Progress towards nucleon-nucleon interactions with stochastic LapH - - PowerPoint PPT Presentation

Progress towards nucleon-nucleon interactions with stochastic LapH

Ben Hörz (LBNL) Frontiers in Lattice QCD and related topics Yukawa Institute for Theoretical Physics, Kyoto University Apr 17, 2019

Hadron interactions from Lattice QCD

[Estabrooks, Martin 1975] [Protopopescu et al. 1973]

nucleon-nucleon interactions nucleon-hyperon interactions NΣ, NΛ

∆

⟨Nπ|Jµ|N⟩

multi-hadron state

• 1. What can we learn about the

QCD spectrum from first principles?

• 2. Lattice QCD as a tool for nuclear

physics

[NuSTEC White Paper: Status and Challenges of Neutrino-Nucleus Scattering 1706.03621]

1/16

Hadron interactions from Lattice QCD

[Estabrooks, Martin 1975] [Protopopescu et al. 1973]

nucleon-nucleon interactions nucleon-hyperon interactions NΣ, NΛ

∆

⟨Nπ|Jµ|N⟩

multi-hadron state

• 1. What can we learn about the

QCD spectrum from first principles?

• 2. Lattice QCD as a tool for nuclear

physics

[NuSTEC White Paper: Status and Challenges of Neutrino-Nucleus Scattering 1706.03621]

1/16

Hadron interactions from Lattice QCD

[Estabrooks, Martin 1975] [Protopopescu et al. 1973]

nucleon-nucleon interactions nucleon-hyperon interactions NΣ, NΛ

∆

⟨Nπ|Jµ|N⟩

multi-hadron state

• 1. What can we learn about the

QCD spectrum from first principles?

• 2. Lattice QCD as a tool for nuclear

physics

[NuSTEC White Paper: Status and Challenges of Neutrino-Nucleus Scattering 1706.03621]

1/16

Hadron interactions from Lattice QCD

[Estabrooks, Martin 1975] [Protopopescu et al. 1973]

nucleon-nucleon interactions nucleon-hyperon interactions NΣ, NΛ

∆

⟨Nπ|Jµ|N⟩

multi-hadron state

• 1. What can we learn about the

QCD spectrum from first principles?

• 2. Lattice QCD as a tool for nuclear

physics

[NuSTEC White Paper: Status and Challenges of Neutrino-Nucleus Scattering 1706.03621]

1/16

Scattering from Lattice QCD

L single particle in a periodic box ⇝ ∆E ∝ e−mL two spinless particles in a periodic box ⇝ ∆E ∝ a0/L3 + O(L−4)

[Lüscher ’86, ’91]

L ⇒ ‘The Lüscher method’

2/16

Review of formalism and results

[Briceño, Dudek, Young 1706.06223] see also [Hansen, Sharpe 1901.00483]

3/16

Two-particle Quantization Condition

det [ M−1(EL) + F(EL, L) ] = 0

2-particle channel partial wave (total angular mom.)

EL – FV spectrum M – 2-to-2 scatt. ampl. F – known functions

group theory worked out and publicly available

• n Github

[Morningstar, Bulava, Singha, Brett, Fallica, Hanlon, BH 1707.05817]

4/16

Two-particle Quantization Condition

det [ M−1(EL) + F(EL, L) ] = 0

2-particle channel partial wave (total angular mom.)

EL – FV spectrum M – 2-to-2 scatt. ampl. F – known functions

group theory worked out and publicly available

• n Github

[Morningstar, Bulava, Singha, Brett, Fallica, Hanlon, BH 1707.05817]

4/16

A simple (yet relevant) resonance: ρ(770)

2.0 2.5 3.0 3.5 4.0 10 50 90 130 170 E∗/mπ δ1/◦ T +

1u(0)

A+

1 (1)

E+ ( 1 ) A+

1 (2)

B+

1 (2)

B+

2 (2)

A+

1 (3)

E+ ( 3 ) 2.0 2.5 3.0 3.5 4.0 E∗/mπ

[plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593]

• elastic ππ scattering neglecting ℓ ≥ 3 partial wave

spectrum ⇔ scattering amplitude

• benchmark system for the lattice

e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, …

• recent interest due to its contribution to (g−2)µ HVP

[Meyer, Wittig 1807.09370]

2.0 2.5 3.0 3.5 4.0 −10 −5 5 10 E∗/mπ (qcm/mπ)3 cot δ1

⇔

difgerent way to plot same data

5/16

A simple (yet relevant) resonance: ρ(770)

2.0 2.5 3.0 3.5 4.0 10 50 90 130 170 E∗/mπ δ1/◦ T +

1u(0)

A+

1 (1)

E+ ( 1 ) A+

1 (2)

B+

1 (2)

B+

2 (2)

A+

1 (3)

E+ ( 3 ) 2.0 2.5 3.0 3.5 4.0 E∗/mπ

[plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593]

• elastic ππ scattering neglecting ℓ ≥ 3 partial wave

spectrum ⇔ scattering amplitude

• benchmark system for the lattice

e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, …

• recent interest due to its contribution to (g−2)µ HVP

[Meyer, Wittig 1807.09370]

2.0 2.5 3.0 3.5 4.0 −10 −5 5 10 E∗/mπ (qcm/mπ)3 cot δ1

⇔

difgerent way to plot same data

5/16

A simple (yet relevant) resonance: ρ(770)

2.0 2.5 3.0 3.5 4.0 10 50 90 130 170 E∗/mπ δ1/◦ T +

1u(0)

A+

1 (1)

E+ ( 1 ) A+

1 (2)

B+

1 (2)

B+

2 (2)

A+

1 (3)

E+ ( 3 ) 2.0 2.5 3.0 3.5 4.0 E∗/mπ

[plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593]

• elastic ππ scattering neglecting ℓ ≥ 3 partial wave

spectrum ⇔ scattering amplitude

• benchmark system for the lattice

e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, …

• recent interest due to its contribution to (g−2)µ HVP

[Meyer, Wittig 1807.09370]

2.0 2.5 3.0 3.5 4.0 −10 −5 5 10 E∗/mπ (qcm/mπ)3 cot δ1

⇔

difgerent way to plot same data

5/16

A simple (yet relevant) resonance: ρ(770)

2.0 2.5 3.0 3.5 4.0 10 50 90 130 170 E∗/mπ δ1/◦ T +

1u(0)

A+

1 (1)

E+ ( 1 ) A+

1 (2)

B+

1 (2)

B+

2 (2)

A+

1 (3)

E+ ( 3 ) 2.0 2.5 3.0 3.5 4.0 E∗/mπ

[plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593]

• elastic ππ scattering neglecting ℓ ≥ 3 partial wave

spectrum ⇔ scattering amplitude

• benchmark system for the lattice

e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, …

• recent interest due to its contribution to (g−2)µ HVP

[Meyer, Wittig 1807.09370]

2.0 2.5 3.0 3.5 4.0 −10 −5 5 10 E∗/mπ (qcm/mπ)3 cot δ1

⇔

difgerent way to plot same data

5/16

A simple (yet relevant) resonance: ρ(770)

2.0 2.5 3.0 3.5 4.0 10 50 90 130 170 E∗/mπ δ1/◦ T +

1u(0)

A+

1 (1)

E+ ( 1 ) A+

1 (2)

B+

1 (2)

B+

2 (2)

A+

1 (3)

E+ ( 3 ) 2.0 2.5 3.0 3.5 4.0 E∗/mπ

[plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593]

• elastic ππ scattering neglecting ℓ ≥ 3 partial wave

spectrum ⇔ scattering amplitude

• benchmark system for the lattice

e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, …

• recent interest due to its contribution to (g−2)µ HVP

[Meyer, Wittig 1807.09370]

2.0 2.5 3.0 3.5 4.0 −10 −5 5 10 E∗/mπ (qcm/mπ)3 cot δ1

⇔

difgerent way to plot same data

5/16

A simple (yet relevant) resonance: ρ(770)

2.0 2.5 3.0 3.5 4.0 10 50 90 130 170 E∗/mπ δ1/◦ T +

1u(0)

A+

1 (1)

E+ ( 1 ) A+

1 (2)

B+

1 (2)

B+

2 (2)

A+

1 (3)

E+ ( 3 ) 2.0 2.5 3.0 3.5 4.0 E∗/mπ

[plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593] mπ=233 MeV

• elastic ππ scattering neglecting ℓ ≥ 3 partial wave

spectrum ⇔ scattering amplitude

• benchmark system for the lattice

e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, …

• recent interest due to its contribution to (g−2)µ HVP

[Meyer, Wittig 1807.09370]

2.0 2.5 3.0 3.5 4.0 −10 −5 5 10 E∗/mπ (qcm/mπ)3 cot δ1

⇔

difgerent way to plot same data

5/16

A simple (yet relevant) resonance: ρ(770)

2.0 2.5 3.0 3.5 4.0 10 50 90 130 170 E∗/mπ δ1/◦ T +

1u(0)

A+

1 (1)

E+ ( 1 ) A+

1 (2)

B+

1 (2)

B+

2 (2)

A+

1 (3)

E+ ( 3 ) 2.0 2.5 3.0 3.5 4.0 E∗/mπ

[plot adapted from Bulava, Fahy, BH, Juge, Morningstar, Wong 1604.05593] mπ=233 MeV

• elastic ππ scattering neglecting ℓ ≥ 3 partial wave

spectrum ⇔ scattering amplitude

• benchmark system for the lattice

e.g. Lang et al. 1105.5636, Aoki et al. 1106.5365, …, Dudek et al. 1212.0830, …

• recent interest due to its contribution to (g−2)µ HVP

[Meyer, Wittig 1807.09370]

2.0 2.5 3.0 3.5 4.0 −10 −5 5 10 E∗/mπ (qcm/mπ)3 cot δ1

⇔

difgerent way to plot same data

5/16

Matrix elements: Timelike pion form factor

µ µ

QCD

γ

• muon anomalous magnetic moment (g − 2)µ
• HVP governed by

Rhad = σ(e+e− → hadrons)/ 4παem(s)2

3s

• two-pion state dominates at low energies

Rhad(s) = 1

4

( 1 − 4m2

π

s

) 3

2

|Fπ(s)|2

γ∗ → ππ

|Fπ(E∗)|2 = gΛ(γ) q ∂(δ1+F )

∂q 3πE∗2 2q5L3

• ⟨0|V (d,Λ)|dΛE∗⟩
• 2

Lellouch, Lüscher hep-lat/0003023 Meyer 1105.1892 Feng et al. 1412.6319

infinite volume finite volume requires scattering amplitude

6/16

Matrix elements: Timelike pion form factor

µ µ

QCD

γ

• muon anomalous magnetic moment (g − 2)µ
• HVP governed by

Rhad = σ(e+e− → hadrons)/ 4παem(s)2

3s

• two-pion state dominates at low energies

Rhad(s) = 1

4

( 1 − 4m2

π

s

) 3

2

|Fπ(s)|2

γ∗ → ππ

|Fπ(E∗)|2 = gΛ(γ) q ∂(δ1+F )

∂q 3πE∗2 2q5L3

• ⟨0|V (d,Λ)|dΛE∗⟩
• 2

Lellouch, Lüscher hep-lat/0003023 Meyer 1105.1892 Feng et al. 1412.6319

infinite volume finite volume requires scattering amplitude

6/16

Vector-correlator reconstruction

mπ = 200 MeV, L = 4.1 fm

[Gérardin, Cè, von Hippel, BH, Meyer, Mohler, Ottnad, Wilhelm, Wittig 1904.03120]

• time-momentum representation

[Bernecker, Meyer 1107.4388]

• clear spectral decomposition

Gl(t) ∼ ∑

x

⟨0|V (x, t)V †(0)|0⟩ ≈

N

∑

n

|⟨0|V |T1u, n⟩|2 e−Ent

7/16

Vector-correlator reconstruction

mπ = 200 MeV, L = 4.1 fm

[Gérardin, Cè, von Hippel, BH, Meyer, Mohler, Ottnad, Wilhelm, Wittig 1904.03120]

• time-momentum representation

[Bernecker, Meyer 1107.4388]

• clear spectral decomposition

Gl(t) ∼ ∑

x

⟨0|V (x, t)V †(0)|0⟩ ≈

N

∑

n

|⟨0|V |T1u, n⟩|2 e−Ent

7/16

Vector-correlator reconstruction

mπ = 200 MeV, L = 4.1 fm

[Gérardin, Cè, von Hippel, BH, Meyer, Mohler, Ottnad, Wilhelm, Wittig 1904.03120]

• time-momentum representation

[Bernecker, Meyer 1107.4388]

• clear spectral decomposition

Gl(t) ∼ ∑

x

⟨0|V (x, t)V †(0)|0⟩ ≈

N

∑

n

|⟨0|V |T1u, n⟩|2 e−Ent

7/16

Vector-correlator reconstruction

mπ = 200 MeV, L = 4.1 fm

[Gérardin, Cè, von Hippel, BH, Meyer, Mohler, Ottnad, Wilhelm, Wittig 1904.03120]

• time-momentum representation

[Bernecker, Meyer 1107.4388]

• clear spectral decomposition

Gl(t) ∼ ∑

x

⟨0|V (x, t)V †(0)|0⟩ ≈

N

∑

n

|⟨0|V |T1u, n⟩|2 e−Ent

7/16

Vector-correlator reconstruction

mπ = 200 MeV, L = 4.1 fm

[Gérardin, Cè, von Hippel, BH, Meyer, Mohler, Ottnad, Wilhelm, Wittig 1904.03120]

• time-momentum representation

[Bernecker, Meyer 1107.4388]

• clear spectral decomposition

Gl(t) ∼ ∑

x

⟨0|V (x, t)V †(0)|0⟩ ≈

N

∑

n

|⟨0|V |T1u, n⟩|2 e−Ent

7/16

Timelike pion form factor results

[Andersen, Bulava, BH, Morningstar 1808.05007]

2.0 2.5 3.0 3.5 5 10 Ecm/mπ |Fπ| t❤r✐❝❡✲s✉❜tr✳ ❞✐s♣✳

• ❙

2.0 2.5 3.0 5 10 15 20 Ecm/mπ |Fπ| a = .076 fm a = .064 fm

mπ = 260 MeV mπ = 280 MeV

allows for a check of HVP FV efgects ⇝ formalism and technology under control for two-meson systems

8/16

Towards meson-baryon systems

∆ resonance (I = 3/2 Nπ scattering)

• combinatorically harder than meson-meson
• typically worse signal-to-noise
• scattering of particles with spin

rotational-symmetry breaking more restricting

[Andersen, Bulava, BH, Morningstar 1710.01557] mπ=280 MeV

• Breit-Wigner shape
• ∆(1232) found on threshold
• more precision required for

matrix elements

(limited by correlator construction) 9/16

NN scattering

[Briceño, Davoudi, Luu 1305.4903]

0+ 0− 1+ 1− 2+ 2− 3+ I = 0 – – {(0, 1), (2, 1)} (1, 0) (2, 1) – (2, 1) I = 1 (0,0) (1, 1) – (1, 1) (2, 0) (1, 1) –

FV

(L, S)

[000] [00n] [0nn] 0+ A1g A1 A1 0− A1u A2 A2 1+ T1g (A2 ⊕ E) (A2 ⊕ B1 ⊕ B2) 1− T1u (A1 ⊕ E) (A1 ⊕ B1 ⊕ B2) 2+ Eg ⊕ T2g (A1 ⊕ B1) ⊕ (B2 ⊕ E) (A1 ⊕ B2) ⊕ (A1 ⊕ A2 ⊕ B1) 2− Eu ⊕ T2u (A2 ⊕ B2) ⊕ (B1 ⊕ E) (A2 ⊕ B1) ⊕ (A1 ⊕ A2 ⊕ B2) 3+ A2g ⊕ T1g ⊕ T2g B1 ⊕ (A2 ⊕ E) ⊕ (B2 ⊕ E) B2 ⊕ (A2 ⊕ B1 ⊕ B2)⊕(A1 ⊕ A2 ⊕ B1

⇝ large degeneracy – many operators – even more correlation functions

10/16

Correlation functions in stochastic laph

[Morningstar et al. 1104.3870]

Γ Γ′

X[r1,l]η[r1,l]† X[r2,m]η[r2,m]† X[r3,n]η[r3,n]† noise index dilution index

• quark propagators outer products

DX[r,l] = η[r,l]

• define baryon-sink function

(analogously for source)

B[z](l,m,n) = ∑

x

e−ipxϵabcΓαβγX[r1,l]

αa

X[r2,m]

βb

X[r3,n]

γc

combined noise, momentum, Γ index

computing correlation functions ≡ tensor contractions

rank-3 tensors B[z](l,m,n) – dilution index range Ndil = O(64)

• factorizes computation of observable

(baryon functions computed timeslice by timeslice)

• reusable for multi-hadron correlators

11/16

Two-nucleon correlation function

48 diagrams (I = 1 NN), 36 diagrams (I = 0 NN)

• ‘unroll’ group-theoretic linear combinations

(many elemental diagrams to compute – a lot of redundancy)

• quark-level optimization investigated before

Doi, Endres 1205.0585 Detmold, Orginos 1207.1452 Wynen et al. 1810.12747

• contraction order matters at tensor level

well-known in quantum chemistry

Hartono et al. Automated Operation Minimization of Tensor Contraction Expressions in Electronic Structure Calculations, 2005 Hartono et al. Identifying Cost-Efgective Common Subexpressions to Reduce Operation Count in Tensor Contraction Evaluations, 2006

more recently in the context of tensor networks

Pfeifer et al. Faster identification of optimal contraction sequences for tensor networks, 1304.6112

12/16

Two-nucleon correlation function (II)

[4] [3] [1] [2] [4] [5] [1] [2]

• single-term optimization

(find the best contraction(s) in a diagram)

• global optimization

(compare utility across all diagrams)

• NN diagrams can be evaluated at
• r

(need to keep track of noise indices etc.)

• a single NN correlator:

w/o CSE 12,992 3,584 25,984 w/ CSE 2,352 64 1,080 13/16

Two-nucleon correlation function (II)

[4] [3] [1] [2]

✗ N 5

dil [4] [5] [1] [2]

• single-term optimization

(find the best contraction(s) in a diagram)

• global optimization

(compare utility across all diagrams)

• NN diagrams can be evaluated at
• r

(need to keep track of noise indices etc.)

• a single NN correlator:

w/o CSE 12,992 3,584 25,984 w/ CSE 2,352 64 1,080 13/16

Two-nucleon correlation function (II)

[4] [3] [1] [2]

✓ N 4

dil [4] [5] [1] [2]

• single-term optimization

(find the best contraction(s) in a diagram)

• global optimization

(compare utility across all diagrams)

• NN diagrams can be evaluated at
• r

(need to keep track of noise indices etc.)

• a single NN correlator:

w/o CSE 12,992 3,584 25,984 w/ CSE 2,352 64 1,080 13/16

Two-nucleon correlation function (II)

[4] [3] [1] [2] [4] [5] [1] [2]

• single-term optimization

(find the best contraction(s) in a diagram)

• global optimization

(compare utility across all diagrams)

• NN diagrams can be evaluated at
• r

(need to keep track of noise indices etc.)

• a single NN correlator:

w/o CSE 12,992 3,584 25,984 w/ CSE 2,352 64 1,080 13/16

Two-nucleon correlation function (II)

[4] [3] [1] [2] [4] [5] [1] [2]

• single-term optimization

(find the best contraction(s) in a diagram)

• global optimization

(compare utility across all diagrams)

• NN diagrams can be evaluated at
• r

(need to keep track of noise indices etc.)

• a single NN correlator:

w/o CSE 12,992 3,584 25,984 w/ CSE 2,352 64 1,080 13/16

Two-nucleon correlation function (II)

[4] [3] [1] [2] [4] [5] [1] [2]

• single-term optimization

(find the best contraction(s) in a diagram)

• global optimization

(compare utility across all diagrams)

• NN diagrams can be evaluated at 2N 4

dil + N 2 dil or 2N 3 dil

(need to keep track of noise indices etc.)

• a single NN correlator:

w/o CSE 12,992 3,584 25,984 w/ CSE 2,352 64 1,080 13/16

Two-nucleon correlation function (II)

[4] [3] [1] [2] [4] [5] [1] [2]

• single-term optimization

(find the best contraction(s) in a diagram)

• global optimization

(compare utility across all diagrams)

• NN diagrams can be evaluated at 2N 4

dil + N 2 dil or 2N 3 dil

(need to keep track of noise indices etc.)

• a single NN correlator:

N 2

dil

N 3

dil

N 4

dil

w/o CSE 12,992 3,584 25,984 w/ CSE 2,352 64 1,080 13/16

New workflow

Quark propagators

4D lattice

Hadron functions

3D lattice

Tune correlator runs

serial code

Correlator construction

multithreaded tensor contractions

• backend-agnostic implementation in C++

with Python interface

• will be publicly available aħter licensing

14/16

Another example: I = 3 πππ

• whole machinery readily applicable

5 10 15 20 25 30 35 40 .002 .004 .006 .008 .01 t/a a∆Eeff mπ = 200 MeV

• at-rest ground state π(0)π(0)π(0)

N 2

dil

N 3

dil

w/o CSE 47,520 60,480 w/ CSE 450 360

• ∆E = E3π − 3mπ

15/16

Summary & Next Steps

• two-hadron interactions from lattice QCD …

(calculations maturing, starting to assess standard lattice systematics)

• …and with practical relevance

(helping to improve (g−2)µ/HVP from lattice QCD)

• addressing (gross) ineffjciencies as we go

(contractions hopefully won’t be a problem again any time soon)

• stay tuned for
• NN scattering
• NΣ, NΛ scattering
• …

16/16