Davies, C

de Water, RSV

DeTar, C

El-Khadra, AX

Gamiz, E

Gottlieb, S

Hatton, D

Kronfeld, A

Laiho, J

Lepage, GP

Liu, Y

Mackenzie, PB

McNeile, C

Neil, ET

Primer, T

Simone, J

Toussaint, D

Van de Water, R

Vaquero, A

Yamamoto, S

Crossref

The hadronic vacuum polarization of the muon fromfour-flavor lattice QCD

LATTICE 2019, 7 pages, 7 figuresWe present an update on the ongoing calculations by the Fermilab Lattice, HPQCD, and MILC Collaboration of the leading-order (in electromagnetism) hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon. Our project employs ensembles with four flavors of highly improved staggered fermions, physical light-quark masses, and four lattice spacings ranging from a ≈ 0.06 to 0.15 fm for most of the results thus far

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PoS(LATTICE2019)070The hadronic vacuum polarization of the muon fromfour-flavor lattice QCDC. DeTar∗ a †, C.T.H. Daviesb, A.X. El-Khadrac f , E. Gámizd , Steven Gottliebe,D. Hattonb, A.S. Kronfeld f , J. Laihog, G.P. Lepageh, Yuzhi Liud , P.B. Mackenzie f ,C. McNeilei, E.T. Neil j, T. Primerk, J.N. Simone f , D. Toussaintk, R.S. Van de Water f ,A. Vaqueroa, and Shuhei Yamamotoaa Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112 USAb SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UKc Department of Physics, University of Illinois, Urbana, IL 61801, USAd CAFPE and Departamento de Física Teórica y del Cosmos, Universidad de Granada, E-18071Granada, Spaine Department of Physics, Indiana University, Bloomington, IN 47405, USAf Fermi National Accelerator Laboratory‡, Batavia, IL 60510 USAg Department of Physics, Syracuse University, Syracuse, NY 13244, USAh Laboratory for Elementary-Particle Physics, Cornell University, Ithaca, New York 14853, USAi Centre for Mathematical Sciences, University of Plymouth, PL4 8AA, UKj Department of Physics, University of Colorado, Boulder, CO 80309, USAk Physics Department, University of Arizona, Tucson, AZ 85721, USAFermilab Lattice, HPQCD, and MILC CollaborationsWe present an update on the ongoing calculations by the Fermilab Lattice, HPQCD, and MILCCollaboration of the leading-order (in electromagnetism) hadronic vacuum polarization contri-bution to the anomalous magnetic moment of the muon. Our project employs ensembles withfour flavors of highly improved staggered fermions, physical light-quark masses, and four latticespacings ranging from a≈ 0.06 to 0.15 fm for most of the results thus far.37th International Symposium on Lattice Field Theory - Lattice201916-22 June 2019Wuhan, China∗Speaker.†email: detar@physics.utah.edu‡Operated by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the US DOE.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/PoS(LATTICE2019)070Hadronic vacuum polarization of the muon C. DeTar1. IntroductionThe measured anomalous magnetic moment of the muon aµ = (gµ −2)/2 disagrees with cur-rent standard-model predictions by more than three standard deviations [1, 2, 3]. This discrepancyhas generated considerable interest as a possible sign of new and as-yet-undiscovered fundamentalprocesses. A concerted effort is now underway to reduce both theoretical and experimental un-certainties to clarify the disagreement. Improved experiments at Fermilab (E989) and planned atJ-PARC (E34) aim to reduce the measured uncertainty by a factor of about four. To match thison the theoretical side requires an uncertainty of approximately 0.3% in the predicted hadroniccontribution, which is the largest source of theoretical uncertainty. The most precise theoreticalprediction for this contribution uses dispersion relations together with the measured cross sectionfor e+e− annihilation to hadrons through a single virtual photon. The uncertainties in that methodbecome difficult to control at the level of precision now sought. Because it is an ab initio method,lattice QCD (plus QED) can provide the needed small uncertainty, but present calculations byseveral groups are still quite far from this goal. Here, we describe progress in the effort of ourcollaboration towards reducing the lattice QCD errors to the target uncertainty. This report updatesRef. [4] and includes new, preliminary results for the quark-line disconnected contribution.2. FormalismWe use both the time-moment method and time-momentum method [5], which start from alattice calculation of the electromagnetic current-density correlator at zero momentum:G(t) =13∫dx∑fq2f Z2V 〈 j fi (x, t) jfi (0,0)〉, (2.1)where the sum is over flavors f with charges q f . In the continuum, the reduced vacuum polarizationΠ̂(ω2) =4π2ω2∫∞0dtG(t)[ω2t2−4sin2(ωt2)](2.2)determines the leading-order hadronic contribution to anomalous magnetic momentaHVP,LOµ =(απ)2 ∫ ∞0dQ2KE(Q2)Π̂(Q2), (2.3)where KE(Q2) is the Schwinger-term loop integrand as formulated by Blum [6]. The time-momentmethod expands the reduced vacuum polarization in powers of Q2Π̂(Q2) =∞∑j=1Q2 jΠ j (2.4)where, on the lattice, the Taylor coefficients are determined by the discrete time-moments of G(t),i.e., q2f G2n = (−)n(2n)!Πn−1, whereG2n ≡ a4 ∑t,x, ft2nq2f Z2V 〈 j fi (x, t) jfi (0,0)〉. (2.5)A Padé approximant is then used to extend the value of Π̂(Q2) to small Q2.Corrections for finite-volume and discretization effects are applied to the Taylor coefficientsΠn using a chiral model before inserting into the integral in Eq. (2.3).1PoS(LATTICE2019)070Hadronic vacuum polarization of the muon C. DeTar1 2 3t* or tc (fm)4005006007008001010aµllNconf. = 997Fit methodBounding method1 2 3 4 5 6n1e-121e-101e-081e-060.00010.01mµ2n |ΠnHVP(conn.)|correctedrawKNT 1802.02995a ~ 0.15 fma ~ 0.12 fma ~ 0.09 fma ~ 0.06 fmFigure 1: Left: Values of allµ obtained with different methods for handling the high t contribution, namely,our “fit method” and the BMW “bounding method” [10]. They agree well for t > 2.3 fm. Right: Raw(open symbols) and corrected (filled symbols) vacuum-polarization Taylor coefficients. They agree wellwith results from the R-ratio method, shown with black lines [3].3. Results3.1 Light quark-line-connected contributionOur calculations are based on gauge-field ensembles at four lattice spacings, namely, approxi-mately 0.15, 0.12, 0.09, and 0.06 fm. These ensembles were generated in the presence of (2+1+1)-flavors of highly-improved staggered sea quarks (HISQ) [8] with masses close to their physicalvalues [9]. Measurements are done with the same physical-mass quarks.The vector-current-density correlator in Eq. 2.1 separates into a larger quark-line-connectedand a smaller, but more difficult to calculate, quark-line-disconnected contributions. We first dis-cuss the large connected light-quark contribution (with degenerate up and down quarks). Thelargest source of statistical uncertainty comes from the rapidly growing noise-to-signal ratio in thecurrent-density correlator at large Euclidean time t. There are a variety of strategies for treating it.We fit the vector-current-density correlator to a set of exponentials and, for t larger than a cutoff t∗,replace the lattice values by an extrapolation of the fit. We show in Fig. 1 (left) that for t∗ > 2.3 fmthe result for the contribution to the anomalous magnetic moment agrees with the BMW boundingmethod [10]. One might be concerned that the extrapolation is risky because our fit model does notinclude the full spectrum of states. We are able to show with a chiral model that the fit method withonly two intermediate states is, in fact, reliable at the level of accuracy we need [4].The lattice results require corrections for finite-size and discretization effects. A large part ofthese corrections are estimated through the use of a chiral model. The corrected Taylor coefficientsΠn are shown in Fig. 1. It is noteworthy that our lattice result agrees with the phenomenologicalresult based on R-ratios [3] at the 1− 2% level. The corrected values of allµ are then extrapolatedto zero lattice spacing, as shown in Fig. 2, with the result a``µ = 637.8(8.8)× 10−10, i.e., with anerror of 1.4%.1 The error breakdown is shown in Table 1. The dominant contributions to the total1This result has been updated since the conference.2PoS(LATTICE2019)070Hadronic vacuum polarization of the muon C. DeTarTable 1: Error budget for the quark-line connected light-quark contributionLattice spacing uncertainty (w0) 0.8%Monte Carlo statistics 0.7Continuum extrapolation 0.7Finite volume and discretization corrections 0.6Current renormalization 0.1Chiral interpolation 0.1Strange sea quark mass adjustment 0.1Total 1.4%0.000 0.005 0.010 0.015 0.020 0.025a2 (fm2)550575600625650675all µ×1010with FV, discretization, mπ correctionsraw values560 580 600 620 640 660 680 700Mainz/CLS (Nf = 2)1705.01775BMW, 1711.04980RBC/UKQCD1801.07224ETM, 1808.00887PACS, 1902.00885Mainz/CLS (Nf = 3)1904.03120Aubin et al., 1905.09307FNAL/HPQCD/MILC20191010allµ(conn.)Figure 2: Left: Continuum extrapolation of our results for the light-quark contribution allµ . Shown arevalues before (open blue symbols) and after (filled red symbols) applying corrections for finite-volume anddiscretization effects. Right: Comparison of our result for the total light-quark contribution with those ofother groups [11, 12, 13, 14, 15, 7, 16].error on a`µ` are the lattice spacing uncertainty, statistics, the continuum extrapolation, and finitevolume and discretization corrections. Our result is compared with recent results of other groups[11, 12, 13, 14, 15, 7, 16] in Fig. 2.3.2 Total leading-order light-quark hadron vacuum polarizationTo obtain the total leading-order up/down quark contribution to aµ we need to include strongisospin-breaking and electromagnetic effects, and we need the quark-line-disconnected contribu-tion. A large part of the strong isospin-breaking comes from the splitting of the pion multiplet in thetwo-pion intermediate state. This contribution can be calculated from the same chiral model used inthe finite-volume and lattice-discretization correction. The same procedure can also be used to es-timate the quark-line-disconnected contribution. (Our more recent, explicit calculation, discussedbelow, was not used in estimating this contribution.) The residual (non-pion-pion) contributions arethen estimated conservatively, as shown in Table 2. In the case of strong isospin breaking, explicitlattice calculations help provide the estimates shown [15, 17, 18]. The largest uncertainty here3PoS(LATTICE2019)070Hadronic vacuum polarization of the muon C. DeTarTable 2: Adjustments to obtain the total up/down quark contribution to audµ . All values are in units of 10−10.M(π0)−M(π+) −4.3π−π disconnected −7.9Total −12(3)disconnected −5(5)Strong isospin breaking 10(10)Electromagnetism 0(5)Total adjustment −7(13)still comes from strong isospin breaking. Altogether, then, the total up-down quark contribution is,then, audµ = 630.8(8.8)(13)× 10−10, where the first error comes from the light-quark value aboveand the second from the corrections in Table 2.610 630 650 670 690 710 730 750no new physicsKeshavarzi et al.1802.02995e+e−Davier et al., 1706.09436e+e−Jegerlehner, 1705.00263e+e− + τBenayoun et al.1507.02943e+e− + τHPQCD/RV1601.03071Mainz/CLS, 1705.01775Nf = 2BMW, 1711.04980RBC/UKQCD1801.07224ETMC, 1808.00887PACS, 1902.00885Mainz/CLS (Nf = 3)1904.03120FNAL/HPQCD/MILC2019Lattice QCDPheno.1010aHVPµ (LO)Figure 3: Comparison of results for the total leading-order hadronic vacuum polarization contribution toaLO,HVPµ [1, 2, 3, 12, 13, 14, 15, 7, 16, 19, 20].Finally, including further contributions from the strange, charm, and bottom quarks [5, 21, 22]gives aµHVP−LO = 699(15)u,d(1)s,c,b×10−10, where the errors are from the light-quark and heavier-quark contributions, respectively. The largest contribution to the uncertainty comes from isospin-breaking and electromagnetic corrections. This value is compared with those of others in Fig. 3.4. Quark-line-disconnected HVPWe conclude by presenting some preliminary results from our explicit lattice calculation of thequark-line-disconnected contribution, which will eventually replace the model estimates described4PoS(LATTICE2019)070Hadronic vacuum polarization of the muon C. DeTar0.000 0.005 0.010 0.015 0.020a2[fm2]−5051015−adisc.µ×1010Fermilab/HPQCD/MILCFermilab/HPQCD/MILC correctedBMW [1711.04980]RBC/UKQCD [1801.07224]Figure 4: Our preliminary results for the quark-line disconnected contribution to aLO,HVPµ , compared withthose of the BMW [7] and RBC/UKQCD [15] collaborations. Our results are shown before and after apply-ing a correction for finite-volume and discretization effects.above. So far we have results for simulations at only two lattice spacings 0.15 and 0.12 fm, up-dated from work described in Ref. [23]. The resulting contributions are summarized in Fig. 4 andcompared with results from the BMW [7] and RBC/UKQCD Collaborations [15]. The results arecorrected for finite-volume and discretization effects, which removes most of the lattice-spacingdependence. We see that the finite-volume plus discretization-error correction is significant. Al-though the extrapolated value is statistically consistent with the model estimates described above,a calculation at a finer lattice spacing is evidently needed in order to meet our goals for a reductionin the overall error.5. OutlookWe continue in our efforts to reduce the lattice uncertainty. Besides increasing statistics andworking at finer lattice spacing, we are also working to reduce the uncertainties in the lattice scale,we are planning calculations of electromagnetic corrections, and we are investigating lattice calcu-lations of contributions from two-hadron intermediate states.AcknowledgmentsThis work is supported by grants from the US Department of Energy, the US National ScienceFoundation, the United Kingdom STFC, MINECO (Spain), the Junta de Andalucía (Spain), theGerman Excellence Initiative and the European Union Seventh Framework Program, the EuropeanUnion’s Marie Curie COFUND program, and the Blue Waters PAID program. Computations werecarried out on USQCD LQCD clusters and at the US DOE NERSC and ALCF centers, the U.K.STFC DiRAC HPC Facility, the US NSF TACC XSEDE center, and computers at the Universityof Colorado and Indiana University.References[1] F. 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The hadronic vacuum polarization of the muon fromfour-flavor lattice QCD

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