Hadronic light-by-light scattering in the muon anomalous magnetic moment on the lattice Nils Asmussen in Collaboration with Antoine G´ erardin, Harvey Meyer, Andreas Nyffeler University of Southampton 30 October 2018 Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 1 / 29
Gyromagnetic Moment (History) gyromagnetic moment: µ = g e 2 m S 1924 1928 Stern-Gerlach experiment Dirac theory: observed µ g = 2 1948 1947 g = 2 × (1 + α 2 π ) g ≈ 2 × (1 + 0 . 00118(3)) ≈ 2 × (1 + 0 . 001161) Foley and Kush Schwinger Where are we today? Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 2 / 29
Hadronic Light-by-Light Contribution anomalous magnetic moment a µ = g µ − 2 2 a µ [10 − 10 ] contribution reference QED 11 658 471 . 895 ± 0 . 008 Aoyama et al ’12 HVP LO 693 . 1 ± 3 . 4 Davier et al ’17 HVP NLO − 9 . 84 ± 0 . 07 Hagiwara et al ’11 HVP NNLO 1 . 24 ± 0 . 01 Kurz et al ’14 HLBL LO 10 . 5 ± 2 . 6 Prades et al ’09 HLBL NLO 0 . 3 ± 0 . 2 Colangelo et al ’14 EW 15 . 36 ± 0 . 10 Gnendiger et al ’13 total 11 659 182 . 3 ± 4 . 3 Davier et al ’17 experimental 11 659 208 . 9 ± 6 . 3 Bennett et al ’06 Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 3 / 29
Anomalous Magnetic Moment of the Muon ≈ 3 to 4 standard deviations discrepancy between a exp and a theo µ µ → new physics? Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 4 / 29
Anomalous Magnetic Moment of the Muon ≈ 3 to 4 standard deviations discrepancy between a exp and a theo µ µ → new physics? reduce uncertainties experiment theory for HLbL J-PARC phenomenology lattice QCD Fermilab reduce model uncertainties model independent estimates for dominant contribution Blum et al. (’05,. . . )’15,. . . ,’17 ( π 0 , η , η ′ ; ππ ) Mainz lattice group using experimental input Colangelo et al. ’14,. . . ,’17 Pauk and Vanderhaeghen ’14 Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 4 / 29
Euclidean position-space approach to a HLbL µ z y 0 x master formula � � � � = me 6 d 4 x ¯ i � a HLbL d 4 y L [ ρ,σ ]; µνλ ( x , y ) Π ρ ; µνλσ ( x , y ) . µ 3 � �� � � �� � QCD QED � � � i � d 4 z z ρ Π ρ ; µνλσ ( x , y ) = − j µ ( x ) j ν ( y ) j σ ( z ) j λ (0) . ¯ L [ ρ,σ ]; µνλ ( x , y ) computed in the continuum & infinite-volume no power-law finite-volume effects from the photons manifest Lorentz covariance Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 5 / 29
Outline Tests of the QED Kernel 1 Tests of the Lattice Gauge Theory Code 2 Lattice QCD 3 Conclusion 4 Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 6 / 29
Stages of the Computation tests of the QED kernel continuum and infinite volume π 0 pole and lepton loop test different choices for the QED kernel tests of the lattice gauge theory code Lattice QED compare to lepton loop results Lattice QCD first results for the fully connected contribution study pion mass dependence and discretisation/finite volume effects Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 7 / 29
Tests of the QED Kernel 1 Tests of the Lattice Gauge Theory Code 2 Lattice QCD 3 Conclusion 4 Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 8 / 29
Tests in Continuum, Infinite Volume master formula � ∞ d | y || y | 3 � � ∞ � π � = me 6 d β sin 2 β ¯ L [ ρ,σ ]; µνλ ( x , y ) i � a HLbL 3 8 π 3 d | x || x | 3 Π ρ ; µνλσ ( x , y ) . µ 0 0 0 � � � i � d 4 z z ρ Π ρ ; µνλσ ( x , y ) = − j µ ( x ) j ν ( y ) j σ ( z ) j λ (0) . � � � d | x | , d | y | and d β evaluated numerically � d 4 z evaluated (semi-)analytically Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 9 / 29
Contribution of the π 0 to a HLbL (Model!) µ m π = 300 MeV m π = 600 MeV 3 ( | y | max ) × 10 10 4 m π = 900 MeV f ( | y | ) × 10 10 fm 2 2 1 a Hlbl m π = 300 MeV µ 0 m π = 600 MeV 0 m π = 900 MeV 0 1 2 3 4 0 1 2 3 4 | y | max / fm | y | / fm dashed line = result from momentum-space integration we reproduce the known result contribution is perhaps surprisingly long-range integrand peaked at short distances Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 10 / 29
Lepton loop integrand contribution to a HLbL µ m l = m µ / 2 8 m l = m µ m l = 2 m µ 6 f ( | y | ) × 10 9 fm we reproduce the known result 4 contribution is long-range 2 integrand sharply peaked at short distances 0 0 2 4 6 8 10 | y | / fm The QED kernel is correct we reproduce the π 0 -pole in VMD model we reproduce the lepton loop Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 11 / 29
What next? achievements challenges in the view of lattice computations method for a HLbL on the lattice µ contributions are quite long range verified the QED kernel integrand peaked at small distances learned about the integrand Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 12 / 29
What next? achievements challenges in the view of lattice computations method for a HLbL on the lattice µ contributions are quite long range verified the QED kernel integrand peaked at small distances learned about the integrand a way to improve do subtractions on the kernel (first proposed by Blum et al. ’17) � � x i ˆ y i ˆ exploit Π( x , y ) = Π( x , y ) = 0 example: L (0) = ¯ L [ ρ,σ ]; µνλ ( x , y ) L (1) = ¯ 2 ¯ 2 ¯ L [ ρ,σ ]; µνλ ( x , y ) − 1 L [ ρ,σ ]; µνλ ( x , x ) − 1 L [ ρ,σ ]; µνλ ( y , y ) = me 6 Π( x , y ) = me 6 a HLbL x , y L (0) ( x , y ) i ˆ x , y L (1) ( x , y ) i ˆ � � Π( x , y ) µ 3 3 Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 12 / 29
Continuum, Infinite Volume master formula � ∞ d | y || y | 3 � � ∞ � π � = me 6 3 8 π 3 d | x || x | 3 d β sin 2 β ¯ L [ ρ,σ ]; µνλ ( x , y ) i � a HLbL Π ρ ; µνλσ ( x , y ) . µ 0 0 0 y integrand lepton loop m l = m µ subtractions on the kernel 0 . 7 we try (short notation): 0 . 6 L (0) = ¯ L (0) L ( x , y ) (standard kernel) L (1) L (1) = ¯ 0 . 5 L ( x , y ) − 1 2 ¯ L ( x , x ) − 1 2 ¯ L ( y , y ) L (2) f ( | y | ) × 10 8 fm L (2) = ¯ 0 . 4 L ( x , y ) − ¯ L (0 , y ) − ¯ L (3) L ( x , 0) L (3) = ¯ L ( x , y ) − ¯ L (0 , y ) − ¯ L ( x , x )+¯ 0 . 3 L (0 , x ) 0 . 2 L (0) (0 , 0) = 0 L (1) ( x , x ) = 0 0 . 1 L (2) (0 , y ) = L (2) ( x , 0) = 0 0 L (3) ( x , x ) = L (3) (0 , y ) = 0 − 0 . 1 0 1 2 3 4 5 | y | fm with all kernels L (0 , 1 , 2 , 3) we can reproduce the known result we expect L (2 , 3) to be advantageous on the Lattice Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 13 / 29
Tests of the QED Kernel 1 Tests of the Lattice Gauge Theory Code 2 Lattice QCD 3 Conclusion 4 Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 14 / 29
Lattice QED Computation master formula a | y | | y | 3 � � 3 2 π 2 � a 4 � = me 6 L [ ρ,σ ]; µνλ ( x , y ) i � ¯ a HLbL Π ρ ; µνλσ ( x , y ) . µ | y | x ∈ Λ � � Π ρ ; µνλσ ( x , y ) = − a 4 � i � j µ ( x ) j ν ( y ) j σ ( z ) j λ (0) . z ρ z ∈ Λ i ˆ Π in Lattice QED goal reproduce known lepton loop result validate Lattice QCD code focus on standard kernel L (0) and subtracted kernel L (2) Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 15 / 29
Lattice QED Computation with Wilson Fermions lattice gauge theory � � O � = 1 ψ ] e − S F [ ψ, ¯ D [ U ] e − S G [ U ] D [ ψ, ¯ ψ, U ] O [ ψ, ¯ ψ, U ] Z � � S G [ U ] = β Re Tr[1 − U µν ( n )] n ∈ Λ µ<ν S F [ ψ, ¯ ψ, U ] = Wilson fermions local vector currents: j l λ ( x ) = ¯ q x γ λ q x conserved vector currents: � � λ ( x ) = 1 λ ( γ λ + 1) U † j c q x +ˆ ¯ λ, x q x + ¯ q x ( γ λ − 1) U λ, x q x +ˆ 2 λ Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 16 / 29
Lattice QED Computation with Wilson Fermions lepton loop − → lattice gauge theory � � O � = 1 ❳❳❳❳❳ ✘ ✘✘✘✘✘ ψ ] e − S F [ ψ, ¯ D [ U ] e − S G [ U ] D [ ψ, ¯ ψ, U ] O [ ψ, ¯ ψ, U ] ❳ Z ✯ 1 � � ✟ Re Tr[1 − ✟✟✟ S G [ U ] = β U µν ( n )] = 0 n ∈ Λ µ<ν S F [ ψ, ¯ ψ, U ] = Wilson fermions QED leading order local vector currents: j l λ ( x ) = ¯ q x γ λ q x conserved vector currents: � � λ ( x ) = 1 λ ( γ λ + 1) U † j c q x +ˆ ¯ λ, x q x + ¯ q x ( γ λ − 1) U λ, x q x +ˆ 2 λ Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 16 / 29
Lattice QED Computation with Wilson Fermions continuum extrapolation lepton loop ( m l = 2 m µ ) L (0) a LbL , ll = 0 . 1659 0 . 1 µ a LbL , cc = 0 . 1596 µ 0 × 10 8 − 0 . 1 − 0 . 2 a LbL µ − 0 . 3 − 0 . 4 L (0) ( x, y ) = L ( x, y ) − 0 . 5 0 0 . 02 0 . 04 0 . 06 0 . 08 0 . 1 0 . 12 am µ dashed line: continuum extrapolation for m µ = 7 . 2 using a quadratic fit solid line: volume extrapolation: curve shifted by the difference between the results for lattice extents m µ L = 7 . 2 and 14 . 4 at fixed a Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 17 / 29
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