Direct calculation of hadronic light-by-light scatering Jeremy Green - PowerPoint PPT Presentation

Direct calculation of hadronic light-by-light scatering Jeremy Green Nils Asmussen, Oleksii Gryniuk, Georg von Hippel, Harvey Meyer, Andreas Nyffeler, Vladimir Pascalutsa Institut für Kernphysik, Johannes Gutenberg-Universität Mainz The 33rd International Symposium on Latice Field Theory July 14–18, 2015

Outline 1. Introduction 2. Latice four-point function 3. Light-by-light scatering amplitude 4. Strategy for g − 2 5. Summary and outlook Some of these results were posted in arXiv:1507.01577 Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 2 / 18

The muon g − 2 One of the most precise tests of the Standard Model  116592080 ( 63 ) × 10 − 11  � g − 2 experiment �  a µ ≡ µ =  116591790 ( 65 ) × 10 − 11 2 theory  δ a µ = ( 290 ± 90 ) × 10 − 11 , a 3 σ deviation ◮ Fermilab 989 has goal to reduce experimental error by factor of 4 ◮ Leading theory errors come from: Hadronic vacuum Hadronic light-by-light polarization, which can (HLbL) scatering, which be improved using is not easily obtained e + e − → hadrons from experiments experiments Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 3 / 18

Light-by-light scatering Before computing a HLbL , start by studying light-by-light scatering by itself. µ This has much more information than just a HLbL . We can: µ ◮ Compare against phenomenology. ◮ Test models used to compute a HLbL . µ Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 4 / 18

Latice four-point function Directly compute four-point function of vector currents ◮ Use one local current Z V J l µ at the source point. ◮ Use three conserved currents J c µ . In position space: Π pos � Z V J l µ 3 ( 0 ) [ J c µ 1 ( x 1 ) J c µ 2 ( x 2 ) J c µ 1 µ 2 µ 3 µ 4 ( x 1 , x 2 , 0 , x 4 ) = µ 4 ( x 4 ) + δ µ 1 µ 2 δ x 1 x 2 T µ 1 ( x 1 ) J c µ 4 ( x 4 ) + δ µ 1 µ 4 δ x 1 x 4 T µ 4 ( x 4 ) J c µ 2 ( x 2 ) + δ µ 2 µ 4 δ x 2 x 4 T µ 4 ( x 4 ) J c µ 1 ( x 1 ) + δ µ 1 µ 4 δ µ 2 µ 4 δ x 1 x 4 δ x 2 x 4 J c � µ 4 ( x 4 ) ] , where T µ ( x ) is a “tadpole” contact operator. This satisfies the conserved-current relations, µ 1 Π pos µ 2 Π pos µ 4 Π pos ∆ x 1 µ 1 µ 2 µ 3 µ 4 = ∆ x 2 µ 1 µ 2 µ 3 µ 4 = ∆ x 4 µ 1 µ 2 µ 3 µ 4 = 0 . Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 5 / 18

Qark contractions Compute only the fully-connected contractions, with fixed kernels summed over x 1 and x 2 : Π pos ′ � f 1 ( x 1 ) f 2 ( x 2 )Π pos µ 1 µ 2 µ 3 µ 4 ( x 4 ; f 1 , f 2 ) = µ 1 µ 2 µ 3 µ 4 ( x 1 , x 2 , 0 , x 4 ) x 1 , x 2 X X 2 X X 2 X X 2 1 1 1 Generically, need the following propagators: ◮ 1 point-source propagator from x 3 = 0 0 X 4 0 X 4 0 X 4 ◮ 8 sequential propagators through x 1 , for each µ 1 and f 1 or f ∗ 1 ◮ 8 sequential propagators through x 2 ◮ 32 double-sequential propagators through x 1 and x 2 , for each ( µ 1 , µ 2 ) and ( f 1 , f 2 ) or ( f ∗ 1 , f ∗ 2 ) Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 6 / 18

Kinematical setup Obtain momentum-space Euclidean four-point function using plane waves: µ 1 µ 2 µ 3 µ 4 ( x 4 ; f 1 , f 2 ) � e − ip 4 · x 4 Π pos ′ � � Π E µ 1 µ 2 µ 3 µ 4 ( p 4 ; p 1 , p 2 ) = � f a ( x ) = e − ipa · x . � x 4 Thus, we can efficiently fix p 1 , 2 and choose arbitrary p 4 . ◮ Full 4-point tensor is very complicated: it can be decomposed into 41 scalar functions of 6 kinematic invariants. ◮ Forward case is simpler: Q 1 ≡ p 2 = − p 1 , Q 2 ≡ p 4 . Then there are 8 scalar functions that depend on 3 kinematic invariants. Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 7 / 18

Latice ensembles Use CLS ensembles: N f = 2 O ( a ) -improved Wilson, with a = 0 . 063 fm. 1. m π = 451 MeV, 64 × 32 3 2. m π = 324 MeV, 96 × 48 3 3. m π = 277 MeV, 96 × 48 3 Keep only u and d quarks in the electromagnetic current, i.e., µ = 2 u γ µ u − 1 ¯ J l 3 ¯ d γ µ d . 3 Study forward case with a few different Q 1 and also more general kinematics. Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 8 / 18

Forward LbL amplitude Take the amplitude for forward scatering of transversely polarized virtual photons, 2 , ν ) = e 4 M TT ( − Q 2 1 , − Q 2 4 R µ 1 µ 2 R µ 3 µ 4 Π E µ 1 µ 2 µ 3 µ 4 ( − Q 2 ; − Q 1 , Q 1 ) , where ν = − Q 1 · Q 2 and R µν projects onto the plane orthogonal to Q 1 , Q 2 . A subtracted dispersion relation at fixed spacelike Q 2 1 , Q 2 2 relates this to the γ ∗ γ ∗ → hadrons cross sections σ 0 , 2 : � ν ′ 2 − q 2 1 q 2 � ∞ 2 , 0 ) = 2 ν 2 2 M TT ( q 2 1 , q 2 2 , ν ) −M TT ( q 2 1 , q 2 d ν ′ ν ′ ( ν ′ 2 − ν 2 − i ϵ ) [ σ 0 ( ν ′ ) + σ 2 ( ν ′ ) ] π ν 0 This is model-independent and will allow for systematically improvable comparisons between latice and experiment. Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 9 / 18

Model for σ ( γ ∗ γ ∗ → hadrons ) (V. Pascalutsa, V. Pauk, M. Vanderhaeghen, Phys. Rev. D 85 (2012) 116001) Include single mesons and π + π − final states: σ ( γ ∗ γ ∗ → M ) + σ ( γ ∗ γ ∗ → π + π − ) � σ 0 + σ 2 = M σ ( γ ∗ γ ∗ → M ) depends on the meson’s: Mesons: ◮ mass m and width Γ ◮ pseudoscalar ( π 0 , η ′ ) ◮ two-photon decay width Γ γγ ◮ scalar ( a 0 , f 0 ) ◮ two-photon transition form factor ◮ axial vector ( f 1 ) F ( q 2 1 , q 2 2 ) ◮ tensor ( a 2 , f 2 ) assume F ( q 2 1 , q 2 2 ) = F ( q 2 1 , 0 ) F ( 0 , q 2 2 ) / F ( 0 , 0 ) Use scalar QED dressed with form factors for σ ( γ ∗ γ ∗ → π + π − ) . Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 10 / 18

Aside: π 0 contribution Leading HLbL contributions to muon g − 2 are expected to come from π 0 exchange diagrams, which dominate at long distances. + + Their contribution to the four-point function: Π E , π 0 µ 1 µ 2 µ 3 µ 4 ( p 4 ; p 1 , p 2 )  F 12 ϵ µ 1 µ 2 αβ F 34 ϵ µ 3 µ 4 στ + F 13 ϵ µ 1 µ 3 ασ F 24 ϵ µ 2 µ 4 βτ  = − p 1 α p 2 β p 3 σ p 4 τ  ( p 1 + p 2 ) 2 + m 2 ( p 1 + p 3 ) 2 + m 2 π π  + F 14 ϵ µ 1 µ 4 ατ F 23 ϵ µ 2 µ 3 βσ   , ( p 2 + p 3 ) 2 + m 2 π where p 3 = − ( p 1 + p 2 + p 4 ) and F ij = F ( p 2 i , p 2 j ) . This is consistent with the dispersion relation using σ ( γ ∗ γ ∗ → π 0 ) . Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 11 / 18

M TT : dependence on ν and Q 2 2 m π = 324 MeV, Q 2 1 = 0 . 377 GeV 2 × 10 − 5 10 1 . 0 ν (GeV 2 ) 2 , 0) For scalar, tensor mesons there 8 0 . 5 1 , − Q 2 is no data from expt; we use 2 , ν ) − M TT ( − Q 2 0 . 0 1 6 F ( q 2 , 0 ) = F ( 0 , q 2 ) = 1 − q 2 / Λ 2 4 1 , − Q 2 with Λ set by hand to 1.6 GeV M TT ( − Q 2 Changing Λ by ± 0 . 4 GeV 2 adjusts curves by up to ± 50 % . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 Q 2 2 (GeV 2 ) Points: latice data. Curves: dispersion relation + model for cross section. Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 12 / 18

M TT : dependence on ν and m π Q 2 1 = Q 2 2 = 0 . 377 GeV 2 × 10 − 5 Points: latice data. 10 2 , 0) Curves: dispersion relation + m π (MeV) 1 , − Q 2 277 model for cross section. 8 324 2 , ν ) − M TT ( − Q 2 In increasing order: 451 6 ◮ π 0 ◮ π 0 + η ′ 4 1 , − Q 2 ◮ full model M TT ( − Q 2 2 ◮ full model + high-energy σ ( γγ → hadrons ) at 0 0 . 00 0 . 05 0 . 10 0 . 15 0 . 20 0 . 25 0 . 30 0 . 35 0 . 40 physical m π ν (GeV 2 ) Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 13 / 18

General kinematics case To study off-forward kinematics, we fix p 2 1 = p 2 2 = ( p 1 + p 2 ) 2 = 0 . 33 GeV 2 and consider contractions of Π E µ 1 µ 2 µ 3 µ 4 ( p 4 ; p 1 , p 2 ) with two different tensors: 1. δ µ 1 µ 2 δ µ 3 µ 4 yields π 0 contribution � ( p 1 · p 2 )( p 3 · p 4 ) − ( p 1 · p 4 )( p 2 · p 3 ) F ( p 2 1 , p 2 3 ) F ( p 2 2 , p 2 − 2 4 ) ( p 1 + p 3 ) 2 + m 2 π � + ( p 1 · p 2 )( p 3 · p 4 ) − ( p 1 · p 3 )( p 2 · p 4 ) F ( p 2 1 , p 2 4 ) F ( p 2 2 , p 2 3 ) , ( p 2 + p 3 ) 2 + m 2 π where F ( 0 , 0 ) = − 1 / ( 4 π 2 F π ) (Wess-Zumino-Witen) and we use vector meson dominance for dependence on p 2 . 2. δ µ 1 µ 2 δ µ 3 µ 4 + δ µ 1 µ 3 δ µ 2 µ 4 + δ µ 1 µ 4 δ µ 2 µ 3 , which is totally symmetric and thus has no π 0 contribution. We also fix p 2 3 = p 2 4 to two different values and plot versus the one remaining kinematic variable. Jeremy Green (Mainz) Direct calculation of hadronic light-by-light scatering Latice 2015 14 / 18