Motivated by the first search for the rare charged-current B decay to four leptons, $$\ell \bar{\nu }_\ell \ell ^{(\prime )} \bar{\ell }^{(\prime )}$$ , we calculate the decay amplitude with factorization methods. We obtain the $$B\rightarrow \gamma ^*$$ form factors, which depend on the invariant masses of the two lepton pairs, at leading power in an expansion in $$\Lambda _\mathrm{QCD}/m_b$$ to next-to-leading order in $$\alpha _s$$ , and at $$\mathcal {O}(\alpha _s^0)$$ at next-to-leading power. Our calculations predict branching fractions of a few times $$10^{-8}$$ in the $$\ell ^{(\prime )} \bar{\ell }^{(\prime )}$$ mass-squared bin up to $$q^2=1~$$ GeV $$^2$$ with $$n_+q>3~$$ GeV. The branching fraction rapidly drops with increasing $$q^2$$ . An important further motivation for this investigation has been to explore the sensitivity of the decay rate to the inverse moment $$\lambda _B$$ of the leading-twist B meson light-cone distribution amplitude. We find that in the small- $$q^2$$ bin, the sensitivity to $$\lambda _B$$ is almost comparable to $$B^- \rightarrow \mathrm {\ell }^- \bar{\nu }_{\mathrm {\ell }}\gamma $$ when $$\lambda _B$$ is small, but with an added uncertainty from the light-meson intermediate resonance contribution. The sensitivity degrades with larger $$q^2$$ .
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"surname": "Beneke",
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"value": "Physik Department T31, Technische Universit\u00e4t M\u00fcnchen, James-Franck-Stra\u00dfe 1, Garching, 85748, Germany",
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"surname": "B\u00f6er",
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"surname": "Vos",
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"title": "QCD factorization of the four-lepton decay $$B^-\\rightarrow \\ell \\bar{\\nu }_\\ell \\ell ^{(\\prime )} \\bar{\\ell }^{(\\prime )}$$ <math> <mrow> <msup> <mi>B</mi> <mo>-</mo> </msup> <mo>\u2192</mo> <mi>\u2113</mi> <msub> <mover> <mrow> <mi>\u03bd</mi> </mrow> <mrow> <mo>\u00af</mo> </mrow> </mover> <mi>\u2113</mi> </msub> <msup> <mi>\u2113</mi> <mrow> <mo>(</mo> <mo>\u2032</mo> <mo>)</mo> </mrow> </msup> <msup> <mover> <mrow> <mi>\u2113</mi> </mrow> <mrow> <mo>\u00af</mo> </mrow> </mover> <mrow> <mo>(</mo> <mo>\u2032</mo> <mo>)</mo> </mrow> </msup> </mrow> </math>"
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"value": "Motivated by the first search for the rare charged-current B decay to four leptons, $$\\ell \\bar{\\nu }_\\ell \\ell ^{(\\prime )} \\bar{\\ell }^{(\\prime )}$$ <math> <mrow> <mi>\u2113</mi> <msub> <mover> <mrow> <mi>\u03bd</mi> </mrow> <mrow> <mo>\u00af</mo> </mrow> </mover> <mi>\u2113</mi> </msub> <msup> <mi>\u2113</mi> <mrow> <mo>(</mo> <mo>\u2032</mo> <mo>)</mo> </mrow> </msup> <msup> <mover> <mrow> <mi>\u2113</mi> </mrow> <mrow> <mo>\u00af</mo> </mrow> </mover> <mrow> <mo>(</mo> <mo>\u2032</mo> <mo>)</mo> </mrow> </msup> </mrow> </math> , we calculate the decay amplitude with factorization methods. We obtain the $$B\\rightarrow \\gamma ^*$$ <math> <mrow> <mi>B</mi> <mo>\u2192</mo> <msup> <mi>\u03b3</mi> <mo>\u2217</mo> </msup> </mrow> </math> form factors, which depend on the invariant masses of the two lepton pairs, at leading power in an expansion in $$\\Lambda _\\mathrm{QCD}/m_b$$ <math> <mrow> <msub> <mi>\u039b</mi> <mi>QCD</mi> </msub> <mo>/</mo> <msub> <mi>m</mi> <mi>b</mi> </msub> </mrow> </math> to next-to-leading order in $$\\alpha _s$$ <math> <msub> <mi>\u03b1</mi> <mi>s</mi> </msub> </math> , and at $$\\mathcal {O}(\\alpha _s^0)$$ <math> <mrow> <mi>O</mi> <mo>(</mo> <msubsup> <mi>\u03b1</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> </math> at next-to-leading power. Our calculations predict branching fractions of a few times $$10^{-8}$$ <math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>8</mn> </mrow> </msup> </math> in the $$\\ell ^{(\\prime )} \\bar{\\ell }^{(\\prime )}$$ <math> <mrow> <msup> <mi>\u2113</mi> <mrow> <mo>(</mo> <mo>\u2032</mo> <mo>)</mo> </mrow> </msup> <msup> <mover> <mrow> <mi>\u2113</mi> </mrow> <mrow> <mo>\u00af</mo> </mrow> </mover> <mrow> <mo>(</mo> <mo>\u2032</mo> <mo>)</mo> </mrow> </msup> </mrow> </math> mass-squared bin up to $$q^2=1~$$ <math> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> <mspace width=\"3.33333pt\"></mspace> </mrow> </math> GeV $$^2$$ <math> <msup> <mrow></mrow> <mn>2</mn> </msup> </math> with $$n_+q>3~$$ <math> <mrow> <msub> <mi>n</mi> <mo>+</mo> </msub> <mi>q</mi> <mo>></mo> <mn>3</mn> <mspace width=\"3.33333pt\"></mspace> </mrow> </math> GeV. The branching fraction rapidly drops with increasing $$q^2$$ <math> <msup> <mi>q</mi> <mn>2</mn> </msup> </math> . An important further motivation for this investigation has been to explore the sensitivity of the decay rate to the inverse moment $$\\lambda _B$$ <math> <msub> <mi>\u03bb</mi> <mi>B</mi> </msub> </math> of the leading-twist B meson light-cone distribution amplitude. We find that in the small- $$q^2$$ <math> <msup> <mi>q</mi> <mn>2</mn> </msup> </math> bin, the sensitivity to $$\\lambda _B$$ <math> <msub> <mi>\u03bb</mi> <mi>B</mi> </msub> </math> is almost comparable to $$B^- \\rightarrow \\mathrm {\\ell }^- \\bar{\\nu }_{\\mathrm {\\ell }}\\gamma $$ <math> <mrow> <msup> <mi>B</mi> <mo>-</mo> </msup> <mo>\u2192</mo> <msup> <mrow> <mi>\u2113</mi> </mrow> <mo>-</mo> </msup> <msub> <mover> <mrow> <mi>\u03bd</mi> </mrow> <mrow> <mo>\u00af</mo> </mrow> </mover> <mi>\u2113</mi> </msub> <mi>\u03b3</mi> </mrow> </math> when $$\\lambda _B$$ <math> <msub> <mi>\u03bb</mi> <mi>B</mi> </msub> </math> is small, but with an added uncertainty from the light-meson intermediate resonance contribution. The sensitivity degrades with larger $$q^2$$ <math> <msup> <mi>q</mi> <mn>2</mn> </msup> </math> ."
}
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