Finitevolume effects to the HVP contribution to the muon g2
The anomalous magnetic moment of the muon, $(g2)_\mu$, 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, $(g2)_\mu^{\text{BSM}} \sim (m_\mu / \Lambda_{\text{BSM}})^2$, where $m_\mu$ is the muon mass and $\Lambda_{\text{BSM}}$ 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 $a_\mu \equiv (g2)_\mu/2$, arise from hadronic contributions, generated via the coupling of QCD fields to the muonphoton vertex. These break into three categories: the leading hadronicvacuumpolarization (HVP) contribution, $a_{\text{HVP}}$, the leading hadroniclightbylight and the subleading 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 lightbylight and the subleading 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 subpercent uncertainty is required to reach the overall $(g2)_\mu$ 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 electromagneticcurrent twopoint function and is thus wellsuited to highprecision lattice determinations. To make progress in practice, a deep theoretical understanding of all uncertainties is crucial, with the dominant sources being discretization effects, scalesetting uncertainty, statistical uncertainty (especially for large separations of the vector currents as well as those arising from quarkdisconnected diagrams) and, finally, uncertainties arising from the effects of working in a finitevolume spacetime.
Projects

Dr. Maxwell T. Hansen (Edinburgh University) and Prof. Dr. Agostino Patella (HumboldtUniversität zu Berlin & IRIS Adlershof & DESY Zeuthen) have calculated [32][33] finitevolume corrections to $a_{\text{HVP}}$ by means of an effective theory of pions with generic local interactions in the isospinsymmetric 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 is generalizing the calculation of finitevolume corrections to $a_{\text{HVP}}$ to the case of Cparity boundary conditions in the spatial directions. She will also study finitevolume corrections to the isospinbreaking corrections to $a_{\text{HVP}}$.
Documents and links
 Agostino Patella’s presentation at APLAT2020 here.
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