Kaon radiative leptonic decay rates from lattice QCD simulations at the physical point

We present a lattice quantum chromodynamics (QCD) calculation of the radiative leptonic decay rates of the kaon, improving upon our previous work in Desiderio [First lattice calculation of radiative leptonic decay rates of pseudoscalar mesons, ]. Our analysis uses gauge ensembles generated by the Extended Twisted Mass Collaboration (ETMC) with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>N</a:mi><a:mi>f</a:mi></a:msub><a:mo>=</a:mo><a:mn>2</a:mn><a:mo>+</a:mo><a:mn>1</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:math> flavors of Wilson-clover twisted mass fermions. For the first time, we go beyond the electroquenched approximation by including quark-disconnected contributions. Several key improvements have been implemented: (i) the simulations are now performed directly at physical light- and strange-quark masses, (ii) finite-size effects are carefully investigated using lattices with spatial extents ranging from <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>L</c:mi><c:mo>≃</c:mo><c:mn>3.8</c:mn><c:mtext> </c:mtext><c:mtext> </c:mtext><c:mi>fm</c:mi></c:math> to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>L</e:mi><e:mo>≃</e:mo><e:mn>7.7</e:mn><e:mtext> </e:mtext><e:mtext> </e:mtext><e:mi>fm</e:mi></e:math>, and (iii) the continuum extrapolation is based on three lattice spacings in the range <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>a</g:mi><g:mo>∈</g:mo><g:mo stretchy="false">[</g:mo><g:mn>0.08</g:mn><g:mo>,</g:mo><g:mn>0.058</g:mn><g:mo stretchy="false">]</g:mo><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>fm</g:mi></g:math>. As a result of the high-precision determination of the relevant correlation functions, we reduce the uncertainties on both the axial and vector form factors by nearly a factor of two compared to our previous analysis. When compared to experimental measurements in the electron channel (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>K</k:mi><k:mo>−</k:mo></k:msup><k:mo stretchy="false">→</k:mo><k:msup><k:mi>e</k:mi><k:mo>−</k:mo></k:msup><k:msub><k:mover accent="true"><k:mi>ν</k:mi><k:mo stretchy="false">¯</k:mo></k:mover><k:mi>e</k:mi></k:msub><k:mi>γ</k:mi></k:math>), our results show a tension—at the level of 2.6 standard deviations—with respect to KLOE data. On the other hand, they are compatible with measurements from the E36 Collaboration at J-PARC. In the muonic decay channel (<p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msup><p:mi>K</p:mi><p:mo>−</p:mo></p:msup><p:mo stretchy="false">→</p:mo><p:msup><p:mi>μ</p:mi><p:mo>−</p:mo></p:msup><p:msub><p:mover accent="true"><p:mi>ν</p:mi><p:mo stretchy="false">¯</p:mo></p:mover><p:mi>μ</p:mi></p:msub><p:mi>γ</p:mi></p:math>), we confirm the tensions, already observed in our previous study, between lattice QCD predictions and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mi>ISTRA</u:mi><u:mo>+</u:mo></u:mrow></u:math> and OKA data, which are both primarily sensitive to the value of the negative-helicity form factor <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msup><w:mi>F</w:mi><w:mo>−</w:mo></w:msup></w:math>.