Leading hadronic contribution to the muon magnetic moment from lattice QCD.

The standard model of particle physics describes the vast majority of experiments and observations involving elementary particles. Any deviation from its predictions would be a sign of new, fundamental physics. One long-standing discrepancy concerns the anomalous magnetic moment of the muon, a measure of the magnetic field surrounding that particle. Standard-model predictions1 exhibit disagreement with measurements2 that is tightly scattered around 3.7 standard deviations. Today, theoretical and measurement errors are comparable; however, ongoing and planned experiments aim to reduce the measurement error by a factor of four. Theoretically, the dominant source of error is the leading-order hadronic vacuum polarization (LO-HVP) contribution. For the upcoming measurements, it is essential to evaluate the prediction for this contribution with independent methods and to reduce its uncertainties. The most precise, model-independent determinations so far rely on dispersive techniques, combined with measurements of the cross-section of electron-positron annihilation into hadrons3-6. To eliminate our reliance on these experiments, here we use ab initio quantum chromodynamics (QCD) and quantum electrodynamics simulations to compute the LO-HVP contribution. We reach sufficient precision to discriminate between the measurement of the anomalous magnetic moment of the muon and the predictions of dispersive methods. Our result favours the experimentally measured value over those obtained using the dispersion relation. Moreover, the methods used and developed in this work will enable further increased precision as more powerful computers become available.

Hadronic Vacuum Polarization Contribution to the Anomalous Magnetic Moments of Leptons from First Principles.

We compute the leading, strong-interaction contribution to the anomalous magnetic moment of the electron, muon, and tau using lattice quantum chromodynamics (QCD) simulations. Calculations include the effects of u, d, s, and c quarks and are performed directly at the physical values of the quark masses and in volumes of linear extent larger than 6 fm. All connected and disconnected Wick contractions are calculated. Continuum limits are carried out using six lattice spacings. We obtain a_{e}^{LO-HVP}=189.3(2.6)(5.6)×10^{-14}, a_{µ}^{LO-HVP}=711.1(7.5)(17.4)×10^{-10} and a_{τ}^{LO-HVP}=341.0(0.8)(3.2)×10^{-8}, where the first error is statistical and the second is systematic.

Review of lattice results concerning low-energy particle physics: Flavour Lattice Averaging Group (FLAG).

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle-physics community. More specifically, we report on the determination of the light-quark masses, the form factor [Formula: see text], arising in the semileptonic [Formula: see text] transition at zero momentum transfer, as well as the decay constant ratio [Formula: see text] and its consequences for the CKM matrix elements [Formula: see text] and [Formula: see text]. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of [Formula: see text] and [Formula: see text] Chiral Perturbation Theory. We review the determination of the [Formula: see text] parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. The latter quantities are an addition compared to the previous review. For the heavy-quark sector, we provide results for [Formula: see text] and [Formula: see text] (also new compared to the previous review), as well as those for D- and B-meson-decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. Finally, we review the status of lattice determinations of the strong coupling constant [Formula: see text].

Up and Down Quark Masses and Corrections to Dashen's Theorem from Lattice QCD and Quenched QED.

In a previous Letter [Borsanyi et al., Phys. Rev. Lett. 111, 252001 (2013)] we determined the isospin mass splittings of the baryon octet from a lattice calculation based on N_{f}=2+1 QCD simulations to which QED effects have been added in a partially quenched setup. Using the same data we determine here the corrections to Dashen's theorem and the individual up and down quark masses. Our ensembles include 5 lattice spacings down to 0.054 fm, lattice sizes up to 6 fm, and average up-down quark masses all the way down to their physical value. For the parameter which quantifies violations to Dashen's theorem, we obtain ϵ=0.73(2)(5)(17), where the first error is statistical, the second is systematic, and the third is an estimate of the QED quenching error. For the light quark masses we obtain, m_{u}=2.27(6)(5)(4) and m_{d}=4.67(6)(5)(4) MeV in the modified minimal subtraction scheme at 2 GeV and the isospin breaking ratios m_{u}/m_{d}=0.485(11)(8)(14), R=38.2(1.1)(0.8)(1.4), and Q=23.4(0.4)(0.3)(0.4). Our results exclude the m_{u}=0 solution to the strong CP problem by more than 24 standard deviations.

Lattice Computation of the Nucleon Scalar Quark Contents at the Physical Point.

We present a QCD calculation of the u, d, and s scalar quark contents of nucleons based on 47 lattice ensembles with N_{f}=2+1 dynamical sea quarks, 5 lattice spacings down to 0.054 fm, lattice sizes up to 6 fm, and pion masses down to 120 MeV. Using the Feynman-Hellmann theorem, we obtain f_{ud}^{N}=0.0405(40)(35) and f_{s}^{N}=0.113(45)(40), which translates into σ_{πN}=38(3)(3) MeV, σ_{sN}=105(41)(37) MeV, and y_{N}=0.20(8)(8) for the sigma terms and the related ratio, where the first errors are statistical and the second errors are systematic. Using isospin relations, we also compute the individual up and down quark contents of the proton and neutron (results in the main text).

Ab initio calculation of the neutron-proton mass difference.

The existence and stability of atoms rely on the fact that neutrons are more massive than protons. The measured mass difference is only 0.14% of the average of the two masses. A slightly smaller or larger value would have led to a dramatically different universe. Here, we show that this difference results from the competition between electromagnetic and mass isospin breaking effects. We performed lattice quantum-chromodynamics and quantum-electrodynamics computations with four nondegenerate Wilson fermion flavors and computed the neutron-proton mass-splitting with an accuracy of 300 kilo-electron volts, which is greater than 0 by 5 standard deviations. We also determine the splittings in the Σ, Ξ, D, and Ξcc isospin multiplets, exceeding in some cases the precision of experimental measurements.

Review of lattice results concerning low-energy particle physics.

We review lattice results related to pion, kaon, [Formula: see text]- and [Formula: see text]-meson physics with the aim of making them easily accessible to the particle-physics community. More specifically, we report on the determination of the light-quark masses, the form factor [Formula: see text], arising in semileptonic [Formula: see text] transition at zero momentum transfer, as well as the decay-constant ratio [Formula: see text] of decay constants and its consequences for the CKM matrix elements [Formula: see text] and [Formula: see text]. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of [Formula: see text] and [Formula: see text] Chiral Perturbation Theory and review the determination of the [Formula: see text] parameter of neutral kaon mixing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, we focus here on [Formula: see text]- and [Formula: see text]-meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant [Formula: see text].

Isospin splittings in the light-baryon octet from lattice QCD and QED.

While electromagnetic and up-down quark mass difference effects on octet baryon masses are very small, they have important consequences. The stability of the hydrogen atom against beta decay is a prominent example. Here, we include these effects by adding them to valence quarks in a lattice QCD calculation based on Nf=2+1 simulations with five lattice spacings down to 0.054 fm, lattice sizes up to 6 fm, and average up-down quark masses all the way down to their physical value. This allows us to gain control over all systematic errors, except for the one associated with neglecting electromagnetism in the sea. We compute the octet baryon isomultiplet mass splittings, as well as the individual contributions from electromagnetism and the up-down quark mass difference. Our results for the total splittings are in good agreement with experiment.

Ab initio determination of light hadron masses.

More than 99% of the mass of the visible universe is made up of protons and neutrons. Both particles are much heavier than their quark and gluon constituents, and the Standard Model of particle physics should explain this difference. We present a full ab initio calculation of the masses of protons, neutrons, and other light hadrons, using lattice quantum chromodynamics. Pion masses down to 190 mega-electron volts are used to extrapolate to the physical point, with lattice sizes of approximately four times the inverse pion mass. Three lattice spacings are used for a continuum extrapolation. Our results completely agree with experimental observations and represent a quantitative confirmation of this aspect of the Standard Model with fully controlled uncertainties.