Long-distance contribution to εK from lattice QCD

A lattice QCD approach to the calculation of the long-distance contributions to <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>ε</a:mi><a:mi>K</a:mi></a:msub></a:math> is presented. This parameter describes indirect <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi>C</c:mi><c:mi>P</c:mi></c:mrow></c:math> violation in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>K</e:mi><e:mo stretchy="false">→</e:mo><e:mi>π</e:mi><e:mi>π</e:mi></e:math> decay. While the short-distance contribution to <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:msub><h:mi>ε</h:mi><h:mi>K</h:mi></h:msub></h:math> can be accurately calculated in terms of standard model parameters and a single hadronic matrix element, <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msub><j:mi>B</j:mi><j:mi>K</j:mi></j:msub></j:math>, there is a long-distance part which is estimated to be approximately 5% of the total and is more difficult to determine. A method for determining this small but phenomenologically important contribution to <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msub><l:mi>ε</l:mi><l:mi>K</l:mi></l:msub></l:math> using lattice QCD is proposed and a complete exploratory calculation of the contribution is presented. This exploratory calculation uses an unphysical light quark mass corresponding to a 339 MeV pion mass and an unphysical charm quark mass of 968 MeV, expressed in the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mover accent="true"><n:mi>MS</n:mi><n:mo stretchy="true">¯</n:mo></n:mover></n:math> scheme at 2 GeV. This calculation demonstrates that future work should be able to determine this long-distance contribution from first principles with a controlled error of 10% or less.