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On the application of Effective Field Theory to finite-volume effects in $a_mu^{rm HVP}$

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 Publication date 2020
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and research's language is English




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One of the more important systematic effects affecting lattice computations of the hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon, $a_mu^{rm HVP}$, is the distortion due to a finite spatial volume. In order to reach sub-percent precision, these effects need to be reliably estimated and corrected for, and one of the methods that has been employed for doing this is finite-volume chiral perturbation theory. In this paper, we argue that finite-volume corrections to $a_mu^{rm HVP}$ can, in principle, be calculated at any given order in chiral perturbation theory. More precisely, once all low-energy constants needed to define the Effective Field Theory representation of $a_mu^{rm HVP}$ in infinite volume are known to a given order, also the finite-volume corrections can be predicted to that order in the chiral expansion.



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The leading finite-volume and thermal effects, arising in numerical lattice QCD calculations of $a^{text{HVP,LO}}_mu equiv (g-2)^{text{HVP,LO}}_mu/2$, are determined to all orders with respect to the interactions of a generic, relativistic effective field theory of pions. In contrast to earlier work based in the finite-volume Hamiltonian, the results presented here are derived by formally summing all Feynman diagrams contributing to the Euclidean electromagnetic-current two-point function, with any number of internal pion loops and interaction vertices. As was already found in our previous publication, the leading finite-volume corrections to $a^{text{HVP,LO}}_mu$ scale as $exp[- m L]$ where $m$ is the pion mass and $L$ is the length of the three periodic spatial directions. In this work we additionally control the two sub-leading exponentials, scaling as $exp[- sqrt{2} m L]$ and $exp[- sqrt{3} m L]$. As with the leading term, the coefficient of these is given by the forward Compton amplitude of the pion, meaning that all details of the effective theory drop out of the final result. Thermal effects are additionally considered, and found to be sub-percent-level for typical lattice calculations. All finite-volume corrections are presented both for $a^{text{HVP,LO}}_mu$ and for each time slice of the two-point function, with the latter expected to be particularly useful in correcting small to intermediate current separations, for which the series of exponentials exhibits good convergence.
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