[PPT] - Systematics in nucleon matrix element calculations Jeremy Green PowerPoint Presentation

SLIDE 1

Systematics in nucleon matrix element calculations

Jeremy Green

NIC, DESY, Zeuthen

The 36th International Symposium on Latice Field Theory East Lansing, MI, USA July 22–28, 2018

SLIDE 2

Motivation

Three reasons for studying structure of protons and neutrons:

◮ Understanding the quark and gluon substructure of a hadron. ◮ Understanding nucleons as tools in experiments. ◮ Validation of latice QCD using “benchmark” observables.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 2

SLIDE 3

Neutron beta decay

W − n p ¯ νe e

Coupling to W boson via axial current depends on nucleon axial charge, p(P,s′)|¯ uγ µγ5d|n(P,s) = дA¯ up(P,s′)γ µγ5un(P,s). Interpreted as isovector quark spin contribution.

→ Yi-Bo Yang plenary

PDG 2018: дA = 1.2724(23).

1970 1980 1990 2000 2010

1.28
1.26
1.24
1.22
1.20
1.18
1.16

WEIGHTED AVERAGE

1.2724±0.0023 (Error scaled by 2.2)

BOPP 86 SPEC 4.3 YEROZLIM... 97 CNTR 11.7 LIAUD 97 TPC 2.5 MOSTOVOI 01 CNTR 0.7 SCHUMANN 08 CNTR MUND 13 SPEC 3.6 MENDENHALL 13 UCNA 1.1 DARIUS 17 SPEC

χ2

23.8 (Confidence Level = 0.0002)

1.29
1.28
1.27
1.26
1.25
1.24

λ ≡ gA / gV

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 3

SLIDE 4

Other observables

Precision β-decay experiments may be sensitive to BSM physics; leading contributions are controlled by the scalar and tensor charges:

T. Bhatacharya et al., Phys. Rev. D 85, 054512 (2012) [1110.6448]

p(P,s′)|¯ ud|n(P,s) = дS ¯ up(P,s′)un(P,s), p(P,s′)|¯ uσ µνd|n(P,s) = дT ¯ up(P,s′)σ µνun(P,s). Sigma terms control the sensitivity of direct-detection dark mater searches to WIMPs that interact via Higgs exchange. σπ N = mudN |¯ uu + ¯ dd|N σq = mqN | ¯ qq|N

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 4

SLIDE 5

Toward precision nucleon structure

Trustworthy latice calculations need control over all systematics:

◮ Physical quark masses ◮ Isolation of ground state ◮ Infinite volume ◮ Continuum limit

Want to transition from this ...

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 0.10 0.15 0.20 0.25 0.30 gA mπ (GeV) Nf=2+1+1 TM (ETMC) Nf=2 TM+clover (ETMC) Nf=2 TM (ETMC) Nf=2+1+1 mixed (PNDME) Nf=2+1+1 mixed (CalLat) Nf=2+1 DW (RBC/UKQCD) Nf=2+1 clover (CSSM) Nf=2+1 clover (LHPC) Nf=2+1 clover (NME) Nf=2 clover (Mainz) Nf=2 clover (RQCD) Nf=2 clover (QCDSF) PDG Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 5

SLIDE 6

Toward precision nucleon structure

Trustworthy latice calculations need control over all systematics:

◮ Physical quark masses ◮ Isolation of ground state ◮ Infinite volume ◮ Continuum limit

Want to transition from this ...to something like this.

22 24 26 28 30 32 34 = + + = + = pheno.

Weinberg 77 Leutwyler 96 Kaiser 98 Narison 06 Oller 07 PDG QCDSF/UKQCD 06 ETM 07 RBC 07 ETM 10B ETM 14D FLAG average for = MILC 04, HPQCD/MILC/UKQCD 04 RBC/UKQCD 08 PACS-CS 08 MILC 09 MILC 09A PACS-CS 09 Blum 10 RBC/UKQCD 10A BMW 10A, 10B Laiho 11 PACS-CS 12 RBC/UKQCD 12 RBC/UKQCD 14B FLAG average for = + ETM 14 FNAL/MILC 14A FLAG average for = + +

/

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 5

SLIDE 7

Outline

1. Methodologies for nucleon matrix elements
2. Excited-state effects: theory and practice
3. Other systematics: L, (a), mq
4. Outlook

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 6

SLIDE 8

Hadron correlation functions

Compute two-point and three-point functions, using interpolator χ and

perator insertion O. In simplest case:

t τ T

C2pt(t) ≡ χ (t)χ†(0) =

n

|Zn|2e−Ent → |Z0|2e−E0t 1 + O(e−∆Et )

,

where Zn = Ω|χ|n, C3pt(τ,T ) ≡ χ (T )O(τ )χ†(0) =

n,n′

Zn′Z ∗

nn′|O|ne−Enτe−En′(T −τ )

→ |Z0|20|O|0e−E0T 1 + O(e−∆Eτ ) + O(e−∆E(T −τ ))

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 7

SLIDE 9

Hadron matrix elements

Ratio method R(τ,T ) ≡ C3pt(τ,T ) C2pt(T ) = 0|O|0 + O(e−∆Eτ ) + O(e−∆E(T −τ )) Midpoint yields R(T

2 ,T ) = 0|O|0 + O(e−∆ET /2).

Summation method

L. Maiani et al., Nucl. Phys. B 293, 420 (1987); дA in S. Güsken et al., Phys. Let. B 227, 266 (1989)

S(T ) ≡

τ

R(τ,T ), d dT S(T ) = 0|O|0 + O(Te−∆ET ) Sum can be over all timeslices or from τ0 to T − τ0. Improved asymptotic behaviour noted in talks at Latice 2010.

S. Capitani et al., PoS LATTICE2010 147 [1011.1358]; J. Bulava et al., ibid. 303 [1011.4393]

In practice noisier than ratio method at same T.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 8

SLIDE 10

Ratio and summation method

1.15 1.20 1.25 1.30 2 4 6 8 10 12 gA (bare) T/a R(T/2, T) S(T + a) − S(T)

physical mπ , a = 0.116 fm

N. Hasan, JG, et al., in preparation

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 9

SLIDE 11

Feynman-Hellmann approach

If we add a term to the Lagrangian L(λ) ≡ L + λO, then ∂ ∂λEn(λ)

λ=0

= n|O|n. Discrete derivatives sometimes used, particularly for sigma terms: σq ≡ mqN | ¯ qq|N = mq ∂mN ∂mq . Exact derivatives lead directly to summation method (neglecting Ω|O|Ω): − ∂ ∂λ logC2pt(t)

λ=0

= S(t).

L. Maiani et al., Nucl. Phys. B 293, 420 (1987)
A. J. Chambers et al. (CSSM and QCDSF/UKQCD), Phys. Rev. D 90, 014510 (2014) [1405.3019]
C. Bouchard et al., Phys. Rev. D 96, 014504 (2017) [1612.06963]

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 10

SLIDE 12

Controlling excited states

Three approaches:

◮ Go to large T. Need ∆ET ≫ 1.

But signal-to-noise decays as e−(E0− 3

2mπ )T .

◮ Improve the interpolating operator. ◮ Fit correlators to remove excited states.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 11

SLIDE 13

Excited-state energies

First approximation: noninteracting stable hadrons in a box. Nπ, Nππ, ...

3.0 3.5 4.0 4.5 5.0 mπL 0.0 0.2 0.4 0.6 0.8 1.0 ∆E (GeV) Nπ

Physical mπ .

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 12

SLIDE 14

Excited-state energies

First approximation: noninteracting stable hadrons in a box. Nπ, Nππ, ...

3.0 3.5 4.0 4.5 5.0 mπL 0.0 0.2 0.4 0.6 0.8 1.0 ∆E (GeV) Nπ Nππ

Physical mπ .

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 12

SLIDE 15

Excited-state energies

First approximation: noninteracting stable hadrons in a box. Nπ, Nππ, ...

3.0 3.5 4.0 4.5 5.0 mπL 0.0 0.2 0.4 0.6 0.8 1.0 ∆E (GeV) Nπ Nππ Nπππ

Physical mπ .

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 12

SLIDE 16

Excited-state energies

First approximation: noninteracting stable hadrons in a box. Nπ, Nππ, ...

3.0 3.5 4.0 4.5 5.0 mπL 0.0 0.2 0.4 0.6 0.8 1.0 ∆E (GeV) Nπ Nππ Nπππ Nππππ

Physical mπ .

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 12

SLIDE 17

Excited-state energies

First approximation: noninteracting stable hadrons in a box. Nπ, Nππ, ...

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 mπ (GeV) 0.0 0.2 0.4 0.6 0.8 1.0 ∆E (GeV) Nπ Nππ Nπππ Nππππ

mπL = 4.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 12

SLIDE 18

Excited-state energies

M. T. Hansen and H. B. Meyer, Nucl. Phys. B 923, 558 (2017) [1610.03843]

Focus on Nπ sector and apply finite-volume quantization using scatering phase shif from experiment.

3.0 3.5 4.0 4.5 5.0 5.5 6.0

Mπ L

1000 1200 1400 1600 1800

E (MeV)

I(JP ) =1/2(1/2 + )

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 13

SLIDE 19

Going to large T

Statistical errors: δstat ∼ N −1/2e(E0− 3

2mπ )T

Excited state systematics: δexc ∼ e−∆ET /2 (ratio method) Suppose we want these to be equal, i.e. δstat = δexc ≡ δ. Required statistics are given by N ∝ δ−

2+ 4E0−6mπ

∆E

.

At the physical point with ∆E = 2mπ , the exponent is ≈ −13. Multi-level methods could potentially improve this.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 14

SLIDE 20

Predicting excited-state contributions

Need to know:

1. energies En
2. matrix elements n′|O|n
3. overlaps Zn

Have been studied in ChPT. B. C. Tiburzi; O. Bär Key insight: at leading order a single LEC controls coupling of local interpolator to N and Nπ states.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 15

SLIDE 21

ChPT prediction of excited-state effects

meff(T )/m

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1.00 1.01 1.02 1.03 1.04

T (fm) physical point mπL = 4 truncated to five Nπ states

O. Bär, Int. J. Mod. Phys. A 32 no. 15, 1730011 (2017) [1705.02806]

generalization to axial FFs → Oliver Bär parallel, Wed. 14:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 16

SLIDE 22

ChPT prediction of excited-state effects

RX (T

2 ,T )/X

hxi∆u−∆d hxiu−d hxiδu−δd gS gA gT

1.0 1.5 2.0 2.5 1.00 1.05 1.10 1.15 1.20 1.25

T (fm) physical point mπL = 4 truncated to five Nπ states need T 2 fm for ChPT to be reliable

O. Bär, Int. J. Mod. Phys. A 32 no. 15, 1730011 (2017) [1705.02806]

generalization to axial FFs → Oliver Bär parallel, Wed. 14:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 16

SLIDE 23

Going beyond ChPT

M. T. Hansen and H. B. Meyer, Nucl. Phys. B 923, 558 (2017) [1610.03843]

Deviations of Ω|χ|Nπ and Nπ |Aµ |N from ChPT in the resonance regime modeled with parameters α and γ.

α = 0, γ = 0 α = 1, γ = 0 α = 1, γ = −4

MπL = 4 MπL = 6

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 17

SLIDE 24

Numerical studies

Available data depends on how C3pt is computed. τ T For connected diagrams:

◮ Fixed sink: most common. Get data at all τ and

any operator insertion. Cost increases with each value of T.

◮ Fixed operator and τ. Get data at all T and any

set of interpolators. variational studies by CSSM

◮ Fixed operator and summed τ. Get summation

data at all T and any set of interpolators.

used by CalLat, NPLQCD

−6 −4 −2 2 4 6 (τ −T/2)/a 1.30 1.35 1.40 1.45 gA (bare) T/a = 14 12 10 8 6 ratio summation

JG et al., PRD 95, 114502 [1703.06703]

21 22 23 24 25 26 27 28 29 30 1.2 1.4 1.6

tS gA

B. J. Owen et al., PLB 723, 217 [1212.4668]

5 10 15 t/a 1.15 1.20 1.25 1.30 1.35 1.40 ˚ geff

A

a09m220

C. C. Chang et al., Nature 558, 91

[1805.12130] Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 18

SLIDE 25

Improving the nucleon interpolator

◮ Standard operator: χ = ϵabc (uT aCγ5db)uc, with smeared quark fields. ◮ Smearing width typically tuned so that meff has early plateau. ◮ Variational approach: χ = i ci χi with optimized ci.

◮ Simple basis: use varying smearing widths for χi. ◮ Can be extended with more local structures (derivatives and Gµν). ◮ More expensive: include nonlocal operators for beter isolation of Nπ

and Nππ states.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 19

SLIDE 26

Variational approach: different smearings — effective mass

3 5 7 9 11 13 15 17 19 21 23

t

0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58

aMN Mass 2exp Variational Comparison

sm32 sm64 sm128 t0 =2 ∆t =2

mπ = 460 MeV, a = 0.074 fm

J. Dragos et al., PRD 94, 074505 (2016) [1606.03195]

0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 4 6 8 10 12 14 16 18 20

MN

t

S3S3 S5S5 S7S7 V357

mπ = 312 MeV, a = 0.081 fm

B. Yoon et al. (NME), PRD 93, 114506 (2016)

[1602.07737]

Small improvement over widest smearing.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 20

SLIDE 27

Variational approach: different smearings — effective mass

3 5 7 9 11 13 15 17 19 21 23

t

0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58

aMN Mass 2exp Variational Comparison

sm32 sm64 sm128 t0 =2 ∆t =2

mπ = 460 MeV, a = 0.074 fm

J. Dragos et al., PRD 94, 074505 (2016) [1606.03195]

0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 4 6 8 10 12 14 16 18 20

MN

t

S5S5 S7S7 S9S9 V579

mπ = 312 MeV, a = 0.081 fm

B. Yoon et al. (NME), PRD 93, 114506 (2016)

[1602.07737]

Small improvement over widest smearing...so use even wider smearings.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 20

SLIDE 28

Variational approach: different smearings — axial charge

8 6 4 2 2 4 6 8

τ−t/2

0.95 1.00 1.05 1.10 1.15 1.20

R(τ,t) gA Variational Comparison

sm32 sm64 sm128 t0 =2 ∆t =2

mπ = 460 MeV, a = 0.074 fm T = 0.96 fm

J. Dragos et al., PRD 94, 074505 (2016) [1606.03195]

1.20 1.25 1.30 1.35 1.40 1.45 1.50

6
4
2

2 4 6

gA

τ - tsep/2

tsep=10 tsep=12 tsep=14 tsep=16 V357 V579

mπ = 312 MeV, a = 0.081 fm variational at T = 0.97 fm versus S5

B. Yoon et al. (NME), PRD 93, 114506 (2016)

[1602.07737]

Improvement over best tuned single smearing seems small.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 21

SLIDE 29

Variational approach: different smearings — axial charge

8 6 4 2 2 4 6 8

τ−t/2

0.95 1.00 1.05 1.10 1.15 1.20

R(τ,t) gA Variational Comparison

sm32 sm64 sm128 t0 =2 ∆t =2

mπ = 460 MeV, a = 0.074 fm T = 0.96 fm

J. Dragos et al., PRD 94, 074505 (2016) [1606.03195]

1.20 1.25 1.30 1.35 1.40 1.45 1.50

6
4
2

2 4 6

gA

τ - tsep/2

tsep=10 tsep=12 tsep=14 tsep=16 V357 V579

mπ = 312 MeV, a = 0.081 fm variational at T = 0.97 fm versus S9

B. Yoon et al. (NME), PRD 93, 114506 (2016)

[1602.07737]

Improvement over best tuned single smearing seems small.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 21

SLIDE 30

Variational approach: different smearings — axial charge

8 6 4 2 2 4 6 8

τ−t/2

0.95 1.00 1.05 1.10 1.15 1.20

R(τ,t) gA Variational Comparison

sm32 sm64 sm128 t0 =2 ∆t =2

mπ = 460 MeV, a = 0.074 fm T = 0.96 fm

J. Dragos et al., PRD 94, 074505 (2016) [1606.03195]

1.20 1.25 1.30 1.35 1.40 1.45 1.50

6
4
2

2 4 6

gA

τ - tsep/2

tsep=10 tsep=12 tsep=14 tsep=16 V357 V579

mπ = 312 MeV, a = 0.081 fm variational at T = 0.97 fm versus S9

B. Yoon et al. (NME), PRD 93, 114506 (2016)

[1602.07737]

Improvement over best tuned single smearing seems small. See also: including negative parity operators in basis when p 0.

F. Stokes et al., in preparation

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 21

SLIDE 31

Variational approach: larger basis — effective mass

Calculation performed using distillation. Operators with derivatives and hybrid operators with Gµν included in basis.

→ C. Egerer parallel, Wed. 14:40

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 22

SLIDE 32

Variational approach: larger basis — tensor charge

Standard operator Variational Calculation performed using distillation. Operators with derivatives and hybrid operators with Gµν included in basis.

→ C. Egerer parallel, Wed. 14:40

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 22

SLIDE 33

Variational approach: larger basis — tensor charge

Standard operator Variational Calculation performed using distillation. Operators with derivatives and hybrid operators with Gµν included in basis.

→ C. Egerer parallel, Wed. 14:40

How does this compare with a standard calculation with tuned smearing?

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 22

SLIDE 34

Fiting approaches

Fit correlators using ansatz based on N-state model.

◮ En and Zn generally determined from C2pt. ◮ n′|O|n determined from C3pt, either in combined or separate fit.

Typical energy gap: 0.5 GeV < E1 − E0 < 1 GeV, usually greater than expected 2mπ .

◮ Each model state approximates contribution from several states. ◮ Fit ansatz should be considered a somewhat uncontrolled model.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 23

SLIDE 35

Fits for axial charge

Fixed sink

1.1 1.2 1.3 1.4 −10 −5 5 10 τ : ∞ 12 14 16 a09m220

R. Gupta et al. (PNDME), 1806.09006

constrained 3-state fit: τ,T − τ ≥ 3a Summed current insertion

5 10 15 t/a 1.15 1.20 1.25 1.30 1.35 1.40 ˚ geff

A

a09m220

C. C. Chang et al., Nature 558, 91 [1805.12130]

data with smeared and point sinks unconstrained 2-state fit: T ≥ 3a

1.22 1.26 ˚ gA/˚ gV a09m220 meff tmin geff

A tmin

geff

V tmin

8 9 10 11 2 3 4 5 5 6 7 8 10−2 10−1 100 P

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 24

SLIDE 36

Fits for axial charge

Fixed sink

1.1 1.2 1.3 1.4 −10 −5 5 10 τ : ∞ 12 14 16 a09m220

R. Gupta et al. (PNDME), 1806.09006

constrained 3-state fit: τ,T − τ ≥ 3a Note: more than 10 states with ∆E < 1 GeV =⇒ e−3a∆E > 0.25. Summed current insertion

5 10 15 t/a 1.15 1.20 1.25 1.30 1.35 1.40 ˚ geff

A

a09m220

C. C. Chang et al., Nature 558, 91 [1805.12130]

data with smeared and point sinks unconstrained 2-state fit: T ≥ 3a

1.22 1.26 ˚ gA/˚ gV a09m220 meff tmin geff

A tmin

geff

V tmin

8 9 10 11 2 3 4 5 5 6 7 8 10−2 10−1 100 P

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 24

SLIDE 37

Simultaneous fits

data at tsep = 1.0fm fit model at tsep = 1.0fm fit result tsep/fm gu−d

A

1 0.5

0.5
1

1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 data at tsep = 1.0fm fit model at tsep = 1.0fm fit result tsep/fm gu−d

S

1 0.5

0.5
1

2 1.5 1 0.5 data at tsep = 1.0fm fit model at tsep = 1.0fm fit result tsep/fm gu−d

T

1 0.5

0.5
1

1.3 1.2 1.1 1 0.9 0.8 data at tsep = 1.0fm fit model at tsep = 1.0fm fit result tsep/fm xu−d 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 data at tsep = 1.0fm fit model at tsep = 1.0fm fit result tsep/fm x∆u−∆d 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 data at tsep = 1.0fm fit model at tsep = 1.0fm fit result tsep/fm xδu−δd 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

mπ = 347 MeV, a = 0.064 fm, T = 16a (all T included in fit). Two-state fit to ratios for six observables in range τ,T − τ ≥ tstart with tstartMπ = 0.4 fixed globally for all ensembles. ∆E determined only from ratios, not C2pt.

→ K. Otnad parallel, Thu 12:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 25

SLIDE 38

Simultaneous fits

data at tsep = 1.2fm fit model at tsep = 1.2fm fit result tsep/fm gu−d

A

1 0.5

0.5
1

1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 data at tsep = 1.2fm fit model at tsep = 1.2fm fit result tsep/fm gu−d

S

1 0.5

0.5
1

2 1.5 1 0.5 data at tsep = 1.2fm fit model at tsep = 1.2fm fit result tsep/fm gu−d

T

1 0.5

0.5
1

1.3 1.2 1.1 1 0.9 0.8 data at tsep = 1.2fm fit model at tsep = 1.2fm fit result tsep/fm xu−d 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 data at tsep = 1.2fm fit model at tsep = 1.2fm fit result tsep/fm x∆u−∆d 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 data at tsep = 1.2fm fit model at tsep = 1.2fm fit result tsep/fm xδu−δd 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

mπ = 347 MeV, a = 0.064 fm, T = 18a (all T included in fit). Two-state fit to ratios for six observables in range τ,T − τ ≥ tstart with tstartMπ = 0.4 fixed globally for all ensembles. ∆E determined only from ratios, not C2pt.

→ K. Otnad parallel, Thu 12:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 25

SLIDE 39

Simultaneous fits

data at tsep = 1.3fm fit model at tsep = 1.3fm fit result tsep/fm gu−d

A

1 0.5

0.5
1

1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 data at tsep = 1.3fm fit model at tsep = 1.3fm fit result tsep/fm gu−d

S

1 0.5

0.5
1

2 1.5 1 0.5 data at tsep = 1.3fm fit model at tsep = 1.3fm fit result tsep/fm gu−d

T

1 0.5

0.5
1

1.3 1.2 1.1 1 0.9 0.8 data at tsep = 1.3fm fit model at tsep = 1.3fm fit result tsep/fm xu−d 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 data at tsep = 1.3fm fit model at tsep = 1.3fm fit result tsep/fm x∆u−∆d 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 data at tsep = 1.3fm fit model at tsep = 1.3fm fit result tsep/fm xδu−δd 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

mπ = 347 MeV, a = 0.064 fm, T = 20a (all T included in fit). Two-state fit to ratios for six observables in range τ,T − τ ≥ tstart with tstartMπ = 0.4 fixed globally for all ensembles. ∆E determined only from ratios, not C2pt.

→ K. Otnad parallel, Thu 12:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 25

SLIDE 40

Simultaneous fits

data at tsep = 1.4fm fit model at tsep = 1.4fm fit result tsep/fm gu−d

A

1 0.5

0.5
1

1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 data at tsep = 1.4fm fit model at tsep = 1.4fm fit result tsep/fm gu−d

S

1 0.5

0.5
1

2 1.5 1 0.5 data at tsep = 1.4fm fit model at tsep = 1.4fm fit result tsep/fm gu−d

T

1 0.5

0.5
1

1.3 1.2 1.1 1 0.9 0.8 data at tsep = 1.4fm fit model at tsep = 1.4fm fit result tsep/fm xu−d 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 data at tsep = 1.4fm fit model at tsep = 1.4fm fit result tsep/fm x∆u−∆d 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 data at tsep = 1.4fm fit model at tsep = 1.4fm fit result tsep/fm xδu−δd 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

mπ = 347 MeV, a = 0.064 fm, T = 22a (all T included in fit). Two-state fit to ratios for six observables in range τ,T − τ ≥ tstart with tstartMπ = 0.4 fixed globally for all ensembles. ∆E determined only from ratios, not C2pt.

→ K. Otnad parallel, Thu 12:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 25

SLIDE 41

Simultaneous fits

data at tsep = 1.5fm fit model at tsep = 1.5fm fit result tsep/fm gu−d

A

1 0.5

0.5
1

1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 data at tsep = 1.5fm fit model at tsep = 1.5fm fit result tsep/fm gu−d

S

1 0.5

0.5
1

2 1.5 1 0.5 data at tsep = 1.5fm fit model at tsep = 1.5fm fit result tsep/fm gu−d

T

1 0.5

0.5
1

1.3 1.2 1.1 1 0.9 0.8 data at tsep = 1.5fm fit model at tsep = 1.5fm fit result tsep/fm xu−d 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 data at tsep = 1.5fm fit model at tsep = 1.5fm fit result tsep/fm x∆u−∆d 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 data at tsep = 1.5fm fit model at tsep = 1.5fm fit result tsep/fm xδu−δd 1 0.5

0.5
1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

mπ = 347 MeV, a = 0.064 fm, T = 24a (all T included in fit). Two-state fit to ratios for six observables in range τ,T − τ ≥ tstart with tstartMπ = 0.4 fixed globally for all ensembles. ∆E determined only from ratios, not C2pt.

→ K. Otnad parallel, Thu 12:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 25

SLIDE 42

Simultaneous fits

∆Nπ 2Mπ Fitted ∆0 Mπtstart ∆0/[GeV] 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 3 2.5 2 1.5 1 0.5

mπ = 347 MeV, a = 0.064 fm Two-state fit to ratios for six observables in range τ,T − τ ≥ tstart with tstartMπ = 0.4 fixed globally for all ensembles. ∆E determined only from ratios, not C2pt.

→ K. Otnad parallel, Thu 12:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 25

SLIDE 43

Fit quality

Can we trust these fits? Ideally we would require that they have good fit

quality. (Not a sufficient condition!)

0.0 0.2 0.4 0.6 0.8 1.0 p 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative distribution function uniform CalLat PNDME

Should expect uniformly distributed p-values. Following

Sz. Borsanyi et al., Science 347, 1452 (2015) [1406.4088]

can use Kolmogorov-Smirnov test. PNDME: p = 0.26 CalLat: p = 0.00077 p-values not provided by Mainz, but reduced χ2 are large.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 χ2 per degree of freedom 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative distribution function Mainz

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 26

SLIDE 44

Fit quality

Can we trust these fits? Ideally we would require that they have good fit

quality. (Not a sufficient condition!)

0.0 0.2 0.4 0.6 0.8 1.0 p 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative distribution function uniform CalLat PNDME

Should expect uniformly distributed p-values. Following

Sz. Borsanyi et al., Science 347, 1452 (2015) [1406.4088]

can use Kolmogorov-Smirnov test. PNDME: p = 0.26 CalLat: p = 0.00077 p-values not provided by Mainz, but reduced χ2 are large.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 χ2 per degree of freedom 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative distribution function Mainz

May be difficult to estimate χ2 due to difficulty inverting large covariance matrix.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 26

SLIDE 45

What about Nπ states?

Expected Nπ states not seen in variational or fiting results. Why?

◮ Volume suppression? Contamination from each state ∝ L−3.

But density of states ∝ L3 compensates.

◮ At short T, higher states dominate?

=⇒ not yet in asymptotic regime. Familiar from meson spectroscopy. Must include nonlocal

perators in variational

basis to identify complete spectrum.

0.10 0.15 0.20

D. J. Wilson et al. (HadSpec), Phys. Rev. D 92, 094502 (2015)

[1507.02599]

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 27

SLIDE 46

Comparison of several approaches

sm32 sm64 sm128 t =10 t =13 t =16 t =19 t =22 δt =0 δt =1 δt =2 δt =3 δt =0 δt =1 δt =2 δt =3 δt =0 δt =1 δt =2 δt =3 δt =2 δt =3 δt =4 δt =2 δt =3 δt =4 δt =2 δt =3 δt =4 δt =2 δt =3 δt =4 δt =2 δt =3 δt =4 t =13 t =16

Methods

0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25

Value

Fits Fits Summation 2exp 2exp 2exp Var Smears Sink Times Method sm32 t =13 All t 10,13 t0 =2 t =13 sm32 All t 10 t 10,13 sm32 sm64 sm128 sm32 sm32 sm32 sm32 ∆t =2

gA Summary

mπ = 460 MeV, a = 0.074 fm

J. Dragos et al., PRD 94, 074505 (2016) [1606.03195]

1.05 1.15 1.25 1.35

gA

Plateau Summation Two-state

0.9 1.0 1.1 1.2 1.3 1.4 ts[fm] 0.50 0.55 0.60 0.65 0.70

gu + d

A

Connected 0.9 1.0 1.1 1.2 1.3

tlow

s

[fm] 1.05 1.15 1.25 1.35

gA

Plateau Summation Two-state

0.9 1.0 1.1 1.2 1.3 1.4 ts[fm] 0.50 0.55 0.60 0.65 0.70

gu + d

A

Connected 0.9 1.0 1.1 1.2 1.3

tlow

s

[fm]

mπ = 130 MeV, a = 0.093 fm

C. Alexandrou et al. (ETMC), PRD 96, 054507 (2017)

[1705.03399]

May be best to estimate systematic uncertainty using multiple different approaches.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 28

SLIDE 47

Finite-volume effects

In general suppressed as e−mπ L. For дA, computed in heavy baryon ChPT including ∆ degrees of freedom:

S. R. Beane and M. J. Savage, Phys. Rev. D 70 074029 (2004) [hep-ph/0404131]

дA(L) − дA дA = m2

π

3π 2f 2

π

д2

AF1(mπL) + д2 ∆N (1 + 25д∆∆ 81дA )F2(mπ, ∆, L)

+ F3(mπL) + д2

∆N F4(mπ, ∆, L)

,

Neglecting loops with ∆ baryons, the leading contribution is дA(L) − дA дA ∼ m2

πд2 A

3π 2f 2

π

π

2mπLe−mπ L

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 29

SLIDE 48

Finite-volume effects: controlled studies

0.80 0.85 0.90 0.95 1.00 1.05 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 gA(L)/gA(Lmax) mπL RQCD 150 MeV RQCD 290 MeV RQCD 420 MeV PNDME 220 MeV CalLat 220 MeV ETMC 130 MeV QCDSF 261 MeV QCDSF 290 MeV LHPC 356 MeV LHPC 254 MeV RBC 420 MeV

Reference data at Lmax indicated with black dot.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 30

SLIDE 49

Finite-volume effects: controlled studies

0.80 0.85 0.90 0.95 1.00 1.05 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 gA(L)/gA(Lmax) (mπ/mphys

π

)2e−mπ L/√mπL RQCD 150 MeV RQCD 290 MeV RQCD 420 MeV PNDME 220 MeV CalLat 220 MeV ETMC 130 MeV QCDSF 261 MeV QCDSF 290 MeV LHPC 356 MeV LHPC 254 MeV RBC 420 MeV

Reference data at Lmax indicated with black dot.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 30

SLIDE 50

Finite-volume effects: global fits

gu−d

A

MπL 1.2 1.3 3 4 5 6 7

R. Gupta et al. (PNDME), 1806.09006

дA(mπ, L,a) = f (mπ,a) + cm2

πe−mπ L

at mphys

π

, mπL = 4: −0.9(5)% effect

0.000 0.005 0.010 0.015 0.020 0.025 e−mπL/(mπL)1/2 1.23 1.25 1.27 1.29 gA NNLO+ct χPT NLO χPT prediction NLO χPT prediction

C. C. Chang et al., Nature 558, 91 [1805.12130]

leading HBChPT expression (no ∆) Larger (few %) effect seen by Mainz group! K. Otnad parallel, Thu 12:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 31

SLIDE 51

Finite-volume effects: global fits

gu−d

A

MπL 1.2 1.3 3 4 5 6 7

R. Gupta et al. (PNDME), 1806.09006

дA(mπ, L,a) = f (mπ,a) + cm2

πe−mπ L

at mphys

π

, mπL = 4: −0.9(5)% effect

0.000 0.005 0.010 0.015 0.020 0.025 e−mπL/(mπL)1/2 1.23 1.25 1.27 1.29 gA NNLO+ct χPT NNLO χPT estimate NNLO χPT estimate

C. C. Chang et al., Nature 558, 91 [1805.12130]

leading HBChPT expression (no ∆) +c(mπ/fπ )3F1(mπL) Larger (few %) effect seen by Mainz group! K. Otnad parallel, Thu 12:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 31

SLIDE 52

Finite-volume effects: global fits

gu−d

A

MπL 1.2 1.3 3 4 5 6 7

R. Gupta et al. (PNDME), 1806.09006

дA(mπ, L,a) = f (mπ,a) + cm2

πe−mπ L

at mphys

π

, mπL = 4: −0.9(5)% effect

0.000 0.005 0.010 0.015 0.020 0.025 e−mπL/(mπL)1/2 1.23 1.25 1.27 1.29 gA model average NNLO χPT estimate NNLO χPT estimate

C. C. Chang et al., Nature 558, 91 [1805.12130]

leading HBChPT expression (no ∆) +c(mπ/fπ )3F1(mπL) Larger (few %) effect seen by Mainz group! K. Otnad parallel, Thu 12:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 31

SLIDE 53

Chiral extrapolation

Increasing number of calculations at physical mπ . Otherwise need extrapolation. Heavy baryon ChPT for дA: дA(mπ ) = д0 − (д0 + 2д3

0)

mπ 4πF 2 log m2

π

µ2 + c1m2

π + c2m3 π + . . .

r use simple polynomials in mπ or m2

π .

Range of convergence is a priori unknown.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 32

SLIDE 54

Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4 gA

2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2 clover (Mainz ’17) 2+1 overlap (JLQCD ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 overlap on domain wall (χQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (PACS ’18) 2+1 clover (PACS ’18b) 2+1 clover (Mainz ’18) 2+1+1 twisted mass clover (ETMC ’18) PDG

filled symbols: published Selection based on quality criteria still missing!

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 33

SLIDE 55

Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4 gA

2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2 clover (Mainz ’17) 2+1 overlap (JLQCD ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 overlap on domain wall (χQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (PACS ’18) 2+1 clover (PACS ’18b) 2+1 clover (Mainz ’18) 2+1+1 twisted mass clover (ETMC ’18) PDG

filled symbols: published Selection based on quality criteria still missing!

1.05 1.15 1.25 1.35

gA

Plateau Summation Two-state

0.9 1.0 1.1 1.2 1.3 1.4 ts[fm] 0.50 0.55 0.60 0.65 0.70

gu + d

A Connected

0.9 1.0 1.1 1.2 1.3 tlow

s

[fm] 1.05 1.15 1.25 1.35 gA

Plateau Summation Two-state

0.9 1.0 1.1 1.2 1.3 1.4 ts[fm] 0.50 0.55 0.60 0.65 0.70

gu + d

A Connected

0.9 1.0 1.1 1.2 1.3 tlow

s

[fm]

One ensemble: mπ = 130 MeV, a = 0.093 fm, mπL = 3.0

C. Alexandrou et al. (ETMC),
Phys. Rev. D 96, 054507 (2017) [1705.03399]

1.1 1.15 1.2 1.25 1.3 1.35 0.8 1 1.2 1.4 1.6 1.8

gA Time separation [fm] L=48, a~0.094, Nf=2 L=64, a~0.094, Nf=2 (preliminary) L=64, a~0.081, Nf=2+1+1 (preliminary)

New Nf = 2 ensemble: larger volume.

C. Lauer poster

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 33

SLIDE 56

Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4 gA

2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2 clover (Mainz ’17) 2+1 overlap (JLQCD ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 overlap on domain wall (χQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (PACS ’18) 2+1 clover (PACS ’18b) 2+1 clover (Mainz ’18) 2+1+1 twisted mass clover (ETMC ’18) PDG

filled symbols: published Selection based on quality criteria still missing!

0.00 0.05 0.10 0.15 0.20 0.25 0.30 ǫπ = mπ/(4πFπ) 1.10 1.15 1.20 1.25 1.30 1.35 gA model average gLQCD

A

(ǫπ, a = 0) gPDG

A

= 1.2723(23) gA(ǫπ, a ≃ 0.15 fm) gA(ǫπ, a ≃ 0.12 fm) gA(ǫπ, a ≃ 0.09 fm) a ≃ 0.15 fm a ≃ 0.12 fm a ≃ 0.09 fm

Bayesian average over fit models for дA(mπ/Fπ,mπL,a2/w2

0).

C. C. Chang et al., Nature 558, 91 (2018)

[1805.12130]

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 33

SLIDE 57

Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4 gA

2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2 clover (Mainz ’17) 2+1 overlap (JLQCD ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 overlap on domain wall (χQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (PACS ’18) 2+1 clover (PACS ’18b) 2+1 clover (Mainz ’18) 2+1+1 twisted mass clover (ETMC ’18) PDG

filled symbols: published Selection based on quality criteria still missing!

1:24 1:28 1:32 gA model avg NNLO fflPT NNLO+ct fflPT NLO Taylor ›2

ı

NNLO Taylor ›2

ı

NLO Taylor ›ı NNLO Taylor ›ı +O(¸sa2) disc. +O(a) disc.

mit FV

NLO FV 2×LO width 2× all widths mı ≤ 350 MeV mı ≤ 310 MeV mı ≥ 220 MeV a ≤ 0:12 fm a ≥ 0:12 fm N3LO fflPT NLO fflPT(∆) 0:0 0:5 1:0 ffl2

aug=dof

0:0 0:5 1:0 Bayes factor

C. C. Chang et al., Nature 558, 91 (2018)

[1805.12130]

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 33

SLIDE 58

Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4 gA

2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2 clover (Mainz ’17) 2+1 overlap (JLQCD ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 overlap on domain wall (χQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (PACS ’18) 2+1 clover (PACS ’18b) 2+1 clover (Mainz ’18) 2+1+1 twisted mass clover (ETMC ’18) PDG

filled symbols: published Selection based on quality criteria still missing!

gu−d

A

a [fm] a15m310 a12m310 a12m220L a12m220 a12m220S a09m310 a09m220 a09m130 a06m310 a06m220 a06m135 extrap. 1-extrap. 1.2 1.3 0.03 0.06 0.09 0.12 0.15 gu−d

A

M2

π [GeV2]

1.2 1.3 0.02 0.04 0.06 0.08 0.1 0.12 gu−d

A

MπL 1.2 1.3 3 4 5 6 7

Main ansatz: дA(mπ, L,a) = c1 + c2a + c3m2

π

+ c4m2

πe−mπ L

R. Gupta et al. (PNDME), 1806.09006
R. Gupta parallel, Thu 12:40

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 33

SLIDE 59

Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4 gA

2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2 clover (Mainz ’17) 2+1 overlap (JLQCD ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 overlap on domain wall (χQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (PACS ’18) 2+1 clover (PACS ’18b) 2+1 clover (Mainz ’18) 2+1+1 twisted mass clover (ETMC ’18) PDG

filled symbols: published Selection based on quality criteria still missing!

5 10 15

t/a

0.8 1 1.2 1.4 1.6

PDG16 mπ=0.146 GeV

gA

ren

One ensemble: mπ = 146 MeV, a = 0.085 fm, mπL = 6.0, T = 1.3 fm.

K.-I. Ishikawa et al. (PACS), 1807.03974

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 33

SLIDE 60

Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4 gA

2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2 clover (Mainz ’17) 2+1 overlap (JLQCD ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 overlap on domain wall (χQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (PACS ’18) 2+1 clover (PACS ’18b) 2+1 clover (Mainz ’18) 2+1+1 twisted mass clover (ETMC ’18) PDG

filled symbols: published Selection based on quality criteria still missing!

5 10 15

t/a

0.8 1 1.2 1.4 1.6

PDG16 mπ=0.146 GeV

gA

ren

One ensemble: mπ = 146 MeV, a = 0.085 fm, mπL = 6.0, T = 1.3 fm.

K.-I. Ishikawa et al. (PACS), 1807.03974

New ensemble: mπ = 135 MeV, L = 10.8 fm. Y. Kuramashi parallel, Thu 9:50

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 33

SLIDE 61

Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4 gA

2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2 clover (Mainz ’17) 2+1 overlap (JLQCD ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 overlap on domain wall (χQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (PACS ’18) 2+1 clover (PACS ’18b) 2+1 clover (Mainz ’18) 2+1+1 twisted mass clover (ETMC ’18) PDG

filled symbols: published Selection based on quality criteria still missing!

lattice data (corrected) value at Mπ,phys chiral + continuum + FS extrapolation Mπ/GeV t0M 2

π

gu−d

A

0.35 0.30 0.25 0.20 0.15 0.10 0.07 0.06 0.05 0.04 0.03 0.02 0.01 1.4 1.3 1.2 1.1 1 0.9 0.8

Ensembles have ms varied such that ms + mu + md = const. Fit ansatz: дA(mπ, L,a) = c1 + c2a2 + c3m2

π

+ c4m2

πe−mπ L

K. Otnad parallel, Thu 12:00

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 33

SLIDE 62

Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4 gA

2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2 clover (Mainz ’17) 2+1 overlap (JLQCD ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 overlap on domain wall (χQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (PACS ’18) 2+1 clover (PACS ’18b) 2+1 clover (Mainz ’18) 2+1+1 twisted mass clover (ETMC ’18) PDG

filled symbols: published Selection based on quality criteria still missing!

1.1 1.15 1.2 1.25 1.3 1.35 0.8 1 1.2 1.4 1.6 1.8

gA Time separation [fm] T wo state fit L=64, a~0.081, Nf=2+1+1 (preliminary)

One ensemble: physical mπ , a = 0.081 fm, mπL = 3.6

C. Lauer poster
M. Constantinou parallel, Fri 17:50

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 33

SLIDE 63

Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4 gA

2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2 clover (Mainz ’17) 2+1 overlap (JLQCD ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 overlap on domain wall (χQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (PACS ’18) 2+1 clover (PACS ’18b) 2+1 clover (Mainz ’18) 2+1+1 twisted mass clover (ETMC ’18) PDG

filled symbols: published Selection based on quality criteria still missing! Other preliminary results:

1 1.1 1.2 1.3 1.4 1.5 8 9 10 11 12 gA/gV T 2:-2 ft 3:-3 ft 4:-4 ft Experiment

Nf = 2 + 1 domain wall (RBC+LHPC) One ensemble: mπ = 139 MeV, a = 0.11 fm, mπL = 3.9

S. Ohta parallel, Thu 11:40

Nf = 2 + 1 + 1 staggered (Fermilab-MILC) Blinded analysis.

Y. Lin parallel, Thu 9:30

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 33

SLIDE 64

Tensor and scalar charges

0.7 0.8 0.9 1.0 1.1 1.2 gT (MS, 2 GeV)

2+1 clover (LHPC ’12) 2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 twisted mass clover (ETMC ’17) 2+1 overlap (JLQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (Mainz ’18)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 gS (MS, 2 GeV)

2+1 clover (LHPC ’12) 2 clover (RQCD ’15) 2+1+1 clover on HISQ (PNDME ’16) 2 TM-clover (ETMC ’17) 2+1 overlap (JLQCD ’18) 2+1+1 clover on HISQ (PNDME ’18) 2+1 clover (Mainz ’18) LQCD average (∆mN/∆mq)QCD (Gonz´ alez-Alonso ’14)

Selection based on quality criteria still missing!

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 34

SLIDE 65

Sigma terms

25 30 35 40 45 50 55 60 65 RQCD ETMC χQCD BMWc

M. Hoferichter et al., PRL 115, 092301

20 40 60 80 100 120 140 160 σπN (MeV) σs (MeV) phenomenology LQCD Feynman-Hellmann LQCD direct

Published results with (near-)physical mπ . Update on BMWc Nf =1 + 1 + 1 + 1 results: L. Varnhorst and C. Hoelbling, Wed. 16:10 and 16:30

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 35

SLIDE 66

Outlook

◮ Excited-state contamination is a major focus.

A full variational study including nonlocal Nπ interpolators would help definitively resolve the issue.

◮ No sign of large finite-volume effect in fully-controlled studies

at low pion mass.

◮ For LQCD averages, individual calculations must be selected using

quality criteria. Fiting approaches for excited states vary; seting standards may be difficult.

◮ One atempt: H.-W. Lin et al., “Parton distributions and latice QCD calculations: A

community white paper” Prog. Part. Nucl. Phys. 100, 107 (2018) [1711.07916]

◮ Some nucleon structure to appear in next FLAG review. Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 36

SLIDE 67

Thanks to all who sent results and replied to my questions: Constantia Alexandrou Chia Cheng Chang Colin Egerer Rajan Gupta Jian Liang Shigemi Ohta Konstantin Otnad Finn Stokes André Walker-Loud Takeshi Yamazaki

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 37

SLIDE 68

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 38

SLIDE 69

Scalar and tensor charges

Precision β-decay experiments may be sensitive to BSM physics; leading contributions are controlled by the scalar and tensor charges:

T. Bhatacharya et al., Phys. Rev. D 85, 054512 (2012) [1110.6448]

p(P,s′)|¯ ud|n(P,s) = дS ¯ up(P,s′)un(P,s), p(P,s′)|¯ uσ µνd|n(P,s) = дT ¯ up(P,s′)σ µνun(P,s). Tensor charges also control contribution from quark electric dipole moment to neutron EDM: LqEDM = − 1

2

q

dq ¯ qiσ µνγ5qFµν =⇒ dn = duдd

T + ddдu T + dsдs T + · · · .

For scalar charge, conserved vector current relation implies дS = (mn − mp)QCD md − mu , up to second-order isospin breaking.

M. González-Alonso and J. Martin Camalich, Phys. Rev. Let. 112, 042501 (2014) [1309.4434]

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 39

SLIDE 70

Sigma terms

The contributions from quark masses to the nucleon mass. σπ N = mudN |¯ uu + ¯ dd|N σq = mqN | ¯ qq|N Control the sensitivity of direct-detection dark mater searches to WIMPs that interact via Higgs exchange. Phenomenological value σπ N = 59.1 ± 3.5 MeV determined based on combining:

◮ Cheng-Dashen low-energy theorem ◮ Roy-Steiner equations to constrain πN scatering amplitude ◮ Precise πN scatering lengths from pionic atoms

M. Hoferichter et al., Phys. Rev. Let. 115, 092301 (2015) [1506.04142]

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 40

SLIDE 71

Variational approach: nonzero momentum

16 18 20 22 24 26 28 30 32 34

t/a

−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2

GM(Q2) / µN dp (PEVA) dp (Conv.) Include operators with negative parity in variational basis.

F. Stokes et al., in preparation

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 41

SLIDE 72

Discretization effects: controlled studies

C. Alexandrou et al. (ETMC),
Phys. Rev. D 83, 045010 (2011) [1012.0857]

Nf = 2 Wilson twisted mass. No significant effect seen.

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

PRELIMINARY

gA a2 [fm2]

Nf = 3, mπ ≈ 420 MeV Expt

G. S. Bali et al. (RQCD), PoS LATTICE2016, 106

[1702.01035]

Nf = 3 clover. Linear in a2/t0 extrapolations.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 42

SLIDE 73

Discretization effects: global fits

gu−d

A

a [fm] a15m310 a12m310 a12m220L a12m220 a12m220S a09m310 a09m220 a09m130 a06m310 a06m220 a06m135 extrap. 1-extrap. 1.2 1.3 0.03 0.06 0.09 0.12 0.15

R. Gupta et al. (PNDME), 1806.09006

Nf = 2 + 1 + 1 clover on HISQ. pink: дA(mπ, L,a) = f (mπ, L) + ca/r1 gray: дA(mπ, L,a) = дA + ca/r1

0.00 0.01 0.02 0.03 0.04 0.05 0.06 ǫ2

a = a2/(4πw2 0 )

1.10 1.15 1.20 1.25 1.30 1.35 gA model average gLQCD

A

(ǫphys.

π

, ǫa) gPDG

A

= 1.2723(23) gA(ǫ(130)

π

, ǫa) gA(ǫ(220)

π

, ǫa) gA(ǫ(310)

π

, ǫa) gA(ǫ(350)

π

, ǫa) gA(ǫ(400)

π

, ǫa)

C. C. Chang et al., Nature 558, 91 [1805.12130]

Nf = 2 + 1 + 1 domain wall on HISQ. дA(mπ, L,a) = f (mπ, L) + ca2/w2

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 43

SLIDE 74

Chiral log?

Preferred fit: дA(mπ ) = c1 + c2m2

π

ChPT: дA(mπ ) = д0 − (д0 + 2д3

0)

mπ 4πF 2 logm2

π + cm2 π + . . .

0.00 0.05 0.10 0.15 0.20 0.25 0.30 ǫπ = mπ/(4πFπ) 1.10 1.15 1.20 1.25 1.30 1.35 gA model average gLQCD

A

(ǫπ, a = 0) gPDG

A

= 1.2723(23) gA(ǫπ, a ≃ 0.15 fm) gA(ǫπ, a ≃ 0.12 fm) gA(ǫπ, a ≃ 0.09 fm) a ≃ 0.15 fm a ≃ 0.12 fm a ≃ 0.09 fm 0.00 0.05 0.10 0.15 0.20 0.25 0.30 ǫπ = mπ/(4πFπ) 1.10 1.15 1.20 1.25 1.30 1.35 gA NNLO χPT gLQCD

A

(ǫπ, a = 0) gPDG

A

= 1.2723(23) gA(ǫπ, a ≃ 0.15 fm) gA(ǫπ, a ≃ 0.12 fm) gA(ǫπ, a ≃ 0.09 fm) a ≃ 0.15 fm a ≃ 0.12 fm a ≃ 0.09 fm

C. C. Chang et al., Nature 558, 91 (2018) [1805.12130]

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 44

SLIDE 75

Chiral log?

Preferred fit: дA(mπ ) = c1 + c2m2

π

ChPT: дA(mπ ) = д0 − (д0 + 2д3

0)

mπ 4πF 2 logm2

π + cm2 π + . . .

0.00 0.05 0.10 0.15 0.20 0.25 0.30 ǫπ = mπ/(4πFπ) 1.10 1.15 1.20 1.25 1.30 1.35 gA NNLO χPT LO NLO NNLO 0.00 0.05 0.10 0.15 0.20 0.25 0.30 ǫπ = mπ/(4πFπ) 1.10 1.15 1.20 1.25 1.30 1.35 gA NNLO χPT gLQCD

A

(ǫπ, a = 0) gPDG

A

= 1.2723(23) gA(ǫπ, a ≃ 0.15 fm) gA(ǫπ, a ≃ 0.12 fm) gA(ǫπ, a ≃ 0.09 fm) a ≃ 0.15 fm a ≃ 0.12 fm a ≃ 0.09 fm

C. C. Chang et al., Nature 558, 91 (2018) [1805.12130]

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 44

SLIDE 76

Chiral log?

Preferred fit: дA(mπ ) = c1 + c2m2

π

ChPT: дA(mπ ) = д0 − (д0 + 2д3

0)

mπ 4πF 2 logm2

π + cm2 π + . . .

0.00 0.05 0.10 0.15 0.20 0.25 0.30 ǫπ = mπ/(4πFπ) 1.10 1.15 1.20 1.25 1.30 1.35 gA NNLO χPT LO NLO NNLO 0.00 0.05 0.10 0.15 0.20 0.25 0.30 ǫπ = mπ/(4πFπ) 1.10 1.15 1.20 1.25 1.30 1.35 gA NNLO χPT gLQCD

A

(ǫπ, a = 0) gPDG

A

= 1.2723(23) gA(ǫπ, a ≃ 0.15 fm) gA(ǫπ, a ≃ 0.12 fm) gA(ǫπ, a ≃ 0.09 fm) a ≃ 0.15 fm a ≃ 0.12 fm a ≃ 0.09 fm

C. C. Chang et al., Nature 558, 91 (2018) [1805.12130]

0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 Axial charge in chiral limit

2+1+1 domain wall on HISQ (CalLat ’18) 2+1 clover (Mainz ’18) 2+1+1 domain wall on HISQ (CalLat ’18) 2+1 clover (Mainz ’18)

Preferred fit ChPT

May need precise data below mphys

π

to see chiral log.

Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 44

SLIDE 77

Calculations with near-physical mπ

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 3.0 4.0 5.0 6.0 7.0 gA (near-physical mπ) mπL Nf=2 TM-clover (ETMC) Nf=2+1+1 TM-clover (ETMC) Nf=2 clover (RQCD) Nf=2+1 clover (PACS) Nf=2+1+1 clover/HISQ (PNDME) Nf=2+1+1 DW/HISQ (CalLat)

Calculations with near-physical mπ

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 gA (near-physical mπ) a (fm) Nf=2 TM-clover (ETMC) Nf=2+1+1 TM-clover (ETMC) Nf=2 clover (RQCD) Nf=2+1 clover (PACS) Nf=2+1+1 clover/HISQ (PNDME) Nf=2+1+1 DW/HISQ (CalLat)