The strange and charm quark contributions to the anomalous magnetic moment of the muon from lattice QCD

We describe a new technique (published in Phys. Rev. D89 114501) to determine the contribution to the anomalous magnetic moment of the muon coming from the hadronic vacuum polarisation using lattice QCD. Our method uses Pad\'e approximants to reconstruct the Adler function from its derivatives at $q^2=0$. These are obtained simply and accurately from time-moments of the vector current-current correlator at zero spatial momentum. We test the method using strange quark correlators calculated on MILC Collaboration's $n_f = 2+1+1$ HISQ ensembles at multiple values of the lattice spacing, multiple volumes and multiple light sea quark masses (including physical pion mass configurations). We find the (connected) contribution to the anomalous moment from the strange quark vacuum polarisation to be $a^s_\mu=53.41(59)\times 10^{-10}$, and the contribution from charm quarks to be $a^c_\mu=14.42(39)\times 10^{-10}$ - 1% accuracy is achieved for the strange quark contribution. The extension of our method to the light quark contribution and to that from the quark-line disconnected diagram is straightforward.

The magnetic moment of the muon can be determined extremely accurately in experiment and the anomaly, a µ = (g µ − 2)/2, is known to 0.5 ppm (Brookhaven E821, [2]).Theoretical calculation of a µ in the Standard Model shows a discrepancy with the experimental result of about 25(9) × 10 −10 , which could be an indication of new virtual particles.Improvements of a factor of 4 in the experimental uncertainty are expected in 2017-18 (Fermilab E989 experiment) -improvements in the theoretical determination would make the discrepancy (if it remains) really compelling.
The theoretical uncertainty is dominated by that from the hadronic vacuum polarisation (HVP) contribution * Speaker µ q q Figure 1: Diagrammatic representation of the hadronic vacuum polarisation contribution to the muon anomalous magnetic moment.The wavy lines are photons and the shaded blob is the hadronic vacuum polarisation contribution with all the quark and gluon interactions.
(Fig. 1).At the moment the most accurate determination of HVP contribution comes from experiment: dispersion relation + cross section for e + e − (and τ) → hadrons gives ∼ 700 × 10 −10 with a 1% error [3,4].Higher order contributions from QCD processes have larger percentage uncertainty but make an order of magnitude smaller contribution, so we set out to calculate the lowest order HVP contribution and aim at 1 − 2% accuracy.

Method
We express the a µ HVP contribution in terms of a small number of derivatives of the hadronic vacuum polarisation function Π evaluated at zero momentum and use lattice current-current correlators to calculate the derivatives [1].
The HVP contribution to the anomalous magnetic moment for quark flavour f is where and α = α QED .The integrand peaks at q 2 ∼ O(m 2 µ ).Previous methods calculated Π(q 2 ) at larger q 2 and extrapolated to zero, which leads to model uncertainties, and calculating directly at small q 2 using "twisted boundary conditions" produces noisy results.We avoid this by working, in effect, from q 2 = 0 upwards.
The vacuum polarisation function is a Fourier transform of the lattice current-current correlator: We calculate the derivatives of the renormalised vacuum polarisation function Π(q 2 ) ≡ Π(q 2 ) − Π(0) from the time moments of the correlator: Here the correlator j i ( x, t) j i (0) is a local spatial vector current at zero spatial momentum.
We define Π(q 2 ) through the series expansion and use Padé approximants for Π(q 2 ) to control the high-q 2 region.High-order Padé approximants converge to the exact result [1].We use 4th, 6th, 8th and 10th time moments (i.e.j = 1, 2, 3 and 4).Only quarkline connected contributions to the lowest order HPV are considered here -disconnected contributions will need to be addressed separately.

Lattice configurations
We use lattice ensembles made by MILC collaboration [5,6] that have u/d, s and c quarks in the sea (i.e.n f = 2 + 1 + 1).The lattice spacings are a ≈ 0.15 fm (very coarse), 0.12 fm (coarse) and 0.09 fm (fine), determined using the Wilson flow parameter w 0 [7].We use Highly Improved Staggered Quark (HISQ) action, which is known for very small discretisation errors.We use ensembles that have different light sea quark masses, including ensembles with physical light quark mass.The strange valence quark is tuned using the η s meson mass m η s = 688.5 MeV [7].We also test tuning effects by deliberately mistuning the strange quark by 5% (set 6).We use large volumes, (5.6 fm) 3 on the finest lattices, and have ensembles with different volumes for testing finite volume effects.Details of the lattice ensembles are listed in Table 1.
The local current used here is not the conserved vector current for this quark action and must be normalised.Renormalisation constant Z V, ss is calculated completely nonperturbatively by demanding that the vector form factor for this current be 1 between two equal mass mesons at rest (q 2 = 0) [8].Accurate normalisation is crucial if the target is total uncertainty at the 1% level.

Meson correlators
The 2-point correlators used in this study are the φ meson correlators made using a local spatial vector operator.As a cross-check and illustration of the accuracy of the correlators we plot the mass difference of the s s vector and pseudoscalar mesons, m φ −m η s , in Fig. 2, and also plot the φ meson decay constant in Fig. 3.Both figures show that the discretisation errors are indeed small and also emphasize the importance of working at the physical light quark masses (here sea quarks, but even more important in the case of light valence quarks).

Fitting the data
We use the [2,2] Padé approximant for each configuration set and then fit these results to a function of the form where Λ QCD = 0.5 GeV and Here the fit parameter c a 2 takes care of the (small) discretisation effects and c sea and c val take care of the dependence on the sea and valence quark masses.The fit results along with lattice data are shown in Fig. 4.More details of the fit can be found in [1].

Results
Our result for the leading order HVP contribution to the muon anomalous magnetic moment from the strange quarks is a s µ = 53.41(59)× 10 −10 (connected pieces only).We also used time moments we calculated in [9,10] to get the charm quark contribution to a µ : a c µ = 14.42(39) × 10 −10 (again, connected pieces only).A preliminary estimate of total light, strange and charm quark connected piece contributions (averaging a The error budget for both strange and charm quark connected contributions is given in Table 2.The dominant error in a s µ is, by far, that coming from the uncertainty in the physical value of the Wilson flow parameter w 0 , which we use to set the lattice spacings.The next largest contribution comes from the uncertainty in the renormalisation factor Z V .For the charm quark contribution this is the dominant error, because a different method for calculating Z V was used in that calculation.This could be improved by using the same method that was used here for the strange quark contribution.More details about the error budget are in [1]. Comparing our results with other lattice QCD results shows good agreement: Fig. 5 shows our results (HPQCD) and European Twisted Mass Collaboration's results plotted against a 2 .This again highlights the fact  µ as a function of a 2 .ETMC results are from [11].
that HISQ action has very small discretisation errors compared to other lattice actions.RBC/UKQCD have also calculated a s µ and agree well with other determinations -see Table 3 for a list of the results.

Connected contribution to a light µ
The signal-to-noise ratio at large t is much worse for light quarks than for strange or charm quarks, which means we will need better statistics and improvements in the fitting method to get the error down to the 1% level.We can get better accuracy by calculating moments from best fit parameters instead of using raw lattice data.5-6% precision is already achieved using 1000 configurations × 12 time sources (very coarse ensemble) and 400 configurations × 4 time sources (coarse ensemble) on physical light quark mass ensembles.1% precision can be achieved by adding more time sources (need 4 times n src ) and up to 10 × configurations.

Conclusions
We have demonstrated in [1] that 1% precision can be achieved for the leading order HVP contribution to a s µ from the connected pieces.The error on a c µ could be pushed down to 1% by using the same method to calculate the renormalisation factor Z V that was used here for the strange quark.However, the charm quark contribution to the total leading order HVP contribution a HVP,LO µ is small compared to contributions from strange and light quarks so this is not a top priority.The main task now is to push down the error coming from the light quark contribution a l µ .We can get good enough statistics to achieve this: we can use more time sources, plus more very coarse and coarse configurations can be made relatively cheaply.Disconnected contributions need to be included in the future.[5,6].The first two sets are "very coarse" (lattice spacing a ∼ 0.15 fm), sets 3 − 8 are "coarse" (a ∼ 0.12 fm) and sets 9 − 10 are "fine" (a ∼ 0.09 fm) ensembles.am sea l and am sea s are the sea light and strange quark masses in lattice units and am ηs is the η s meson mass.Z V, ss is the vector current renormalisation constant.L and T are the spatial and temporal extents of the lattice.n cfg is the number of configurations and n src is the number of time sources used in this study.1.0% 0.6% Uncertainty in Z V : 0.4%   [3] and [12]; HPQCD results are from [1] (moments used for a c µ were calculated in [9,10]); ETMC results are from [11]; RBC/UKQCD results are from [13].

Figure 2 :
Figure 2: Mass difference m φ − m ηs as a function of a 2 .Comparing results on ensembles that have different light sea quark masses shows the advantage of working at the physical point.

Figure 3 :
Figure 3: φ meson decay constant as a function of a 2 , comparing results on ensembles that have different light sea quark masses (m lat l = m s /5 and m lat l = m phys l).

Figure 4 :
Figure 4: Continuum extrapolation fit.Here we compare results from ensembles with m lat l = m s /5 (blue points) to results from ensembles with physical light quark mass (red points).The shaded error band is our final result.
light µ on very coarse and coarse physical m l ensembles) is a HVP,LO µ = a light µ + a s µ + a c µ ∼ 662(35) × 10 −10 .This is to be compared with the dispersion relation + e + e − → hadrons cross section result of ∼ 700×10 −10 [3, 4] mentioned earlier.Note that our result does not include the disconnected diagrams.

Figure 5 :
Figure 5: Comparison with other lattice results: a sµ as a function of a 2 .ETMC results are from[11].

Table 1 :
Lattice ensembles used in this study, made by MILC collaboration

Table 2 :
[1]or budgets for connected contributions to the muon anomaly a µ from vacuum polarization of s and c quarks.See[1]for more detailed discussion on the estimation of the errors.

Table 3 :
Comparison with other results.The dispersion relation + experiment results are from