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Few-body resonances from finite-volume calculations Sebastian K - - PowerPoint PPT Presentation

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Few-body resonances from finite-volume calculations Sebastian K onig FRIB TA Workshop Connecting bound state calculations with scattering and reactions NSCL, Michigan State University June 19, 2018 P. Klos, SK, J. Lynn, H.-W. Hammer,

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SLIDE 1

Few-body resonances from finite-volume calculations

Sebastian K¨

  • nig

FRIB TA Workshop “Connecting bound state calculations with scattering and reactions” NSCL, Michigan State University

June 19, 2018

  • P. Klos, SK, J. Lynn, H.-W. Hammer, and A. Schwenk, arXiv:1805.02029 [nucl-th]

Few-body resonances from finite-volume calculations –

  • p. 1
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SLIDE 2

Motivation

terra incognita at the doorstep. . .

? ? ?

bound dineutron state not excluded by pionless EFT

Hammer + SK, PLB 736 208 (2014)

recent indications for a three-neutron resonance state. . .

Gandolfi et al., PRL 118 232501 (2017)

. . . although excluded by previous theoretical work

Offermann + Gl¨

  • ckle, NPA 318, 138 (1979); Lazauskas + Carbonell, PRC 71 044004 (2005)

possible evidence for tetraneutron resonance

Kisamori et al., PRL 116 052501 (2016) Few-body resonances from finite-volume calculations –

  • p. 2
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SLIDE 3

Short (recent) history of tetraneutron states

1 2002: experimental claim of bound tetraneutron Marques et al., PRC 65 044006 2 2003: several studies indicate unbound four-neutron system Bertulani et al.. JPG 29 2431; Timofeyuk, JPG 29 L9; Pieper, PRL 90 252501 3 2005: observable tetraneutron resonance excluded Lazauskas PRC 72 034003

Γ (MeV)

1 2 3 4

ER (MeV)

2 4 6 8

Few-body resonances from finite-volume calculations –

  • p. 3
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SLIDE 4

Short (recent) history of tetraneutron states

1 2002: experimental claim of bound tetraneutron Marques et al., PRC 65 044006 2 2003: several studies indicate unbound four-neutron system Bertulani et al.. JPG 29 2431; Timofeyuk, JPG 29 L9; Pieper, PRL 90 252501 3 2005: observable tetraneutron resonance excluded Lazauskas PRC 72 034003 4 2016: RIKEN experiment: possible tetraneutron resonance

ER = (0.83 ± 0.65stat. ± 1.25syst.) MeV, Γ 2.6 MeV

Kisamori et al., PRL 116 052501 5

following this: several new theoretical investigations complex scaling → need unphys. T = 3/2 3N force or strong rescaling

Hiyama et al., PRC 93 044004 (2016),; Deltuva, PLB 782 238 (2018)

incompatible predictions:

Γ (MeV)

1 2 3 4

ER (MeV)

2 4 6 8

Gandolfi et al., PRL 118 232501 (2017) Shirokov et al. PRL 117 182502(2016) Fossez et al., PRL 119 032501 (2017) Few-body resonances from finite-volume calculations –

  • p. 3
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SLIDE 5

Short (recent) history of tetraneutron states

1 2002: experimental claim of bound tetraneutron Marques et al., PRC 65 044006 2 2003: several studies indicate unbound four-neutron system Bertulani et al.. JPG 29 2431; Timofeyuk, JPG 29 L9; Pieper, PRL 90 252501 3 2005: observable tetraneutron resonance excluded Lazauskas PRC 72 034003 4 2016: RIKEN experiment: possible tetraneutron resonance

ER = (0.83 ± 0.65stat. ± 1.25syst.) MeV, Γ 2.6 MeV

Kisamori et al., PRL 116 052501 5

following this: several new theoretical investigations complex scaling → need unphys. T = 3/2 3N force or strong rescaling

Hiyama et al., PRC 93 044004 (2016),; Deltuva, PLB 782 238 (2018)

incompatible predictions:

Γ (MeV)

1 2 3 4

ER (MeV)

2 4 6 8

Gandolfi et al., PRL 118 232501 (2017) Shirokov et al. PRL 117 182502(2016) Fossez et al., PRL 119 032501 (2017)

−4.0 −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 V0 (MeV) −7.0 −6.0 −5.0 −4.0 −3.0 −2.0 −1.0 0.0 1.0 2.0 3.0 E (MeV)

RWS = 7.5 fm RWS = 6.0 fm RWS = 4.5 fm

4 neutrons 3 neutrons

−3.0 −2.0 −1.0 −6.0 −4.0 −2.0 0.0 2.0 4 neutrons RWS = 6.0 fm LO NLO N2LO

Gandolfi et al., PRL 118 232501 (2017)

indications for three-neutron resonance. . . . . . lower in energy than tetraneutron state

Few-body resonances from finite-volume calculations –

  • p. 3
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SLIDE 6

How to tackle resonances?

Resonances

metastable states decay width ↔ lifetime

1 2 3 4 5 r 0.5 1.0 1.5 2.0 V(r)

1 Look for jump by π in scattering phase shift:

simple possibly ambiguous (background), need 2-cluster system

0.5 1.0 1.5 2.0 2.5 3.0E 50 100 150 δ(E}

Few-body resonances from finite-volume calculations –

  • p. 4
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SLIDE 7

How to tackle resonances?

Resonances

metastable states decay width ↔ lifetime

1 2 3 4 5 r 0.5 1.0 1.5 2.0 V(r)

1 Look for jump by π in scattering phase shift:

simple possibly ambiguous (background), need 2-cluster system

0.5 1.0 1.5 2.0 2.5 3.0E 50 100 150 δ(E}

↔

2 Find complex poles in S-matrix: e.g., Gl¨

  • ckle, PRC 18 564 (1978); Borasoy et al., PRC 74 055201 (2006); . . .

direct, clear signature technically challenging, needs analytic pot.

Few-body resonances from finite-volume calculations –

  • p. 4
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SLIDE 8

How to tackle resonances?

Resonances

metastable states decay width ↔ lifetime

1 2 3 4 5 r 0.5 1.0 1.5 2.0 V(r)

1 Look for jump by π in scattering phase shift:

simple possibly ambiguous (background), need 2-cluster system

0.5 1.0 1.5 2.0 2.5 3.0E 50 100 150 δ(E}

↔

2 Find complex poles in S-matrix: e.g., Gl¨

  • ckle, PRC 18 564 (1978); Borasoy et al., PRC 74 055201 (2006); . . .

direct, clear signature technically challenging, needs analytic pot.

3 Put system into periodic box! Few-body resonances from finite-volume calculations –

  • p. 4
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SLIDE 9

Finite periodic boxes

physical system enclosed in finite volume (box) typically used: periodic boundary conditions volume-dependent energies

Few-body resonances from finite-volume calculations –

  • p. 5
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SLIDE 10

Finite periodic boxes

physical system enclosed in finite volume (box) typically used: periodic boundary conditions volume-dependent energies L¨ uscher formalism

Physical properties encoded in the L-dependent energy levels! infinite-volume S-matrix governs discrete finite-volume spectrum PBC natural for lattice calculations. . . . . . but can also be implemented with other methods

Few-body resonances from finite-volume calculations –

  • p. 5
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SLIDE 11

General bound-state volume dependence

volume dependence ↔ overlap of asymptotic wave functions

L¨ uscher, Commun. Math. Phys. 104 177 (1986); . . .

κA|N−A =

  • 2µA|N−A(BN−BA−BN−A)

Volume dependence of N-body bound state

∆BN(L) ∝ (κA|N−AL)1−d/2 Kd/2−1(κA|N−AL) ∼ exp

  • −κA|N−AL
  • /L(d−1)/2 as L → ∞

(L = box size, d no. of spatial dimensions, Kn = Bessel function)

SK and D. Lee, PLB 779, 9 (2018)

channel with smallest κA|N−A determines asymptotic behavior

Few-body resonances from finite-volume calculations –

  • p. 6
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SLIDE 12

Numerical results

SK and D. Lee, PLB 779, 9 (2018)

N = 2 N = 3 N = 4 N = 5

D = 1, alatt = 1/3, k = 2 log(ΔB) −30 −25 −20 −15 −10 −5 L 10 20 30 40 50 N = 2 N = 3

D = 3, alatt = 1/2, k = 2 log(L ΔB) −12.5 −10 −7.5 −5 −2.5 2.5 L 5 10 15 20 25

֒ → straight lines ↔ excellent agreement with prediction

N BN Lmin . . . Lmax κfit κ1|N−1 d = 1, V0 = −1.0, R = 1.0 2 0.356 20 . . . 48 0.59536(3) 0.59625 3 1.275 15 . . . 32 1.1062(14) 1.1070 4 2.859 12 . . . 24 1.539(3) 1.541 5 5.163 12 . . . 20 1.916(21) 1.920 d = 3, V0 = −5.0, R = 1.0 2 0.449 15 . . . 24 0.6694(2) 0.6700 3 2.916 4 . . . 14 1.798(3) 1.814

Few-body resonances from finite-volume calculations –

  • p. 7
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SLIDE 13

Finite-volume resonance signatures

L¨ uscher formalism: phase shift ↔ box energy levels

p cot δ0(p) = 1 πLS(η) , η =

Lp

2π

  • 2

, p = p

E(L)

  • L¨

uscher, Nucl. Phys. B 354 531 (1991); . . .

resonance contribution avoided level crossing

Wiese, Nucl. Phys. B (Proc. Suppl.) 9, 609 (1989); . . . Few-body resonances from finite-volume calculations –

  • p. 8
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SLIDE 14

Finite-volume resonance signatures

L¨ uscher formalism: phase shift ↔ box energy levels

p cot δ0(p) = 1 πLS(η) , η =

Lp

2π

  • 2

, p = p

E(L)

  • L¨

uscher, Nucl. Phys. B 354 531 (1991); . . .

resonance contribution avoided level crossing

Wiese, Nucl. Phys. B (Proc. Suppl.) 9, 609 (1989); . . . 2 4 6 8 10 L 2 4 6 8 10 12 p

no interaction, δ(p) = 0 ֒ → free levels ∼ 1/L

Few-body resonances from finite-volume calculations –

  • p. 8
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SLIDE 15

Finite-volume resonance signatures

L¨ uscher formalism: phase shift ↔ box energy levels

p cot δ0(p) = 1 πLS(η) , η =

Lp

2π

  • 2

, p = p

E(L)

  • L¨

uscher, Nucl. Phys. B 354 531 (1991); . . .

resonance contribution avoided level crossing

Wiese, Nucl. Phys. B (Proc. Suppl.) 9, 609 (1989); . . . 2 4 6 8 10 L 2 4 6 8 10 12 p

1 2 3 4 5 6 p

  • 0.4
  • 0.3
  • 0.2
  • 0.1

δ(p}

Few-body resonances from finite-volume calculations –

  • p. 8
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SLIDE 16

Finite-volume resonance signatures

L¨ uscher formalism: phase shift ↔ box energy levels

p cot δ0(p) = 1 πLS(η) , η =

Lp

2π

  • 2

, p = p

E(L)

  • L¨

uscher, Nucl. Phys. B 354 531 (1991); . . .

resonance contribution avoided level crossing

Wiese, Nucl. Phys. B (Proc. Suppl.) 9, 609 (1989); . . . 2 4 6 8 10 L 2 4 6 8 10 12 p

1 2 3 4 5 6 p 0.5 1.0 1.5 2.0 2.5 δ(p}

Few-body resonances from finite-volume calculations –

  • p. 8
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SLIDE 17

Finite-volume resonance signatures

L¨ uscher formalism: phase shift ↔ box energy levels

p cot δ0(p) = 1 πLS(η) , η =

Lp

2π

  • 2

, p = p

E(L)

  • L¨

uscher, Nucl. Phys. B 354 531 (1991); . . .

resonance contribution avoided level crossing

Wiese, Nucl. Phys. B (Proc. Suppl.) 9, 609 (1989); . . . 2 4 6 8 10 L 2 4 6 8 10 12 p

1 2 3 4 5 6 p 0.5 1.0 1.5 2.0 2.5 δ(p}

Effect can be very subtle in practice. . .

Bernard et al., JHEP 0808 024 (2008); D¨

  • ring et al., EPJA 47 139 (2011); . . .

Few-body resonances from finite-volume calculations –

  • p. 8
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SLIDE 18

Discrete variable representation

Needed: calculation of several few-body energy levels

difficult to achieve with QMC methods

Klos et al., PRC 94 054005 (2016)

direct discretization possible, but not very efficient

֒ → use a Discrete Variable Representation (DVR)

well established in quantum chemistry, suggested for nuclear physics by Bulgac+Forbes, PRC 87 87, 051301 (2013)

Main features

basis functions localized at grid points potential energy matrix diagonal kinetic energy matrix sparse (in d > 1). . . . . . or implemented via Fast Fourier Transform

periodic boundary condistions ↔ plane waves as starting point

  • 6
  • 4
  • 2

2 4 6

  • 0.2

0.2 0.4 0.6 0.8 1.0 Few-body resonances from finite-volume calculations –

  • p. 9
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SLIDE 19

DVR construction

start with some initial basis; here: φi(x) =

1 √ L exp

  • i2πi

L x

  • consider (xk, wk) such that

N/2−1

  • k=−N/2

wk φ∗

i (xk)φj(xk) = δij

  • 6
  • 4
  • 2

2 4 6

  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3

unitary trans. − →

Uki = √wkφi(xk)

  • 6
  • 4
  • 2

2 4 6

  • 0.2

0.2 0.4 0.6 0.8 1.0

DVR states

ψk(x) localized at xk, ψk(xj) = δkj/√wk note: momentum mode φi ↔ spatial mode ψk

Few-body resonances from finite-volume calculations –

  • p. 10
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SLIDE 20

DVR features

1 potential energy is diagonal!

ψk|V |ψl =

  • dx ψk(x) V (x) ψl(x)

≈

N/2−1

  • n=−N/2

wn ψk(xn) V (xn) ψl(xn) = V (xk)δkl

no need to evaluate integrals number N of DVR states controls quadrature approximation        

Few-body resonances from finite-volume calculations –

  • p. 11
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SLIDE 21

DVR features

1 potential energy is diagonal!

ψk|V |ψl =

  • dx ψk(x) V (x) ψl(x)

≈

N/2−1

  • n=−N/2

wn ψk(xn) V (xn) ψl(xn) = V (xk)δkl

no need to evaluate integrals number N of DVR states controls quadrature approximation        

2 kinetic energy is simple (via FFT) or sparse (in d > 1)!

plane waves φi are momentum eigenstates ˆ T |ψk ∼ F−1⊗ ˆ p2 ⊗ F |ψk ψk| ˆ T|ψl = known in closed form ֒ → replicated for each coordinate, with Kronecker deltas for the rest

Few-body resonances from finite-volume calculations –

  • p. 11
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SLIDE 22

General DVR basis states

construct DVR basis in simple relative coordinates. . . . . . because Jacobi coord. would complicate the boundary conditions separate center-of-mass energy (choose P = 0) mixed derivatives in kinetic energy operator xi =

n

  • i=1

Uijri Uij =

      

δij for i, j < n −1 for i < n, j = n 1/n for i = n

General DVR state

|s = |(k1,1, · · · , k1,d), · · · , (kn−1,1, · · · ); spins ∈ B basis size: dim B = (2S + 1)n × Nd×(n−1)

Few-body resonances from finite-volume calculations –

  • p. 12
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SLIDE 23

(Anti-)symmetrization and parity

Permutation symmetry

for each |s ∈ B, construct |s

A = N

  • p∈Sn

sgn(p) Dn(p) |s then |sA is antisymmetric: A |s

A = |s A

for bosons, leave out sgn(p) symmetric state Dn(p) |s = some other |s′ ∈ B — modulo PBC

Few-body resonances from finite-volume calculations –

  • p. 13
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SLIDE 24

(Anti-)symmetrization and parity

Permutation symmetry

for each |s ∈ B, construct |s

A = N

  • p∈Sn

sgn(p) Dn(p) |s then |sA is antisymmetric: A |s

A = |s A

for bosons, leave out sgn(p) symmetric state Dn(p) |s = some other |s′ ∈ B — modulo PBC

This operation partitions the orginal basis, i.e., each state appears in at most one (anti-)symmetric combination.

efficient reduction to (anti-)symmetrized orthonormal basis ֒ → no need for numerically expensive diagonalization! B → Breduced, significantly smaller: N → Nreduced ≈ N/n! Note: parity (with projector P± = 1 ± P) can be handled analogously.

Few-body resonances from finite-volume calculations –

  • p. 13
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SLIDE 25

DVR computational aspects

DVR basis size N = Nspin ( × Nisospin) × Nndim×(nbody−1)

DVR

Nspin = (2S + 1)nbody, Nisospin = 1 for neutrons only 3n: 8 × N6

DVR, 4n: 16 × N9 DVR large-scale calculation

Few-body resonances from finite-volume calculations –

  • p. 14
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SLIDE 26

DVR computational aspects

DVR basis size N = Nspin ( × Nisospin) × Nndim×(nbody−1)

DVR

Nspin = (2S + 1)nbody, Nisospin = 1 for neutrons only 3n: 8 × N6

DVR, 4n: 16 × N9 DVR large-scale calculation

diagonalization via distributed Lanczos algorithm (PARPACK) large matrix-vector products kinetic part (via FFT) in original basis (before reduction) ֒ → expansion/reduction via sparse matrices

        =

reduce

  ×

  • F−1⊗ ˆ

p2 ⊗ F

  • ×

expand

           (note: kinetic matrix diagonal in spin-configurations space)

Few-body resonances from finite-volume calculations –

  • p. 14
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SLIDE 27

DVR computational aspects

DVR basis size N = Nspin ( × Nisospin) × Nndim×(nbody−1)

DVR

Nspin = (2S + 1)nbody, Nisospin = 1 for neutrons only 3n: 8 × N6

DVR, 4n: 16 × N9 DVR large-scale calculation

diagonalization via distributed Lanczos algorithm (PARPACK) large matrix-vector products kinetic part (via FFT) in original basis (before reduction) ֒ → expansion/reduction via sparse matrices

        =

reduce

  ×

  • F−1⊗ ˆ

p2 ⊗ F

  • ×

expand

           (note: kinetic matrix diagonal in spin-configurations space)

Few-body resonances from finite-volume calculations –

  • p. 14
slide-28
SLIDE 28

DVR computational aspects

DVR basis size N = Nspin ( × Nisospin) × Nndim×(nbody−1)

DVR

Nspin = (2S + 1)nbody, Nisospin = 1 for neutrons only 3n: 8 × N6

DVR, 4n: 16 × N9 DVR large-scale calculation

diagonalization via distributed Lanczos algorithm (PARPACK) large matrix-vector products kinetic part (via FFT) in original basis (before reduction) ֒ → expansion/reduction via sparse matrices

        =

reduce

  ×

  • F−1⊗ ˆ

p2 ⊗ F

  • ×

expand

           (note: kinetic matrix diagonal in spin-configurations space)

potential part still diagonal in symmetry-reduced basis

Few-body resonances from finite-volume calculations –

  • p. 14
slide-29
SLIDE 29

Broken symmetry

The finite volume breaks the symmetry of the system: rotation group SO(3)

→

cubic group O Irreducible representations of SO(3) are reducible with respect to O!

finite subgroup of SO(3) number of elements = 24 five irreducible representations

Γ A1 A2 E T1 T2 dim Γ 1 1 2 3 3

Few-body resonances from finite-volume calculations –

  • p. 15
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SLIDE 30

Cubic projection

Cubic projector

PΓ = dim Γ 24

  • R∈O

χΓ(R)Dn(R) , χΓ(R) = character

Johnson, PLB 114 147 (1982)

Dn(R) realizes a cubic rotation R on the n-body DVR basis permutation/inversion of relative coordinate components indices are wrappen back into range −N/2, . . . , N/2 − 1 e.g. − →

Few-body resonances from finite-volume calculations –

  • p. 16
slide-31
SLIDE 31

Cubic projection

Cubic projector

PΓ = dim Γ 24

  • R∈O

χΓ(R)Dn(R) , χΓ(R) = character

Johnson, PLB 114 147 (1982)

Dn(R) realizes a cubic rotation R on the n-body DVR basis permutation/inversion of relative coordinate components indices are wrappen back into range −N/2, . . . , N/2 − 1 e.g. − → numerical implementation: ˆ H → ˆ H + λ(1 − PΓ) , λ ≫ E

Few-body resonances from finite-volume calculations –

  • p. 16
slide-32
SLIDE 32

Two-body check: anything goes

V (r) = V0 exp

  • −

r − a R0 2 ֒ → use barrier to produce S-wave resonance

2 4 6 8 10 r 1 2 3 4 5 6 V(r)

V0 = 6.0 V0 = 2.0 δ(k) [deg] −60 −120 −180 −240 −300 −360 −420 E 2 4 6 8 10

Few-body resonances from finite-volume calculations –

  • p. 17
slide-33
SLIDE 33

Two-body check: anything goes

V (r) = V0 exp

  • −

r − a R0 2 ֒ → use barrier to produce S-wave resonance

2 4 6 8 10 r 1 2 3 4 5 6 V(r)

V0 = 6.0 V0 = 2.0 δ(k) [deg] −60 −120 −180 −240 −300 −360 −420 E 2 4 6 8 10

phase shifts

Few-body resonances from finite-volume calculations –

  • p. 17
slide-34
SLIDE 34

Two-body check: anything goes

V (r) = V0 exp

  • −

r − a R0 2 ֒ → use barrier to produce S-wave resonance

2 4 6 8 10 r 1 2 3 4 5 6 V(r)

V0 = 6.0 V0 = 2.0 δ(k) [deg] −60 −120 −180 −240 −300 −360 −420 E 2 4 6 8 10

phase shifts

i −0.039 1.592 Im E

−0.15 −0.12 −0.10 −0.08 −0.05 −0.03 0.00 Re E 1.00 1.25 1.50 1.75 2.00 2.25

S-matrix pole

Few-body resonances from finite-volume calculations –

  • p. 17
slide-35
SLIDE 35

Two-body check: anything goes

V (r) = V0 exp

  • −

r − a R0 2 ֒ → use barrier to produce S-wave resonance

2 4 6 8 10 r 1 2 3 4 5 6 V(r)

V0 = 6.0 V0 = 2.0 δ(k) [deg] −60 −120 −180 −240 −300 −360 −420 E 2 4 6 8 10

phase shifts

i −0.039 1.592 Im E

−0.15 −0.12 −0.10 −0.08 −0.05 −0.03 0.00 Re E 1.00 1.25 1.50 1.75 2.00 2.25

S-matrix pole finite-volume spectra

5 6 7 8 9 10

L

2 4 6 8 10

E A+

1 rep.

5 6 7 8 9 10

L

2 4 6 8 10

E A+

1 rep.

Few-body resonances from finite-volume calculations –

  • p. 17
slide-36
SLIDE 36

Three-body check

Take established three-body resonance from literature:

Fedorov et al., Few-Body Syst. P 33 153 (2003); Blandon et al., PRA 75 042508 (2007)

V (r) = V0 exp

  • −

r R0 2 + V1 exp

  • −

r − a R1 2

V0 = −55 MeV, V1 = 1.5 MeV, R0 = √ 5 fm, R1 = 10 fm, a = 5 fm

5 10 15 20 r

  • 50
  • 40
  • 30
  • 20
  • 10

10 V(r)

three spinless bosons with mass m = 939.0 MeV two- and three-body bound states at −6.76 MeV and −37.22 MeV three-body resonance at −5.31 − i0.12 MeV (Blandon et al.), −5.96 − i0.40 MeV (Fedorov et al.)

Few-body resonances from finite-volume calculations –

  • p. 18
slide-37
SLIDE 37

Three-body check

Take established three-body resonance from literature:

Fedorov et al., Few-Body Syst. P 33 153 (2003); Blandon et al., PRA 75 042508 (2007)

V (r) = V0 exp

  • −

r R0 2 + V1 exp

  • −

r − a R1 2

V0 = −55 MeV, V1 = 1.5 MeV, R0 = √ 5 fm, R1 = 10 fm, a = 5 fm

5 10 15 20 r

  • 50
  • 40
  • 30
  • 20
  • 10

10 V(r)

three spinless bosons with mass m = 939.0 MeV two- and three-body bound states at −6.76 MeV and −37.22 MeV three-body resonance at −5.31 − i0.12 MeV (Blandon et al.), −5.96 − i0.40 MeV (Fedorov et al.)

20 25 30 35 40 45

L [fm]

−7 −6 −5 −4 −3 −2 −1 1

E [MeV] ↓ ground state at −37.3 MeV

A+

1

E+ T +

2

fit inflection point(s) to extract resonance energy ER = −5.32(1) MeV

Few-body resonances from finite-volume calculations –

  • p. 18
slide-38
SLIDE 38

Three bosons with shifted Gaussian interaction

three-boson system

shifted Gaussian 2-body potential note: no 2-body bound state!

1 2 3 4 5 r 0.5 1.0 1.5 2.0 V(r)

5 6 7 8 9 10

box size

1 2 3 4 5 6 7 8

energy

A+

1 rep.

Few-body resonances from finite-volume calculations –

  • p. 19
slide-39
SLIDE 39

Three bosons with shifted Gaussian interaction

three-boson system

shifted Gaussian 2-body potential note: no 2-body bound state! add short-range 3-body force

1 2 3 4 5 r 0.5 1.0 1.5 2.0 V(r)

+ V3 = 0

5 6 7 8 9 10

box size

1 2 3 4 5 6 7 8

energy

A+

1 rep.

Few-body resonances from finite-volume calculations –

  • p. 19
slide-40
SLIDE 40

Three bosons with shifted Gaussian interaction

three-boson system

shifted Gaussian 2-body potential note: no 2-body bound state! add short-range 3-body force

1 2 3 4 5 r 0.5 1.0 1.5 2.0 V(r)

+ V3 = −1.0

5 6 7 8 9 10

box size

1 2 3 4 5 6 7 8

energy

A+

1 rep.

V3 = −1

Few-body resonances from finite-volume calculations –

  • p. 19
slide-41
SLIDE 41

Three bosons with shifted Gaussian interaction

three-boson system

shifted Gaussian 2-body potential note: no 2-body bound state! add short-range 3-body force

1 2 3 4 5 r 0.5 1.0 1.5 2.0 V(r)

+ V3 = −2.0

5 6 7 8 9 10

box size

1 2 3 4 5 6 7 8

energy

A+

1 rep.

V3 = −2

Few-body resonances from finite-volume calculations –

  • p. 19
slide-42
SLIDE 42

Three bosons with shifted Gaussian interaction

three-boson system

shifted Gaussian 2-body potential note: no 2-body bound state! add short-range 3-body force

1 2 3 4 5 r 0.5 1.0 1.5 2.0 V(r)

+ V3 = −4.0

5 6 7 8 9 10

box size

1 2 3 4 5 6 7 8

energy

A+

1 rep.

V3 = −4

֒ → possible to move three-body state ↔ spatially localized wf.

Few-body resonances from finite-volume calculations –

  • p. 19
slide-43
SLIDE 43

Three fermions

Consider same shifted Gaussian potential for three fermions. . . add spin d.o.f., but no spin dependence in potential total spin S good quantum number (fix Sz to determine) also: can still consider simple cubic irreps.

6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0

L

4 5 6 7 8 9

E

T −

1 rep. S = 1/2 S = 3/2 1 2 3 4 5 r 0.5 1.0 1.5 2.0 V(r)

V0 = 2.0, a = 3.0, R = 1.5

all lowest states found to be in T −

1 irrep. (∼ P-wave state)

some remaining volume dependence (box not very large) extracted S = 1/2 resonance energy: ER = 5.7(2)

Few-body resonances from finite-volume calculations –

  • p. 20
slide-44
SLIDE 44

Four-boson resonance

Still same potential, look at four bosons. . .

6.5 7.0 7.5 8.0 8.5

L

4 5 6 7 8 9 10 11 12

E

A+

1 rep.

֒ → (supposedly) narrow resonance at ER = 7.31(8)

Few-body resonances from finite-volume calculations –

  • p. 21
slide-45
SLIDE 45

Summary and outlook

method established for up to four particles handle large NDVR for three-body systems (current record: 32) efficient symmetrization and antisymmetrization projection onto cubic irreps. (H → H + λ(1 − PΓ), λ large)

Few-body resonances from finite-volume calculations –

  • p. 22
slide-46
SLIDE 46

Summary and outlook

method established for up to four particles handle large NDVR for three-body systems (current record: 32) efficient symmetrization and antisymmetrization projection onto cubic irreps. (H → H + λ(1 − PΓ), λ large)

Work in progress

chiral interactions (non-diagonal due to spin dependence!) application to few-neutron systems further optimization (especially for spin-dep. potentials) ֒ → need to reach decent NDVR for four-neutron calculation! isospin degrees of freedom treat general nuclear systems different boundary conditions (e.g., antiperiodic)

Few-body resonances from finite-volume calculations –

  • p. 22
slide-47
SLIDE 47

Summary and outlook

method established for up to four particles handle large NDVR for three-body systems (current record: 32) efficient symmetrization and antisymmetrization projection onto cubic irreps. (H → H + λ(1 − PΓ), λ large)

Work in progress

chiral interactions (non-diagonal due to spin dependence!) application to few-neutron systems further optimization (especially for spin-dep. potentials) ֒ → need to reach decent NDVR for four-neutron calculation! isospin degrees of freedom treat general nuclear systems different boundary conditions (e.g., antiperiodic)

*** Thank you! ***

Few-body resonances from finite-volume calculations –

  • p. 22