Hadronic light-by-light scatering from latice QCD Jeremy Green in collaboration with Nils Asmussen, Oleksii Gryniuk, Georg von Hippel, Harvey Meyer, Andreas Nyffeler, Vladimir Pascalutsa, Hartmut Witig Institut für Kernphysik, Johannes Gutenberg-Universität Mainz Humboldt Universität – NIC, DESY Zeuthen Join Latice Seminar June 27, 2016
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 published in Phys. Rev. Let. 115 , 222003 (2015) [1507.01577] PoS(Latice 2015)109 [1510.08384] Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 2 / 30
Precision low-energy physics ◮ Direct way to search for new physics: try to create new particles at high energy in a collider. E.g., Higgs boson at LHC. ◮ Indirect way: measure a low-energy observable very precisely, and look for small deviations from a similarly precise Standard Model prediction. This can impose strong constraints on new physics that are complementary to direct searches. Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 3 / 30
Magnetic moment of the electron In an external magnetic field, a particle will have potential energy µ · � U = − � B , where � µ is its magnetic moment. An electron’s magnetic moment is given by − e � � µ = g e S . 2 m e A classical rotating charged body has gyromagnetic ratio g = 1, whereas the Dirac equation predicts g = 2. QED and the Standard Model produce deviations from g = 2: the anomalous magnetic moment , a e ≡ g e − 2 + . . . = 2 = α 2 π + . . . Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 4 / 30
Anomalous magnetic moment of the electron T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, Phys. Rev. Let. 109 , 111807 (2012) � n � α � a ( 2 n ) + a ( had ) + a ( EW ) a SM = e e e e π n � 2 � 3 � 4 = α � α � α � α 2 π − ( 0 . 328 . . . ) + ( 1 . 181 . . . ) − ( 1 . 910 . . . ) π π π � 5 � α + 1 . 68 ( 2 ) × 10 − 12 + 0 . 0297 ( 5 ) × 10 − 12 + ( 9 . 16 . . . ) π Using α from atomic physics experiments yields = 1 159 652 181 . 78 ( 6 )( 4 )( 2 )( 77 ) × 10 − 12 [0.67 ppb] , a SM e where the dominant uncertainty comes from α . This agrees well with the experimental value, = 1 159 652 180 . 73 ( 28 ) × 10 − 12 [0.24 ppb] . a expt e D. Hanneke, S. Fogwell, G. Gabrielse, Phys. Rev. Let. 100 , 120801 (2008) This can be reversed: use the experiment as a determination of α , which results in the current most precise value of the fine structure constant. Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 5 / 30
Sensitivity to heavy particles ◮ The Standard Model prediction for a e is dominated by QED diagrams containing only photons and electrons. ◮ Loops with heavier particles ( µ , τ , hadrons) contribute only ∼ 5 × 10 − 12 . ◮ Generically the contribution from a heavy particle with mass M is suppressed by ( m e M ) 2 . ◮ In the muon g − 2, contributions from heavy particles will therefore be enhanced by ( m µ m e ) 2 ≈ 40 000, relative to their contributions to a e . Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 6 / 30
Muon g − 2: experiment BNL E821: G. W. Bennet et al. (Muon g − 2 Collaboration), Phys. Rev. D 73 , 072003 (2006) ◮ Polarized muons injected into a 14 m diameter storage ring. ◮ The muon’s spin precesses relative to its � � velocity: � � ω a = e 1 a µ � v × � � � B − a µ − E . γ 2 − 1 m µ ◮ “Magic” muon energy E ≈ 3 . 098 GeV used to eliminate the dependence on � E . a µ = 116 592 089 ( 63 ) × 10 − 11 ◮ The electron produced in muon decay [0.54 ppm] has direction correlated with the muon spin. BNL storage ring moved to Fermilab for new muon g − 2 experiment: E989. Goal: reduce uncertainty by a factor of four. New experiment using ultra-cold muons also planned by J-PARC E34. Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 7 / 30
Muon g − 2: theory T. Blum et al. (US Physics “Snowmass” Self Study), 1311.2198 F. Jegerlehner, A. Nyffeler, Phys. Rept. 477 (2009) 1–110 QED O ( α ) 116 140 973 . 32 ( 8 ) = 0 . 5 ( α / π ) QED O ( α 2 ) 413 217 . 63 ( 1 ) = 0 . 765 857 425 ( 17 )( α / π ) 2 QED O ( α 3 ) 30 141 . 90 ( 0 ) = 24 . 050 509 96 ( 32 )( α / π ) 3 QED O ( α 4 ) 381 . 01 ( 2 ) = 130 . 8796 ( 63 )( α / π ) 4 QED O ( α 5 ) 5 . 09 ( 1 ) = 753 . 29 ( 1 . 04 )( α / π ) 5 QED combined 116 584 718 . 95 ( 8 ) T. Aoyama et al. , Phys. Rev. Let. 109 , 111808 (2012) Electroweak 154 ( 1 ) J. P. Miller et al. , Ann. Rev. Nucl. Part. Sci. 62 (2012) 237–264 HVP (LO) 6949 ( 43 ) K. Hagiwara et al. , J. Phys. G 38 (2011) 085003 HVP (HO) − 98 . 4 ( 7 ) HLbL 105 ( 26 ) “Glasgow consensus” or 116(39) × 10 − 11 Standard Model 116 591 828 ( 50 ) Jegerlehner + Nyffeler × 10 − 11 ∆ a µ ( expt − SM ) 261 ( 78 ) Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 8 / 30
Muon g − 2: theory uncertainty The two dominant sources of uncertainty are hadronic effects: Hadronic vacuum polarization: a HVP,LO = 6949 ( 43 ) × 10 − 11 . µ ◮ Determined using experimental data on cross section for e + e − → hadrons. ◮ Very active field for latice QCD calculations working toward an ab initio prediction with competitive uncertainty. Hadronic light-by-light scatering: a HLbL = 105 ( 26 ) or 116 ( 39 ) × 10 − 11 . µ ◮ Determined using models that include meson exchange terms, charged meson loops, etc. ◮ Could benefit significantly with reliable input from the latice. Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 9 / 30
π 0 contribution to HLbL scatering comes from π 0 exchange diagrams, About 2 / 3 of theory prediction for a HLbL µ which dominate at long distances. Large contributions also come from η , η ′ . + + 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 ) = − p 1 α p 2 β p 3 σ p 4 τ � F 12 ϵ µ 1 µ 2 α β 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 + 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 j ) is the π 0 γ ∗ γ ∗ form factor. where p 3 = − ( p 1 + p 2 + p 4 ) and F ij = F ( p 2 i , p 2 Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 10 / 30
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) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 11 / 30
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) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 12 / 30
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) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 13 / 30
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) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 14 / 30
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