Hadronic contribution to the muon $g-2$: A Dyson–Schwinger perspective

Abstract

We summarize our results for hadronic contributions to the anomalous magnetic moment of the muon $\left(a_{\mu }\right)$, the one from hadronic vacuum-polarization (HVP) and the light-by-light scattering contribution (LBL), obtained from the Dyson–Schwinger equations (DSEs) of QCD. In the case of HVP we find good agreement with model independent determinations from dispersion relations for $a_{\mu }^{\mathrm{HV\; P}}$ as well as for the Adler function with deviations well below the ten percent level. From this we conclude that the DSE approach should be capable of describing $a_{\mu }^{\mathrm{LBL}}$ with similar accuracy. We also present results for LBL using a resonance expansion of the quark–anti-quark $T$-matrix. Our preliminary value is $a_{\mu }^{\mathrm{LBL}}=(217\pm 91)\times 10^{-11}$.

Introduction

In the search for new physics beyond the standard model the anomalous magnetic moment of the muon ($a_{\mu }$) is one of the most interesting observables. Compared to the corresponding electron anomaly ($a_e$) it is more sensitive to contributions from high lying scales. These include the weak interactions, QCD and potential new physics [1]. Especially the contributions from soft QCD desire highest attention because, due to their non-perturbative nature and the resulting technical complications, they dominate the theoretical standard model (SM) uncertainty.

The efforts of the E821 experiment at Brookhaven National Lab [2], [3] as well as theoretical efforts of more than a decade [4] culminated in a determination of $a_{\mu }$ down to a level where significant deviations have been found

$$\text{Experiment: }116592089\left(63\right)\times 10^{-11}\text{,}$$$$\text{Theory: }116591828\left(49\right)\times 10^{-11}\text{,}$$

where the theoretical number is taken from Ref. [5]. Comparing theory and experiment the deviation amounts to $a_{\mu }^{\exp }-a_{\mu }^{\mathrm{theo}}=261\left(80\right)$ which corresponds to a $3.2\sigma$ effect. In order to confirm this result the uncertainties have to be reduced further.

There are two hadronic contributions that dominate the SM uncertainty. There is the hadronic vacuum polarization (HVP) contribution which gives rise to the leading hadronic contribution as well as the leading SM uncertainty contribution [5]

$$a_{\mu }^{\mathrm{HV\; P,DR}}=6949.1\left(42.7\right)\times 10^{-11}\text{.}$$

The relevant diagram, involving the hadronic one-particle irreducible (1PI) photon self-energy ${\Pi }_{\mu \nu }$ is shown in Fig. 1(a). The next to leading uncertainty contribution comes from the light-by-light (LBL) scattering contribution that is shown in Fig. 1(b). Estimates from the viewpoint of effective field theory (EFT) from different approaches were recently combined into a single number [6]

$$a_{\mu }^{\mathrm{LBL}}=105\left(26\right)\times 10^{-11}\text{.}$$

The uncertainty given here is rather small compared to most estimates. In fact our results indicate that this error may be far too optimistic.

Our strategy to determine these quantities is the following. We work with the Dyson–Schwinger and Bethe–Salpeter equations (DSEs/BSEs) of QCD [7], [8]. With these we calculate the HVP contribution to $a_{\mu }$ where we use a parameter set, among others, that is completely fixed by meson phenomenology. The HVP contribution can be compared to essentially model independent result from dispersion relations [9] such that the calculation serves as a non-trivial cross check of our methods. Afterwards we approach the LBL contribution using exactly the same truncation such that we have reasons to believe that we can ultimately reach a similar precision as in the case of HVP.

This proceedings contribution is organized as follows. First of all we summarize the employed truncation in Section 2. The HVP contribution will be discussed in Section 3 and LBL in Section 4. Afterwards we discuss our results for both of these contributions in Section 5 and conclude.