Hadronic contributions to д − 2 from latice QCD Jeremy Green NIC, DESY, Zeuthen Second annual symposium Helmholtz Programme “Mater and the Universe” December 12–13, 2016
Outline 1. Brief overview of latice activities at DESY 2. Introduction — muon д − 2 3. Hadronic vacuum polarization 4. Hadronic light-by-light scatering 5. Outlook Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 2
Latice QCD ...is a regularization of Euclidean-space QCD such that the path integral can be done fully non-perturbatively. ◮ Euclidean spacetime becomes a periodic hypercubic latice, with spacing a and box size L 3 s × L t . ◮ Path integral over fermion degrees of freedom is done analytically, for each gauge configuration. Solving the Dirac equation with a fixed source yields a source-to-all quark propagator. ◮ Path integral over gauge degrees of freedom is done numerically using Monte Carlo methods to generate an ensemble of gauge configurations . The a → 0 and L s , L t → ∞ extrapolations need to be taken by using multiple ensembles. Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 3
Latice activities at DESY ◮ Broad research in latice field theory ◮ latice QCD ◮ algorithm and conceptual developments ◮ new approaches (e.g. tensor network techniques) ◮ Multi-level algorithm → reduce exponential signal-to-noise problem ◮ Group plays leading and central role in two large European efforts: ◮ European Twisted Mass Collaboration (ETMC) ◮ Coordinated Latice Simulations (CLS) ∼ M Z ∼ M τ ∼ 200 MeV ALPHA Collaboration: 1-loop 0 2-loop Schrödinger Functional − 0 . 5 Gradient Flow ◮ HQET, B-physics − 1 0 ◮ final determination of strong − 1 . 5 − 0 . 05 coupling constant based on − 2 β ( g ) − 0 . 1 − 2 . 5 3-flavour calculations − 0 . 15 − 3 − 0 . 2 ◮ latice ensembles pushing − 3 . 5 A LPHA − 0 . 25 toward physical quark Collaboration − 4 − 0 . 3 1 1 . 1 1 . 2 1 . 3 1 . 4 1 . 5 1 . 6 masses − 4 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 g Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 4
Latice activities at DESY Phys. Rev. Let. 116 , 252001 (2016), 1601.01624 ETMC: c QCD '13 Pavan '02 This work ◮ Nucleon sigma terms Alarcón '12 s QCDSF-UKQCD '12 ◮ computed on one Hoferichter '15 QCDSF-UKQCD '12 BMWc '16 N f = 2 ensemble with ETMC '14 QCDSF '12 physical m π BMWc '16 QCD '13 ◮ relevant for dark mater QCDSF '12 QCD '15 QCD '15 searches This work This work ◮ most precise results for 0 25 50 75 100 0 25 50 75 100 125 150 σ s and σ c N [MeV] s , c [MeV] ◮ Nucleon structure: ◮ form factors ◮ moments of parton distribution functions ◮ direct calculation of parton distribution functions ◮ Muon anomalous magnetic moment ◮ Future outlook: more-physical calculations ◮ N f = 2 + 1 + 1 with correct π , K and D meson masses ◮ include QED and isospin breaking Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 5
Muon anomalous magnetic moment 2 m µ � e A muon has magnetic moment � S . The Dirac equation predicts µ = д µ д µ = 2 , but quantum effects produce a small deviation, 116 592 089 ( 63 ) × 10 − 11 a µ ≡ д µ − 2 experiment BNL E821, PRD 73 , 072003 (2006) = 116 591 828 ( 50 ) × 10 − 11 2 theory US "Snowmass" Self Study, 1311.2198 ∆ a µ = ( 261 ± 78 ) × 10 − 11 , a 3 σ discrepancy. ◮ New experiments promise to reduce the uncertainty fourfold: ◮ Fermilab E989, using the same storage ring from BNL. ◮ J-PARC E34, using a new method with ultra-cold muons. ◮ The theoretical uncertainty should likewise be reduced. ◮ Hadronic effects are the dominant contributions. Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 6
Muon д − 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 | DESY, Zeuthen | MUTAG 2016 | Page 7
Hadronic vacuum polarization (HVP) on the latice � � � 2 � ∞ � α a HVP,LO dQ 2 f ( Q 2 ) Π( Q 2 ) − Π( 0 ) , = µ π 0 where f ( Q 2 ) is a known kernel and the integrand peaks near Q 2 = ( m µ / 2 ) 2 . Two main strategies: 2. Time-momentum representation 1. Momentum space D. Bernecker and H. B. Meyer, Eur. Phys. J. A 47 , 148 (2011) � 3 � G ( x 0 ) ≡ − 1 d 4 x e iQ · x � J µ ( x ) J ν ( 0 ) � � Π µν ( Q ) ≡ d 3 x � J k ( x 0 , x ) J k ( 0 ) � , 3 k = 1 д µν Q 2 − Q µ Q ν � � Π( Q 2 ) � ∞ = Π( Q 2 ) − Π( 0 ) = dx 0 G ( x 0 ) x 2 0 д ( Qx 0 ) , Cannot directly obtain Π( 0 ) . 0 y 2 sin 2 ( y / 2 ) д ( y ) ≡ 1 − 4 Limited resolution at low Q 2 , where f ( Q 2 ) is peaked, so constrained Challenge is understanding large- x 0 fiting is needed. behaviour of G ( x 0 ) . Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 8
Timelike pion form factor The time-momentum correlator has a spectral representation, � ∞ σ ( e + e − → hadrons ) 1 dω ω 2 ρ ( ω 2 ) e − ω | x 0 | , G ( x 0 ) = ρ ( s ) = . 12 π 2 4 πα ( s ) 2 / ( 3 s ) 0 At low energies, this is given by σ ( e + e − → π + π − ) , which depends on the timelike pion form factor | F π ( √ s ) | 2 . For 2 m π ≤ √ s ≤ 4 m π , this can be computed from finite-volume energy levels and matrix elements. H. B. Meyer, Phys. Rev. Let. 107 , 072002 (2011) By separately computing | F π | 2 and fiting it with a curve, we can replace the discrete low-energy finite-volume spectrum in G ( x 0 ) with a π + π − continuum, and improve the approach to the infinite-volume limit. δ π First step: compute the p -wave ππ scatering phase shif. Exploratory π Preliminary CMF 2 d 2 = 1 : A 1 study with m π = 437 MeV. χ 2 / d . o . f . = 23 . 18 / 12 d 2 = 1 : E 2 d 2 = 2 : A 1 F. Erben, JG, D. Mohler, H. Witig, d 2 = 2 : B 1 d 2 = 2 : B 2 poster at Latice 2016, 1611.06805 d 2 = 3 : A 1 0 0.30 0.35 0.40 0.45 0.50 aE cm Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 9
Hadronic contributions to the muon д − 2 O ( α 2 ) : Leading order hadronic vacuum polarization. O ( α 3 ) : Higher-order Leading-order Hadronic light-by-light. contributions from contribution from O ( α ) leading order HVP. correction to HVP. Included in the phenomenological leading-order HVP. Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 10
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 | DESY, Zeuthen | MUTAG 2016 | Page 11
Latice calculation of the π 0 → γ ∗ γ ∗ form factor A. Gérardin, H. B. Meyer, A. Nyffeler, Phys. Rev. D 94 , 074507 (2016), 1607.08174 In Minkowski space: � 1 q β d 4 x e iq 1 x � 0 | T { J µ ( x ) J ν ( 0 ) }| π 0 ( p ) � = ϵ µνα β q α 2 F π 0 γ ∗ γ ∗ ( q 2 1 , q 2 M µν ( p , q 1 ) = i 2 ) , where p = q 1 + q 2 . In Euclidean space on the latice, compute � � M E dτ e ω 1 τ d 3 z e − i � q 1 � z � 0 | T { J µ ( � z , τ ) J ν ( � µν ≡ − 0 , 0 ) }| π ( p ) � . Different models were fit to the latice data, of which only LMD+V has the correct behaviour at large Q 2 of F ( − Q 2 , 0 ) and F ( − Q 2 , − Q 2 ) . Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 12
Latice calculation of the π 0 → γ ∗ γ ∗ form factor A. Gérardin, H. B. Meyer, A. Nyffeler, Phys. Rev. D 94 , 074507 (2016), 1607.08174 0.12 0.12 0.1 0.1 0.08 0.08 Doubly virtual [GeV] [GeV] 0.06 0.06 (on one ensemble) Q 2 | F πγ ∗ γ ∗ ( − Q 2 , − Q 2 ) | Q 2 | F πγ ∗ γ ∗ ( − Q 2 , − Q 2 ) | 0.04 0.04 0.02 VMD 0.02 LMD+V 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Q 2 [GeV 2 ] Q 2 [GeV 2 ] Q 2 | F πγ ∗ γ ∗ ( − Q 2 , 0) | Preferred LMD+V fit model used to 0.2 estimate the π 0 exchange 0.15 contribution to д − 2 : [GeV] 0.1 a HLbL , π 0 = ( 65 . 0 ± 8 . 3 ) × 10 − 11 , VMD CELLO 0.05 LMD CLEO µ LMD+V BL 0 0 0.5 1 1.5 2 Q 2 [GeV 2 ] which fits well into the range of Singly virtual (extrapolated model calculations, ( 50 − 80 ) × 10 − 11 . to m phys , a = 0 ) π Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 13
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 Some of these results were published in JG, O. Gryniuk, G. von Hippel, H. B. Meyer, V. Pascalutsa, Phys. Rev. Let. 115 , 222003 (2015) [1507.01577] Jeremy Green | DESY, Zeuthen | MUTAG 2016 | Page 14
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