Hadronic vacuum polarization in QCD and its evaluation in Euclidean spacetime

Abstract

We discuss a new technique to evaluate integrals of QCD Green's functions in the Euclidean based on their Mellin-Barnes representation. We present as a first application the evaluation of the lowest order Hadronic Vacuum Polarization (HVP) contribution to the anomalous magnetic moment of the muon 1 2 (gµ − 2)HVP ≡ a HVP µ . It is shown that with a precise determination of the slope and curvature of the HVP function at the origin from lattice QCD (LQCD), one can already obtain a result for a HVP µ which may serve as a test of the determinations based on experimental measurements of the e + e − annihilation cross-section into hadrons.

The hadronic vacuum polarization contribution to the muon g − 2 from lattice QCD

“…Recently it was proposed to extract the photon vacuum polarization in the spacelike region from Bhabha and µe scattering data [8,9], which would allow for a direct comparison with lattice results. Other approaches that combine phenomenological constraints with information from lattice QCD employ expansions of a hvp µ in terms of Mellin-Barnes moments [10][11][12] or finite energy sum rules [13,14]. The hadronic light-by-light scattering contribution has so far only been determined via model estimates (as reviewed in [5,[15][16][17]), although efforts have been undertaken to move towards a data-driven [18][19][20][21][22][23][24][25][26][27] approach as well.…”

Mellin-Barnes approach to hadronic vacuum polarization and gμ−2

“…A simple example of this procedure was discussed in Ref. [2] in the case of vacuum polarization in QED at the one loop level where, in that case, the corresponding Mellin transform is exactly reproduced from its knowledge at just three s values: e.g., s ¼ 1, 0, and −1.…”

“…[2]). However, when inserted in a dispersion relation integral, they reproduce the predicted smooth behaviour of the successive self-energy functions Π N ðQ 2 Þ and Adler A N ðQ 2 Þ functions.…”

“…The answer that we propose follows the way initiated in Ref. [2]. It is based on MellinBarnes techniques which we shall describe below and which we shall illustrate with several examples.…”

“…[1][2][3] to evaluate the hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon a HVP μ . Our motivation is threefold: (1) The persistent discrepancy at the ∼4σ level between the experimental determination of the anomalous magnetic moment of the muon [4] a μ ðE821 − BNLÞ ¼ 116592089ð54Þ stat ð33Þ syst × 10 −11 ½0.54 ppm; ð1:1Þ and the standard model prediction [5] a μ ðSMÞ ¼ 116 591 805ð42Þ × 10 −11 : ð1:2Þ…”