Hadronic light-by-light contribution to the muon g-2 from lattice QCD+QED

Transcription

1 Hadronic light-by-light contribution to the muon g-2 from lattice QCD+QED Tom Blum (University of Connecticut, RIKEN BNL Research Center) with Christopher Aubin (Fordham) Saumitra Chowdhury (UConn) Masashi Hayakawa (Nagoya) Taku Izubuchi (BNL/RBRC) Norikazu Yamada (KEK) Takeshi Yamazaki (Nagoya) Hadronic Light-by-light Workshop, INT, Seattle, Mar 1, 2011

2 Outline of the talk Method Tests in pure QED Towards full QCD+QED result Summary 1

3 The hadronic light-by-light contribution (O(α 3 )) Model estimates put this O(α 3 ) contribution at about (10 12) with a 25-40% uncertainty Blob contains all possible hadronic states No dispersion relation a la vacuum polarization Lattice regulator: model independent, approximations systematically improvable 2

4 Conventional approach (QCD only on the lattice) Correlation of 4 currents Two independent momenta +external mom q Compute for all possible values of p 1 and p 2, four index tensor several q, (extrap. q 0), plug into perturbative QED two-loop integrals Pursued by Rakow, et al, (QCDSF collaboration) 3

5 New approach QCD and QED on the lattice Average over combined gluon and photon gauge configurations Quarks coupled to gluons and photons muon coupled to photons [hep-lat/ ; Chowdhury et al. (2008); Chowdhury Ph. D. thesis (2009)] 4

6 New approach... Attach one photon by hand (see why in a minute) Correlation of hadronic loop and muon line [hep-lat/ ; Chowdhury et al. (2008); Chowdhury Ph. D. thesis (2009)] 5

7 The leading and next-leading contributions in α to magnetic part of correlation function come from 6

8 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-by-light term which is absent in the subtraction 7

9 connect with each other. Hereby, we achieve Fig We calculate this diagram non-perturbatively, which ensures the inclusion of the other two photons to get Test calculation in pure QED the target Fig (Compare to diagram well known PT result) [Chowdhury thesis] Incoming muon has p ~ = 0, outgoing p ~0 = ~ pop = (1, 0, 0) (+ perms). Fig. 4.5: Lattice implementation of lbl external muons put on shell in usual way (t1 top t2 ) t1 = 0, top = 4, 6, 8, t2 = 12 If we consider the path integral represented in Fig. 4.5, and expand it pertur Single lattice size, batively in the QED coupling constant, we can see how the O(αem ) terms arise, (4d) Fourier transform loop and line separately 2 and can dominate our calculations. The QCD contribution starts at O(αem ) to enhance signal, take e = 1, or α = 1/4π through vacuum polarization effects of the photon propagator [60]. Hence, it is Subtraction difficult in practice since need to average loop and line sep2. arately over gauge sum over rather warranted to getfields rid offirst, thesethen contributions to qextract our target diagram in Fig Therefore, we propose the following method given by 8

10 more details - Domain wall fermions (to match 2+1 QCD ensembles) (L s = 8) - loop/line masses degenerate: m µ, m e = 0.4 (heavy), and loop mass = Non-compact, quenched QED (easy to generate, fewer lattice artifacts) - Fix to Feynman gauge (photon propagator is simple) - O( ) configurations (measurements) 9

11 Extracting the form factors G µ (t, t) = ψ(t, p ) J µ (t, q) ψ (0, p). Insert two complete sets of states, take t t 0, G µ (t, t) = s,s 0 χ N p, s p, s J µ p, s p, s χ N 0 1 2E 2E e E (t t) e Et +... = G µ (q 2 ) f(t, t, E, E ) +..., (like LHZ reduction, but in Euclidean space) For example, J µ = J x, Similarly, trp xy G x (q 2 ) = p y m(f 1 (q 2 ) + F 2 (q 2 )) P xy = i 1 + γ t γ y γ x 4 2 trp t G t (q 2 ) = m (E + m) P t = γ t, 4 2 ) (F 1 (q 2 ) + q2 (2m) 2F 2(q 2 ), 10

12 Check Ward-Takahashi Identities Use conserved (lattice) currents WTI satisfied on each configuration The loop is checked using q µ Π µν = 0, same as for vacuum polarization The muon-line is a little more complicated iq ρ ψ( p 2, t 2 )J ρ ψ( p 1, t 1 ) = e iq 4t 2 ψ( q + p 2, t 2 ) ψ( p 1, t 1 ) δ( q + p 2 p 1 ) e iq 4t 1 ψ( p 2, t 2 ) ψ( q p 1, t 1 ) δ( p 2 q p 1 ) Checked for free case and non-trivial gauge field 11

13 F 2 (0) (QED only, degenerate leptons) 8e-05 6e-05 4e configs (Coulomb) 500 configs (Coulomb) 300 configs (Feynmann) 2e-05 F 2 0-2e-05-4e-05-6e-05-8e t op F 2 = ( 0.50 ± 0.37) 10 5 (lowest non-zero momentum, stat error only) Fig. 5.30: Anomalous magnetic moment (F 2 ) of electron as a function of time slices of the external vertex (t op ) on lattice volume of with Continuum PT result: 0.36(α/π) 3 = (e = 1) loop mass =0.4, line mass = 0.4, and electron charge = 1. Statistical error same order as PT result 12

14 Large m µ /m e enhancement seen in perturbation theory Try m = 0.01 in the loop, or m µ /m e = 40 Finite volume effects could be large [Aldins, Brodsky, Duffner, Kinoshita (1970)] 13

15 F 2 m µ /m e = 40 (QED only) configs F t op F 2 = (1.32 ± 0.13) 10 4 (lowest non-zero momentum, stat error only) Fig. 5.31: Anomalous magnetic moment (F 2 ) of muon as a function of time slices Continuum PT result: 10(α/π) 3 = (e = 1) of the external vertex (t op ) on lattice volume of with loop roughly consistent with PT result mass =0.01, line mass = 0.4, charge = 1 (for both electron and muon). 14

16 110 Finite volume effects (QED only, Schwinger term) Large finite volume effect in O(α) Schwinger term ( e = 1, m µ = 0.2): Local Conserved Direct Ratio; Local Direct Ratio; Conserved Continuum Curve F q 2 /m 2 Fig. 5.24: F 2 as a function of q 2 /m 2 for both local and conserved currents (continuum curve is also shown in black), where results from three lattice 15 volume of (top data points), (middle data

17 F 2 in 2+1 flavor QCD+QED Include hadronic part in the loop only (same in subtraction) 2+1 flavors of DWF (RBC/UKQCD) a = fm, ( 16), a 1 = 1.73 GeV m q (m π 400 MeV) 1000 configurations (one QED conf. for each QCD conf.) F 2 = ( 5.3 ± 6.0) 10 5 (lowest non-zero momentum, e = 1) Magnitude of error is about 13 model estimates model calculations (physical mass and charge) about 200 times smaller than QED light-by-light contribution. Signal has disappeared, but statistical error stayed about the same (m µ /m π 2) 16

18 Low mode average Need large improvement in statistics Current method uses point source at external vertex Gauge (ensemble) average enforces momentum conservation Volume average (fourier transform) would improve statistics by O(V ) (project onto q = p p) V point source propagators (D 1 s) too expensive Low-mode average instead 17

19 Low mode average Just usual spectral decomposition of the (Hermitian) Dirac operator /D H = γ 5 /D /D H ψ λ = (λ + m)ψ λ /D 1 = /D 1 H γ 5 = λ ψ λ ψ λ γ 5 λ + m Still too expensive to calculate all the eigenmodes. Compute the low-modes, and treat the high(er) ones in the usual way with a stochastic source(s) generate all-to-all propagator 18

20 Low mode average Test calculation underway, uses same parameters as before Low-modes only (O(100)), high-mode (stochastic) part later Have O(150) low modes on larger 24 3 lattices, m π 300 MeV, QCD only Use them to compute QCD four-point function (momentum sums) Directly compare the two methods 19

21 Disconnected diagram(s) (connected by gluons) Haven t tried these yet... 20

22 Summary/Outlook: light-by-light contributions (O(α 3 )) Pure QED calculation on the lattice roughly reproduces the perturbative result. Encouraging. Full hadronic contribution is O(10 2 ) times smaller, still swamped by the statistical noise Small volumes, poor statistics. Try Volume (low-mode) averaging for the loop Larger volumes More statistics, i.e. more QED configurations per QCD configuration conventional calculation using all-to-all propagator multi-quark loops not yet attempted Acknowledgments: This research was supported by the US DOE and RIKEN BNL Research Center. Computations done on the QCDOC supercomputers at BNL and Columbia University. 21

23 were forced to agree at 2 GeV by imposing a simple additive shift to the perturbation theory curve. The lattice results for 3 q 2 8 GeV 2 agree impressively with perturbation theory. For lower values of q 2 and m l 0:0031, the lattice value increases faster until about 0:5 GeV 2 when the diverging perturbation theory result overtakes it again. For the heavier mass, the two results coincide until about 0.11 one of the physically motivated fitting functions. For fits A and B, we note that there is little difference in the final result for a HLO, as expected from the fit results themselves. As mentioned in the last section, there is no difference in a HLO for fits B and C since the one-loop corrections in Eq. (35) only rescale the tree-level value of f V. The statis- for the S PT fits are much smaller than The order α 2 tical errors a hadronic contribution HLO to g-2 Fits 0.11 m fu re po fo er co ga er to ac a H -Π(q 2 ) q 2 (GeV 2 ) FIG. 12 (color online). Cubic (dashed line) and quartic (solid line) fits to ^q 2 for am l 0:0031 (diamonds), (squares), and (circles). The strange quark mass is fixed to in each case. -Π(q 2 ) q 2 (GeV 2 ) FIG. 13 (color online). S PT fits to ^q 2 for the three light masses, am l 0:0031 (diamonds), (squares), and (circles). The strange quark mass is fixed to in each case. The solid lines correspond to fit B, and the dashed lines to Fit A, as described in the text. ou pr co TA sc on m Fi Po Po A B C

24 The order α 2 hadronic contribution to g-2 a µ x disp. relation Proposed am l Extrapolate m l m u,d Simple linear and quadratic chiral extrapolations consistent with e + e hadrons result a HLO a HLO µ = (713 ± 15) (linear) µ = (742 ± 21) (quad) (statistical errors only). [Aubin, Blum, Phys. Rev. D, 2006] Fit quenched am l = am l = am l = Poly (63) 370(49) 445(43) 542(24) Poly (142) 410(91) 639(123) 729(59) A (7.0) (7.8) (9.5) (8.1) B (7.8) (9.5) (8.1) C (7.8) (9.5) (8.1) 23

25 New results for vacuum polarization a = 0.09 and 0.06 fm, m π down to 170 MeV, volume < (6 fm)3 24

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28 Backup Slides 25

29 Introduction to muon g-2 Classical interaction of particle with static magnetic field V ( x) = µ B The magnetic moment µ is proportional to its spin ( e ) µ = g S 2m The Landé g-factor is predicted from the free Dirac eq. to be for elementary fermions g = 2 26

30 In the quantum (field) theory g receives radiative corrections k p 1 q p 2 p 1 q p 2 γ µ Γ µ (q) = ( γ µ F 1 (q 2 ) + i ) σµν q ν 2m F 2(q 2 ) which results from Lorentz invariance and current-conservation (Ward-Takahashi identity) when the muon is on-mass-shell. F 2 (0) = g 2 2 (the anomalous magnetic moment) a µ 27

31 Compute these corrections order-by-order in perturbation theory by expanding Γ µ (q 2 ) in QED coupling constant α = e2 4π = Corrections begin at O(α); Schwinger term = α 2π = Hadronic contribution times smaller 28

32 Status of the experimental measurement (Muon (g 2) Collaboration, BNL- E821) of a µ µ - µ Avg. [τ] a µ [e e ] Experiment Theory a µ (exp) = (6) (accurate to about 0.5 ppm) 29

33 Theory calculation (Summary from D. W. Hertzog [E821 Collaboration], hep-ex/ ) Table 1: Comparison of a µ (SM) with a µ (Exp) a µ a µ QED QCD Weak Theory Experiment a µ (EXP) a µ (SM) Table 2: QCD contribution to the muon g 2 a µ a µ hadronic vacuum polarization (O(α 2 em )) hadronic vacuum polarization (O(α 3 em )) hadronic light-by-light (O(α 3 em )) Total QCD

34 O(α 2 ) hadronic contribution hadronic vacuum polarization: (Π(k 2 )) Q Q e e p 1 q p 2 The blob, which represents all possible intermediate hadronic states, is not calculable in perturbation theory, but can be calculated from dispersion relation + experimental cross-section for e + e hadrons first principles using lattice QCD 31

35 Dispersive method for the vacuum polarization contribution [Bouchiat and Michel (1961); Durand (1962);...] The vacuum polarization is an analytic function. Π(q 2 ) = 1 ds IΠ(s) π 0 (s q 2 ) σ total (e + e hadrons) = 4π2 α 1 s π IΠ(s) (by the optical theorem) which leads to a had(2) µ = 1 4π 2 where where K(s) is a known function 4m 2 π dsk(s)σ total (s) K(s) is strongly weighted to low energy region: roughly 91% from s < 1.8 GeV, 73% from two pion final state which is dominated by the ρ(770) resonance. Can get part of σ total from τ π ± π 0 ν decay (needs isospin correction) 32

36 Recent theory updates hadronic vacuum polarization (a exp µ a theory µ ) June 2009 new τ-based evaluation incl. Belle data: 1.8 σ (Davier, et al.) Aug 2009 new e + e -based, incl. Babar ISR data: 3.1 σ (Davier, et al.) Oct 2009 new e + e -based (Phipsi conf., Beijing): 4.0 σ (Hagiwara, et al.) hadronic light-by-light 2009 (10.5 ± 2.6) (Prades-de Rafael-Vainshtein, arxiv: ) 2009 (11.6 ± 4.0) (Nyffeler, arxiv: ) 33

37 The precise measurement at Brookhaven (E821) allows a precision test of the standard model O(α 2 ) hadronic contribution dominates theory error: Dispersion relation for Π(q 2 ) + optical theorem + exp. e + e hadrons cross section e + e : 2.7 σ deviation with experiment τ decay: 1.4 σ deviation with experiment Lattice calculation (purely theoretical) provides an important check of the dispersive calculation, but required precision poses a great challenge. Lattice calculation of hadronic light-by-light contribution important since only model calculations exist 34

38 Q Q e p 1 q p 2 e Inserting the quark loop (blob) into the one-loop diagram is easy since only the photon propagator is modified (c.f., charge renormalization in QED), 1 k 2 i ɛ 1 k 2 i ɛ (1 + Π(k2 ) +... ) The diagram boils down to [B.E.Lautrup and E. de Rafael (1969), Blum (2002)] ( α 2 a (2)had µ = dk π) 2 f(k 2 ) Π(K 2 ) where K 2 is Euclidean (space-like) momentum-squared Note: kernel f(k 2 ) diverges as K 2 0 Ok, since renormalized Π(K 2 ) vanishes at K 2 = 0 But it means the integral is dominated by the low momentum region. 0 35

39 Lattice calculation of Π(q 2 ) The continuum vacuum polarization is defined as Π µν (q) = d 4 x e i q (x y) J µ (x)j ν (y) (J µ (x) = ψγ µ ψ(x)) = (q 2 g µν q µ q ν )Π(q 2 ) and satisfies the Ward-Takahashi identity (charge conservation) q µ Π µν (q) = 0 ( µ J µ = 0) On the lattice this also holds, provided conserved current is used J µ (x) = 1 ( ψ(x + ˆµ)U (x)(1 + γ µ )ψ(x) ψ(x)u(x)(1 γ µ )ψ(x + ˆµ) ) 2 µ J µ J µ (x) J µ (x aˆµ) (x) = a µ 37

40 Fourier transformation of the two point function yields the continuum form of W-T identity ˆq µ Π µν (ˆq 2 ) = 0 ˆq µ = 2 ( ) aq µ a sin from which we compute Π(ˆq 2 ) 2 q µ = 2πn µ a L µ, n µ = 0, ±1, ±2,..., ±(L µ 1) Π µν (ˆq) = (ˆq µˆq ν ˆq 2 δ µν )Π(ˆq 2 ) The W-T Identity provides a strong check on the calculation since it must be satisfied exactly (up to machine precision) in the numerical calculation 38

41 Anatomy of a lattice calculation In practice, calculate the twopoint correlation function of the electromagnetic current in coordinate space on a discrete Euclidean space-time lattice, ψ(y)γ µ ψ(y) ψ(x)γ ν ψ(x) x y Wick contract quark fields into propagators (M 1 xy ) (M xy is lattice Dirac op, large sparse matrix) C µν (x, y) = trm 1 xy Then compute the Fourier transform γµ M 1 yx γν Π µν (q) = x e i q(x y) C µν (x, y), and average over many gauge-field configurations (do the path intergral) 39

42 2+1 flavor QCD configuration summary a (fm) am l am s β size m π L # lats. avail. 0.09* * * MILC Asqtad ensembles lattices labeled with * have been used in [Aubin and Blum (2006)] New MILC ensembles: lightest mass on the fine (a 0.09 fm) ensemble and superfine (a 0.06 fm) ensemble, marked in bold. 40

43 Hadronic vacuum polarization: 2+1 flavors of quarks -Π(q 2 ) large lattice, L 3.5 fm Small q 2 : q = sin(2πn/l) Quark masses (0.1, 0.2, 1.0) m s slope increases as m u,d single spacing a = fm q 2 (GeV 2 ) [Aubin, Blum, Phys. Rev. D, 2006] 41

44 The order α 2 hadronic contribution to g-2 Fit Π(q 2 ) to obtain smooth function of q 2, plug into a µ formula ( α 2 a (2)had µ = dk π) 2 f(k 2 ) Π(K 2 ) χpt + vector meson (resonance χpt) [Aubin and Blum (2006)] 0 High momentum part of integral done with 3-loop continuum PT [Chetyrkin, et al. (1996)] (small contribution 1 2%) Π(q 2 ) Π(q 2 ) q 2 (GeV 2 ) q 2 (GeV 2 ) 42

45 The order α 2 hadronic contribution to g-2 a µ x disp. relation Proposed am l Extrapolate m l m u,d Simple linear and quadratic chiral extrapolations consistent with e + e hadrons result a HLO a HLO µ = (713 ± 15) (linear) µ = (742 ± 21) (quad) (statistical errors only). [Aubin, Blum, Phys. Rev. D, 2006] Fit quenched am l = am l = am l = Poly (63) 370(49) 445(43) 542(24) Poly (142) 410(91) 639(123) 729(59) A (7.0) (7.8) (9.5) (8.1) B (7.8) (9.5) (8.1) C (7.8) (9.5) (8.1) 43

46 Summary/Outlook: vacuum polarization contribution(o(α 2 ) Previous results (2006) consistent with e + e /τ decay results, but need to reduce systematic errors (momentum and quark mass extrapolations) new configurations available from MILC (staggered) and RBC/UKQCD (DWF) New results for lighter quark mass and bigger volume soon (ρ above ππ threshold) Finer lattice spacing soon Still need disconnected quark loop diagrams, (vanish in the SU(3) limit) Take continuum, infinite volume limits Several groups (RBC/UKQCD, LSD, DESY-Zeuthen,...) now calculating vacuum polarization to calculate S-parameter and g 2. 44