pion scattering to (g-2) at physical pion mass - cern indico

Pion Scattering to (g − 2)µ

at Physical Pion Mass

Aaron S. Meyer ([email protected])

Brookhaven National Laboratory

August 2, 2019

Advances in Lattice Gauge Theory 2019

The RBC & UKQCD collaborations

BNL and BNL/RBRC

Ryan AbbotNorman ChristDuo GuoChristopher KellyBob MawhinneyMasaaki TomiiJiqun Tu

University of Connecticut

Peter BoyleLuigi Del DebbioFelix ErbenVera GülpersTadeusz JanowskiJulia KettleMichael MarshallFionn Ó hÓgáinAntonin PortelliTobias TsangAndrew YongAzusa Yamaguchi

Nicolas Garron

Nils AsmussenJonathan FlynnRyan HillAndreas JüttnerJames RichingsChris Sachrajda

Julien Frison

Xu Feng

Bigeng WangTianle WangYidi Zhao

UC Boulder

Yasumichi Aoki (KEK)Taku IzubuchiYong-Chull JangChulwoo JungMeifeng LinAaron MeyerHiroshi OhkiShigemi Ohta (KEK)Amarjit Soni

Oliver Witzel

Columbia University

Tom BlumDan Hoying (BNL)Luchang Jin (RBRC)Cheng Tu

Edinburgh University

University of Southampton

Peking University

University of Liverpool

KEK

Stony Brook University

Jun-Sik YooSergey Syritsyn (RBRC)

MIT

David Murphy

Mattia BrunoCERN

Christoph Lehner (BNL)University of Regensburg

Outline

I Muon g − 2 ExperimentI Motivation from muon g − 2I Tensions in ππ ScatteringI Error Budget and LQCD Strategy

I Correlation Function Spectrum & OverlapI Lattice ParametersI GEVP Spectrum & OverlapsI ππ Scattering Phase ShiftI 4π Correlation Functions

I Bounding Method and the Muon HVPI Correlation Function ReconstructionI (Improved) Bounding MethodI Results

I ππ Scattering Phase ShiftsI I = 2, ` = 0I I = 1, ` = 1I I = 2, ` = 2

I Conclusions/Outlook

Aaron S. Meyer Pion Scattering to (g − 2)µ at Physical Pion Mass 1/ 46

Introduction


Muon Anomalous Magnetic Moment Experiment

[Bennett et al., 0602035[hep-ex]]~p

~s

See talk by C.Lehner for detailed introduction

High-precision experiment of spin precessionrelative to momentum direction in storage ring

Anomalous frequency ωa = g−22

eBm = aµ eB

m

Sensitive to new physics, and also discrepant with experiment!

Aaron S. Meyer Section: Introduction 3/ 46

Fermilab Muon g − 2 Experiment

Experiment has come a long way (and so has theory!)Aiming for a 4× improvement in uncertainty over the BNL result


Muon g − 2 Theory Error BudgetContribution Value ×1010 Uncertainty ×1010

QED 11 658 471.895 0.008EW 15.4 0.1HVP LO 692.5 2.7HVP NLO −9.84 0.06HVP NNLO 1.24 0.01Hadronic light-by-light 10.5 2.6Total SM prediction 11 659 181.7 3.8BNL E821 result 11 659 209.1 6.3Fermilab E989 target ≈ 1.6

Experiment-Theory difference is 27.4(7.3) =⇒ 3.7σ tension!

Target measurement:Hadronic VacuumPolarization (HVP)

=⇒Lattice results havelarger uncertainty, butsystematically improve

=⇒Dispersive (“R-ratio”)results more precise,but static

No new physicsKNT 2018

Jegerlehner 2017DHMZ 2017DHMZ 2012

HLMNT 2011RBC/UKQCD 2018

Mainz 2019FNAL/HPQCD/MILC 2019

SK 2019ETMC 2018

RBC/UKQCD 2018BMW 2017Mainz 2017

HPQCD 2016ETMC 2013

610 630 650 670 690 710 730 750

Lattice + R-ratio

Lattice

R-ratio

aµ × 1010[1904.09479[hep-lat]]Aaron S. Meyer Section: Introduction 5/ 46

Tensions in Experiment

[Zhang, EPS (2019)]

e

π

R-ratio data for ee → ππ exclusive channel,√

s = 0.6− 0.9 GeV regionTension between most precise measurements (BABAR/KLOE)R-ratio aHVP

µ uncertainty < difference in this channel

Avoid tension by computing precise lattice-only estimate of aHVPµ

Use lattice QCD to inform experiment, resolve discrepancy


Error Budget

[Blum et al., (2018)]

Full program of computations to reduce uncertainties:


Error Budget

[Blum et al., (2018)]


Reduce statistical uncertainties on light connected contribution


Error Budget

[Blum et al., (2018)]


Reduce statistical uncertainties on light connected contributionCompute QED contribution


Error Budget

[Blum et al., (2018)]


Reduce statistical uncertainties on light connected contributionCompute QED contributionImprove lattice spacing determination


Error Budget

[Blum et al., (2018)]


Reduce statistical uncertainties on light connected contributionCompute QED contributionImprove lattice spacing determinationFinite volume and continuum extrapolation study


Exclusive Channels in the HVP

0 5 10 15 20 25 30t/a

10 9

10 7

10 5

10 3

10 1

C(t)

Long distance

π

γ γ

C(t) = 13∑

i

⟨ [ψγiψ

]t

[ψγiψ

]0

⟩

≈∑

n

∣∣ 〈Ω|ψγiψ|n〉∣∣2e−Ent

Correlator has large statistical error in long-distance region,but contributions from high energy states are exponentially suppressed

Long distance correlator dominated by two-pion states,but overlap of vector current with two-pion states is minimal

Strategy:I Construct & measure operators that overlap strongly with ππ statesI Correlate these operators with the local vector currentI aHVP

µ computed by integrating with time-momentum representation kernel,aHVPµ =

∑t wtC(t) [Bernecker et al., 1107.4388 [hep-lat]]


Exclusive Channels in the HVP

0 5 10 15 20 25 30t/a

10 9

10 7

10 5

10 3

10 1

C(t)

Long distance

π

γ γ

C(t) = 13∑

i

⟨ [ψγiψ

]t

[ψγiψ

]0

⟩≈∑

n

∣∣ 〈Ω|ψγiψ|n〉∣∣2e−Ent

Correlator has large statistical error in long-distance region,but contributions from high energy states are exponentially suppressed

Long distance correlator dominated by two-pion states,but overlap of vector current with two-pion states is minimal

Strategy:I Construct & measure operators that overlap strongly with ππ statesI Correlate these operators with the local vector currentI aHVP

µ computed by integrating with time-momentum representation kernel,aHVPµ =

∑t wtC(t) [Bernecker et al., 1107.4388 [hep-lat]]


Pion Scattering

Extraction of ππ exclusive channels requires many operatorsOperator basis generated with distillation, perambulators

I Can access pion scattering phase shifts with dataI Study ρ resonance region at physical pion massI Low 4π threshold, study effects on spectrum, HVP

π

γ γ


LQCD Computation Setup


Computation Details

0.00 0.05 0.10 0.15 0.20 0.25 0.30a[fm]

012345678

ML

243 × 64323 × 64

483 × 96643 × 128963 × 192

483 × 64

Computed on 2 + 1 flavor Mobius Domain Wall Fermions for valance and sea,Mπ at physical value on all ensembles

Computations using distillation setup with Neig eigenvectorsResults in this talk include four ensembles:

I “24ID”: 243 × 64 (4.8 fm), a ≈ 0.194 fm ≈ 1.015 GeV−1

I “32ID”: 323 × 64 (6.2 fm), a ≈ 0.194 fm ≈ 1.015 GeV−1

I “48I”: 483 × 96 (5.5 fm), a ≈ 0.114 fm ≈ 1.730 GeV−1

I “64I”: 643 × 128 (5.3 fm), a ≈ 0.084 fm ≈ 2.359 GeV−1

Future work including other ensembles for better control of extrapolations

Aaron S. Meyer Section: LQCD Computation Setup 16/ 46

OperatorsOperators constructed in I = 1, P-wave channel to impact upon HVPµDesigned to have strong overlap with specific target states,

but all operators unavoidably couple to all states in HVP spectrum

Vector current operators:I Local OJµ =

∑x ψ(x)γµψ(x), µ ∈ 1, 2, 3

I Smeared Ojµ =∑

xyz ψ(x)f (x − z)γµf (z − y)ψ(y)

2π operators with On given by ~pπ ∈ 2πL × (1, 0, 0), (1, 1, 0), (1, 1, 1), (2, 0, 0)

On =∣∣∣∑xyz ψ(x)f (x − z)e−i~pπ·~zγ5f (z − y)ψ(y)

∣∣∣2Also test a 4π operator with ~pπ = 2π

L × (1, 0, 0):

O4π =∣∣∣∑xyz ψ(x)f (x − z)e−i~pπ·~zγ5f (z − y)ψ(y)

∣∣∣2 ∣∣∣∑xy ψ(x)f (x − y)γ5ψ(y)∣∣∣2

Correlators arranged in a N × N symmetric matrix:

⊗ OJµ Ojµ O2π O4π

OJµ CJµJµ CJµ jµ CJµ2π CJµ4π

Ojµ Cjµ jµ Cjµ2π Cjµ4π

O2π C2π2π C2π4πO4π C4π4π

→ C(t)


Generalized EigenValue Problem (GEVP)

Generalized EigenValue Problem to estimate overlap with vector current & energies

C(t) V = C(t + δt) V Λ(δt)

Λnn(δt) ∼ e+Enδt , Vim ∝ 〈Ω| Oi |m〉

C(t) is the matrix of correlation functions from previous slideCompute at fixed δt, vary t: plateau for large t

From result, reconstruct exponential dependence of local vector correlation function

C latt.ij (t) =

N∑n

〈Ω| Oi |n〉〈n| Oj |Ω〉 e−Ent

In theory, infinite number of states contribute to correlation functionIn practice, only finite N necessary to model correlation function=⇒ finite GEVP basis is sufficient


GEVP Results - Jµ + 2π Operators only

0 2 4 6 8 10 12 14t0/a

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

aEn|

t=3

E0E1E2E3E4

PRELIMINARY0 2 4 6 8 10 12 14

t0/a0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||

i|n

| t=

3

n = 0n = 1n = 2n = 3

PRELIMINARY

6-operator basis on 48I ensemble: local+smeared vector, 4×(2π)Data points from solving GEVP at fixed δt

C(t0) V = C(t0 + δt) V Λ(δt) , Λnn(δt) ∼ e+Enδt

Excited state contaminations decay as t0, δt →∞moving right on plot =⇒ asymptote to lowest states’ spectrum & overlaps

Left: Spectrum; Right: Overlap with local vector current


Group Theory & Contraction Engine

An automated group theory engine has been anintegral part of RBC-UKQCD’s automated setup fortwo-pion diagrams in exclusive channel study

Code builds a text representation of operators by performingtensor products and irrep decompositions of lattice operatorswith arbitrary spin & momentum

0 5 10 15 20 25 30t/a

10 4

10 2

100

102

104

106

C(t)

4 B

4 B 4 A

4 B 4 B

4 B 2 (p2 = 1)4 B 2 (p2 = 2)4 B 2 (p2 = 3)

This has resulted in a world-first computation of4π to 4π correlation functions in I = 1 channel

+~p

−~p





0 5 10 15 20 25 30t/a

10 4

10 2

100

102

104

106

C(t)

4 B

4 B 4 A

4 B 4 B

4 B 2 (p2 = 1)4 B 2 (p2 = 2)4 B 2 (p2 = 3)


+~p

−~p


4π Contractions


4π Contractions cont...


4π Contractions cont... cont...


From 4π to Nπ (and Beyond)Group theory engine handles bosonic & fermionic octahedral irreps, any momentumEasily build N-particle operators from single-particle building blocks

Strong need for multiparticle operators, as seen over past two weeks:I Difficulty extracting scattering states without bilocal operatorsI Large multiplicities of Xπ + N states in nucleon correlatorsI When simulating close to physical point, multiparticle thresholds are relevant

Nucleon matrix elements vital for neutrino community (discussion?)Incredible work done by many people, could start tackling these problems soon


GEVP Results - 4π Operators

Jµ + 2π

Jµ + 2π

0 2 4 6 8 10 12 14t0/a

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

aEn|

t=3

E0E1E2E3E4

0 2 4 6 8 10 12 14t0/a

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||

i|n

| t=

3

n = 0n = 1n = 2n = 3

PRELIMINARY

0 2 4 6 8 10 12 14t0/a

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||

i|n

| t=

3

n = 0n = 1n = 2n = 3

PRELIMINARY

Jµ + 2π + 4π

0 2 4 6 8 10 12 14t0/a

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

aEn|

t=3

E0E1E2E3E4E5

0 2 4 6 8 10 12 14t0/a

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||

i|n

| t=

3

n = 0n = 1n = 2n = 3n = 4

0 2 4 6 8 10 12 14t0/a

0

10

20

30

40

50

60

||

4|n

| t=

3

n = 0n = 1n = 2n = 3n = 4

PRELIMINARY

Breakdown of formalism for phase shifts +FVC could occur at 4π thresholdCompute 2π → 4π, 4π → 4π correlation functions and check explicitlySpectrum unaffected by inclusion of 4π operator, but state is resolvable

Overlap of 4π state with local vector current unresolvableOverlap of state with 4π operator significant=⇒ 4π state safely negligible in local vector current



Jµ + 2π

Jµ + 2π

0 2 4 6 8 10 12 14t0/a

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

aEn|

t=3

E0E1E2E3E4

0 2 4 6 8 10 12 14t0/a

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||

i|n

| t=

3

n = 0n = 1n = 2n = 3

PRELIMINARY

0 2 4 6 8 10 12 14t0/a

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||

i|n

| t=

3

n = 0n = 1n = 2n = 3

PRELIMINARY

Jµ + 2π + 4π

0 2 4 6 8 10 12 14t0/a

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

aEn|

t=3

E0E1E2E3E4E5

0 2 4 6 8 10 12 14t0/a

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||

i|n

| t=

3

n = 0n = 1n = 2n = 3n = 4

0 2 4 6 8 10 12 14t0/a

0

10

20

30

40

50

60

||

4|n

| t=

3

n = 0n = 1n = 2n = 3n = 4

PRELIMINARY

Breakdown of formalism for phase shifts +FVC could occur at 4π thresholdCompute 2π → 4π, 4π → 4π correlation functions and check explicitlySpectrum unaffected by inclusion of 4π operator, but state is resolvableOverlap of 4π state with local vector current unresolvable

Overlap of state with 4π operator significant=⇒ 4π state safely negligible in local vector current



Jµ + 2π

Jµ + 2π

0 2 4 6 8 10 12 14t0/a

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

aEn|

t=3

E0E1E2E3E4

0 2 4 6 8 10 12 14t0/a

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||

i|n

| t=

3

n = 0n = 1n = 2n = 3

PRELIMINARY

0 2 4 6 8 10 12 14t0/a

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||

i|n

| t=

3

n = 0n = 1n = 2n = 3

PRELIMINARY

Jµ + 2π + 4π

0 2 4 6 8 10 12 14t0/a

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

aEn|

t=3

E0E1E2E3E4E5

0 2 4 6 8 10 12 14t0/a

0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||

i|n

| t=

3

n = 0n = 1n = 2n = 3n = 4

0 2 4 6 8 10 12 14t0/a

0

10

20

30

40

50

60

||

4|n

| t=

3

n = 0n = 1n = 2n = 3n = 4

PRELIMINARY

Breakdown of formalism for phase shifts +FVC could occur at 4π thresholdCompute 2π → 4π, 4π → 4π correlation functions and check explicitlySpectrum unaffected by inclusion of 4π operator, but state is resolvableOverlap of 4π state with local vector current unresolvableOverlap of state with 4π operator significant=⇒ 4π state safely negligible in local vector current


Correlator Reconstruction andBounding


Correlation Function Reconstruction - 48I

0 10 20 30 40t/a

0

10

20

30

40

50

w(t)

C(t)

local vector current1-state reconstruction2-state reconstruction3-state reconstruction4-state reconstruction

PRELIMINARY0 5 10 15 20 25 30

t/a0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

C n.r

ec(t)

/Cl.v

ec(t)

local vector uncertainty1-state reconstruction2-state reconstruction3-state reconstruction4-state reconstructionPRELIMINARY

Plotted: (weight kernel) × (correlation function); integral → aHVPµ

GEVP results to reconstruct long-distance behavior oflocal vector correlation function needed to compute connected HVP

Explicit reconstruction good estimate of correlation function at long-distance,missing excited states at short-distance

More states =⇒ better reconstruction, can replace C(t) at shorter distances

Aaron S. Meyer Section: Correlator Reconstruction and Bounding 31/ 46

Improved Bounding MethodUse known results in spectrum to make a precise estimate of

upper & lower bound on aHVPµ [RBC (2017)]

C(t; tmax,E) =

C(t) t < tmax

C(tmax)e−E(t−tmax) t ≥ tmax

Upper bound: E ≤ E0, lowest state in spectrum

Lower bound: E ≥ log[ C(tmax)C(tmax+1) ]

BMW Collaboration [K.Miura, Lattice2018] takes E →∞

With good control over lower states in spectrum from exclusive reconstruction,improve bounding method [RBC/UKQCD 2018 (CL@KEK Feb 2018)]:

Replace C(t)→ C(t)−∑N

n |cn|2e−Ent and apply bounding procedure for aµ − δaµ

=⇒ Long distance convergence now ∝ e−EN+1t , lower bound falls faster=⇒ Smaller overall contribution from neglected states

After bounding, add back δaµ =∑∞

t=tmaxwt∑N

n |cn|2e−Ent


Bounding Method Results - 48I

0 10 20 30 40t

580

600

620

640

660

680

700T/

2 tw

(t)C(

t)/10

10

upper boundlower bound

PRELIMINARY

No bounding method: aHVPµ = 646(38)

Bounding method tmax = 3.3 fm, no reconstruction: aHVPµ = 631(16)

Bounding method tmax = 3.0 fm, 1 state reconstruction: aHVPµ = 631(12)


Bounding method tmax = 2.2 fm, 3 state reconstruction: aHVPµ = 624.3(7.5)


Bounding method gives factor of 2 improvement over no bounding method

Improving the bounding method increases gain to factor of 7, including systematicsImprovement should make all-lattice computation of aHVP

µ

competitive with R-ratio by 2020



0 10 20 30 40t

580

600

620

640

660

680

700T/

2 tw

(t)C(

t)/10

10


PRELIMINARY









µ




0 10 20 30 40t

580

600

620

640

660

680

700T/

2 tw

(t)C(

t)/10

10


PRELIMINARY







Bounding method gives factor of 2 improvement over no bounding methodImproving the bounding method increases gain to factor of 7, including systematicsImprovement should make all-lattice computation of aHVP

µcompetitive with R-ratio by 2020


Error Budget and Timeline

No new physicsKNT 2018

Jegerlehner 2017DHMZ 2017DHMZ 2012

HLMNT 2011RBC/UKQCD 2018

Mainz 2019FNAL/HPQCD/MILC 2019

SK 2019ETMC 2018

RBC/UKQCD 2018BMW 2017Mainz 2017

HPQCD 2016ETMC 2013

610 630 650 670 690 710 730 750

Lattice + R-ratio

Lattice

R-ratio

aµ × 1010

Update to RBC-UKQCD calculation including exclusive study in preparation=⇒ could expect precision improvement ×3, error on aHVP

µ at 5× 10−10

=⇒ to be included in g − 2 Theory WP before release of Fermilab first resultsFurther reduction will require full RBC-UKQCD program of computationsWork on the exclusive channel study using bounding method has led to

world-first estimation of finite volume corrections to aHVPµ at physical Mπ

Complete analysis with full suite of systematic improvements ongoing=⇒ precision improvement ×10 over original, target error on aHVP

µ at 1× 10−10

Compare to dispersive (3− 5)× 10−10


ππ Scattering Phase Shifts


ππ Scattering Phase Shifts in LQCDPhase shifts in continuum, infinite volume classified by partial wave, isospin channelBose symmetry over ππ states: (−1)L+I = +1

=⇒Angular momentum conservation violated by lattice symmetriesOur analysis assumes all but lowest partial wave negligible

det[e2iδ(k) − U(k)] = 0 `=0=⇒ tanδ(q) =qπ3/2

Z(1, q2), q =

kL2π

~p(

L2π

)Iso. LQCD irrep Cont. L ≤ 4

(0, 0, 0) 2 A+1 0, 4

E+ 2, 4T +

2 2, 4(0, 0, 0) 1 T −

1 1, 3(1, 0, 0) 2 A1 0, 2, 4

B1 2, 4B2 2, 4E2 2, 4

(1, 1, 0) 2 A1 0, 2, 4A2 2, 4B1 2, 4B2 2, 4

Aaron S. Meyer Section: ππ Scattering Phase Shifts 40/ 46

I = 2 ` = 0 Phase Shift (48I)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0s /GeV

40

30

20

10

0

10

20(k

2 CM)/°

p2 = 0p2 = 1p2 = 2SU(2) PTRoy equation

PRELIMINARY

χPT credit: D.MurphyRoy Eqn credit: M.Bruno

No resonances in I = 2 =⇒ cleanData requires subtraction periodic BC effects in time=⇒ handled through dedicated computation of 2π matrix elements

Breakdown of SU(2)χPT expected at around 500 MeVGood agreement with phenoAaron S. Meyer Section: ππ Scattering Phase Shifts 41/ 46

Phase Shift

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0s /GeV

0

25

50

75

100

125

150

175

11(k

2 CM)/°

24ID 12032ID 15048I 6064I 60Roy equationBreit-Wigner

0.40 0.45 0.50 0.55 0.60 0.65 0.700

5

10

15

20

25

30

35

40

PRELIMINARYBW: cot δ11 = − 2

Γρ(E − Eρ)

Compute ππ scattering phase shifts in I = 1 channel from spectrumStatistics + systematics includedCompare to simple Breit-Wigner parametrization and pheno (courtesy of M.Bruno)Good agreement with pheno for 32ID, 48I, 64I24ID: remnant excited state contaminations, still to be removed


Phase Shift

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0s /GeV

0

25

50

75

100

125

150

175

11(k

2 CM)/°

24ID 12032ID 15048I 6064I 60Roy equationBreit-Wigner

0.40 0.45 0.50 0.55 0.60 0.65 0.700

5

10

15

20

25

30

35

40PRELIMINARY

BW: cot δ11 = − 2Γρ

(E − Eρ)

Compute ππ scattering phase shifts in I = 1 channel from spectrumStatistics + systematics includedCompare to simple Breit-Wigner parametrization and pheno (courtesy of M.Bruno)Good agreement with pheno for 32ID, 48I, 64I24ID: remnant excited state contaminations, still to be removed


I = 2 ` = 2 Phase Shift (48I)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0s /GeV

10.0

7.5

5.0

2.5

0.0

2.5

5.0

7.5

10.0

22(k

2 CM)/°

E(p2 = 0)T2(p2 = 0)B1(p2 = 1)B2(p2 = 1)E2(p2 = 1)A2(p2 = 2)B2(p2 = 2)PRELIMINARY

Other lattice irreps probe higher partial wavesPhase shift small, unresolved =⇒ limit only


Conclusions


Conclusions

Pion scattering is a simple playground to learn aboutmultiparticle scattering in LQCD while addressing aHVP

µ :I Computed 2π → 4π, 4π → 4π correlation functions to show explicitly

that 4π state has negligible effect on HVPI Study of exclusive channels able to significantly reduce statistical uncertainty

on an all-lattice computation of muon HVP=⇒ factor of 4 more statistics on 48I available now=⇒ expect to reach precision of O(5× 10−10) by the end of year=⇒ can target O(1× 10−10) for all-lattice calculation

I Part of ongoing lattice study to address all lattice systematics inRBC+UKQCD HVP computation (see talk by C.Lehner)

I Computed scattering phase shifts in I = 1, 2=⇒ first calculation of ρ resonance at physical Mπ!

I Scattering phase shifts to appear as part of series of papers by RBC+UKQCDWith experience from ππ scattering, will move on to tackle

more challenging problems with Nπ states and beyond

Thank you!

Aaron S. Meyer Section: Conclusions 46/ 46

BACKUP


Neutrino Oscillation Experiments

ArgoNeuT [1405.4261[hep-ex]]

Neutrino oscillation experiments are of great interest to the physics community,seeking to do a high-precision measurement of neutrino oscillation parameters

Large program of research goals addressing fundamental physics questions:I Precision measurements of oscillation parameters,

including δCP & sign[∆m213]

I Measure supernova neutrinosI Search for proton decay

DUNE due to start installation of first detector in 2022, data collection in 2026

This experiment sets a timescale for making theory contributions!

Aaron S. Meyer Section: BACKUP 48/ 46

Flux ⊗ Cross Section

(GeV)E-110 1 10 210

/ G

eV)

2 c

m-3

8 (1

0 c

ross

sec

tion

/ E

0

0.2

0.4

0.6

0.8

1

1.2

1.4

(GeV)E-110 1 10 210

/ G

eV)

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8 (1

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sec

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/ E

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TOTAL

QE

DIS

RES

[1305.7513[hep-ex]]νµ flux [arb.unit]

HyperK [1805.04163[physics.ins-det]]DUNE [1512.06148[physics.ins-det]]

Neutrino interactions classified by their interaction products3 general classes of interaction types:

Quasielastic (QE), Resonance (Res), Deep Inelastic Scattering (DIS)

DUNE events will be a mixture of all three; ∼ 13 events will be resonant

A dominant contribution to systematics in DUNE will be cross section uncertainties=⇒ Stringent requirements on cross section uncertainties, lots at stake=⇒ QE was dominant before; coming under control now =⇒ next step is resonant


Discrepancies with Monte Carlo

[Mahn et al.,1803.08848[hep-ex]]

Resonant interactions typically result in CC1π (charge current with 1 pion) final statesCurrent state of affairs for CC1π interactions is confusingLepton kinematics under control, consistent and agree well with Monte Carlo (GENIE)

Pion kinematics systematically disagree with shapeDifficult to change pion kinematics without breaking other data

=⇒ Need another handle on pion kinematics!Ideally a high-statistics H or D bubble chamber experiment; not likely to happen. . .


Discrepancies with Monte Carlo

[Mahn et al.,1803.08848[hep-ex]]

Resonant interactions typically result in CC1π (charge current with 1 pion) final statesCurrent state of affairs for CC1π interactions is confusingLepton kinematics under control, consistent and agree well with Monte Carlo (GENIE)Pion kinematics systematically disagree with shapeDifficult to change pion kinematics without breaking other data

=⇒ Need another handle on pion kinematics!Ideally a high-statistics H or D bubble chamber experiment; not likely to happen. . .


Nucleon to Delta TransitionsKnowledge of the N → ∆ transition is vital for neutrino oscillation experiments:

I Resonance region for neutrino scattering dominated by transitions to ∆ baryonI Four N → ∆ transition form factors, only C5A is relevantI Assuming a 10% uncertainty on normalization of C5A:

×1/2 for near/far detector ratio×1/3 for resonant/total interactions

=⇒ 1− 2% total theory uncertainty on oscillation observablesCompared to DUNE total error budget of 1− 3%=⇒ 10% knowledge of C5A norm would saturate the entire error budget of DUNE

Added difficulty: transition amplitudes needed to analyze same data that is usedto constrain transition amplitudes =⇒ likely underestimated uncertainties

Room for improvement with Lattice QCD:I Independent measurement to break

theory/experiment degeneracyI Providing realistic uncertainty

estimatesI Improving constraints on form factor

& amplitude uncertainty

∆N

νµ

N′

π

µ−


Nucleon to Delta TransitionsKnowledge of the N → ∆ transition is vital for neutrino oscillation experiments:

I Resonance region for neutrino scattering dominated by transitions to ∆ baryonI Four N → ∆ transition form factors, only C5A is relevantI Assuming a 10% uncertainty on normalization of C5A:

×1/2 for near/far detector ratio×1/3 for resonant/total interactions

=⇒ 1− 2% total theory uncertainty on oscillation observablesCompared to DUNE total error budget of 1− 3%=⇒ 10% knowledge of C5A norm would saturate the entire error budget of DUNE

Added difficulty: transition amplitudes needed to analyze same data that is usedto constrain transition amplitudes =⇒ likely underestimated uncertainties

Room for improvement with Lattice QCD:I Independent measurement to break

theory/experiment degeneracyI Providing realistic uncertainty

estimatesI Improving constraints on form factor

& amplitude uncertainty∆

N

νµ

N′

π

µ−


Multiparticle States in Lattice QCDExtracting properties of resonances in LQCD is complicated by finite volume:

I Resonance properties must be extracted from decay product statesI Particles in multiparticle states are not asymptotic statesI Need to enforce energy & momentum conservation with discretized momentaI Energy eigenstates are superpositions of states with definite particle content

=⇒ avoided level crossings & complicated spectrumArgument can be turned on its head:

Finite volume as a probe of multiparticle scattering states

[Nuc.Phys.B 364, p237.]

Mρ

Mπ= 3.0

ρ Unstable N

νµ

N′

π

µ−


Neutrino Physics to g − 2Studying N → ∆ resonances requires study of Nπ final statesTechnically challenging:

I Fermionic spin states, unequal massesI Many Wick contractions =⇒ large computational costI Signal to noise degrades rapidlyI More than two particle scattering states, e.g. Nππ, open up quickly

This is a long-term goal that we are developing the methodology to address

In the short term, use this formalism to address a simpler but timely physics question:=⇒ ππ scattering in the isospin-1 channel =⇒ (g − 2)µ HVP contribution

This is an ideal starting point:I Simplest channel with resonanceI Above issues significantly mitigated/eliminatedI Consistency checks against other LQCD calculations & experiment available

π

γ γ


Lattice QCD Checklist

Lattice QCD

Experiment

MC Nucleon

NuclearLattice QCD is ideal tool for filling in missing piecesTo have the greatest impact, must satisfy the checklist:

I Process is important for meeting experimental goals X

I Current precision not sufficient X

I Difficult/impractical to measure experimentally X

I Accessible to Lattice QCD X


pion scattering to (g-2) at physical pion mass - cern indico

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