Theoretical aspects of quantum electrodynamics in a finite volume with periodic boundary conditions

Abstract

First-principles studies of strongly-interacting hadronic systems using lattice quantum chromodynamics (QCD) have been complemented in recent years with the inclusion of quantum electrodynamics (QED). The aim is to confront experimental results with more precise theoretical determinations, e.g. for the anomalous magnetic moment of the muon and the CP-violating parameters in the decay of mesons. Quantifying the effects arising from enclosing QED in a finite volume remains a primary target of investigations. To this end, finite-volume corrections to hadron masses in the presence of QED have been carefully studied in recent years. This paper extends such studies to the self-energy of moving charged hadrons, both on and away from their mass shell. In particular, we present analytical results for leading finite-volume corrections to the self-energy of spin-0 and spin-1 2 particles in the presence of QED on a periodic hypercubic lattice, once the spatial zero mode of the photon is removed, a framework that is called QED L . By altering modes beyond the zero mode, an improvement scheme is introduced to eliminate the leading finite-volume corrections to masses, with potential applications to other hadronic quantities. Our analytical results are verified by a dedicated numerical study of a lattice scalar field theory coupled to QED L . Further, this paper offers new perspectives on the subtleties involved in applying low-energy effective field theories in the presence of QED L , a theory that is rendered non-local with the exclusion of the spatial zero mode of the photon, clarifying recent discussions on this matter.

Renormalizing vector currents in lattice QCD using momentum-subtraction schemes

“…Because QED is a long-range interaction it is important to test finite-volume effects, although we do not expect them to be large here since we studying renormalisation of electrically neutral currents. The finite-volume effects in the self-energy function of fermions has been studied in [44] with the result that for off-shell quarks the finite-volume effects start at order 1/L 2 s where L s is lattice spatial extent. This implies that even the finitevolume effects for quantities such as Z q should be small.…”

QED-corrected Lellouch-Luescher formula for $K \rightarrow \pi\pi$ decay

“…All these diagrams are evaluated in a finite cubic volume with PBCs and with the zero mode of the photon removed, using the techniques developed in Refs. [13]. The results will be presented in an upcoming publication.…”

The path from finite to infinite volume: Hadronic observables from lattice QCD

“…These corrections have been recently evaluated in Ref. [102] for the self-energy of point-like charged particles on and off the mass shell and with a general boost vector v. The first two orders are universal corrections, i.e., are independent of spin or structure of the hadron [103], and can be written as:…”

“…While explanations, such as the need to introduce antiparticle degrees of freedom to the low-energy theory, are proposed [106,107], a more natural explanation consistent with the spirit of an EFT is provided recently in Ref. [102]. Since the removal of the zero mode is equivalent to introducing a uniform charge density over the entire volume, even though the high-energy degrees of freedom are integrated out in the EFT, fields still interact at short distances with the background charge.…”