Abstract
The vacuum polarization tensor and the corresponding vacuum polarization function are the basis for calculations of numerous observables in lattice QCD. Examples are the hadronic contributions to lepton anomalous magnetic moments, the running of the electroweak and strong couplings and quark masses. Quantities which are derived from the vacuum polarization tensor often involve a summation of current correlators over all distances in position space leading thus to the appearance of short-distance terms. The mechanism of O(a) improvement in the presence of such short-distance terms is not directly covered by the usual arguments of on-shell improvement of the action and the operators for a given quantity. If such short-distance contributions appear, the property of O(a) improvement needs to be reconsidered. We discuss the effects of these short-distance terms on the vacuum polarization function for twisted mass lattice QCD and find that even in the presence of such terms automatic O(a) improvement is retained if the theory is tuned to maximal twist.
Highlights
We discuss the effects of these short-distance terms on the vacuum polarization function for twisted mass lattice QCD and find that even in the presence of such terms automatic O(a) improvement is retained if the theory is tuned to maximal twist
Employing Symanzik’s effective theory [15, 16], we show in the following that with our definition of the hadronic vacuum polarization function and at maximal twist these short-distance contributions do not spoil the automatic O (a)-improvement of the vacuum polarization function in the twisted mass formulation of lattice QCD
A crucial element in obtaining accurate results from lattice QCD calculations is the suppression of lattice spacing artefacts and a controlled approach towards the continuum limit
Summary
Contact to the physical quark content is made by identifying (u, d) ↔ (χ+l , χ−l ), s ↔ χ−s and c ↔ χ+c We choose these fields to initially construct the electromagnetic current operator as a Noether current resulting from the infinitesimal vector variation δV χ = iα(x) Qem χ(x) δV χ = −iα(x) χ(x) Qem , with Qem = diag (+2/3, −1/3, +2/3, 0, 0, −1/3) related to the electromagnetic charge matrix taking into account our choice of physical fields. In contrast to the conserved point-split vector current in eq (2.3), the local vector current is not exactly conserved at non-zero lattice spacing and the polarization tensor ΠLμν is not transverse. The latter will have to be potentially additively and multiplicatively renormalized. In the heavy valence sector analogous arguments are used and the latter will be covered in a more general framework in [24]
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