GoKawiilThe muon is a short-lived elementary particle with spin 1/2 and a mass 207 times larger than that of the electron. Both particles create a magnetic field around them, characterized by a magnetic dipole moment. This moment is proportional to the spin and charge of the particle and inversely proportional to twice its mass. Dirac’s relativistic quantum mechanics predicts that the constant of proportionality, g μ , known as the Landé factor, is precisely 2. Relativistic quantum field theory introduces further small corrections induced not only by all particles and interactions of the standard model but also potentially by yet undiscovered ones. Because muons are more massive than electrons, quantum corrections associated with heavy particles are generically much larger for the former than for the latter5. This increased sensitivity to the effects of possible unknown particles is the reason for the present focus on the muon. The corrections to g μ are commonly called the anomalous magnetic moment and are quantified as a μ = (g μ − 2)/2.
When calculating a μ , the uncertainty comes almost exclusively from the strong interaction, described in the standard model by QCD. In particular, the dominant source of uncertainty comes from hadronic vacuum polarization (HVP) at leading order in the fine-structure constant (LO-HVP). More generally, HVP induces a modification in the propagation of a virtual photon in the vacuum, caused by the strong interaction.
Here we present a calculation of this LO-HVP contribution to a μ (\({a}_{\mu }^{\text{LO-HVP}}\)) with unprecedented accuracy. To that end, we apply numerical lattice quantum field theory techniques that allow QCD predictions to be made in the highly nonlinear regime that is relevant here. Mathematically, QCD is a generalized version of quantum electrodynamics (QED). However, QCD predicts physical phenomena that are very different from those described by QED. The Euclidean Lagrangian for a quark of mass m and charge q (in units of the positron charge, e), subject to strong and electromagnetic interactions, can be written as \({\mathcal{L}}=1/(4{e}^{2}){F}_{\mu
u }{F}_{\mu
u }+1/(2{g}^{2}){\rm{Tr}}{G}_{\mu
u }{G}_{\mu
u }+\bar{\psi }[{\gamma }_{\mu }({\partial }_{\mu }+{\rm{i}}q{A}_{\mu }+{\rm{i}}{G}_{\mu })+m]\psi \), in which F μν = ∂ μ A ν − ∂ ν A μ , G μν = ∂ μ G ν − ∂ ν G μ + i[G μ , G ν ] and g is the QCD coupling constant. The fermionic quark fields ψ have an extra ‘colour’ index in QCD, which runs from 1 to 3. Different ‘flavours’ of quarks are represented by independent fermionic fields, with different masses and charges. In QED, the gauge potential A μ is a real-valued field, whereas in QCD, G μ is a 3 × 3 traceless Hermitian matrix field acting in ‘colour’ space. In the present work, we include both QCD and QED as well as four nondegenerate quark flavours (up, down, strange and charm) in a fully dynamical, staggered-fermion formulation. We also consider the tiny contribution of the bottom quark. Its error is subdominant and we repeat the treatment of our earlier analysis1.
To calculate the LO-HVP contribution to a μ , we start with the zero-three-momentum, two-point function of the quark electromagnetic current in Euclidean time t (ref. 6). In this so-called time-momentum representation, it is given by
$$G(t)=-\frac{1}{3{e}^{2}}\sum _{\mu =1,2,3}\int {{\rm{d}}}^{3}x\langle \,{J}_{\mu }(\overrightarrow{x},t){J}_{\mu }(0)\rangle ,$$ (1)
in which J μ is the quark electromagnetic current with \({J}_{\mu }/e\,=\) \(\frac{2}{3}\bar{{\rm{u}}}{\gamma }_{\mu }{\rm{u}}-\frac{1}{3}\bar{{\rm{d}}}{\gamma }_{\mu }{\rm{d}}-\frac{1}{3}\bar{{\rm{s}}}{\gamma }_{\mu }{\rm{s}}+\frac{2}{3}\bar{{\rm{c}}}{\gamma }_{\mu }{\rm{c}}\). u, d, s and c are the up, down, strange and charm quark fields, respectively. The angle brackets stand for the QCD + QED expectation value to order e2. It is convenient to decompose G(t) into light (u and d), strange, charm and disconnected components, which have very different statistical and systematic uncertainties. Performing a weighted integral of the one-photon-irreducible part, G 1γI (t), of G(t) from t = 0 to infinity yields the LO-HVP contribution to a μ (ref. 6). The weight is a known kinematic function, K(tm μ ) (refs. 6,7,8,9). Thus:
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