HVP lattice finite-volume Giusti corrections OUTLINE Motivations - - PowerPoint PPT Presentation

HVP lattice finite-volume corrections

OUTLINE ▪ Motivations ▪ Current status from Collaborations

Second Plenary Workshop of the Muon g-2 Theory Initiative Helmholtz Institut Mainz 18th - 22nd June 2018

Davide Giusti

Motivations

HVP of the muon

3

e+e- data

100%

lattice data

100% ates

550 600 650 700 750

a

µ HVP * 10 10

ETMC 18 RBC/UKQCD 18 BMW 17 CLS/Mainz 17 HPQCD 16 KNT18 no New Physics DHMZ 17 FJ 17 RBC/UKQCD 18

lattice + e+e-

~ 30% + 70%

µ q q

≳ 2%

FVEs cannot be neglected

≳ 0.4%

LO-HVP FVEs

4

Chiral Perturbation Theory Gounaris-Sakurai parameterisation + Lüscher formalism Time-momentum representation

Izubuchi et al. 2018 Della Morte et al. 2017 Blum et al. 2018; Borsanyi et al. 2017; Chakraborty et al. 2017 Bijnens and Relefors 2017; Aubin et al. 2016 ETMC, talk by S. Simula

RBC/UKQCD talk by C. Lehner (new updates)

Current status from Collaborations

6

𝝍PT Groups

Aubin et al. 2016

aµ

LO,HVP Qmax 2

⎡ ⎣ ⎤ ⎦ = 4α em

2

dQ2 f Q2

( )

Qmax

2

∫

Π Q2

( )− Π 0

( )

⎡ ⎣ ⎤ ⎦

Πµν Q

( ) = Q2δ µν −QµQν

( )Π Q2

( )

0.00 0.05 0.10 0.15 0.20 0.000 0.002 0.004 0.006 0.008 0.010 0.012

Q2 GeV2 ⎡ ⎣ ⎤ ⎦

Q2 ! mµ

2 4

π π

• ΠChPT

µν

(Q) = 4 1 L3T X

p

sin (p + Q/2)µ sin (p + Q/2)ν (2 P

κ(1 − cos pκ) + m2 π) (2 P κ(1 − cos (p + Q)κ) + m2 π)

− 2 δµν 1 L3T X

p

✓ cos pµ (2 P

κ(1 − cos pκ) + m2 π)

◆ (10/9) x [

]

10 9 4πα em

NLO 𝝍PT; PBCs

MILC ensemble

Staggered 𝝍PT weighted average taste-split pion spectrum

a = 0.059 fm mπ = 220 MeV

L = 3.8 fm

connected contribution

7

𝝍PT Groups

GeV2 Blue laEce data Red NLO ChPT

GeV2 Blue laEce data Red NLO ChPT

Aubin et al. 2016

Red A1 subtracted Blue A1

44 unsubtracted

Black A1 infinite volume

0.00 0.05 0.10 0.15 0.20

• 0.014
• 0.012
• 0.010
• 0.008

Red A1 unsubtracted Blue A1

44 unsubtracted

Black A1 infinite volume

0.00 0.05 0.10 0.15 0.20

• 0.014
• 0.012
• 0.010
• 0.008

ˆ Q2 GeV2 ⎡ ⎣ ⎤ ⎦ ˆ Q2 GeV2 ⎡ ⎣ ⎤ ⎦ ˆ Q2 GeV2 ⎡ ⎣ ⎤ ⎦ ˆ Q2 GeV2 ⎡ ⎣ ⎤ ⎦

A1 : Πii

i

∑

; A1

44 :Π44

10-15% FVEs

mπL = 4.2

8

𝝍PT Groups

• 0.0001
• 8e-05
• 6e-05
• 4e-05
• 2e-05

2e-05 4e-05

• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

∆VΠµν part twist p4+p6 q2 ♦ sinθu µν=00 µν=11 µν=22 µν=33

• 0.0001
• 8e-05
• 6e-05
• 4e-05
• 2e-05

2e-05 4e-05

• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

∆VΠµν part twist p4+p6 q2 ♦ sinθx

u

µν=00 µν=11 µν=12 µν=33

q

Bijnens and Relefors 2017

NNLO 𝝍PT PQ𝝍PT + twisted BCs

mπL = 4

FV corrections: different twist angles at same q2 Small corrections with respect to NLO FVEs sizeable (few %) for present lattices

q = ✓ 0, p

p

◆

q = ⇣ 0, p q2, 0, 0 ⌘

9

RBC/UKQCD Collaboration

mπL = 3.8 ÷ 3.9

L = 5.4 ÷ 5.5 fm

Physical mass point

FVEs corrected with NLO 𝝍PT

Two ensembles

T = 10.7 ÷11 fm

a = 0.084 ÷ 0.114 fm

Systematic uncertainty from the largest ratio of p6 to p4

Blum et al. 2018

ΔFVEsaµ conn ud

( ) = 15.9 3.7 ( )⋅10−10

ΔFVEsaµ conn ud

( ) = 20 3 ( )⋅10−10

Talk by C. Lehner (updates)

GSL approach:

10

640 660 680 700 720 740

BMWc + FV + IB

BMWc + FV BMWc (L=6fm)

RBC/UKQCD 18 HPQCD 16 ETM 14 Jegerlehner 17 DHMZ 17 KNT 18 RBC/UKQCD 18 No new physics

aµ

LO-HVP . 1010

LQCD (Nf ≥2+1) Pheno. Pheno+LQCD

BMW Collaboration

L = 6.1÷ 6.6 fm T = 8.6 ÷11.3 fm

Physical mass point

FVEs corrected with NLO SU(2) S𝝍PT (I=1 channel only)

ΔFVEsaµ

I=1 ud

( ) = 15.0 15.0 ( )⋅10−10

extrapolated to the continuum limit (six lattice spacings ranging from 0.064 to 0.134 fm)

Borsanyi et al. 2017

β a [fm] (T×L/a2) 3.7000 0.134 64 × 48 3.7500 0.118 96 × 56 3.7753 0.111 84 × 56 3.8400 0.095 96 × 64 3.9200 0.078 128 × 80 4.0126 0.064 144 × 96

mπL ! 4.1

fixed

mπL = 4.2 ÷ 4.5

500 550 600 650 700 0.005 0.01 0.015 0.02 aµ,ud

a2[fm2]

Fig.S4 (FV + taste) crr.

11

HPQCD Collaboration

Chakraborty et al. 2017

Combined FV and discretisation effects (pion tastes)

mπL = 3.2 ÷ 5.4

mπ = 134 ÷ 311 MeV

3 lattice volumes @:

mπ ! 220 MeV

a = 0.12 fm

L = 2.4 ÷ 5.8 fm

T = 7.2 ÷ 8.6 fm

γ − ρ 0 −π +π −

mixing to all orders in leading interactions

NLO S𝝍PT +

×

π

ρ

ˆ Π( q2

E, fρ, mρ, mπ) = ˆ

Σ(q2

E, mπ, mπ)

+ f 2

ρ

2m2

ρ

q2

E

⇣ 1 + gρgρππ ˆ Σ(q2

E, mπ, mπ)

⌘2 q2

E

⇣ 1 + g2

ρππ ˆ

Σ(q2

E, mπ, mπ)

⌘ + m2

ρ

5 times larger r

π 2

Largest correction for lightest pion masses: 7%

Uncertainty: ±0.7%

aµ

LO,HVP conn ud

( ) ! 610 9 ( )⋅10−10

small FVEs+discr. for s quark contribution

Preliminary Talk by R. S. Van de Water

12

Mainz Group

a = 0.085 fm

mπ = 280 MeV

0.0005 0.001 0.0015 0.002 0.5 1 1.5 2 2.5 3

x0 [fm] G(x0) K(x0)/mµ

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

s quark

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] G(x0) K(x0)/mµ

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

u/d quarks

Della Morte et al. 2017

mπL = 4.0 ÷ 6.0 mπ = 185 ÷ 495 MeV

L = 2.1÷ 4.2 fm T = 4.2 ÷ 8.4 fm

xcut xcut x0 [fm] −0.005 0.005 0.01 0.015 0.5 1 1.5 2 2.5 3 G(x0)f K(x0)/mµ Data 1-Exp GS(L) GS(∞) −0.005 0.005 0.01 0.015

FVEs corrected with Gounaris-Sakurai parameterisation + Lüscher formalism

mρ

exp fit Γρ

GS

parameters:

mπL = 4.0

mπ = 185 MeV mπ = 268 MeV

mπL = 4.2

FVEs: 5% shift in for and near-phys. point

aµ

mπL ≈ 4

ΔFVEsaµ ! 20.4 4.1

( )⋅10−10

mπ = 140 MeV mπL = 4.0

interactions

ππ

important for

t >1 fm

Preliminary Talk by H. Meyer

13

ETM Collaboration

Talk by S. Simula

u- and d-quark (connected) contributions 250 300 350 400 450 2 3 4 5 6 7 8

a

M

A40.XX M

a ~ 0.09 fm

FVE ~ few %

s-quark contribution

300 350 400 2 3 4 5 6 7 8 9 10 data dual + π−π FVE corr. (π−π only) FVE corr. (dual + π−π)

a

M

A40.XX

Mπ ~ 320 MeV a ~ 0.09 fm

e he representation

A40.XX

Mπ ~ 320 MeV a ~ 0.09 fm

mπL = 3.0 ÷ 5.8 mπ = 223÷ 495 MeV

L = 1.8 ÷ 3.5 fm T = 3.5 ÷ 7.1 fm

FVEs corrected with 2𝝆 Lüscher formalism and GS + dual pQCD contribution

DG et al. 2017

mρ gρππ parameters:

Rdual Edual

aµ

LO,HVP conn ud

( ) = 622.8 12.8 ( )⋅10−10

pure FVEs: ∼5% correction to aµ

mπL ≈ 4 mπ = 135 MeV a2 → 0

@

π

π

F

π

• 80
• 70
• 60
• 50
• 40
• 30
• 20
• 10

2 4 6 8 10 non-interacting π−π (M

π = 135 MeV)

interacting π−π (M

π = 135 MeV)

interacting π−π (M

π = 300 MeV)

[a

µ HVP(L) - a µ HVP(∞)] * 10 10

M

π L

continuum limit

1% L=4.5 fm 4 levels L=6.0 fm 8 levels L=8.0 fm 12 levels

the same as ChPT @ NLO non-interacting π-π: Vππ

L

( ) t

( )−Vππ

∞

( ) t

( )= Mπ

4

3π 2t K2 Mπ L

2!

n2+4t 2 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Mπ

2 L 2!

n2+4t 2

( )

− 1 MπL ! n dy K0 Mπ y L

2!

n2+4t 2 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

1 ∞

∫

sinh MπL ! n y −1

( )

⎡ ⎣ ⎤ ⎦ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪

! n≠0

∑

interacting π-π: dual + π-π representation [note that ΔaμHVP(L) depends approximately on MπL only] [Francis et al. ’13]

FVE correction @ a2 → 0

[Aubin et al. ’16, Bijnens&Relefors ’16]

Thursday, June 21, 18

14

15

PACS Collaboration

Izubuchi et al. 2018

mπL = 3.8 ÷ 5.8

L = T = 5.4 ÷ 8.1 fm

near-phys. mass point Two ensembles

a = 0.085 fm

Backward state propagation (2T=10.8 fm) 4% @ tcut=2.6 fm positive contribution to aµ

FVEs estimated using TMR

comparison between two volumes

aµ

LO,HVP on L=5.4 fm is

10 ± 26

( )⋅10−10

from L=8.1 fm @ mπ = 146 MeV

0.5 1 1.5 2 2.5 3

tcut fm

• 40
• 30
• 20
• 10

10 20 30 40

aµ 1010

ChPT(L/a=64,T/a=64) ChPT(L/a=64,T/a=128) (L/a=96,T/a=96)146 MeV - (L/a=64,T/a=64)146 MeV (L/a=96,T/a=96)146 MeV - (L/a=64,T/a=128)146 MeV

Light

IB contribution: FVEs

16

Talk by A. Portelli Blum et al. 2018 expected to start at O(1/L^3) (IR safe, neutral meson states, vanishing charge radius) DG et al. 2017

s/c contribution only

GL(x) = 1 V X

k

1 ˆ k2 eikx ,

QEDL prescription for

zero mode subtraction photon propagator

QED∞

G1(x) = Z π

π

d4k (2π)4 1 ˆ k2 eikx

Analytical calculation

h h )

0.01 0.02 0.03 0.04 0.05

mud (GeV)

0.005 0.01 0.015 0.02

δaµ

HVP(ud) / aµ HVP(ud)

β=1.90, L=20 β=1.90, L=24 β=1.90, L=32 β=1.90, L=40 β=1.95, L=24 β=1.95, L=32 β=2.10, L=48 physical point

Talk by S. Simula

Discussion points

17

Discrepancy between 𝝍PT predictions and lattice determinations with interacting pions New systematic lattice study with several volumes has been performed Small QED FVEs

π π

Sizeable FVEs for present lattices in aµ

LO,HVP ud

( )

Most Collaborations adopt model estimates so far (new recent efforts for first-principles determinations)

Acknowledgments

Special thanks to

• J. Bijnens; A. X. El-Khadra; A. Gérardin; M. Golterman;
• C. Lehner; L. Lellouch; A. Portelli;
• R. S.

Van de Water; G. von Hippel