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

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.

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Muon Anomalous Magnetic Moment a ℓ = e ,µ,τ Dirac Eq. with B: i � ∂ψ � � � � + β c 2 m ℓ + eA 0 ∂ t = α · − i � c ∇ − e A ψ , Nonlelativistic Limit, Pauli Eq.: μ B � ( − i � c ∇ − e A ) 2 i � ∂φ � ∂ t = − M ℓ · B + eA 0 φ , 2 m ℓ c Muon Strorage e � σ Magnetic Moment: M ℓ = g ℓ 2 , 2 m ℓ c In Dirac Theory: p g ℓ = 2 , a ℓ ≡ ( g ℓ − 2 ) / 2 = 0 , ω cyc = ω prec . s In QFT (with Loops) for Electron (M.Knecht ,NPPP2015): = 1 159 652 180 . 07 ( 6 )( 4 )( 77 ) × 10 − 12 ( O ( α 5 )) , a SM e exp = 1 159 652 180 . 73 ( 0 . 28 ) × 10 − 12 a [ 0 . 24 ppb ] . e a exp . = a SM µ ? µ

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Muon Anomalous Magnetic Moment a ℓ = e ,µ,τ Dirac Eq. with B: i � ∂ψ � � � � + β c 2 m ℓ + eA 0 ∂ t = α · − i � c ∇ − e A ψ , Nonlelativistic Limit, Pauli Eq.: μ B � ( − i � c ∇ − e A ) 2 i � ∂φ � ∂ t = − M ℓ · B + eA 0 φ , 2 m ℓ c Muon Strorage e � σ Magnetic Moment: M ℓ = g ℓ 2 , 2 m ℓ c In Dirac Theory: p g ℓ = 2 , a ℓ ≡ ( g ℓ − 2 ) / 2 = 0 , ω cyc = ω prec . s In QFT (with Loops) for Electron (M.Knecht ,NPPP2015): = 1 159 652 180 . 07 ( 6 )( 4 )( 77 ) × 10 − 12 ( O ( α 5 )) , a SM e exp = 1 159 652 180 . 73 ( 0 . 28 ) × 10 − 12 a [ 0 . 24 ppb ] . e a exp . = a SM µ ? µ

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective a exp . vs. a SM µ µ a contrib . × 10 10 SM contribution Ref. µ 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] | NoNewPhys × 10 10 ≃ 720 ± 7, a LO-HVP µ FNAL E989: 0.14-ppm (first data 0.5-ppm: 2019-Dec.?)), J-PARC E34: 0.1-ppm

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective a ℓ in QFT QFT Def. for a ℓ : � ¯ ℓ − ( p ) |J µ | ℓ − ( p ′ ) � = ¯ u ( p )Γ µ ( p , p ′ ) u ( p ′ ) = (1) Γ µ ( q = p − p ′ ) = γ µ F 1 ( q 2 ) + i σ µν q ν 2 m µ F 2 ( q 2 ) + · · · , (2) F 2 ( 0 ) = a ℓ = ( g ℓ − 2 ) / 2 . (3) Standard Model, Loop Corr.: a ℓ = α/ ( 2 π ) + · · · . BSM = MSSM (Padley et.al.’15) or TC (Kurachi et.al. ’13) etc.: γ ∝ ( m ℓ / Λ BSM ) 2 . µ µ Technicolor

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Really a exp . � = a SM µ ? µ The Hadronic Vacuum Polarization (HVP) contributions to a µ is a bottle-neck to answer for this question. γ µ µ HAD

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Phenomenology of HVP

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Pion Contributions to a µ from Experimental Data ± ± ± KLOE 08 368.1 0.4 2.3 2.2 ± ± BaBar 09 376.7 2.0 1.9 ± ± ± 365.3 0.9 2.3 2.2 KLOE 10 ± ± ± KLOE 12 366.7 1.2 2.4 0.8 ± ± BESIII 368.2 2.5 3.3 360 365 370 375 380 385 390 395 π π ,LO -10 a (600 - 900 MeV) [10 ] µ Figure: Borrowed by BESIII, PLB’16: Some tension among experiments on pion contributions to a µ .

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.

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Table of Contents Introduction 1 Results 2 Setup Continuum Extrapolations Comparison among LQCDs Discussions: Lattice vs Pheno 3 Summary and Perspective 4

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Table of Contents Introduction 1 Results 2 Setup Continuum Extrapolations Comparison among LQCDs Discussions: Lattice vs Pheno 3 Summary and Perspective 4

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Simulation Setup (BMWc. PRD-2017 and PRL-2018) 27.5 BMW Ensemble PRD2017 and PRL2018 27.0 6- β , 15 simulation with all physical 2 -1 26.5 masses. 2 /M π 26.0 Nf=(2+1+1) staggered quarks. 2M K 3.7000 25.5 3.7500 Large Volume: ( L , T ) ∼ ( 6 , 9 − 12 ) fm . 3.7753 3.8400 25.0 3.9200 AMA with 6000-9000 random-source 4.0126 phys meas. for disconnected. 24.5 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 2 /F π 2 M π β a [fm] N t N s #traj. M π [MeV] M K [MeV] #SRC (l,s,c,d) 3 . 7000 0 . 134 ∼ 131 ∼ 479 ( 768 , 64 , 64 , 9000 ) 64 48 10000 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 , − )

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Observables and Objectives 1e-01 20000 time-moment rep. a = 0.064 [fm] Π ud (Q 2 ) x 10 10 1e-02 lattice data 15000 1e-03 1e-04 C ud (t) 10000 1e-05 µ ) ^ 1e-06 ω (Q 2 /m 2 5000 1e-07 (m µ /2) 2 1e-08 0 1e-09 0 0.05 0.1 0.15 0.2 0 1 2 3 4 Q 2 [GeV 2 ] t [fm] � Π µν ( Q ) = ( Q µ Q ν − δ µν Q 2 )Π( Q 2 ) = d 4 x e iQx � j µ ( x ) j ν ( 0 ) � , (4) u γ µ u − ( 1 / 3 )¯ j µ = ( 2 / 3 )¯ d γ µ d − ( 1 / 3 )¯ s γ µ s + ( 2 / 3 )¯ c γ µ c + · · · , (5) � 2 � 1 � sin[ Qt / 2 ] 3 � Π( Q 2 ) = Π( Q 2 ) − Π( 0 ) = ˆ � t 2 � 1 − � j i ( t ) j i ( 0 ) � . (6) Qt / 2 3 t i = 1 � ∞ ℓ = e ,µ,τ = α 2 Q 2 dQ 2 ω � � Π( Q 2 ) . ˆ a LO-HVP (7) π 2 m 2 0 ℓ = e ,µ,τ

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Observables and Objectives 1e-01 20000 time-moment rep. a = 0.064 [fm] Π ud (Q 2 ) x 10 10 1e-02 lattice data 15000 1e-03 1e-04 C ud (t) 10000 1e-05 µ ) ^ 1e-06 ω (Q 2 /m 2 5000 1e-07 (m µ /2) 2 1e-08 0 1e-09 0 0.05 0.1 0.15 0.2 0 1 2 3 4 Q 2 [GeV 2 ] t [fm] � Π µν ( Q ) = ( Q µ Q ν − δ µν Q 2 )Π( Q 2 ) = d 4 x e iQx � j µ ( x ) j ν ( 0 ) � , (4) u γ µ u − ( 1 / 3 )¯ j µ = ( 2 / 3 )¯ d γ µ d − ( 1 / 3 )¯ s γ µ s + ( 2 / 3 )¯ c γ µ c + · · · , (5) � 2 � 1 � sin[ Qt / 2 ] 3 � Π( Q 2 ) = Π( Q 2 ) − Π( 0 ) = ˆ � t 2 � 1 − � j i ( t ) j i ( 0 ) � . (6) Qt / 2 3 t i = 1 � ∞ ℓ = e ,µ,τ = α 2 Q 2 dQ 2 ω � � Π( Q 2 ) . ˆ a LO-HVP (7) π 2 m 2 0 ℓ = e ,µ,τ

Introduction Results Discussions: Lattice vs Pheno Summary and Perspective Bounding [BMW PRD2017 and PRL2018] 1.0e-04 data The connected-light correlator C ud ( t ) loses 2-pi corr. 1.0e-05 signal for t > 3 fm . To control statistical error, consider C ud ( t > t c ) → C ud 1.0e-06 up / low ( t , t c ) , where C ud (t) C ud up ( t , t c ) = C ud ( t c ) ϕ ( t ) /ϕ ( t c ) , 1.0e-07 C ud low ( t , t c ) = 0 . 0 , 1.0e-08 with ϕ ( t ) = cosh[ E 2 π ( T / 2 − t )] , 1.0e-09 and E 2 π = 2 ( M 2 π + ( 2 π/ L ) 2 ) 1 / 2 . 1.0e-10 1 2 3 4 Similarly, C disc ( t ) → C disc up / low ( t , t c ) , t [fm] − C disc up ( t > t c ) = 0 . 1 C ud ( t c ) ϕ ( t ) /ϕ ( t c ) , − C disc low ( t > t c ) = 0 . 0 . Figure shows 3 C ud ( t ) = 5 1 By construction, � � � j ud x , t ) j ud ( � ( 0 ) � , C ud , disc ( t , t c ) ≤ C ud , disc ( t ) ≤ C ud , disc i i ( t , t c ) . 9 3 up low � i = 1 x by BMW Ensemble with a = 0 . 064 [fm] used in PRD2017/PRL2018.