We find that the phase appearing in the unitarity relation between $$ \mathcal{B}\left({K}_L\to {\mu}^{+}{\mu}^{-}\right) $$ and $$ \mathcal{B}\left({K}_L\to \gamma \gamma \right) $$ is equal to the phase shift in the interference term of the time- dependent K → μ + μ − decay. A probe of this relation at future kaon facilities constitutes a Standard Model test with a theory precision of about 2%. The phase has further importance for sensitivity studies regarding the measurement of the time-dependent K → μ + μ − decay rate to extract the CKM matrix element combination $$ \mid {V}_{ts}{V}_{td}\sin \left(\beta +{\beta}_s\right)\mid \approx {A}^2{\lambda}^5\overline{\eta} $$ . We find a model-independent theoretically clean prediction, cos2 φ 0 = 0.96 ± 0.03. The quoted error is a combination of the theoretical and experimental errors, and both of them are expected to shrink in the future. Using input from the large-N C limit within chiral perturbation theory, we find a theory preference towards solutions with negative cos φ 0, reducing a four-fold ambiguity in the angle φ 0 to a two-fold one.
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"title": "A precision relation between \u0393(K \u2192 \u03bc + \u03bc \u2212 )(t) and <math> <mi>B</mi> <mfenced> <mrow> <msub> <mi>K</mi> <mi>L</mi> </msub> <mo>\u2192</mo> <msup> <mi>\u03bc</mi> <mo>+</mo> </msup> <msup> <mi>\u03bc</mi> <mo>\u2212</mo> </msup> </mrow> </mfenced> <mo>/</mo> <mi>B</mi> <mfenced> <mrow> <msub> <mi>K</mi> <mi>L</mi> </msub> <mo>\u2192</mo> <mi>\u03b3\u03b3</mi> </mrow> </mfenced> </math> $$ \\mathcal{B}\\left({K}_L\\to {\\mu}^{+}{\\mu}^{-}\\right)/\\mathcal{B}\\left({K}_L\\to \\gamma \\gamma \\right) $$"
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"value": "We find that the phase appearing in the unitarity relation between <math> <mi>B</mi> <mfenced> <mrow> <msub> <mi>K</mi> <mi>L</mi> </msub> <mo>\u2192</mo> <msup> <mi>\u03bc</mi> <mo>+</mo> </msup> <msup> <mi>\u03bc</mi> <mo>\u2212</mo> </msup> </mrow> </mfenced> </math> $$ \\mathcal{B}\\left({K}_L\\to {\\mu}^{+}{\\mu}^{-}\\right) $$ and <math> <mi>B</mi> <mfenced> <mrow> <msub> <mi>K</mi> <mi>L</mi> </msub> <mo>\u2192</mo> <mi>\u03b3\u03b3</mi> </mrow> </mfenced> </math> $$ \\mathcal{B}\\left({K}_L\\to \\gamma \\gamma \\right) $$ is equal to the phase shift in the interference term of the time- dependent K \u2192 \u03bc + \u03bc \u2212 decay. A probe of this relation at future kaon facilities constitutes a Standard Model test with a theory precision of about 2%. The phase has further importance for sensitivity studies regarding the measurement of the time-dependent K \u2192 \u03bc + \u03bc \u2212 decay rate to extract the CKM matrix element combination <math> <mo>\u2223</mo> <msub> <mi>V</mi> <mi>ts</mi> </msub> <msub> <mi>V</mi> <mi>td</mi> </msub> <mo>sin</mo> <mfenced> <mrow> <mi>\u03b2</mi> <mo>+</mo> <msub> <mi>\u03b2</mi> <mi>s</mi> </msub> </mrow> </mfenced> <mo>\u2223</mo> <mo>\u2248</mo> <msup> <mi>A</mi> <mn>2</mn> </msup> <msup> <mi>\u03bb</mi> <mn>5</mn> </msup> <mover> <mi>\u03b7</mi> <mo>\u00af</mo> </mover> </math> $$ \\mid {V}_{ts}{V}_{td}\\sin \\left(\\beta +{\\beta}_s\\right)\\mid \\approx {A}^2{\\lambda}^5\\overline{\\eta} $$ . We find a model-independent theoretically clean prediction, cos2 \u03c6 0 = 0.96 \u00b1 0.03. The quoted error is a combination of the theoretical and experimental errors, and both of them are expected to shrink in the future. Using input from the large-N C limit within chiral perturbation theory, we find a theory preference towards solutions with negative cos \u03c6 0, reducing a four-fold ambiguity in the angle \u03c6 0 to a two-fold one."
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