Consistency of hadronic vacuum polarization between lattice QCD and the R-ratio
CConsistency of hadronic vacuum polarization between lattice QCD and the R-ratio
Christoph Lehner ∗ Universit¨at Regensburg, Fakult¨at fr Physik, 93040, Regensburg, Germany andPhysics Department, Brookhaven National Laboratory, Upton, New York 11973, USA
Aaron S. Meyer † Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA (Dated: March 10, 2020)There are emerging tensions for theory results of the hadronic vacuum polarization contribu-tion to the muon anomalous magnetic moment both within recent lattice QCD calculations andbetween some lattice QCD calculations and R-ratio results. In this paper we work towards scruti-nizing critical aspects of these calculations. We focus in particular on a precise calculation of Eu-clidean position-space windows defined by RBC/UKQCD that are ideal quantities for cross-checkswithin the lattice community and with R-ratio results. We perform a lattice QCD calculationusing physical up, down, strange, and charm sea quark gauge ensembles generated in the stag-gered formalism by the MILC collaboration. We study the continuum limit using inverse latticespacings from a − ≈ . . et al. and similar to the recent study of BMW. Our calculation exhibits a tension forthe particularly interesting window result of a ud , conn ., isospin , W µ from 0 . . a ud , conn ., isospin µ , a s , conn ., isospin µ , a SIB , conn .µ for the total contribution and a large set of windows. Forthe total contribution, we find a HVP LO µ = 714(27)(13)10 − , a ud , conn ., isospin µ = 657(26)(12)10 − , a s , conn ., isospin µ = 52 . − , and a SIB , conn .µ = 9 . . . − , where the first uncertaintyis statistical and the second systematic. We also comment on finite-volume corrections for thestrong-isospin-breaking corrections. I. INTRODUCTION
The established theory result for the muon anomalousmagnetic moment, a µ , exhibits a 3 . σ [1] to a 3 . σ [2] ten-sion with the results of the BNL experiment [3]. Withinthis year, we expect the Fermilab g − g − ∗ [email protected] † [email protected] tension between theory and experiment.For the HVP contribution, however, tensions existwithin lattice QCD calculations [50] as well as betweenlattice QCD calculations and R-ratio results [27, 50].At this point, the lattice calculations exhibiting a ten-sion with R-ratio results share some aspects. They areperformed at physical pion mass, with staggered seaquarks and a conserved valence vector current, and useinverse lattice spacings in the range from a − ≈ . a − ≈ . a − = 1 . . a ud , conn ., isospin , W µ for times t = 0 . t =1 . .
15 fm as defined by RBC/UKQCD [21]between Aubin et al. [50] on one side and RBC/UKQCD[21] and a combined R-ratio/Lattice result [21, 27, 50] onthe other side. The second is a tension between the total a ud , conn ., isospin µ with high values for BMW [27] and lowervalues for FNAL/HPQCD/MILC [24] and ETMC [30].In this work, we focus on scrutinizing the first tension.To this end, we use the same lattice QCD ensem-bles as Aubin et al. [50] but use a site-local current in-stead of a conserved current. Within this framework, wethen consider different approaches towards the contin-uum limit for windows in the staggered formalism and a r X i v : . [ h e p - l a t ] M a r provide an analysis with minimal input from effectivetheories. In our analysis, we find a substantially lowervalue for a ud , conn ., isospin , W µ compared to Ref. [50]. Thisis particularly noteworthy since the same sea-quark sec-tor is used and may indicate difficulties with properlyestimating uncertainties associated with the continuumlimit.Within our numerical framework, we can also accessthe connected strong-isospin breaking contribution aswell as the strange-quark connected contribution. Weprovide results also for these contributions including awide range of different windows. We hope that these re-sults will prove useful to further understand the currenttensions.This manuscript is organized as follows: Section II dis-cusses the main methods used in the analysis, includingthe window method and a correlator smoothing tech-nique to reduce the unwanted parity partner contribu-tions. Section III gives information about the ensemblesused for this study and the computational setup for ourdata generation. In Section IV, we describe our anal-ysis and uncertainty estimates. In this section we inparticular also comment on finite-volume corrections tothe strong-isospin breaking contributions. In Section V,we summarize and give some concluding remarks. Ap-pendix A provides additional tables of results for cross-comparisons with other analyses. II. METHODOLOGYA. General setup
In this work we perform a calculation of the HVP con-tribution to a µ using the Euclidean time-momentum rep-resentation [51] a HVP µ = ∞ (cid:88) t =0 w t C ( t ) (1)with sum over Euclidean time t and C ( t ) = 112 π (cid:90) ∞ d ( √ s ) R ( s ) se −√ st (2)with R-ratio R ( s ) = (3 s/ πα ) σ ( s, e + e − → had . ). Wecan also relate C ( t ) to vacuum expectation values of vec-tor currents V µ that we compute in lattice QCD+QEDas C ( t ) = 13 (cid:88) i,(cid:126)x (cid:104) V i ( (cid:126)x, t ) V i ( (cid:126) , (cid:105) , (3)where the sum is over spatial indices i and all points (cid:126)x in the spatial volume and V µ = 23 i (¯ uγ µ u + ¯ cγ µ c ) − i ( ¯ dγ µ d + ¯ sγ µ s + ¯ bγ µ b ) (4) with quark flavors u, d, s, c, b . In the absence of QED butthe presence of a quark-mass splitting between up anddown quarks with individual quark masses m u = m l − ∆ m ,m d = m l + ∆ m , (5)the total up, down, and strange contributions can bewritten as a uds µ = a ud , conn ., isospin µ + a s , conn ., isospin µ + a uds , disc ., isospin µ + a SIB , conn .µ + a SIB , disc .µ . (6)In this work, we focus on the connected contributions a ud , conn ., isospin µ , a s , conn ., isospin µ , and a SIB , conn .µ , which we ex-press as a ud , conn ., isospin µ = 5 a vµ ( m l ) , (7) a s , conn ., isospin µ = a vµ ( m s ) , (8)where m v denotes the mass of the valence quark and a vµ ( m v ) = 19 c ( m v ) (9)in terms of the diagrams of Fig. 1. The connected strong-isospin breaking (SIB) contribution can be written as a SIB , conn .µ = 23 ∆ mM ( m v = m l )= 3 λ κ − κ + 1 lim λ → λ ∂∂λ a vµ ( λ = m v /m s ) (10)where diagram M and O of Fig. 2 are related to diagramsc and d of Fig. 1 by ∂∂m v c ( m v ) = − M ( m v ) , (11) ∂∂m v d ( m v ) = − O ( m v ) , (12)and λ ≡ m l m s , κ ≡ m u m d . (13)Both κ and λ are obtained from FLAG 2019 [52], where λ is taken from 2+1+1 flavor simulations and κ is from2+1 flavor simulations. Only one 2+1+1 flavor resultfor κ is available, so we choose to use the 2+1 flavorsimulation so as not to tie our results to a single externalmeasurement. The values for these quantities are κ = 0 . , λ − = 27 . . (14)The diagrams R and R d do not contribute in the defi-nition of the isospin symmetric point given in Eq. (5). (a) Diagram c (b) Diagram d FIG. 1. Feynman diagrams for the isospin symmetric con-tribution to the HVP. The dots represent the vector currentscoupling to external photons. These diagrams represent gluoncontributions to all orders. x xx (a) Diagram M x xx (b) Diagram O x xx (c) Diagram R x (d) Diagram R d FIG. 2. Connected and disconnected strong-isospin breaking(SIB) diagrams. The cross denotes the insertion of a scalaroperator. Also here each diagram represents gluon contribu-tions to all orders.
The weighting kernel in Eq. (1) is determined as [51,53] w ft = 8 α m µ (cid:90) ∞ dsK ( s, m µ ) f ( t, √ s ) , (15) K ( s, m µ ) = sZ ( s, m µ ) (1 − sZ ( s, m µ ))1 + m µ sZ ( s, m µ ) , (16) Z ( s, m µ ) = (cid:113) s + 4 m µ s − s m µ s , (17)where we will use two alternative choices for the function f , f p ( t, q ) = cos( qt ) − q + 12 t , (18) f ˆ p ( t, q ) = cos( qt ) − q/ + 12 t . (19)We refer to the choice f = f p as the p prescription andto the choice of f = f ˆ p as the ˆ p description. Both arewell-motivated within a lattice calculation and differ onlydue to discretization errors. We will provide results forboth and scrutinize the difference when considering un-certainties associated with the continuum limit. B. Window Method
It is instructive to isolate specific ranges of Euclideantime in order to better understand their contributionsto a v µ . This can be accomplished by constructing win-dows that suppress contributions outside of the windowregion [21]. Rather than using Heaviside step functionsto isolate these ranges, which would have significant de-pendence on the lattice cutoff near the boundary of thewindow, a smoothed step is considered [16, 51]:Θ( t, µ, ∆) = [1 + tanh[( t − µ ) / ∆]] / . (20)This step function suppresses all values below µ and hasa width parameterized by ∆. From these step functions,windows into specific regions of a v µ Euclidean time canbe studied by instead summing the integral relation a v , W µ ( t , t , ∆)= ∞ (cid:88) t =0 w t C ( t )[Θ( t, t , ∆) − Θ( t, t , ∆)] . (21)We will quote results both for the total contribution, cor-responding to t → −∞ and t → ∞ , as well as specificwindows. It should be noted that windows of a v µ that iso-late specific Euclidean distance scales can be related tospecific windows of time-like s in the experimental dataused in the R-ratio [21]. C. Parity Improvement
When performing computations with staggeredquarks, parity projections are not possible and cor-relation functions receive contributions from paritypartner states. These parity partners have different spinand taste quantum numbers and constitute unwantedcontributions to the correlation function. The unwantedcontributions come as oscillating terms with a prefactorproportional to ( − t/a . To suppress these contributionsto the correlation functions, we also study the ImprovedParity Averaging (IPA) procedure which averagesneighboring timeslices [54], C IPA ( t ) = e − m ρ t (cid:20) C ( t − e − m ρ ( t − + 2 C ( t ) e − m ρ ( t ) + C ( t + 1) e − m ρ ( t +1) (cid:21) . (22)The correlation function times are weighted by exponen-tial factors that reflect the falloff of the correlation func-tion in order to better enforce the cancellation of oscil-lating parity partner contributions. The exponent usedis the ρ meson mass, obtained from PDG [55], which isexpected to give the best cancellation in the ρ resonancepeak. The ρ resonance region accounts for the majorityof the contribution to a v µ , and so cancellation in this re-gion would be most beneficial. In the continuum limitthe choice of m ρ is irrelevant, however, the IPA prescrip-tion using the rho mass is not well-motivated for veryshort or long distances or for heavier quark-masses. III. NUMERICAL SETUP
The computation in this work is performed with theHighly-Improved Staggered Quark (HISQ) action forboth valence and sea quarks. The ensembles were gener-ated by the MILC collaboration [56], and details aboutthese ensembles are given in Table I. For the staggered
Ens L × T w /a Z V (¯ ss ) M π M π L
48c 48 ×
64 1 . . . ×
96 1 . . . ×
192 3 . . . w = 0 . quark action, the vector current operator is written V i ( x ) = (cid:88) (cid:126)x ( − x i /a (cid:15) ( x ) ¯ χ ( (cid:126)x, x ) χ ( (cid:126)x, x ) (23)where χ ( (cid:126)x, x ) is the staggered one-component spinorand (cid:15) ( x ) is the usual staggered sign phase (cid:15) ( x ) = ( − (cid:80) µ x µ /a . (24)This vector current bilinear has the advantage of beinglocal to a single site. Vector currents of other tastes maybe formed by distributing the quark and antiquark overthe unit hypercube, but these bilinear combinations re-quire extra inversions and so were not explored.Sources are inverted on random noise vectors that solvethe Green’s function equation (cid:88) y /D abxy G bcy ; t = η acx δ x ,t (25)with η satisfying the condition (cid:104) η abx ( η bcy ) † (cid:105) = 18 δ ac δ xy (cid:89) i =1 (1 − ( − x i /a ) . (26)The phase factor on the RHS of Eq. (26) results fromprojecting out sites where x i /a is odd for at least one i . When the propagator obtained from Eq. (25) is con-tracted with its Hermitian conjugate at the source, theconstruction produces an operator that couples to manystaggered spin-taste meson irreducible representations.The vector current of Eq. (23) is contracted explicitly atthe sink, projecting out the unwanted spin-taste irrepsat the source and reproducing the correlation function ofEq. (3) up to a factor of 8. In the propagator solutionsfor the Dirac equation, the Naik epsilon term set to zero.Results are computed on three ensembles with 2+1+1flavors of sea quarks and up to 7 choices of valence quarkmass per ensemble. The parameters for the 3 ensemblesused in this study are given in Table I. The sea quark masses are given in Table II along with retuned quarkmasses for the strange quarks. The valence quark massesused in this study are rational fractions λ times the tunedstrange quark masses from Table II. The list of rationalfractions is given in Table III along with the number oftime sources per configurations and the number of config-urations used for each ensemble and mass combination. Ens m (cid:96) m s m c m s, tuned
48c 0.00184 0.0507 0.628 0 . . . t src / conf N conf [ λ ]1 / / / / / / / ∗ ) 32 32 32 64 64 192 20 / ∗
96c 24 32 − − − − − TABLE III. Number of configurations and time sources usedfor each ensemble and valence quark mass combination. Thevalence quark masses are quoted as ratios of the valence quarkmass to the tuned strange quark mass, λ ≡ m valence /m s, tuned ,obtained from Table II. The 64c λ = 1 /
12 mass point was in-verted on double the number of time sources compared to theother 64c mass points, for a total of 48 time sources. Thisensemble/mass point combination was computed using thetruncated solver method [58] in an AMA setup [59] with 20configurations solved with full precision and 1540 configura-tions solved with a residual of 10 − . The 20 full-precisionsolves are used to correct the bias introduced by this pro-cedure, which was tuned to have negligible impact on theresults. IV. RESULTSA. Bounding method
For the two lightest masses λ = 1 /
12 and λ = 1 /
6, wealso employ the bounding method [20, 21, 60] to createstrict upper and lower bounds for a vµ . We show resultsfor the total a vµ in Fig. 3. In the bounding method, onereplaces the correlator C ( t ) by˜ C ( t ; T, ˜ E ) = (cid:40) C ( t ) t < T ,C ( T ) e − ( t − T ) ˜ E t ≥ T (27)which then defines a strict upper or lower bound of C ( t )for each t given an appropriate choice of ˜ E . For the upperbound we use ˜ E equals to the free two-pion ground stateenergy and for the lower bound we use ˜ E = ∞ [21]. Weselect the data points of T = T , where upper and lowerbounds agree. For λ = 1 /
12, we use T /a = 21 for the48c ensemble and T /a = 34 for the 64c ensemble. For λ = 1 /
6, we use T /a = 24 for the 48c ensemble and T /a = 36 for the 64c ensemble. B. Continuum Extrapolation
The first part of the analysis consists of taking thecontinuum limit of each individual mass point. This ex-trapolation is applied before considering extrapolationsin quark mass or volume.Figs. 4 and 5 show the extrapolation of the m v /m s =1 / / λ = 1 / λ = 1 / t − t = 0 . m v /m s = 1 / t = 0 . a . Beyond t = 0 . t ≥ . ρ resonance peak, and has larger discretization effects forshort-distance windows. In the long-distance windows,IPA becomes identical to the unimproved data.Fig. 8 demonstrates the effect of the window smearingparameter ∆ on the continuum extrapolation. If ∆ issmaller than the lattice spacing, the window turns on oroff rapidly and can resolve contributions from individualtimeslices. These discretization effects are clearly visiblefor the coarsest ensembles when ∆ is too small. Thisis cleaned up by increasing the window smearing. TheIPA also smears neighboring timeslices, which reducesthe effect of discretizations for even the smallest windowsmearings.Fig. 9 shows the difference between the IPA proce-dure of Eq. (22) and the unimproved total result forseveral choices of valence quark mass and lattice spac-ing. The improvement with the ρ resonance mass inEq. (22) is only well motivated for quark masses closeto the isospin-symmetric valence quark mass limit. Sig-nificant deviations from the unimproved data are seen inthe m v /m s = 1 data, while the m v /m s = 1 / − σ level. Good agree- FIG. 3. We show 10 a v, BND µ ( T ) with a v, BND µ ( T ) = (cid:80) ∞ t =0 w t C ( t, T, ˜ E ) for both the upper bound ˜ E equals to thefree two-pion ground state energy as well as the lower bound˜ E = ∞ . From top to bottom, the results are for λ = 1 /
12 onthe 48c ensemble, λ = 1 /
12 on the 64c ensemble, λ = 1 / λ = 1 / / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved Total 51.5 52 52.5 53 53.5 54 54.5 55 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA Total 26.6 26.7 26.8 26.9 27 27.1 27.2 27.3 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.4, t /fm=1.0, ∆ /fm=0.15 26.8 27 27.2 27.4 27.6 27.8 28 28.2 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.4, t /fm=1.0, ∆ /fm=0.15 FIG. 4. Continuum extrapolation of the m v /m s = 1 / a v µ contribution and for the windowwith ( t , t ) = (0 . , .
0) fm. We show the unimproved dataand as well as data with the parity improvement described inSection II C. The data include both statistical and systematicuncertainties. The extrapolations with either 48c and 64c orwith 64c and 96c ensembles are shown as shaded bands.
74 76 78 80 82 84 86 88 90 92 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved Total 78 80 82 84 86 88 90 92 94 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA Total / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.4, t /fm=1.0, ∆ /fm=0.15 35.4 35.6 35.8 36 36.2 36.4 36.6 36.8 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.4, t /fm=1.0, ∆ /fm=0.15 FIG. 5. Same as Fig. 4, but for m v /m s = 1 / ment between the two procedures is observed for the m v /m s = 1 /
12 data. / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.0, t /fm=0.1, ∆ /fm=0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.0, t /fm=0.1, ∆ /fm=0.15 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.2, t /fm=0.3, ∆ /fm=0.15 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.2, t /fm=0.3, ∆ /fm=0.15 / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.4, t /fm=0.5, ∆ /fm=0.15 4.6 4.65 4.7 4.75 4.8 4.85 4.9 4.95 5 5.05 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.4, t /fm=0.5, ∆ /fm=0.15 5.9 5.95 6 6.05 6.1 6.15 6.2 6.25 6.3 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.6, t /fm=0.7, ∆ /fm=0.15 6.06 6.08 6.1 6.12 6.14 6.16 6.18 6.2 6.22 6.24 6.26 6.28 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.6, t /fm=0.7, ∆ /fm=0.15 FIG. 6. Windows with t − t = 0 . m v /m s = 1 / t , ≤ . t and t . We therefore also show results for t − t = 0 . ρ resonance region, and so is not expected to work well for shorter distances. / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.8, t /fm=0.9, ∆ /fm=0.15 6.35 6.4 6.45 6.5 6.55 6.6 6.65 6.7 6.75 6.8 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.8, t /fm=0.9, ∆ /fm=0.15 / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=1.0, t /fm=1.1, ∆ /fm=0.15 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=1.0, t /fm=1.1, ∆ /fm=0.15 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=1.2, t /fm=1.3, ∆ /fm=0.15 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=1.2, t /fm=1.3, ∆ /fm=0.15 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=1.4, t /fm=1.5, ∆ /fm=0.15 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=1.4, t /fm=1.5, ∆ /fm=0.15 FIG. 7. Same as Fig. 6, but for windows with t , ≥ . / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.4, t /fm=1.0, ∆ /fm=0.05 36.6 36.8 37 37.2 37.4 37.6 37.8 38 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.4, t /fm=1.0, ∆ /fm=0.05 35.2 35.4 35.6 35.8 36 36.2 36.4 36.6 36.8 37 37.2 37.4 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.4, t /fm=1.0, ∆ /fm=0.1 36 36.2 36.4 36.6 36.8 37 37.2 37.4 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.4, t /fm=1.0, ∆ /fm=0.1 34.5 35 35.5 36 36.5 37 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.4, t /fm=1.0, ∆ /fm=0.15 35.4 35.6 35.8 36 36.2 36.4 36.6 36.8 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.4, t /fm=1.0, ∆ /fm=0.15 33.5 34 34.5 35 35.5 36 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)Unimproved t /fm=0.4, t /fm=1.0, ∆ /fm=0.2 34.6 34.8 35 35.2 35.4 35.6 35.8 0 0.005 0.01 0.015 0.02a / fm Continuum Extrapolation (64c, 48c)Continuum Extrapolation (96c, 64c)IPA t /fm=0.4, t /fm=1.0, ∆ /fm=0.2 FIG. 8. Continuum extrapolations with m v /m s = 1 / t , t ) = (0 . , .
0) fm. The left and right columns are for the unimproved data and for the data with the parity improvement,respectively. The continuum extrapolation errors are enhanced when ∆ is too small, but the IPA softens this issue.
90 95 100 105 110 115 120 0 0.005 0.01 0.015 0.02a / fm Unimproved Total IPA Total 92 94 96 98 100 102 104 106 0 0.005 0.01 0.015 0.02a / fm Unimproved Total IPA Total 78 80 82 84 86 88 90 92 0 0.005 0.01 0.015 0.02a / fm Unimproved Total IPA Total
50 50.5 51 51.5 52 52.5 53 53.5 54 0 0.005 0.01 0.015 0.02a / fm Unimproved Total IPA Total
FIG. 9. Unimproved versus parity improved data as a function of lattice spacing. The upper-left, upper-right, lower-left, andlower-right plots have m v /m s = 1 / , / , / , and 1, respectively. The parity improved data with ρ parameters is not wellmotivated for the strange quark masses and is not used for the final analysis. C. Valence Mass Extrapolation
After the continuum extrapolation the valence quarkmasses are extrapolated to the isospin-symmetric lightquark mass. The value obtained from this extrapolationgives the connected light-quark contribution to the HVPfrom Eq. (7). In addition to the intercept of this extrap-olation, the slope also provides information about thestrong isospin-breaking given in Eq (10). No extrapola-tion is needed to get to m v = m s since this calculation isperformed explicitly.Fig. 10 shows the extrapolation in valence quark massfor both the total contribution and for the window with( t , t ) = (0 . , .
0) fm. Both statistical and systematicerrors are included, and all data have been extrapolatedto the continuum. In this figure, the IPA procedure is notperformed and the ˆ p prescription is used in all windows.The mass dependence of the short-distance ρ resonancestates is very linear over most of the range between thelight and strange quark masses, while the long-distance ππ states give a noticeable curvature. The fit parametersfor the extrapolation in valence quark masses for each ofthe windows is given in Table IV.Fig. 11 shows the valence quark data extrapolated tothe isospin-symmetric limit for a few choices of t − t =0 . t = 0 . t = 0 . t .Table V shows the results for a series of time ranges andwidths after extrapolation to the light quark mass. Theseresults are repeated in Tables X-XIII in Appendix Awith all systematic uncertainties shown in detail. For allwindows, the p and ˆ p prescriptions give results that are1 σ consistent. The parity improvement also gives con-sistent results with unimproved data for windows with t ≥ . t and t close to 0 fmare also subject to much larger discretization errors fromthe continuum extrapolation than wider window regionsor those farther from 0 fm. For instance, the window with( t , t ) = (0 . , .
2) fm has a smaller relative systematicuncertainty than the window with ( t , t ) = (0 . , .
1) fm.The parity improvement is not well motivated for veryshort distance windows and we give the results only forcompleteness. t /fm t /fm ∆/fm d n kTotal 2 6 50.0 0.1 0.15 1 4 30.1 0.2 0.15 1 4 30.2 0.3 0.15 1 4 30.3 0.4 0.15 1 4 30.4 0.5 0.15 1 4 30.5 0.6 0.15 1 4 30.6 0.7 0.15 1 4 30.7 0.8 0.15 1 4 30.8 0.9 0.15 1 4 30.9 1.0 0.15 1 4 31.0 1.1 0.15 2 5 41.1 1.2 0.15 2 5 41.2 1.3 0.15 2 5 41.3 1.4 0.15 2 5 41.4 1.5 0.15 2 5 41.5 1.6 0.15 2 5 41.6 1.7 0.15 2 7 61.7 1.8 0.15 2 7 61.8 1.9 0.15 2 7 61.9 2.0 0.15 2 7 60.3 1.0 0.15 1 4 30.3 1.3 0.15 2 5 40.3 1.6 0.15 2 5 40.4 1.0 0.15 1 4 30.4 1.3 0.15 2 5 40.4 1.6 0.15 2 5 40.4 1.0 0.05 1 4 30.4 1.0 0.1 1 4 30.4 1.0 0.2 1 4 3TABLE IV. List of fit parameters for the valence mass extrap-olation in each window. The fits are parameterized as degree d polynomials of m v /m s , including the n lightest masses. Afit is repeated with the k lightest masses to estimate the sys-tematic error due to the valence extrapolation and is addedas a systematic uncertainty. This uncertainty is included inthe systematic error band in Figs. 10 and 11.
50 60 70 80 90 100 110 120 130 140 0 0.2 0.4 0.6 0.8 1m v / m s Valence mass extrapolationUnimproved Total 24 26 28 30 32 34 36 38 40 42 0 0.2 0.4 0.6 0.8 1m v / m s Valence mass extrapolationUnimproved t /fm=0.4, t /fm=1.0, ∆ /fm=0.15 FIG. 10. Valence mass extrapolation of a vµ for the total time extent (left) and for the window with ( t , t ) = (0 . , .
0) fm(right). The points are the the continuum-extrapolated data and the shaded region is the mass extrapolation. For the rathershort-distance window, a linear extrapolation is sufficient, while a quadratic fit is needed for the total result including longdistances. The fit parameters for each window are given in Table IV and the points included in the central fit are highlightedin the figure. The extrapolated band must be multiplied by a factor of 5 when comparing to the light quark results in Table V,see Eq. (7). v / m s Valence mass extrapolationUnimproved t /fm=0.4, t /fm=0.5, ∆ /fm=0.15 v / m s Valence mass extrapolationUnimproved t /fm=0.9, t /fm=1.0, ∆ /fm=0.15 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1m v / m s Valence mass extrapolationUnimproved t /fm=1.4, t /fm=1.5, ∆ /fm=0.15 v / m s Valence mass extrapolationUnimproved t /fm=1.9, t /fm=2.0, ∆ /fm=0.15 FIG. 11. Valence mass extrapolation for four different windows. The points are the the continuum-extrapolated data and theshaded region is the mass extrapolation. For short-distance windows, a linear extrapolation is sufficient, while a quadratic fitis needed for long-distance windows. The fit parameters for each window are given in Table IV and the points included in thecentral fit are bold. t /fm t /fm ∆/fm l ,ˆ p U l ,ˆ p I l , p U l , p I s ,ˆ p U s , p UTotal 627(26)(08) 632(27)(07) 628(26)(07) 634(27)(07) 52.83(22)(65) 53.08(22)(82)0.0 0.1 0.15 3.59(00)(59) 4.60(00)(31) 4.32(00)(20) 5.69(00)(24) 0.81(00)(12) 0.887(00)(40)0.1 0.2 0.15 8.633(03)(73) 9.93(00)(52) 9.29(00)(46) 11.0(0.0)(1.3) 1.666(01)(12) 1.728(01)(86)0.2 0.3 0.15 14.24(01)(82) 15.4(0.0)(1.3) 14.7(0.0)(1.2) 16.0(0.0)(1.8) 2.57(00)(16) 2.59(00)(24)0.3 0.4 0.15 18.62(02)(35) 20.2(0.0)(1.1) 18.71(02)(27) 20.3(0.0)(1.3) 3.448(05)(65) 3.451(05)(49)0.4 0.5 0.15 24.552(35)(60) 24.71(04)(24) 24.518(36)(59) 24.65(04)(21) 4.170(07)(20) 4.169(08)(24)0.5 0.6 0.15 29.38(06)(29) 29.42(06)(26) 29.36(06)(31) 29.36(06)(31) 4.666(10)(59) 4.665(11)(64)0.6 0.7 0.15 33.72(10)(36) 33.87(10)(26) 33.70(10)(39) 33.82(10)(29) 4.866(13)(74) 4.866(13)(79)0.7 0.8 0.15 37.54(14)(14) 37.30(15)(19) 37.54(15)(14) 37.28(15)(20) 4.799(16)(39) 4.799(16)(39)0.8 0.9 0.15 39.32(20)(20) 39.52(21)(18) 39.33(21)(20) 39.52(21)(18) 4.505(17)(44) 4.504(18)(44)0.9 1.0 0.15 40.47(27)(29) 40.43(28)(28) 40.47(27)(30) 40.44(28)(28) 4.058(19)(65) 4.058(19)(65)1.0 1.1 0.15 40.47(44)(39) 40.56(46)(37) 40.49(45)(39) 40.57(46)(37) 3.527(19)(76) 3.527(19)(76)1.1 1.2 0.15 39.34(54)(39) 39.47(56)(43) 39.35(55)(39) 39.48(57)(44) 2.973(19)(75) 2.973(19)(75)1.2 1.3 0.15 37.53(65)(49) 37.54(67)(48) 37.55(66)(49) 37.56(68)(48) 2.441(18)(77) 2.440(18)(77)1.3 1.4 0.15 34.88(77)(49) 34.98(79)(51) 34.89(77)(49) 35.00(80)(52) 1.955(17)(67) 1.955(17)(67)1.4 1.5 0.15 31.94(88)(52) 31.98(91)(53) 31.96(89)(52) 31.99(92)(53) 1.534(15)(60) 1.534(15)(60)1.5 1.6 0.15 28.66(100)(52) 28.7(1.0)(0.5) 28.7(1.0)(0.5) 28.7(1.0)(0.5) 1.181(13)(52) 1.181(13)(52)1.6 1.7 0.15 24.58(81)(61) 24.64(82)(62) 24.59(81)(61) 24.65(83)(62) 0.894(12)(44) 0.894(12)(44)1.7 1.8 0.15 21.20(85)(60) 21.28(86)(62) 21.21(85)(60) 21.20(86)(55) 0.667(10)(37) 0.667(10)(37)1.8 1.9 0.15 18.13(86)(59) 18.22(88)(63) 18.13(87)(60) 18.23(88)(63) 0.491(08)(30) 0.491(08)(30)1.9 2.0 0.15 15.49(89)(66) 15.42(91)(50) 15.37(89)(46) 15.57(91)(70) 0.357(07)(24) 0.357(07)(25)0.0 0.2 0.15 12.22(00)(52) 14.53(00)(21) 13.61(00)(26) 16.7(0.0)(1.5) 2.48(00)(11) 2.615(02)(47)0.2 0.4 0.15 32.87(03)(48) 35.6(0.0)(2.4) 33.41(03)(94) 36.3(0.0)(3.0) 6.02(01)(10) 6.05(01)(19)0.4 0.6 0.15 53.93(10)(28) 54.12(10)(13) 53.88(10)(33) 54.00(10)(17) 8.837(18)(74) 8.834(18)(85)0.6 0.8 0.15 71.26(24)(37) 71.16(25)(44) 71.23(24)(40) 71.11(25)(49) 9.666(29)(91) 9.665(29)(97)0.8 1.0 0.15 79.80(47)(42) 79.96(49)(44) 79.81(48)(42) 79.96(49)(44) 8.56(04)(10) 8.56(04)(10)0.3 1.0 0.15 223.6(0.8)(1.1) 225.5(0.8)(1.2) 223.6(0.8)(1.1) 225.4(0.8)(1.2) 30.51(08)(25) 30.51(09)(26)0.3 1.3 0.15 340.7(2.6)(1.9) 343.4(2.7)(2.7) 340.7(2.6)(1.9) 343.3(2.7)(2.7) 39.45(13)(35) 39.45(13)(35)0.3 1.6 0.15 436.2(5.1)(3.1) 439.0(5.3)(4.2) 436.3(5.1)(3.2) 439.5(5.3)(4.0) 44.12(17)(49) 44.12(17)(49)0.4 1.0 0.15 204.99(79)(85) 205.25(82)(77) 204.93(80)(90) 205.08(83)(83) 27.06(08)(21) 27.06(08)(22)0.4 1.3 0.15 322.2(2.6)(1.8) 322.8(2.7)(1.9) 322.2(2.6)(1.8) 321.6(2.7)(2.1) 36.01(13)(36) 36.00(13)(35)0.4 1.6 0.15 417.7(5.1)(3.2) 418.5(5.3)(3.4) 417.9(5.1)(3.2) 418.3(5.3)(3.3) 40.68(17)(51) 40.67(17)(50)0.4 1.0 0.05 215.5(0.8)(6.2) 208.5(0.8)(1.5) 215.8(0.8)(6.4) 208.4(0.8)(1.6) 27.9(0.1)(1.1) 27.9(0.1)(1.1)0.4 1.0 0.1 208.85(77)(74) 207.6(0.8)(1.1) 208.76(78)(70) 207.4(0.8)(1.3) 27.70(08)(21) 27.69(08)(20)0.4 1.0 0.2 201.08(82)(85) 201.86(85)(77) 201.10(84)(86) 201.83(86)(76) 26.24(08)(21) 26.24(08)(21)TABLE V. Results for a ud , conn ., isospin µ , labelled by ”l”, and a s , conn ., isospin µ , labelled by ”s”, for the total contribution as wellas different windows. We compare results obtained from the ˆ p and p prescriptions as well as unimproved (U) and IPA (I)results. The IPA procedure is defined in Section II C. The value for the full time range is given in the first row, labeled“Total”. All windows with t ≥ . t , discretization effects account for significant differences in the window sums. The IPA prescription, however, is notwell motivated for small distances. We use the ˆ p and unimproved prescriptions for our central values further discussed in thefollowing. We notice that broader windows with t − t = 0 . t compared to the narrower windows. D. Corrections for Finite Volume
The finite-volume correction (FVC) is a correctionto the long-distance physics in the correlation functiondue to the finite spatial extent of the lattice. The lat-tice states in the long-distance region are mostly com-posed of two-pion scattering states with zero center-of-mass momentum, up to mixing with other states thatshare the same quantum numbers. The finite spatialextent imposes a lower limit on the size of a unit ofmomentum, p = 2 π/L , which discretizes the spectrumof states that satisfy the periodic boundary conditions.The lowest-energy state in the spectrum of the connectedisospin-1 channel is a two-pion state with both pionshaving one unit of momentum back-to-back. In infinite-volume, where there is no minimum momentum, two-pion states would contribute all the way down to thetwo-pion threshold energy s = 4 M π , significantly chang-ing the long-distance exponential tail of the correlationfunction. The FVC attempts to fix this mismatch of thefinite- and infinite-volume.The estimate of the FVC is obtained from the Lellouch-L¨uscher-Gounaris-Sakurai procedure [61–64]. This is car-ried out by combining the pion form factor with estimatesof the finite- and infinite-volume spectrum and matrix el-ements. In the infinite volume, the pion form factor isrelated to R ( s ) via R ( s ) = 14 (cid:18) − M π s (cid:19) / | F π ( s ) | (28)and R ( s ) is inserted into Eq. (2) to obtain the ππ contri-bution to the correlation function C ( t ). The pion formfactor F π ( s ) used in R ( s ) is obtained from the Gounaris-Sakurai (GS) parameterization [61]. The pion scatteringphase shift for obtaining the finite-volume spectrum andmatrix elements is computed with the GS parameteriza-tion as well, which has the simple relation F π ( s ) = f (0) /f ( s ) (29)with f ( s ) = ( − i + cot δ ( s )) k π ( s ) √ s , (30)where k π ( s ) = (cid:114) s − M π (31)is the pion momentum for the center of mass energy √ s .The phase shift δ ( s ) is related to the finite-volume spec-trum according to the relation [62] δ ( s ) + φ ( q ) = nπ, n ∈ Z (32)where q ( s ) = ( L/ π ) k π ( s ) and φ ( q ) is determined fromtan φ ( q ) = − π / q Z (1; q ) (33) with the analytic continuation of the zeta function Z ( s ; q ) = 1 √ π (cid:88) (cid:126)n ∈ Z | (cid:126)n | − q ) s . (34)For a set of finite-volume states obtained by solvingEq. (32), the corresponding vector current amplitudesare obtained from [63, 64] (cid:12)(cid:12) (cid:104) | V i | ππ ( √ s = E ππ ) (cid:105) (cid:12)(cid:12) = | F π ( E ππ ) | k π πE ππ (cid:20) ∂∂k π ( δ + φ ) (cid:21) − . (35)The C ( t ) obtained from both the infinite-volume (IV)parameterization of F π and from the explicit reconstruc-tion of a finite number of states N , C FV ( t ) = N (cid:88) n (cid:12)(cid:12) (cid:104) | V i | ππ n (cid:105) (cid:12)(cid:12) e − E ππn t , (36)are both summed with the Bernecker-Meyer kernel w t asin Eq. (1). The finite volume correction for the connecteddiagram is then∆ a FVC µ = 109 ∞ (cid:88) t =0 w t [ C IV ( t ) − C FV ( t )] , (37)where C IV denotes the infinite-volume correlaion func-tion. The factor 10 / C FV ( t ) is reconstructed with 12 states. The differencebetween the 11- and 12-state reconstructions is added asa systematic error. An additional 30% uncertainty on thecorrection is applied to cover other uncontrolled system-atic effects associated with the finite volume corrections.The size of the contribution from each light and strangewindow and the size of the finite-volume correction forthe light quark mass for each of the t − t = 0 . t = t − .
05 fm to t = t + 0 .
05 fmwith ∆ = 0 .
15 fm for both the isospin-symmetric con-nected light-quark as well as strange quark contribution.The lower panel shows the size of the FVC to the lightquark mass contribution for each of the windows. Thesenumbers are provided in Table VI. For the strange quarkcontribution, we assume FVC to be negligible. We com-bine these FVC with the finite-volume isospin-symmetriclight-quark connected windows to our final results for a ud , conn ., isospin µ and a s , conn ., isospin µ in Table VII.Since at leading order in ∆ m the pion-mass splittingis a pure QED effect, it is expected that two-pion con-tributions to the total SIB contribution largely cancel.This is, however, not true for the connected and discon-nected pieces (diagrams M and O of Fig. 2) separately.5 t /fm t /fm ∆/fm ∆ a ud , conn ., isospin µ ∆ a SIB , conn .µ Total 29.9(9.0) 3.8(1.1)0.0 0.1 0.15 0.0068(21) 0.000103(32)0.1 0.2 0.15 0.0162(50) 0.000320(100)0.2 0.3 0.15 0.0299(92) 0.00085(26)0.3 0.4 0.15 0.046(14) 0.00195(60)0.4 0.5 0.15 0.065(20) 0.0039(12)0.5 0.6 0.15 0.089(27) 0.0069(21)0.6 0.7 0.15 0.123(38) 0.0112(34)0.7 0.8 0.15 0.169(52) 0.0168(52)0.8 0.9 0.15 0.229(70) 0.0238(73)0.9 1.0 0.15 0.301(93) 0.0320(98)1.0 1.1 0.15 0.38(12) 0.041(13)1.1 1.2 0.15 0.47(15) 0.051(16)1.2 1.3 0.15 0.57(17) 0.062(19)1.3 1.4 0.15 0.66(20) 0.072(22)1.4 1.5 0.15 0.76(23) 0.083(25)1.5 1.6 0.15 0.85(26) 0.093(28)1.6 1.7 0.15 0.93(28) 0.102(31)1.7 1.8 0.15 1.00(30) 0.110(34)1.8 1.9 0.15 1.05(32) 0.117(36)1.9 2.0 0.15 1.10(33) 0.123(38)0.0 0.2 0.15 0.0230(71) 0.00042(13)0.2 0.4 0.15 0.076(23) 0.00280(87)0.4 0.6 0.15 0.154(47) 0.0108(33)0.6 0.8 0.15 0.293(90) 0.0279(86)0.8 1.0 0.15 0.53(16) 0.056(17)0.3 1.0 0.15 1.02(31) 0.096(30)0.3 1.3 0.15 2.45(75) 0.251(77)0.3 1.6 0.15 4.7(1.4) 0.50(15)0.4 1.0 0.15 0.98(30) 0.094(29)0.4 1.3 0.15 2.40(74) 0.249(76)0.4 1.6 0.15 4.7(1.4) 0.50(15)0.4 1.0 0.05 0.92(28) 0.088(27)0.4 1.0 0.1 0.94(29) 0.091(28)0.4 1.0 0.2 1.02(31) 0.099(31)TABLE VI. Finite volume corrections for each window. Thenumbers for the light-quark connected isospin-symmetric con-tribution are plotted in the bottom panel of Fig. 12.
In particular, when connected and disconnected SIB cor-rections are compared between different lattice collabo-rations, performed at different volumes, this is importantto take into account.In order to address this issue, we use NLO PQChPT[50, 65], which yields a correlator for the connected anddisconnected diagrams of Fig. 1, C NLO , PQ χ PT , conn . ( t ) = 109 13 1 L (cid:88) (cid:126)p (cid:126)p ( E vlp ) e − E vlp t , (38) C NLO , PQ χ PT , disc . ( t ) = −
19 13 1 L (cid:88) (cid:126)p (cid:126)p ( E vvp ) e − E vvp t , (39)with E vlp = (cid:113) ( m vlπ ) + (cid:126)p , E vvp = (cid:112) ( m vvπ ) + (cid:126)p , ( m vlπ ) = B ( m l + m v ) , ( m vvπ ) = 2 Bm v . (40) =t-0.05fm and t =t+0.05fm FIG. 12. The top panel shows the continuum, infinite-volumelimit of the connected isospin-symmetric light and strangewindows with t = t − .
05 fm and t = t + 0 .
05 fm and ∆ =0 .
15 fm. The bottom panel shows the FVC for light valencequark mass in the isospin-symmetric limit. The continuumlimit of the very short-distance windows is difficult to controlas described in Sec. II.
In these expressions L is the spatial volume. We thenuse Eqs. (11) and (12), to relate this correlator to diagramM and O, for which we can the compute finite-volumecorrections. We find ∂∂m v C NLO , PQ χ PT , conn . = 59 ∂∂m v c = − M , (41) ∂∂m v C NLO , PQ χ PT , disc . = − ∂∂m v d = 29 O . (42)From Eq. (38) it then follows that ∂∂m v C NLO , PQ χ PT , conn . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m v = m l = − ∂∂m v C NLO , PQ χ PT , disc . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m v = m l (43)and therefore that within NLO PQChPT M = O . (44)6 t /fm t /fm ∆/fm a ud , conn ., isospin µ a s , conn ., isospin µ Total 657(26)(12) 52.83(22)(65)0.0 0.1 0.15 3.60(00)(59) 0.81(00)(12)0.1 0.2 0.15 8.649(03)(73) 1.666(01)(12)0.2 0.3 0.15 14.27(01)(82) 2.57(00)(16)0.3 0.4 0.15 18.67(02)(35) 3.448(05)(65)0.4 0.5 0.15 24.617(35)(63) 4.170(07)(20)0.5 0.6 0.15 29.47(06)(29) 4.666(10)(59)0.6 0.7 0.15 33.85(10)(37) 4.866(13)(74)0.7 0.8 0.15 37.71(14)(15) 4.799(16)(39)0.8 0.9 0.15 39.55(20)(21) 4.505(17)(44)0.9 1.0 0.15 40.77(27)(31) 4.058(19)(65)1.0 1.1 0.15 40.86(44)(41) 3.527(19)(76)1.1 1.2 0.15 39.81(54)(42) 2.973(19)(75)1.2 1.3 0.15 38.10(65)(51) 2.441(18)(77)1.3 1.4 0.15 35.54(77)(53) 1.955(17)(67)1.4 1.5 0.15 32.70(88)(56) 1.534(15)(60)1.5 1.6 0.15 29.50(100)(58) 1.181(13)(52)1.6 1.7 0.15 25.51(81)(66) 0.894(12)(44)1.7 1.8 0.15 22.20(85)(66) 0.667(10)(37)1.8 1.9 0.15 19.18(86)(67) 0.491(08)(30)1.9 2.0 0.15 16.59(89)(75) 0.357(07)(24)0.0 0.2 0.15 12.25(00)(52) 2.48(00)(11)0.2 0.4 0.15 32.95(03)(48) 6.02(01)(10)0.4 0.6 0.15 54.08(10)(29) 8.837(18)(74)0.6 0.8 0.15 71.55(24)(38) 9.666(29)(91)0.8 1.0 0.15 80.33(47)(44) 8.56(04)(10)0.3 1.0 0.15 224.6(0.8)(1.1) 30.51(08)(25)0.3 1.3 0.15 343.1(2.6)(2.0) 39.45(13)(35)0.3 1.6 0.15 441.0(5.1)(3.4) 44.12(17)(49)0.4 1.0 0.15 205.97(79)(90) 27.06(08)(21)0.4 1.3 0.15 324.6(2.6)(1.9) 36.01(13)(36)0.4 1.6 0.15 422.4(5.1)(3.5) 40.68(17)(51)0.4 1.0 0.05 216.5(0.8)(6.2) 27.9(0.1)(1.1)0.4 1.0 0.1 209.80(77)(79) 27.70(08)(21)0.4 1.0 0.2 202.10(82)(91) 26.24(08)(21)TABLE VII. Final results, including finite-volume correc-tions, for connected isospin-symmetric light and strange quarkcontributions.Contribution Result × FromTotal 714(27)(13)ud, conn., isospin 657(26)(12) Table VIIs, conn., isospin 52.83(22)(65) Table VIIc, conn., isospin 14.3(0.0)(0.7) Ref. [21]uds, disc., isospin -11.2(3.3)(2.3) Ref. [21]SIB, conn. 9.0(0.8)(1.2) Table IXSIB, disc. -6.9(0.0)(3.5) Eq. (45)QED, conn. 5.9(5.7)(1.7) Ref. [21]QED, disc. -6.9(2.1)(2.0) Ref. [21]TABLE VIII. We combine new results obtained in this paperwith results for the missing contributions from RBC/UKQCD[21] to our total result for a HVP LO µ . Since the connected plus disconnected SIB enters as M − O , indeed the total two-pion contributions cancel.In this work, we use the separate expressions for the con-nected and disconnected SIB FVC and quote the appro-priate infinite-volume result for a SIB , conn .µ in addition tothe finite-volume result a SIB , conn ., fv µ for different windows t /fm t /fm ∆/fm a SIB , conn ., fv µ a SIB , conn .µ Total 5.25(76)(29) 9.0(0.8)(1.2)0.0 0.1 0.15 -0.002(00)(17) -0.002(00)(17)0.1 0.2 0.15 0.0015(01)(23) 0.0019(01)(23)0.2 0.3 0.15 0.007(00)(23) 0.008(00)(23)0.3 0.4 0.15 0.009(01)(11) 0.011(01)(11)0.4 0.5 0.15 0.0266(10)(16) 0.0305(10)(22)0.5 0.6 0.15 0.0462(16)(91) 0.0531(16)(93)0.6 0.7 0.15 0.077(02)(11) 0.088(02)(12)0.7 0.8 0.15 0.1159(35)(66) 0.1327(35)(90)0.8 0.9 0.15 0.1502(46)(76) 0.174(05)(11)0.9 1.0 0.15 0.189(06)(14) 0.221(06)(18)1.0 1.1 0.15 0.255(20)(19) 0.296(20)(24)1.1 1.2 0.15 0.296(24)(22) 0.348(24)(28)1.2 1.3 0.15 0.331(27)(29) 0.393(27)(36)1.3 1.4 0.15 0.348(31)(29) 0.420(31)(38)1.4 1.5 0.15 0.356(34)(30) 0.439(34)(41)1.5 1.6 0.15 0.351(37)(27) 0.443(37)(41)1.6 1.7 0.15 0.297(18)(26) 0.399(18)(42)1.7 1.8 0.15 0.270(18)(25) 0.381(18)(43)1.8 1.9 0.15 0.243(18)(24) 0.361(18)(44)1.9 2.0 0.15 0.219(18)(26) 0.342(18)(47)0.0 0.2 0.15 -0.001(00)(14) -0.000(00)(14)0.2 0.4 0.15 0.016(01)(13) 0.019(01)(13)0.4 0.6 0.15 0.0729(26)(83) 0.0836(26)(91)0.6 0.8 0.15 0.193(06)(12) 0.221(06)(16)0.8 1.0 0.15 0.339(10)(19) 0.395(10)(27)0.3 1.0 0.15 0.615(19)(35) 0.711(19)(49)0.3 1.3 0.15 1.47(12)(10) 1.72(12)(13)0.3 1.6 0.15 2.53(21)(17) 3.03(21)(24)0.4 1.0 0.15 0.606(18)(31) 0.700(18)(46)0.4 1.3 0.15 1.47(12)(10) 1.72(12)(13)0.4 1.6 0.15 2.53(21)(18) 3.03(21)(25)0.4 1.0 0.05 0.63(02)(19) 0.72(02)(20)0.4 1.0 0.1 0.603(18)(35) 0.693(18)(48)0.4 1.0 0.2 0.615(19)(31) 0.715(19)(47)TABLE IX. We provide results for the connected SIB con-tribution both at finite volume ( a SIB , conn ., fv µ ) and at infinitevolume a SIB , conn .µ . While the sum of the connected and discon-nected SIB contribution have likely small finite-volume cor-rections, a SIB , conn .µ itself receives a significant correction. Thisis important for comparisons of this contribution between dif-ferent lattice results. in Tab. IX.It is instructive to consider the infinite-volume NLOPQChPT results a SIB , conn ., NLOPQ χ PT µ = − a SIB , disc ., NLOPQ χ PT µ (45)= 6 . . − , (46)where we add a 50% systematic error. In these expres-sions, we use m π = 135 MeV since in this context we areinterested in the evaluation of mass-derivatives at theisospin symmetric limit ( m v = m l ). V. DISCUSSION AND CONCLUSION
We summarize our results for the total contributions to a HVP LO µ in Tab. VIII and compare this result in Fig. 147to results by other collaborations. In Fig. 13, we com-pare our results for a ud , conn ., isospin µ , a SIB , conn .µ , and thewindow a ud , conn ., isospin , W µ with t = 0 . t = 1 . .
15 fm to other collaborations. We would liketo stress in particular the difference between the win-dow results from Aubin et al. and this work, which isespecially noteworthy since they were performed on thesame gauge configurations. Apart from the small va-lence mass-extrapolation, which we suggest to be mildfor the window, the main difference between this workand Aubin et al. is the choice of a site-local comparedto a conserved current. This suggests that properly esti-mating the uncertainties associated with the continuumlimit may be challenging. We note that in this work,Aubin et al. , as well as the recent BMW result, resultsat similar inverse lattice spacings from a − ≈ . a − ≈ . a HVP LO µ with uncertaintiesclose to 5 × − , which may shed further light on theemerging tensions. It will be particularly important thatsuch results include different lattice discretizations. ACKNOWLEDGMENTS
We thank our colleagues in the RBC & UKQCD collab-orations for interesting discussions. We thank the MILCcollaboration for the ensembles used in this analysis. In-versions and contractions were performed with the MILCcode version 7. This work was supported by resourcesprovided by the Scientific Data and Computing Center(SDCC) at Brookhaven National Laboratory (BNL), aDOE Office of Science User Facility supported by theOffice of Science of the US Department of Energy. TheSDCC is a major component of the Computational Sci-ence Initiative at BNL. We gratefully acknowledge com-puting resources provided through USQCD clusters atBNL and Jefferson Lab. CL and ASM are supported inpart by US DOE Contract DESC0012704(BNL) and bya DOE Office of Science Early Career Award.
LM 2020BMW 2020ETMC 2019 UpdateAubin et al. 2019Mainz 2019FNAL/HPQCD/MILC 2019SK 2019ETMC 2018RBC/UKQCD 2018BMW 2017Mainz 2017HPQCD 2016540 560 580 600 620 640 660 680 700a µ , ud, conn, isospin × LM 2020LM 2020 FVBMW 2020ETMC 2019RBC/UKQCD 2018FNAL/HPQCD/MILC 2017 0 5 10 15 20a µ , SIB, conn × KNT 2018/LatticeLM 2020BMW 2020Aubin et al. 2019RBC/UKQCD 2018195 200 205 210 215a µ , ud, conn, isospin, W-0.4-1.0-0.15 × FIG. 13. Overview of results for a ud , conn ., isospin µ , a SIB , conn .µ ,and the window a ud , conn ., isospin , W µ with t = 0 . t = 1 . .
15 fm. The referenced contributions are listedin the caption of Fig. 14 apart from ETMC 2019 [29] andFNAL/HPQCD/MILC 2017 [17]. The result of this work islabelled “LM 2020”. For a ud , conn ., isospin µ the precise BMW2020 is higher in particular compared to values by ETMC2019 Update as well as FNAL/HPQCD/MILC 2019. For a SIB , conn .µ , we provide also a finite-volume result “LM 2020FV” to compare to the other results obtained at similar vol-ume. “LM 2020” is the value corrected to infinite volume. Forthe window a ud , conn ., isospin , W µ there is a clear tension betweenAubin et al. et al. No new physicsKNT 2019DHMZ 2019KNT 2018Jegerlehner 2017DHMZ 2017DHMZ 2012HLMNT 2011RBC/UKQCD 2018LM 2020BMW 2020ETMC 2019 UpdateMainz 2019FNAL/HPQCD/MILC 2019SK 2019ETMC 2018RBC/UKQCD 2018BMW 2017Mainz 2017HPQCD 2016ETMC 2013610 630 650 670 690 710 730 750Lattice + R-ratioLatticeR-ratioa µ × FIG. 14. Overview of total results for a HVP µ . The refer-enced contributions are: ETMC 2013 [11], HPQCD 2016 [14],Mainz 2017 [19], BMW 2017 [20], RBC/UKQCD 2018 [21],ETMC 2018 [22], SK 2019 [25], FNAL/HPQCD/MILC2019 [24], Mainz 2019 [26], ETMC 2019 Update [30], BMW2020 [27], HLMNT 2011 [6], DHMZ 2012 [7], DHMZ 2017 [8],Jegerlehner 2017 [9], KNT 2018 [10], DHMZ 2019 [1], andKNT 2019 [2]. The result of this work is labelled “LM 2020”. Appendix A: Results
This section contains tables with a detailed breakdownof systematic errors from the window data that appear inTable V. Tables X and XI give the systematic error break- down for the ˆ p prescription, while Tables XII and XIIIgive the p prescription. Tables X and XII both containunimproved data while Tables XI and XIII both use theIPA procedure of Section II C. All of Tables X-XIII usethe light quark mass data. Tables XIV and XV give theunimproved data with the ˆ p and p prescriptions for thestrange quark mass, respectively. [1] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Anew evaluation of the hadronic vacuum polarisation con-tributions to the muon anomalous magnetic moment andto α ( m ), (2019), arXiv:1908.00921 [hep-ph].[2] A. Keshavarzi, D. Nomura, and T. Teubner, g − α ( M Z ) , and the hyperfine split-ting of muonium, Phys. Rev. D101 , 014029 (2020),arXiv:1911.00367 [hep-ph].[3] G. W. Bennett et al. (Muon g-2), Final Report of theMuon E821 Anomalous Magnetic Moment Measurementat BNL, Phys. Rev.
D73 , 072003 (2006), arXiv:hep-ex/0602035 [hep-ex].[4] J. Grange et al. (Muon g-2), Muon (g-2) Technical DesignReport, (2015), arXiv:1501.06858 [physics.ins-det].[5] M. Abe et al. , A New Approach for Measuring the MuonAnomalous Magnetic Moment and Electric Dipole Mo-ment, PTEP , 053C02 (2019), arXiv:1901.03047[physics.ins-det].[6] K. Hagiwara, R. Liao, A. D. Martin, D. Nomura,and T. Teubner, ( g − µ and α ( M Z ) re-evaluated us-ing new precise data, J. Phys. G38 , 085003 (2011),arXiv:1105.3149 [hep-ph].[7] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang,Reevaluation of the Hadronic Contributions to the Muong-2 and to α ( M Z ), Eur. Phys. J. C71 , 1515 (2011), [Er-ratum: Eur. Phys. J.C72,1874(2012)], arXiv:1010.4180[hep-ph].[8] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang,Reevaluation of the hadronic vacuum polarisation contri-butions to the Standard Model predictions of the muon g − α ( m Z ) using newest hadronic cross-section data,Eur. Phys. J. C77 , 827 (2017), arXiv:1706.09436 [hep-ph].[9] F. Jegerlehner, Muon g-2 theory: The hadronic part,
Pro-ceedings, KLOE-2 Workshop on e + e − Collision Physicsat 1 GeV: Frascati, Italy, October 26-28, 2016 , EPJ WebConf. , 00022 (2018), arXiv:1705.00263 [hep-ph].[10] A. Keshavarzi, D. Nomura, and T. Teubner, Muon g-2and α ( M Z ): a new data-based analysis, Phys. Rev. D97 ,114025 (2018), arXiv:1802.02995 [hep-ph].[11] F. Burger, X. Feng, G. Hotzel, K. Jansen, M. Petschlies,and D. B. Renner (ETM), Four-Flavour Leading-OrderHadronic Contribution To The Muon Anomalous Mag-netic Moment, JHEP , 099, arXiv:1308.4327 [hep-lat].[12] T. Blum, P. A. Boyle, T. Izubuchi, L. Jin, A. J¨uttner,C. Lehner, K. Maltman, M. Marinkovic, A. Portelli, andM. Spraggs, Calculation of the hadronic vacuum polar-ization disconnected contribution to the muon anomalousmagnetic moment, Phys. Rev. Lett. , 232002 (2016),arXiv:1512.09054 [hep-lat].[13] T. Blum et al. (RBC/UKQCD), Lattice calcula-tion of the leading strange quark-connected contribu- tion to the muon g −
2, JHEP , 063, [Erratum:JHEP05,034(2017)], arXiv:1602.01767 [hep-lat].[14] B. Chakraborty, C. T. H. Davies, P. G. de Oliviera,J. Koponen, G. P. Lepage, and R. S. Van de Water,The hadronic vacuum polarization contribution to a µ from full lattice QCD, Phys. Rev. D96 , 034516 (2017),arXiv:1601.03071 [hep-lat].[15] M. A. Clark, C. Jung, and C. Lehner, Multi-GridLanczos,
Proceedings, 35th International Symposium onLattice Field Theory (Lattice 2017): Granada, Spain,June 18-24, 2017 , EPJ Web Conf. , 14023 (2018),arXiv:1710.06884 [hep-lat].[16] C. Lehner (RBC, UKQCD), A precise determination ofthe HVP contribution to the muon anomalous magneticmoment from lattice QCD,
Proceedings, 35th Interna-tional Symposium on Lattice Field Theory (Lattice 2017):Granada, Spain, June 18-24, 2017 , EPJ Web Conf. ,01024 (2018), arXiv:1710.06874 [hep-lat].[17] B. Chakraborty et al. (Fermilab Lattice, LATTICE-HPQCD, MILC), Strong-Isospin-Breaking Correction tothe Muon Anomalous Magnetic Moment from LatticeQCD at the Physical Point, Phys. Rev. Lett. , 152001(2018), arXiv:1710.11212 [hep-lat].[18] P. Boyle, V. G¨ulpers, J. Harrison, A. J¨uttner, C. Lehner,A. Portelli, and C. T. Sachrajda, Isospin breaking cor-rections to meson masses and the hadronic vacuumpolarization: a comparative study, JHEP , 153,arXiv:1706.05293 [hep-lat].[19] M. Della Morte, A. Francis, V. G¨ulpers, G. Herdoza,G. von Hippel, H. Horch, B. Jger, H. B. Meyer, A. Nyf-feler, and H. Wittig, The hadronic vacuum polarizationcontribution to the muon g − , 020, arXiv:1705.01775 [hep-lat].[20] S. Borsanyi et al. (Budapest-Marseille-Wuppertal),Hadronic vacuum polarization contribution to theanomalous magnetic moments of leptons from firstprinciples, Phys. Rev. Lett. , 022002 (2018),arXiv:1711.04980 [hep-lat].[21] T. Blum, P. A. Boyle, V. G¨ulpers, T. Izubuchi, L. Jin,C. Jung, A. J¨uttner, C. Lehner, A. Portelli, and J. T.Tsang (RBC, UKQCD), Calculation of the hadronic vac-uum polarization contribution to the muon anomalousmagnetic moment, Phys. Rev. Lett. , 022003 (2018),arXiv:1801.07224 [hep-lat].[22] D. Giusti, F. Sanfilippo, and S. Simula, Light-quark con-tribution to the leading hadronic vacuum polarizationterm of the muon g − D98 , 114504 (2018), arXiv:1808.00887 [hep-lat].[23] T. Izubuchi, Y. Kuramashi, C. Lehner, and E. Shintani(PACS), Finite-volume correction on the hadronic vac-uum polarization contribution to the muon g-2 in latticeQCD, Phys. Rev.
D98 , 054505 (2018), arXiv:1805.04250 t /fm t /fm ∆/fm a ud , conn ., isospin , W µ × Total 627(26) S (00) Z V (00) Z V (02) Z V (03) C (00) C (03) m l (00) mlms (02) mvms (00) w a (00) w a (01) w a (06) w . S (00) Z V (00) Z V (00) Z V (06) C (52) C (00) m l (00) mlms (26) mvms (00) w a (00) w a (00) w a (01) w . S (04) Z V (13) Z V (02) Z V (06) C (63) C (01) m l (00) mlms (32) mvms (00) w a (01) w a (00) w a (10) w . S (01) Z V (02) Z V (01) Z V (09) C (73) C (00) m l (00) mlms (36) mvms (00) w a (00) w a (00) w a (02) w . S (01) Z V (03) Z V (01) Z V (03) C (30) C (00) m l (00) mlms (16) mvms (00) w a (00) w a (00) w a (01) w . S (20) Z V (49) Z V (03) Z V (03) C (19) C (14) m l (00) mlms (10) mvms (01) w a (01) w a (00) w a (07) w . S (02) Z V (05) Z V (01) Z V (03) C (25) C (02) m l (00) mlms (13) mvms (00) w a (00) w a (00) w a (03) w . S (03) Z V (06) Z V (01) Z V (03) C (32) C (04) m l (00) mlms (15) mvms (01) w a (01) w a (00) w a (02) w . S (03) Z V (07) Z V (01) Z V (00) C (03) C (06) m l (00) mlms (04) mvms (01) w a (01) w a (00) w a (08) w . S (03) Z V (07) Z V (01) Z V (02) C (09) C (08) m l (00) mlms (02) mvms (01) w a (02) w a (00) w a (14) w . S (03) Z V (07) Z V (02) Z V (01) C (12) C (10) m l (00) mlms (16) mvms (02) w a (02) w a (01) w a (17) w . S (04) Z V (10) Z V (03) Z V (01) C (20) C (14) m l (00) mlms (11) mvms (03) w a (05) w a (02) w a (25) w . S (04) Z V (09) Z V (02) Z V (00) C (17) C (16) m l (01) mlms (14) mvms (04) w a (06) w a (01) w a (26) w . S (03) Z V (09) Z V (01) Z V (00) C (19) C (18) m l (01) mlms (23) mvms (04) w a (06) w a (01) w a (32) w . S (03) Z V (07) Z V (01) Z V (00) C (12) C (19) m l (01) mlms (23) mvms (04) w a (06) w a (01) w a (35) w . S (02) Z V (06) Z V (00) Z V (00) C (14) C (19) m l (01) mlms (26) mvms (04) w a (06) w a (00) w a (37) w . S (02) Z V (05) Z V (00) Z V (00) C (13) C (19) m l (01) mlms (23) mvms (04) w a (06) w a (00) w a (40) w . S (01) Z V (02) Z V (03) Z V (12) C (00) C (16) m l (01) mlms (42) mvms (02) w a (02) w a (05) w a (39) w . S (01) Z V (02) Z V (03) Z V (10) C (01) C (15) m l (00) mlms (43) mvms (02) w a (02) w a (05) w a (37) w . S (01) Z V (02) Z V (02) Z V (04) C (00) C (13) m l (00) mlms (47) mvms (02) w a (02) w a (04) w a (33) w . S (01) Z V (01) Z V (02) Z V (06) C (03) C (12) m l (00) mlms (58) mvms (02) w a (02) w a (04) w a (30) w . S (00) Z V (02) Z V (00) Z V (06) C (46) C (00) m l (00) mlms (23) mvms (00) w a (00) w a (00) w a (02) w . S (02) Z V (06) Z V (01) Z V (05) C (43) C (01) m l (00) mlms (20) mvms (00) w a (00) w a (00) w a (01) w . S (04) Z V (10) Z V (02) Z V (02) C (23) C (04) m l (00) mlms (11) mvms (00) w a (00) w a (00) w a (02) w . S (06) Z V (14) Z V (02) Z V (03) C (29) C (10) m l (00) mlms (10) mvms (01) w a (02) w a (00) w a (10) w . S (06) Z V (15) Z V (03) Z V (02) C (04) C (18) m l (01) mlms (14) mvms (03) w a (04) w a (01) w a (30) w . S (18) Z V (42) Z V (06) Z V (08) C (79) C (33) m l (01) mlms (23) mvms (05) w a (06) w a (01) w a (41) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (0 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (1 . w . . S (0 . Z V (1 . Z V (0 . Z V (0 . C (0 . C (1 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (2 . w . S (16) Z V (39) Z V (05) Z V (05) C (48) C (33) m l (01) mlms (07) mvms (05) w a (06) w a (01) w a (42) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (0 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (1 . w . . S (0 . Z V (1 . Z V (0 . Z V (0 . C (0 . C (1 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (2 . w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (5 . C (0 . m l (0 . mlms (2 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (17) Z V (40) Z V (05) Z V (02) C (23) C (33) m l (01) mlms (25) mvms (05) w a (06) w a (01) w a (36) w . S (16) Z V (38) Z V (06) Z V (06) C (46) C (33) m l (01) mlms (03) mvms (05) w a (07) w a (01) w a (46) w TABLE X. A detailed breakdown of the systematic uncertainties for the data in the column labeled l, ˆ p U in Table V. Theseare the unimproved data with the ˆ p prescription applied. The subscripts on each uncertainty denote the different sources ofuncertainty. In cases where each ensemble has a different uncertainty, a superscript of 48, 64, or 96 is included to indicatethe 48c, 64c, or 96c ensemble, respectively. The subscript S denotes the statistical error. The subscript Z V indicates theuncertainty on the vector current renormalization factors, which are given in Table I. The subscript C indicates the continuumlimit uncertainty on the m v /m s = 1 / C . We noticesignificant fluctuations of the short-distance window continuum error estimates since for the t = 0 . t = 0 . m l m s denotes the uncertainty propagated from λ , m l denotes the uncertainty fromthe light sea-quark mistuning, and m v m s denotes the light quark mass extrapolation uncertainty. The subscript w /a denotesthe scale setting uncertainty from the corresponding values in Table I, and the subscript w from the value in the caption ofTable I. t /fm t /fm ∆/fm a ud , conn ., isospin , W µ × Total 632(27) S (00) Z V (00) Z V (02) Z V (00) C (00) C (03) m l (00) mlms (01) mvms (00) w a (00) w a (01) w a (06) w . S (00) Z V (01) Z V (00) Z V (03) C (28) C (00) m l (00) mlms (14) mvms (00) w a (00) w a (00) w a (01) w . S (01) Z V (01) Z V (00) Z V (06) C (46) C (00) m l (00) mlms (23) mvms (00) w a (00) w a (00) w a (00) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (02) Z V (05) Z V (01) Z V (02) C (21) C (01) m l (00) mlms (09) mvms (00) w a (00) w a (00) w a (01) w . S (02) Z V (06) Z V (01) Z V (02) C (22) C (03) m l (00) mlms (11) mvms (00) w a (00) w a (00) w a (01) w . S (03) Z V (06) Z V (01) Z V (02) C (22) C (04) m l (00) mlms (10) mvms (00) w a (01) w a (00) w a (04) w . S (03) Z V (07) Z V (01) Z V (01) C (13) C (06) m l (00) mlms (04) mvms (01) w a (01) w a (00) w a (08) w . S (03) Z V (07) Z V (01) Z V (01) C (01) C (08) m l (00) mlms (04) mvms (01) w a (02) w a (00) w a (13) w . S (03) Z V (07) Z V (02) Z V (00) C (08) C (10) m l (00) mlms (14) mvms (02) w a (02) w a (01) w a (18) w . S (04) Z V (10) Z V (03) Z V (01) C (19) C (14) m l (00) mlms (12) mvms (03) w a (05) w a (01) w a (23) w . S (04) Z V (10) Z V (02) Z V (01) C (20) C (16) m l (01) mlms (17) mvms (04) w a (06) w a (01) w a (27) w . S (03) Z V (09) Z V (01) Z V (00) C (18) C (18) m l (01) mlms (22) mvms (04) w a (06) w a (01) w a (32) w . S (03) Z V (07) Z V (01) Z V (00) C (15) C (19) m l (01) mlms (26) mvms (04) w a (06) w a (01) w a (35) w . S (02) Z V (06) Z V (00) Z V (00) C (14) C (19) m l (01) mlms (27) mvms (04) w a (06) w a (00) w a (38) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (0 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (01) Z V (02) Z V (03) Z V (13) C (00) C (16) m l (01) mlms (43) mvms (02) w a (02) w a (05) w a (39) w . S (01) Z V (02) Z V (03) Z V (10) C (01) C (15) m l (00) mlms (46) mvms (02) w a (02) w a (05) w a (37) w . S (01) Z V (01) Z V (03) Z V (04) C (00) C (13) m l (00) mlms (51) mvms (02) w a (02) w a (04) w a (34) w . S (01) Z V (01) Z V (02) Z V (08) C (03) C (12) m l (00) mlms (37) mvms (02) w a (02) w a (04) w a (31) w . S (01) Z V (02) Z V (01) Z V (02) C (19) C (00) m l (00) mlms (09) mvms (00) w a (00) w a (00) w a (01) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (2 . C (0 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (05) Z V (11) Z V (01) Z V (00) C (01) C (04) m l (00) mlms (02) mvms (00) w a (00) w a (00) w a (03) w . S (06) Z V (14) Z V (02) Z V (03) C (36) C (10) m l (00) mlms (14) mvms (01) w a (02) w a (00) w a (11) w . S (06) Z V (14) Z V (03) Z V (01) C (07) C (18) m l (01) mlms (18) mvms (03) w a (04) w a (01) w a (31) w . S (18) Z V (42) Z V (07) Z V (07) C (74) C (34) m l (01) mlms (49) mvms (05) w a (06) w a (02) w a (47) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (1 . w . . S (0 . Z V (1 . Z V (0 . Z V (0 . C (2 . C (1 . m l (0 . mlms (2 . mvms (0 . w a (0 . w a (0 . w a (2 . w . S (17) Z V (39) Z V (06) Z V (04) C (29) C (33) m l (01) mlms (01) mvms (05) w a (06) w a (01) w a (45) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (0 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (1 . w . . S (0 . Z V (1 . Z V (0 . Z V (0 . C (0 . C (1 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (2 . w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (17) Z V (40) Z V (06) Z V (09) C (84) C (32) m l (01) mlms (29) mvms (05) w a (06) w a (01) w a (43) w . S (16) Z V (38) Z V (07) Z V (01) C (08) C (33) m l (01) mlms (23) mvms (05) w a (07) w a (02) w a (48) w TABLE XI. Same as Table X, but with the parity improvement.[hep-lat].[24] C. T. H. Davies et al. (Fermilab Lattice, LATTICE-HPQCD, MILC), Hadronic-vacuum-polarization contri-bution to the muons anomalous magnetic moment fromfour-flavor lattice QCD, Phys. Rev.
D101 , 034512(2020), arXiv:1902.04223 [hep-lat].[25] E. Shintani and Y. Kuramashi (PACS), Hadronic vac-uum polarization contribution to the muon g − latticeat the physical point, Phys. Rev. D100 , 034517 (2019),arXiv:1902.00885 [hep-lat].[26] A. Grardin, M. C, G. von Hippel, B. Hrz, H. B. Meyer,D. Mohler, K. Ottnad, J. Wilhelm, and H. Wittig, The leading hadronic contribution to ( g − µ fromlattice QCD with N f = 2 + 1 flavours of O( a ) im-proved Wilson quarks, Phys. Rev. D100 , 014510 (2019),arXiv:1904.03120 [hep-lat].[27] S. Borsanyi et al. , Leading-order hadronic vacuum polar-ization contribution to the muon magnetic momentfromlattice QCD, (2020), arXiv:2002.12347 [hep-lat].[28] C. Lehner et al. (USQCD), Opportunities for LatticeQCD in Quark and Lepton Flavor Physics, Eur. Phys.J.
A55 , 195 (2019), arXiv:1904.09479 [hep-lat].[29] D. Giusti, V. Lubicz, G. Martinelli, F. Sanfilippo, andS. Simula, Electromagnetic and strong isospin-breakingcorrections to the muon g − t /fm t /fm ∆/fm a ud , conn ., isospin , W µ × Total 628(26) S (00) Z V (00) Z V (02) Z V (02) C (00) C (03) m l (00) mlms (02) mvms (00) w a (00) w a (01) w a (06) w . S (00) Z V (01) Z V (00) Z V (02) C (18) C (00) m l (00) mlms (09) mvms (00) w a (00) w a (00) w a (01) w . S (01) Z V (02) Z V (00) Z V (05) C (40) C (00) m l (00) mlms (20) mvms (00) w a (00) w a (00) w a (00) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (02) Z V (04) Z V (01) Z V (03) C (23) C (00) m l (00) mlms (13) mvms (00) w a (00) w a (00) w a (01) w . S (22) Z V (50) Z V (04) Z V (02) C (13) C (14) m l (00) mlms (03) mvms (01) w a (00) w a (00) w a (08) w . S (02) Z V (06) Z V (01) Z V (03) C (27) C (02) m l (00) mlms (14) mvms (00) w a (00) w a (00) w a (03) w . S (03) Z V (06) Z V (01) Z V (04) C (34) C (04) m l (00) mlms (17) mvms (01) w a (01) w a (00) w a (02) w . S (03) Z V (07) Z V (01) Z V (00) C (03) C (06) m l (00) mlms (04) mvms (01) w a (01) w a (00) w a (08) w . S (03) Z V (08) Z V (01) Z V (02) C (09) C (08) m l (00) mlms (02) mvms (01) w a (02) w a (00) w a (14) w . S (03) Z V (07) Z V (02) Z V (01) C (12) C (10) m l (00) mlms (17) mvms (02) w a (02) w a (01) w a (16) w . S (04) Z V (10) Z V (03) Z V (01) C (21) C (14) m l (00) mlms (11) mvms (03) w a (05) w a (02) w a (25) w . S (04) Z V (09) Z V (02) Z V (00) C (17) C (16) m l (01) mlms (14) mvms (04) w a (06) w a (01) w a (26) w . S (03) Z V (09) Z V (01) Z V (00) C (20) C (18) m l (01) mlms (23) mvms (04) w a (06) w a (01) w a (32) w . S (03) Z V (07) Z V (01) Z V (00) C (13) C (19) m l (01) mlms (24) mvms (04) w a (06) w a (01) w a (35) w . S (02) Z V (06) Z V (00) Z V (00) C (14) C (19) m l (01) mlms (26) mvms (04) w a (06) w a (00) w a (37) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (0 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (01) Z V (02) Z V (03) Z V (12) C (00) C (16) m l (01) mlms (42) mvms (02) w a (02) w a (05) w a (39) w . S (01) Z V (02) Z V (03) Z V (11) C (01) C (15) m l (00) mlms (43) mvms (02) w a (02) w a (05) w a (37) w . S (01) Z V (02) Z V (02) Z V (05) C (00) C (13) m l (00) mlms (47) mvms (02) w a (02) w a (04) w a (33) w . S (01) Z V (01) Z V (02) Z V (06) C (03) C (11) m l (00) mlms (32) mvms (02) w a (02) w a (04) w a (30) w . S (01) Z V (02) Z V (00) Z V (03) C (23) C (00) m l (00) mlms (12) mvms (00) w a (00) w a (00) w a (01) w . S (03) Z V (06) Z V (01) Z V (10) C (84) C (01) m l (00) mlms (40) mvms (00) w a (00) w a (00) w a (02) w . S (05) Z V (10) Z V (02) Z V (03) C (28) C (04) m l (00) mlms (14) mvms (00) w a (00) w a (00) w a (02) w . S (06) Z V (14) Z V (02) Z V (03) C (32) C (10) m l (00) mlms (11) mvms (01) w a (02) w a (00) w a (10) w . S (06) Z V (15) Z V (03) Z V (01) C (04) C (18) m l (01) mlms (14) mvms (03) w a (04) w a (01) w a (30) w . S (19) Z V (43) Z V (06) Z V (08) C (80) C (33) m l (01) mlms (23) mvms (05) w a (06) w a (01) w a (41) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (0 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (1 . w . . S (0 . Z V (1 . Z V (0 . Z V (0 . C (0 . C (1 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (2 . w . S (17) Z V (39) Z V (06) Z V (06) C (56) C (33) m l (01) mlms (11) mvms (05) w a (06) w a (01) w a (42) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (0 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (1 . w . . S (0 . Z V (1 . Z V (0 . Z V (0 . C (0 . C (1 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (2 . w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (5 . C (0 . m l (0 . mlms (3 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (17) Z V (40) Z V (05) Z V (01) C (13) C (33) m l (01) mlms (21) mvms (05) w a (06) w a (01) w a (36) w . S (17) Z V (38) Z V (06) Z V (06) C (48) C (33) m l (01) mlms (03) mvms (05) w a (07) w a (01) w a (46) w TABLE XII. Same as Table X, but with the p prescription.Phys. Rev. D99 , 114502 (2019), arXiv:1901.10462 [hep-lat].[30] D. Giusti and S. Simula, Lepton anomalous magnetic mo-ments in Lattice QCD+QED, , PoS
LATTICE2019 ,104 (2019), arXiv:1910.03874 [hep-lat].[31] J. Prades, E. de Rafael, and A. Vainshtein, The HadronicLight-by-Light Scattering Contribution to the Muon andElectron Anomalous Magnetic Moments, Adv. Ser. Di-rect. High Energy Phys. , 303 (2009), arXiv:0901.0306[hep-ph]. [32] T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin,C. Jung, and C. Lehner, Hadronic light-by-light con-tribution to the muon anomalous magnetic momentfrom lattice QCD, in (2019)arXiv:1907.00864 [hep-lat].[33] First plenary workshop of the muon g-2 theory initiative, https://indico.fnal.gov/event/13795/ (2017).[34] Workshop on hadronic vacuum polarization contribu-tions to muon g-2, https://kds.kek.jp/indico/event/26780/ (2018). t /fm t /fm ∆/fm a ud , conn ., isospin , W µ × Total 634(27) S (00) Z V (00) Z V (01) Z V (02) C (01) C (03) m l (00) mlms (01) mvms (00) w a (00) w a (01) w a (06) w . S (00) Z V (01) Z V (00) Z V (03) C (21) C (00) m l (00) mlms (11) mvms (00) w a (00) w a (00) w a (00) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (02) Z V (05) Z V (01) Z V (02) C (19) C (01) m l (00) mlms (08) mvms (00) w a (00) w a (00) w a (02) w . S (03) Z V (06) Z V (01) Z V (03) C (27) C (03) m l (00) mlms (14) mvms (00) w a (00) w a (00) w a (01) w . S (03) Z V (06) Z V (01) Z V (03) C (25) C (04) m l (00) mlms (12) mvms (00) w a (01) w a (00) w a (04) w . S (03) Z V (07) Z V (01) Z V (01) C (15) C (06) m l (00) mlms (05) mvms (01) w a (01) w a (00) w a (08) w . S (03) Z V (07) Z V (01) Z V (01) C (01) C (08) m l (00) mlms (04) mvms (01) w a (02) w a (00) w a (13) w . S (03) Z V (07) Z V (02) Z V (00) C (08) C (10) m l (00) mlms (14) mvms (02) w a (02) w a (01) w a (18) w . S (04) Z V (10) Z V (03) Z V (01) C (19) C (14) m l (00) mlms (12) mvms (03) w a (05) w a (01) w a (23) w . S (04) Z V (10) Z V (02) Z V (01) C (21) C (16) m l (01) mlms (17) mvms (04) w a (06) w a (01) w a (27) w . S (03) Z V (09) Z V (01) Z V (00) C (18) C (18) m l (01) mlms (23) mvms (04) w a (06) w a (01) w a (32) w . S (03) Z V (07) Z V (01) Z V (00) C (16) C (19) m l (01) mlms (27) mvms (04) w a (06) w a (01) w a (35) w . S (02) Z V (06) Z V (00) Z V (00) C (15) C (19) m l (01) mlms (27) mvms (04) w a (06) w a (00) w a (38) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (0 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (01) Z V (02) Z V (03) Z V (13) C (00) C (16) m l (01) mlms (43) mvms (02) w a (02) w a (05) w a (39) w . S (01) Z V (02) Z V (03) Z V (10) C (01) C (14) m l (00) mlms (37) mvms (02) w a (02) w a (05) w a (37) w . S (01) Z V (01) Z V (03) Z V (04) C (00) C (13) m l (00) mlms (51) mvms (02) w a (02) w a (04) w a (34) w . S (01) Z V (01) Z V (02) Z V (08) C (03) C (12) m l (00) mlms (61) mvms (02) w a (02) w a (04) w a (30) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (2 . C (0 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (05) Z V (11) Z V (01) Z V (00) C (09) C (04) m l (00) mlms (05) mvms (00) w a (00) w a (00) w a (03) w . S (06) Z V (14) Z V (02) Z V (04) C (40) C (10) m l (00) mlms (17) mvms (01) w a (02) w a (00) w a (11) w . S (06) Z V (15) Z V (03) Z V (01) C (07) C (18) m l (01) mlms (18) mvms (03) w a (04) w a (01) w a (31) w . S (19) Z V (43) Z V (08) Z V (07) C (73) C (34) m l (01) mlms (49) mvms (05) w a (06) w a (02) w a (47) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (1 . w . . S (0 . Z V (1 . Z V (0 . Z V (0 . C (2 . C (1 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (2 . w . S (17) Z V (40) Z V (06) Z V (05) C (42) C (33) m l (01) mlms (05) mvms (05) w a (07) w a (01) w a (45) w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (0 . C (0 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (1 . w . . S (0 . Z V (1 . Z V (0 . Z V (0 . C (0 . C (1 . m l (0 . mlms (1 . mvms (0 . w a (0 . w a (0 . w a (2 . w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . . S (0 . Z V (0 . Z V (0 . Z V (0 . C (1 . C (0 . m l (0 . mlms (0 . mvms (0 . w a (0 . w a (0 . w a (0 . w . S (17) Z V (38) Z V (07) Z V (01) C (05) C (33) m l (01) mlms (22) mvms (05) w a (07) w a (02) w a (48) w TABLE XIII. Same as Table XII, but with the parity improvement.[35] Muon g-2 theory initiative hadronic light-by-light work-ing group workshop, https://indico.phys.uconn.edu/event/1/ (2018).[36] Second plenary workshop of the muon g-2 theory ini-tiative, https://indico.him.uni-mainz.de/event/11/overview (2018).[37] Third plenary workshop of the muon g-2 theory initia-tive, https://indico.fnal.gov/event/21626/overview (2019).[38] Muon g-2 Theory Initiative, Muon g-2 theory initiativewhite paper, In preparation.[39] G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer,Dispersion relation for hadronic light-by-light scattering: theoretical foundations, JHEP , 074, arXiv:1506.01386[hep-ph].[40] G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer,Dispersion relation for hadronic light-by-light scattering:two-pion contributions, JHEP , 161, arXiv:1702.07347[hep-ph].[41] G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer,Rescattering effects in the hadronic-light-by-light contri-bution to the anomalous magnetic moment of the muon,Phys. Rev. Lett. , 232001 (2017), arXiv:1701.06554[hep-ph].[42] G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, andP. Stoffer, Longitudinal short-distance constraints for the t /fm t /fm ∆/fm a s , conn . isospin , W µ × Total 52 . S (04) Z V (20) Z V (30) C (03) w a (13) w a (53) w . S (00) Z V (00) Z V (12) C (00) w a (00) w a (00) w . S (01) Z V (06) Z V (10) C (00) w a (00) w a (01) w . S (00) Z V (01) Z V (16) C (00) w a (00) w a (00) w . S (02) Z V (13) Z V (63) C (00) w a (01) w a (04) w . S (03) Z V (16) Z V (08) C (00) w a (02) w a (08) w . S (03) Z V (18) Z V (53) C (01) w a (04) w a (16) w . S (04) Z V (18) Z V (67) C (01) w a (06) w a (24) w . S (03) Z V (18) Z V (08) C (02) w a (08) w a (33) w . S (03) Z V (17) Z V (03) C (02) w a (10) w a (39) w . S (03) Z V (15) Z V (43) C (02) w a (11) w a (45) w . S (03) Z V (13) Z V (57) C (02) w a (12) w a (46) w . S (02) Z V (11) Z V (57) C (02) w a (12) w a (46) w . S (02) Z V (09) Z V (62) C (02) w a (11) w a (44) w . S (01) Z V (07) Z V (52) C (02) w a (10) w a (40) w . S (01) Z V (06) Z V (47) C (02) w a (09) w a (35) w . S (01) Z V (05) Z V (41) C (01) w a (08) w a (30) w . S (01) Z V (03) Z V (35) C (01) w a (06) w a (25) w . S (00) Z V (03) Z V (30) C (01) w a (05) w a (21) w . S (00) Z V (02) Z V (25) C (01) w a (04) w a (17) w . S (00) Z V (01) Z V (20) C (01) w a (03) w a (13) w . S (00) Z V (01) Z V (11) C (00) w a (00) w a (00) w . S (00) Z V (02) Z V (10) C (00) w a (00) w a (01) w . S (06) Z V (34) Z V (61) C (01) w a (06) w a (24) w . S (07) Z V (37) Z V (59) C (03) w a (14) w a (57) w . S (06) Z V (33) Z V (45) C (04) w a (21) w a (84) w . S (02) Z V (12) Z V (14) C (01) w a (04) w a (17) w . S (03) Z V (15) Z V (04) C (01) w a (08) w a (31) w . S (03) Z V (17) Z V (18) C (02) w a (10) w a (41) w . S (02) Z V (10) Z V (08) C (01) w a (04) w a (16) w . S (03) Z V (14) Z V (10) C (01) w a (08) w a (30) w . S (03) Z V (15) Z V (24) C (02) w a (10) w a (41) w . . S (0 . Z V (0 . Z V (1 . C (0 . w a (0 . w a (0 . w . S (02) Z V (11) Z V (05) C (01) w a (04) w a (17) w . S (02) Z V (10) Z V (06) C (01) w a (04) w a (16) w TABLE XIV. Same as Table X, but for the strange quark mass data.hadronic light-by-light contribution to ( g − µ with large- N c Regge models, (2019), arXiv:1910.13432 [hep-ph].[43] G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, andP. Stoffer, Short-distance constraints on hadronic light-by-light scattering in the anomalous magnetic moment ofthe muon, (2019), arXiv:1910.11881 [hep-ph].[44] T. Blum, S. Chowdhury, M. Hayakawa, and T. Izubuchi,Hadronic light-by-light scattering contribution to themuon anomalous magnetic moment from lattice QCD,Phys. Rev. Lett. , 012001 (2015), arXiv:1407.2923[hep-lat].[45] T. Blum, N. Christ, M. Hayakawa, T. Izubuchi,L. Jin, and C. Lehner, Lattice Calculation of Hadronic Light-by-Light Contribution to the Muon AnomalousMagnetic Moment, Phys. Rev.
D93 , 014503 (2016),arXiv:1510.07100 [hep-lat].[46] T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin,C. Jung, and C. Lehner, Connected and Leading Dis-connected Hadronic Light-by-Light Contribution to theMuon Anomalous Magnetic Moment with a Physi-cal Pion Mass, Phys. Rev. Lett. , 022005 (2017),arXiv:1610.04603 [hep-lat].[47] T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin,C. Jung, and C. Lehner, Using infinite volume, contin-uum QED and lattice QCD for the hadronic light-by-lightcontribution to the muon anomalous magnetic moment, t /fm t /fm ∆/fm a s , conn . isospin , W µ × Total 53 . S (04) Z V (20) Z V (58) C (03) w a (13) w a (53) w . S (01) Z V (03) Z V (40) C (00) w a (00) w a (00) w . S (01) Z V (06) Z V (86) C (00) w a (00) w a (02) w . S (00) Z V (01) Z V (24) C (00) w a (00) w a (00) w . S (03) Z V (13) Z V (47) C (00) w a (01) w a (04) w . S (03) Z V (16) Z V (15) C (00) w a (02) w a (08) w . S (03) Z V (18) Z V (59) C (01) w a (04) w a (16) w . S (04) Z V (19) Z V (73) C (01) w a (06) w a (24) w . S (04) Z V (18) Z V (06) C (02) w a (08) w a (33) w . S (03) Z V (17) Z V (02) C (02) w a (10) w a (39) w . S (03) Z V (16) Z V (43) C (02) w a (11) w a (45) w . S (03) Z V (14) Z V (58) C (02) w a (12) w a (46) w . S (02) Z V (11) Z V (57) C (02) w a (12) w a (46) w . S (02) Z V (09) Z V (62) C (02) w a (11) w a (44) w . S (01) Z V (08) Z V (52) C (02) w a (10) w a (40) w . S (01) Z V (06) Z V (47) C (02) w a (09) w a (35) w . S (01) Z V (05) Z V (41) C (01) w a (08) w a (30) w . S (01) Z V (03) Z V (35) C (01) w a (06) w a (25) w . S (00) Z V (03) Z V (31) C (01) w a (05) w a (21) w . S (00) Z V (02) Z V (25) C (01) w a (04) w a (17) w . S (00) Z V (01) Z V (21) C (01) w a (03) w a (13) w . S (02) Z V (10) Z V (46) C (00) w a (00) w a (02) w . S (00) Z V (02) Z V (19) C (00) w a (00) w a (01) w . S (07) Z V (34) Z V (74) C (01) w a (06) w a (24) w . S (07) Z V (37) Z V (67) C (03) w a (14) w a (57) w . S (06) Z V (33) Z V (44) C (04) w a (21) w a (84) w . S (02) Z V (12) Z V (14) C (01) w a (04) w a (17) w . S (03) Z V (15) Z V (03) C (02) w a (08) w a (31) w . S (03) Z V (17) Z V (17) C (02) w a (10) w a (41) w . S (02) Z V (10) Z V (10) C (01) w a (04) w a (16) w . S (03) Z V (14) Z V (08) C (01) w a (08) w a (30) w . S (03) Z V (16) Z V (22) C (02) w a (10) w a (41) w . . S (0 . Z V (0 . Z V (1 . C (0 . w a (0 . w a (0 . w . S (02) Z V (11) Z V (03) C (01) w a (04) w a (17) w . S (02) Z V (10) Z V (07) C (01) w a (04) w a (16) w TABLE XV. Same as Table XIV, but for the p prescription.Phys. Rev. D96 , 034515 (2017), arXiv:1705.01067 [hep-lat].[48] N. Asmussen, J. Green, H. B. Meyer, and A. Nyffeler,Position-space approach to hadronic light-by-light scat-tering in the muon g − Proceedings, 34thInternational Symposium on Lattice Field Theory (Lat-tice 2016): Southampton, UK, July 24-30, 2016 , PoS
LATTICE2016 , 164 (2016), arXiv:1609.08454 [hep-lat].[49] J. Green, O. Gryniuk, G. von Hippel, H. B. Meyer, andV. Pascalutsa, Lattice QCD calculation of hadronic light-by-light scattering, Phys. Rev. Lett. , 222003 (2015),arXiv:1507.01577 [hep-lat]. [50] C. Aubin, T. Blum, C. Tu, M. Golterman, C. Jung, andS. Peris, Light quark vacuum polarization at the physicalpoint and contribution to the muon g −
2, Phys. Rev.
D101 , 014503 (2020), arXiv:1905.09307 [hep-lat].[51] D. Bernecker and H. B. Meyer, Vector Correlators in Lat-tice QCD: Methods and applications, Eur. Phys. J.
A47 ,148 (2011), arXiv:1107.4388 [hep-lat].[52] S. Aoki et al. (Flavour Lattice Averaging Group),FLAG Review 2019, Eur. Phys. J.
C80 , 113 (2020),arXiv:1902.08191 [hep-lat].[53] C. Lehner and T. Izubuchi, Towards the large volumelimit - A method for lattice QCD + QED simulations,
Proceedings, 32nd International Symposium on Lattice Field Theory (Lattice 2014): Brookhaven, NY, USA,June 23-28, 2014 , PoS
LATTICE2014 , 164 (2015),arXiv:1503.04395 [hep-lat].[54] J. A. Bailey et al. , The B → π(cid:96)ν semileptonic form fac-tor from three-flavor lattice QCD: A Model-independentdetermination of | V ub | , Phys. Rev. D79 , 054507 (2009),arXiv:0811.3640 [hep-lat].[55] M. Tanabashi et al. (Particle Data Group), Review ofParticle Physics, Phys. Rev.
D98 , 030001 (2018).[56] A. Bazavov et al. (Fermilab Lattice, MILC), Charmedand Light Pseudoscalar Meson Decay Constants fromFour-Flavor Lattice QCD with Physical Light Quarks,Phys. Rev.
D90 , 074509 (2014), arXiv:1407.3772 [hep-lat].[57] R. J. Dowdall, C. T. H. Davies, G. P. Lepage, and C. Mc-Neile, Vus from pi and K decay constants in full latticeQCD with physical u, d, s and c quarks, Phys. Rev.
D88 ,074504 (2013), arXiv:1303.1670 [hep-lat].[58] S. Collins, G. Bali, and A. Schafer, Disconnected contri-butions to hadronic structure: a new method for stochas-tic noise reduction,
Proceedings, 25th International Sym-posium on Lattice field theory (Lattice 2007): Regens-burg, Germany, July 30-August 4, 2007 , PoS
LAT-TICE2007 , 141 (2007), arXiv:0709.3217 [hep-lat]. [59] E. Shintani, R. Arthur, T. Blum, T. Izubuchi, C. Jung,and C. Lehner, Covariant approximation averaging,Phys. Rev.
D91 , 114511 (2015), arXiv:1402.0244 [hep-lat].[60] C. Lehner, The hadronic vacuum polarization contribu-tion to the muon anomalous magnetic moment (2016),RBRC Workshop on Lattice Gauge Theories.[61] G. J. Gounaris and J. J. Sakurai, Finite width correctionsto the vector meson dominance prediction for rho —¿ e+e-, Phys. Rev. Lett. , 244 (1968).[62] M. Luscher, Two particle states on a torus and their re-lation to the scattering matrix, Nucl. Phys. B354 , 531(1991).[63] L. Lellouch and M. Luscher, Weak transition matrix ele-ments from finite volume correlation functions, Commun.Math. Phys. , 31 (2001), arXiv:hep-lat/0003023 [hep-lat].[64] H. B. Meyer, Lattice QCD and the Timelike PionForm Factor, Phys. Rev. Lett. , 072002 (2011),arXiv:1105.1892 [hep-lat].[65] M. Della Morte and A. Juttner, Quark disconnected di-agrams in chiral perturbation theory, JHEP , 154,arXiv:1009.3783 [hep-lat]. LM 2020BMW 2020ETMC 2019 UpdateAubin et al. 2019Mainz 2019FNAL/HPQCD/MILC 2019SK 2019ETMC 2018RBC/UKQCD 2018BMW 2017Mainz 2017HPQCD 2016540 560 580 600 620 640 660 680 700a µ , ud, conn, isospin × LM 2020BMW 2020Mainz 2019SK 2019RBC/UKQCD 2018BMW 2017ETMC 2017Mainz 2017HPQCD 201446 48 50 52 54 56 58a µ , s, conn, isospin × BMW 2020Mainz 2019SK 2019RBC/UKQCD 2018BMW 2017ETMC 2017Mainz 2017HPQCD 201413 13.5 14 14.5 15 15.5 16a µ , c, conn, isospin × BMW 2020Mainz 2019RBC/UKQCD 2018BMW 2017RBC/UKQCD 2015-30 -25 -20 -15 -10 -5 0a µ , uds, disc, isospin × LM 2020LM 2020 FVBMW 2020ETMC 2019RBC/UKQCD 2018FNAL/HPQCD/MILC 2017 0 5 10 15 20a µ , SIB, conn × BMW 2020-10 -8 -6 -4 -2 0a µ , SIB, disc × KNT 2018/LatticeLM 2020BMW 2020Aubin et al. 2019RBC/UKQCD 2018195 200 205 210 215a µ , ud, conn, isospin, W-0.4-1.0-0.15 × BMW 2020ETMC 2019RBC/UKQCD 2018-15 -10 -5 0 5 10 15a µ , QED, conn × BMW 2020RBC/UKQCD 2018-15 -10 -5 0 5 10 15a µ , QED, disc × FIG. 15. Overview of individual contributions to a HVP µµ