Introduction
ResultSummary and Perspective

HVP contributions to anomalous magnetic momentsof all leptons from first principle
At Physical Point Mass with Full Systematics

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 Aug. 2017

Budapest-Marseille-Wuppertal Collaboration
1612.02364 [hep-lat] and in preparation

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Motivation
Our Challenges

Muon Anomalous Magnetic Moment aµ

μ

B
Muon
Stroragegµ − 2,2e~ = gµ
M
S~ .
2mµ

aµ ≡

p
s

aµexp. = aµSM ?

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

(1)
(2)

Introduction
ResultSummary and Perspective

Motivation
Our Challenges

aµexp. vs. aµSM
SM contribution
QED [5 loops]
HVP-LO (pheno.)
HVP-NLO

contrib. × 1010
aµ11658471.8951 ± 0.0080
692.3 ± 4.2
694.9 ± 4.3
681.5 ± 4.2−9.84 ± 0.07

Ref.
[Aoyama et al ’12]
[Davier et al ’11]
[Hagiwara et al ’11]
[Benayoun et al ’16]
[Hagiwara et al ’11]
[Kurz et al ’11]

HVP-NNLO
HLbyL
Weak (2 loops)
SM tot [0.42 ppm]
[0.43 ppm]
[0.51 ppm]
Exp [0.54 ppm]
Exp − SM1.24 ± 0.01
10.5 ± 2.6
15.36 ± 0.1011659180.2 ± 4.911659182.8 ± 5.011659184.0 ± 5.911659208.9 ± 6.3
28.7 ± 8.0
26.1 ± 7.8
24.9 ± 8.7[Kurz et al ’11]
[Prades et al ’09]
[Gnendiger et al ’13]
[Davier et al ’11]
[Hagiwara et al ’11]
[Aoyama et al ’12]
[Bennett et al ’06]
[Davier et al ’11]
[Hagiwara et al ’11]
[Aoyama et al ’12]

FNAL E989 (2017): 0.14-ppm, J-PARC E34: 0.1-ppm

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Motivation
Our Challenges

Motivation

exp.
SM
Really aµ6= aµ?
SUSY (MSSM, Padley et.al.’15)?
Technicolor (Kurachi et.al. ’13)?.

A rigorous determination of aµ by SM is anecessary step to attacking BSM. A
bottle-neck is Hadronic vacuum polarization(HVP) contribution due to its
non-perturvative feature.

γ

In phenomenological estimates of aµ , the HVP
is evaluated based on dispersion relations withexperimental data inputs (e + e− → had. etc.)
Some tension among expr. (BESIII, PLB’16).

µµTechnicolor

Purely theoretical estimates for HVP effects
are demanded for a rigorous test of SM. The
Lattice QCD meets the requirement. We(BMWc) are at the forefront of this approach.
360

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

365

370

375

380

KLOE 08368.1 ± 0.4 ± 2.3 ± 2.2

BaBar 09376.7 ± 2.0 ± 1.9

KLOE 10365.3 ± 0.9 ± 2.3 ± 2.2

KLOE 12366.7 ± 1.2 ± 2.4 ± 0.8

BESIII

368.2 ± 2.5 ± 3.3aπµπ,LO(600 - 900 MeV) [10-10]

PPP 2017, YITP, 02 August 2017

385

390

395

Introduction
ResultSummary and Perspective

Motivation
Our Challenges

Motivation

exp.
SM
Really aµ6= aµ?
SUSY (MSSM, Padley et.al.’15)?
Technicolor (Kurachi et.al. ’13)?.

A rigorous determination of aµ by SM is anecessary step to attacking BSM. A
bottle-neck is Hadronic vacuum polarization(HVP) contribution due to its
non-perturvative feature.

γ

In phenomenological estimates of aµ , the HVP
is evaluated based on dispersion relations withexperimental data inputs (e + e− → had. etc.)
Some tension among expr. (BESIII, PLB’16).

µµTechnicolor

Purely theoretical estimates for HVP effects
are demanded for a rigorous test of SM. The
Lattice QCD meets the requirement. We(BMWc) are at the forefront of this approach.
360

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

365

370

375

380

KLOE 08368.1 ± 0.4 ± 2.3 ± 2.2

BaBar 09376.7 ± 2.0 ± 1.9

KLOE 10365.3 ± 0.9 ± 2.3 ± 2.2

KLOE 12366.7 ± 1.2 ± 2.4 ± 0.8

BESIII

368.2 ± 2.5 ± 3.3aπµπ,LO(600 - 900 MeV) [10-10]

PPP 2017, YITP, 02 August 2017

385

390

395

Introduction
ResultSummary and Perspective

Motivation
Our Challenges

Motivation

exp.
SM
Really aµ6= aµ?
SUSY (MSSM, Padley et.al.’15)?
Technicolor (Kurachi et.al. ’13)?.

A rigorous determination of aµ by SM is anecessary step to attacking BSM. A
bottle-neck is Hadronic vacuum polarization(HVP) contribution due to its
non-perturvative feature.

γ

In phenomenological estimates of aµ , the HVP
is evaluated based on dispersion relations withexperimental data inputs (e + e− → had. etc.)
Some tension among expr. (BESIII, PLB’16).

µµTechnicolor

Purely theoretical estimates for HVP effects
are demanded for a rigorous test of SM. The
Lattice QCD meets the requirement. We(BMWc) are at the forefront of this approach.
360

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

365

370

375

380

KLOE 08368.1 ± 0.4 ± 2.3 ± 2.2

BaBar 09376.7 ± 2.0 ± 1.9

KLOE 10365.3 ± 0.9 ± 2.3 ± 2.2

KLOE 12366.7 ± 1.2 ± 2.4 ± 0.8

BESIII

368.2 ± 2.5 ± 3.3aπµπ,LO(600 - 900 MeV) [10-10]

PPP 2017, YITP, 02 August 2017

385

390

395

Introduction
ResultSummary and Perspective

Motivation
Our Challenges

Objective in This Work
Leading-Order (LO) HVP contribution to anomalous magnetic moments for all
leptons (A few % precision now, and a per-mil within few years):
LO-HVP,fa`=e,µ,τ= α 2 Z ∞πγ

dQ 2 ω(Q 2 /m`2 )Π̂f (Q 2 ) .

0µwhere suffix f stands for a flavor f = l(u, d), s, c, disc, andΠ̂f (Q 2 ) = Πf (Q 2 ) − Πf (0) =HAD3 sin(z/2) 2 
X 2
1X f
t 1−
Cii (t) , (3)z/2z=Qt 3 i=1t

with
f =l,s,cCµν(t) = qf2=l,s,c

X fh jµ (x)jνf (0) i|conn ,~x

f =disc(t) = qf2=discCµνX¯ µ l − s̄γµ s)(lγ¯ ν l − s̄γν s)i|disc .
h(lγ
~
x

2
Here, charge factors are given by (ql2 , qs2 , qc2 , qdisc) = (5/9, −1/9, 4/9, 1/9).

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017µ

Introduction
ResultSummary and Perspective

Motivation
Our Challenges

Our Challenges
The integrand kurnel ω(Q 2 /mµ2 ) is known
and makes a peak aroundQ 2 ∼ (mµ /2)2 ∼ (0.05GeV )2 → 4fm:
Our Lattice: (L, T ) ∼ (6, 9 − 12)fm.

20000^ l(Q2) x 1010
ω(Q2/m2µ) Π

The Pion/Kaon dynamics precisely:
Our simulations are performed with
Physical Pion/Kaon Masses.
Large distance signal:
104 Traj., 768 (9000) random sources for
ud-conn. (uds-disc.) correlators.
Need controled continuum limit:
15 lattice spacings (a ∼ 0.064 − 0.134 fm).

15000

10000

5000

(mµ/2)200

0.02

For a few % precision, we take account of:c quark w. matching onto perturb. theory.

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

0.04
Q2 GeV2

0.06

0.08

Introduction
ResultSummary and Perspective

Motivation
Our Challenges

Simulation Setup

Nf=(2+1+1) stout-smeared staggered
quarks (mc /ms = 11.85).
Scale setting fπ = 130.41 MeV via
scale w0 .
Rational Hybrid Monte Carlo.

β3.70003.75003.77533.84003.92004.0126

a[fm]
0.134
0.118
0.111
0.095
0.078
0.064

Nt
64
96
84
96
128
144

Ns
48
56
56
64
80
96

27.5
27.0
2MK2/Mπ2-1

Tree-level improved Symanzik gauge
action.

26.5
26.0
25.5
25.0

3.70003.75003.77533.84003.92004.0126phys

24.5
1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1
Mπ2/Fπ2#traj.
10000
15000
15000
25000
35000
04500

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

Mπ [MeV]
∼ 131
∼ 132
∼ 133
∼ 133
∼ 133
∼ 133

MK [MeV]
∼ 479
∼ 483
∼ 483
∼ 488
∼ 488
∼ 490

PPP 2017, YITP, 02 August 2017

#SRC (l,s,c,d)(768, 64, 64, 9000)(768, 64, 64, 6000)(768, 64, 64, 6144)(768, 64, 64, 3600)(768, 64, 64, 6144)(768, 64, 64, −)Introduction
ResultSummary and Perspective

Motivation
Our Challenges

Table of Contents

1

Introduction
Motivation
Our Challenges

2

ResultLong Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

3

Summary and Perspective

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

Table of Contents

1

Introduction
Motivation
Our Challenges

2

ResultLong Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

3

Summary and Perspective

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

Light-Conn. (and Disc.) Correlator: An Example

The connected-light correlator C ud (t)loses signal for t > 3fm. To control
statistical error, considerud
C ud (t > tc ) → Cup/low(t, tc ), whereudud
Cup (t, tc ) = C (tc ) ϕ(t)/ϕ(tc ),ud
Clow(t, tc ) = 0.0,
with ϕ(t) = cosh[E2π (T /2 − t)],and E2π = 2(Mπ2 + (2π/L)2 )1/2 .

1.0e-01
1.0e-021.0e-03Cl(t)1.0e-04
1.0e-05
1.0e-06
1.0e-071.0e-08
1.0e-09
0

1

2

3

4

t fm

Figure:PP
C ud (t) = 59 ~x 13 3i=1 hjiud (~x , t)jiud (0)i

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

disc
Similarly, C disc (t) → Cup/low(t, tc ),disc
ud−Cup (t > tc ) = 0.1C (tc ) ϕ(t)/ϕ(tc ),
disc
−Clow
(t > tc ) = 0.0.
ud,discud,discClow(t, tc ) ≤ C ud,disc (t) ≤ Cup(t, tc ).

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

LO-HVP
IR-CUT (tc ) Deps. of Light/Disc. Component: a`,ud/disc

ud,discCorresponding to Cup/low(tc ), we obtain
upper/lower bounds for g − 2:
ud,disca`,up/low(tc ).

11-aLO-HVPµ, disc x 10

aLO-HVP
x 1010
µ, ud

700
650
600

Two bounds meet around tc = 3fm.
Consider the average of bounds:
ud,discud,discā`ud,disc (tc ) = 0.5(a`,up+ a`,low)(tc ),
which is stable around tc = 3fm.

550

150

2-pion
zero
avg

100
50
0
2.0

2.5

3.0
3.5
tc [fm]4.0

4.5

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

We pick up such averages ā`ud,disc (tc ) with
4 − 6 kinds of tc around 3fm. The average
LO-HVPof average is adopted as a`,ud/discto beanalysed, and a fluctuation over selected tc
is incorporated into the systematic error.

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

LO-HVP
Continuum Extrap. of Light Conn. Component: aµ,ud

650

data
fit0
fit1
fit2
fit3

aLO-HVP
x 1010
µ,l

625
600
575
550
525
0

0.005

0.01

0.015

0.02

a2 fm2
LO-HVPLO-HVPF (aµ,ud, A, Cπ , · · ·) = aµ,ud(1 + Aa2 )+Cπ ∆Mπ2 + · · · .

LO-HVPaµ,ud= 634.11(8.10)(8.24) ,Kohtaroh Miura (CPT, Aix-Marseille Univ.)

χ2 /dof = 7.8/12 (fit1 case).

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

LO-HVP
Continuum Extrap. of Strange Conn. Component: aµ,s

aLO-HVP
x 1010
µ,s

54.0

dat
fit0
fit1
fit2
fit3

53.5

53.0

52.5
0

0.005

0.01

0.015

0.02

a2 fm2
LO-HVPLO-HVP(1 + Aa2 ) + CK ∆MK2 .
F (aµ,s, A, CK ) = aµ,s

LO-HVPaµ,s= 53.60(05)(13) ,Kohtaroh Miura (CPT, Aix-Marseille Univ.)

χ2 /dof = 16.7/11 (fit1 case).

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

LO-HVP
Continuum Extrap. of Charm Conn. Component: aµ,c

aLO-HVP
x 1010
µ,c

16.0

12.0

8.0

4.0

dat
fit1
0

0.005

0.01

0.015

0.02

a2 fm2
LO-HVPLO-HVPF (aµ,c, A, Cπ,K ,ηc ) = aµ,c
(1 + Aa2 ) + Cπ ∆Mπ2 + CM ∆MK2 + Cηc ∆Mηc .

LO-HVPaµ,c= 14.99(08)(08) ,Kohtaroh Miura (CPT, Aix-Marseille Univ.)

χ2 /dof = 1/7 (fit1 case).

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

LO-HVP
Continuum Extrap. of Disc. Component: aµ,disc10− aLO-HVPµ,disc x 10

12.5

dat
fit0
fit1
fit2

10.0
7.5
5.0
2.5
0

0.005

0.01

0.015

0.02

a2 fm2
LO-HVPLO-HVPF (aµ,disc, A, Cπ , · · ·) = aµ,disc(1 + Aa2 )+Cπ ∆Mπ2 + · · · .

LO-HVPaµ,disc= −10.9(1.1)(0.7) ,Kohtaroh Miura (CPT, Aix-Marseille Univ.)

χ2 /dof = 2.4/10 (fit1 case).

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

Perturbative Corrections

Consider separation of momentum integral range as α 2 Z Qmax Z ∞  dQ  Q 2 
LO-HVPωa`,f=+Π̂f (Q 2 ) ,πm`20Qmax m`LO-HVPLO-HVP= a`,f(Q ≤ Qmax ) + a`,f
(Q > Qmax ) ,

(4)
(5)

where
LO-HVPa`,f(Q ≤ Qmax ) : lattice simulations, investigated so far ,LO-HVPLO-HVPa`,f(Q > Qmax ) : γl (Qmax )Π̂f (Qmax ) + ∆pert a`,f
(Q > Qmax ) .
LO-HVPFor muon and electron, a`=µ,e(Q > Qmax ) is small. For example,

Q

=2GeV

aµLO-HVP (Q > Qmax ) −−max−−−−−→ 0.678(1)(1) , (0.1%).

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017(6)

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

Perturbative Corrections

Consider separation of momentum integral range as α 2 Z Qmax Z ∞  dQ  Q 2 
LO-HVPωa`,f=+Π̂f (Q 2 ) ,πm`20Qmax m`LO-HVPLO-HVP= a`,f(Q ≤ Qmax ) + a`,f
(Q > Qmax ) ,

(4)
(5)

where
LO-HVPa`,f(Q ≤ Qmax ) : lattice simulations, investigated so far ,LO-HVPLO-HVPa`,f(Q > Qmax ) : γl (Qmax )Π̂f (Qmax ) + ∆pert a`,f
(Q > Qmax ) .
LO-HVPFor muon and electron, a`=µ,e(Q > Qmax ) is small. For example,

Q

=2GeV

aµLO-HVP (Q > Qmax ) −−max−−−−−→ 0.678(1)(1) , (0.1%).

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017(6)

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

Perturbative Corrections

Consider separation of momentum integral range as α 2 Z Qmax Z ∞  dQ  Q 2 
LO-HVPωa`,f=+Π̂f (Q 2 ) ,πm`20Qmax m`LO-HVPLO-HVP= a`,f(Q ≤ Qmax ) + a`,f
(Q > Qmax ) ,

(4)
(5)

where
LO-HVPa`,f(Q ≤ Qmax ) : lattice simulations, investigated so far ,LO-HVPLO-HVPa`,f(Q > Qmax ) : γl (Qmax )Π̂f (Qmax ) + ∆pert a`,f
(Q > Qmax ) .
LO-HVPFor muon and electron, a`=µ,e(Q > Qmax ) is small. For example,

Q

=2GeV

aµLO-HVP (Q > Qmax ) −−max−−−−−→ 0.678(1)(1) , (0.1%).

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017(6)

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

Perturbative Corrections for tau

aτ,f(Q ≤ Qmax )f+γl (Qmax )Π̂ (Qmax )+LO-HVP∆pert aτ,f
(Q > Qmax ) .

For tau, Q > Qmax effects arelarge, while the total is stable.
The fluctuation over2
Qmax
= 2.0 − 5.0GeV 2 areincorporated into systematic
errors.

matching
perturb.

30
20
10

20
10

2
1
0
0.0

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

lattice100

aLO-HVP
x 108
τ,s

=

total
200

aLO-HVP
x 108
τ,c

aτ,f

LO-HVP

8− aLO-HVP
τ,disc x 10

LO-HVP

aLO-HVP
x 108
τ,ud

300

1.0

2.0
3.0
Q2max [GeV2]

PPP 2017, YITP, 02 August 2017

4.0

5.0

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

LO-HVP
The aµcomes from Euclidean
mommenta; Exponentially suppressed
FV with LMπ ∼ 4 for L ∼ 6fm.

We work with L ∼ fixed, and FV
effects cannot be estimated fromsimulations and need model.
Long-distance I = 1 (I = 0)
contribution dominated by 2-pions(3-pions). The dominant FV in I = 1
channel could be estimated by XPT
for π + π − loop (Aubin et al ’16).

Kohtaroh Miura (CPT, Aix-Marseille Univ.)( aLO-HVP,l(16fm) - aLO-HVP,l(L) ) x 1010µµ

LO-HVP
FV for aµ,ud
by XPT

0
-2.5
-5
-7.5
-10
-12.5
-15
6

8

10

12

14

L fm

LO-HVPLO-HVP(aµ,I=1 (∞) − aµ,I =1 (6fm))|XPT= 13.42(13.42) × 10−10 , (1.9%).

PPP 2017, YITP, 02 August 201716

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

Isospin breaking effects (Preliminary)
Get missing effects from phenomenology
Effectρ−ω mix.
FSR
Mπ → Mπ±
π0 γ
ηγ
Total

LO-HVP × 1010
corr. to aµ
2.71 ± 1.36
4.22 ± 2.11
−4.47 ± 4.47
4.64 ± 0.04
0.65 ± 0.01
7.8 ± 5.1

Thanks to F.Jegerlehner (& M. Benayoun) for correspondance and numbers
The ρ − ω and FSR based on Gounaris-Sakurai fit to e + e − , from 2Mπ to 1 GeV,
ascribed conservative 50% error.
EM modes from M. Benayoun et al ’12
Correction by Mπ → Mπ± from XPT: given 100% error to account for othermissing effects.
Thus: ∆IB aµLO-HVP = (7.8 ± 5.6) × 10−10 (Preliminary )Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

Summary on aµLO-HVP (Preliminary)aµLO-HVP BMWc, Preliminary
I =1
I =0
total

584(8)st (8)acut (0)tcut (0)qcut (13)fv
121(4)st (4)acut (0)tcut (0)qcut
713(9)st (9)acut (0)tcut (0)qcut (13)fv (5)iso

Total error is 2.6%,
dominated FV effects.
Our results are consistent
with both “No New Physics”
and Dispersive Method.
There is some tension with
HPQCD 16.

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

More detailed comparison
LO-HVP

aµ,s

HPQCD 16

575

600

625

650

Mainz 17 (TMR)

RBC/UKQCD 16

Nf = 2+1+1This work

550

675

HPQCD 14This work

50

BMWc ’17 is only calculation performed directly at
physical quark masses with 6 β’s to fully control
continuum extrapolation.
LO-HVP .
→ contributes only 0.2% to error on aµ
Kohtaroh Miura (CPT, Aix-Marseille Univ.)

52

53

54

Nf = 2aLO-HVP. 1010µ,cMainz 17 (TMR)

Nf = 2+1+1

BMWc ’17 c contribution is slightly smaller thanother Nf=2+1+1 results

51

HPQCD 14This work

14

14.5

15

15.5

10
aLO-HVPµ,disc . 10

Nf = 2+1+1 Nf = 2+1

BMWc ’17 ud contribution is signficantly larger
than other Nf=2+1+1 results→ difference with HPQCD ’14/Mainz ’17 is∼ 2.4/1.5σ.

LO-HVP = 1.5 × 10−10
BMWc ’17 δaµ,
disc

10

Nf = 2

Nf = 2+1+1

Mainz 17
(TMR+FV). 10

Nf = 2+1

Nf = 2aLO-HVP. 1010µ,ud

RBC/UKQCD 15

This work

-14

-12

-10

PPP 2017, YITP, 02 August 2017

-8

-6

-4

16

Introduction
ResultSummary and Perspective

Long Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

LO-HVP
Summary on ae,τ(Preliminary)aeLO-HVP BMWc, Preliminary
I =1
I =0
total

157.0(2.8)st (2.3)acut (0.0)tcut (0.0)qcut (4.6)fv
30.8(1.4)st (1.2)acut (0.1)tcut (0.0)qcut
189.0(3.1)st (2.6)acut (0.1)tcut (0.0)qcut (4.6)fv (1.0)isoaτLO-HVP BMWc, Preliminary
I =1
I =0
total

253(1)st (2)acut (0)tcut (0)qcut (2)fv
85(0)st (1)acut (0)tcut (1)qcut
342(1)st (2)acut (0)tcut (1)qcut (2)fv (1)isoBurger et.al.(’15): aeLO-HVP = 178.2(6.4)(8.6), aτLO-HVP = 341(8)(6)
HPQCD (’16): aeLO-HVP = 177.9(3.9)
Jeherlehner(’16): aeLO-HVP = 185.11(1.24).
Eidelman et.al.(’07): aτLO-HVP = 338(4).
Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Table of Contents

1

Introduction
Motivation
Our Challenges

2

ResultLong Distance Mng. for Light/Disc. Correlators
Continuum Extrapolation
Corrections: Perturb, FV, and Isospin Breaking
Short Summary on a` and Discussion

3

Summary and Perspective

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Introduction
ResultSummary and Perspective

Summary of Summary and Perspective

We have obtained aµLO-HVP (as well as slope/curvature of Π̂(Q 2 = 0),1612.02364 [hep-lat]) directly at physical point masses.
Full controlled continuum extrapolation and matching to perturbation
theory.
Model/pheno. assumptions are put on only for small corrections from FV,
QED and isospin breaking.
Our results are consistent with both “No New Physics” and Dispersive
Methods with a conservative systematic errors. There is some tension with
LO-HVP
HPQCD 16 on aµ,ud.
Total error is 2.6%, dominated by FV effects.
Need ∼ 0.2% precision to match forthcoming experiments!!
1
2
3

increase statistics by 50 − 100 times.
control FV effects directly with simulations.
simulations with QED and isospin breaking corrections taken account.

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Table of Contents

4

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Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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LO-HVP
Continuum Extrap. of Light Component: aµ,ud

700

BMWc
HPQCD (no corr.)
HPQCD

aLO-HVP
x 1010
µ,l

650

600

550

500
0

0.005

0.01

0.015

0.02

a2 fm2

Figure: Red-squares = Our data. Blue-triangles = HPQCD, 1403.1778.
LO-HVPaµ,ud= 634.11(8.10)(8.24) ,Kohtaroh Miura (CPT, Aix-Marseille Univ.)

χ2 /d.o.f. = 7.8/12 (fit1 case).

PPP 2017, YITP, 02 August 2017

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Continuum Extrapolation of Πln=1 I
0.18
our dataΠln=1 GeV-2

0.17
0.16
0.15
0.14
0.13
0.12
0

0.005

0.01

0.015

0.02

a2 fm2(2)(2)

CΠ 1 + Aa2 + · · ·1 + CMπ ∆Mπ + CMK ∆MK .
2
2
a1+Ba + · · ·Πln=1 |a2 →0 = 0.1652(31) , χ2 /d.o.f. = 24.3/20

F (CΠ , A, B, CMπ , CMK ) =Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Continuum Extrapolation of Πln=2-0.1
our dataΠln=2 GeV-4

-0.15
-0.2
-0.25
-0.3
-0.35
-0.4
0

0.005

0.01

0.015

0.02

a2 fm2(4)(4)

CΠ 1 + Aa2 + · · ·1 + CMπ ∆Mπ + CMK ∆MK .
42
a1+Ba + · · ·Πln=2 |a2 →0 = −0.306(23) .

F (CΠ , A, B, CMπ , CMK ) =Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Continuum Extrapolation of Πsn=1,2

0.068-0.051-0.052our data, w. MK crr.
our data, wo. MK crr.

hpqcd

0.067
Πsn=2 GeV-4

Πsn=1 GeV-2-0.0530.066
0.065-0.054-0.055-0.0560.064

our data, w. MK crr.
our data, wo. MK crr.

-0.057

hpqcd

0.063

-0.058
0

0.005

0.01
2

a fm

0.015

0.02

0.025

2

0

0.005

0.01

0.015

0.02

0.025

a2 fm2

Figure: Red-squares = Our data. Green-triangles = HPQCD, 1403.1778.

(2,4)

F (CΠ

, A, B, CMπ , CMK ) =Πsn=1 |a2 →0 = 0.0658(1) ,CΠ 1 + Aa2 + · · ·1 + CMπ ∆Mπ + CMK ∆MK .
a2,4 1+Ba2 + · · ·Πsn=2 |a2 →0 = −0.0534(2) , χ2 /d.o.f. = 20.9/18

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Continuum Extrapolation of Πcn=1,20.0045-0.0002
our data, w. MK crr.
our data, wo. MK crr.

our data, w. MK crr.
our data, wo. MK crr.

-0.00025

hpqcd

0.004

Πcn=2 GeV-4

Πcn=1 GeV-2

0.00425

0.00375
0.00350.00325

hpqcd-0.0003-0.00035-0.0004-0.00045

0.003-0.0005
0

0.005

0.01
2

a fm

0.015

0.02

0

0.005

2

0.01

0.015

0.02

a2 fm2

Figure: Red-squares = Our data. Green-triangles = HPQCD, 1208.2855.

(2,4)

F (CΠ


CΠ 1 + Aa2 + · · ·1 + CMπ ∆Mπ + CMK ∆MK .
a2,4 1+Ba2 + · · ·Πcn=1 |a2 →0 = 0.00403(2) , Πcn=2 |a2 →0 = −2.73(2) × 10−4 .

, A, B, CMπ , CMK ) =Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Disconnected Contributions

Figure: From Presentaion of T.Kawanai in Lattice 2016.

Πdisc
1
Πdisc
2

=
=

−1.5(2)(1) × 10−3 GeV−2 ,−4.6(1.0)(0.4) × 10

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

−3

−4

GeV

(7)
.

PPP 2017, YITP, 02 August 2017(8)

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Summary Table of Moments (Preliminary)lightstrangecharm
disconnected
I =0
I =1
total

Π1 [GeV−2 ]0.1657(16)(18)
6.57(1)(2) × 10−2
4.04(1)(1) × 10−3
−1.5(2)(1) × 10−20.0166(2)(2)0.0828(8)(9)0.0995(9)(10)

Π2 [GeV−4 ]−0.297(10)(05)−5.32(1)(3) × 10−2
−2.68(1)(4) × 10−4
4.6(1.0)(0.4) × 10−2−0.017(1)(1)−0.148(5)(2)−0.166(6)(3)Table: Preliminary results on the first two moments of the HVP function.

TOTAL ERROR: 1.4% for Π1 , and 4.0% for Π2 .

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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FV via Box Asymmetry, XPT Estimate for Various L I
c.f. Aubin et.al., PRD (2016).
Mπ = 135 MeV, T = 1.5L

Mπ = 135 MeV, T = 1.5L
0.05
xpt
lat
(Πxpt
n = 2 (L) - Πn = 2 (∞)) / Πn = 2 (6fm)

xpt
lat
(Πxpt
n = 1 (L) - Πn = 1 (∞)) / Πn = 1 (6fm)

0.04
0.02
0
-0.02
-0.04
-0.06

ss
ts
st

-0.08
0

2

4

6

8

10

12

0
-0.05
-0.1
-0.15
-0.2

ss
ts
st

-0.25
14

L fm

16

0

2

4

6

8

10

12

14

16

L fm

∆in=1,2 (L)=∆in (L = 6fm)
Πlat,in∼xptΠxpt,ii = ss, ts, st ,n=1,2 (L) − Πn=1,2 (∞) ,(2%(for the 1st moment, n = 1) ,10% (for the 2nd moment, n = 2) .
Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017(9)

Backups

FV via Box Asymmetry, XPT Estimate for Various L II
c.f. Aubin et.al., PRD (2016).
Mπ = 135 MeV, T = 1.5L

Mπ = 135 MeV, T = 1.5L
0.05
xpt
lat
(Πxpt
n = 2 (L) - Πn = 2 (∞)) / Πn = 2 (6fm)

xpt
lat
(Πxpt
n = 1 (L) - Πn = 1 (∞)) / Πn = 1 (6fm)

0.04
0.02
0
-0.02
-0.04
-0.06

ss
ts
st

-0.08
0

2

4

6

8

10

12

0
-0.05
-0.1
-0.15
-0.2

ss
ts
st

-0.25
14

L fm

16

0

2

4

6

8

10

12

14

16

L fm



xpt
∆in=1,2 (L) = Πxpt,ii = ss, ts, st ,n=1,2 (L) − Πn=1,2 (∞) ,FV .(L) ± dFV .(L) = max{∆i } + min{∆i } /2 ± max{∆i } − min{∆i } /2(
0.0006(22) (for the 1st moment, n = 1) ,
−−−−→
(10)
L→6fm
−0.015(19) (for the 2nd moment, n = 2) .
Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Summary Table of Moments with FV I (Preliminary)lightstrangecharm
disconnected
I =0
I =1
total
I = 1 FV corr.
I = 1 + FV corr.
total + FV corr.

Π1 [GeV−2 ]0.1657(16)(18)
6.57(1)(2) × 10−2
4.04(1)(1) × 10−3
−1.5(2)(1) × 10−20.0166(2)(2)0.0828(8)(9)0.0995(9)(10)0.0006(23)0.0834(8)(9)(23)0.1001(9)(10)(23)

Π2 [GeV−4 ]−0.297(10)(05)−5.32(1)(3) × 10−2
−2.68(1)(4) × 10−4
4.6(1.0)(0.4) × 10−2−0.017(1)(1)−0.148(5)(2)−0.166(6)(3)−0.015(10)−0.164(5)(2)(10)−0.182(6)(3)(10)

Table: Preliminary results on the first two moments of the HVP function.

c.f. HPQCD(arXiv:1601.03071 and PRD2014):Πl1 = 0.1606(25) GeV−2 , Πl2 = −0.368(16) GeV−4 ,Πs1 = 0.06625(74) GeV−2 , Πs2 = −0.0526(11) GeV−4 .
Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Summary Table of Moments with FV II (Preliminary)lightstrangecharm
disconnected
I =0
I =1
total
I = 1 FV corr.
I = 1 + FV corr.
total + FV corr.

Π1 [GeV−2 ]0.1657(16)(18)
6.57(1)(2) × 10−2
4.04(1)(1) × 10−3
−1.5(2)(1) × 10−20.0166(2)(2)0.0828(8)(9)0.0995(9)(10)0.0006(23)0.0834(8)(9)(23)0.1001(9)(10)(23)

Π2 [GeV−4 ]−0.297(10)(05)−5.32(1)(3) × 10−2
−2.68(1)(4) × 10−4
4.6(1.0)(0.4) × 10−2−0.017(1)(1)−0.148(5)(2)−0.166(6)(3)−0.015(10)−0.164(5)(2)(10)−0.182(6)(3)(10)

Table: Preliminary results on the first two moments of the HVP function.

c.f. Phenomenology(Benayoun et.al.1605.04474):
Π1 = 0.990(7) GeV−2 , Π2 = −0.206(2) GeV−4 .

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Backups

Why Π̂f ?

LO-HVPa`,f= α 2 Z Qmax
π

0

+

Z ∞ 
Qmax

dQ  Q 2  f 2
ωΠ̂ (Q ) ,m`m`2

LO-HVPLO-HVP= a`,f(Q ≤ Qmax ) + a`,f
(Q > Qmax ) ,

(11)
(12)

LO-HVPa`,f(Q ≤ Qmax ) : computed by lattice simulations ,LO-HVPa`,f(Q > Qmax ) : computed by lattice Π̂f (Qmax ) and perturbations .

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Backups

Why Π̂f ?

LO-HVPa`,f= α 2 Z Qmax
π

0

+

Z ∞ 
Qmax

dQ  Q 2  f 2
ωΠ̂ (Q ) ,m`m`2

LO-HVPLO-HVP= a`,f(Q ≤ Qmax ) + a`,f
(Q > Qmax ) ,

(11)
(12)

LO-HVPa`,f(Q ≤ Qmax ) : computed by lattice simulations ,LO-HVPa`,f(Q > Qmax ) : computed by lattice Π̂f (Qmax ) and perturbations .

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Backups

Why Π̂f ?

 α 2 Z ∞ dQ  Q 2 ωΠ̂f (Q 2 ) ,
(13)
2
π
m
m
`
Qmax
`
Z
h
 α 2 ∞ dQ  Q 2 i22
4π 2 qf2 Πf (Q 2 ) − Πf (Qmax) + Πf (Qmax) − Πf (0) ,
ω
=
2
π
m`
Qmax m` α 2 Z ∞ dQ  Q 2 2
LO-HVP=ω
4π 2 qf2 Πf (Q 2 ) − Πf (Qmax) (→ ∆a`,f
)
2
π
m
m
`
Qmax
`
h α 2 Z ∞ dQ  Q 2 i222Πf (Qmax) − Πf (0) (→ γ` (Qmax+ω)Π̂f (Qmax)) .(14)
2
π
m
m
`
Qmax
`
LO-HVPa`,f(Q > Qmax ) =Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Why Π̂f ?

 α 2 Z ∞ dQ  Q 2 ωΠ̂f (Q 2 ) ,
(13)
2
π
m
m
`
Qmax
`
Z
h
 α 2 ∞ dQ  Q 2 i22
4π 2 qf2 Πf (Q 2 ) − Πf (Qmax) + Πf (Qmax) − Πf (0) ,
ω
=
2
π
m`
Qmax m` α 2 Z ∞ dQ  Q 2 2
LO-HVP=ω
4π 2 qf2 Πf (Q 2 ) − Πf (Qmax) (→ ∆a`,f
)
2
π
m
m
`
Qmax
`
h α 2 Z ∞ dQ  Q 2 i222Πf (Qmax) − Πf (0) (→ γ` (Qmax+ω)Π̂f (Qmax)) .(14)
2
π
m
m
`
Qmax
`
LO-HVPa`,f(Q > Qmax ) =Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

Backups

Why Π̂f ?

 α 2 Z ∞ dQ  Q 2 ωΠ̂f (Q 2 ) ,
(13)
2
π
m
m
`
Qmax
`
Z
h
 α 2 ∞ dQ  Q 2 i22
4π 2 qf2 Πf (Q 2 ) − Πf (Qmax) + Πf (Qmax) − Πf (0) ,
ω
=
2
π
m`
Qmax m` α 2 Z ∞ dQ  Q 2 2
LO-HVP=ω
4π 2 qf2 Πf (Q 2 ) − Πf (Qmax) (→ ∆a`,f
)
2
π
m
m
`
Qmax
`
h α 2 Z ∞ dQ  Q 2 i222Πf (Qmax) − Πf (0) (→ γ` (Qmax)Π̂f (Qmax)) .(14)
+
ω
2
π
m
m
`
Qmax
`
LO-HVPa`,f(Q > Qmax ) =Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Continuum Extrap. of Light Component: Π̂l
1.9
data
fit0
fit1
fit2
fit3

^ l(2 GeV2)
Π

1.85
1.8
1.75
1.7
1.65
1.6
0

0.005

0.01

0.015

0.02

a2 fm2F (Π̂l , A, B, CMπ , CMK ) = Π̂l
1 + Aa2 + · · ·1 + CMπ ∆Mπ + CMK ∆MK .
1+Ba2 + · · ·Π̂l = 1.8318(42)(60) ,χ2 /d.o.f. = 8.2/12 (fit1 case).

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Continuum Extrap. of Strange Component: Π̂s
0.242
data
fit0
fit1
fit2
fit3

0.24

^ s(2 GeV2)
Π

0.238
0.236
0.234
0.232
0.23
0.228
0

0.005

0.01

0.015

0.02

a2 fm2F (Π̂s , A, B, CMπ , CMK ) = Π̂s
1 + Aa2 + · · ·1 + CMπ ∆Mπ + CMK ∆MK .
1+Ba2 + · · ·Π̂s = 0.2406(1)(2) ,χ2 /d.o.f. = 13.6/11 (fit1 case).

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Continuum Extrap. of Charm Component: Π̂c
0.13
data
fit0
fit1
fit2
fit3

^ c(2 GeV2)
Π

0.12
0.11
0.1
0.09
0.08
0

0.005

0.01

0.015

0.02

a2 fm2F (Π̂c , A, B, CMπ , CMK ) = Π̂c
1 + Aa2 + · · ·1 + CMπ ∆Mπ + CMK ∆MK .
1+Ba2 + · · ·Π̂c = 0.1246(9)(7) ,χ2 /d.o.f. = 26.1/9 (fit1 case).

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017

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Continuum Extrap. of Disc. Component: Π̂d0^ d(2 GeV2)Π-0.0025-0.005-0.0075
-0.01
data
fit0
fit1
fit2

-0.0125-0.015
0

0.005

0.01

0.015

0.02

a2 fm2F (Π̂d , A, B, CMπ , CMK ) = Π̂d
1 + Aa2 + · · ·1 + CMπ ∆Mπ + CMK ∆MK .
1+Ba2 + · · ·Π̂d = −0.0126(6)(7) ,χ2 /d.o.f. = 3.5/9 (fit1 case).

Kohtaroh Miura (CPT, Aix-Marseille Univ.)

PPP 2017, YITP, 02 August 2017