systematics in nucleon matrix element calculations€¦ · motivation three reasons for studying...

Systematics in nucleon matrix element calculations

Jeremy Green

NIC, DESY, Zeuthen

The 36th International Symposium on Laice Field TheoryEast Lansing, MI, USA

July 22–28, 2018

Motivation

Three reasons for studying structure of protons and neutrons:I Understanding the quark and gluon substructure of a hadron.I Understanding nucleons as tools in experiments.I Validation of laice QCD using “benchmark” observables.

Jeremy Green | DESY, Zeuthen | Laice 2018 |

Neutron beta decay

W−

n

pνee Coupling toW boson via axial current depends on nucleonaxial charge,

〈p (P , s ′) |uγ µγ5d |n(P , s )〉 = дAup (P , s ′)γ µγ5un (P , s ).

Interpreted as isovector quark spin contribution.→ Yi-Bo Yang plenary

PDG 2018: дA = 1.2724(23).

History plots 1

1960 1970 1980 1990 2000 2010850

900

950

1000

1050

1100

1960 1970 1980 1990 2000 201080

85

90

95

100

105

1990 2000 20100.8

1.0

1.2

1.4

1.6

1.8

2.0

1970 1980 1990 2000 2010-1.28

-1.26

-1.24

-1.22

-1.20

-1.18

-1.16

1970 1980 1990 2000 20106

8

10

12

14

16

1990 2000 20100.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

1970 1980 1990 2000 2010547.0

547.5

548.0

548.5

549.0

549.5

550.0

1970 1980 1990 2000 20100.52

0.53

0.54

0.55

0.56

0.57

0.58

0.59

1990 2000 201077

78

79

80

81

82

83

84

1960 1970 1980 1990 2000 20101115.0

1115.2

1115.4

1115.6

1115.8

1980 1990 2000 20101770

1780

1790

1800

1810

1820

1830

1970 1980 1990 2000 2010-20

0

20

40

60

80

100

Figure 1: A historical perspective of values of a few particle properties tabulated in this Review as a function of date of publication of the

Review. A full error bar indicates the quoted error; a thick-lined portion indicates the same but without the “scale factor.”

Citation: M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018)

1DARIUS 17 calculates this value from the measurement of the a parameter (see below).2MENDENHALL 13 gets A = −0.11954 ± 0.00055 ± 0.00098 and λ = −1.2756 ±0.0030. We quote the nearly identical values that include the earlier UCNA measurement(PLASTER 12), with a correction to that result.

3This MUND 13 value includes earlier PERKEO II measurements (ABELE 02 andABELE 97D).

4MOSTOVOI 01 measures the two P-odd correlations A and B, or rather SA and SB,where S is the n polarization, in free neutron decay.

5YEROZOLIMSKY 97 makes a correction to the EROZOLIMSKII 91 value.6This PLASTER 12 value is identical with that given in LIU 10, but the experiment isnow described in detail.

7 This is the combined result of ABELE 02 and ABELE 97D.8 These experiments measure the absolute value of gA/gV only.9KROHN 75 includes events of CHRISTENSEN 70.

10KROPF 74 reviews all data through 1972.

WEIGHTED AVERAGE-1.2724±0.0023 (Error scaled by 2.2)

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

χ2

23.8(Confidence Level = 0.0002)

-1.29 -1.28 -1.27 -1.26 -1.25 -1.24

λ ≡ gA / gV

e− ASYMMETRY PARAMETER Ae− ASYMMETRY PARAMETER Ae− ASYMMETRY PARAMETER Ae− ASYMMETRY PARAMETER AThis is the neutron-spin electron-momentum correlation coefficient. Unless otherwisenoted, the values are corrected for radiative effects and weak magnetism. In theStandard Model, A is related to λ ≡ gA/gV by A = −2 λ (λ + 1) / (1 + 3λ2); thisassumes that gA and gV are real.

VALUE DOCUMENT ID TECN COMMENT

−0.1184 ±0.0010 OUR AVERAGE−0.1184 ±0.0010 OUR AVERAGE−0.1184 ±0.0010 OUR AVERAGE−0.1184 ±0.0010 OUR AVERAGE Error includes scale factor of 2.4. See the ideogrambelow.−0.11952±0.00110 1 MENDENHALL13 UCNA Ultracold n, polarized

−0.11926±0.00031+0.00036−0.00042

2 MUND 13 SPEC Cold n, polarized

−0.1160 ±0.0009 ±0.0012 LIAUD 97 TPC Cold n, polarized

−0.1135 ±0.0014 3 YEROZLIM... 97 CNTR Cold n, polarized

−0.1146 ±0.0019 BOPP 86 SPEC Cold n, polarized

HTTP://PDG.LBL.GOV Page 11 Created: 6/5/2018 19:00


Other observables

Precision β-decay experiments may be sensitive to BSM physics; leadingcontributions are controlled by the scalar and tensor charges:T. Bhaacharya 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 maer searchesto WIMPs that interact via Higgs exchange.

σπN =mud 〈N |uu + dd |N 〉 σq =mq〈N |qq |N 〉


Toward precision nucleon structure

Trustworthy laice calculations need control over all systematics:I Physical quark massesI Isolation of ground stateI Infinite volumeI 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


Toward precision nucleon structure

Trustworthy laice calculations need control over all systematics:I Physical quark massesI Isolation of ground stateI Infinite volumeI Continuum limit

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

22 24 26 28 30 32 34

=+

+=

+=

pheno.

Weinberg 77Leutwyler 96Kaiser 98Narison 06Oller 07PDG

QCDSF/UKQCD 06ETM 07RBC 07ETM 10BETM 14D

FLAG average for =

MILC 04, HPQCD/MILC/UKQCD 04RBC/UKQCD 08PACS-CS 08MILC 09MILC 09APACS-CS 09Blum 10RBC/UKQCD 10ABMW 10A, 10BLaiho 11PACS-CS 12RBC/UKQCD 12RBC/UKQCD 14B

FLAG average for = +

ETM 14FNAL/MILC 14A

FLAG average for = + +

/

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

Outline

1. Methodologies for nucleon matrix elements

2. Excited-state eects: theory and practice

3. Other systematics: L, (a),mq

4. Outlook


Hadron correlation functions

Compute two-point and three-point functions, using interpolator χ andoperator insertion O. In simplest case:

t0

τ0 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∗n〈n′ |O|n〉e−Enτ e−En′ (T−τ )

→ |Z0 |2〈0|O|0〉e−E0T(1 +O (e−∆Eτ ) +O (e−∆E (T−τ ) )

)Jeremy Green | DESY, Zeuthen | Laice 2018 | Page 7

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 (T2 ,T ) = 〈0|O|0〉 +O (e−∆ET /2).

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

S (T ) ≡∑τ

R (τ ,T ),d

dTS (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 Laice 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 .


Ratio and summation method

1.15

1.20

1.25

1.30

2 4 6 8 10 12

g A(bare)

T/a

R(T/2, T )

S(T + a)− S(T )

physicalmπ , a = 0.116 fmN. Hasan, JG, et al., in preparation


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 ≡mq〈N |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]


Controlling excited states

Three approaches:I Go to large T . Need ∆ET 1.

But signal-to-noise decays as e−(E0− 32mπ )T .

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


Excited-state energies

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

3.0 3.5 4.0 4.5 5.0mπL

0.0

0.2

0.4

0.6

0.8

1.0∆E

(GeV

)

Nπ

Physicalmπ .




3.0 3.5 4.0 4.5 5.0mπL

0.0

0.2

0.4

0.6

0.8

1.0∆E

(GeV

)

Nπ Nππ

Physicalmπ .




3.0 3.5 4.0 4.5 5.0mπL

0.0

0.2

0.4

0.6

0.8

1.0∆E

(GeV

)

Nπ Nππ Nπππ

Physicalmπ .




3.0 3.5 4.0 4.5 5.0mπL

0.0

0.2

0.4

0.6

0.8

1.0∆E

(GeV

)

Nπ Nππ Nπππ Nππππ

Physicalmπ .




0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50mπ (GeV)

0.0

0.2

0.4

0.6

0.8

1.0∆E

(GeV

)

Nπ Nππ Nπππ Nππππ

mπL = 4.



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

Focus on Nπ sector and apply finite-volume quantization usingscaering phase shi from experiment.

3.0 3.5 4.0 4.5 5.0 5.5 6.0MπL

1000

1200

1400

1600

1800

E (

MeV

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


Going to largeT

Statistical errors:

δstat ∼ N −1/2e (E0− 32mπ )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.


Predicting excited-state contributions

Need to know:

1. energies En2. 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 localinterpolator to N and Nπ states.


ChPT prediction of excited-state eectsm

e(T)/m

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.01.00

1.01

1.02

1.03

1.04

T (fm)

physical pointmπL = 4

truncated to fiveNπ states

need T & 2 fm forChPT 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


ChPT prediction of excited-state eectsRX(T

2,T

)/X

Results: Midpoint estimate for all charges and moments

hxiud

hxiud

hxiud

gS

gA

gT

1.0 1.5 2.0 2.51.00

1.05

1.10

1.15

1.20

1.25

T (fm)

physical pointmπL = 4

truncated to fiveNπ states

need T & 2 fm forChPT 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


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 resonanceregime modeled with parameters α and γ .

↵ = 0, = 0↵ = 1, = 0↵ = 1, = 4

ML = 4ML = 6


Numerical studies

Available data depends onhow C3pt is computed.

τ0 T

For connected diagrams:I Fixed sink: most common. Get data at all τ and

any operator insertion. Cost increases with eachvalue of T .

I Fixed operator and τ . Get data at all T and anyset of interpolators. variational studies by CSSM

I Fixed operator and summed τ . Get summationdata at all T and any set of interpolators.used by CalLat, NPLQCD

−6 −4 −2 0 2 4 6(τ−T/2)/a

1.30

1.35

1.40

1.45

g A(b

are)

T/a =

14121086

ratiosummation

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]

0 5 10 15t/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]


Improving the nucleon interpolator

I Standard operator: χ = ϵabc (uTaCγ5db )uc , with smeared quark fields.I Smearing width typically tuned so thatme has early plateau.I Variational approach: χ =

∑i ci χi with optimized ci .

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

and Nππ states.


Variational approach: dierent smearings — eective 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

aM

N

Mass 2exp Variational Comparison

sm32

sm64

sm128

t0 =2 ∆t=2

mπ = 460 MeV, a = 0.074 fmJ. 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 fmB. Yoon et al. (NME), PRD 93, 114506 (2016)

[1602.07737]

Small improvement over widest smearing.


Variational approach: dierent smearings — eective 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

aM

N

Mass 2exp Variational Comparison

sm32

sm64

sm128

t0 =2 ∆t=2


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

[1602.07737]

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


Variational approach: dierent smearings — axial charge

8 6 4 2 0 2 4 6 8

τ−t/20.95

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 fmT = 0.96 fmJ. 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 0 2 4 6

gA

τ - tsep/2

tsep=10tsep=12tsep=14

tsep=16

V357V579

mπ = 312 MeV, a = 0.081 fmvariational at T = 0.97 fm versus S5B. 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


Variational approach: dierent smearings — axial charge

8 6 4 2 0 2 4 6 8

τ−t/20.95

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 fmT = 0.96 fmJ. 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 0 2 4 6

gA

τ - tsep/2

tsep=10tsep=12tsep=14

tsep=16

V357V579

mπ = 312 MeV, a = 0.081 fmvariational at T = 0.97 fm versus S9B. 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


Variational approach: larger basis — eective mass

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?


Variational approach: larger basis — tensor charge

Standard operator VariationalCalculation 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?


Fiing approaches

Fit correlators using ansatz based on N -state model.I En and Zn generally determined from C2pt.I 〈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 .I Each model state approximates contribution from several states.I Fit ansatz should be considered a somewhat uncontrolled model.


Fits for axial charge

Fixed sink

1.1

1.2

1.3

1.4

−10 −5 0 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

0 5 10 15t/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 sinksunconstrained 2-state fit: T ≥ 3a

1.22

1.26

g A/

g V

a09m220

meff tmin geffA tmin geff

V tmin

8 9 10 11 2 3 4 5 5 6 7 810−2

10−1

100

P


Simultaneous fits

data at tsep = 1.0fmfit model at tsep = 1.0fm

fit result

tsep/fm

gu−d

A

10.50-0.5-1

1.3

1.25

1.2

1.15

1.1

1.05

1

0.95

0.9data at tsep = 1.0fm

fit model at tsep = 1.0fmfit result

tsep/fm

gu−d

S

10.50-0.5-1

2

1.5

1

0.5

0data at tsep = 1.0fm


tsep/fm

gu−d

T

10.50-0.5-1

1.3

1.2

1.1

1

0.9

0.8


fit result

tsep/fm

〈x〉 u

−d

10.50-0.5-1

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05



tsep/fm

〈x〉 ∆

u−∆d

10.50-0.5-1

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05



tsep/fm

〈x〉 δ

u−δd

10.50-0.5-1

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

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 withtstartMπ = 0.4 fixed globally for all ensembles.∆E determined only from ratios, not C2pt.→ K. Onad parallel, Thu 12:00


Simultaneous fits


fit result

tsep/fm

gu−d

A

10.50-0.5-1

1.3

1.25

1.2

1.15

1.1

1.05

1

0.95



tsep/fm

gu−d

S

10.50-0.5-1

2

1.5

1

0.5



tsep/fm

gu−d

T

10.50-0.5-1

1.3

1.2

1.1

1

0.9

0.8


fit result

tsep/fm

〈x〉 u

−d

10.50-0.5-1

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05



tsep/fm

〈x〉 ∆

u−∆d

10.50-0.5-1

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05



tsep/fm

〈x〉 δ

u−δd

10.50-0.5-1

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0




Simultaneous fits

∆Nπ

2Mπ

Fitted ∆0

Mπtstart

∆0/[G

eV]

0.80.70.60.50.40.30.20.1

3

2.5

2

1.5

1

0.5

0

mπ = 347 MeV, a = 0.064 fm



Fit qualityCan we trust these fits? Ideally we would require that they have good fitquality. (Not a suicient condition!)

0.0 0.2 0.4 0.6 0.8 1.0p

0.0

0.2

0.4

0.6

0.8

1.0

Cum

ulat

ive

dist

ribu

tion

func

tion

uniformCalLatPNDME

Should expect uniformly distributed p-values.Following

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

can use Kolmogorov-Smirnov test.

PNDME: p = 0.26CalLat: p = 0.00077

p-values not provided by Mainz,but reduced χ 2 are large.

0.0 0.5 1.0 1.5 2.0 2.5 3.0χ2 per degree of freedom

0.0

0.2

0.4

0.6

0.8

1.0

Cum

ulat

ive

dist

ribu

tion

func

tion

Mainz


Fit qualityCan we trust these fits? Ideally we would require that they have good fitquality. (Not a suicient condition!)

0.0 0.2 0.4 0.6 0.8 1.0p

0.0

0.2

0.4

0.6

0.8

1.0

Cum

ulat

ive

dist

ribu

tion

func

tion

uniformCalLatPNDME

Should expect uniformly distributed p-values.Following

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

can use Kolmogorov-Smirnov test.

PNDME: p = 0.26CalLat: p = 0.00077

p-values not provided by Mainz,but reduced χ 2 are large.

0.0 0.5 1.0 1.5 2.0 2.5 3.0χ2 per degree of freedom

0.0

0.2

0.4

0.6

0.8

1.0

Cum

ulat

ive

dist

ribu

tion

func

tion

Mainz

May be diicult to estimate χ 2 due to diiculty inverting large covariancematrix.


What about Nπ states?

Expected Nπ states not seen in variational or fiing results. Why?I Volume suppression? Contamination from each state ∝ L−3.

But density of states ∝ L3 compensates.I At short T , higher states dominate?=⇒ not yet in asymptotic regime.

Familiar from mesonspectroscopy.

Must include nonlocaloperators in variationalbasis to identify completespectrum.

0.10

0.15

0.20

D. J. Wilson et al. (HadSpec), Phys. Rev. D 92, 094502 (2015)[1507.02599]


Comparison of several approachessm

32sm

64sm

128

t=

10t=

13t=

16t=

19t=

22

δt=

0δt

=1

δt=

2δt

=3

δt=

0δt

=1

δt=

2δt

=3

δt=

0δt

=1

δt=

2δt

=3

δt=

2δt

=3

δt=

4δt

=2

δt=

3δt

=4

δt=

2δt

=3

δt=

4

δt=

2δt

=3

δt=

4

δt=

2δt

=3

δt=

4

t=

13t=

16

Methods

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

Valu

e

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 sm32sm32 sm32sm32 ∆t=2

gA Summary


1.05

1.15

1.25

1.35

g A

Plateau Summation Two-state

0.9 1.0 1.1 1.2 1.3 1.4ts[fm]

0.50

0.55

0.60

0.65

0.70

gu+

dA

Connected

0.9 1.0 1.1 1.2 1.3tlow

s [fm]

1.05

1.15

1.25

1.35

g A


0.9 1.0 1.1 1.2 1.3 1.4ts[fm]

0.50

0.55

0.60

0.65

0.70

gu+

dA

Connected

0.9 1.0 1.1 1.2 1.3tlow

s [fm]

mπ = 130 MeV, a = 0.093 fmC. Alexandrou et al. (ETMC), PRD 96, 054507 (2017)

[1705.03399]

May be best to estimate systematic uncertainty using multiple dierentapproaches.


Finite-volume eects

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) − дAдA

=m2π

3π 2 f 2π

[д2AF1 (mπL) + д

2∆N (1 + 25д∆∆

81дA )F2 (mπ ,∆,L)

+ F3 (mπL) + д2∆N F4 (mπ ,∆,L)

],

Neglecting loops with ∆ baryons, the leading contribution is

дA (L) − дAдA

∼ m2πд

2A

3π 2 f 2π

√π

2mπLe−mπ L


Finite-volume eects: 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 MeVRQCD 290 MeVRQCD 420 MeV

PNDME 220 MeVCalLat 220 MeVETMC 130 MeV

QCDSF 261 MeVQCDSF 290 MeVLHPC 356 MeVLHPC 254 MeVRBC 420 MeV

Reference data at Lmax indicated with black dot.


Finite-volume eects: 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/mphysπ )2e−mπL/

√mπL

RQCD 150 MeVRQCD 290 MeVRQCD 420 MeVPNDME 220 MeVCalLat 220 MeVETMC 130 MeV

QCDSF 261 MeVQCDSF 290 MeVLHPC 356 MeVLHPC 254 MeVRBC 420 MeV

Reference data at Lmax indicated with black dot.


Finite-volume eects: global fitsgu−d

A

MπL

1.2

1.3

3 4 5 6 7


дA (mπ ,L,a) = f (mπ ,a) + cm2πe−mπ L

atmphysπ ,mπL = 4: −0.9(5)% eect

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

g A

NNLO+ct χPT

NLO χPT predictionNLO χPT prediction


leading HBChPT expression (no ∆)

+c (mπ /fπ )3F1 (mπL)

Larger (few %) eect seen by Mainz group! K. Onad parallel, Thu 12:00



A

MπL

1.2

1.3

3 4 5 6 7




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

g A

NNLO+ct χPT

NNLO χPT estimateNNLO χPT estimate


leading HBChPT expression (no ∆)+c (mπ /fπ )

3F1 (mπL)




A

MπL

1.2

1.3

3 4 5 6 7




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

g A

model average

NNLO χPT estimateNNLO χPT estimate


leading HBChPT expression (no ∆)+c (mπ /fπ )

3F1 (mπL)



Chiral extrapolation

Increasing number of calculations at physicalmπ .Otherwise need extrapolation.

Heavy baryon ChPT for дA:

дA (mπ ) = д0 − (д0 + 2д30 )

( mπ

4πF

)2log

m2π

µ2 + c1m2π + c2m

3π + . . .

or use simple polynomials inmπ orm2π .

Range of convergence is a priori unknown.


Axial charge: extrapolated and physical-point results

0.8 0.9 1.0 1.1 1.2 1.3 1.4gA

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 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 criteriastill missing!



0.8 0.9 1.0 1.1 1.2 1.3 1.4gA













PDG



1.05

1.15

1.25

1.35

g A


0.9 1.0 1.1 1.2 1.3 1.4ts[fm]

0.50

0.55

0.60

0.65

0.70

gu+

dA

Connected

0.9 1.0 1.1 1.2 1.3tlow

s [fm]

1.05

1.15

1.25

1.35

g A


0.9 1.0 1.1 1.2 1.3 1.4ts[fm]

0.50

0.55

0.60

0.65

0.70

gu+

dA

Connected

0.9 1.0 1.1 1.2 1.3tlow

s [fm]

One ensemble: mπ = 130 MeV,a = 0.093 fm,mπL = 3.0C. 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=2L=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



0.8 0.9 1.0 1.1 1.2 1.3 1.4gA













PDG



0.00 0.05 0.10 0.15 0.20 0.25 0.30ǫπ = mπ/(4πFπ)

1.10

1.15

1.20

1.25

1.30

1.35

g A

model average gLQCDA (ǫπ , a = 0)

gPDGA = 1.2723(23)

gA(ǫπ , a ≃ 0.15 fm)gA(ǫπ , a ≃ 0.12 fm)gA(ǫπ , a ≃ 0.09 fm)

a ≃ 0.15 fma ≃ 0.12 fma ≃ 0.09 fm

Bayesian average over fit models forдA (mπ /Fπ ,mπL,a

2/w20 ).

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



0.8 0.9 1.0 1.1 1.2 1.3 1.4gA













PDG



1:24 1:28 1:32

gA

model avg

NNLO fflPT

NNLO+ct fflPT

NLO Taylor ›2ı

NNLO Taylor ›2ı

NLO Taylor ›ı

NNLO Taylor ›ı

+O(¸sa2) disc.

+O(a) disc.

omit 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

ffl2aug=dof

0:0 0:5 1:0

Bayes factor

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



0.8 0.9 1.0 1.1 1.2 1.3 1.4gA













PDG



gu−d

A

a [fm]

a15m310a12m310a12m220La12m220a12m220S

a09m310a09m220a09m130a06m310a06m220

a06m135extrap.1-extrap.

1.2

1.3

0 0.03 0.06 0.09 0.12 0.15

gu−d

A

M2π [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 + c3m2π

+ c4m2πe−mπ L


R. Gupta parallel, Thu 12:40



0.8 0.9 1.0 1.1 1.2 1.3 1.4gA













PDG



0 5 10 15t/a

0.8

1

1.2

1.4

1.6

PDG16m

π=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



0.8 0.9 1.0 1.1 1.2 1.3 1.4gA













PDG



0 5 10 15t/a

0.8

1

1.2

1.4

1.6

PDG16m

π=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



0.8 0.9 1.0 1.1 1.2 1.3 1.4gA













PDG



lattice data (corrected)value at Mπ,phys

chiral + continuum + FS extrapolation

Mπ/GeV

t0M2π

gu−d

A

0.350.300.250.200.150.100

0.070.060.050.040.030.020.010

1.4

1.3

1.2

1.1

1

0.9

0.8

Ensembles havems varied such thatms +mu +md = const.Fit ansatz:

дA (mπ ,L,a) = c1 + c2a2 + c3m

2π

+ c4m2πe−mπ L

K. Onad parallel, Thu 12:00



0.8 0.9 1.0 1.1 1.2 1.3 1.4gA













PDG



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]

Two state fitL=64, a~0.081, Nf=2+1+1 (preliminary)

One ensemble: physicalmπ ,a = 0.081 fm,mπL = 3.6C. Lauer posterM. Constantinou parallel, Fri 17:50



0.8 0.9 1.0 1.1 1.2 1.3 1.4gA













PDG



Other preliminary results:

1

1.1

1.2

1.3

1.4

1.5

8 9 10 11 12

gA/gV

T

2:-2 ft3:-3 ft4:-4 ft

Experiment

Nf = 2 + 1 domain wall (RBC+LHPC)One ensemble: mπ = 139 MeV,a = 0.11 fm,mπL = 3.9S. Ohta parallel, Thu 11:40

Nf = 2 + 1 + 1 staggered(Fermilab-MILC)Blinded analysis.Y. Lin parallel, Thu 9:30


Tensor and scalar charges

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

2+1 clover (LHPC ’12)







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

2+1 clover (LHPC ’12)



2 TM-clover (ETMC ’17)




LQCD average (∆mN/∆mq)QCD(Gonzalez-Alonso ’14)

Selection based on quality criteria still missing!


Sigma terms

25 30 35 40 45 50 55 60 65

RQCD

ETMC

χQCD

BMWc

M. Hoferichter et al., PRL 115, 092301

0 20 40 60 80 100 120 140 160σπN (MeV) σs (MeV)

phenomenologyLQCD Feynman-HellmannLQCD direct

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


Outlook

I Excited-state contamination is a major focus.A full variational study including nonlocal Nπ interpolators would helpdefinitively resolve the issue.

I No sign of large finite-volume eect in fully-controlled studiesat low pion mass.

I For LQCD averages, individual calculations must be selected usingquality criteria. Fiing approaches for excited states vary; seingstandards may be diicult.I One aempt: H.-W. Lin et al., “Parton distributions and laice QCD calculations: A

community white paper” Prog. Part. Nucl. Phys. 100, 107 (2018) [1711.07916]I Some nucleon structure to appear in next FLAG review.


Thanks to all who sent results and replied to my questions:

Constantia AlexandrouChia Cheng Chang

Colin EgererRajan Gupta

Jian LiangShigemi Ohta

Konstantin OnadFinn Stokes

André Walker-LoudTakeshi Yamazaki


Scalar and tensor charges

Precision β-decay experiments may be sensitive to BSM physics; leadingcontributions are controlled by the scalar and tensor charges:T. Bhaacharya 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 momentto neutron EDM:

LqEDM = − 12

∑q

dqqiσµνγ5qFµν =⇒ dn = duд

dT + ddд

uT + dsд

sT + · · · .

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. Le. 112, 042501 (2014) [1309.4434]


Sigma terms

The contributions from quark masses to the nucleon mass.

σπN =mud 〈N |uu + dd |N 〉 σq =mq〈N |qq |N 〉Control the sensitivity of direct-detection dark maer searches to WIMPsthat interact via Higgs exchange.

Phenomenological value σπN = 59.1 ± 3.5 MeV determined based oncombining:I Cheng-Dashen low-energy theoremI Roy-Steiner equations to constrain πN scaering amplitudeI Precise πN scaering lengths from pionic atoms

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


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(Q

2)/µN

dp (PEVA) dp (Conv.)

Include operators with negative parity in variational basis.F. Stokes et al., in preparation Jeremy Green | DESY, Zeuthen | Laice 2018 | Page 41

Discretization eects: controlled studies

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

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

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

PRELIMINARY

g A

a2 [fm2]

Nf = 3, mπ ≈ 420 MeVExpt

G. S. Bali et al. (RQCD), PoS LATTICE2016, 106[1702.01035]

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


Discretization eects: global fitsgu−d

A

a [fm]

a15m310a12m310a12m220La12m220a12m220S

a09m310a09m220a09m130a06m310a06m220

a06m135extrap.1-extrap.

1.2

1.3

0 0.03 0.06 0.09 0.12 0.15


Nf = 2 + 1 + 1 clover on HISQ.pink: дA (mπ ,L,a) = f (mπ ,L) + ca/r1gray: дA (mπ ,L,a) = дA + ca/r1

0.00 0.01 0.02 0.03 0.04 0.05 0.06ǫ2

a = a2/(4πw20 )

1.10

1.15

1.20

1.25

1.30

1.35

g A

model average gLQCDA (ǫphys.

π , ǫa)

gPDGA = 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)


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

2/w20


Chiral log?

Preferred fit: дA (mπ ) = c1 + c2m2π

ChPT: дA (mπ ) = д0 − (д0 + 2д30 )

( mπ

4πF

)2logm2

π + cm2π + . . .

0.00 0.05 0.10 0.15 0.20 0.25 0.30ǫπ = mπ/(4πFπ)

1.10

1.15

1.20

1.25

1.30

1.35

g A

model average gLQCDA (ǫπ , a = 0)

gPDGA = 1.2723(23)


a ≃ 0.15 fma ≃ 0.12 fma ≃ 0.09 fm

0.00 0.05 0.10 0.15 0.20 0.25 0.30ǫπ = mπ/(4πFπ)

1.10

1.15

1.20

1.25

1.30

1.35

g A

NNLO χPT gLQCDA (ǫπ , a = 0)

gPDGA = 1.2723(23)


a ≃ 0.15 fma ≃ 0.12 fma ≃ 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 1.35Axial charge in chiral limit





Preferred fit

ChPT May need precise data belowmphysπ

to see chiral log.


Chiral log?

Preferred fit: дA (mπ ) = c1 + c2m2π

ChPT: дA (mπ ) = д0 − (д0 + 2д30 )

( mπ

4πF

)2logm2

π + cm2π + . . .

0.00 0.05 0.10 0.15 0.20 0.25 0.30ǫπ = mπ/(4πFπ)

1.10

1.15

1.20

1.25

1.30

1.35

g A

NNLO χPT

LONLO

NNLO

0.00 0.05 0.10 0.15 0.20 0.25 0.30ǫπ = mπ/(4πFπ)

1.10

1.15

1.20

1.25

1.30

1.35

g A

NNLO χPT gLQCDA (ǫπ , a = 0)

gPDGA = 1.2723(23)


a ≃ 0.15 fma ≃ 0.12 fma ≃ 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 1.35Axial charge in chiral limit





Preferred fit

ChPT May need precise data belowmphysπ

to see chiral log.


Calculations with near-physicalmπ

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-ph

ysical

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)

see also: Nf = 2 + 1 domain wall (RBC+LHPC) → S. Ohta parallel, Thu 11:40

Nf = 2 + 1 + 1 HISQ staggered → Y. Lin parallel, Thu 9:30Jeremy Green | DESY, Zeuthen | Laice 2018 | Page 45

Calculations with near-physicalmπ

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-ph

ysical

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)

see also: Nf = 2 + 1 domain wall (RBC+LHPC) → S. Ohta parallel, Thu 11:40

Nf = 2 + 1 + 1 HISQ staggered → Y. Lin parallel, Thu 9:30Jeremy Green | DESY, Zeuthen | Laice 2018 | Page 45

systematics in nucleon matrix element calculations€¦ · motivation three reasons for studying...

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