Introduction Challenges and Progresses Discussion Summary and Conclusions Review on Lattice Muon g-2 HVP Calculation Kohtaroh Miura (GSI Helmholtz-Instute Mainz) Lattice 2018, 36th International Symposium on Lattice Field Theory, Michigan State University USA, 22 − 28 July 2018
Introduction Challenges and Progresses Discussion Summary and Conclusions Hadron Vaccum Polarization (HVP) Contribution to Muon g - 2 γ µ µ HAD ˆ Π( Q 2 )
Introduction Challenges and Progresses Discussion Summary and Conclusions 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 (2017): 0.14-ppm, J-PARC E34: 0.1-ppm
Introduction Challenges and Progresses Discussion Summary and Conclusions Really a exp . � = a SM µ ? µ
Introduction Challenges and Progresses Discussion Summary and Conclusions Motivation HVP in Phenomenology The HVP in Pheno. is: � ∞ Q 2 Π( Q 2 ) = ˆ Im Π( s ) ds 0 s ( s + Q 2 ) π � ∞ = ( Q 2 / ( 12 π 2 )) ds R had ( s ) s ( s + Q 2 ) , 0 with R-ratio [right fig. Jegerlehner EPJ-Web2016] given by R had ( s ) ≡ σ ( e + e − → had . ) , 4 πα 2 ( s ) / ( 3 s ) where the systematics is challenging to control(next talk). Some tension among experiments in σ ( e + e − → π + π − ) . Requirement for Lattice QCD: Independent cross-check of Hadronic Vauccum Polarization Contribution to muon g-2 ( a HVP µ ), Permil-Level determination of a HVP w.r.t. FNAL/J-PARC expr. µ
Introduction Challenges and Progresses Discussion Summary and Conclusions Motivation HVP in Phenomenology The HVP in Pheno. is: � ∞ Q 2 Π( Q 2 ) = ˆ Im Π( s ) ds 0 s ( s + Q 2 ) π � ∞ = ( Q 2 / ( 12 π 2 )) ds R had ( s ) s ( s + Q 2 ) , 0 with R-ratio [right fig. Jegerlehner EPJ-Web2016] given by R had ( s ) ≡ σ ( e + e − → had . ) , 4 πα 2 ( s ) / ( 3 s ) where the systematics is challenging to control(next talk). Some tension among experiments in σ ( e + e − → π + π − ) . Requirement for Lattice QCD: Independent cross-check of Hadronic Vauccum Polarization Contribution to muon g-2 ( a HVP µ ), Permil-Level determination of a HVP w.r.t. FNAL/J-PARC expr. µ
Introduction Challenges and Progresses Discussion Summary and Conclusions Objective in This Work Hadron Vacuum Polarization (HVP): d 4 x e iQx � j µ ( x ) j ν ( 0 ) � Π µν ( Q ) = � = ( Q µ Q ν − δ µν Q 2 )Π( Q 2 ) , j µ = 2 u γ µ u − 1 3 ¯ d γ µ d − 1 s γ µ s + 2 3 ¯ 3 ¯ 3 ¯ c γ µ c + · · · . γ Leading-Order(LO) HVP Contr. to Muon g-2: µ µ HAD = ( α/π ) 2 � ∞ dQ 2 ω ( Q 2 / m 2 µ )ˆ a LO-HVP Π( Q 2 ) , µ 0 ˆ Π( Q 2 ) = Π( Q 2 ) − Π( 0 ) . 20000 Π l (Q 2 ) x 10 10 15000 HVP Time-Moments: 10000 Π( Q 2 ) = � ˆ n = 1 Q 2 n Π n , µ ) ^ ω (Q 2 /m 2 5000 d n ˆ Π( Q 2 ) x 2 ν ) n + 1 � ( − ˆ Π n = 1 Q 2 → 0 = � ( 2 n + 2 )! � j µ ( x ) j µ ( 0 ) � . � (m µ /2) 2 x n ! ( dQ 2 ) n � 0 0 0.02 0.04 0.06 0.08 Q 2 GeV 2
Introduction Challenges and Progresses Discussion Summary and Conclusions Access to Deep IR: Pade and Time-Moment Rep. Model Independent Approximants Pade Approximant For Q 2 < Q 2 cut , lattice HVP data are fitted to A 2 Q 2 + · · · ˆ Π( Q 2 ) = 1 + B 2 Q 2 + · · · . (1) � ∞ Q 2 The dispersion relation ˆ Π( Q 2 ) = Im Π( s ) ds is seen as so-called 0 s ( s + Q 2 ) π Stieltjes Integral [Aubin et.al., PRD2012] , which guarantees a finite conversion radius. Time-Momentum Representation (TMR) For Q 2 < Q 2 uv - cut , define [Bernecker and Meyera, EPJA2011] , 3 � � sin[ Qt / 2 ] � 2 � 1 Π( Q 2 ) = ˆ � t 2 � 1 − � j i ( t ) j i ( 0 ) � . (2) Qt / 2 3 t i = 1 The momentum Q is Continuous . The Sine-Cardinal sin[ Qt / 2 ] / ( Qt / 2 ) accounts for a pediodic feature of lattice correlators � j i ( t ) j i ( 0 ) � .
Introduction Challenges and Progresses Discussion Summary and Conclusions Access to Deep IR: Pade and Time-Moment Rep. Model Independent Approximants Pade Approximant For Q 2 < Q 2 cut , lattice HVP data are fitted to A 2 Q 2 + · · · ˆ Π( Q 2 ) = 1 + B 2 Q 2 + · · · . (1) � ∞ Q 2 The dispersion relation ˆ Π( Q 2 ) = Im Π( s ) ds is seen as so-called 0 s ( s + Q 2 ) π Stieltjes Integral [Aubin et.al., PRD2012] , which guarantees a finite conversion radius. Time-Momentum Representation (TMR) For Q 2 < Q 2 uv - cut , define [Bernecker and Meyera, EPJA2011] , 3 � � sin[ Qt / 2 ] � 2 � 1 Π( Q 2 ) = ˆ � t 2 � 1 − � j i ( t ) j i ( 0 ) � . (2) Qt / 2 3 t i = 1 The momentum Q is Continuous . The Sine-Cardinal sin[ Qt / 2 ] / ( Qt / 2 ) accounts for a pediodic feature of lattice correlators � j i ( t ) j i ( 0 ) � .
Introduction Challenges and Progresses Discussion Summary and Conclusions Example of TMR 20000 time-moment rep. Π ud (Q 2 ) x 10 10 lattice data 15000 10000 µ ) ^ ω (Q 2 /m 2 5000 (m µ /2) 2 0 0 0.05 0.1 0.15 0.2 Q 2 [GeV 2 ] Figure: From BMW Ensemble ( a = 0 . 064 fm) used in PRD2017 and PRL2018.
Introduction Challenges and Progresses Discussion Summary and Conclusions Table of Contents Introduction 1 Challenges and Progresses 2 Large Distance Systematics Continuum Extrapolation SIB/QED Corrections Discussion 3 Comparisons Lattice QCD Combined with Phenomenology Summary and Conclusions 4
Introduction Challenges and Progresses Discussion Summary and Conclusions Table of Contents Introduction 1 Challenges and Progresses 2 Large Distance Systematics Continuum Extrapolation SIB/QED Corrections Discussion 3 Comparisons Lattice QCD Combined with Phenomenology Summary and Conclusions 4
Introduction Challenges and Progresses Discussion Summary and Conclusions Multi-Exponential Fits [HPQCD PRD2017] Left: HPQCD PRD2017, vector-current correlator with ud-quarks and a fit line t > t ∗ : G ud ( t , t ∗ ) = G ud data ( t < t ∗ ) or ( G fit ( t > t ∗ ) + G ππ ( t > t ∗ )) , where t ∗ ∈ [ 0 . 5 , 1 . 5 ] fm . Multi( N = 5)-Exponential Ansatz are adopted and ρ -meson dominates. Right: From a slide of Van de Water at Mainz Workshop 2018. Diagrams in effective theory to correct missing effects in the fits. Taste-spliting and finite volume corrections are also taken account.
Introduction Challenges and Progresses Discussion Summary and Conclusions Multi-Exponential Fits [FNAL/HPQCD/MILC Preliminary] a~0.15 fm ensemble with 9362 configurations 900 800 a~0.15 fm (997 configurations) a~0.15 fm (7746 configurations) G data (t<t*) + G fit (t>t*)+ G ππ (t>t*) average of upper and lower bounds 800 700 HVP a µ 700 10 a µ 600 10 600 500 500 0 1 2 3 4 5 400 * (fm) 1 2 3 t t* or t cut (fm) Left: The t ∗ dependence of � α � 2 � ∞ dQ 2 ω ( Q 2 / m 2 µ, ud ( t ∗ ) = µ ) FT [ G ud ( t , t ∗ ) , Q 2 ] with Pade . a LO-HVP (3) π 0 get stable at larger t ∗ . For t ∗ � 2 fm, low-(used in With high-statistics, a LO-HVP µ, ud PRD2017) and high-statistics are consistent. Right: The high-statistics in the left-panel is compared with Bounding Method (next page).
Introduction Challenges and Progresses Discussion Summary and Conclusions Bounding [BMW PRD2017 and PRL2018] 1.0e-01 The connected-light correlator C ud ( t ) loses 1.0e-02 signal for t > 3 fm . To control statistical error, 1.0e-03 consider C ud ( t > t c ) → C ud up / low ( t , t c ) , where 1.0e-04 C ud up ( t , t c ) = C ud ( t c ) ϕ ( t ) /ϕ ( t c ) , C l (t) 1.0e-05 C ud low ( t , t c ) = 0 . 0 , 1.0e-06 with ϕ ( t ) = cosh[ E 2 π ( T / 2 − t )] , 1.0e-07 and E 2 π = 2 ( M 2 π + ( 2 π/ L ) 2 ) 1 / 2 . 1.0e-08 1.0e-09 Similarly, C disc ( t ) → C disc up / low ( t , t c ) , 0 1 2 3 4 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 By construction, C ud ( t ) = 5 1 � � � 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 . 078 [fm] used in PRD2017/PRL2018.
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