Strange nucleon form factors and isoscalar charges with Nf ...€¦ · Strange nucleon form factors and isoscalar charges with N f = 2 + 1 O(a)-improved Wilson fermions Dalibor Djukanovic,

motivation simulation results outlook

Strange nucleon form factors and isoscalarcharges with Nf = 2 + 1 O(a)-improved Wilson

fermions

Dalibor Djukanovic, Konstantin Ottnad, Harvey Meyer,Georg von Hippel, Jonas Wilhelm, Hartmut Wittig

21 June, 2019

1 / 32


table of contents

1 motivation

2 simulationform factorsensemblessetuprenormalizationz-expansion

3 resultsstrange vector form factorsstrange axial vector form factorsstrange tensor chargequark flavor contributions to gA and gT

4 outlook

2 / 32


Standard Model observables

weak mixing angle ΘW1

APV =GµQ

2

4πα√

2

{QW (p)− F (Ei ,Q

2)}

QW (p) = 1− 4 sin2 Θw

→ F (Ei ,Q2) contains G s

E , G sM and G s

A

1Becker et al., 1802.047593 / 32


Standard Model observables

contribution of intrinsic spin of quark flavor f

G fA(0) = g f

A = ∆f

1

2=∑f

(1

2∆f + Lf

)+ Jg

1

neutron electric dipole moment

dn =∑f

dγf gfT

⇒ appearance of quark-disconnected contributions

1X.-D. Ji, Phys. Rev. Lett. 78, 610 (1997), arXiv:hep-ph/96032494 / 32


Previous Work

Strange nucleon form factors with Nf = 2 + 1O(a)-improved Wilson fermions

D. Djukanovic, H. Meyer, K. Ottnad, G. von Hippel,J. Wilhelm, H. Wittig, arXiv:1810.10810

Strange electromagnetic form factors of the nucleon withNf = 2 + 1 O(a)-improved Wilson fermions

D. Djukanovic, K. Ottnad, J. Wilhelm, H. Wittig,arXiv:1903.12566

Nucleon isovector charges and twist-2 matrix elements withNf = 2 + 1 dynamical Wilson quarks

T. Harris, G. v. Hippel, P. Junnarkar, H. Meyer, K. Ottnad,J. Wilhelm, H. Wittig, L. Wrang, arXiv:1905.01291

5 / 32


form factors

construct a ratio

RJµ(~q;~p′; Γν) =CN

3,Jµ(~q, z0;~p′, y0; Γν)

CN2 (~p′, y0)

√CN

2 (~p′, y0)CN2 (~p′, z0)CN

2 (~p′-~q, y0-z0)

CN2 (~p′-~q, y0)CN

2 (~p′-~q, z0)CN2 (~p′, y0-z0)

s.d.= M1

νµ(~q, ~p′)G1(Q2) + M2νµ(~q, ~p′)G2(Q2)

(overdetermined) system of equations at each Q2

M ~G = ~R ; M =

M11

...M1

N

M21

...M2

N

, ~G =

(G1

G2

), ~R =

R1...

RN

minimizing the least-squares function

χ2 = (~R −M~G )T C−1 (~R −M~G )

6 / 32


form factor parametrization

⟨N, ~p, s |Jµ(x)|N, ~p′, s ′

⟩= us(~p) Jµ us

′(~p′)e iq·x

axial vector current

Aµ = γµγ5GA(Q2) + γ5qµ2m

GP(Q2)

vector current

Vµ = γµF1(Q2) + iσµνqν

2mF2(Q2)

F1,F2 ↔ GE,GM

tensor current at Q2 = 0

Tµν = iσµνgT

7 / 32


ensembles

Coordinated Lattice Simulations (CLS)1,2

Nf = 2 + 1 O(a)-improved Wilson fermionstree-level improved Luscher-Weisz gauge actionopen boundary conditions in timetr Mq = const

β a [fm] N3s × Nt mπ[MeV] mK [MeV] Ndis

cfg Nconcfg

H102 3.40 0.08636 323 × 96 352 436 997 997

H105 3.40 0.08636 323 × 96 278 461 1019 1019

C101 3.40 0.08636 483 × 96 223 473 489 1956

N401 3.46 0.07634 483 × 128 289 462 701 701

N203 3.55 0.06426 483 × 128 345 441 768 1536

N200 3.55 0.06426 483 × 128 282 462 852 852

D200 3.55 0.06426 643 × 128 203 482 234 936

N302 3.70 0.04981 483 × 128 359 458 1177 1177

1CLS, 1411.39822CLS, 1608.08900

8 / 32


setup

nucleon interpolator

Nα(x) = εabc

(uaβ(x) (Cγ5)βγ db

γ (x))

ucα(x)

smeared-smeared quark propagators

truncated solver method for two- and three-point functions1,2

CN2/3 =

1

NLPsrc

NLPsrc∑

n=1

CN2/3(xn)LP︸︷︷︸

biased estimate

+

1

NHPsrc

NHPsrc∑

n=1

(CN

2/3(xn)HP − CN2/3(xn)LP

)︸︷︷︸

bias correction

1G. Bali et al., PoS (LATTICE 2015) 3502E. Shintani et al., Phys. Rev. D91 114511 (2015)

9 / 32


lattice currents

improved local axial current

Aµ(~z , z0)Imp. = Aµ(~z , z0) + acA ∂µP(~z , z0)

improved conserved vector current

Vµ(~z , z0)Imp. = Vµ(~z , z0) + acV ∂νTνµ(~z , z0)

non-perturbative determination of cA and cV1,2

local tensor current

1Alpha Collaboration, Nucl. Phys. B 896 (2015) 5552Gerardin et al., Phys. Rev. D 99, 014519 (2019)

10 / 32


contributions

disconnected: G l(Q2) and G s(Q2)

ux y

d

u

zl/s

connected: Gu+dcon. (Q2)

ux y

d

u

z

+ · · ·

11 / 32


contributions

disconnected:

three-point function factorizes

CN,l/s3,Jµ

(~q, z0;~p′, y0; Γν) =⟨Ll/sJµ

(~q, z0) · CN2 (~p′, y0; Γν)⟩G

parity and spin projectors

Γ0 =1

2(1 + γ0) , Γk = Γ0 iγ5γk

connected:

sequential propagators for three-point functiondrop A0 due to large excited-state contaminationparity and spin projector

Γ =1

2(1 + γ0)(1 + iγ5γ3)

12 / 32


source positions

disconnected:

NHPsrc = 1 or 4, NLP

src = 32on 7 timeslices evenly distributed around Nt/2separated by seven timeslices without sources

connected:

NHPsrc = 1 or 4, NLP

src = 16− 48on 1 timeslicessame sources for two- and three-point functionsalready used in Mainz isovector charges paper1

1K. Ottnad et al., 1905.0129113 / 32


quark loopsLl/sΓµ

(~q, z0) = −∑~z∈Λ

e i~q·~z⟨tr[S l/s(z ; z) Γµ

]⟩G

= −∑~z∈Λ

e i~q·~z⟨η†(z) Γµ s l/s(z)

⟩G ,η

hierarchical probing1

ηn → hn � η

4D noise vectors, 2×512 Hadamard vectors/configuration

1Stathopoulos et al., 1302.401814 / 32


excited-state contamination

G eff(Q2, z0, y0) = G (Q2) + A1(Q2) e−∆z0 + B1(Q2) e−∆′(y0−z0)

+ C1(Q2) e−∆z0−∆′(y0−z0)

disconnected:plateau method y0 ∼ 1 fmsummation method y0 ∈ [0.5, 1.3] fm

S(Q2, y0) =

y0−1∑z0=1

G eff(Q2, z0, y0)

= K (Q2) + y0G (Q2) + O(e−∆y0 , e−∆′y0

)

connected + disconnected:summation method with 3 or 4 y0 ∈ [1.0, 1.5] fmtwo-state fits

15 / 32


renormalization

Nf = 3 periodic boundary condition ensembles

β ∈ {3.40, 3.46, 3.55}RI′-MOM scheme in Landau gauge

ZO 〈p |OΓ| p〉|p2=µ2 = 〈p |OΓ| p〉tree|p2=µ2 .

extrapolation to the chiral limitperturbative subtraction of leading-order lattice artifacts1

conversion to RGI scheme

MS as intermediate schemeMS β- and γ-functionsfit residual µ-dependence

convert to MS at µ = 2.0 GeV

1G. von Hippel et al., PoS LATTICE2016 (2016) 19416 / 32


renormalization

start with flavor-diagonal basis

OaΓ(x) = ψ(x)Γλaψ(x) , ψ = (u, d , s)T , a ∈ {3, 8, 0} ,

ZΓ =

Z 33Γ 0 00 Z 88

Γ 00 0 Z 00

Γ

, Z 33Γ = Z 88

Γ , Z iq = Zq .

basis transformation Ou−dΓ (x)R

Ou+dΓ (x)ROs

Γ(x)R

=

Zu−d ,u−dΓ 0 0

0 Zu+d ,u+dΓ Zu+d ,s

Γ

0 Z s,u+dΓ Z s,s

Γ

Ou−d

Γ (x)

Ou+dΓ (x)Os

Γ(x)

Zu−d ,u−d

Γ already used in Mainz isovector charges paper1

1K. Ottnad et al., 1905.0129117 / 32


renormalization

linear extrapolation to β = 3.7

error multiplied with factor 10checks performed for gu−d

A in Mainz isovector charges paper1

⇒ consistent results compared to using ZSFA

0.75

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

3.4 3.45 3.5 3.55 3.6 3.65 3.7

(Z33 A)R

GI

β

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

3.4 3.45 3.5 3.55 3.6 3.65 3.7

(Z00 A)R

GI

β

1K. Ottnad et al., 1905.0129118 / 32


z-expansion

fit the Q2-dependence

G (Q2) =5∑

k=0

akz(Q2)k

z(Q2) =

√tcut + Q2 −

√tcut√

tcut + Q2 +√tcut

, tcut = (2mK )2

Gaussian priors

ak = 0± 5 max{|a0|, |a1|} ∀ k > 1 (1)

for electric form factor: a0 = 0

charge radius

r2 ≡ −6dG

dQ2

∣∣∣∣Q2=0

= − 3

2tcuta1 (2)

19 / 32


strange vector form factors

mK ≈ 460 MeV

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0 0.2 0.4 0.6 0.8 1 1.2

Gs E(Q

2)

Q2 [GeV2]

a = 0.08636 fma = 0.07634 fma = 0.06426 fm

−0.035

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0 0.2 0.4 0.6 0.8 1 1.2

Gs M(Q

2)

Q2 [GeV2]

a = 0.08636 fma = 0.07634 fma = 0.06426 fm

β = 3.55

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0 0.2 0.4 0.6 0.8 1

Gs E(Q

2)

Q2 [GeV2]

mK = 482MeVmK = 462MeVmK = 441MeV

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0 0.2 0.4 0.6 0.8 1

Gs M(Q

2)

Q2 [GeV2]


20 / 32


extrapolation

SU(3) HBChPT1

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

−1.14 −1.12 −1.1 −1.08 −1.06 −1.04 −1.02 −1 −0.98

χ2/dof = 0.84

( r2) s E/t

0

log(√t0mK)

β = 3.40

H102

H105

C101

β = 3.46

N401

β = 3.55

N203 N200

D200

β = 3.70

N302

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

2.7 2.75 2.8 2.85 2.9 2.95 3 3.05 3.1

χ2/dof = 1.13

( r2) s M/t

0

1/(√t0mK)

β = 3.40

H102

H105C101

β = 3.46

N401

β = 3.55

N203

N200

D200

β = 3.70

N302

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0.32 0.325 0.33 0.335 0.34 0.345 0.35 0.355 0.36 0.365 0.37

χ2/dof = 2.67

µs

√t0mK

β = 3.40

H102H105

C101

β = 3.46

N401

β = 3.55

N203

N200

D200

β = 3.70N302

(r2)sE (mK ) = a0 + a1 log(mK )

µs(mK ) = a2 + a3mK

(r2)sM(mK ) = a4 + a5/mK

1T. R. Hemmert et al., nucl-th/990407621 / 32


error budget

(r2)sE [fm2] µs (r2)sM [fm2] χ2/DOFMain result -0.0039(13) -0.020(5) -0.010(5) 0.84, 2.67, 1.13Variations:

Doubling prior width -0.0049(18) -0.022(6) -0.015(9) 0.68, 1.83, 0.69Plateau method ∼ 1 fm -0.0040(9) -0.009(6) -0.0026(53) 1.12, 2.54, 1.58Including O(a2) -0.0030(16) -0.016(6) -0.007(6) 0.75, 2.98, 2.05No cut in Q2 -0.0034(9) -0.016(4) -0.007(4) 2.36, 2.77, 1.69

(r2)sE = −0.0039(13)(18) fm2

µs = −0.020(5)(12)

(r2)sM = −0.010(5)(11) fm2

22 / 32


comparison

−0.01 −0.005 0 −0.1 −0.05 0 −0.04 −0.02 0

this work

LHPC

ETMC

χQCD

1505.01803

1801.09581

1705.05849

(r2)sE [fm2] µs (r2)sM [fm2]

(r2)sE = −0.0039(13)(18) fm2

µs = −0.020(5)(12)

(r2)sM = −0.010(5)(11) fm2

23 / 32


strange axial vector form factors

mK ≈ 460 MeV

−0.055

−0.05

−0.045

−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

−0.01

0 0.2 0.4 0.6 0.8 1 1.2

Gs A(Q

2)

Q2 [GeV2]

a = 0.08636 fma = 0.07634 fma = 0.06426 fm

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0 0.2 0.4 0.6 0.8 1 1.2

Gs P(Q

2)

Q2 [GeV2]

a = 0.08636 fma = 0.07634 fma = 0.06426 fm

β = 3.55

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0 0.2 0.4 0.6 0.8 1

Gs A(Q

2)

Q2 [GeV2]


−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

Gs P(Q

2)

Q2 [GeV2]


24 / 32


extrapolation

linear in m2π

−0.05

−0.045

−0.04

−0.035

−0.03

−0.025

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

χ2/dof = 0.43

gs A

t0m2π

β = 3.40

H102

H105

C101

β = 3.46

N401

β = 3.55

N203

N200

D200

β = 3.70

N302

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

χ2/dof = 0.23( r2) s A

/t0

t0m2π

β = 3.40

H102H105

C101

β = 3.46

N401

β = 3.55

N203

N200

D200

β = 3.70

N302

25 / 32


comparison

−0.075 −0.05 −0.025 0

this work

ETMC

PNDME

χQCD

JLQCD

1703.08788

1806.10604

1806.08366

1805.10507

gsA

g sA = −0.035(4)(8)

(r2)sA = −0.005(5)(10) fm2

variations: same as for strange vector form factors,but O(a) lattice artifacts

26 / 32


strange tensor charge

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

χ2/dof = 1.44

gs T

t0m2π

β = 3.40

H102

H105

C101

β = 3.46

N401

β = 3.55

N203

N200

D200 β = 3.70

N302

−0.05 −0.025 0 0.025 0.05

this work

ETMC

PNDME

JLQCD

1703.08788

1812.03573

1805.10507

gsT

g sT = −0.0037(79)(441)

variations: plateau method ∼ 1 fm, including O(a) latticeartifacts

27 / 32


quark flavor contributions to gA and gT

guA/T =

1

2

(gu+d−2sA/T + 2g s

A/T + gu−dA/T

)gdA/T =

1

2

(gu+d−2sA/T + 2g s

A/T − gu−dA/T

)u + d − 2s renormalizes as u − d

additional noise cancellation for disconnected l − s1

gu−dA/T determined in Mainz isovector charges paper2

g sA/T already determined in this work

determine gu+d−2sA/T with two-state fit

gO(z0, y0) = gO + AO

(e−∆z0 + e−∆(y0−z0)

)+ BOe

−∆y0

gap ∆ determined as in Mainz isovector charges paper1

1V. Gulpers et al., PoS (LATTICE2014) 1282K. Ottnad et al., 1905.01291

28 / 32


two-state fit

N302

0.6

0.65

0.7

0.75

0.8

0 0.5

y0/a = 20

0 0.5

y0/a = 22

0 0.5

y0/a = 24

0 0.5

y0/a = 26

(gu

+d−

2sA

)eff bar

e(z 0,y

0)

z0/y0 z0/y0 z0/y0 z0/y0

29 / 32


extrapolation

linear in m2π

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

χ2/dof = 0.93

gu+d−2s

A

t0m2π

β = 3.40

H102

H105

C101

β = 3.46

N401

β = 3.55

N203

N200D200

β = 3.70 N302

0.45

0.5

0.55

0.6

0.65

0.7

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

χ2/dof = 2.35

gu+d−2s

Tt0m

2π

β = 3.40

H102

H105C101

β = 3.46N401β = 3.55

N203

N200

D200

β = 3.70

N302

variations: summation method, including O(a) lattice artifacts

30 / 32


comparison

0.6 0.8 1 1.2 −0.6 −0.35 −0.1

this work

ETMC

χQCD

PNDME

1706.02973

1806.08366

1806.10604

guA gdA

0.6 0.8 1 1.2 −0.3 −0.1 0.1

this work

JLQCD

ETMC

PNDME

1805.10507

1703.08788

1812.03573

guT gdT

guA = 0.84(3)(5)

gdA = −0.40(3)(5)

guT = 0.77(4)(5)

gdT = −0.19(4)(5)

31 / 32


outlook

improve renormalization at β = 3.7

more configurations and ensembles

electric and magnetic charge radius and magnetic moment ofproton and neutron

(r2)p/n =1

2

[1

3(r2)u+d−2s ± (r2)u−d

]

µp/n =1

2

[1

3µu+d−2s ± µu−d

]

32 / 32

Documents

Strange nucleon form factors and isoscalar charges with Nf ...€¦ · Strange nucleon form factors and isoscalar charges with N f = 2 + 1 O(a)-improved Wilson fermions Dalibor Djukanovic,