchmark Calogero Hamiltonian (3-bosons, 1D, HO-Interactions): Ref: F. Calogero, Jour. of Math. Phys. 10, Num. 12, 2191 (1969) Calogero Hamiltonian (3-bosons, 1D, HO-Interactions): Analytic solution: Our results: 𝐸 𝑛, 𝑙 = 3 ℏ𝜔(2𝑛+𝑙)

Three-body Model: 6Be = alpha + p + p Core-p potential = Woods-Saxon + Coulomb forces → reproduce the resonance of alpha + p in (p3/2)-orbit. Pairing potential = Effective Nuclear Pairing + Coulomb forces → reproduce the empirical Q-value = <H3b> = 1.37 MeV. 密度依存性による核力の引力強化を、ここでは手で入れた。 本来、Coreから遠ざかったら、真空中の核力に戻す必要がある。 あまり現実的ではないが、モデルの可動性を先に確認したかった。

Toward 3D calculation I will explain details in the following. But summarizing important points, … OK.

= Time-dependent 3-body model Our method = Time-dependent 3-body model This model can take into account, many-body properties, and quantum resonance (meta-stability) in e.g., two-nucleon emission, cold-atomic resonance, etc. I will explain details in the following. But summarizing important points, … OK.

Ref: T.O., M. Kortelainen and A. Pastore, arXiv: 1606.03111.

Time-Dependent Method (TDM) The initial state is confined inside the potential barrier. → The state at t>0 is an out-going wave. Krylov-Fockの定理は、離散化＆cutoff がある場合でも問題無く成り立つ。 生き残り率がexponentialに近ければ、用意したICが適切な近似である。 利点； （１）FSIsとICとが明瞭かつ厳密に区別して取り扱える（ただし、今回の計算では、この長所はあまり活かせていない）。 （２）崩壊過程を可視化できるので、どんな物理が潜んでいるのかが分かり易い。

Initial 2p-state of 6Be Vppについて、``full’’以外ではもはや非現実的な対相関ポテンシャルを使わなければ、Q-valueを再現できない。 実験との合致を至上とするなら、実験方法に即したICを採用するべき。 しかし、今回は理論のテストも兼ねているのと、単純化したモデルでの描写を重視している。 Localization of 2p (diproton-like configuration) ↑ Pairing interaction

Time-dependent 2p-emission Note: Decay-state is mainly an out-going state.

Decay-width (1) Exponential decay with the constant decay-width.

Decay-width (2) Changes of Pairing Strength

Outline of my talk 2p-emission with time-dep. 3-body model Recent project I: proton-neutron emission Recent project II: 1d 3-boson problem Summary I will explain details in the following. But summarizing important points, … OK.

Quantum Resonance (Meta-stability) Keywords Complex Energy Non-Hermitian Method Few-body 2p-decay Quantum Resonance (Meta-stability) Tunneling Time-dependent Pairing Open Quantum Systems Efimov State Alpha-decay Gamov State

Recent project I: proton-neutron emission

? Li-6 nucleus E (MeV) p n α α+p+n α+p α+n α E=2.57, Γ≈2.0 1 + E=1.59, Γ=1.29 E=0.77, Γ=0.67 3 − 2 E=0.29, Γ=0.09 3 − 2 α 𝐽 𝜋 = 1 + 0 + α+p+n α+p α+n α

Bound & Resonant States of 6Li Ground state  1+, spin-triplet pairing should be dominant. Spn(6Li)= Sp(6Li) + Sn(5He) = 3.70 MeV, at ground state. https://www.nndc.bnl.gov/chart/getdataset.jsp?nucleus=6LI&unc=nds

Results (1) Confining potentials:

Results (3) At t=0, pn-W.F. should be confined.  Density of pn-probabilities:

Results (4) Decay-state: