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

Davide HVP lattice finite-volume Giusti corrections OUTLINE ▪ Motivations Second Plenary Workshop of the Muon g-2 Theory Initiative Helmholtz Institut Mainz ▪ Current status from Collaborations 18th - 22nd June 2018

Motivations

HVP of the muon HPQCD 16 CLS/Mainz 17 ≳ 2% lattice data BMW 17 q 100% µ RBC/UKQCD 18 q ETMC 18 lattice + e + e - RBC/UKQCD 18 ~ 30% + 70% FJ 17 ≳ 0.4% e + e - data DHMZ 17 100% KNT18 no New Physics ates 550 600 650 700 750 HVP * 10 10 a µ FVEs cannot be neglected 3

LO-HVP FVEs Chiral Perturbation Theory Blum et al. 2018; Borsanyi et al. 2017; Chakraborty et al. 2017 Bijnens and Relefors 2017; Aubin et al. 2016 Gounaris-Sakurai parameterisation + Lüscher formalism RBC/UKQCD Della Morte et al. 2017 talk by C. Lehner ETMC, talk by S. Simula (new updates) Time-momentum representation Izubuchi et al. 2018 4

Current status from Collaborations

𝝍 PT Groups Aubin et al. 2016 0.012 Q 2 ! m µ 2 4 0.010 ( ) ( ) − Π 0 2 ( ) LO,HVP Q max Q max ∫ ⎡ ⎤ MILC ensemble ⎡ ⎤ ⎦ = 4 α em Π Q 2 2 2 dQ 2 f Q 2 a µ ⎣ 0.008 ⎣ ⎦ 0 a = 0.059 fm 0.006 ( ) Π Q 2 m π = 220 MeV ( ) ( ) = Q 2 δ µ ν − Q µ Q ν Π µ ν Q 0.004 L = 3.8 fm 0.002 0.000 0.00 0.05 0.10 0.15 0.20 NLO 𝝍 PT; PBCs ⎡ ⎤ Q 2 GeV 2 ⎣ ⎦ connected (10/9) x [ contribution 10 9 4 πα em Π ChPT ( Q ) = π µ ν sin ( p + Q/ 2) µ sin ( p + Q/ 2) ν 1 X 4 π L 3 T (2 P κ (1 − cos p κ ) + m 2 π ) (2 P κ (1 − cos ( p + Q ) κ ) + m 2 π ) p ] ✓ ◆ 1 cos p µ X − 2 δ µ ν L 3 T (2 P κ (1 − cos p κ ) + m 2 π ) p weighted average Staggered 𝝍 PT taste-split pion spectrum 6 �

𝝍 PT Groups ∑ m π L = 4.2 44 : Π 44 Π ii Aubin et al. 2016 A 1 : ; A 1 i - 0.008 - 0.008 - 0.010 - 0.010 - 0.012 - 0.012 Red A 1 subtracted Red A 1 unsubtracted Blue A 1 44 unsubtracted Blue A 1 44 unsubtracted Black A 1 infinite volume Black A 1 infinite volume - 0.014 - 0.014 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 ˆ ⎡ ⎤ ˆ ⎡ ⎤ Q 2 GeV 2 Q 2 GeV 2 ⎣ ⎦ ⎣ ⎦ Blue laEce data Blue laEce data Red NLO ChPT Red NLO ChPT GeV 2 GeV 2 10-15% FVEs ˆ ⎡ ⎤ Q 2 GeV 2 ⎣ ⎦ ˆ ⎡ ⎤ Q 2 GeV 2 7 ⎣ ⎦

𝝍 PT Groups NNLO 𝝍 PT PQ 𝝍 PT + twisted BCs Bijnens and Relefors 2017 p p ✓ ◆ ⇣ ⌘ − q 2 − q 2 p � q 2 , 0 , 0 q = 0 , q = 0 , 2 , 0 √ √ 2 , 4e-05 4e-05 q 2e-05 2e-05 ∆ V Π µ ν part twist p 4 +p 6 ∆ V Π µ ν part twist p 4 +p 6 0 0 sin θ u sin θ x u µ ν =00 -2e-05 -2e-05 µ ν =00 µ ν =11 µ ν =11 -4e-05 -4e-05 µ ν =22 µ ν =12 µ ν =33 -6e-05 -6e-05 µ ν =33 -8e-05 -8e-05 ♦ ♦ -0.0001 -0.0001 -0.1 -0.08 -0.06 -0.04 -0.02 0 -0.1 -0.08 -0.06 -0.04 -0.02 0 q 2 q 2 m π L = 4 FV corrections: different twist angles at same q 2 Small corrections with respect to NLO FVEs sizeable (few %) for present lattices 8

RBC/UKQCD Collaboration Blum et al. 2018 L = 5.4 ÷ 5.5 fm a = 0.084 ÷ 0.114 fm Two ensembles T = 10.7 ÷ 11 fm m π L = 3.8 ÷ 3.9 Physical mass point FVEs corrected with NLO 𝝍 PT Systematic uncertainty from the largest ratio of p 6 to p 4 ( ) = 15.9 3.7 ( ) ⋅ 10 − 10 Δ FVEs a µ conn ud Talk by C. Lehner ( ) = 20 3 ( ) ⋅ 10 − 10 Δ FVEs a µ conn ud GSL approach: (updates) 9

BMW Collaboration Borsanyi et al. 2017 LO-HVP . 10 10 a µ L = 6.1 ÷ 6.6 fm BMWc + FV + IB BMWc + FV BMWc (L=6fm) ( T × L/a 2 ) β a [fm] RBC/UKQCD 18 T = 8.6 ÷ 11.3 fm 3.7000 0.134 64 × 48 HPQCD 16 3.7500 0.118 96 × 56 ETM 14 3.7753 0.111 84 × 56 Physical mass point 3.8400 0.095 96 × 64 Jegerlehner 17 3.9200 0.078 128 × 80 DHMZ 17 m π L = 4.2 ÷ 4.5 4.0126 0.064 144 × 96 KNT 18 RBC/UKQCD 18 FVEs corrected with LQCD (N f ≥ 2+1) Pheno. NLO SU(2) S 𝝍 PT No new physics Pheno+LQCD 640 660 680 700 720 740 ( I =1 channel only) 700 Fig.S4 m π L ! 4.1 (FV + taste) crr. fixed Fig.S4 cont.lim. + FV 650 LO-HVP x 10 10 ( ) = 15.0 15.0 ( ) ⋅ 10 − 10 600 I = 1 ud Δ FVEs a µ a µ ,ud 550 extrapolated to the continuum limit 500 0 0.005 0.01 0.015 0.02 a 2 [fm 2 ] (six lattice spacings ranging from 0.064 to 0.134 fm) 10

HPQCD Collaboration Chakraborty et al. 2017 L = 2.4 ÷ 5.8 fm Combined FV and discretisation effects (pion tastes) T = 7.2 ÷ 8.6 fm mixing to all orders γ − ρ 0 − π + π − NLO S 𝝍 PT + in leading interactions m π = 134 ÷ 311 MeV 2 r π m π L = 3.2 ÷ 5.4 × ρ π 3 lattice volumes @: 5 times larger m π ! 220 MeV ˆ E , f ρ , m ρ , m π ) = � ˆ Π ( � q 2 Σ ( � q 2 E , m π , m π ) a = 0.12 fm ⌘ 2 ⇣ 1 + g ρ g ρππ ˆ q 2 Σ ( � q 2 E , m π , m π ) + f 2 E ρ 2 m 2 Preliminary ⇣ ⌘ ρππ ˆ q 2 Σ ( � q 2 1 + g 2 + m 2 E , m π , m π ) ρ ρ E ( ) ! 610 9 ( ) ⋅ 10 − 10 LO,HVP a µ conn ud small FVEs+discr. for Talk by R. S. Van de Water s quark contribution Uncertainty: ±0.7% Largest correction for lightest pion masses: 7% 11

Mainz Group Della Morte et al. 2017 G ( x 0 ) � L = 2.1 ÷ 4.2 fm K ( x 0 ) /m µ G ( x 0 ) f K ( x 0 ) /m µ 0 . 013 T = 4.2 ÷ 8.4 fm Data 0 . 015 1-Exp 0 . 012 GS( L ) u/d quarks GS( ∞ ) 0 . 011 0 . 01 m π = 185 ÷ 495 MeV 0 . 01 0 . 005 0 . 009 m π L = 4.0 ÷ 6.0 0 0 . 008 m π = 185 MeV m π L = 4.0 0 . 007 − 0 . 005 FVEs corrected with x cut L/a = 32 ( L = 2 . 70 fm) 0 0 . 006 L/a = 48 ( L = 4 . 10 fm) L/a = 32 with FSE 0 . 015 Gounaris-Sakurai 0 . 005 L/a = 48 with FSE 0 . 004 0 . 01 parameterisation 0 . 6 0 . 8 1 . 2 1 . 4 1 . 6 1 . 8 1 2 m π = 280 MeV x 0 [fm] 0 . 005 + a = 0.085 fm m π = 268 MeV G ( x 0 ) � 0 K ( x 0 ) /m µ Lüscher formalism m π L = 4.2 0 . 002 − 0 . 005 x cut exp fit Γ ρ 0 L = 2 . 05 fm 0 0 . 5 1 1 . 5 2 2 . 5 3 parameters: GS m ρ L = 2 . 70 fm x 0 [fm] 0 . 0015 L = 4 . 10 fm FVEs: 5% shift in ππ interactions m π L ≈ 4 for a µ s quark 0 . 001 important for and near- phys. point t > 1 fm 0 . 0005 ( ) ⋅ 10 − 10 Δ FVEs a µ ! 20.4 4.1 Preliminary 0 0 0 . 5 1 1 . 5 2 2 . 5 3 Talk by H. Meyer x 0 [fm] m π = 140 MeV m π L = 4.0 12

ETM Collaboration Talk by S. Simula L = 1.8 ÷ 3.5 fm u- and d-quark (connected) contributions T = 3.5 ÷ 7.1 fm 450 m π = 223 ÷ 495 MeV 400 400 FVE ~ 5 % M π ~ 320 MeV a ~ 0.09 fm 10 m π L = 3.0 ÷ 5.8 HVP (ud) * 10 HVP (ud) A40.XX A40.XX 350 M π ~ 320 MeV µ 350 a data a ~ 0.09 fm FVEs corrected with µ a FVE ~ 25 % dual + π−π 300 2 𝝆 Lüscher formalism FVE corr. ( π−π only) FVE corr. (dual + π−π ) 300 and GS F 250 π π 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9 10 π + M π L M π L DG et al. 2017 dual pQCD contribution s-quark contribution representation 50 M π ~ 320 MeV A40.XX m ρ g ρππ a ~ 0.09 fm parameters: R dual E dual 400 FVE ~ few % 45 M π ~ 320 MeV a ~ 0.09 fm 10 HVP (ud) HVP (s) * 10 A40.XX pure FVEs: 40 e µ 350 a data he µ a full FVE corr. ∼ 5% correction to a µ 35 partial FVE corr. 300 m π = 135 MeV m π L ≈ 4 @ 2 3 4 5 6 7 8 9 10 30 a 2 → 0 M π L 2 3 4 5 6 7 8 M π L ( ) = 622.8 12.8 ( ) ⋅ 10 − 10 LO,HVP a µ conn ud 13

FVE correction @ a 2 → 0 0 L=8.0 fm 1% -10 12 levels 10 -20 HVP ( ∞ )] * 10 the same as ChPT @ NLO L=6.0 fm 8 levels [Aubin et al. ’16, Bijnens&Relefors ’16] -30 non-interacting π−π (M π = 135 MeV) -40 µ HVP (L) - a interacting π−π (M π = 135 MeV) -50 L=4.5 fm interacting π−π (M π = 300 MeV) 4 levels -60 µ [a continuum limit -70 -80 2 4 6 8 10 M π L [Francis et al. ’13] 2 ! ⎧ ⎫ ⎡ ⎤ n 2 + 4 t 2 K 2 M π L 4 ( ) = M π 1 2 ! sinh M π L ! ⎪ ⎢ ⎥ ∞ ⎪ ( ) t ( ) t ⎣ ⎦ ⎡ ⎤ L ⎡ ⎤ ∞ n 2 + 4 t 2 non-interacting π - π : V ππ ( ) − V ππ 3 π 2 t dy K 0 M π y L n y − 1 ( ) ∑ ∫ ⎨ − M π L ! ⎬ 2 ! 2 L ⎢ ⎥ ⎣ ⎦ ( n 2 + 4 t 2 ) ⎣ ⎦ M π n ! ⎪ ⎪ n ≠ 0 1 ⎩ ⎭ interacting π - π : dual + π - π representation [note that Δ a μ HVP (L) depends approximately on M π L only] 14 Thursday, June 21, 18

PACS Collaboration Izubuchi et al. 2018 Two ensembles FVEs estimated using TMR L = T = 5.4 ÷ 8.1 fm comparison between two volumes LO,HVP on L=5.4 fm is a = 0.085 fm ( ) ⋅ 10 − 10 10 ± 26 a µ from L=8.1 fm @ m π = 146 MeV near- phys. mass point m π L = 3.8 ÷ 5.8 40 Light 30 Backward state propagation 20 (2T=10.8 fm) 10 a µ 10 10 0 -10 positive contribution to a µ -20 4% @ t cut =2.6 fm 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 -30 (L/a=96,T/a=96) 146 MeV - (L/a=64,T/a=128) 146 MeV -40 15 0 0.5 1 1.5 2 2.5 3 t cut fm

IB contribution: FVEs DG et al. 2017 Blum et al. 2018 ) s/c contribution only photon propagator h QED L prescription for expected to start at O( 1/L^3 ) zero mode subtraction (IR safe, neutral meson states, h vanishing charge radius) 0 G L ( x ) = 1 1 k 2 e ikx , X ˆ V Talk by S. Simula k 0.02 β =1.90, L=20 QED ∞ β =1.90, L=24 β =1.90, L=32 β =1.90, L=40 HVP (ud) 0.015 β =1.95, L=24 β =1.95, L=32 Z π d 4 k 1 β =2.10, L=48 HVP (ud) / a µ k 2 e ikx G 1 ( x ) = physical point 0.01 ˆ (2 π ) 4 � π δ a µ Analytical calculation 0.005 Talk by A. Portelli 0 0 0.01 0.02 0.03 0.04 0.05 16 m ud (GeV)