Christine Davies University of Glasgow HPQCD Collaboration

Turning the screws on the Standard Model: theory predictions for the anomalous magnetic moment of the muon Christine Davies University of Glasgow Birmingham February 2019 HPQCD collaboration Outline

1) Introduction : what is the anomalous magnetic moment (a µ ) of the muon and how is it determined (so accurately) in experiment? (recap)

2) Theory calculations in the Standard Model: QED/EW perturbation theory

3) Pinning down QCD effects, using experimental data and using Lattice QCD calculations.

4) Conclusions and prospects e, µ, ⌧ have electric charge and spin Interaction with an external em field p0 has a magnetic component: q 0 µ ! ieu(p0) (p, p0)u(p)A (q) µ µ⌫ µ µ 2 i q⌫ 2 (p, p0)= F1(q )+ F2(q ) p 2m

Electric field interaction (charge consvn): F1(0) = 1 Magnetic field intn, equiv. to scattering from potential : V (x)= µ~ B~ (x) h i · e ~ e µ~ = [F (0) + F (0)] g S~ m 1 2 2 ⌘ 2m Peskin + Schroeder ⇣ ⌘g =2+2F2(0) 1 Introduction

The Standard-Model (SM) value of the muon anomaly can be calculated with sub-parts- per-million precision. The comparison between the measured and the SM prediction provides a test of the completeness of the Standard Model. At present, there appears to be a three- to four-standard deviation between these two values, which has motivated extensive theoretical and experimental work on the hadronic contributions to the muon 1 Introduction anomaly. A lepton (` = e, µ, ⌧)hasamagneticmomentwhichisalongitsspin,givenbythe The Standard-Model (SM) value of the muon anomaly can be calculated with sub-parts- relationship per-million precision. The comparison between the measured and the SM prediction Qe g` 2 provides a test of the completeness of the Standard Model. At present,µ~ ` = thereg` appears~s, g` =2(1 + a`),a` = (1) to be a three- to four-standard deviation between these two values, which has2 motivatedm` 2 Dirac extensive theoretical and experimental work on the hadronic contributions to the muon where Q = 1, e>0andm is the lepton mass. Dirac theory predicts that g 2, anomaly. ± ` | {z } ⌘ A lepton (` = e, µ, ⌧)hasamagneticmomentwhichisalongitsspin,givenbythebut experimentally, it is known to be greater than 2. The small number a,theanomaly, relationship arises from quantum fluctuations, with the largest contribution coming from the mass- Qe independentg single-loop` 2 diagram in Fig. 1(a). With his famous calculation that obtained µ~ ` = g` ~s, g` =2(1 + a`),a` = (1) 2m` a =(↵/2⇡)=02.00116 , Schwinger [1] started an “industry”, which required Aoyama, Dirac Hayakawa, Kinoshita and··· Nio to calculate more than 12,000 diagrams to evaluate the where Q = 1, e>0andm is the lepton mass. Dirac theory predicts that g 2, ± ` | {z } tenth-order (five loop) contribution⌘ [2]. but experimentally, it is known to be greater than 2. The smallAnomalous number amagnetic,theanomaly, moment g 2 arises from quantum fluctuations, with the largest contribution coming from the mass-ae,µ,⌧ = = F2(0) independent single-loop diagram in Fig. 1(a). With his famous calculationγ that obtained 2 γ γ a =(↵/2⇡)=0.00116 , Schwinger [1] started an “industry”, which requiredµ Aoyama, X X ··· µ LO contribn is leptonµ masse− independentµ Hayakawa, Kinoshita and Nio to calculate more than 12,000 diagrams to evaluate the Schwinger 1948 µ γ ↵ µ µ Y tenth-order (five loop) contribution [2]. =0.00116 ...+ γ 2⇡ γ e (a) (b) (c) many higher order pieces ….. γ γ γ New physics could appear in loops Figure 1: The FeynmanX graphs for: (a) The lowest-order (Schwinger) contribution to µ µ X 2 µ e− µ new physics m the lepton anomaly ; (b) Thea vacuum polarization` 1 TeV? contribution, which is one of five µ γ µ µ Y2 ` / m2 γ e+ γ fourth-order, (↵/⇡) , terms; (c) The schematicX contribution of new particles X and Y (a) (b) flavour,CP-conserving(c) Motivates study of µ rather than e that couple to the muon. 8 13 10 10 chirality flipping ⇡ ⇡ Figure 1: The Feynman graphs for: (a) The lowest-order (Schwinger) contribution to the lepton anomaly ; (b) The vacuum polarizationThe contribution, interaction which shown is one in of Fig. five 1(a) is a chiral-changing, flavor-conserving process, fourth-order, (↵/⇡)2, terms; (c) The schematic contribution of new particles X and Y which gives it a special sensitivity to possible new physics [3, 4]. Of course heavier that couple to the muon. particles can also contribute, as indicated by the diagram in Fig. 1(c). For example, 0 X = W ± and Y = ⌫µ, along with X = µ and Y = Z , are the lowest-order weak The interaction shown in Fig. 1(a) is a chiral-changing,contributions. flavor-conserving In the Standard-Model, process, aµ gets measureable contributions from QED, the which gives it a special sensitivity to possible newstrong physics interaction, [3, 4]. and Of course from the heavier electroweak interaction, particles can also contribute, as indicated by the diagram in Fig. 1(c). For example, 0 SM QED Had Weak X = W ± and Y = ⌫µ, along with X = µ and Y = Z , are the lowest-ordera weak= a + a + a . (2) contributions. In the Standard-Model, aµ gets measureable contributions from QED, the strong interaction, and from the electroweak interaction,In this document we present the latest evaluations of the SM value of aµ, and then discuss expected improvements that will become available over the next five to seven years. SM QED Had Weak a = a + a The+ a uncertainty. in this evaluation(2) is dominated by the contribution of virtual hadrons in loops. A worldwide e↵ort is under way to improve on these hadronic contributions. In this document we present the latest evaluations of the SM value of a , and then dis- By the time that theµ Fermilab muon (g 2) experiment, E989, reports a result later cuss expected improvements that will become available over the next five to seven years. The uncertainty in this evaluation is dominatedin by this the contribution decade, the of uncertainty virtual hadrons should be significantly reduced. We emphasize that the in loops. A worldwide e↵ort is under way to improveexistence on these of E821 hadronic at Brookhaven contributions. motivated significant work over the past thirty years By the time that the Fermilab muon (g 2) experiment, E989, reports a result later in this decade, the uncertainty should be significantly reduced. We emphasize that the 2 existence of E821 at Brookhaven motivated significant work over the past thirty years

2 CURRENT STATUS

SM 10 Keshavarzi aµ = 11659182.0(3.6) 10 et al, ⇥ 1802.02995 tantalising 3.7σ discrepancy! details to follow …

higher accuracy small-scale experiments possible (Penning trap) but discrepancies will be tiny …

very hard since decays in 0.3picoseconds …. 2 + + + a =5 10 (LEP) e e e e⌧ ⌧ ⌧ ⇥ ! New determination of α (2018) : Mueller et al (h/MCs) Now SM expt SM 14 a a a = 87(36) 10 e ⌘ Twoe ge-2 anomalies ⇥

2.4σ ‘tension’ and opposite sign to discrepancy for µ potentially adds excitement to the story!

Davoudiasl+Marciano, 1806.10252 Aoyama, Kinoshita and Nio, 1712.06060 for QED calc. http://resonaances.blogspot.co Davoudiasl & Marciano, m/2018/06/alpha-and-g-minus- arXiv:1806.10252 two.html Accurate experimental results + theory calculations needed Muon g-2: spin 0 Ion traps unsuitable:both helicity -1 in rest frame p ⇡+ ⌫ + µ+ ⇡ ! ! µCyclotron radius =so 2 m,get impossible polarised to load, requireµ beam relativistic pulse speeds (lifetime = 2 us). B field perpendicular to ring, µ spin precesses

measure frequency difference !S !C Lepton Dipole Moments Adam West, PIC 2015 Qe m 2 ~ E~ ~! ~! = a B~ + a ⇥ + .. S C m µ µ p c " ✓ ◆ ! # Q = 1,µ± ± from directly gives a need uniform µ possible stable B, measure electric field term vanishes to sub-ppm with EDM at ‘magic momentum’ NMR probes ~ B~ calibrated using gp p =3.094 GeV/c / ⇥ measure spin direction from e produced in weak decay µ+ e+ + ⌫ + ⌫ ! e µ direction of highest energy e correlated with µ spin so N e oscillates at ! ! S C NS62CH10-Roberts ARI 17 September 2012 11:40

a SM Theory

(9.4 ppm) CERN μ+ (10 ppm) CERN μ– (13 ppm) E821(97) μ+ (5 ppm) E821(98) μ+ (1.3 ppm) E821(99) μ+ (0.7 ppm) E821(00) μ+ (0.7 ppm) E821(01) μ– World average 116 590 000 116 591 000 116 592 000 116 593 000 116 594 000 116 595 000 −11 aμ (× 10 )

b Fermilab Status of experiment goal Muon g-2 E989

α 4 Brookhaven 2004 ⎛ ⎞ + hadrionic + weak + ? 2013: E821 ring moved ⎝ πE821⎠ - 0.7ppm α 3 CERN III 1979 ⎛ ⎞ + hadrionic to Fermilab ⎝ π ⎠ α 3 CERN II 1968 ⎛ ⎞ ⎝ π ⎠ Experiment α 2 CERN I 1962 ⎛ ⎞ ⎝ π ⎠

α Nevis 1960 2π

10 100 1E7 1,000 10,000 100,000 1,000,000 −11 σαμ (× 10 ) Figure 2

(a) Measurements of aµ.(b)Theuncertaintyonaµ and the physics reach as the uncertainty decreased.

by Glasgow University on 03/24/14. For personal use only. Abbreviation: SM, Standard Model. Involvement from

that a (m/p)2,sotheEﬁelddidnotcontributetoGermany,ω .3 ByItaly, measuring UK the frequency ω ,and becomes µ = a a ω the magnetic ﬁeld B⃗ calibrated to the Larmor frequency of the free proton p , one can determine

Annu. Rev. Nucl. Part. Sci. 2012.62:237-264. Downloaded from www.annualreviews.org E989 aµ from the relationship ωa /ωp aµ R , 3. = λ ωa /ωp = λ + − + − R where λ µµ /µp 3.183 345 137 (85) is the muon-to-proton magnetic moment ratio (27). Aim: Much higher statistics+ with= + cleaner= injection to ring, more Assuming CPT invariance (aµ aµ ; λ λ ), the result is (28, 29) uniform B field + temp. control : 0.15ppm+ = −i.e+ = − 10 E821 aµ =2 11 10 a 116 592 089(54) (33) (63) 10− ( 0.54 ppm). 4. µ = stat syst tot × ⇥ ± The muon anomaly is one of the most precisely calculated quantities in the SM. To match the experimental precision, it is necessary to calculate QED contributions to ﬁve loops and

3The magic momentum was ﬁrst employed by the third CERN collaboration (25).

240 Miller et al. Muon g-2 now running at Fermilab Run 2018 for 1-3 x E821, first results summer 2019 2017 commissioning run: 0.001% of final stats

t/t N (t)=N e µ e 0 ⇥ [1 + Acos(!at + )]

6 ⌧µ = 60 10 s J-PARC future plan: ⇥ slow µ in 1m ring - no need for ‘magic momentum’ Hadronic corrections to the muon g 2 from lattice QCD T. Blum

Table 1: Standard Model contributions to the muon anomaly. The QED contribution is through a5, EW a2, and QCD a3. The two QED values correspond to different values of a, and QCD to lowest order (LO) contributions from the hadronic vacuum polarization (HVP) using e+e hadrons and t hadrons, higher Hadronic corrections to the muon g 2 from lattice QCD T. Blum ! ! order (HO) from HVP and an additional photon, and hadronic light-by-light (HLbL) scattering.

10 QED 11658471.8845(9)(19)(57)(30) 10 [2] Table 1: Standard Model contributions to the muon anomaly. The QED contribution is through a , EW ⇥ a2, and QCD a3. The two QED values correspond to different values11658471 of a, and. QCD8951 to( lowest9)(19 order)(7 (LO))(77) 10 10 [2] + contributions from the hadronic vacuum polarization (HVP) using e e hadrons and t hadrons, higher ⇥ EW 15.4!(2) 10 10 ! [5] order (HO) from HVP and an additional photon, and hadronic light-by-light (HLbL)⇥ scattering. QCD LO (e+e ) 692.3(4.2) 10 10, 694.91(3.72)(2.10) 10 10 [3, 4] 10 QED 11658471.8845(9)(19)(7)(30) 10 ⇥ 10[2] ⇥ LO (t) 701⇥.5(410.7) 10 [3] 11658471.8951(9)(19)(7)(77) 10 ⇥ [2] 10 ⇥ 10 EW 15.4(2) 10HO HVP 9.79(9) 10 [5] [6] + ⇥ 10 ⇥ 1010 QCD LO (e e) 692.3(4.2) 10 , 694.91(3.72.)((2.10. ) 10 [3, 4] ⇥HLbL 10 5 2 6 ⇥ 10 [9] LO (t) 701.5(4.7) 10 10 ⇥ [3] ⇥ HO HVP 9.79(9) 10 10 [6] ⇥ HLbL 10.5(2.6) 10 10 [9] ⇥ The HVP contribution to the muon anomaly has been computed using the experimentally + measured cross-section for the reaction e e hadrons and a dispersion relation to relate the real The HVP contribution to the muon anomaly has been computed using! the experimentally and imaginary parts of P(Q2). The current quoted precision on such calculations is a bit more than measured cross-section for the reaction e+e hadrons and a dispersion relation to relate the real ! and imaginary parts ofone-halfP(Q2). The of current one percent quoted precision [3, 4]. on The such HVP calculations contributions is a bit more can than also be calculated from first principles one-half of one percentin [lattice3, 4]. The QCD HVP [ contributions8]. While can the also current be calculated precision from is first significantly principles higher for the dispersive method, in latticeAOYAMA QCDet [8 al.]. Whilelattice the calculations current precision are is poised significantlyPHYSICAL to reduceREVIEW higher D 91, for033006 errors the (2015) dispersive significantly method, in next one or two years. These will in the small-q limit. The g 2 term MG is projected out I-operation that deal with the IR divergences [23,47]. See lattice calculationsfrom either the lhs are or the poised rhs− of Eq. to(19) reduce. Considerable errorsSecs. significantlyIII D and III E for in more next details. one or two years. These will numerical cancellationprovide is expected among important the nine terms on checksIn practice, of it is the very difficult dispersive to carry out such method a before the new Fermilab experiment. Unlike provide importantthe lhs of Eq. (19) checks. In fact, the of rhs exhibits the dispersive the consequence methodcalculation before without making the mistakes new because Fermilab of the gigantic experiment. Unlike of such a cancellationthe at the case algebraic forlevel. Thusa starting(HVPsize) of, thea integrals(HLbL and the large) can number of not terms requiredbe computed from experimental data and a dispersion the case forfrom theaµ rhs(HVP enables) us, toa reduceµ (HLbL the amount) can of computingµ not befor computed renormalization.µ from To deal experimental with this problem, we data devel- and a dispersion time substantially (by at least a factor 5), and also to oped an automatic code-generating algorithm, GENCODEN relation (theresignificantly are improve manyrelation the off-shell precision of (there numerical form results. factors are many that[22,23] enter, in off-shell which whichN implies can form that itnot works factors be for measured). the q-type that enter While modelwhich can not be measured). While model calculationsSince exist these (see integrals [9 have] for UV-divergent a summary), subdiagrams, theydiagrams are not of any systematically order N in the perturbation improvable. theory of QED. A determination they must be regularizedcalculations by some means. For the exist diagrams (see [9] for a summary), they are not systematically improvable. A determination of Set V the Feynman cutoff, which is a sort of “mass” for A. Diagram generation using latticethe virtual QCD photons, where works fine all as errors the regulator. are We controlled suppose is therefore desirable. that all of the integrals,using including renormalizationlattice QCD terms, are whereThe Feynman all diagrams errors of Set V are have the controlled structure that ten is therefore desirable. In Sec.initially2 regularizedwe review by the Feynman the status cutoff. Of of course, lattice the calculationsvertices along the electron of aµ line(HVP are connected). Section by the virtual3 is a presentation final renormalizedAccurate result is finite experimental and well defined in the photons resultsa, b, c, d, and e ,+ and thustheory are specified bycalculations the pairing needed of our resultslimit of for infiniteaµ cutoff(HLbL mass. In) computed Sec. 2 inwe the review frameworkpatterns of the how these of status lattice vertices are of QCD+QED. connected. lattice Excluding calculations Section 4 gives of ouraµ (HVP). Section 3 is a presentation patterns that are not one-particle irreducible, and taking conclusions and outlook for future calculations.time-reversal invariance into account, we obtain 389 differ- III. FORMULATION a ( ) of our results for µ entHLbL patterns which arecomputed represented by the diagrams in the of Fig. framework1. of lattice QCD+QED. Section 4 gives our MostQED of these diagrams corrections are so huge and complicated dominate that They are denoted - bycalculate Xnnn, nnn 001; 002 in; … ;Perturbation389. theory numerical integrationconclusions is currently the only viable and option. outlookThe diagram for X001 future represents the calculations.¼ diagram in the upper left However, in order to evaluate them on a computer, which corner of Fig. 1. Subsequent expressions represent diagrams requires that every step of the computation↵ is finite, it ishigherplaced below X001orders until X025, anddepend X026 corresponds on to the ratios necessary to remove all sources0. of5 divergence of an diagram placed to the right of X001, and so on. Diagrams integrand before carrying out the integration. This isofX001 lepton to X072 are time-reversalmasses: symmetric and diagrams achieved by the introduction of K-operation⇡ that deals X073 to X389 are asymmetric. Within each group they are with the UV divergences [22,47], and R-subtraction and arranged in a lexicographical order. ... Z Z W subset of 5 ... Z diagrams at ↵ Z

integrationW Figure 1: Representative diagrams, up to order a3, in the Standard Model that contributechallenging- to the muon use anomaly. The rows, from to top to bottom, correspond to QED, EW, and QCD. HorizontalVEGAS solid lines represent the muon, wiggly lines denote photons unless otherwise labeled, other solid lines are leptons, ﬁlled loops denote quarks (hadrons), and the dashed line represents the higgs boson.Aoyama, Kinoshita et al PRD91:033006(2015), Figure 1: Representative diagrams, up to ordererr:PRD96:019901(2017)a3, in the Standard Model that contribute to the muon anomaly. The rows, from to top to bottom,For correspond α use a e to or QED, Rb/Cs EW, and QCD. Horizontal solid lines represent the muon, wiggly3 lines denote photons unless<0.5ppb otherwise labeled, other solid lines are leptons, FIG. 1. Overview of 389 diagrams which represents 6354 vertex diagrams of Set V. The horizontal solid lines represent the electron propagators in a constantﬁlled weak magnetic loops field. denoteSemicircles stand quarks for photon propagators. (hadrons), The left-most and figures theare denoted dashed as X001– line represents the higgs boson. X025 from the top to the bottom. The top figure in the second column from the left is denoted as X026, and so on.

033006-4

3 62. The muon anomalous magnetic moment 1 62. The Muon Anomalous Magnetic Moment

Updated August 2017 by A. Hoecker (CERN) and W.J. Marciano (BNL). e The Dirac equation predicts a muon magnetic moment, M⃗ = gµ S⃗,with 2mµ gyromagnetic ratio gµ =2.Quantumloopeffects lead to a small calculable deviation from gµ =2,parameterizedbytheanomalousmagneticmoment gµ 2 aµ − . (62.1) ≡ 2 That quantity can be accurately measured and, within the Standard Model (SM) framework, precisely predicted. Hence, comparison of experiment and theory tests the exp SM at its quantum loop level. A deviation in aµ from the SM expectation would signal effects of new physics, with current sensitivity reaching up to mass scales of (TeV) [1,2]. For recent and thorough muon g 2reviews,seeRefs.[3–5]. O − The E821 experiment at Brookhaven National Lab (BNL) studiedtheprecessionof µ+ and µ− in a constant external magnetic field as they circulated in a confining storage ring. It found [7] 1 − aexp =11659204(6)(5) 10 10 , µ+ × exp −10 a − =11659215(8)(3) 10 , (62.2) µ × where the first errors are statistical and the second systematic. Assuming CPT invariance and taking into account correlations between systematic uncertainties, one finds for their average [6,7] − aexp =11659209.1(5.4)(3.3) 10 10 . (62.3) µ × These results represent about a factor of 14 improvement overtheclassicCERN experiments of the 1970’s [8]. Improvement of the measurement by a factor of four by setting up the E821 storage ring at Fermilab, and utilizing a cleaner and more intense muon beam is in progress with the commissioning of the experiment having started in 2017. SM The SM prediction for aµ is generally divided into three parts (see Fig. 62.1 for representative Feynman diagrams) SM QED EW Had aµ = aµ + aµ + aµ . (62.4) The QED part includes all photonic and leptonic (e, µ, τ)loopsstartingwiththeclassic α/2π Schwinger contribution. It has been computed through 5 loops[9] α α 2 α 3 aQED = +0.765 857 425(17) +24.050 509 96(32) µ 2π π π α 4 ! " α 5 ! " +130.879 6(6 3) +753.3(1.0) + (62.5) π π ··· ! " ! " Hoecker+ 1 The original results reported by the experiment have been upMarcianodated in Eqs. (62.2) RPP 2017 and (62.3) to the newest value for the absolute muon-to-proton magnetic ratio λ = QED exp 3.183 345 107(84)aµ [6].= The 0.00116 change + induced 0.00000413 in aµ …with + 0.000000301 respect to the value of λ = 3.183 345 39(10) used in Ref. + 7 0.00000000381 amounts to +1.12 + 0.000000000050910−10. + … × using Rb α = 11,658,471.895(8) x 10-10 C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40,100001(2016)and2017update December 1, 2017uncertainty 09:36 from error in α but missing α6 (light-by- light) also this size Hadronic corrections to the muon g 2 from lattice QCD T. Blum

Table 1: Standard Model contributions to the muon anomaly. The QED contribution is through a5, EW a2, and QCD a3. The two QED values correspond to different values of a, and QCD to lowest order (LO) contributions from the hadronic vacuum polarization (HVP) using e+e hadrons and t hadrons, higher ! ! order (HO) from HVP and an additional photon, and hadronic light-by-light (HLbL) scattering.

QED 11658471.8845(9)(19)(7)(30) 10 10 [2] ⇥ 11658471.8951(9)(19)(7)(77) 10 10 [2] ⇥ EW 15.4(2) 10 10 [5] ⇥ QCD LO (e+e ) 692.3(4.2) 10 10, 694.91(3.72)(2.10) 10 10 [3, 4] ⇥ ⇥ LO (t) 701.5(4.7) 10 10 [3] ⇥ HO HVP 9.79(9) 10 10 [6] ⇥ HLbL 10.5(2.6) 10 10 [9] ⇥

The HVP contribution to the muon anomaly has been computed using the experimentally measured cross-section for the reaction e+e hadrons and a dispersion relation to relate the real ! and imaginary parts of P(Q2). The current quoted precision on such calculations is a bit more than one-half of one percent [3, 4]. The HVP contributions can also be calculated from first principles in lattice QCD [8]. While the current precision is significantly higher for the dispersive method, lattice calculations are poised to reduce errors significantly in next one or two years. These will provide important checks of the dispersive method before the new Fermilab experiment. Unlike the case for aµ (HVP), aµ (HLbL) can not be computed from experimental data and a dispersion relation (there are many off-shell form factors that enter which can not be measured). While model calculations exist (see [9] for a summary), they are not systematically improvable. A determination using lattice QCD where all errors are controlled is therefore desirable. In Sec. 2 we review the status of lattice calculations of aµ (HVP). Section 3 is a presentation of our results for aµ (HLbL) computed in the framework of lattice QCD+QED. Section 4 gives our conclusions and outlook for future calculations.

Electroweak contributions from Z, W, H Gnendiger et al, H ⌫ 1306.5546 ... Z Z

W 2 EW is small - suppressed by powers of mµ aµ m2 H piece tiny 2 W at 1-loop; GF mµ 5 1 aEW(1) = + (1 4s2 )2 2-loops µ 2 W p28⇡ 3 3 γ γ γ γ  3 (a) (b) (c) (d) Figure 1: Representative diagrams, up to order a , in157 the Standard Model that contribute to the muon 10 f f µµ = 19.480(1) 10 156 W anomaly. The rows, from to top to bottom,⇥ correspond to QED, EW,H and QCD. HorizontalH γ, Z solid linesZ γ 155 f " µµµµµµµµZ γ represent the muon, wiggly lines denote photons unless11 otherwise labeled, other solid lines are leptons, EW(2) 10 " 154 10 !

a = 4.12(10) 10 µ 153

ﬁlled loops denote quarks (hadrons), and the dashed lineEW represents the higgs boson. ⇥ Μ Figure 1: Sample two-loop diagrams: Higgs-dependent bosonic (a) and a 152 fermionic (b) diagram, diagram with γγZ-fermion triangle (c) and γ–Z mixing EW 10 a = 15.36(10) 10 151 (d). µ 150 ⇥ 50 100 150 200 250 300 350

M H ! GeV"

3 The Standard Model electroweak contributions are split up into one-loop, EW two-loop and higher orders as Figure 5: Numerical result for aµ as a function of the Higgs boson mass. The vertical band indicates the measured value of MH .Thedashedlinescorrespond aEW = aEW(1) + aEW(2) + aEW(2) + aEW(≥3), (7) to the uncertainty of the final result, quoted in Eq. (25). µ µ µ;bos µ;ferm µ where the two-loop contributions are further split into bosonic and fermionic contributions, as discussed below. parametrization at the two-loop level. We provide exact numerical results for The one-loop contribution is given by [4,5]3 the full bosonic and the Higgs-dependent fermionic two-loopcontributions,for the latter also analytical results. These results are supplemented2 by updates EW(1) GF mµ 5 1 2 2 −11 aµ = + (1 4sW ) =(194.80 0.01) 10 , (8) of the most advanced available results on all other electrow√2eak8π2 contributions.3 3 − ± × Our final result obtained from Eqs. (8), (9), (19), (21), (22),(23),(24)reads! " 2 2 2 where sW =1 MW /MZ is the square of the weak mixing angle in the on- EW −11 − 2 2 aµ =(153.6 1.shell0) renormalization10 scheme. One-loop(25) contributions suppressed by mµ/MZ or ± 2 × 2 −13 mµ/MH are smaller than 10 and hence neglected here. The parametrization and is illustrated in Fig. 5. We assess thein final terms theory of GF erroralready ofthesecontribu- absorbs important higher-order contributions. Theerror tions to be 1.0 10−11.ThisisthesamevalueastheonegiveninRef.[14]in Eq. (8) is due to the uncertainty of the input parameters, inparticularof ± × for the overall hadronic uncertainty fromthe the W-boson diagrams mass. of Fig. 1(c), which is now by far the dominant source of error of theBefore electroweak discussing con higher-ordertributions. The contributions we briefly explain possible parametrizations in terms of G and α.Theone-loopcontributioninEq.(8) error from unknown three-loop contributions and neglected two-loop termsF sup- 2 2 2 has been parametrized in terms of GF .Generally,n-loop contributions are pressed by M /m and (1 4s )issignificantlysmallerandtheerrordueto(n−1) Z t − W proportional to GF α ,anditispossibletoreparametrizeα in terms of the experimental uncertainty of the Higgs boson,other quantities. W-boson, a Possibilitiesnd top-quark are mass to replace α by a running α at the scale −12 is well below 10 and thus negligible. of the muon mass or the Z-boson mass, or to replace α α(GF ), where 2 2 → Our result is consistent with the previousα(GF ) √ evaluations2GF s M /π of= α theelec-(1 + ∆r). The quantity ∆r summarizes radia- ≡ W W × troweak contributions in Refs. [4, 5, 14], whosetive corrections central val toues muon range decay. between Different choices amount to differences which (153 ...154) 10−11,butthelargeuncertaintyduetotheunknownHiggsbo-are formally of the order n +1. Wewillalwayschoose α in the Thomson limit, × son mass has been reduced. In comparison,i.e. the given recent by 5-loop Eq. (5b). calculation [6] has shifted the QED result by +0.8 10−11.WecannowcombineEq.(25)andtheWe now turn to the first set of contributions with noticeable dependence × on the Higgs boson mass: the bosonic two-loop contributions aEW(2).They result of Ref. [6] with the hadronic contributions. We take the recent leading µ;bos are defined by two-loop and associated counterterm diagrams without a closed order evaluations of Refs. [7] and [8] and the higher order results of Refs. [8,11]. fermion loop, see Fig. 1(a) for a sample diagram. They are conceptually The resulting difference between the experimental result Eq. (1) and the full 3 Standard Model prediction is: In the literature sometimes the experimental value for MW instead of the theory value is used. If we would use the current value of MW =80.385 ± 0.015 GeV [17] instead of Eq. (6), EW(1) ± × −11 the result− would11 be shifted to aµ =(194.81 0.01) 10 . exp SM (287 80) 10 [7], aµ aµ = ± × −11 (26) − !(261 80) 10 [8]. ± × 3 The Standard Model theory error remains dominated by the non- electroweak hadronic contributions. The QED and electroweak contributions

9 Hadronic corrections to the muon g 2 from lattice QCD T. Blum

Table 1: Standard Model contributions to the muon anomaly. The QED contribution is through a5, EW a2, and QCD a3. The two QED values correspond to different values of a, and QCD to lowest order (LO) contributions from the hadronic vacuum polarization (HVP) using e+e hadrons and t hadrons, higher ! ! order (HO) from HVP and an additional photon, and hadronic light-by-light (HLbL) scattering.

2 QCD contributions to a µ start at α , nonpert.... in QCD Z Z Blum et al, W 1301.2607 q `

3 Figure 1: Representative diagrams, up to order a , in the StandardHigher Model thatorder contribute Hadronic to the muon anomaly. The rows, from to top to bottom, correspond to QED, EW, and QCD. Horizontal solid lines represent theLO muon, Hadronic wiggly lines vacuum denote photons unless otherwise labeled,vacuum other polarisation solid lines are leptons, ﬁlled loops denotepolarisation quarks (hadrons), (HVP) and the dashedHadronic line represents light- the(HOHVP) higgs boson. dominates uncertainty by-light, not well in SM result known but small Since QED, EW known accurately,3 subtract from expt and compare QCD calculations to remainder E821 10 aµ = 11659209.1(6.3) 10 QED ⇥ 10 EW 10 a = 11658471.895(8) 10 aµ = 15.36(10) 10 µ ⇥ ⇥ Hadronic (and other) contributions = EXPT - QED - EW E821 QED EW 10 a a a = 721.9(6.3) 10 µ µ µ ⇥ HV P HOHV P HLBL new physics = aµ + aµ + aµ + aµ Focus on lowest order hadronic vacuum polarisation (HVP), so take: “consensus” value HLbL 10 a = 10.5(2.6) 10 will return to this µ ⇥ HOHV P 10 NLO+NNLO aµ = 8.85(9) 10 Kurz et al, ⇥ 1403.6400 HVP,no new physics 10 a = 720.2(6.8) 10 µ ⇥ EW Note: much larger than aµ Leading order of hadronic contribution (HVP) Dispersion relations and VP insertions in g 2 ! Hadronic vacuum polarization (HVP) Starting point: 2 2 Optical TheoremVμ (unitarity)Vν =( for theq g photonµ propagatorqµq )V (q ) s ¯ HVP + Vµ = Qf fµf quark’sIm⇥⇤ ( sEM) = current⌅tot(e e : anything) 4⇤ ⇥ f How to calculate a µ - Two approaches: X ! Optical Theorem + Analyticity (causality), may be expressed in form of a so–called (subtracted) 1) ( e e hadrons) dispersion + dispersion relation relations.s + Im (s)= ⇤ (e e X) ! V 4⇥ tot ! ! Analycity 2) lattice QCD - “first principles” 2 ⌅ 2Im (s) 2 k k ⇥⇤1 ImV (s) ⇥⇤ (k ) ⇥⇤ (0) = ds . V (s) V (0)⇤ = s (s k2 i⇧ds) 2 Analyticity+optical 0 ⇥ theorem4m2 s(s k i) + Z ⇡ (e e hadrons) 2 1) ! had HV P 1 1 0 ⇥ aµ = dshad(s)K(s) had F.%Jegerlehner’s%lecture 3 had 2 had 2 4⇡ 2 (q ) ⇥tot (q ) Zm⇡ ⇡0 F. Jegerlehner SFB/TR 09 Meeting, Aachen, November 14, 2011 68 + e e ⇤ hadrons threshold ! ! q µ K(s) kernel emphasises low s - integral q + dominated by ⇢ , ⇡ ⇡ . Use pert. QCD at high s. 0 is ‘bare’, with running α effects removed . Final state em radiation IS included - γ inside hadron bubble Need to combine multiple sets of experimental data from many hadronic channels (+ inclusive) inc. correlations New data sets from KLOE, BESIII, SND(Novosibirsk) ..

New results Keshavarzi, Nomura, Teubner ps 1802.02995 : 70% redn in uncty since 2011. New data, more channels, correlations

had, LO VP (a) Fractional contributions to aµ . HVP 10 KNT18 a = 693.3(2.5) 10 agree well - µ ⇥ Davier et al, HVP 10 0.4% uncty a = 693.1(3.4) 10 1706.09436 µ ⇥ 3.5σ from no HVP 10 Jegerlehner aµ = 688.8(3.4) 10 new physics. 1705.00263 ⇥

(5) 2 (b) Fractional contributions to ↵had(MZ )

Figure 20: Pie charts showing the fractional contributions to the total mean value (left pie chart) and 2 had, LO VP (5) 2 (error) (right pie chart) of both aµ (upper panel) and ↵had(MZ ) (lower panel) from various had, LO VP energy intervals. The energy intervals for aµ are defined by the boundaries m⇡, 0.6, 0.9, 1.43, (5) 2 2.0 and GeV. For ↵had(MZ ), the intervals are defined by the energy boundaries m⇡, 0.6, 0.9, 1.43, 2.0, 4.0,1 11.2 and GeV. In both cases, the (error)2 includes all experimental uncertainties (including 1 2 2 all available correlations) and local min/d.o.f. inflation. The fractional contribution to the (error) from the radiative correction uncertainties are shown in black and indicated by ‘rad.’.

analysis is

had, LO VP 10 a = (693.26 1.19 2.01 0.22 0.71 ) 10 µ ± stat ± sys ± vp ± fsr ⇥ 10 = (693.26 2.46 ) 10 , (3.28) ± tot ⇥ where the uncertainties include all available correlations and local 2 inﬂation as discussed in had, LO VP Section 2.2.2. Using the same data compilation as described for the calculation of aµ , had,VP the next-to-leading order (NLO) contribution to aµ is determined here to be

had, NLO VP 10 a =( 9.82 0.02 0.03 0.01 0.02 ) 10 µ ± stat ± sys ± vp ± fsr ⇥ 10 =( 9.82 0.04 ) 10 . (3.29) ± tot ⇥

27 2) Lattice QCD Blum, hep-lat/0212018 ↵ 1 q HV P,i 2 2 2 ˆ 2 q aµ = dq f(q )(4⇡↵ei )⇧i(q ) ⇡ 0 µ Z q ‘connected’ contribution for flavour i Integrate over Euclidean q2 – f(q2) diverges at small q2 with 2 scale set by m µ so q 0 dominates ⇡ Renormalised vacuum polarisation function J J ⇧ˆ(q2)=⇧(q2) ⇧(0) vanishes at q2=0 This is (Fourier transform of) vector meson correlators. Can perform q2 integral using time-moments of standard correlatorrs calculated in lattice QCD to determine meson masses. HPQCD,1403.1778 Lattice QCD: fields defined on 4-d discrete space-(Euclidean) time. Lagrangian parameters: ↵s,mqa 1) Generate sets of gluon fields for Monte Carlo integrn of Path Integral (inc effect of u, d, s, (c) sea quarks) 2) Calculate valence quark propagators and combine for “hadron correlators” . Average results over gluon fields. Fit for hadron masses and amplitudes • Determine a to convert results in lattice units to physical units. Fix m q from hadron mass *numerically extremely challenging* • cost increases as a 0 ,m phys ! u/d ! a and with statistics, volume. Using Darwin@Cambridge,

www.dirac.ac.uk Allows us to calculate quark propagators rapidly and store them for flexible re-use.

Inversion of 107 x 107 sparse matrix solves the Dirac equation for the quark propagator on a given gluon field configuration. Must repeat thousands of times for statistical precision. ‘2nd generation’ gluon field configs generated by MILC including HPQCD’s HISQ sea quarks. Physical u/d quark masses now possible. u/d (same mass), s and c sea quarks m = m MILC HISQ, 2+1+1 u d 0.14 = ml HISQ = Highly 0.12 mass of u,d improved staggered quarks - quarks 0.1 very accurate 2 discretisation 0.08 E.Follana, et al, / GeV

2 HPQCD, hep-lat/ 0610092. m 0.06 real mu,d ms/10 0.04 ⇡ world physical 0 m⇡ = 0.02 mu,d ms/27 135 MeV ⇡ 0 Volume: 0 0.005 0.01 0.015 0.02 0.025 0.03 m⇡L>3 a2 / fm2 5

2j TABLE II: Columns 2-5 give the Taylor coecients ⇧j (Eq. 6), in units of 1/GeV , for each of the lattice data sets in Table I. The errors given include statistics and the (correlated) uncertainty from setting the lattice spacing using w0,whichdominates. Estimates of the connected contribution from s-quarks to aµ,HVP are given for each of the [1, 0], [1, 1], [2, 1] and [2, 2] Pad´e 10 approximants‘connected’ in columns s quark 6-9; results contribution are multiplied by 10 to. HVP Chakraborty et al, a 10 10 10 10 Set ⇧1 ⇧2 ⇧3 ⇧4 µ [1, 0] 10HPQCD[1 ,1403.17781] 10 [2, 1] 10 [2, 2] 10 ⇥ ⇥ ⇥ ⇥ 10HISQ.06598(76) quarks0 on.0516(11) configs 0.0450(15) 55.00.0403(19) 58.11(67) 53.80(59) 53.95(59) 53.90(59) 20.06648(75) 0.0523(11) 0.0458(15) 0.0408(18) 58.55(66) 54.19(58) 54.33(59) 54.29(59) 30with.06618(75) u, d, s and0.0523(11) c sea. 0 .0466(15) 54.05.0425(20) 58.28(66) 53.93(58) 54.09(58) 54.04(58) 40.06614(74) 0.0523(11) 0.0467(15) 0.0427(19) 58.25(65) 53.90(57) 54.06(58) 54.01(57)

50.06626(74) 0.0527(11) 0.0473(15)10 0.0438(19) 58.36(65) 53.99(57) 54.15(57) 54.10(57) Local Jv - nonpert. Zv. 54.0

60.06829(77) 0.0557(12) 0.0514(17)10 0.0490(22) 60.14(67) 55.55(59) 55.73(59) 55.67(59) 70.06619(74) 0.0524(11) 0.0468(15) 0.0430(19) 58.29(65) 53.93(57) 54.10(57) 54.05(57) multiple a (fixed by w0), ⇥

80.06625(74) 0.0526(11) 0.0470(15)s µ 53.05.0429(19) 58.34(65) 53.98(57) 54.14(57) 54.09(57)

a 90ml (inc..06616(77) phys.),0.0531(12) volumes. 0.0483(17) 0.0450(22) 58.27(68) 53.87(59) 54.04(60) 53.99(59) 10 0.06630(72) 0.0534(11) 0.0487(16) 53.00.0458(20) 58.39(64) 53.98(56) 54.15(56) 54.10(56) Tune s from ⌘s HV P,s 10 52.5 aµ = 53.4(4) 10 0.005 0.010 0.015 0.020 0.025 TABLE III: Error budgets for connected contributions to the ⇥ 2 2 muon anomalyallowingaµ forfrom missing vacuum polarization QED of s and c quarks. a (fm )

s c aµ aµ HPQCD 1403.1778 Uncertainty in lattice spacing (w0, r1): 0.4% 0.6% Uncertainty in ZV :0.4%2.5% ETMC 1411.0705 MonteCarlostatistics: 0.1% 0.1% a2 0extrapolation: 0.1% 0.4% BMW 1711.04980 ! QED corrections: 0.1% 0.3%u,d,s,c sea Quarkmasstuning: 0.4% 0.4% Finite lattice volume: < 0.1% 0.0% RBC/UKQCD 1606.01767 Padéapproximants: < 0.1% 0.0% Total: 0.7% 2.7%u,d,s sea 50 51 52 53 54 55 56 57 58 10 aHVP,s 1010 Re+e < 55 10 µ ⇥ ⇡ ⇥ mistuning of the sea and valence light-quark bare masses: FIG. 4: Lattice QCD results for the connected contribution to msea mphys x q q (10) the muon anomaly aµ from vacuum polarization of s quarks. sea ⌘ phys Results are for three lattice spacings, and two light-quark q=u,d,s ms lat lat phys X masses: m` = ms/5 (lower, blue points), and m` = m` val phys ms ms (upper, red points). The dashed lines are the corresponding xs phys . (11) values from the fit function, with the best-fit parameter val- ⌘ ms ues: ca2 =0.29(13), csea = 0.020(6) and cval = 0.61(4). 10 The gray band shows our final result, 53.41(59) 10 ,with For our lattices with physical u/d sea masses xsea is very ⇥ 2 lat phys small. a errors from staggered ‘taste-changing’ e↵ects m` = m` , after extrapolation to a =0. will remain and they are handled by ca2 . The four fit 2 parameters are aµ, ca2 , csea and cval ; we use the following (broad) Gaussian priors for each: TABLE IV: Contributions to aµ from s and c quark vacuum polarization. Only connected parts of the vacuum polariza- s 10 tion are included. Results, multiplied by 1010, are shown for aµ =0 100 10 ± ⇥ each of the Padéapproximants. ca2 = 0(1) csea = 0(1) cval = 0(1). (12) Quark [1, 0] 1010 [1, 1] 1010 [2, 1] 1010 [2, 2] 1010 ⇥ ⇥ ⇥ ⇥ Our final result for the connected contribution for s 57.63(67) 53.28(58) 53.46(59) 53.41(59) s quarks to g 2 is: c 14.58(39) 14.41(39) 14.42(39) 14.42(39) s 10 a = 53.41(59) 10 . (13) µ ⇥ The fit to [2, 2] Padéresults from all 10 of our configu- The error budget for our result is given in Table III. ration sets is excellent, with a 2 per degree of freedom The dominant error, by far, comes from the uncertainty of 0.22 (p-value of 0.99). In Fig. 4 we compare our fit in the physical value of the Wilson flow parameter w0, with the data from configurations with ms/m` equal 5 which we use to set the lattice spacings. We estimate the and with the physical mass ratio. uncertainty from QED corrections to the vacuum polar- ‘connected’ c quark contribution to aHVP HPQCD, µ 1208.2855, + 1403.1778 (e e hadrons via cc) 5 7 4.5 ! 4 3.5 3 1/2 1/4 1/6 1/8 GV GV GV GV 2.5 ▲ BES (2001)

4 6 8 10 R(s) Set mca 2 2 2 4 2 6 2 8 2 , ❍ MD-1 Z a Z a Z a Z a 1.5 J/ψψ pQCD ▼ CLEO 10.6220.5399(1)1.2162(1)1.7732(1)2.2780(1) 1.4 1 ■ BES (2006) ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ 0.5 20.630.5339(1)1.2054(1)1.7581(1)2.2584(1) 0 n = 10 2 3 4 5 6 7 8 9 10 20.660.5135(1)1.1692(1)1.7081(1)2.1941(1) 1.2 √ s (GeV) 30.6170.5434(1)1.2223(1)1.7817(1)2.2888(1) ) 1 5

40.4130.7586(1)1.6351(1)2.3887(2)3.0952(2) 4.5 4 50.2731.0681(1)2.2705(2)3.3454(3)4.3601(4) For c case3.5 can directly 1.0 n = 8 3

60.1931.4323(3)3.0397(5)4.4990(7)5.8738(8) (GeV 2.5 R(s)

) compare2 lattice, correlator 2 1.5 ψ

1 n

TABLE IV: Results in lattice units for time moments of the ( 0.8 0.5

/ time-moments0 to e+e- expt J/⇤ correlator as deﬁned in eq. (10). We give results for n=4, 1 3.8 4 4.2 4.4 4.6 4.8 ) 6, 8 and 10. n = 6 - agree√ s (GeV) to 1.5% 0.6 5 4.5 V 1/2 V 1/4 V 1/6 V 1/8 4

(G4 ) (G6 ) (G8 ) (G10) moment 3.5 2 3 (amc) extrapolation 0.18 0.18 0.16 0.16 th 0.4 2.5 n Lattice QCD, gives: R(s) ( 2 , statistics 0.05 0.04 0.03 0.03 1.5 ψ lattice spacing 0.32 0.51 0.43 0.30 1 n = 4 HVP,c0.5 10 seaquarkextrapolation 0.14 0.13 0.12 0.12 0.2 0 a 3.65= 3.675 3.714 3.725. 3.754(4) 3.775 3.8 3.82510 3.85 3.875 3.9 M tuning 0.15 0.18 0.17 0.16 µ c √ s (GeV)⇥ Z1.230.610.410.31 0.0 Figure 3: R(s)fordiﬀerent energy intervals around the charm threshold region. The electromagnetism 0.3 0.2 0.1 0.05 0.0 0.1 0.2 0.3 0.4 0.5 solid line corresponds to the theoretical prediction, the uncertainties obtained from the Total (%) 1.3 0.9 0.7 0.5 variation of the input parameters and of µ are indicated by the dashed curves. The inner 2 and outerHVP error,b bars give the statistical and systematical uncertainty,10 respectively. (amc) aµ = 0.27(4) 10 TABLE V: Complete error budget for the time moments of bottom case are obvious. ⇥ the J/⇤ correlator as a percentage of the ﬁnal answer. Below 3.73 GeV only u, d and s quarks are produced. To allow for a smooth transition

FIG. 4: Results for the 4th, 6th, 8th and 10th time moments 6 of the charmonium vector correlator shown as blue points and + + Re e = ⌃(e e hadrons)/⌃pt [22, 23]. The values, plotted as a function of lattice spacing. The errors shown (the extracted from experiment by [22] and appropriately nor- same size or smaller than the points) include (and are domi- malised for the comparison to ours, are: nated by) uncertainties from the determination of the current renormalization factor, Z, that are correlated between the exp 2 2 1/2 1 (M 4!/(12⇧ e )) =0.3142(22) GeV points. The data points have been corrected for c quark mass 1 c mistuning and sea quark mass e⇥ects, but the corrections are exp 2 2 1/4 1 (M2 6!/(12⇧ ec )) =0.6727(30) GeV smaller than the error bars (the value for the deliberately exp 2 2 1/6 1 mistuned c mass on set 2 is not shown). The blue dashed (M 8!/(12⇧ e )) =1.0008(34) GeV 3 c line with grey error band displays our continuum/chiral ﬁt. exp 2 2 1/8 1 (M 10!/(12⇧ e )) =1.3088(35) GeV . (12) Experimental results determined from R + (eq. (12)) are 4 c e e plotted as the black points at the origin o⇥set slightly from Our results from lattice QCD have approximately double the y-axis for clarity. the error of the experimental values but together these results provide a further test of QCD to better than 1.5%. ⌅c. The form factor is related to the matrix element of the vector current between the two mesons by:

C. (J/⇤ ⇥c) 2 µ 2V (q ) µ⇥⇤ ⌅c(p⇥) c⇥ c J/⌥(p) = p⇥ p⇥⇤J/⌃,⇤ ⇥ | | ⇤ (MJ/⌃ + M⌅c ) The radiative decay of the J/⌥ meson to the ⌅c re- (13) quires the emission of a photon from either the charm Note that the right-hand-side vanishes unless all the vec- quark or antiquark and a spin-ﬂip, so it is an M1 transi- tors are in di⇥erent directions. Here we use a normalisa- tion. Because it is sensitive to relativistic corrections this tion for V (q2) appropriate to a lattice QCD calculation rate is hard to predict in nonrelativistic e⇥ective theories in which the vector current is inserted in one c quark line and potential models (see, for example, [24, 25]) Here only and the quark electric charge (2e/3) is taken as a we use a fully relativistic method in lattice QCD with separate factor. The decay rate is then given by [8]: a nonperturbatively determined current renormalisation and so none of these issues apply. In addition, of course, 64 q 3 (J/⌥ ⌅ ⇥)= V (0) 2, (14) the lattice QCD result is free from model-dependence. c QED | | 2 27(M⌅c + MJ/⌃) | | The quantity that parameterises the nonperturbative QCD information (akin to the decay constant of the pre- where it is the form factor at q2 = 0 that contributes be- vious section) is the vector form factor, V (q2), where q2 cause the real photon is massless. q is the corresponding | | is the square of the 4-momentum transfer from J/⌥ to momentum of the ⌅c in the J/⌥ rest-frame. 10

HVP TABLE III. Light-quark connected contribution to aµ and the slope and curvature of the renormalized vacuum polarization beforeUP/DOWN and after applying ﬁnite-volume,contribution, discretization, largest and M ⇡andcorrections. most difficult 10 ll ll 2 t ll 4 - signal/noise10 a µworse(conn.) and results ⇧1 (conn.)(GeV) ⇧2 (conn.)(GeV) a (fm) raw corrected raw corrected raw corrected ⇡ 0.15sensitive 572(12) to u/d mass 624(13) 0.0814(18) 0.0916(28) 0.1250(54) 0.217(20) 0.15 570(6) 623(8) 0.08117(94) 0.0913(22) 0.1271(30) 0.216(18) J J 0.12*NEW* 580(9) HPQCD/Fermilab/MILC 627(10) 0.0828(14) 0.0919(24) 0.1308(45) 0.216(19) 0.09 605(9) 634(10) 0.0868(15) 0.0929(21) 0.1463(51) 0.217(17) 0.06 608(15) 629(15) 0.0871(24) 0.0915(25) 0.1438(73) 0.196(13) result (updating HPQCD 1601.03071) u/d physical mu/d only - high stats. large-t correlator dominated 2 686 with a /dof = 0.34 and p =0.71. The ﬁt posterior for 660 with FV + discretization corrections by ρ but also has ππ - fit to 687 cs is tiny because the sea-quark masses are well tuned, and Mπ adjustment 688constrainwhile ca2 is smalldata and has a 100% uncertainty because we 640 ππ689 mangledalready removed on thecoarse dominant taste-breaking discretiza- 690 tion e↵ects from our data. Note that c 2 = 5(1) for the a lattices691 raw values and in in Fig. finite-vol.9. ll

µ 620

a 692 To study the stability of the values and errors in

10 Correct with chi.pt. 693 Eq. (3.2), we consider a number of ﬁt variations including

10 600 2 HVP694 adding,u/d higher-order terms in a and10mf , doubling the raw values aµ695 prior widths= on 630(8) the fit parameters,10 and omitting the two 580 696 coarsest ensembles. All of the alternate fits yield results ll ⇥ connected,697 for aµ(conn.) m thatu=m ared statistically, no QED consistent with our ll 560 698 central fit. Further, the uncertainty on aµ(conn.)does 699 not change except when the prior widths are doubled; in 0 0.005 0.01 0.015 0.02 0.025 700 this case the fitgives uncertainties +~1.5% are still only 20% larger a2 2 ll ⇠ (fm ) HPQCD/Fermilab/MILC:1710.11212701 than in Eq. (3.2). Because aµ(conn.) is robust against 702 reasonable fit variations, we conclude that the fit error FIG. 9. (color online.)Lattice-spacingdependenceof703 captures the systematic uncertainty associated with the ll aµ (conn.) before (open blue circles) and after (filled dark- 704 continuum extrapolation. pink squares) finite-volume, taste-breaking, and M⇡ correc- 705 We follow the same approach for the slope and cur- tions are applied. The dark-pink line shows the fit of all 706 vature of the renormalized vacuum polarization, first corrected data points to the function in Eq. (3.1). The hor- 707 applying finite-volume and taste-breaking discretiza- izontal light-pink band shows the continuum-limit result for ll 708 tion corrections, and then extrapolating to the con- aµ (conn.) from this fit. 709 tinuum limit using Eq. (3.1). We obtain for the ll 2 710 continuum-limit values ⇧1 (conn.)=0.0920(15) GeV ll 4 711 and ⇧2 (conn.)= 0.2089(82)GeV .Thep values of the where m m mphys, and ⇤ =0.5 GeV is of order the f ⌘ f f 712 fits are 0.83 and 0.41, respectively. The lattice-spacing QCD scale. This is similar to the fit function employed 713 dependence of the corrected Taylor coecients is even in Ref. [22], except that we no longer include terms to 714 milder than for aµ, and cannot be statistically resolved. extrapolate in the valence-quark mass because all of our 715 For ⇧ , we obtain c 2 = 0.12(90), which is consis- 1 a data are at the physical light-quark mass. The first term 716 tent with the result from the continuum extrapolation in parentheses adjusts for small sea-quark mass mistun- ll 717 of aµ(conn.) because the quantities are proportional at ing, while the second removes generic discretization er- 718 lowest order in the Taylor expansion. For ⇧2, we obtain rors; we employ priors for the coecients cs =0.0(3) 719 ca2 =0.15(99), which is again consistent in magnitude, ll and ca2 = 0(1). The values of aµ(conn.) on each en- 720 but is opposite in sign because ⇧2 is negative. As for semble are statistically independent; we include in our ll 721 aµ(conn.) , the sea-quark mass dependence of ⇧1 and ⇧2 fit correlations between the two a 0.15 fm ensembles ll ⇡ 722 is tiny. Finally, the continuum-limit values ⇧1 (conn.) from using the same ZV , and between all ensembles from ll 723 and ⇧2 (conn.) are both stable against the fit variations the common value of w0 used to convert lattice-spacing ll 724 discussed above for a (conn.). units to GeV. We obtain from fitting our full data set to µ Eq. (3.1)

ll 725 IV. RESULTS aµ(conn.) = 630.1(8.3),

cs =0.01(30), ll ll ll 726 Here we present our ﬁnal results for aµ(conn.), ⇧1 , ⇧2 , ca2 = 0.50(79), (3.2) HVP 2 727 and aµ (LO) and the slope and curvature of ⇧(Q )with

b PHYSICAL REVIEW D 93, 000000 (XXXX) 9 16 1 Estimate of the hadronic vacuum polarization disconnected contribution to LO-HVP 10 + 2 the anomalous magnetic moment of the muon from lattice QCD aµ,ud . 10 [19] F. Jegerlehner, in KLOE-2 workshop on e e collider 180 650 200physics at 1 GeV, INFN-Laboratori Nazionali di Frascati, 1 1,* 1 2 3 3 8 Italy, 26-28 October 2016 (2017) arXiv:1705.00263 [hep- 14 10 Mainz 17

3 Bipasha Chakraborty, C. T. H. Davies, J. Koponen, G. P. Lepage, M. J. Peardon, and S. M. Ryan = 2 f 170 (TMR+FV) ph]. 4 1SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom 600 190 [20] T. Blum, Phys. Rev. Lett. 91, 052001 (2003), arXiv:hep- 2 x 10 5 Laboratory for Elementary-Particle Physics, Cornell University, Ithaca, x 10 New 160 York 14853, USA x 10 3 lat/0212018 [hep-lat]. 6 School of Mathematics, Trinity College, Dublin 2, Ireland HPQCD 16 550 180[21] C. Aubin and T. Blum, Phys.Rev.D 75:114502,2007 ,ud ,ud

7 (Received 11 December 2015) LO-HVP τ

LO-HVP e,ud 150 LO-HVP µ

This work a (Phys.Rev.D75:114502,2007), hep-lat/0608011. a a = 2+1+1 N f

8 1 The quark-line disconnected diagram is a potentially important ingredient in lattice QCD calculations of N [22] X. Feng, K. Jansen, M. Petschlies, and D. B. Ren- 9 the hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon. It is also a 550 575 600 625 650 675 ner, Phys.Rev.Lett. 107, 081802 (2011), arXiv:1103.4818

8 10 notoriously difficult one to evaluate. Here, for the first time, we give an estimate14 of this contribution based 10 LO-HVP 10 [hep-lat]. 13.6 aµ,s . 10 11 on lattice QCD results that have a statistically significant signal, albeit at one value of the lattice spacing and 53.5 22.5[23] M. Della Morte, B. Jager, A. Juttner, and H. Wittig, x 10 x 10 x 10 JHEP 03, 055 (2012), arXiv:1112.2894 [hep-lat]. 12 an unphysically heavy value of the u=d quark mass. We use HPQCD’s method13.5 of determining the = 2 f 9 53.0 Mainz 17 (TMR) 22.0[24] F. Burger et al. (ETM), JHEP 1402,099(2014),

13 anomalous magnetic moment by reconstructing the Adler function from time moments of the current- ,s ,s

LO-HVP τ arXiv:1308.4327 [hep-lat].

LO-HVP e,s 13.4 LO-HVP µ a 14 current correlator at zero spatial momentum. Our results lead to a total (includinga u, d and s quarks) quark- a [25] T. Blum, M. Hayakawa, and T. Izubuchi, Proceedings, 15 line disconnected contribution to a μ of −0.15% of the u=d hadronic vacuumOther polarization 180 recent contribution withresults, also mu=md 650 RBC/UKQCD 16 20030th International Symposium on Lattice Field Theory = 2+1 N f

16 an uncertainty which is 1% of that contribution. 8 8 (Lattice 2012): Cairns, Australia, June 24-29, 2012 ,PoS 14 10 14 170 10 10.0 BMW(1711.04980): 3.0 ~1million 12.0600 190LATTICE2012, 022 (2012), arXiv:1301.2607 [hep-lat]. 17 DOI: HPQCD 14 [26] E. B. Gregory, Z. Fodor, C. Hoelbling, S. Krieg, L. Lel- x 10 x 10 x 10 x 10 x 10 160 x 10 7.5louch, R. Malak, C. McNeile, and K. Szabo (Budapest- correlators per point, bound ETM 17 2.0 5508.0 180Marseille-Wuppertal), Proceedings, 31st International ,ud ,c = 2+1+1 N f ,ud ,c LO-HVP τ LO-HVP τ

LO-HVP e,ud 150 LO-HVP µ LO-HVP +e,c LO-HVP µ This work 5.0Symposium on Lattice Field Theory (Lattice 2013),PoS a a a a 18 I. INTRODUCTION effectively including⇡ a ⇡ all possibilities from fordata. final states Large that a-45 a N LATTICE2013, 302 (2014), arXiv:1311.4446 [hep-lat]. would be seen if the two diagrams were cut in half. 46 50 51 52 53 54 55 56 [27] R. Malak, Z. Fodor, C. Hoelbling, L. Lellouch, A. Sas- 19 2 The high accuracy with which the magnetic moment of 8 14 10

8 a can alsodependence14 be determined from lattice(handled QCD calcu- by47 10 LO-HVP tre, and K. Szabo (Budapest-Marseille-Wuppertal), Pro- 20 the muon can be determined in experiment makes it a very μ;HVP 13.6 3.0 10.0 a . 1010 lations using a determination of the vacuum polarization 48 53.5 µ,c 22.5 1.5ceedings, 32nd International Symposium on Lattice Field

21 useful quantity in the search for new physics beyond the x 10 x 10 x 10

2 x 10 function at Euclidean-extrapolation, x 10 13.5q 2.0values [7]. It is important rather that this than49 x 10 7.5 Theory (Lattice 2014),PoSLATTICE2014,161(2015), 22 Standard Model. Its anomaly, defined as the fractional Mainz 17 (TMR) is done to at least a comparable level of uncertainty to that 50 = 2 f arXiv:1502.02172 [hep-lat]. ,disc ,disc 53.0 22.0 1.0 LO-HVP τ 23 difference of its gyromagnetic ratio from the naive ,s ,s

LO-HVP e,disc LO-HVP µ [28] B. Chakraborty et al. (HPQCD), Phys.Rev. D89,114501 correcting 1.0 ). 5.0 LO-HVP τ a

LO-HVP e,s 13.4 LO-HVP µ a a a − 24 value of 2 [a g − 2 =2]isknownto0.5ppm[1]. a a (2014), arXiv:1403.1778 [hep-lat]. μ − − HPQCD 14 25 The anomaly arises¼ð fromÞ muon interactions with a cloud [29] T. Blum, P. A. Boyle, T. Izubuchi, L. Jin, A. J¨uttner, 0.0 2.5 ETM 17 0.5 26 of virtual particles and can therefore probe the existence of 0.000 0.005 0.010 0.015 0.020 0.000 0.005 0.010 0.015 0.020 C.0.000 Lehner,0.005 K. Maltman,0.010 M. Marinkovic,0.015 0.020 A. Portelli, 8 14 10

= 2+1+1 N f This work 2 2 2 2 10.0and M. Spraggs, Phys.2 Rev.2 Lett. 116,232002(2016), 27 particles that have not been seen directly. The theoretical N 3.0 a [fm ] 12.0 a [fm ] arXiv:1512.09054 [hep-lat].a [fm ]

28 calculation of a μ in the Standard Model shows a discrep- x 10 Also x 10 calculate small -ve x 10 14 14.5 15 15.5 [30]7.5B. Chakraborty, C. T. H. Davies, J. Koponen, G. P. Lep- 29 −10 ancy with the experimental result of about 25 8 × 10 aLO-HVP . 1010 ð Þ 2.0 8.0 µ,disc LO-HVP age, M. J. Peardon, and S. M. Ryan, Phys. Rev. D93, ,c 30 [2–4] which could be an exciting indication of new ,c LO-HVP τ FIG.‘disconnectedLO-HVP e,c S4. Continuum contribn’ extrapolation of theLO-HVP variousµ ﬂavor contributions to a` (Q 2GeV)5.0074509 (2016), obtainedusing arXiv:1512.03270t [hep-lat].c =(3.000 a 31 physics. Improvements by a factor of 4 in the experi- a a [31] G. Bali and G. Endr˝odi, Phys. Rev. D92,054506(2015),± 32 mental uncertainty are expected and improvements in the 0.134) fm for the ud contribution and tc =(2.600 0.134) fm for the disconnectedRBC/UKQCD 15 one. From left to right, ` = e, µ, ⌧. From = 2+1 f ± arXiv:1506.08638 [hep-lat]. 8 33 theoretical determination would make the discrepancy (if top14 to bottom, the connected light, strange, 10 charm, and disconnected contributions. The[32] redT. Blum openet al. circles (RBC/UKQCD), with errors JHEP 04 are,063(2016),the 3.0 10.0 34 it remains) really compelling [5]. results from our 15 simulations for the ud and s, 13 for the charm and 12 for the disconnected 1.5[Erratum: contributions, JHEP05,034(2017)], with arXiv:1602.01767 statistical [hep- x 10 35 The current theoretical uncertainty is dominated by that x 10 x 10 lat]. 2.0 ‘disc’ has u, d, s on 7.5 This work 36 from the lowest order (α2 ) hadronic vacuum polarization uncertainties. These points have been interpolated to the physicalBMW17 mass point using the ﬁts[33] B. to Chakraborty, all lattice C. T. spacings H. Davies, P. (solid G. de Oliviera, lines). J. Ko-

QED = 2+1+1 N f ,disc ,disc 1.0ponen, G. P. Lepage, and R. S. Van de Water, Phys. Rev. N each side, suppressed LO-HVP τ 37 (HVP) contribution, in which the virtual particles are LO-HVP e,disc 1.0 LO-HVP µ 5.0

The di↵erent lines represent the ﬁts obtained by imposing cuts in a (solid for no cut,a dashed for a 0.118 fm, dotted for a a -14 -12 -10 -8 -6 -4 D96, 034516 (2017), arXiv:1601.03071 [hep-lat]. −

38 strongly interacting, depicted in Fig. 1. This contribution, − −  a 0.111 fm, dot-dashedby q masses for a 0since.095fm). The fact that a few of the lines do not appear[34] S. to Borsanyi, fit the Z. redFodor, points T. Kawanai, is due S. Krieg, to the L. Lel- 39 which we denote a μ;HVP, is currently determined most 0.0 2.5 0.5   FIG. S7. Comparison of our results for the individual flavor louch, R. Malak, K. Miura, K. K. Szabo, C. Torrero, 40 accurately from experimental results on e e− → hadrons dependence0.000 on other0.005 lattice0.010 parameters0.015 0.020 in those fits,0.000 which0.005 is slightly0.010 di↵erent0.015 from0.020 the one corresponding0.000 0.005 to0.010 the solid0.015 line.0.020 The þ contributions to aLO-HVP with those from other lattice collab- and B. Toth (Budapest-Marseille-Wuppertal), (2016), 41 or from τ decay to be of order 700 × 10−10 with a 1% 2 2 Qf =0 *We findµ disc.2 contrib.2 2 2 green squaresg are thea continuum[fm ] extrapolatedorations. results The for reference the for given thea [fm resultsQ] max plottedand are HPQCDtc, with 14 statisticalarXiv:1612.02364 and continuum [hep-lat].a [fm ] extrapolation 42 uncertainty or better [3,4,6]. This method for determining FIG. 1. The hadronicerrors vacuum only. polarization contributionu,d,s to the F1:1 sensitive[28], RBC/UKQCD to 15 [29], RBC/UKQCD 16 [32], HPQCD [35] M. Della Morte, A. Francis, V. Gülpers, G. Herdo´ıza, 43 a μ;HVP does not distinguish the two diagrams of Fig. 1 muon anomalous magnetic moment is represented as a shadedX F1:2 16 [33], Mainz 17 [35] and ETM [36]. For Mainz 17 latter, G. von Hippel, H. Horch, B. Jäger, H. B. Meyer, A. Nyf- 44 because it uses experimental cross-section information, LO-HVP feler, and H. Wittig, (2017), arXiv:1705.01775 [hep-lat]. blob insertedFIG. into the photon S4. propagator Continuum (represented extrapolation by a wavy F1:3 of the variousTMR and flavor TMR+FV contributions stand for the particular to a results` of Ta-(Q 2GeV) obtainedusing tc =(3.000 line) that corrects the pointlike photon-muon coupling at the top F1:4 ble 7 in that paper. [36] D. Giusti, V. Lubicz, G. Martinelli, F. Sanfilippo,± and S. Simula, 1707.03019v2. * of each diagram.0.134) The top fm diagram for is the the connectedud contribution contribution F1:5 and tc =(2.600 0.134) fm for the disconnected one. From left to right, ` = e, µ, ⌧. From [email protected] ± and the lowertopall diagram toof thethesebottom, quark-line situations the disconnected connected in (but the con- light, sameF1:6 strange, way and charm, discuss and them disconnectedOur contributions. results for a TheLO-HVP[37] redP. Boyle,(Q open V.2 Gülpers, GeV) circles J. Harrison, have with large errors A. Jüttner, lattice are C. Lehner, the nected by gluons denoted by curly lines) contribution that is F1:7 `,disc,A.lat Portelli, and C. T. Sachrajda, 1706.05293v1. Published by the American Physical Society under the terms of Eur. Phys. J. C71, 1515 (2011), [Erratum: Eur. Phys. [38] B. Chakraborty, C. T. H. Davies, C. DeTar, A. X. El- the Creative Commons Attribution 3.0 License. Further distri- discussed here.resultstogether. The shaded from box in our the lower 15 diagram simulations indicates forF1:8 the ud and s, 13 for the charmartefacts, and 12 for as shown the disconnected in the bottom contributions, panels of withFig. S4. statistical This Khadra, E. Gámiz, S. Gottlieb, D. Hatton, J. Kopo- bution of this work must maintain attribution to the author(s) and strong interactionuncertainties. effects that could occur These between points the two quark haveF1:9 been interpolatedJ.C72,1874(2012)], to the physical arXiv:1010.4180 mass [hep-ph]. point using the fits to all lattice spacings (solid lines). the published article’s title, journal citation, and DOI. loops. For the light-quarkF1:10 contribution[17] K. Hagiwara, to R. Liao,is A.because D. Martin, the D. taste No- violationsnen, A. S. of Kronfeld, the ud J. Laiho,contribution G. P. Lepage, en- Y. Liu, TheLO-HVP di↵erent lines represent the fits obtained bymura, imposing and T. Teubner, cuts J. in Phys.a (solidG38,085003(2011), for no cut, dashedP. B. Mackenzie, for a C. McNeile,0.118 E. fm, T. Neil, dotted J. N. Simone, for a (Q 2 GeV), the dependence on mesonarXiv:1105.3149 masses [hep-ph].hance the SU(3)-flavor cancellationR. Sugar, D. Toussaint, against R. S. V. the de Water,s contri- and A. Va- a ` 0.111 fm, dot-dashed for a 0.095fm). The fact that a few of the linesLO-HVP do not appear2 to fit the red points is due to the  [18] J. P. Miller, E. de Rafael,bution B. L. in Roberts,a and, as a quero,is increased. 1710.11212v1. In these results we is not significant statistically and the termsD. associated Stöckinger, Ann. Rev. Nucl. Part. Sci. 62µ,,237(2012).disc,lat [39] B. Colquhoun, R. J. Dowdall, C. T. H. Davies, K. Horn- 1dependence Published by on the American other Physical lattice Society parameters in those fits, which is slightlyneglect di↵erent the from charm the contribution one corresponding which to we the find solid to line. be less The greenwith squaresthis dependence are the continuum can be ignored. extrapolated However, results as can for the be given Q and t , with statistical and continuum extrapolation thanmax 1% of thec total disconnected contribution on our errorsseen in only. the upper panels of Fig. S4, the dependence on a2 is strong, due to the sensitivity of this contribution to coarsest lattice, i.e. much smaller than the disconnected, low-energy, two-pion states which, in turn, are sensitive statistical error. In addition, because statistical errors to taste splittings. The fact that the anomalous moment are quite large, no dependenceLO-HVP on quark mass is required all of these situations in the same way and discuss them Our results for a`,disc,lat(Q 2 GeV) have large lattice together.of the lighter e is more sensitive to these states than toartefacts, describe as the shown lattice in the data. bottom panels of Fig. S4. This that of the µ that is, in turn, more sensitive than that is because the taste violations of the ud contribution en- For the light-quarkLO-HVP contribution to ofLO-HVP the ⌧, explains the fact that a (Q 2 GeV) has a (Q 2 GeV), the dependencee on meson masses hance the SU(3)-flavor cancellation against the s contri- the` strongest a2 dependence while aLO-HVP(Q 2GeV) LO-HVP 2 is not significant statistically and the⌧ terms associated bution in a , as a is increased. In these results we has the weakest.  µ,disc,lat with this dependence can be ignored. However, as can be neglect the charm contribution which we find to be less seenThe in situation the upper is panels di↵erent of for Fig. the S4, strange the dependence contribution, on thanAs explained 1% of the in total the main disconnected text, the systematiccontribution error on as- our amuch2 is strong, less a↵ dueected to by the taste sensitivity violations. of this As the contribution second pan- to sociatedcoarsest lattice,with these i.e. continuum much smaller limits than and the physical-point disconnected, low-energy,els of Fig. S4 two-pion show, the states continuum which, in limits turn, are are very sensitive mild. interpolationsstatistical error. are obtained In addition, by imposing because fourstatistical cuts on errors the toThey taste are splittings. much less The so fact for thatthe charm, the anomalous as shown moment in the latticeare quite spacing large, in no the dependence quark-connected on quark case mass and is three required for ofthird the panels, lighter duee is to more the large sensitive value oftom thesec in lattice states units. than theto describe disconnected the lattice contributions. data. The results of these cuts thatHere of it the is theµ magneticthat is, in moments turn, more of the sensitive more massive than that lep- are then combined, as detailed in the main text, to give a tons which are steeper, due to theirLO-HVP sensitivity to larger central value and statistical and systematic errors. The of the ⌧, explains the fact that ae 2 (Q 2 GeV) has Q. In addition2 to the dependence on aLO-HVP, a linear depen- results for the various contributions to the magnetic mo- the strongest2 a dependence while a⌧ (Q 2GeV) hasdence the on weakest.MK is needed for both contributions and one ments of all three leptons with the four values of the on M⌘c is required to correct a slight mistuning of the momentum cut Qmax considered are summarized in Ta- charmThe situation mass in that is di quark’s↵erent for contribution. the strange contribution, bleAs S2. explained in the main text, the systematic error as- much less a↵ected by taste violations. As the second pan- sociated with these continuum limits and physical-point els of Fig. S4 show, the continuum limits are very mild. interpolations are obtained by imposing four cuts on the They are much less so for the charm, as shown in the lattice spacing in the quark-connected case and three for third panels, due to the large value of mc in lattice units. the disconnected contributions. The results of these cuts Here it is the magnetic moments of the more massive lep- are then combined, as detailed in the main text, to give a tons which are steeper, due to their sensitivity to larger central value and statistical and systematic errors. The Q. In addition to the dependence on a2, a linear depen- results for the various contributions to the magnetic mo- 2 dence on MK is needed for both contributions and one ments of all three leptons with the four values of the

on M⌘c is required to correct a slight mistuning of the momentum cut Qmax considered are summarized in Ta- charm mass in that quark’s contribution. ble S2. 16

TABLE VI. Individual flavor contributions to the leading Taylor coecients of the vacuum-polarization function and the muon anomaly. The first error quoted for the u/d contributions is from the lattice analysis; the second comes from uncertainties in our estimates of the e↵ects of strong isospin-breaking, electromagnetism, and quark disconnected diagrams. Results for strange and heavier quarks include only the quark-connected contributions and are not new, but come from earlier HPQCD calculations [16–18]; disconnected contributions are expected to be negligible. The definitions of the Taylor coecients include the factor of the quark’s electric charge squared.

10 HVP HVP 2 HVP 4 Contribution 10 aµ (LO) ⇧1 (LO)(GeV ) ⇧2 (LO)(GeV ) light 623.1(8.3)(13) 0.0907(15)(19) 0.2029(82)(71) strange 53.40(60) 0.007291(78) 0.00587(12) charm 14.40(40) 0.001840(49) 0.0001240(43) bottom 0.270(40) 0.0000342(48) 2.28(37)e 07 + - TotalTotal LO HVP contribution 691(15) - compare lattice 0 .0999(24)QCD and e e 0.209(11) equivalent to testing vs

1132 = 0.15fm) resolves issues around how to handle statisti- no new physics 1133addcal u/d, uncertainties s and at c: large Euclidean times. Such a sam- 1134 ple is numerically expensive to obtain on finer lattices, HVP 10 Lattice QCD FNAL/HPQCD/MILC a1135 although= 691(15) tripling the10 statistics is certainly feasible us- 2019 µ 1136 ing the truncated⇥ solver method. We estimate that this ETMC, 1808.00887 11372%would uncty reduce from our total systs. uncertainty to 1%. To get be- RBC/UKQCD 1138 low 1% requires a reduction in the uncertainty on the 1801.07224 1139 physical value of w0 that determines the lattice spacing BMW, 1711.04980 1140Lattice(w0/a is QCD determined future: very precisely, see Table I). This un- Nf =2 Mainz/CLS, 1705.01775 1141•certainty QED,currently m =reliesm on a determination of the pion HPQCD/RV 1142 decay constant,uf⇡, on thed lattice [36]. The error budget u, d, s, c sea 1601.03071 6 1143mustin [36] showsbe inc. that thefully dominant uncertainties are related u, d sea 1144 to statistical precision and extrapolation to the physical Pheno. + Keshavarzi et al. e e 1802.02995 1145inpoint conn. where +w0 fdisc.⇡ is fixed against experiment (assuming + 1146 a value of V from nuclear physics). An improvement by e e Davier et al.,1706.09436 HVP ud sum rules + 1147 a factor of two in this uncertainty seems feasible with the e e + ⌧ Jegerlehner, 1705.00263 1148 higher statistics gluon-field ensembles now available with + Benayoun et al. • clarify large-t e e + ⌧ 1507.02943 1149 physical mu/d on finer lattices. Analysis on QCD+QED 1150 gluonbehaviour field ensembles (with will be important in the long-term 610 630 650 670 690 710 730 1151 here too to take into account fully the fact that that f⇡ 10 HVP 1152 isstats the decay and/or constant ππ of ) an electrically-charged particle. 10 aµ (LO) 1153 Other quantities should also be investigated for fixing the 1154 lattice spacing. FIG. 12. (color online.) Comparison of our result in Eq. (5.1) 1155 Given the above discussion, a reduction in uncertainty for the leading-order hadronic-vacuum-polarization contribu- 1156 on the lattice-QCD result for the hadronic vacuum po- tion to the muon anomalous magnetic moment (magenta 1157 larization contribution to the muon g 2to 0.5% square) with results from Nf 2latticeQCD[14, 15, 20, 50] ⇡ + 1158 is certainly feasible on the timescale of the new exper- (blue and purple squares), and from experimental e e cross- section data [5–7, 51] (red and orange triangles). The filled 1159 iments. This would give precision comparable to that HVP,LO 1160 currently available from using experimental information black circle shows the value of aµ that is implied by the + measurement of a by BNL experiment E821 [1]assuming1161 on e e hadrons and would allow lattice-QCD re- µ ! no contributions beyond the Standard Model; vertical dashed 1162 sults to play a significant role in the unfolding story of lines denote the 1 range [22]. 1163 the search for new physics in the anomalous magnetic ± 1164 moment of the muon.

ll 1123 the light-quark connected contribution aµ(conn.)in 1165 ACKNOWLEDGMENTS 1124 Eq. (5.4). The error budget (Table IV) is dominated 1125 by the lattice-spacing uncertainty, statistical errors and 1126 the continuum extrapolation. The last two can be re- 1166 We thank Bob Sugar for his scientiﬁc leadership and 1127 duced by increasing statistics, so that the results at each 1167 tireless e↵ort to obtain computational resources, with- 1128 lattice spacing value are more precise and the extrapo- 1168 out which the MILC physics program would never have 1129 lation has more accurate information to work with. We 1169 been realized. We thank B. Chakraborty and J. Koponen 1130 have demonstrated here that a calculation with nearly 1170 for generating data employed in this analysis. We thank 1131 0.5 million correlators (our high statistics sample at a 1171 Thom Primer for initial studies of, and parameter-tuning MITP/16-079

0 Lattice calculation of the pion transition form factor ⇡ ⇤⇤ ! 1, 1, 2, 1, Antoine Gérardin, ⇤ Harvey B. Meyer, † and Andreas Nyffeler ‡ 1PRISMA Cluster of Excellence and Institut für Kernphysik, Johannes Gutenberg-UniversitätMainz, 55099 Mainz, Germany 2Helmholtz Institute Mainz, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany

0 2 2 We calculate the ⇡ ⇤⇤ transition form factor 0 (q ,q )inlatticeQCDwithtwo ! F⇡ ⇤⇤ 1 2 ﬂavors of quarks. Our main motivation is to provide the input to calculate the ⇡0-pole contribution HLbL;⇡0 to hadronic light-by-light scattering in the muon (g 2), aµ . We therefore focus on the region where both photons are spacelike up to virtualities of about 1.5GeV2, which has so far not been experimentally accessible. Results are obtained in the continuum at the physical pion mass by a combined extrapolation. We reproduce the prediction of the chiral anomaly for real photons with an accuracy of about 8 9%. We also compare to various recently proposed models and ﬁnd reasonable agreement for the parameters of some of these models with their phenomenological values. Finally, HLbL;⇡0 we use the parametrization of our lattice data by these models to calculate aµ .

I. INTRODUCTION

The anomalous magnetic moment of the muon provides one of the most precise tests of the Standard Model of particle physics [1, 2]. It is known to comparable precision in experiment [3] and theory but the results disagree by about 3 4 standard deviations [4] depending on the theoretical estimate. To interpret this tension as a sign of new physics, improving the accuracy is of primary importance. On the experimental side, new experiments at Fermilab and J-PARC are expected to reduce the error by a factor of four [5]. Therefore, a corresponding theoretical e↵ort is necessary to fully beneﬁt from the increased experimental precision. The theory error of (g 2)µ is dominated by hadronic contributions: the hadronic vacuum polarization (HVP) and hadronic light-by-light scattering (HLbL). + The ﬁrst contribution can be related to the cross section e e hadrons using a dispersion relation such that the estimate can, in principle, be improved by accumulating more! data. Also, in recent years, more and more precise lattice QCD calculations of the HVP have become available but are not yet competitive with the dispersive approach [6–9]. However, the HLbL contribution to the muon g 2 cannot fully be related to direct experimental information and current determinations usually rely on model assumptions where systematic errors are dicult to estimate [1, 10, 11]. However, recently a dispersive approach was proposed [12] which relates the, presumably, numerically dominant pseudoscalar-poleElephant contribution, in the room? as depicted inhadronic Fig. 1, and the pionlight-by-light loop in HLbL with on-shell contribution 0 intermediate pseudoscalar states to measurable form factors and cross sections with o↵-shell photons: ⇤⇤ ⇡ , ⌘, ⌘0 + 0 0 ! and ⇤⇤ ⇡ ⇡, ⇡ ⇡ .Not Furthermore, simply increasingly related realistic latticeto experiment, calculations of the HLbL values contribution obtained to the muon use g 2 have! been carried out recently [13, 14]. Also, the hadronic light-by-light scattering amplitude per se has been calculated on the lattice inlarge [15]. Nc, chiral pert. th. etc. Within the dispersive framework, the pseudoscalar-pole contribution requires as hadronic input the transition form 2 2 0 factor (q ,q ) describing the interaction of an on-shell pseudoscalar meson,HLbL P = ⇡ , ⌘, ⌘0,withtwoo↵-shell 10 FP⇤⇤ 1 2 ‘Glasgow Consensus’ 2009: a = 10.5(2.6) 10 µ ⇥ dominated by π0 exchange : arXiv:1607.08174v2 [hep-lat] 15 Dec 2016 0 ⇡ , ⌘ , ⌘0 there also OPE constraints + ... ⇡ 10% possible? with improved

11 dispersive approaches (with Figure 1. Pseudoscalar-pole contribution to hadronic light-by-light scattering in the muonimp.g 2. expt The blobs for on e.g. the right-hand 0 side represent the P ⇤⇤ transition form factors with P = ⇡ , ⌘, ⌘0. ! 0.3 2 2 F 0 (q ,q ) Nyffeler, 1602.03398 ⇡ ⇤⇤ 1 2 0.25 Colangelo et al, 1702.07347 ⇤ [email protected]

] 0.2 † [email protected] 1 ny↵[email protected] ‡ 0.15 Lattice QCD calcs of [GeV 0.1 n2 =1 n2 =6 n2 =2 n2 =8 can test these approaches 0.05 n2 =3 n2 =9 n2 =4 n2 =10 n2 =5 n2 =11 0 Mainz, 00.10.20.30.40.50.60.7 ✓ [rad] 1607.08174,1712.00421

2 2 2 Figure 6. Left: Sampling of our data in the (q1 ,q2 ) plane. Right: The form factor for di↵erent values of ~q1 .Foreachvalueof 2 2 2 2 2 2 2 2 2 | | 2 2 ~q1 = n (2⇡/L) =(q2 q1 m⇡) /(4m⇡) q1 , one gets a curve by varying continuously the value of tan(✓ ⇡/4) = q1 /q2 . Data| | correspond to the lattice ensemble O7.

2. Fits in four-momentum space

In this section, we propose to compare our results with the phenomenological models introduced in Sec. II. In particular, since we are using Wilson fermions, the chiral symmetry is lost even in the chiral limit and is recovered only once the results are extrapolated to the continuum and chiral limit. It is then important to check that our results are in agreement with the ABJ anomaly. On the lattice, the form factor is obtained as a continuous function of !1 for each value of the discretized spatial momentum ~q 2 and a typical example for the lattice ensemble F6 is depicted in Fig. 6. Therefore, to fit the form | 1| factor, we first have to sample our data. We have selected values of !1 such that data points are regularly distributed 2 2 along each curve in the (q1,q2) plane as depicted in the left panel of Fig. 6. However, as discussed in Sec. IV C, no significant di↵erence has been observed by using di↵erent samplings.

We first compare our data with the VMD model. Two fitting procedures have been used. In the first method, each lattice ensemble is fitted independently using Eq. (6) with ↵ and MV treated as free parameters. Then, in a second step, the two parameters are extrapolated to the chiral and continuum limit assuming a linear dependence in both 2 2 2 the lattice spacing a/a=5.3 and y = m⇡/8⇡ F⇡ . The results are summarized in Table VIII (Appendix B). In the second fitting procedure, a global fit is performed where all lattice ensembles are fitted simultaneously assuming a linear dependence in both a/a=5e.3 and y for each parameter of the model. In this case, we are left with only six fit parameters and the results are given in Table IX (Appendix B). Both methods give similar results and choosing the second method, with a reduced number ofe fit parameters, as our preferred estimate, we obtain at the physical point VMD 1 VMD ↵ =0.243(18) GeV ,MV =0.944(34) GeV , (38) where the covariance matrix is (in appropriate units of GeV)

4 4 VMD +3.16 10 3.62 10 ij (↵,MV )= ⇥ 4 ⇥ 3 . (39) 3.62 10 +1.14 10 ✓ ⇥ ⇥ ◆ The covariance matrix is estimated from a jackknife procedure and used in Sec. V for error propagation, but the fits are uncorrelated fits. As can be seen in Fig. 7 (top panel), the VMD model leads to a poor description of our data (2/d.o.f. =2.94), especially in the double virtual case and at large Euclidean momenta. It is a direct evidence that the wrong asymptotic behavior of this model, compared to the OPE prediction in Eq. (5), already matters at Euclidean momenta of order Q2 1 GeV2. In particular we do not recover the anomaly result in the chiral and continuum ⇠ 2 2 1 limit. However, fitting our data with the constraint Qi < 0.5 GeV (i =1, 2) leads to ↵ =0.268(21) GeV and | | 1 MV =0.870(45) GeV where ↵ is now compatible with the theoretical prediction ↵th =0.274 GeV . Also, in the latter case we get a much better chi-squared 2/d.o.f. =1.29. It confirms that the VMD model is unable to describe our data in the whole kinematical range studied here. We have repeated the same analysis for the LMD model (7) using ↵, and MV as free parameters and the results are summarized in Tables VIII and IX (Appendix B). The first fitting procedure suggests that lattice artifacts for the 2

The HVP contribution is the largest hadronic contri- xop, ν bution and can be computed from a dispersion relation + and experimental e e annihilation data. This is a well- z,κ developed method with a fractional-percent error. The y,σ x, ρ 10 leading-order HVP contribution is 692.3(4.2) 10 [8] 10 ⇥ or 694.9(4.3) 10 [9]. This dispersive approach is an active research⇥ area and results with reduced errors xsrc z′, κ′ ′ ′ x′, ρ′ xsnk should be expected [10]. The HVP contribution can also y , σ be calculated with lattice QCD. With recently developed Figure 3. Leading-order disconnected diagram which is non- methods and increased computational power, similar or zero in SU(3) limit. There are additional diagrams which can even higher precision results may be possible [11–15]. be obtained from permutation of the photon vertices on the In contrast, the HLbL contribution is at present only muon line. estimated by model calculations which give a result of 10 10 10.5(2.6) 10 [16, 17] or 11.6(3.9) 10 [1]. This xop, ν xop, ν method is⇥ dicult to improve further although⇥ it is possible to compare the model result for hadronic light-by- z,κ z,κ light scattering with a lattice result for this scattering y,σ x, ρ y,σ x, ρ amplitude [18]. A dispersion relation analysis of the

HLbL contribution is not available although work is un- xsrc ′ ′ ′ ′ ′ ′ xsnk x ′ ′ ′ ′ ′ ′ x y , σ x , ρ z , κ src y , σ z , κ x , ρ snk derway in this direction [19–24].

Combining these results gives the standard model pre- Figure 4. Disconnected diagrams of order ms ml. There are sm 10 diction aµ = 11659184.0(5.9) 10 which di↵ers from additional diagrams which can be obtained from permutation ⇥ exp sm the experimental value above by aµ aµ = 24.0(6.9) of the photon vertices on the muon line. 10 ⇥ 10 , about twice the estimate for the HLbL contribu-

tion. Thus, a systematically improvable, lattice determi- xop, ν xop, ν nation of the HLbL contribution is needed to resolve or

ﬁrmly establish the discrepancy. z,κ y,σ x, ρ z,κ y,σ x, ρ The complete set of HLbL diagrams include the connected diagrams in Fig. 2 and the disconnected diagrams

x ′ ′ ′ ′ ′ ′ x xsrc ′ ′ ′ ′ ′ ′ xsnk in Fig. 3, 4, and 5. Only quark loops that are directly src z , κ y , σ x , ρ snk y , σ x , ρ z , κ

connected to photons are drawn in the figures. This is be- xop, ν cause only these quark propagators need to be explicitly calculated on the lattice. The e↵ects of gluons and other z,κ y,σ x, ρ quark loops are included automatically through the evaluation of these explicit quark propagators and the use xsrc ′ ′ ′ ′ ′ ′ xsnk of an unquenched gauge ensemble. Although there are z , κ y , σ x , ρ many di↵erent types of disconnected diagrams, only one 2 Figure 5. Disconnected diagrams of order (ms ml) and type, shown in Fig. 3, survives in the SU(3) limit. The higher. There are additional diagrams which can be obtained other diagrams, shown in Figs. 4 and 5, vanish in SU(3) from permutation of the photon vertices on the muon line. limit because they contain quark loops that couple only to one photon and the sum of the charges of the u, d, s quarks is zero. Also, because the strange quark carries The first attempt using lattice QCD to compute the only 1/3 of the electron charge, diagrams that are sup- connected contribution to HLbL was made by Blum, Directpressed bycomputation the di↵erence between of the strange in and lattice light QCDChowdhury, Hayakawa, and Izubuchi [25], which demon- quark masses are suppressed by their charge factors too. strated the2 possibility of performing such calculation. ARBC series 1610.04603 of improvements in methodology were made x , ν The HVP contribution is the largest hadronic contri-xop, ν xopop, ν in Ref. [26], eliminating two sources of systematic ef- bution and can be computed from a dispersion relation fects arising from the use of larger-than-physical electric + and experimental e e annihilation data. This is a well- charge and non-zero momentum transfer. The methods z,κ z,z,κκ y,σ x, ρ developed method with a fractional-percent error.y,σ Thex, ρ y,σ x, ρ Note: gluons 10 presented in Ref. [26] also lead to a substantial reduction leading-order HVP contribution is 692.3(4.2) 10 [8] 10 ⇥ in the statistical noise making a direct lattice calcula- or 694.9(4.3) 10 [9]. This dispersive approach is NOT shown ⇥ tion with a physical pion mass possible. Here, we report an active research area and results with reduced errors x ′ ′ ′ ′ ′ ′ x xsrc ′ ′ ′ ′ ′ ′ xsnk xsrc ′ src′ z , κ′ ′ ′ ′ xx,snkρ snk should be expected [10]. The HVP contributiony , σ canz , alsoκ x , ρ y , σ x , ρ y , σz , κ the result of the first connected HLbL lattice calculation be calculated with lattice QCD. With recently developed with physical pion mass. In addition to the connected Figure 2. Connected hadronicFigure 3. light-by-light Leading-order diagrams. disconnected There diagram which is non- methods and increased computational power,‘connected’ similar or zero in SU(3) limit.leading There are additional ‘disconnected’ diagramsHLbL which calculation, can we extended the methods of Ref. [26] even higher precision results mayare be four possible additional [11–15]. diagramsbe obtained resulting from from permutation further of permuta- the photon verticesand compute on the the leading disconnected diagrams shown In contrast, the HLbL contributiontions is at of present the photon only verticesmuon on line. the muon line. estimated by model calculations whichCalculate give a result 4 quark of propagators and combinein Fig. with 3 using factors the same set of configurations. This is the 10 10 10.5(2.6) 10 [16, 17] or 11.6(3.9) 10 [1]. This first disconnected HLbL calculation and the result sug- xop, ν xop, ν method is⇥ dicult to improve furtherfrom although⇥ muon it is pos- and photon propagators, sum over points. sible to compare the model result for hadronic light-by- z,κ z,κ light scattering with a lattice resultMassless for this scattering photon meansy,σ x, ρthat finite volumey,σ isx, ρ an issue. amplitude [18]. A dispersion relation analysis of the stat. HLbL contribution is not available although work is un- xsrc ′ ′ ′ ′ ′ ′ xsnk x ′ ′ ′ ′ ′ ′ x First result: y , σ x , ρ z , κ src y , σ z , κ x , ρ snk derway in this direction [19–24]. errors Combining these results gives the1 standard lattice model spacing pre- Figure 4. Disconnected diagrams of order ms ml. There are sm 10 diction aµ = 11659184.0(5.9) 10 which di↵ers from additionalconnected: diagrams which can be11.6 obtained ; disc. from permutation : -6.3 ⇥ exp sm only the experimental value above by aµ physicalaµ = 24.0(6 .9) of the photon vertices on the muon line. 10 ⇥ 10 , about twice the estimate for the HLbL contribu- improving finite-volume systematics: tion. Thus, a systematically improvable, lattice determi- xop, ν Mainz, 1711.02466;xop, ν RBC 1705.01067 nation of the HLbL contribution is needed to resolve or

ﬁrmly establish the discrepancy. z,κ y,σ x, ρ z,κ y,σ x, ρ The complete set of HLbL diagrams include the connected diagrams in Fig. 2 and the disconnected diagrams

x ′ ′ ′ ′ ′ ′ x xsrc ′ ′ ′ ′ ′ ′ xsnk in Fig. 3, 4, and 5. Only quark loops that are directly src z , κ y , σ x , ρ snk y , σ x , ρ z , κ connected to photons are drawn in the figures. This is be- xop, ν cause only these quark propagators need to be explicitly calculated on the lattice. The e↵ects of gluons and other z,κ y,σ x, ρ quark loops are included automatically through the evaluation of these explicit quark propagators and the use xsrc ′ ′ ′ ′ ′ ′ xsnk of an unquenched gauge ensemble. Although there are z , κ y , σ x , ρ many di↵erent types of disconnected diagrams, only one 2 Figure 5. Disconnected diagrams of order (ms ml) and type, shown in Fig. 3, survives in the SU(3) limit. The higher. There are additional diagrams which can be obtained other diagrams, shown in Figs. 4 and 5, vanish in SU(3) from permutation of the photon vertices on the muon line. limit because they contain quark loops that couple only to one photon and the sum of the charges of the u, d, s quarks is zero. Also, because the strange quark carries The first attempt using lattice QCD to compute the only 1/3 of the electron charge, diagrams that are sup- connected contribution to HLbL was made by Blum, pressed by the di↵erence between the strange and light Chowdhury, Hayakawa, and Izubuchi [25], which demon- quark masses are suppressed by their charge factors too. strated the possibility of performing such calculation. A series of improvements in methodology were made xop, ν xop, ν in Ref. [26], eliminating two sources of systematic effects arising from the use of larger-than-physical electric charge and non-zero momentum transfer. The methods z,κ z,κ y,σ x, ρ y,σ x, ρ presented in Ref. [26] also lead to a substantial reduction in the statistical noise making a direct lattice calculation with a physical pion mass possible. Here, we report x ′ ′ ′ ′ ′ ′ x xsrc ′ ′ ′ ′ ′ ′ xsnk src y , σ z , κ x , ρ snk y , σ x , ρ z , κ the result of the first connected HLbL lattice calculation with physical pion mass. In addition to the connected Figure 2. Connected hadronic light-by-light diagrams. There HLbL calculation, we extended the methods of Ref. [26] are four additional diagrams resulting from further permuta- and compute the leading disconnected diagrams shown tions of the photon vertices on the muon line. in Fig. 3 using the same set of configurations. This is the first disconnected HLbL calculation and the result sug- Beyond the Standard Model explanations for the aµ deviation in gdiscrepancy2 cannot be simultaneously in explained? to- e gether with the 3.7 anomaly in gµ 2 in the simplest versions of thoseSUSY⇠ models, still even a ifviable one could explanation circumvent existing experimental constraints. In this paper,- more we would constrained like to point out thatnow a minimalby LHC µ µ model based onsearches a single light since real scalar need, canrelatively in princi- light ple explain the deviations“Invisible” of both Darkgµ 2 and Photonge 2, in a relatively economicalsmuon fashion. and more We will fine-tuning. show that a two- loop Barr-Zee diagram [16, 17] might explain a while simple GeV-scale ‘dark dark with 2 and Br(e ¯) 1 X mX

–6– Conclusion E821 10 • aµ = 11659209.1(6.3) 10 SM ⇥ 10 aµ = 11659182.0(3.6) 10 expt SM⇥ 10 disagreement a a = 27(7) 10 µ µ ⇥ • SM uncertainty dominated by HVP. Methods using have improved to 0.4%; lattice QCD results now at 2-3% - aim is <1% with QED and isospin-breaking included. A key issue is ππ . • HLbL determination will also improve - first direct lattice QCD results now available. It seems clearly small. • Muon g-2 @FNAL will report its first new exptl result in 2019 - final aim is to reduce uncty by factor of 4. If central value remains, this will be evidence for BSM