[Paper Review] Finite-volume effects in the hadronic vacuum polarization

Christopher Aubin, Thomas Blum+4 more|arXiv (Cornell University)|Oct 18, 2015

Particle physics theoretical and experimental studies6 references3 citations

TL;DR

This paper investigates finite-volume effects in the hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment, showing that these effects are significant even at moderate volumes (mπL ≈ 4) and can induce 9–13% discrepancies in aμ^HVP when using different irreducible representations of the cubic group. The authors demonstrate that subtracting Πμν(0) drastically reduces these effects, and chiral perturbation theory confirms that leading-order ChPT provides a surprisingly accurate description of finite-volume corrections, despite its poor description of the HVP itself.

ABSTRACT

We investigate finite-volume effects in the hadronic vacuum polarization, with an eye toward the corresponding systematic error in the muon anomalous magnetic moment. While it is well known that leading-order chiral perturbation theory does not provide a good description of the hadronic vacuum polarization, it turns out that it gives a much better representation of finite-volume effects. Indications are that finite-volume effects cannot be ignored when the aim is a few percent level accuracy for the hadronic contribution to the muon anomalous magnetic moment, even when $m_πL \sim 4$ and $m_π\sim 200$ MeV.

Motivation & Objective

• To quantify finite-volume effects in the hadronic vacuum polarization (HVP), a key source of systematic error in lattice QCD calculations of the muon anomalous magnetic moment.
• To assess whether standard chiral perturbation theory (ChPT) can reliably describe finite-volume corrections, despite its known limitations in describing the HVP itself.
• To evaluate the impact of finite-volume effects on the extracted value of aμ^HVP, particularly in the low-Q² region where the integrand is most sensitive.
• To examine the role of irreducible representations (irreps) of the cubic group in lattice data analysis and their sensitivity to finite-volume artifacts.
• To determine whether the subtraction of Πμν(0) significantly reduces finite-volume effects, improving the accuracy of aμ^HVP estimates.

Proposed method

• The authors define a subtracted vacuum polarization tensor, Pμκ(Q)[Πκλ(Q)−Πκλ(0)]Pλν(Q), to ensure gauge invariance and reduce finite-volume effects.
• They project the vacuum polarization onto irreducible representations (irreps) of the cubic group—A₁, A₁⁴⁴, T₁, T₂, and E—to analyze how finite-volume effects break rotational symmetry.
• Using leading-order chiral perturbation theory (ChPT), they compute finite-volume corrections to Π(Q²) in a periodic L³×T box, focusing on pion contributions.
• They compare lattice data projected onto different irreps (e.g., A₁ vs. A₁⁴⁴) and find systematic differences of 9–13% in aμ^HVP, attributed to finite-volume effects.
• They employ Padé and conformally-mapped polynomial fits to lattice data in the Q² < 1 GeV² region to extract aμ^HVP values and assess systematic uncertainties.
• They validate their findings by comparing ChPT predictions with lattice data, showing that ChPT accurately describes finite-volume effects despite its poor description of the full HVP.

Experimental results

Research questions

• RQ1 How significant are finite-volume effects in the hadronic vacuum polarization for lattice QCD calculations of the muon anomalous magnetic moment?
• RQ2 Can leading-order chiral perturbation theory accurately describe finite-volume corrections to the HVP, even though it fails to describe the HVP itself?
• RQ3 To what extent do different irreducible representations of the cubic group in lattice data lead to inconsistent values of aμ^HVP, and can this be attributed to finite-volume effects?
• RQ4 Does the subtraction of Πμν(0) significantly reduce finite-volume artifacts in the HVP, and if so, why?
• RQ5 Are finite-volume effects in the HVP equally problematic for other interpolation methods, such as the moment method?

Key findings

• Finite-volume effects in the HVP can induce a 9–13% discrepancy in aμ^HVP when comparing results extracted from different irreducible representations (e.g., A₁ vs. A₁⁴⁴), even at mπL ≈ 4 and mπ ≈ 200 MeV.
• The subtraction of Πμν(0) reduces finite-volume effects dramatically, particularly in the low-Q² region, bringing the HVP closer to the infinite-volume limit.
• Leading-order chiral perturbation theory provides a surprisingly accurate description of finite-volume effects in the HVP, despite its poor description of the HVP itself.
• The ChPT computation shows that after subtraction, the A₁ and A₁⁴⁴ components of the HVP straddle the infinite-volume result, indicating that the subtraction effectively removes the dominant finite-volume artifacts.
• The difference between A₁ and A₁⁴⁴ results—about 9–13%—is consistent across different fitting functions (Padé and conformally-mapped polynomials), indicating a robust finite-volume systematic error.
• Finite-volume effects are expected to affect other interpolation methods, such as the moment method, since the t² moment of the vector current correlator in finite volume is a linear combination of Π(Q²) at non-zero momenta, which remain sensitive to volume effects.