The scalar radius of the pion from Lattice QCD in the continuum limit

We extend our study of the pion scalar radius in two-flavour lattice QCD to include two additional lattice spacings as well as lighter pion masses, enabling us to perform a combined chiral and continuum extrapolation. We find discretisation artefacts to be small for the radius, and confirm the importance of the disconnected diagrams in reproducing the correct chiral behaviour. Our final result for the scalar radius of the pion at the physical point is $\left\langle r^2\right\rangle^\pi_{\rm S}=0.600\pm0.052$ fm$^2$, corresponding to a value of $\overline{\ell}_4=4.54\pm0.30$ for the low-energy constant $\overline{\ell}_4$ of NLO chiral perturbation theory.Comment: 4 pages, 4 figures, uses svjour.cl

Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment

We present a first-principles lattice QCD+QED calculation at physical pion mass of the leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment. The total contribution of up, down, strange, and charm quarks including QED and strong isospin breaking effects is found to be $a_\mu^{\rm HVP~LO}=715.4(16.3)(9.2) \times 10^{-10}$, where the first error is statistical and the second is systematic. By supplementing lattice data for very short and long distances with experimental R-ratio data using the compilation of Ref. [1], we significantly improve the precision of our calculation and find $a_\mu^{\rm HVP~LO} = 692.5(1.4)(0.5)(0.7)(2.1) \times 10^{-10}$ with lattice statistical, lattice systematic, R-ratio statistical, and R-ratio systematic errors given separately. This is the currently most precise determination of the leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment. In addition, we present the first lattice calculation of the light-quark QED correction at physical pion mass.Comment: 12 pages, 11 figure

An update of Euclidean windows of the hadronic vacuum polarization

We compute the standard Euclidean window of the hadronic vacuum polarization using multiple independent blinded analyses. We improve the continuum and infinite-volume extrapolations of the dominant quark-connected light-quark isospin-symmetric contribution and address additional sub-leading systematic effects from sea-charm quarks and residual chiral-symmetry breaking from first principles. We find $a_\mu^{\rm W} = 235.56(65)(50) \times 10^{-10}$, which is in $3.8\sigma$ tension with the recently published dispersive result of Colangelo et al., $a_\mu^{\rm W} = 229.4(1.4) \times 10^{-10}$, and in agreement with other recent lattice determinations. We also provide a result for the standard short-distance window. The results reported here are unchanged compared to our presentation at the Edinburgh workshop of the g-2 Theory Initiative in 2022.Comment: 24 pages, 15 figure

The hadronic vacuum polarization contribution to the muon g − 2 from lattice QCD

We present a calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment, $a_\mu^{\mathrm hvp}$, in lattice QCD employing dynamical up and down quarks. We focus on controlling the infrared regime of the vacuum polarization function. To this end we employ several complementary approaches, including Pad\'e fits, time moments and the time-momentum representation. We correct our results for finite-volume effects by combining the Gounaris-Sakurai parameterization of the timelike pion form factor with the L\"uscher formalism. On a subset of our ensembles we have derived an upper bound on the magnitude of quark-disconnected diagrams and found that they decrease the estimate for $a_\mu^{\mathrm hvp}$ by at most 2%. Our final result is $a_\mu^{\mathrm hvp}=(654\pm32\,{}^{+21}_{-23})\cdot 10^{-10}$, where the first error is statistical, and the second denotes the combined systematic uncertainty. Based on our findings we discuss the prospects for determining $a_\mu^{\mathrm hvp}$ with sub-percent precision.Comment: 42 pages, 7 figures, version published in JHE

Isospin breaking corrections to meson masses and the hadronic vacuum polarization: a comparative study

We calculate the strong isospin breaking and QED corrections to meson masses and the hadronic vacuum polarization in an exploratory study on a $64\times24^3$ lattice with an inverse lattice spacing of $a^{-1}=1.78$ GeV and an isospin symmetric pion mass of $m_\pi=340$ MeV. We include QED in an electro-quenched setup using two different methods, a stochastic and a perturbative approach. We find that the electromagnetic correction to the leading hadronic contribution to the anomalous magnetic moment of the muon is smaller than $1\%$ for the up quark and $0.1\%$ for the strange quark, although it should be noted that this is obtained using unphysical light quark masses. In addition to the results themselves, we compare the precision which can be reached for the same computational cost using each method. Such a comparison is also made for the meson electromagnetic mass-splittings.Comment: 49 pages, 20 figure

The anomalous magnetic moment of the muon in the Standard Model

194 pages, 103 figures, bib files for the citation references are available from: https://muon-gm2-theory.illinois.eduWe review the present status of the Standard Model calculation of the anomalous magnetic moment of the muon. This is performed in a perturbative expansion in the fine-structure constant $\alpha$ and is broken down into pure QED, electroweak, and hadronic contributions. The pure QED contribution is by far the largest and has been evaluated up to and including $\mathcal{O}(\alpha^5)$ with negligible numerical uncertainty. The electroweak contribution is suppressed by $(m_\mu/M_W)^2$ and only shows up at the level of the seventh significant digit. It has been evaluated up to two loops and is known to better than one percent. Hadronic contributions are the most difficult to calculate and are responsible for almost all of the theoretical uncertainty. The leading hadronic contribution appears at $\mathcal{O}(\alpha^2)$ and is due to hadronic vacuum polarization, whereas at $\mathcal{O}(\alpha^3)$ the hadronic light-by-light scattering contribution appears. Given the low characteristic scale of this observable, these contributions have to be calculated with nonperturbative methods, in particular, dispersion relations and the lattice approach to QCD. The largest part of this review is dedicated to a detailed account of recent efforts to improve the calculation of these two contributions with either a data-driven, dispersive approach, or a first-principle, lattice-QCD approach. The final result reads $a_\mu^\text{SM}=116\,591\,810(43)\times 10^{-11}$ and is smaller than the Brookhaven measurement by 3.7$\sigma$. The experimental uncertainty will soon be reduced by up to a factor four by the new experiment currently running at Fermilab, and also by the future J-PARC experiment. This and the prospects to further reduce the theoretical uncertainty in the near future-which are also discussed here-make this quantity one of the most promising places to look for evidence of new physics

Simulating rare kaon decays K+→π+ℓ+ℓ−K^{+}\to\pi^{+}\ell^{+}\ell^{-} using domain wall lattice QCD with physical light quark masses

We report the first calculation using physical light-quark masses of the electromagnetic form factor V(z) describing the long-distance contributions to the K+→π+ℓ+ℓ- decay amplitude. The calculation is performed on a 2+1 flavor domain wall fermion ensemble with inverse lattice spacing a-1=1.730(4)  GeV. We implement a Glashow-Iliopoulos-Maiani cancellation by extrapolating to the physical charm-quark mass from three below-charm masses. We obtain V(z=0.013(2))=-0.87(4.44), achieving a bound for the value. The large statistical error arises from stochastically estimated quark loops.We report the first calculation using physical light-quark masses of the electromagnetic form factor $V(z)$ describing the long-distance contributions to the $K^+\to\pi^+\ell^+\ell^-$ decay amplitude. The calculation is performed on a 2+1 flavor domain wall fermion ensemble with inverse lattice spacing $a^{-1}=1.730(4)$GeV. We implement a Glashow-Iliopoulos-Maiani cancellation by extrapolating to the physical charm-quark mass from three below-charm masses. We obtain $V(z=0.013(2))=-0.87(4.44)$, achieving a bound for the value. The large statistical error arises from stochastically estimated quark loops