Simulating rare kaon decays $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ using domain wall lattice QCD with physical light quark masses

We report the first calculation using physical light-quark masses of the electromagnetic form factor <math display="inline"><mi>V</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> describing the long-distance contributions to the <math display="inline"><msup><mi>K</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mo>ℓ</mo><mo>+</mo></msup><msup><mo>ℓ</mo><mo>-</mo></msup></math> decay amplitude. The calculation is performed on a <math display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor domain wall fermion ensemble with inverse lattice spacing <math display="inline"><msup><mi>a</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>1.730</mn><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. We implement a Glashow-Iliopoulos-Maiani cancellation by extrapolating to the physical charm-quark mass from three below-charm masses. We obtain <math display="inline"><mi>V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>=</mo><mn>0.013</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mn>0.87</mn><mo stretchy="false">(</mo><mn>4.44</mn><mo stretchy="false">)</mo></math>, achieving a bound for the value. The large statistical error arises from stochastically estimated quark loops.