Lattice QCD Approach to HVP and Muon g-2 Kohtaroh Miura (GSI - - PowerPoint PPT Presentation

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Lattice QCD Approach to HVP and Muon g-2 Kohtaroh Miura (GSI - - PowerPoint PPT Presentation

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Lattice QCD Approach to HVP and Muon g-2 Kohtaroh Miura (GSI Helmholtz-Instute Mainz, Nagoya-Univ. KMI) RIKEN Seminar August 27, 2019, RIKEN-KOBE

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Lattice QCD Approach to HVP and Muon g-2

Kohtaroh Miura (GSI Helmholtz-Instute Mainz, Nagoya-Univ. KMI) RIKEN Seminar August 27, 2019, RIKEN-KOBE Budapest-Marseille-Wuppertal (BMW) Collab. Refs:

  • Phys. Rev. Lett. 121, no. 2, 022002 (2018).
  • Phys. Rev. D 96, no. 7, 074507 (2017).

With some updates and preliminary results.

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Muon Anomalous Magnetic Moment aℓ=e,µ,τ

Dirac Eq. with B: i∂ψ ∂t =

  • α ·
  • −ic∇ − eA
  • + βc2mℓ + eA0
  • ψ ,

Nonlelativistic Limit, Pauli Eq.: i∂φ ∂t = (−ic∇ − eA)2 2mℓc − Mℓ · B + eA0

  • φ ,

Magnetic Moment: Mℓ = gℓ

e 2mℓc σ 2 ,

In Dirac Theory: gℓ = 2 , aℓ ≡ (gℓ − 2)/2 = 0 , ωcyc = ωprec. In QFT (with Loops) for Electron (M.Knecht ,NPPP2015): aSM

e

= 1 159 652 180.07(6)(4)(77) × 10−12 (O(α5)) , a

exp

e

= 1 159 652 180.73(0.28) × 10−12 [0.24ppb] .

B

p s

Muon Strorage

μ

aexp.

µ

= aSM

µ ?

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Muon Anomalous Magnetic Moment aℓ=e,µ,τ

Dirac Eq. with B: i∂ψ ∂t =

  • α ·
  • −ic∇ − eA
  • + βc2mℓ + eA0
  • ψ ,

Nonlelativistic Limit, Pauli Eq.: i∂φ ∂t = (−ic∇ − eA)2 2mℓc − Mℓ · B + eA0

  • φ ,

Magnetic Moment: Mℓ = gℓ

e 2mℓc σ 2 ,

In Dirac Theory: gℓ = 2 , aℓ ≡ (gℓ − 2)/2 = 0 , ωcyc = ωprec. In QFT (with Loops) for Electron (M.Knecht ,NPPP2015): aSM

e

= 1 159 652 180.07(6)(4)(77) × 10−12 (O(α5)) , a

exp

e

= 1 159 652 180.73(0.28) × 10−12 [0.24ppb] .

B

p s

Muon Strorage

μ

aexp.

µ

= aSM

µ ?

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

aexp.

µ

  • vs. aSM

µ

SM contribution acontrib.

µ

× 1010 Ref. QED [5 loops] 11658471.8951 ± 0.0080

[Aoyama et al ’12]

HVP-LO (pheno.) 692.6 ± 3.3

[Davier et al ’16]

694.9 ± 4.3

[Hagiwara et al ’11]

681.5 ± 4.2

[Benayoun et al ’16]

688.8 ± 3.4

[Jegerlehner ’17]

HVP-NLO (pheno.) −9.84 ± 0.07

[Hagiwara et al ’11] [Kurz et al ’11]

HVP-NNLO 1.24 ± 0.01

[Kurz et al ’11]

HLbyL 10.5 ± 2.6

[Prades et al ’09]

Weak (2 loops) 15.36 ± 0.10

[Gnendiger et al ’13]

SM tot [0.42 ppm] 11659180.2 ± 4.9

[Davier et al ’11]

[0.43 ppm] 11659182.8 ± 5.0

[Hagiwara et al ’11]

[0.51 ppm] 11659184.0 ± 5.9

[Aoyama et al ’12]

Exp [0.54 ppm] 11659208.9 ± 6.3

[Bennett et al ’06]

Exp − SM 28.7 ± 8.0

[Davier et al ’11]

26.1 ± 7.8

[Hagiwara et al ’11]

24.9 ± 8.7

[Aoyama et al ’12]

aLO-HVP

µ

|NoNewPhys × 1010 ≃ 720 ± 7, FNAL E989: 0.14-ppm (first data 0.5-ppm: 2019-Dec.?)), J-PARC E34: 0.1-ppm

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

aℓ in QFT

QFT Def. for aℓ: = ¯ ℓ−(p)|J µ|ℓ−(p′) = ¯ u(p)Γµ(p, p′)u(p′) (1) Γµ(q = p − p′) = γµF1(q2) + iσµνqν 2mµ F2(q2) + · · · , (2) F2(0) = aℓ = (gℓ − 2)/2 . (3) Standard Model, Loop Corr.: aℓ = α/(2π) + · · · . BSM = MSSM (Padley et.al.’15) or TC (Kurachi et.al. ’13) etc.:

γ µ µ Technicolor

∝ (mℓ/ΛBSM)2.

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Really aexp.

µ

= aSM

µ ?

The Hadronic Vacuum Polarization (HVP) contributions to aµ is a bottle-neck to answer for this question.

HAD

µ µ γ

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Phenomenology of HVP

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Pion Contributions to aµ from Experimental Data

]

  • 10

(600 - 900 MeV) [10

,LO π π µ

a

360 365 370 375 380 385 390 395

BaBar 09 KLOE 12 KLOE 10 KLOE 08 BESIII 1.9 ± 2.0 ± 376.7 0.8 ± 2.4 ± 1.2 ± 366.7 2.2 ± 2.3 ± 0.9 ± 365.3 2.2 ± 2.3 ± 0.4 ± 368.1 3.3 ± 2.5 ± 368.2

Figure: Borrowed by BESIII, PLB’16: Some tension among experiments on pion contributions to aµ.

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

THIS TALK Lattice QCD for Muon g − 2 First Principle Crosschecks of the dispersive results. First Principle Predictions for assessing SM with measurements by FermiLab/J-PARC experiments (0.1-ppm). THIS TALK: Report BMW-Collab. results for muon g − 2. Compare/Discuss various results from lattice QCD as well as dispersive method.

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Table of Contents

1

Introduction

2

Results Setup Continuum Extrapolations Comparison among LQCDs

3

Discussions: Lattice vs Pheno

4

Summary and Perspective

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Table of Contents

1

Introduction

2

Results Setup Continuum Extrapolations Comparison among LQCDs

3

Discussions: Lattice vs Pheno

4

Summary and Perspective

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Simulation Setup (BMWc. PRD-2017 and PRL-2018)

BMW Ensemble PRD2017 and PRL2018 6-β, 15 simulation with all physical masses. Nf=(2+1+1) staggered quarks. Large Volume: (L, T) ∼ (6, 9 − 12)fm. AMA with 6000-9000 random-source

  • meas. for disconnected.

24.5 25.0 25.5 26.0 26.5 27.0 27.5 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 2MK

2/Mπ 2-1

Mπ

2/Fπ 2

3.7000 3.7500 3.7753 3.8400 3.9200 4.0126 phys

β a[fm] Nt Ns #traj. Mπ[MeV] MK [MeV] #SRC (l,s,c,d) 3.7000 0.134 64 48 10000 ∼ 131 ∼ 479 (768, 64, 64, 9000) 3.7500 0.118 96 56 15000 ∼ 132 ∼ 483 (768, 64, 64, 6000) 3.7753 0.111 84 56 15000 ∼ 133 ∼ 483 (768, 64, 64, 6144) 3.8400 0.095 96 64 25000 ∼ 133 ∼ 488 (768, 64, 64, 3600) 3.9200 0.078 128 80 35000 ∼ 133 ∼ 488 (768, 64, 64, 6144) 4.0126 0.064 144 96 04500 ∼ 133 ∼ 490 (768, 64, 64, −)

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Observables and Objectives

1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 1 2 3 4 a = 0.064 [fm] Cud(t) t [fm] 5000 10000 15000 20000 0.05 0.1 0.15 0.2 (mµ/2)2 ω(Q2/m2

µ) ^

Πud(Q2) x 1010 Q2 [GeV2] time-moment rep. lattice data

Πµν(Q) = (QµQν − δµνQ2)Π(Q2) =

  • d4x eiQxjµ(x)jν(0) ,

(4) jµ = (2/3)¯ uγµu − (1/3)¯ dγµd − (1/3)¯ sγµs + (2/3)¯ cγµc + · · · , (5) ˆ Π(Q2) = Π(Q2) − Π(0) =

  • t

t2

  • 1 −

sin[Qt/2] Qt/2 2 1 3

3

  • i=1

ji(t)ji(0) . (6) aLO-HVP

ℓ=e,µ,τ = α2

π2 ∞ dQ2 ω

  • Q2

m2

ℓ=e,µ,τ

  • ˆ

Π(Q2) . (7)

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Observables and Objectives

1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 1e-03 1e-02 1e-01 1 2 3 4 a = 0.064 [fm] Cud(t) t [fm] 5000 10000 15000 20000 0.05 0.1 0.15 0.2 (mµ/2)2 ω(Q2/m2

µ) ^

Πud(Q2) x 1010 Q2 [GeV2] time-moment rep. lattice data

Πµν(Q) = (QµQν − δµνQ2)Π(Q2) =

  • d4x eiQxjµ(x)jν(0) ,

(4) jµ = (2/3)¯ uγµu − (1/3)¯ dγµd − (1/3)¯ sγµs + (2/3)¯ cγµc + · · · , (5) ˆ Π(Q2) = Π(Q2) − Π(0) =

  • t

t2

  • 1 −

sin[Qt/2] Qt/2 2 1 3

3

  • i=1

ji(t)ji(0) . (6) aLO-HVP

ℓ=e,µ,τ = α2

π2 ∞ dQ2 ω

  • Q2

m2

ℓ=e,µ,τ

  • ˆ

Π(Q2) . (7)

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Bounding [BMW PRD2017 and PRL2018]

1.0e-10 1.0e-09 1.0e-08 1.0e-07 1.0e-06 1.0e-05 1.0e-04 1 2 3 4 Cud(t) t [fm] data 2-pi corr.

Figure shows Cud(t) = 5 9

  • x

1 3

3

  • i=1

jud

i

( x, t)jud

i

(0) , by BMW Ensemble with a = 0.064 [fm] used in PRD2017/PRL2018. The connected-light correlator Cud(t) loses signal for t > 3fm. To control statistical error, consider Cud(t > tc) → Cud

up/low(t, tc), where

Cud

up (t, tc) = Cud(tc) ϕ(t)/ϕ(tc),

Cud

low(t, tc) = 0.0,

with ϕ(t) = cosh[E2π(T/2 − t)], and E2π = 2(M2

π + (2π/L)2)1/2.

Similarly, Cdisc(t) → Cdisc

up/low(t, tc),

−Cdisc

up (t > tc) = 0.1Cud(tc) ϕ(t)/ϕ(tc),

−Cdisc

low (t > tc) = 0.0.

By construction, Cud,disc

low

(t, tc) ≤ Cud,disc(t) ≤ Cud,disc

up

(t, tc).

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Bounding [BMW PRL2018]

550 600 650 700 aµ, ud

LO-HVP x 1010

50 100 150 2.0 2.5 3.0 3.5 4.0 4.5

  • aµ, disc

LO-HVP x 1011

tc [fm]

2-pion zero avg

Figure: BMW, PRL2018.

Corresponding to Cud,disc

up/low (tc), we obtain

upper/lower bounds for muon g-2: aud,disc

µ,up/low(tc).

Two bounds meet around tc = 3fm. Consider the average of bounds: ¯ aud,disc

µ

(tc) = 0.5(aud,disc

µ,up

+ aud,disc

µ,low )(tc),

which is stable around tc = 3fm. We pick up such averages ¯ aud,disc

µ

(tc) with 4 − 6 kinds of tc around 3fm. The average

  • f average is adopted as aLO-HVP

µ,ud/disc to be

analysed, and a fluctuation over selected tc gives systematic error. A similar method is proposed by C.Lehner in Lattice2016 and used in RBC/UKQCD-PRL2018. Improved bounding method with GEVP:

[A. Meyer/C. Lehner, 27 Fri Hadron Structure].

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Controlled Continuum Extrap. [BMW PRL2018]

550 600 650 aµ,ud

LO-HVP x 1010

53.0 53.5 aµ,s

LO-HVP x 1010

8.0 12.0 aµ,c

LO-HVP x 1010

2.5 5.0 7.5 10.0 0.000 0.005 0.010 0.015 0.020 −aµ,disc

LO-HVP x 1010

a2[fm2]

With 6 β′s = 15 a2[fm2] simulations, allowing full control over continuum limit. Get systematic uncertainty from various cuttings: no-cut, or cutting a ≥ 0.134, 0.111, or 0.095. Get good χ2/dof with extrapolation linear in a2, and interpolation linear in M2

K (strange) or M2 π and Mηc (charm).

Strong a2 dependences for aLO-HVP

µ,ud/disc

due to taste violations, and for aLO-HVP

µ,c

due to large mc.

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Crosscheck of Continuum Extrapolation [BMW PRL2018]

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

LO-HVP x 1010

a2[fm2]

Fig.S4 (FV + taste) crr.

Fig.S4 cont.lim. + FV

1

Red open-circles are raw lattice data and continuum-extrapolated (red filled-circle). Then finite-volume correction using XPT is added to get the green-square point.

2

Similarly to HPQCD-PRD2017, raw data (red-circles) are first corrected with finite-volume and taste-partner effects to get blue open-triangles, which are continuum-extrapolated to get blue filled-triangle.

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Various Corrections

High Q2 Control: The lattice data have enough overlap to perturbative regime even in tau case. aLO-HVP

ℓ,f

= aLO-HVP

ℓ,f

(Q ≤ Qmax)+ (γℓ ˆ Πf)(Qmax) + ∆pertaLO-HVP

ℓ,f

(Q > Qmax) . Isospin/QED Collections: Model estimates amounts to 1.1% corrections (table thanks to F.Jegerlehner (& M. Benayoun)). FV Collections: The dominant FV in I = 1, π+π− loop channel is estimated by XPT (Aubin et al ’16):

  • aLO-HVP

µ,I=1 (∞) − aLO-HVP µ,I=1 (6fm)

  • |XPT

= 13.42(13.42) × 10−10 , (1.9%) .

100 200 300 400 1 2 3 4 5 lattice matching perturb. total aτ

LO-HVP x 108

Qmax

2 [GeV2]

Effect δaLO-HVP

µ

× 1010 ρ−ω mix. 2.71 ± 1.36 FSR 4.22 ± 2.11 Mπ → Mπ± −4.47 ± 4.47 π0γ 4.64 ± 0.04 ηγ 0.65 ± 0.01 Total 7.8 ± 5.1

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Summary on aLO-HVP

µ

PRL2018

aLO-HVP

µ

BMWc I = 1 582.9(6.7)st(7.2)acut(0.1)tcut(0.0)qcut(4.5)da(13.5)fv I = 0 120.5(3.4)st(3.5)acut(0.2)tcut(0.0)qcut(1.0)da total 711.1(7.5)st(8.0)acut(0.2)tcut(0.0)qcut(5.5)da(13.5)fv(5.1)iso Remarks Our Lattice QCD results are consistent with both “No New Physics” and Dispersive Method. Total error in our LQCD result is 2.6%, dominated FV effects.

640 660 680 700 720 740 760

ETM-14 HPQCD-16 BMW-17 RBC/UKQCD-18 ETM-18 FHM-19 Mainz/CLS-19 PACS-19 Jegerlehner-17 DHMZ-17 KNT-18 RBC/UKQCD-18 No new physics

aµ

HLO . 1010

LQCD Pheno. Pheno+LQCD

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

aLO-HVP

µ

: flavor by flavor comparison

550 575 600 625 650 675 700

FHM-19 Aubin et.al.-19 ETM-18 BMW-17 HPQCD-16 RBC/UKQCD-18 PACS-19 Mainz/CLS-19 Mainz-17 (TMR+FV) Nf=2+1+1 Nf=2+1 Nf=2

aµ

HLO (ud) . 1010

50 51 52 53 54 55 56

BMW-17 ETM-17 HPQCD-14 RBC/UKQCD-18 PACS-19 Mainz/CLS-19 Mainz-17 (TMR)

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

aµ

HLO (s) . 1010

10 11 12 13 14 15 16

BMW-17 ETM-17 HPQCD-14 RBC/UKQCD-18 PACS-19 Mainz/CLS-19 Mainz-17 (TMR)

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

aµ

HLO (c) . 1010

  • 30
  • 25
  • 20
  • 15
  • 10
  • 5

FHM-19 BMW-17 RBC/UKQCD-18 Mainz/CLS-19

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

[aµ

HLO]disconn. . 1010

The results do not yet converge in all flavors... “Disagreement” is particularly on aLO-HVP

µ, ud

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Table of Contents

1

Introduction

2

Results Setup Continuum Extrapolations Comparison among LQCDs

3

Discussions: Lattice vs Pheno

4

Summary and Perspective

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

ˆ Πlat(Q2) vs ˆ Πpheno(Q2) for Various Q2 Preliminary

5000 10000 15000 20000 0.0001 0.001 0.01 0.1 1 w(Q2 / mµ

2) ^

Π(Q2) x 1010 Q2 [GeV2] lattice w. FV pheno.

ˆ Πlat(ω2) = lim

a→0 T/2

  • t=0

t2 1 − sinc2[ωt/2]

  • C(t) ,

ˆ Πpheno(Q2) = Q2 12π2 ∞ ds Rhad(s) s(s + Q2) . Lat (BMWc) vs Pheno (alphaQEDc17 by Jegerlehner) for ω(Q2/m2

µ)ˆ

Π(Q2) The contributions at Q2 ∼ (mµ/2)2 are dominant, and the lattice and phemenology are consistent within the error-bars there. However, the lattice error gets larger at Q2 ∼ (mµ/2)2. More precise estimates are demanded and in progress.

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Integrand of aLO-HVP

µ,ud

I

100 200 300 400 500 1 2 3 4 5 W(t,mµ) Ctot(t) x 1010 [fm-1] t [fm] a = 0.134 fm a = 0.118 fm a = 0.111 fm a = 0.095 fm a = 0.079 fm pheno.

aLO-HVP

µ,ud

=

  • t

W(t, mµ)Ctot(t) , (8) c.f. Cpheno

tot

(t) = ∞ ds √ sRhad(s)e−√s|t| . (9)

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Integrand of aLO-HVP

µ,ud

II

100 200 300 400 1 2 3 4 W(t,mµ) Cud(t) x 1010 [fm-1] t [fm] a = 0.134 fm a = 0.064 fm

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Integrand of aLO-HVP

µ,ud

III

100 200 300 400 1 2 3 4 UV IM IR W(t,mµ) Cud(t) x 1010 [fm-1] t [fm] a = 0.134 fm a = 0.064 fm

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Window Method

100 200 300 400 1 2 3 4 UV IM IR W(t,mµ) Cud(t) x 1010 [fm-1] t [fm] a = 0.134 fm a = 0.064 fm 0.25 0.5 0.75 1 1 2 3 4 UV IM IR Smeared Step Functions t [fm]

UV: SUV(t) = 1.0 − (1.0 + tanh

  • (t − t0)/∆
  • )/2 ,

(10) IM: SIM(t) = 1 2

  • tanh
  • (t − t0)/∆
  • − tanh
  • (t − t1)/∆
  • ,

(11) IR: SIR(t) = (1.0 + tanh

  • (t − t1)/∆
  • )/2 ,

(12) We shall adopt t0 = 0.6fm , t1 = 1.5fm , ∆ = 0.3fm . (13)

c.f. RBC-UKQCD (PRL2018), Aubin et.al. (1905.09307)

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Continuum Extrapolation in Dominant Window Preliminary

305 310 315 320 325 0.005 0.01 0.015 0.02 aµ,ud

LO-HVP(0.6-1.5fm) x 1010

a2[fm2]

fit0 fit1 fit2 fit3

For the most important window (0.6 − 1.5 fm), the lattice QCD provides very precise data with per-mil level precision.

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Table of Contents

1

Introduction

2

Results Setup Continuum Extrapolations Comparison among LQCDs

3

Discussions: Lattice vs Pheno

4

Summary and Perspective

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Introduction Results Discussions: Lattice vs Pheno Summary and Perspective

Summary and Perspective

We have obtained aLO-HVP

µ

directly at physical point masses: aLO-HVP

µ

= 711.1(7.5)(17.4) × 10−10. Full controlled continuum extrapolation and matching to perturbation

  • theory. Model assumptions are put on only for small corrections from

FV/QED/isospin breaking. Total error is 2.6%, dominated by FV. Our Lattice QCD results are consistent with “No New Physics” as well as Phenomenological Dispersive Methods with a conservative systematic errors. Lat-Pheno. comparisons are made for HVP: consistent at small Q2, but lattice tends to be larger, leading to larger aLO-HVP

µ,lat .

Need ∼ 0.2% precision to match Fermilab/J-PARC experiments!!

1

lat-pheno combined analyses: window method (on going, per-mil level precision at present statistics).

2

QED/SIB based on lattice QCD (on going, correction to Dashen’s theorem as an exercise).

3

control FV effects directly based on the first-principle.