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This paper studies a practical way to include the electromagnetic effects of sea quarks in lattice simulations of QCD+QED at leading order in the fine‑structure constant α. Sea quarks are the virtual quark–antiquark pairs that pop in and out of the vacuum. Their electric charges create “quark‑line disconnected” contributions that are noisy and costly to compute. The authors test variance‑reduction tricks so these contributions can be computed more precisely and cheaply in the perturbative RM123 expansion (an expansion in the electric charge and small quark mass differences).
The team focuses on the two kinds of diagrams that appear at this order. One is a disconnected diagram (called W1) made of two separate quark loops that are joined by a photon line. The other (W2) is a connected quark loop with two electromagnetic currents. To avoid adding extra stochastic noise from the photon field, they use stochastic estimators only for the quark lines and insert the exact photon propagator in the QED_L formulation evaluated in Feynman gauge. They also compute the volume convolution with the photon propagator efficiently using a Fast Fourier Transform (FFT). The numerical tests use an ensemble of Nf=2+1 domain‑wall fermions generated by the RBC/UKQCD collaboration (lattice size 24×64, pion mass about 340 MeV, spatial extent mπL≈4.9, lattice spacing ≈0.12 fm, 50 configurations).
For the disconnected diagram W1 the authors exploit a favourable cancellation between light and strange quark contributions. Writing the light–strange difference of propagators as S_ud−S_s=(m_s−m_ud)S_ud S_s makes explicit a suppression by the SU(3) symmetry‑breaking factor (m_s−m_ud). This reduces the expected fluctuations and in particular cancels short‑distance divergences in the variance. Two stochastic estimators are compared: the standard estimator and a “split‑even” estimator (also called the split‑even or one‑end trick in related work). On the tested ensemble the split‑even estimator lowers the variance by about a factor of 10^4 compared with the standard choice. At small numbers of stochastic samples the variance falls roughly like 1/Ns^2, but for Ns around 100 the remaining variance is dominated by gauge‑field fluctuations, so no further reduction is possible without more independent gauge configurations.
The connected diagram W2 is harder. There is no light–strange cancellation and power counting indicates its variance is dominated by short distances and grows like a^−4 as the lattice spacing a goes to zero. For this contribution translation averaging and so‑called all‑to‑all estimators are effective ways to reduce noise, but they come at a high computational cost because they require many quark propagator inversions. The authors show numerical tests of variance versus separation between the currents and discuss how different estimator choices trade off noise and cost.
Why this matters: many modern lattice QCD calculations aim for sub‑percent precision in quantities used to test the Standard Model, such as contributions to the muon g−2, decay‑constant ratios, and hadronic scales. Electromagnetic effects from sea quarks can be a non‑negligible source of isospin breaking. Past work sometimes used an “electroquenched” approximation that omits these dynamical effects and can carry uncontrolled systematics. The methods tested here are steps toward including sea‑quark charges systematically at O(α) without prohibitive cost.
Important caveats: the results shown are preliminary and come from a single lattice ensemble with domain‑wall fermions. The strategy works at leading order in the RM123 perturbative expansion; higher‑order electromagnetic effects are not treated here. For W1 the split‑even estimator greatly reduces stochastic noise, but once stochastic noise is no longer dominant, reducing the total error requires more independent gauge configurations. The connected contribution W2 remains challenging because of short‑distance dominance and the need for many inversions. Overall, the work shows promising variance reductions for some pieces of the problem, but more work is needed to make full, precise unquenched QCD+QED calculations routine.