Factorizing the position-space photon propagator in QED corrections to lattice QCD correlators

Abstract

Electromagnetic corrections to the $n$-point functions of lattice QCD can be evaluated using a position-space photon propagator defined in infinite volume. Here we address the computational challenge arising from the volume-squared sum over the endpoints of the photon propagator. We consider a class of integral representations of the photon propagator that lead to a factorization of the two volume-sums, the Fourier representation being one instance thereof. An alternative choice is based on expressing the free scalar propagator as the autoconvolution of the corresponding five-dimensional propagator. We compare the performance of three different choices in the context of electromagnetic corrections to the hadronic vacuum polarization, on a gauge ensemble of size $48^3\times128$ with a pion mass of 286 MeV. As an outlook, we discuss more generally the factorization of sums over internal vertices, taking as an example the hadronic light-by-light contribution to the muon $(g-2)$.

Figures (12)

• Figure 1: Feynman diagrams depicting all Wick contractions of the four-point function in eq. \ref{['equ::atotvio']}.
• Figure 2: The diagrams discussed in this work as one would calculate them with the 2PS method. The filled black dots represent the vertices from which the one-to-all propagators start, while the unfilled dots represent the vertices which get summed over. Each color is a separate propagator. Only diagram (4)b includes a sequential propagator, which goes over one of the internal vertices and includes the photon propagator.
• Figure 3: The diagrams discussed in this work as one would calculate them with the factorization methods. The filled black dots represent the point source from which the one-to-all propagators are calculated, while the unfilled dots represent the vertices which get summed over. Each color is a separate propagator. If a propagator goes over an unfilled dot it is a (double) sequential propagator over that vertex.
• Figure 4: Final integrands for the different methods with $\Lambda = 3\, m_\mu$ and the XcYlZc discretization on the gluon-less ensemble with $L=64$. The lepton in the loop has mass $m_\ell=m_\mu$. We show the (4)a diagram (blue) and the (4)b diagram (orange) separately as well as their sum (black).
• Figure 5: Continuum extrapolation of the (4)a diagram, (a)--(c), (4)b diagram (d)--(f) and the sum of them, (4)a + (4)b, (g)--(i), for the 'gluon-less' ensembles and $\Lambda=3\, m_\mu$. The three windows are the three different implementations of the photon propagator; left window 2PS method, middle window Fourier method, right window 5D propagator method. The fits are of the form given in eq. \ref{['equ::Gluonless_Fit_Function']}. The dashed lines correspond to fits performed directly on the data points, while the solid lines represent the volume-corrected fits. The different colors correspond to the different discretizations. For (g)--(i) we also include the continuum prediction indicated by a black triangle, which was computed following the methodology outlined in Appendix A of Erb:2025nxk.