The anomalous magnetic moment of the muon: status of lattice QCD calculations

Abstract

In recent years, the anomalous magnetic moment of the muon has triggered a lot of activity in the lattice QCD community because a persistent tension of about is observed between the phenomenological estimate and the Brookhaven measurement. The current best phenomenological estimate has an uncertainty comparable to the experimental one and the error is completely dominated by hadronic effects: the leading order hadronic vacuum polarization (HVP) contribution and the hadronic light-by-light (HLbL) scattering contribution. Both are accessible via lattice simulations and a reduction of the error by a factor 4 is required in view of the forthcoming experiments at Fermilab and J-PARC whose results, expected in the next few years, should reduce the experimental precision down to the level of 0.14 ppm. In this article, I review the status of lattice calculations of those quantities, starting with the HVP. This contribution has now reached sub-percent precision and requires a careful understanding of all sources of systematic errors. The HLbL contribution, that is much smaller, still contributes significantly to the error. This contribution is more challenging to compute, but rapid progress has been made on the lattice in the last few years.

Leading isospin breaking effects in the HVP contribution to $a_μ$ and to the running of $α$

“…Neglecting isospin breaking effects in the scale setting we match the proxies for the average light and strange quark masses in both theories and set the light quark mass difference and the electromagnetic coupling to their physical values. Applying the leading-order perturbative expansion 𝑎𝑚 𝐻 = (𝑎𝑚 𝐻 ) (0) + 𝑙 Δ𝜀 𝑙 (𝑎𝑚 𝐻 ) (1) 𝑙 + 𝑂 (Δ𝜀 2 ) for 𝐻 = 𝜋 0 , 𝜋 + , 𝐾 0 , 𝐾 + this scheme translates into a system of linear equations that determines the expansion coefficients Δ𝜀 = (𝑎Δ𝑚 u , 𝑎Δ𝑚 d , 𝑎Δ𝑚 s , Δ𝛽, 𝑒 2 ):…”

“…, which fulfils the lattice vector Ward identity in QCD+QED for diagonal Λ 𝑖 and which depends on the combined QCD+QED gauge links 𝑊 𝑥 𝜇 = 𝑈 𝑥 𝜇 𝑒 i𝑎𝑒𝑄 𝐴 𝑥 𝜇 . Treating isospin breaking perturbatively, correlation functions are expanded according to 𝐶 = 𝐶 (0) + 𝑙 Δ𝜀 𝑙 𝐶 (1) 𝑙 +𝑂 (Δ𝜀 2 ). As a consequence, operators also have to be expanded in…”

“…𝑥 0 1 0 in the limit of large time separations, where lattice artefacts become small. We further perform a perturbative expansion (1) 𝑙 + 𝑂 (Δ𝜀 2 ). For the results of the extracted renormalisation factors we refer to [5].…”

“…For both quantities the hadronic vacuum polarisation (HVP) contribution is a main source of uncertainty in the SM prediction. In particular, at the desired level of accuracy isospin breaking effects in the HVP contribution have to be taken into account [1,2]. In this work, we continue the investigation of isospin breaking effects [3][4][5] making use of Coordinated Lattice Simulations (CLS) 𝑁 f = 2 + 1 QCD ensembles [6][7][8][9] with open and (anti-)periodic temporal boundary conditions [10].…”

Probing the dark axion portal with muon anomalous magnetic moment

“…The first two are the QED and electroweak (EW) predictions, respectively, while a HVP μ is the hadronic vacuum polarization (HVP) and a HLbL μ the hadronic light-by-light (HLbL) contribution. Although the biggest source of uncertainty comes from the hadronic part [9][10][11][12], the most recent calculations [13][14][15] have included the updated measurement of the hadronic contributions [16][17][18]. The latest theoretical calculation [21] gives a SM μ = 116591810 (43) × 10 −11 , (…”