HVP contributions to anomalous magnetic moments of all leptons from first principle

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1 Introduction HVP contributions to anomalous magnetic moments of all leptons from first principle At Physical Point Mass with Full Systematics Kohtaroh Miura (CPT, Aix-Marseille Univ.) PPP 2017, YITP, 02 Aug Budapest-Marseille-Wuppertal Collaboration [hep-lat] and in preparation

2 Introduction Motivation Our Challenges Muon Anomalous Magnetic Moment a µ B Muon Strorage μ a µ gµ 2, (1) 2 e M = g µ S. (2) 2m µ p s a exp. µ = a SM µ?

3 a exp. µ vs. a SM µ Introduction Motivation Our Challenges SM contribution aµ contrib Ref. QED [5 loops] ± [Aoyama et al 12] HVP-LO (pheno.) ± 4.2 [Davier et al 11] ± 4.3 [Hagiwara et al 11] ± 4.2 [Benayoun et al 16] HVP-NLO 9.84 ± 0.07 [Hagiwara et al 11] [Kurz et al 11] HVP-NNLO 1.24 ± 0.01 [Kurz et al 11] HLbyL 10.5 ± 2.6 [Prades et al 09] Weak (2 loops) ± 0.10 [Gnendiger et al 13] SM tot [0.42 ppm] ± 4.9 [Davier et al 11] [0.43 ppm] ± 5.0 [Hagiwara et al 11] [0.51 ppm] ± 5.9 [Aoyama et al 12] Exp [0.54 ppm] ± 6.3 [Bennett et al 06] Exp SM 28.7 ± 8.0 [Davier et al 11] 26.1 ± 7.8 [Hagiwara et al 11] 24.9 ± 8.7 [Aoyama et al 12] FNAL E989 (2017): 0.14-ppm, J-PARC E34: 0.1-ppm

4 Motivation Introduction Motivation Our Challenges Really a exp. µ a SM µ? SUSY (MSSM, Padley et.al. 15)? Technicolor (Kurachi et.al. 13)?. A rigorous determination of a µ by SM is a necessary step to attacking BSM. A bottle-neck is Hadronic vacuum polarization (HVP) contribution due to its non-perturvative feature. In phenomenological estimates of a µ, the HVP is evaluated based on dispersion relations with experimental data inputs (e + e had. etc.) Some tension among expr. (BESIII, PLB 16). Purely theoretical estimates for HVP effects are demanded for a rigorous test of SM. The Lattice QCD meets the requirement. We (BMWc) are at the forefront of this approach. γ µ µ Technicolor KLOE 08 BaBar 09 KLOE 10 KLOE ± 0.4 ± 2.3 ± ± 2.0 ± ± 0.9 ± 2.3 ± ± 1.2 ± 2.4 ± 0.8 BESIII ± 2.5 ± ππ,lo -10 a µ ( MeV) [10 ]

5 Motivation Introduction Motivation Our Challenges Really a exp. µ a SM µ? SUSY (MSSM, Padley et.al. 15)? Technicolor (Kurachi et.al. 13)?. A rigorous determination of a µ by SM is a necessary step to attacking BSM. A bottle-neck is Hadronic vacuum polarization (HVP) contribution due to its non-perturvative feature. In phenomenological estimates of a µ, the HVP is evaluated based on dispersion relations with experimental data inputs (e + e had. etc.) Some tension among expr. (BESIII, PLB 16). Purely theoretical estimates for HVP effects are demanded for a rigorous test of SM. The Lattice QCD meets the requirement. We (BMWc) are at the forefront of this approach. γ µ µ Technicolor KLOE 08 BaBar 09 KLOE 10 KLOE ± 0.4 ± 2.3 ± ± 2.0 ± ± 0.9 ± 2.3 ± ± 1.2 ± 2.4 ± 0.8 BESIII ± 2.5 ± ππ,lo -10 a µ ( MeV) [10 ]

6 Motivation Introduction Motivation Our Challenges Really a exp. µ a SM µ? SUSY (MSSM, Padley et.al. 15)? Technicolor (Kurachi et.al. 13)?. A rigorous determination of a µ by SM is a necessary step to attacking BSM. A bottle-neck is Hadronic vacuum polarization (HVP) contribution due to its non-perturvative feature. In phenomenological estimates of a µ, the HVP is evaluated based on dispersion relations with experimental data inputs (e + e had. etc.) Some tension among expr. (BESIII, PLB 16). Purely theoretical estimates for HVP effects are demanded for a rigorous test of SM. The Lattice QCD meets the requirement. We (BMWc) are at the forefront of this approach. γ µ µ Technicolor KLOE 08 BaBar 09 KLOE 10 KLOE ± 0.4 ± 2.3 ± ± 2.0 ± ± 0.9 ± 2.3 ± ± 1.2 ± 2.4 ± 0.8 BESIII ± 2.5 ± ππ,lo -10 a µ ( MeV) [10 ]

7 Objective in This Work Introduction Motivation Our Challenges Leading-Order (LO) HVP contribution to anomalous magnetic moments for all leptons (A few % precision now, and a per-mil within few years): ( a LO-HVP,f α ) 2 l=e,µ,τ = π 0 dq 2 ω(q 2 /m 2 l)ˆπ f (Q 2 ). where suffix f stands for a flavor f = l(u, d), s, c, disc, and with ˆΠ f (Q 2 ) = Π f (Q 2 ) Π f (0) = t C f =l,s,c µν (t) = q 2 f =l,s,c C f =disc µν (t) = q 2 f =disc t 2 [1 jµ(x)j f ν(0) f conn, x ( sin(z/2) ) ] 2 z/2 z=qt ( lγ µl sγ µs)( lγ νl sγ νs) disc. x 1 3 µ µ HAD 3 i=1 γ C f ii (t), (3) Here, charge factors are given by (q 2 l, q 2 s, q 2 c, q 2 disc) = (5/9, 1/9, 4/9, 1/9).

8 Our Challenges Introduction Motivation Our Challenges The integrand kurnel ω(q 2 /m 2 µ) is known and makes a peak around Q 2 (m µ/2) 2 (0.05GeV ) 2 4fm: Our Lattice: (L, T ) (6, 9 12)fm. The Pion/Kaon dynamics precisely: Our simulations are performed with Physical Pion/Kaon Masses. Large distance signal: 10 4 Traj., 768 (9000) random sources for ud-conn. (uds-disc.) correlators. Need controled continuum limit: 15 lattice spacings (a fm). For a few % precision, we take account of: c quark w. matching onto perturb. theory. ω(q 2 /m 2 µ ) ^Π l (Q 2 ) x (m µ /2) Q 2 GeV 2

9 Simulation Setup Introduction Motivation Our Challenges Tree-level improved Symanzik gauge action. Nf=(2+1+1) stout-smeared staggered quarks (m c/m s = 11.85). Scale setting f π = MeV via scale w 0. 2M K 2 /Mπ 2-1 Rational Hybrid Monte Carlo M π /Fπ phys β a[fm] N t N s #traj. M π[mev] M K [MeV] #SRC (l,s,c,d) (768, 64, 64, 9000) (768, 64, 64, 6000) (768, 64, 64, 6144) (768, 64, 64, 3600) (768, 64, 64, 6144) (768, 64, 64, )

10 Table of Contents Introduction Motivation Our Challenges 1 Introduction Motivation Our Challenges 2 Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion 3

11 Table of Contents Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion 1 Introduction Motivation Our Challenges 2 Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion 3

12 Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion Light-Conn. (and Disc.) Correlator: An Example C l (t) 1.0e e e e e e e e e t fm Figure: C ud (t) = 5 9 x i=1 jud i ( x, t)ji ud (0) The connected-light correlator C ud (t) loses signal for t > 3fm. To control statistical error, consider C ud (t > t c) Cup/low(t, ud t c), where Cup(t, ud t c) = C ud (t c) ϕ(t)/ϕ(t c), Clow(t, ud t c) = 0.0, with ϕ(t) = cosh[e 2π(T /2 t)], and E 2π = 2(Mπ 2 + (2π/L) 2 ) 1/2. Similarly, C disc (t) C disc up/low(t, t c), C disc C disc up (t > t c) = 0.1C ud (t c) ϕ(t)/ϕ(t c), low (t > t c) = 0.0. C ud,disc low (t, t c) C ud,disc (t) Cup ud,disc (t, t c).

13 Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion IR-CUT (t c ) Deps. of Light/Disc. Component: a LO-HVP l,ud/disc LO-HVP a µ, ud x LO-HVP -a µ, disc x pion zero avg t c [fm] Corresponding to C ud,disc up/low (tc), we obtain upper/lower bounds for g 2: a ud,disc l,up/low (tc). Two bounds meet around t c = 3fm. Consider the average of bounds: ā ud,disc l (t c) = 0.5(a ud,disc l,up + a ud,disc l,low )(tc), which is stable around t c = 3fm. We pick up such averages ā ud,disc l (t c) with 4 6 kinds of t c around 3fm. The average of average is adopted as a LO-HVP l,ud/disc to be analysed, and a fluctuation over selected t c is incorporated into the systematic error.

14 Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion Continuum Extrap. of Light Conn. Component: a LO-HVP µ,ud LO-HVP a µ,l x data fit0 fit1 fit2 fit a 2 fm 2 F (aµ,ud LO-HVP, A, C π, ) = aµ,ud LO-HVP (1 + Aa 2 )+C π Mπ 2 +. a LO-HVP µ,ud = (8.10)(8.24), χ 2 /dof = 7.8/12 (fit1 case).

15 Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion Continuum Extrap. of Strange Conn. Component: a LO-HVP µ,s LO-HVP a µ,s x dat fit0 fit1 fit2 fit a 2 fm 2 F (aµ,s LO-HVP, A, C K ) = aµ,s LO-HVP (1 + Aa 2 ) + C K MK 2. a LO-HVP µ,s = 53.60(05)(13), χ 2 /dof = 16.7/11 (fit1 case).

16 Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion Continuum Extrap. of Charm Conn. Component: a LO-HVP µ,c 16.0 LO-HVP a µ,c x dat fit a 2 fm 2 F (aµ,c LO-HVP, A, C π,k,ηc ) = aµ,c LO-HVP (1 + Aa 2 ) + C π Mπ 2 + C M MK 2 + C ηc M ηc. a LO-HVP µ,c = 14.99(08)(08), χ 2 /dof = 1/7 (fit1 case).

17 Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion Continuum Extrap. of Disc. Component: a LO-HVP µ,disc LO-HVP a µ,disc x dat fit0 fit1 fit a 2 fm 2 F (a LO-HVP µ,disc, A, C π, ) = a LO-HVP µ,disc (1 + Aa 2 )+C π M 2 π +. a LO-HVP µ,disc = 10.9(1.1)(0.7), χ 2 /dof = 2.4/10 (fit1 case).

18 Perturbative Corrections Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion Consider separation of momentum integral range as where ( α ) 2 ( Q max al,f LO-HVP = + π 0 Q max ) dq ( Q 2 )ˆΠf ω (Q 2 ), (4) m l = al,f LO-HVP (Q Q max) + al,f LO-HVP (Q > Q max), (5) a LO-HVP l,f (Q Q max) : lattice simulations, investigated so far, a LO-HVP l,f (Q > Q max) : γ l (Q max)ˆπ f (Q max) + pert a LO-HVP l,f (Q > Q max). For muon and electron, a LO-HVP l=µ,e (Q > Q max) is small. For example, aµ LO-HVP Qmax =2GeV (Q > Q max) 0.678(1)(1), (0.1%). (6) m 2 l

19 Perturbative Corrections Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion Consider separation of momentum integral range as where ( α ) 2 ( Q max al,f LO-HVP = + π 0 Q max ) dq ( Q 2 )ˆΠf ω (Q 2 ), (4) m l = al,f LO-HVP (Q Q max) + al,f LO-HVP (Q > Q max), (5) a LO-HVP l,f (Q Q max) : lattice simulations, investigated so far, a LO-HVP l,f (Q > Q max) : γ l (Q max)ˆπ f (Q max) + pert a LO-HVP l,f (Q > Q max). For muon and electron, a LO-HVP l=µ,e (Q > Q max) is small. For example, aµ LO-HVP Qmax =2GeV (Q > Q max) 0.678(1)(1), (0.1%). (6) m 2 l

20 Perturbative Corrections Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion Consider separation of momentum integral range as where ( α ) 2 ( Q max al,f LO-HVP = + π 0 Q max ) dq ( Q 2 )ˆΠf ω (Q 2 ), (4) m l = al,f LO-HVP (Q Q max) + al,f LO-HVP (Q > Q max), (5) a LO-HVP l,f (Q Q max) : lattice simulations, investigated so far, a LO-HVP l,f (Q > Q max) : γ l (Q max)ˆπ f (Q max) + pert a LO-HVP l,f (Q > Q max). For muon and electron, a LO-HVP l=µ,e (Q > Q max) is small. For example, aµ LO-HVP Qmax =2GeV (Q > Q max) 0.678(1)(1), (0.1%). (6) m 2 l

21 Introduction Perturbative Corrections for tau a LO-HVP τ,f = aτ,f LO-HVP (Q Q max) + γ l (Q max)ˆπ f (Q max) + pert a LO-HVP τ,f (Q > Q max). For tau, Q > Q max effects are large, while the total is stable. The fluctuation over Q 2 max = GeV 2 are incorporated into systematic errors. Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion LO-HVP a τ,ud x 10 8 LO-HVP a τ,s x 10 8 LO-HVP a τ,c x 10 8 LO-HVP a τ,disc x total lattice matching perturb Q max [GeV 2 ]

22 FV for a LO-HVP µ,ud by XPT Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion The a LO-HVP µ comes from Euclidean mommenta; Exponentially suppressed FV with LM π 4 for L 6fm. We work with L fixed, and FV effects cannot be estimated from simulations and need model. Long-distance I = 1 (I = 0) contribution dominated by 2-pions (3-pions). The dominant FV in I = 1 channel could be estimated by XPT for π + π loop (Aubin et al 16). (L) ) x (16fm) - a µ LO-HVP,l ( a µ LO-HVP,l L fm (aµ,i LO-HVP =1 ( ) alo-hvp µ,i =1 (6fm)) XPT = 13.42(13.42) 10 10, (1.9%).

23 Introduction Isospin breaking effects (Preliminary) Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion Get missing effects from phenomenology Effect corr. to aµ LO-HVP ρ ω mix ± 1.36 FSR 4.22 ± 2.11 M π M π± 4.47 ± 4.47 π 0 γ 4.64 ± 0.04 ηγ 0.65 ± 0.01 Total 7.8 ± 5.1 Thanks to F.Jegerlehner (& M. Benayoun) for correspondance and numbers The ρ ω and FSR based on Gounaris-Sakurai fit to e + e, from 2M π to 1 GeV, ascribed conservative 50% error. EM modes from M. Benayoun et al 12 Correction by M π M π± from XPT: given 100% error to account for other missing effects. Thus: IB a LO-HVP µ = (7.8 ± 5.6) (Preliminary)

24 Introduction Summary on a LO-HVP µ (Preliminary) Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion a LO-HVP µ BMWc, Preliminary I = 1 I = 0 total 584(8) st(8) acut(0) tcut(0) qcut(13) fv 121(4) st(4) acut(0) tcut(0) qcut 713(9) st(9) acut(0) tcut(0) qcut(13) fv (5) iso Total error is 2.6%, dominated FV effects. Our results are consistent with both No New Physics and Dispersive Method. There is some tension with HPQCD 16.

25 More detailed comparison Introduction Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion LO-HVP 10 a µ,ud. 10 LO-HVP a µ,s N f = N f = Mainz 17 (TMR+FV) HPQCD 16 This work N f = N f = 2+1 N f = 2 Mainz 17 (TMR) RBC/UKQCD 16 HPQCD 14 This work BMWc 17 ud contribution is signficantly larger than other Nf=2+1+1 results difference with HPQCD 14/Mainz 17 is 2.4/1.5σ. BMWc 17 c contribution is slightly smaller than other Nf=2+1+1 results BMWc 17 is only calculation performed directly at physical quark masses with 6 β s to fully control continuum extrapolation. BMWc 17 δa LO-HVP µ, disc = contributes only 0.2% to error on a LO-HVP µ. N f = N f = 2 N f = N f = LO-HVP a µ,c Mainz 17 (TMR) HPQCD 14 This work LO-HVP 10 a µ,disc. 10 This work RBC/UKQCD 15

26 Summary on a LO-HVP e,τ Introduction (Preliminary) Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion a LO-HVP e BMWc, Preliminary I = 1 I = 0 total 157.0(2.8) st(2.3) acut(0.0) tcut(0.0) qcut(4.6) fv 30.8(1.4) st(1.2) acut(0.1) tcut(0.0) qcut 189.0(3.1) st(2.6) acut(0.1) tcut(0.0) qcut(4.6) fv (1.0) iso a LO-HVP τ BMWc, Preliminary I = 1 I = 0 total 253(1) st(2) acut(0) tcut(0) qcut(2) fv 85(0) st(1) acut(0) tcut(1) qcut 342(1) st(2) acut(0) tcut(1) qcut(2) fv (1) iso Burger et.al.( 15): ae LO-HVP = 178.2(6.4)(8.6), aτ LO-HVP = 341(8)(6) HPQCD ( 16): ae LO-HVP = 177.9(3.9) Jeherlehner( 16): ae LO-HVP = (1.24). Eidelman et.al.( 07): aτ LO-HVP = 338(4).

27 Table of Contents Introduction 1 Introduction Motivation Our Challenges 2 Long Distance Mng. for Light/Disc. Correlators Continuum Extrapolation Corrections: Perturb, FV, and Isospin Breaking Short Summary on a l and Discussion 3

28 Introduction Summary of We have obtained a LO-HVP µ (as well as slope/curvature of ˆΠ(Q 2 = 0), [hep-lat]) directly at physical point masses. Full controlled continuum extrapolation and matching to perturbation theory. Model/pheno. assumptions are put on only for small corrections from FV, QED and isospin breaking. Our results are consistent with both No New Physics and Dispersive Methods with a conservative systematic errors. There is some tension with HPQCD 16 on aµ,ud LO-HVP. Total error is 2.6%, dominated by FV effects. Need 0.2% precision to match forthcoming experiments!! 1 increase statistics by times. 2 control FV effects directly with simulations. 3 simulations with QED and isospin breaking corrections taken account.

29 Backups Table of Contents 4 Backups

30 Backups Continuum Extrap. of Light Component: a LO-HVP µ,ud BMWc HPQCD (no corr.) HPQCD LO-HVP a µ,l x a 2 fm 2 Figure: Red-squares = Our data. Blue-triangles = HPQCD, a LO-HVP µ,ud = (8.10)(8.24), χ 2 /d.o.f. = 7.8/12 (fit1 case).

31 Backups Continuum Extrapolation of Π l n=1 I our data Π l n=1 GeV a 2 fm 2 F (C (2) Π, A, B, C M π, C MK ) = C (2) Π 1 + Aa 2 + ( ) 1 + a 2 1+Ba 2 CMπ M π + C MK M K. + Π l n=1 a 2 0 = (31), χ 2 /d.o.f. = 24.3/20

32 Backups Continuum Extrapolation of Π l n= our data Π l n=2 GeV a 2 fm 2 F (C (4) Π, A, B, C M π, C MK ) = C (4) Π 1 + Aa 2 + ( ) 1 + a 4 1+Ba 2 CMπ M π + C MK M K. + Π l n=2 a 2 0 = 0.306(23).

33 Backups Continuum Extrapolation of Π s n=1, our data, w. M K crr. our data, wo. M K crr. hpqcd Π s n=1 GeV Π s n=2 GeV our data, w. M K crr. our data, wo. M K crr. hpqcd a 2 fm a 2 fm 2 Figure: Red-squares = Our data. Green-triangles = HPQCD, F (C (2,4) Π, A, B, C Mπ, C MK ) = C Π 1 + Aa 2 + ( ) 1 + a 2,4 1+Ba 2 CMπ M π + C MK M K. + Π s n=1 a 2 0 = (1), Π s n=2 a 2 0 = (2), χ 2 /d.o.f. = 20.9/18

34 Backups Continuum Extrapolation of Π c n=1, our data, w. M K crr. our data, wo. M K crr. hpqcd our data, w. M K crr. our data, wo. M K crr. hpqcd Π c n=1 GeV Π c n=2 GeV a 2 fm a 2 fm 2 Figure: Red-squares = Our data. Green-triangles = HPQCD, F (C (2,4) Π, A, B, C Mπ, C MK ) = C Π 1 + Aa 2 + ( ) 1 + a 2,4 1+Ba 2 CMπ M π + C MK M K. + Π c n=1 a 2 0 = (2), Π c n=2 a 2 0 = 2.73(2) 10 4.

35 Backups Disconnected Contributions Figure: From Presentaion of T.Kawanai in Lattice Π disc 1 = 1.5(2)(1) 10 3 GeV 2, (7) Π disc 2 = 4.6(1.0)(0.4) 10 3 GeV 4. (8)

36 Backups Summary Table of Moments (Preliminary) Π 1[GeV 2 ] Π 2[GeV 4 ] light (16)(18) 0.297(10)(05) strange 6.57(1)(2) (1)(3) 10 2 charm 4.04(1)(1) (1)(4) 10 4 disconnected 1.5(2)(1) (1.0)(0.4) 10 2 I = (2)(2) 0.017(1)(1) I = (8)(9) 0.148(5)(2) total (9)(10) 0.166(6)(3) Table: Preliminary results on the first two moments of the HVP function. TOTAL ERROR: 1.4% for Π 1, and 4.0% for Π 2.

37 Backups FV via Box Asymmetry, XPT Estimate for Various L I c.f. Aubin et.al., PRD (2016) M π = 135 MeV, T = 1.5L 0.05 M π = 135 MeV, T = 1.5L (6fm) 0.02 (6fm) 0 lat ( )) / Π n = 1 xpt (L) - Π n = lat ( )) / Π n = 2 xpt (L) - Π n = xpt (Π n = ss ts st L fm xpt (Π n = ss ts st L fm i n=1,2(l) = [ Π xpt,i n=1,2 (L) Πxpt n=1,2 ( )], i = ss, ts, st, { i n(l = 6fm) 2% (for the 1st moment, n = 1), Π lat,i n 10% (for the 2nd moment, n = 2). (9)

38 Backups FV via Box Asymmetry, XPT Estimate for Various L II c.f. Aubin et.al., PRD (2016) M π = 135 MeV, T = 1.5L 0.05 M π = 135 MeV, T = 1.5L (6fm) 0.02 (6fm) 0 lat ( )) / Π n = 1 xpt (L) - Π n = lat ( )) / Π n = 2 xpt (L) - Π n = xpt (Π n = ss ts st L fm xpt (Π n = ss ts st L fm i n=1,2(l) = [ Π xpt,i n=1,2 (L) Πxpt n=1,2 ( )], i = ss, ts, st, FV.(L) ± dfv.(l) = [ max{ i } + min{ i } ] /2 ± [ max{ i } min{ i } ] /2 { (22) (for the 1st moment, n = 1), L 6fm 0.015(19) (for the 2nd moment, n = 2). (10)

39 Backups Summary Table of Moments with FV I (Preliminary) Π 1[GeV 2 ] Π 2[GeV 4 ] light (16)(18) 0.297(10)(05) strange 6.57(1)(2) (1)(3) 10 2 charm 4.04(1)(1) (1)(4) 10 4 disconnected 1.5(2)(1) (1.0)(0.4) 10 2 I = (2)(2) 0.017(1)(1) I = (8)(9) 0.148(5)(2) total (9)(10) 0.166(6)(3) I = 1 FV corr (23) 0.015(10) I = 1 + FV corr (8)(9)(23) 0.164(5)(2)(10) total + FV corr (9)(10)(23) 0.182(6)(3)(10) Table: Preliminary results on the first two moments of the HVP function. c.f. HPQCD(arXiv: and PRD2014): Π l 1 = (25) GeV 2, Π l 2 = 0.368(16) GeV 4, Π s 1 = (74) GeV 2, Π s 2 = (11) GeV 4.

40 Backups Summary Table of Moments with FV II (Preliminary) Π 1[GeV 2 ] Π 2[GeV 4 ] light (16)(18) 0.297(10)(05) strange 6.57(1)(2) (1)(3) 10 2 charm 4.04(1)(1) (1)(4) 10 4 disconnected 1.5(2)(1) (1.0)(0.4) 10 2 I = (2)(2) 0.017(1)(1) I = (8)(9) 0.148(5)(2) total (9)(10) 0.166(6)(3) I = 1 FV corr (23) 0.015(10) I = 1 + FV corr (8)(9)(23) 0.164(5)(2)(10) total + FV corr (9)(10)(23) 0.182(6)(3)(10) Table: Preliminary results on the first two moments of the HVP function. c.f. Phenomenology(Benayoun et.al ): Π 1 = 0.990(7) GeV 2, Π 2 = 0.206(2) GeV 4.

41 Backups Why ˆΠ f? ( α ) 2 ( Q max ) dq ( Q 2 al,f LO-HVP = + ω (Q 2 ), π 0 Q max m l ml 2 (11) = al,f LO-HVP (Q Q max) + al,f LO-HVP (Q > Q max), (12) al,f LO-HVP (Q Q max) : computed by lattice simulations, a LO-HVP l,f (Q > Q max) : computed by lattice ˆΠ f (Q max) and perturbations.

42 Backups Why ˆΠ f? ( α ) 2 ( Q max ) dq ( Q 2 al,f LO-HVP = + ω (Q 2 ), π 0 Q max m l ml 2 (11) = al,f LO-HVP (Q Q max) + al,f LO-HVP (Q > Q max), (12) al,f LO-HVP (Q Q max) : computed by lattice simulations, a LO-HVP l,f (Q > Q max) : computed by lattice ˆΠ f (Q max) and perturbations.

43 Backups Why ˆΠ f? ( α ) 2 dq al,f LO-HVP (Q > Q max) = π Q max ( α ) 2 ( Q 2 π = Q max ( α ) 2 = π Q max [( α ) 2 + π dq m l ω dq m l ω Q max m 2 l ( Q 2 m 2 l dq ( Q 2 ω m l ( Q 2 ω )ˆΠ f (Q 2 ), (13) m l ml 2 ) [ (Π 4π 2 qf 2 f (Q 2 ) Π f (Qmax) 2 ) + ( Π f (Qmax) 2 Π f (0) )], ) 4π 2 qf 2 ( Π f (Q 2 ) Π f (Qmax) 2 ) ( al,f LO-HVP ) m 2 l )] (Π f (Q 2 max) Π f (0) ) ( γ l (Q 2 max)ˆπ f (Q 2 max)).(14)

44 Backups Why ˆΠ f? ( α ) 2 dq al,f LO-HVP (Q > Q max) = π Q max ( α ) 2 ( Q 2 π = Q max ( α ) 2 = π Q max [( α ) 2 + π dq m l ω dq m l ω Q max m 2 l ( Q 2 m 2 l dq ( Q 2 ω m l ( Q 2 ω )ˆΠ f (Q 2 ), (13) m l ml 2 ) [ (Π 4π 2 qf 2 f (Q 2 ) Π f (Qmax) 2 ) + ( Π f (Qmax) 2 Π f (0) )], ) 4π 2 qf 2 ( Π f (Q 2 ) Π f (Qmax) 2 ) ( al,f LO-HVP ) m 2 l )] (Π f (Q 2 max) Π f (0) ) ( γ l (Q 2 max)ˆπ f (Q 2 max)).(14)

45 Backups Why ˆΠ f? ( α ) 2 dq al,f LO-HVP (Q > Q max) = π Q max ( α ) 2 ( Q 2 π = Q max ( α ) 2 = π Q max [( α ) 2 + π dq m l ω dq m l ω Q max m 2 l ( Q 2 m 2 l dq ( Q 2 ω m l ( Q 2 ω )ˆΠ f (Q 2 ), (13) m l ml 2 ) [ (Π 4π 2 qf 2 f (Q 2 ) Π f (Qmax) 2 ) + ( Π f (Qmax) 2 Π f (0) )], ) 4π 2 qf 2 ( Π f (Q 2 ) Π f (Qmax) 2 ) ( al,f LO-HVP ) m 2 l )] (Π f (Q 2 max) Π f (0) ) ( γ l (Q 2 max)ˆπ f (Q 2 max)).(14)

46 Backups Continuum Extrap. of Light Component: ˆΠ l ^Πl (2 GeV 2 ) data fit0 fit1 fit2 fit a 2 fm 2 F (ˆΠ l, A, B, C Mπ, C MK ) = ˆΠ l 1 + Aa 2 + ( ) Ba 2 CMπ M π + C MK M K. + ˆΠ l = (42)(60), χ 2 /d.o.f. = 8.2/12 (fit1 case).

47 Backups Continuum Extrap. of Strange Component: ˆΠ s data fit0 fit1 fit2 fit3 ^Πs (2 GeV 2 ) a 2 fm 2 F (ˆΠ s, A, B, C Mπ, C MK ) = ˆΠ s 1 + Aa 2 + ( ) Ba 2 CMπ M π + C MK M K. + ˆΠ s = (1)(2), χ 2 /d.o.f. = 13.6/11 (fit1 case).

48 Backups Continuum Extrap. of Charm Component: ˆΠ c ^Πc (2 GeV 2 ) data fit0 fit1 fit2 fit a 2 fm 2 F (ˆΠ c, A, B, C Mπ, C MK ) = ˆΠ c 1 + Aa 2 + ( ) Ba 2 CMπ M π + C MK M K. + ˆΠ c = (9)(7), χ 2 /d.o.f. = 26.1/9 (fit1 case).

49 Backups Continuum Extrap. of Disc. Component: ˆΠ d ^Πd (2 GeV 2 ) data fit0 fit1 fit a 2 fm 2 F (ˆΠ d, A, B, C Mπ, C MK ) = ˆΠ d 1 + Aa 2 + ( ) Ba 2 CMπ M π + C MK M K. + ˆΠ d = (6)(7), χ 2 /d.o.f. = 3.5/9 (fit1 case).

Muon g 2 Hadronic Vacuum Polarization from flavors of sea quarks using the HISQ action

Muon g 2 Hadronic Vacuum Polarization from 2+1+1 flavors of sea quarks using the HISQ action Jack Laiho Syracuse University April 31, 2015 Motivation The muon anomalous magnetic moment is currently measured