Lattice calculation of hadronic light-by-light scattering contribu

Transcription

1 Lattice calculation of hadronic light-by-light scattering contribution to the muon g-2 Lepton Moments, Cape Cod July 10, 2014 Based on arxiv: , TB, Saumitra Chowdhury, Masashi Hayakawa, and Taku Izubuchi For HVP see Mainz work shop mini-proceedings arxiv: and Lattice 2014 talks

2 Collaborators Work on g-2 done in collaboration with HVP Christopher Aubin (Fordham U) Maarten Golterman (SFSU) Santiago Peris (Barcelona) RBC/UKQCD Collaboration HLbL Saumitra Chowdhury (UConn) Norman Christ (Columbia) Masashi Hayakawa (Nagoya) Taku Izubuchi (BNL/RBRC) Luchang Jin (Columbia) Christoph Lehner (BNL) Norikazu Yamada (KEK)

3 The magnetic moment of the muon Compute corrections to g-2 in pert. theory in α = e2 4π = (leading) Schwinger term = α 2π = hadronic contributions , 10 6 times smaller dominate error, 0.4 ppm (exp 0.54 ppm) QED (9)(19)(7)(77) Aoyama (2012) EW 15.4 (2) Czarnecki (2002) QCD LO HVP (4.2) Davier (2010) (3.72) (2.10) Hagiwara (2011) (4.7) Davier (2010) NLO HVP (9) Hagiwara (2006), Kurz (2014) HLbL 10.5 (2.6) Prades (2009) NNLO HVP 1.24 (1) Kurz (2014)

4 The hadronic light-by-light amplitude Blobs: all possible hadronic states Model estimates: about (10 12) with a 25-40% uncertainty (difficult to quantify) Lattice calculation: model independent, approximations (non-zero a, finite V,... ) systematically improvable Compute directly on lattice, using QCD and QED

5 Lattice field theory calculation reminder Compute QFT path integrals numerically and stochastically Fields live on finite 4d (Euclidean) space-time lattice Generate ensemble of field configurations using monte carlo Average over configurations Typically compute correlation function of fields, extract (Minkowski) matrix element or amplitude Dominated by quark propagators, inverse of large, sparse matrix. Extrapolate to continuum, infinite volume, physical quark masses (now directly accessible)

6 Lattice QCD: conventional approach (c.f., HVP) Correlation of 4 EM currents Π µνρσ (q, p 1, p 2 ) Two independent momenta +external mom q Compute for all possible values of p 1 and p 2 (O(V 2 )) four index tensor several q (extrap q 0), fit, plug into perturbative QED two-loop integrals

7 Alternate approach: Lattice QCD+QED Average over combined gluon and photon gauge configurations Quarks coupled to gluons and photons muon coupled to photons [Hayakawa, et al. hep-lat/ ; Chowdhury et al. (2008); Chowdhury Ph. D. thesis (2009)]

8 Alternate approach: Lattice QCD+QED Attach one photon by hand (see why in a minute) Correlation of hadronic loop and muon line [Hayakawa, et al. hep-lat/ ; Chowdhury et al. (2008); Chowdhury Ph. D. thesis (2009)]

9 Formally expand in α electromagnetic The leading and next-to-leading contributions in α to magnetic part of correlation function come from

10 Subtraction of lowest order piece Subtraction term is product of separate averages of the loop and line Gauge configurations identical in both, so two are highly correlated In PT, correlation function and subtraction have same contributions except the light-bylight term which is absent in the subtraction

11 Subtraction of lowest order piece: two photons? absent in subtraction term, but vanishes due to Furry s theorem Only after averaging over gauge fields, potentially large error (O(α 2 ) compared to signal of O(α 3 )) Exact symmetry under p p e e on muon line only If e unchanged, only effect is to flip the sign of all diagrams with two photons, so these cancel on each configuration. Observe large reductions in statistical errors after ± momentum averaging

12 LbL contribution from lattice QED a test LbL calculation: quenched (no vac. pol.), non-compact QED m lepton = 0.1, loop and line 16 3 and ( 8) Domain Wall Fermions (L/4) 3 = 64 and 216 propagators for the lepton loop to enhance statistics incoming muon at rest, p = ±(2π/L, 0, 0) T, and permutations, at external vertex several source/sink separations for muon (8-32) to project on to ground states

13 LbL contribution from lattice QED a test F 2 ((2π/L) 2 ) QED (m loop =m µ =0.1, 24 3 ) QED, (m loop =m µ =0.1, 16 3 ) QED pert. theory, F 2 (0) t sep lowest non-trivial momentum only Stat. errors only Several source/sink separations for muon (loop is same, only line differs) Significant excited state contamination m µ = 0.1 loop and line Blum, Chowdhury, Hayakawa, and Izubuchi (arxiv: )

14 HLbL contribution from lattice QCD+QED Calculation is almost the same, just take U µ (x) = U QED µ (x)u µ (x) QCD for the combined gauge field HLbL calculation: RBC/UKQCD 2+1f DWF ensemble m π = 329 MeV (light quarks heavier than u,d; s is physical) lattice size ( 16) DWF lattice spacing a = fm quenched QED for now (sea quarks not charged)

15 HLbL contribution from lattice QCD+QED F 2 ((2π/L) 2 ) QCD+QED (m π =330 MeV) hadronic models, F 2 (0) t sep Stat. errors only, lowest non-trivial momentum Several source/sink separations for muon Significant excited state contamination m π = 329 MeV Model value/error is Glasgow Consensus (arxiv: [hep-ph]) Blum, Chowdhury, Hayakawa, and Izubuchi (arxiv: )

16 HLbL contribution from lattice QCD+QED Momentum dependence F 2 (Q 2 ) Models t sep =0-10 (m π =330 MeV) Q 2 (GeV 2 ) t sep = 10 Stat. errors only, lowest non-trivial momentum m π = 329 MeV Model value/error is Glasgow Consensus (arxiv: [hep-ph]) Blum, Chowdhury, Hayakawa, and Izubuchi (arxiv: )

17 Disconnected diagrams (and similar) not calculated yet Omission due to use of quenched QED, i.e., sea quarks not electrically charged. Two possibilities, 1. Re-weight in α (T. Ishikawa, et al., Phys.Rev.Lett. 109 (2012) ) or 2. dynamical QED(+QCD) in HMC Use same non-perturbative method as for quenched QED

18 quark loop with four electromagnetic (EM) verties, called LBL(4). Below, I list up all diagrams containing more than one quark loop having EM vertices Disconnected quark loop diagrams (with no lattice-artifact interactions) 1. The hadronic light-by-light scattering diagrams with two quark loops having EM vertices I call the contributions (1), (2) and (3) as LBL(1,3), LBL(2,2) and LBL 2 The hadronic light-by-light diagrams with three quark loops having QCD QCD I call the contributions (1), (2) and (3) as LBL(1,3), LBL(2,2) and LBL(3,1), respectively The hadronic, light-by-light diagrams with (1) three quark loops having EM vertices, QCD QCD QCD., (2), (4) I call the contributions (4) and (5) as LBL(1,1,2) and LBL(2,1,1), respe The hadronic light-by-light diagrams with four quark loops having QCD QCD QCD. (3). (5), 1 All figures are brought from M.H. s slide usedi at call Lattice the contributions Sorry for difference (4) and (5) of notations as LBL(1,1,2) used I call in and the contribution LBL(2,1,1), (6) respectively. as LBL(1,1,1,1). Sec. II The hadronic light-by-light diagrams with four quark loops having EM vertices 2 Individual photon lines emanated from quark loops should be contracted with those attatched on the muon lines Tomin all Blum possible (UConn ways. / RIKEN BNL Research Center) Lattice calculation of hadronic light-by-light scattering contribu

19 M Disconnected quark loop diagrams D =. in non-pert. method II. NONPERTURBATIVE QED METHOD WITH FULL QED SIMULATION The subtraction term for the connected component with the int loop different from the external vertex The main terms that we compute by lattice simulation are 3 QCD+f-QED is S C = f-qed. M C = The, subtraction term for the connected QCD+f-QED component (7) with photon e S C = QCD+f-QED vertex is f-qed. The subtraction term for the connected component with photon M C =, (8) vertex is QCD+f-QED QCD+f-QED The subtraction S C = term for the disconnected component with photon emitted from th M D = loop different from the external vertex is vertex is 3 The second contribution (8) arises fromqcd+f-qed the lattice-artifact interaction. It S. C QCD+f-QED gauge invariance = at finite lattice spacing a. SD = (9) f-qed.. QCD+f-QED f-qed. QCD+f-QED The subtraction term for the connected component with the internal vertices on the quark Our new version of non-perturbative QED method, loop different from the external vertex is 3 The second contribution (8) arises from the lattice-artifact 4 which incorporates a interaction. It hadronic light-by-light scattering conitributions is gauge invariance at finite lattice spacing a. h-lbl + O(α 4 ) = 1 [MC + MC + MD SC SC SD]. 3 A new finding Lattice herecalculation is that although of hadronic individuals light-by-light arise fromscattering MC + MC contribu and/or

20 3 Disconnected quark loop diagrams in our non-pert. method is that although individuals arise from M C + M C and/or M D with es, as shown in Table I, all hadronic light-by-light diagrams arise with y in M C + M C + M D. Diagrams in non-perturbative method have various multiplicities TABLE I: Origin of degeneracy factor M C + M C M D LBL(4) 3 0 LBL(1,3) 0 3 LBL(2,2) 1 2 LBL(3,1) 2 1 But, physical linear combination, M C + M C + M D has overall factor of 3 LBL(1,1,2) 0 3 LBL(2,1,1) 1 2 LBL(1,1,1,1) 0 3 O CALCULATE THE DISCONNECTED CONTRIBUTION (9)

21 more systematic errors Need to address quark and muon masses quenched QED Finite volume continuum limit a 0 q 2 0 exptrap QED renormalization excited states/ around the world effects

22 Alternative non-pert. method (Luchang Jin) see talk at Lattice 2014 No subtraction, use 2 independent stochastic photons, one exact 4 Light by Light Evaluation Strategy 4.2 Evaluation Formula z, ν x op, µ xop, µ xop, µ z, ν x op, µ A m1 ρ (x) z, Bν σ m2 (y) A m1 ρ (x) Bσ m2 (y) A m1 ρ (x) z, ν Bσ m2 (y) A m1 ρ (x) Bσ m2 (y) x src A m1 ρ (x ) xsrc A m1 ρ (x ) z, ν B m2 B m2 σ (y ) xsnk σ (y ) x snk x src z, ν A m1 A m1 ρ ρ (x ) B m2 σ (y ) xsnk (x ) B m2 σ (y ) x snk Figure M 11. LbL µ = Light ( ie) by Light 6 1 M 1 δνν diagrams calculated with one exact photon and two stochastic photon. M There are 4 other possible 2 V k permutations. 2 m1,m2=1 k [ ] [ ] tr M = 12 stochastic γµ m1 Sq(xop; x)γρa ρ (x)sq(x; z) γνe photon fields for both A and B. ik z m2 Sq(z; y)γσb σ (y)sq(y, xop) z x y S = 18 random [ ] [ ] wall sources for the external local current. S(xsrc; x )γρ A m1 ρ (x )S(x ; z ) γν e ik z S(z ; y )γσ B m2 σ (y )S(y ; xsnk) z 4.1 Computation x Cost y +S(xsrc; z 2 S M times inversion )γν e ik z for ( S(z the quark ; x )γρ loop. A m1 ρ (x ) S(x ; y )γσ B m2 σ (y )S(y ; xsnk)) x y 8 M 2 times inversion for muon line. +other 4 permutations Statistics roughly proportion to S M 2 (8) Cost Tom grows Blum as O(Volume) (UConn / RIKEN not O(Volume BNL Research 2 ). Center) z, ν xsrc z, ν

23 Summary and Outlook First lattice QCD calulation of HLbL contribution to g-2 appears promising Checked QCD+QED method in QED Crucial to leverage FNAL E989, J-PARC E34 experiments, search for new physics Next HLbL calculation: RBC/UKQCD 2+1f DWF ensemble m π = 170 MeV, And/or next HLbL calculation: 2+1f DWF QCD+QED ensemble m π = 300 MeV, Next-to-Next HLbL calc: RBC/UKQCD 2+1f Möbius-DWF m π = 140 MeV,