Hadronic light-by-light contribution to the muon anomalous magnetic - PowerPoint PPT Presentation

Hadronic light-by-light contribution to the muon anomalous magnetic moment from lattice QCD Tom Blum(UCONN/RBRC), Norman Christ (Columbia), Masashi Hayakawa (Nagoya), Taku Izubuchi (BNL/RBRC), Luchang Jin (UConn/RBRC), Chulwoo Jung (BNL), Christoph Lehner (BNL), (RBC and UKQCD Collaborations) Mainz Institute of Theoretical Physics June 18, 2018 1 / 31

Outline I 1 Hadronic light-by-light (HLbL) scattering contribution 2 Summary 3 References 2 / 31

Point source method in QCD+pQED (L. Jin) [Blum et al., 2016] The desired amplitude + + · · · is obtained from a Euclidean space lattice calculation e � i ~ q e E q / 2 ( t snk � t src ) X 2 · ( ~ x snk + ~ x src ) e i ~ q · ~ x op M ⌫ ( x snk , x op , x src ) , M ⌫ ( ~ q ) = lim t src !�1 x snk , ~ ~ x src t snk !1 where ⌦ ↵ � e M ⌫ ( x src , x op , x snk ) = µ ( x snk ) J ⌫ ( x op ) µ ( x src ) X X F ⌫ ( x , y , z , x 0 , y 0 , z 0 , x op , x snk , x src ) . = � e x , y , z x 0 , y 0 , z 0 and q + + m µ q � + m µ ⇣ � i / F 1 ( q 2 ) � ⌫ + i F 2 ( q 2 ) ⌘⇣ � i / ⌘� ⌘⇣ ⇣ ⌘ [ � ⌫ , � ⇢ ] q ⇢ = M ⌫ ( ~ q ) ↵� , 2 E q / 2 4 m 2 E q / 2 ↵� 3 / 31

Point source method in QCD+pQED (L. Jin) [Blum et al., 2016] x op , ν y, σ x, ρ z, κ x src x snk y � , σ � z � , κ � x � , ρ � F C q ; x , y , z , x op ) = ( � ie ) 6 G ⇢ , � ,  ( ~ q ; x , y , z ) H C ⌫ ( ~ ⇢ , � ,  , ⌫ ( x , y , z , x op ) i 4 H C ⇢ , � ,  , ⌫ ( x , y , z , x op ) ( e q / e ) 4 X = ⌦ tr ⇥ � i � ⇢ S q ( x , z ) i �  S q ( z , y ) i � � S q ( y , x op ) i � ⌫ S q ( x op , x ) ⇤↵ QCD + 5 permutations 6 q = u , d , s i 3 G ⇢ , � ,  ( ~ q ; x , y , z ) p m 2 + ~ q 2 / 4( t snk � t src ) X = G ⇢ , ⇢ 0 ( x , x 0 ) G � , � 0 ( y , y 0 ) G  ,  0 ( z , z 0 ) e x 0 , y 0 , z 0 X e � i ~ q / 2 · ( ~ x snk + ~ x src ) S � x snk , x 0 � i � ⇢ 0 S ( x 0 , z 0 ) i �  0 S ( z 0 , y 0 ) i � � 0 S � y 0 , x src � ⇥ + 5 permutations ~ x snk , ~ x src 4 / 31

Point source method in QCD+pQED (L. Jin) [Blum et al., 2016] Do all sums in the QED part exactly (using FFT’s), QCD part done stochastically Key idea: contribution exponentially suppressed with r = | x � y | , so importance sample, concentrate on r < ⇠ � compton ⇡ space-time translational invariance allows coordinates relative to the hadronic loop 8 9 ✓ q , r 2 , � r ◆ < = X X e i ~ q · ~ x op M ⌫ ( ~ q ) = F ⌫ ~ 2 , z , x op r : z , x op ; where r = x � y , z ! z � w , x op ! x op � w and w = ( x + y ) / 2 We sum all the internal points over the entire space-time except we fix x + y = 0. ( x , y ) pairs stochastically sampled, z and x op sums exact 5 / 31

Point source method in QCD+pQED (L. Jin) [Blum et al., 2016] F 1 ( q 2 ) � ⌫ + i F 2 ( q 2 ) ✓ ◆ p 0 ) | J ⌫ (0) | µ ( ~ p 0 ) h µ ( ~ p ) i = � e ¯ u ( ~ [ � ⌫ , � ⇢ ] q ⇢ u ( ~ p ) 4 m implies F 2 (0) only accessible by extrapolation q ! 0. Form is due to Ward Identity, or charge conservation need WI to be exact on each config, or error blows up as ~ q ! 0 To enforce WI compute average of diagrams with all possible insertions of J ⌫ ( x op ) x op , ν y, σ x, ρ y, σ x, ρ y, σ x, ρ z, κ z, κ z, κ x op , ν x op , ν x src y � , σ � x � , ρ � x snk x src y � , σ � x � , ρ � x snk x src y � , σ � x � , ρ � x snk z � , κ � z � , κ � z � , κ � 6 / 31

Point source method in QCD+pQED (L. Jin) [Blum et al., 2016] x op , ν y, σ x, ρ y, σ x, ρ y, σ x, ρ z, κ z, κ z, κ x op , ν x op , ν x src y � , σ � x � , ρ � x snk x src y � , σ � x � , ρ � x snk x src y � , σ � x � , ρ � x snk z � , κ � z � , κ � z � , κ � WI allows a moment method that projects directly to q = 0 q , r 2 , � r ⇣ ⌘� x op � 1 X F C e i ~ q · ~ � M ⌫ ( ~ q ) = 2 , z , x op ~ ⌫ r , z , x op q , r 2 , � r ⇣ ⌘ X F C ⇡ ~ 2 , z , x op ( i ~ q · ~ x op ) ⌫ r , z , x op @ ⇣ ⌘ X F C M ⌫ ( ~ q ) | ~ = q = 0 , r , � r , z , x op ( x op ) i i ~ q =0 ⌫ @ q i r , z , x op 7 / 31

Point source method in QCD+pQED (L. Jin) [Blum et al., 2016] q ) between positive energy Dirac spinors u ( ~ u ( ~ Sandwich M ⌫ ( ~ 0 , s ), ¯ 0 , s ) ✓ F 2 ( q 2 = 0) i ◆ 0 , s 0 ) @ u ( ~ 0 , s 0 ) u ( ~ 0 , s ) = u ( ~ 0 u ( ~ 2[ � i , � j ] M i ( ~ q ) | ~ 0 , s ) q = ~ 2 m µ @ q j multiply both sides by 1 2 ✏ ijk , sum over i and j , 2 3 ~ F 2 (0) 1 0; x = � r 2 , y = + r Σ F C ⇣ ⌘ u s 0 ( ~ 2 u s ( ~ X 4X u s 0 ( ~ 0) i ~ ~ u s ( ~ ¯ 0) 0) = x op ⇥ ¯ 2 , z , x op 0) 2 ~ 5 m r z , x op where Σ i = 1 4 i ✏ ijk [ � j , � k ]. 8 / 31

Lattice setup Photons: Feynman gauge, QED L [Hayakawa and Uno, 2008] (omit all modes with ~ q = 0) Gluons: Iwasaki (I) gauge action (RG improved, plaquette+rectangle) muons: L s = 1 free domain-wall fermions (DWF) quarks: M¨ obius-DWF 2+1f M¨ obius-DWF, I and I-DSDR physical point QCD ensembles (RBC/UKQCD) [Blum et al., 2014] 48I 64I 24D 32D 32D fine 48D a � 1 (GeV) 1.73 2.36 1.0 1.0 1.38 1.0 a (fm) 0.114 0.084 0.2 0.2 0.14 0.2 L (fm) 5.47 5.38 4.8 6.4 4.6 9.6 48 64 24 24 24 24 L s m ⇡ (MeV) 139 135 140 140 140 140 m µ (MeV) 106 106 106 106 106 106 meas (con,disco) 65,65 43,44 33,32 42,20 8,7 62,0 9 / 31

Continuum and 1 volume limits in QED [Blum et al., 2016] Test method in pure QED QED systematics large, O ( a 4 ), O (1 / L 2 ), but under control 0.4 0.35 0.35 0.3 0.3 0.25 F 2 (0) / ( α / π ) 3 F 2 (0) / ( α / π ) 3 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5 3 3.5 4 a 2 (GeV − 2 ) 1 / ( m µ L ) 2 Limits quite consistent with well known PT result 10 / 31

Physical point cHLbL contribution, 48 3 , 1.73 GeV lattice [Blum et al., 2017a] Measurements on 65 configurations, separated by 20 trajectories ignore strange quark contribution (down by 1/17 plus mass suppressed) exponentially suppressed with distance most of contribution by about 1 fm 0.03 0.12 48I 0.025 0.1 F 2 (0) / ( α / π ) 3 0.02 0.08 0.015 F 2 (0) / ( α / π ) 3 0.01 0.06 0.005 0.04 0 0.02 -0.005 0 5 10 15 20 0 0 5 10 15 20 25 | r | ceiling distance in lattice unit = 11 . 60 ± 0 . 96 ⇥ 10 � 10 a cHLbL µ 11 / 31

Disconnected contributions SU(3) flavor: x op , ν x op , ν x op , ν z, κ z, κ z, κ y, σ y, σ x, ρ y, σ x, ρ x, ρ x src x snk x src z ′ , κ ′ x ′ , ρ ′ x snk y ′ , σ ′ x ′ , ρ ′ z ′ , κ ′ x src y ′ , σ ′ x ′ , ρ ′ x snk y ′ , σ ′ z ′ , κ ′ Leading O ( m s � m u , d ) x op , ν x op , ν x op , ν z, κ z, κ y, σ x, ρ y, σ x, ρ z, κ y, σ x, ρ x src x snk x src z ′ , κ ′ x ′ , ρ ′ x snk y ′ , σ ′ x ′ , ρ ′ z ′ , κ ′ y ′ , σ ′ x src z ′ , κ ′ x ′ , ρ ′ x snk y ′ , σ ′ O ( m s � m u , d ) 2 and higher Gluons within and connecting quark loops have not been drawn To ensure loops are connected by gluons, explicit “vacuum” subtraction is required 12 / 31

Leading disconnected contribution x op , ν z, κ y, σ x, ρ x src x snk z ′ , κ ′ x ′ , ρ ′ y ′ , σ ′ We use two point sources at y and z , chosen randomly. The points sinks x op and x are summed over exactly on lattice. Only point source quark propagators are needed. We compute M point source propagators and all M 2 combinations are used to perform the stochastic sum over r = z � y ( M 2 trick). F D ( � ie ) 6 G ⇢ , � ,  ( x , y , z ) H D ⌫ ( x , y , z , x op ) = ⇢ , � ,  , ⌫ ( x , y , z , x op ) ⌧ 1 ⇤� H D Π ⇢ , � ( x , y ) � Π avg ⇢ , � ,  , ⌫ ( x , y , z , x op ) = 2 Π ⌫ ,  ( x op , z ) ⇥ ⇢ , � ( x � y ) QCD ( e q / e ) 2 Tr[ � ⇢ S q ( x , y ) � � S q ( y , x )] . X Π ⇢ , � ( x , y ) = � q 13 / 31

Leading disconnected contribution x op , ν z, κ y, σ x, ρ x src x snk z ′ , κ ′ y ′ , σ ′ x ′ , ρ ′ F dHLbL (0) ( � s 0 , s ) i 1 2 X X u s 0 ( ~ 0) F D k ( x , y = r , z = 0 , x op ) u s ( ~ = 2 ✏ i , j , k ( x op ) j · i ¯ 0) 2 m r , x x op ⌧ 1 ⇤� H D Π ⇢ , � ( x , y ) � Π avg ⇥ ⇢ , � ,  , ⌫ ( x , y , z , x op ) = 2 Π ⌫ ,  ( x op , z ) ⇢ , � ( x � y ) QCD 1 1 X X 2 ✏ i , j , k ( x op ) j h Π ⇢ , � ( x op , 0) i QCD = 2 ✏ i , j , k ( � x op ) j h Π ⇢ , � ( � x op , 0) i QCD = 0 x op x op Because of parity, the expectation value for the (moment of) left loop averages to zero. Π ⇢ , � ( x , y ) � Π avg is only a noise reduction technique. Π avg ⇥ ⇤ ⇢ , � ( x � y ) ⇢ , � ( x � y ) should remain constant through out the entire calculation. 14 / 31

Physical point dHLbL contribution [Blum et al., 2017a] Use AMA with 2000 low-modes of the Dirac operator and randomly choose 256 “spheres” of radius 6 lattice units Uniformly sample 4 (unique) points in each do half as many strange quark props Construct (1024 + 512) 2 point-pairs per configuration 15 / 31