K019 $K_L \to e^{\pm}\mu^{\mp}$

This is a SECONDARY this catalog wave entry; see flavor\_catalog/PRIORITY\_TIERS.md for the tier policy. The process was deferred because the quark-only RS flavor pipeline does not naturally produce an \(s\to d e\mu\) amplitude. It becomes relevant once bulk leptons, lepton-flavor-nonuniversal gauge couplings, Pati--Salam leptoquark gauge bosons, or other lepton-sector flavor spurions are included. In such extensions, \(K_L\to e\mu\) constrains products of \(s\)-\(d\) and \(e\)-\(\mu\) flavor-violating couplings. Neutral KK gauge exchange or a flavor-changing \(Z\) can generate vector/axial semileptonic operators, while scalar or Higgs-mediated structures probe chirality-enhanced pseudoscalar matrix elements proportional to the kaon decay constant and quark masses. Dipole effects are less direct for this two-body mode but become part of the same lepton-bulk RS flavor story when the charged-lepton misalignment is not minimal-flavor-violating.

No newer direct \(K_L\to e\mu\) branching-fraction limit appears in the PDG 2025 \(K_L^0\) listing; the BNL E871 result remains the canonical bound. Post-2008 work matters mainly through model interpretation. Buras et al. (arXiv:1006.5356) included \(K_L\to\mu e\) and \(K_L\to\pi^0\mu e\) in a broad charged-LFV pattern study, providing a useful template for comparing kaon LFV with charged-lepton and \(B\)-meson LFV channels. Beneke, Moch, and Rohrwild (arXiv:1508.01705) updated the RS charged-lepton LFV calculation in a full 5D framework, emphasizing dipoles, \(\mu\to 3e\), and muon conversion rather than kaon modes, but fixing the lepton-bulk flavor ingredients needed before K019 can be predicted. D'Ambrosio and Iyer (arXiv:1712.08122) studied warped custodial models with partially composite leptons, rare \(K\) decays, and lepton-sector flavor violation. Blanke and Crivellin (arXiv:1801.07256) are the most explicit RS-adjacent post-2008 handle for this entry: their Pati--Salam-in-RS setup notes that conventional vector-leptoquark scenarios are stringently constrained by \(K_L\to\mu e\) and \(K\to\pi\mu e\). The LHCb strange-hadron prospects report (arXiv:1808.03477) is included only as experimental/prospective context, not as a replacement for the \(K_L\) limit.

The experimental limit is robust and clean as a bound on the physical branching fraction. Its use as an RS constraint is model-dependent. A minimal quark-only RS implementation should not impose this as a primary constraint, because the \(e\)-\(\mu\) flavor transition is absent. A lepton-bulk or leptoquark-extended implementation can use it as a strong constraint after specifying the charged-lepton flavor basis, the relevant semileptonic Wilson coefficients, and whether vector, axial, scalar, pseudoscalar, or Higgs-mediated operators dominate.

NO. Exact K019 searches over quarkConstraints/, qcd/, flavorConstraints/, neutrinos/, yukawa/, warpConfig/, solvers/, scanParams/, and tests/ found no \(K_L\to e^{\pm}\mu^{\mp}\), \(s\to d e\mu\), or kaon semileptonic LFV implementation. The nearby code surfaces are different observables: flavorConstraints/muToEGamma.py:1 and flavorConstraints/muToEGamma.py:15--21 document a \(\mu\to e\gamma\) dipole constraint, scanParams/scan.py:523--535 consumes only that LFV check, and quarkConstraints/deltaf2.py:610--624 defines neutral-kaon \(\Delta F=2\) inputs rather than a charged-LFV kaon decay.

MEDIUM. The observable itself is experimentally clean, but the repo would need a new \(\Delta S=1\) charged-LFV semileptonic operator interface, matching from the chosen lepton-bulk or leptoquark RS extension, and a kaon decay-rate evaluator with vector/axial and scalar/pseudoscalar matrix elements. No new long-distance SM subtraction is the main simplifying feature, but the necessary operator basis is not currently present.

Process-local source keys before bibliography consolidation: PDG2025\_KL\_emu, Ambrose1998\_KL\_emu, PDG2025\_ConservationLaws\_CLFV, CsakiFalkowskiWeiler2008\_RsFlavor, PerezRandall2008\_WarpedNeutrino, BurasDulingEtAl2010\_LFVFourthGeneration, BenekeMochRohrwild2015\_RsLeptonLFV, DAmbrosioIyer2018\_WarpedRareK, BlankeCrivellin2018\_PatiSalamRS, and LHCbStrange2018\_Prospects.