Lattice Field Theory

Finite-volume effects to the HVP contribution to the muon g-2

Last Modified: 1 March 2024

The anomalous magnetic moment of the muon, $(g-2{)}_{\mu }$, has become a central focus in the broader particle physics community, due to significant tension between the best experimental [1][2] and theoretical determinations [3][4][5]. The $\sim 3$ to $4$ sigma discrepancy represents a real opportunity to discover new physics beyond the Standard Model (BSM), especially given the incredibly clean determinations on both the experimental and theoretical sides, together with the quadratic sensitivity to BSM effects, $(g-2{)}_{\mu }^{\text{BSM}}\sim (m_{\mu }/{\mathrm{\Lambda }}_{\text{BSM}}{)}^2$, where $m_{\mu }$ is the muon mass and ${\mathrm{\Lambda }}_{\text{BSM}}$ the scale of the putative new physics.

Experiments are underway at both Fermilab [6][7][8][9] and JPARC [10][11], with an update expected some time this year, and at least a factor of 2 uncertainty reduction targeted in the coming years. The theory community is committed to maximizing the impact of this update, by providing a Standard Model determination of comparable overall uncertainty. The effort on this side is also very advanced and is summarized in detail in a forthcoming theory white paper [12].

On the theoretical side, the leading uncertainties in $a_{\mu }\equiv (g-2{)}_{\mu }/2$, arise from hadronic contributions, generated via the coupling of QCD fields to the muon-photon vertex. These break into three categories: the leading hadronic-vacuum-polarization (HVP) contribution, $a_{\text{HVP}}$, the leading hadronic-light-by-light and the sub-leading corrections to the HVP. Outstanding progress has been made in the determination of the latter two contributions such that these are well in line to reach the overall target uncertainty [13][14][15]. Since both the hadronic light-by-light and the sub-leading HVP are suppressed relative to the leading HVP, the targeted relative uncertainties here are $\sim 10$ and, despite the complicated nature of these quantities, well within reach; see again ref. [12]. In this work we restrict attention to the $a_{\text{HVP}}$ contribution, for which sub-percent uncertainty is required to reach the overall $(g-2{)}_{\mu }$ precision target.

Numerical lattice QCD (LQCD) provides an ideal tool in the determination of $a_{\text{HVP}}$, and many leading collaborations have already presented very advanced calculations [16][17][18][19][20][21][22][23][24][25][26][14][27][28][29][30][31]. The observable can be directly extracted from a Euclidean electromagnetic-current two-point function and is thus well-suited to high-precision lattice determinations. To make progress in practice, a deep theoretical understanding of all uncertainties is crucial, with the dominant sources being discretization effects, scale-setting uncertainty, statistical uncertainty (especially for large separations of the vector currents as well as those arising from quark-disconnected diagrams) and, finally, uncertainties arising from the effects of working in a finite-volume spacetime.

Projects

• Dr. Maxwell T. Hansen (Edinburgh University) and Prof. Dr. Agostino Patella (Humboldt-Universität zu Berlin & IRIS Adlershof & DESY Zeuthen) have calculated [32][33] finite-volume corrections to $a_{\text{HVP}}$ by means of an effective theory of pions with generic local interactions in the isospin-symmetric limit. The interaction Lagrangian can be arbitrarily complicated, and in order to make sense of the Feynman integrals, an ultraviolet cutoff that preserves all relevant symmetries is assumed. However, the resulting formulae and their proof are insensitive to these details.

• Sofie Martins (CP3-Origins) has generalized the calculation of finite-volume corrections to $a_{\text{HVP}}$ to the case of C-parity boundary conditions in the spatial directions. She is also studying finite-volume corrections to the isospin-breaking corrections to $a_{\text{HVP}}$.

Documents and links

• Agostino Patella’s presentation at APLAT2020 here.
• Sofie Martins’ presentation at Lattice2022 here.

Bibliography

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