HVP contribution of the light quarks Davide Giusti to (g -2) - PowerPoint PPT Presentation

HVP contribution of the light quarks Davide Giusti to (g 𝝂 -2) including QED corrections with twisted-mass fermions OUTLINE Isospin breaking effects on the lattice XXXVI International (RM123 method) Symposium on Lattice Field Theory Results for the light quark East Lansing contribution to a 𝝂 HVP 22 nd - 28 th July In collaboration with: V. Lubicz, G. Martinelli, S. Romiti, F. Sanfilippo, S. Simula, C. Tarantino

Muon magnetic anomaly  � γ µ F 1 ( q 2 ) + i σ µ ν q ν F 2 ( q 2 ) u ( p 0 ) = ( − i e ) ¯ u ( p ) 2 m a µ ≡ g µ − 2 ( ) = F muon anomalous magnetic moment: 2 0 2 is generated by quantum loops; receives contribution from QED, EW and QCD effects in the SM; (c) (e) γ is a sensitive probe of new physics µ Z Z f (c) 3b PDG 2018 SM = 116 591 823 1 ( ) 34 ( ) 26 ( ) ⋅ 10 − 11 0.4ppm a µ NLO/NNLO QED+EW dispersion relations Had. LO Had. e + e - hadrons HVP ab-initio LQCD q µ 2 error budget q

Phenomenological motivations The determination of some hadronic observables in flavor physics has reached such an accurate degree of experimental and theoretical precision that electromagnetic and strong isospin breaking effects cannot be neglected anymore 3

ISOSPIN BREAKING EFFECTS Isospin symmetry is an almost exact property of the strong interactions Isospin breaking effects are induced by: m u ≠ m d : O[ (m d -m u )/ Λ QCD ] ≈ 1/100 “Strong” “Electromagnetic” O( α e.m. ) ≈ 1/100 Q u ≠ Q d : Since electromagnetic interactions renormalise quark masses the two corrections are intrinsically related Though small, IB effects can play a very important role (quark masses, M n - M p , leptonic decay constants, vector form factor) 4

Hadronic Vacuum Polarisation HPQCD 16 CLS/Mainz 17 ≳ 2% lattice data BMW 17 100% RBC/UKQCD 18 ETMC 18 q lattice + e + e - RBC/UKQCD 18 µ ~ 30% + 70% q 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 Given the present exper. and theor. (LQCD) accuracy, an important source of uncertainty are long distance electromagnetic and SU(2) breaking corrections ) estimate in sQED ( ) ⋅ 10 − 11 HVP ~ 39 1 δ a µ h 19) 20) 21) h K. Melnikov, 2001 5

Isospin breaking effects on the lattice RM123 method

A strategy for Lattice QCD: The isospin breaking part of the Lagrangian is treated as a perturbation + Expand in: α em m d – m u arXiv:1110.6294 arXiv:1303.4896 RM123 Collaboration 7

The (m d -m u ) expansion - Identify the isospin breaking term in the QCD action 1 ) − 1 ⎡ ⎤ ( ( ) ( ) uu + dd ( ) uu − dd ∑ ∑ S m u uu + m d dd 2 m u + m 2 m d − m ⎡ ⎤ m = = = ⎢ ⎥ ⎣ ⎦ d u ⎣ ⎦ x x ˆ ( ) − Δ m uu − dd ( ) Ŝ = Σ x ( ū u- ƌ d) = ∑ ⎡ ⎤ m ud uu + dd = S 0 − Δ m S ⎣ ⎦ x Advantage: - Expand the functional integral in powers of Δ m factorised out 0 + Δ m O ˆ 0 1 + Δ m ˆ 0 + Δ m ˆ ( ) S − S S − S S O D φ O e D φ O e ∫ ∫ 1 st 0 + Δ m O ˆ ! ! 0 S O = = O 0 1 + Δ m ˆ 0 + Δ m ˆ 1 + Δ m ˆ ( ) S − S − S S S 0 D φ e D φ e ∫ ∫ 0 for isospin symmetry - At leading order in Δ m the corrections only appear in the valence quark propagators: (disconnected contractions of ū u and ƌ d vanish due to isospin symmetry) 8

The QED expansion for the quark propagator In the electro-quenched ( qQED ) approximation: � 9

Results for the light quark contribution to a HVP a µ

HVP from LQCD q µ q ( ) ( ) = δ µ ν Q 2 − Q µ Q ν ( ) = ( ) J ν 0 ∫ ⎡ ⎤ Π µ ν Q d 4 x e iQ ⋅ x ⎦ Π Q 2 J µ x ⎣ 20000 pheno. ⎦⋅ 10 10 ⎛ ⎞ ∞ Q 2 ( ) − Π 0 HVP = 4 α em 1 ⎤ 15000 ( ) ∫ ( ) ⎡ ⎤ Q 2 ! m µ ⎟ Π Q 2 ) − Π 0 2 4 2 dQ 2 a µ 2 f ⎜ ⎣ ⎦ 2 ⎝ ⎠ m µ m µ ⎟ Π Q 2 10000 0 ( ⎡ ⎣ B. E. Lautrup and E. de Rafael, 1969; T. Blum, 2002 ⎞ ⎠ m µ Q 2 2 5000 ⎛ ⎜ ⎝ f 0 Time-Momentum Representation 0.0001 0.001 0.01 0.1 1 Q 2 [GeV 2 ] ∞ HVP = 4 α em ( ) ( ) ∫ ! t 2 F. Jegerlehner, “alphaQEDc17” a µ dt f V t d ! ! ( ) ≡ 1 ∑ ( ) J i 0 ( ) ∫ 0 V t x J i x , t D. Bernecker and H. B. Meyer, 2011 3 i = 1, 2, 3 ⎧ ⎫ ∞ ⎪ ⎪ T data f ( ) G V t ≤ ∑ ∑ ∑ T data < T/2 (avoid bw signals) ( ) V f t ( ) + HVP = 2 ! t ! t f e − M V 4 α em f t ⎨ ⎬ a µ f f ⎪ ⎪ 2 M V ⎩ ⎭ t > T data > t min (ground-state dom.) f = u , d , s , c t = 0 t = T data + a quark-connected analytic representation lattice data terms only up to 10% for light quarks 11 local vector currents

Details of the lattice simulation We have used the gauge field configurations generated by ETMC, European Twisted Mass Collaboration, in the pure isosymmetric QCD theory with Nf=2+1+1 dynamical quarks - Gluon action: Iwasaki V/a 4 ensemble � aµud aµ � aµ � Ncf aµs M ⇡ MK - Quark action: twisted mass at maximal twist (MeV) (MeV) 403 · 80 A 40 . 40 1 . 90 0 . 0040 0 . 15 0 . 19 100 0 . 02363 317(12) 576(22) (automatically O(a) improved) 323 · 64 A 30 . 32 0 . 0030 150 275(10) 568(22) A 40 . 32 0 . 0040 100 316(12) 578(22) OS for s and c valence quarks A 50 . 32 0 . 0050 150 350(13) 586(22) 243 · 48 A 40 . 24 0 . 0040 150 322(13) 582(23) A 60 . 24 0 . 0060 150 386(15) 599(23) Pion masses in the range 220 - 490 MeV A 80 . 24 0 . 0080 150 442(17) 618(14) A 100 . 24 0 . 0100 150 495(19) 639(24) a ! 0.09 fm 4 volumes @ and M π ! 320 MeV 203 · 48 A 40 . 20 0 . 0040 150 330(13) 586(23) 323 · 64 B 25 . 32 1 . 95 0 . 0025 0 . 135 0 . 170 150 0 . 02094 259 (9) 546(19) M π L ! 3.0 ÷ 5.8 B 35 . 32 0 . 0035 150 302(10) 555(19) B 55 . 32 0 . 0055 150 375(13) 578(20) B 75 . 32 0 . 0075 80 436(15) 599(21) 243 · 48 B 85 . 24 0 . 0085 150 468(16) 613(21) 483 · 96 D 15 . 48 2 . 10 0 . 0015 0 . 1200 0 . 1385 100 0 . 01612 223 (6) 529(14) D 20 . 48 0 . 0020 100 256 (7) 535(14) D 30 . 48 0 . 0030 100 312 (8) 550(14) 12

Light quark contribution 650 0.6 β = 1.90, L/a = 20 β = 1.95, L/a = 24 600 β = 1.90, L/a = 24 β = 1.95, L/a = 32 0.5 β = 1.90, L/a = 32 β = 2.10, L/a = 48 550 β = 1.90, L/a = 40 0.4 10 HVP (ud) * 10 500 a 3 V u , d eff 0.3 450 q µ q 0.2 A80.24 400 µ a aM eff B55.32 0.1 350 D30.48 0.0 300 0 5 10 15 20 25 30 35 t / a t / a t / a 250 ( ) t − M ρ − M π 0.1 0.2 0.3 0.4 0.5 0.6 StN: ∝ e 160 stoch. sources / gauge conf. M π (GeV) G. Parisi, 1984; G. P . Lepage, 1989 preliminary ETMC 14 ( ) = 619.4 12.7 ( ) stat + fit 6.8 ( ) chir 6.2 ( ) FVE 5.4 ( ) disc ⋅ 10 − 10 HVP ud HPQCD 16 a µ CLS/Mainz 17 ( ) ⋅ 10 − 10 = 619.4 16.6 talk by S. Simula BMW 17 @ (g 𝝂 -2) plenary workshop, Mainz 2018 RBC/UKQCD 18 quark-connected ETMC 18 ( ) = 53.1 2.5 ( ) ⋅ 10 − 10 ( ) = 14.75 56 ( ) ⋅ 10 − 10 HVP s HVP c terms only a µ a µ 450 500 550 600 650 700 DG et al. , 2017 HVP (ud) * 10 10 13 a µ

LIB corrections ) h h HVP = δ a µ ( ) + δ a µ ( ) HVP QCD HVP QED quark-connected δ a µ terms only photon zero-mode: QED L M. Hayakawa and S. Uno, 2008 ∞ ∞ ∑ ∑ ∑ ( ) = 4 α em ( ) δ V QCD t ( ) ( ) = 4 α em ( ) ( ) HVP QCD ! t HVP QED ! t QED t δ a µ δ V f δ a µ 2 2 f f t = 0 f = u , d , s , c t = 0 RM123 method G. M. de Divitiis et al. , 2012; 2013 ( ) = m f ( ) qQED approximation 0 MS ,2 GeV QCD/QED separation is m f MS ,2 GeV scheme and scale dependent J. Gasser et al. , 2003 ∞ ∑ ( ) = 4 α em ( ) δ V QCD t ( ) HVP QCD ! t δ a µ 2 D 20.48 f 400 M π ! 260 MeV a ! 0.06 fm t = 0 10 { ! } QCD *10 ( ) q d 200 ) Z P ( ∑ ∑ ( ) † x m d − m u ⎡ 2 ψ d ψ d − q u 2 ψ u ψ u ⎤ 0 T J i , t ⎦ J i 0 0 ⎣ ! 3 i = 1,2,3 x , y 3 δ V 0 a ( ) = 2.38 18 [ ] MS ,2 GeV ( ) MeV m d − m u -200 DG et al. , 2017 0 10 20 30 40 50 14 t / a tad

LIB corrections ) h h ∞ ∑ ∑ ( ) = 4 α em ( ) ( ) HVP QED ! t QED t δ a µ δ V f 2 f t = 0 f = u , d , s , c ( ) + δ V self t ( ) + δ V tad t ( ) + δ V PS t ( ) + δ V S t ( ) + δ V Z A t ( ) δ V exch t UV and IR finite e.m. shift of the critical mass G. M. de Divitiis et al. , 2013 u -, d -quark contributions Mtm-LQCD setup + local J i (x) with opposite 600 Wilson r -parameters D 20.48 M π ! 260 MeV a ! 0.06 fm 400 10 QED *10 ( ) + O α em ( ) ( ) 1 − 2.51406 α em q f 2 Z A Z A = Z A 0 fact 2 3 δ V 200 perturbative estimate at LO a G. Martinelli and Y.-C. Zhang, 1982 ( ) = − 2.51406 α em q f ( ) 2 Z A fact V t δ V Z A t 0 preliminary fact = 0.95 5 10 20 30 40 ( ) ( ) t / a Z A O α em α s n RI-MOM 15 pol

LIB corrections ) h h ⎧ ⎫ ⎡ ⎤ ∞ ⎪ ⎪ T data f HVP = 4 α em G V ∑ ∑ ( ) δ V t ( ) + ( ) δ 2 ! t ! t f e − M V δ a µ f t ⎨ ⎬ f f ⎢ ⎥ ⎪ ⎪ ⎣ 2 M V ⎦ ⎩ ⎭ t = 0 t = T data + a lattice data analytic repr. RM123 method ( ) G. M. de Divitiis et al. , 2012; 2013 δ V t δ G V − δ M V ( ) ⎯ ⎯ → 1 + M V t ( ) t ≫ a V t G V M V u -, d -quark contributions δ V QCD δ V QED V V D 20.48 D 20.48 M π ! 260 MeV a ! 0.06 fm M π ! 260 MeV a ! 0.06 fm t / a t / a 16