from 2+1 flavours of O ( a ) improved Wilson quarks Hartmut Wi6g - - PowerPoint PPT Presentation

36th Interna+onal Symposium on La6ce Field Theory — La6ce 2018 East Lansing, MI — USA 22–28 July 2018

Hartmut Wi6g

PRISMA Cluster of Excellence, Ins6tute for Nuclear Physics and Helmholtz Ins6tute Mainz

The hadronic vacuum polarisa+on contribu+on to (g–2)μ from 2+1 flavours of O(a) improved Wilson quarks

Hartmut Wittig

The muon anomalous magne+c moment

2

aµ = aQED

µ

+ aweak

µ

+ astrong

µ

QED+EW HLbL HVP E821 @ BNL SM predic6on aµ ≡ 1

2(g − 2)µ =

       116 592 080(54)(33) · 10−11 116 591 825(34)(26)(1) · 10−11

QED Strong SM es6mate dominated by QED; error dominated by QCD

Hartmut Wittig

The muon anomalous magne+c moment

2

QED+EW HLbL HVP E821 @ BNL SM predic6on aµ ≡ 1

2(g − 2)µ =

       116 592 080(54)(33) · 10−11 116 591 825(34)(26)(1) · 10−11

QED Strong SM es6mate dominated by QED; error dominated by QCD

µ µ γ

?

Contribu6on from “New Physics”?

aµ = aQED

µ

+ aweak

µ

+ astrong

µ

+ aNP?

µ

Hartmut Wittig 3

The Mainz (g – 2)μ project

Collaborators:

• N. Asmussen, M. Cè, A. Gérardin, O. Gryniuk, G. von Hippel, B. Hörz,


H.B. Meyer, A. Nyffeler, K. O]nad, V. Pascalutsa, A. Risch, T. San José Perez, HW

• J. Green, B. Jäger, G. Herdoíza
• Direct determina6ons

• f LO
• Running of α and sin2θW

ahvp

µ

• Exact QED kernel
• Forward sca]ering

amplitude

• Transi6on form factor

for π0 → γ∗γ∗

Hartmut Wittig

La6ce QCD approach to HVP

4

Convolu6on integral over Euclidean momenta:

[Lautrup & de Rafael; Blum]

ahvp

µ

= ✓α π ◆2 Z ∞ dQ2 f(Q2) ˆ Π(Q2), ˆ Π(Q2) = 4π2 ⇣ Π(Q2) − Π(0) ⌘

Πµν(Q) = i Z d4x eiQ·(x−y) D Jµ(x)Jν(y) E ≡ (QµQν − δµνQ2)Π(Q2)

Vacuum polarisa6on tensor: Electromagne6c current: Jµ = 2

3uγµu − 1 3dγµd − 1 3 sγµs + . . .

Weight func6on f(Q2) strongly peaked near muon mass

[Meyer & HW, arXiv:1807.09370]

Hartmut Wittig

La6ce QCD approach to HVP

5

Time-momentum representa+on (TMR):

[Bernecker & Meyer]

ahvp

µ

= ✓↵ ⇡ ◆2 Z 1 dx0 ˜ K(x0)G(x0), G(x0) = a3 X

~ x

hJk(x)Jk(0)i

˜ K(x0) = 4π2 Z ∞ dQ2 f(Q2) " x2

0 − 4

Q2 sin2 ⇣ 1

2Qx0

⌘#

1 2 3 4 5 6 0.0 0.1 0.2 0.3 0.4 0.5 t @fmD 103◊t3 GHtL K é HtL

41% 45% 11% 3%

Hartmut Wittig

La6ce QCD approach to HVP

5

Time-momentum representa+on (TMR):

[Bernecker & Meyer]

ahvp

µ

= ✓↵ ⇡ ◆2 Z 1 dx0 ˜ K(x0)G(x0), G(x0) = a3 X

~ x

hJk(x)Jk(0)i

˜ K(x0) = 4π2 Z ∞ dQ2 f(Q2) " x2

0 − 4

Q2 sin2 ⇣ 1

2Qx0

⌘#

1 2 3 4 5 6 0.0 0.1 0.2 0.3 0.4 0.5 t @fmD 103◊t3 GHtL K é HtL

41% 45% 11% 3%

Significant contribu6on from tail of G(x0)

Hartmut Wittig

Past and present calcula+ons

6

Extension to QCD with Nf = 2+1: ahvp

µ

= (654 ± 32 stat ± 17 syst ± 10 scale ± 7 FV

+ 0 −10 disc) · 10−10

Published result in two-flavour QCD:

[Della Morte et al., JHEP 10 (2017) 020]

Use O(a) improved vector currents ⟶ con6nuum extrapola6on in a2 Dedicated calcula6on of the I = ℓ = 1 sca]ering phase shii
 and 6meline pion form factor

[H.B. Meyer, EPJC 77 (2017) 616] [Bruno et al., PRD 95 (2017) 074504]

p 8t0 = 0.415(4)(2) fm

Scale sejng: Disconnected diagrams:
 Hierarchical probing; covariant coordinate-space method

Hartmut Wittig 7

CLS ensembles

Tree-level Symanzik gauge ac6on Four values of the lajce spacing:
 a = 0.085, 0.077, 0.065, 0.050 fm Pion masses and volumes: mmin

π

≈ 135 MeV, mπL > 4 Quark mass trajectory: Tr M = const.

Open boundary condi6ons — topology freezing Nf = 2+1 flavours of O(a) improved Wilson fermions

Hartmut Wittig 8

Current data sets

“Landscape” of CLS ensembles for HVP determina6on

100 150 200 250 300 350 400 450 0.0502 0.0652 0.0772 0.0852

mπ [MeV] a2 [fm]

H101 H102 N101 (H105) C101 B450 S400 N401 N202 (H200) N203 N200 D200 E250 N300 N302 J303

Hartmut Wittig 8

Current data sets

“Landscape” of CLS ensembles for HVP determina6on

100 150 200 250 300 350 400 450 0.0502 0.0652 0.0772 0.0852

mπ [MeV] a2 [fm]

H101 H102 N101 (H105) C101 B450 S400 N401 N202 (H200) N203 N200 D200 E250 N300 N302 J303

Hartmut Wittig 9

Vector currents and correlators

Can be applied with open boundary condi6ons O(a) improved local and conserved currents: cA

V :

determined non-perturba6vely ZV determined from ra6o of three- and two-point func6ons;

[Della Morte et al., arXiv:1705.01775]

Quark-connected TMR correlators: (VA

k )imp(x) = (VA k )(x) + a cA V ˜

∂0Tk0(x), A = loc, con

G f (x0) = −a3 3 X

k

X

x

q2

f (Z f V)2 D

Vloc

k, f (x0, x)Vloc k, f (0)

E ,

G f (x0) = −a3 3 X

k

X

x

q2

f Z f V

D Vcon

k, f (x0, x)Vloc k, f (0)

E ,

f = ud, s, c

Hartmut Wittig 10

Preliminary results: ud, s, c

ud, s and c contribu6ons on “D200”, i.e. mπ = 200 MeV, a = 0.065 fm

0.004 0.008 0.012 0.016 0.5 1 1.5 2 2.5 3 3.5

x0 [fm] G(x0) K(x0)/mµ

Light Strange (x6) Charm (x6)

Hartmut Wittig 11

Preliminary results: charm

Chiral and con6nuum extrapola6on Simultaneous fits using two discre6sa6ons work well Physical pion mass to be added — determine κcharm

5 10 15 20 25 30 0.02 0.04 0.06 0.08 0.1 0.12

• y

µ,c

β = 3.40 β = 3.46 β = 3.55 β = 3.70

loc-loc loc-con

= m2

π/(16π2 f 2 π )

(ahvp

µ )c · 1010

Hartmut Wittig 12

Preliminary results: charm

Nf = 2 + 1 + 1 BMW 17 ETMC 17 HPQCD 14 ETMC 13 Nf = 2 + 1 RBC/UKQCD 18 CLS preliminary 13.5 14.0 14.5 15.0 15.5 (ahvp

µ

)c · 10−10 Nf = 2 CLS/Mainz 17

Scale sejng error dominates Systema6c error from pion mass cut at mπ = 360 MeV (ahvp

µ )c = 14.95(47)stat(11)χ · 10−10

Hartmut Wittig 13

Preliminary results: strange

0.0005 0.001 0.0015 0.002 0.5 1 1.5 2 2.5 3

x0 [fm]

L = 2.05 fm L = 2.70 fm L = 4.10 fm

0.0005 0.001 0.0015 0.002 0.5 1 1.5 2 2.5 3

x0 [fm]

loc-loc cons-loc

Physical pion mass FV effects @ mπ = 280 MeV Small remaining discre6sa6on effects Finite-volume effects negligible
 for L ≥ 2.7 fm

L [fm] loc-con loc-loc 2.05 65.8(0.6) 69.5(0.6) 2.70 68.0(0.4) 71.8(0.4) 4.10 68.0(0.3) 71.9(0.3)

Hartmut Wittig 14

Preliminary results: strange

Simultaneous chiral and con6nuum extrapola6on

= m2

π/(16π2 f 2 π )

20 30 40 50 60 70 80 90 100 0.02 0.04 0.06 0.08 0.1 0.12

• y

β = 3.40 β = 3.46 β = 3.55 β = 3.70

(ahvp

µ )s · 1010

(ahvp

µ )s = γ1 + γ2m2 π + γ3m2 π ln m2 π + γ4 a2

Apply pion mass cut to es6mate systema6c error

Hartmut Wittig 15

Preliminary results: strange

Nf = 2 + 1 + 1 BMW 17 ETMC 17 HPQCD 14 ETMC 13 Nf = 2 + 1 CLS preliminary RBC/UKQCD 18 RBC/UKQCD 16 50 52 54 56 (ahvp

µ

)s · 10−10 Nf = 2 CLS/Mainz 17

Scale sejng error dominates (ahvp

µ )s = 53.6(2.5)stat(0.8)χ · 10−10

Hartmut Wittig 16

Preliminary results: light quarks

Long-distance regime of iso-vector TMR correlator Gρρ(x0, L)

x0→∞

= X

n

|An|2 e−ωnx0, ωn = 2 q m2

π + k2

0.004 0.008 0.012 0.016 0.5 1 1.5 2 2.5 3

xcut x0 [fm]

1 exp fit loc-cons n=4 n=3 n=2 n=1

mπ = 200 MeV

Hartmut Wittig 16

Preliminary results: light quarks

Long-distance regime of iso-vector TMR correlator Gρρ(x0, L)

x0→∞

= X

n

|An|2 e−ωnx0, ωn = 2 q m2

π + k2

0.004 0.008 0.012 0.016 0.5 1 1.5 2 2.5 3

xcut x0 [fm]

1 exp fit loc-cons n=4 n=3 n=2 n=1

mπ = 200 MeV

Hartmut Wittig 17

Preliminary results: light quarks

Finite-volume correc6on

Gρρ(x0, ∞) = Z ∞ dω ω2ρ(ω2)e−ω|x0|, ρ(ω2) = 1 48π2 1 − 4m2

π

ω2 !3/2 |Fπ(ω)|2 Gρρ(x0, L)

x0→∞

= X

n

|An|2 e−ωnx0,

|An|2 = 2k5 3πω2 |Fπ(ω)|2 n kφ0(k) + kδ0

1(k)

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Use Gounaris-Sakurai parameterisa6on of Fπ(ω) in terms of (mρ, Γρ) Iso-vector correlator in infinite volume Finite volume: Isospin decomposi6on:

ahvp

µ (∞) − ahvp µ (L) =

✓α π ◆2 Z ∞ dx0 ˜ K(x0) [G(x0, ∞) − G(x0, L)]

G(x0) = Gρρ(x0) + GI=0(x0)

[Francis et al., PRD 88 (2013) 054502; Della Morte et al., JHEP 10 (2017) 020]

Hartmut Wittig 18

Preliminary results: light quarks

Cross check at mπ = 280 MeV (H105, N101: mπ L = 3.8, 5.8)

0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x0 [fm]

L/a = 32 (L = 2.70 fm) L/a = 48 (L = 4.10 fm) L/a = 32 with FSE L/a = 48 with FSE

Hartmut Wittig 18

Preliminary results: light quarks

Cross check at mπ = 280 MeV (H105, N101: mπ L = 3.8, 5.8)

0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x0 [fm]

L/a = 32 (L = 2.70 fm) L/a = 48 (L = 4.10 fm) L/a = 32 with FSE L/a = 48 with FSE

Hartmut Wittig 18

Preliminary results: light quarks

Cross check at mπ = 280 MeV (H105, N101: mπ L = 3.8, 5.8)

0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x0 [fm]

L/a = 32 (L = 2.70 fm) L/a = 48 (L = 4.10 fm) L/a = 32 with FSE L/a = 48 with FSE

Finite-size effects well described by GS parameterisa6on of Fπ(ω)

Hartmut Wittig 19

Preliminary results: light quarks

Representa6on of Fπ(ω) via 3-subtracted Omnès rela6on

[Erben, Green, Mohler, HW, arXiv:1710.03529 and in prep.]

F⇡(t) = exp Pn−1(t) + tn ⇡ Z ∞

4m2

⇡

ds 11(s) sn(s − t − i✏) !

(Nf = 2)

Hartmut Wittig 20

Preliminary results: light quarks

TMR integrand at the physical pion mass with current sta6s6cs

0.004 0.008 0.012 0.016 0.5 1 1.5 2 2.5 3

xcut x0 [fm]

1 exp fit loc-cons

Hartmut Wittig 20

Preliminary results: light quarks

TMR integrand at the physical pion mass with current sta6s6cs Es6mated FV correc6on at mπ = 140 MeV, mπL = 4: (20.4 ± 4.1) · 10–10

0.004 0.008 0.012 0.016 0.5 1 1.5 2 2.5 3

xcut x0 [fm]

1 exp fit loc-cons

For mπL = 6 the correc6on is smaller by factor ≈ 10

Hartmut Wittig 21

Preliminary results: light quarks

Chiral and con6nuum extrapola6on: (ahvp

µ )ud · 1010

= m2

π/(16π2 f 2 π )

300 400 500 600 700 800 0.02 0.04 0.06 0.08 0.1 0.12 Fit 1 : aHVP,LO

µ

= 592(17) × 10−10 Fit 2 : aHVP,LO

µ

= 643(21) × 10−10

• y

β = 3.40 β = 3.46 β = 3.55 β = 3.70

Hartmut Wittig 21

Preliminary results: light quarks

Chiral and con6nuum extrapola6on: Chiral fit in good agreement with direct calcula6on at physical pion mass (ahvp

µ )ud · 1010

= m2

π/(16π2 f 2 π )

300 400 500 600 700 800 0.02 0.04 0.06 0.08 0.1 0.12 Fit 1 : aHVP,LO

µ

= 592(17) × 10−10 Fit 2 : aHVP,LO

µ

= 643(21) × 10−10

• y

β = 3.40 β = 3.46 β = 3.55 β = 3.70

(ahvp

µ )ud = 643(21)stat(xx)syst

Hartmut Wittig 22

Disconnected diagrams

Stochas6c noise cancella6on: Asympto6c behaviour: −Gdisc(x0) Gρρ(x0)

x0→∞

−→ −1 9

mπ = 280 MeV

ratio of disc and conn −1/9 1.5 1 0.5 0.2 0.15 0.1 0.05

• 0.05
• 0.1
• 0.15
• 0.2

−Gdisc(x0)/Gρρ(x0)

x0 [fm]

Employ hierarchical probing,
 Hadamard vectors

[Statopoulos & Orginos, 1302.4077]

Gdisc(x0) = − 1

9

D⇣ ∆ud(x0) − ∆s(x0) ⌘ ⇣ ∆ud(0) − ∆s(0) ⌘E

∆ f (x0) = a3 X

x

D Tr S f (x, x)γk E

[Francis, Gülpers et al., 1411.7592]

Hartmut Wittig

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 mµ|x| −0.00010 −0.00008 −0.00006 −0.00004 −0.00002 0.00000 0.00002 0.00004 CCS TMR −5 −4 −3 −2 −1 1 2 1010ahvp

µ

0.0 0.5 1.0 1.5 2.0 2.5 |x| [fm]

23

Disconnected diagrams

Lorentz-covariant coordinate space representa6on Analy6c expression for Hμν(x)
 in terms of Bessel func6ons ahvp

µ

= lim

R→∞

Z

|x|<R

d4xGµν(x) Hµν(x), Gµν(x) ≡ D Jµ(x)Jν(0) E

[H.B. Meyer, EPJC 77 (2017) 616]

mπ = 200 MeV

Expect higher sta6s6cal
 precision Applica6on to running of
 sin2 θW

[Marco Cè, FRI 15:40]

Hartmut Wittig 24

Summary & Outlook

Preliminary result for HVP contribu+on from CLS Nf = 2+1 ensembles

• Excellent control over long-distance regime: iso-vector correlator
• O(a) improved vector currents
• Error dominated by lajce scale

Refinements

• Increase sta6s6cs
• Include quark-disconnected diagrams
• Include isospin-breaking correc6ons

ahvp

µ

= 711 · 10−10 ± 3%