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---
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title: Already defined observables in FlavorKit
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permalink: /Already_defined_observables_in_FlavorKit/
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---
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# Already defined observables in FlavorKit
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[Category:FlavorKit](/Category:FlavorKit "wikilink") Many observables are already implemented in FlavorKit and can be used out-of-the-box
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Many observables are already implemented in FlavorKit and can be used out-of-the-box
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Lepton flavor observables
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-------------------------
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... | ... | @@ -16,7 +13,7 @@ We discuss in the following the implementation of the most important LFV observa |
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The decay width is given by
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$\\Gamma \\left( \\ell_\\alpha \\to \\ell_\\beta \\gamma \\right) = \\frac{\\alpha m_{\\ell_\\alpha}^5}{4} \\left( |K_2^L|^2 + |K_2^R|^2 \\right) \\, ,$
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$`\Gamma \left( \ell_\alpha \to \ell_\beta \gamma \right) = \frac{\alpha m_{\ell_\alpha}^5}{4} \left( |K_2^L|^2 + |K_2^R|^2 \right) \, ,`$
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where *α* is the fine structure constant and the dipole Wilson coefficients *K*<sub>2</sub><sup>*L*, *R*</sup> are defined in Eq..
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... | ... | @@ -68,13 +65,7 @@ where *α* is the fine structure constant and the dipole Wilson coefficients *K* |
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The decay width is given by
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$\\begin{aligned}
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\\Gamma \\left( \\ell_\\alpha \\to 3 \\ell_\\beta \\right) &=& \\frac{m_{\\ell_\\alpha}^5}{512 \\pi^3} \\left\[ e^4 \\, \\left( \\left| K_2^L \\right|^2 + \\left| K_2^R \\right|^2 \\right) \\left( \\frac{16}{3} \\log{\\frac{m_{\\ell_\\alpha}}{m_{\\ell_\\beta}}} - \\frac{22}{3} \\right) \\right. \\label{L3Lwidth} \\\\
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&+& \\frac{1}{24} \\left( \\left| A_{LL}^S \\right|^2 + \\left| A_{RR}^S \\right|^2 \\right) + \\frac{1}{12} \\left( \\left| A_{LR}^S \\right|^2 + \\left| A_{RL}^S \\right|^2 \\right) \\nonumber \\\\
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&+& \\frac{2}{3} \\left( \\left| \\hat A_{LL}^V \\right|^2 + \\left| \\hat A_{RR}^V \\right|^2 \\right) + \\frac{1}{3} \\left( \\left| \\hat A_{LR}^V \\right|^2 + \\left| \\hat A_{RL}^V \\right|^2 \\right) + 6 \\left( \\left| A_{LL}^T \\right|^2 + \\left| A_{RT}^T \\right|^2 \\right) \\nonumber \\\\
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&+& \\frac{e^2}{3} \\left( K_2^L A_{RL}^{S \\ast} + K_2^R A_{LR}^{S \\ast} + c.c. \\right) - \\frac{2 e^2}{3} \\left( K_2^L \\hat A_{RL}^{V \\ast} + K_2^R \\hat A_{LR}^{V \\ast} + c.c. \\right) \\nonumber \\\\
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&-& \\frac{4 e^2}{3} \\left( K_2^L \\hat A_{RR}^{V \\ast} + K_2^R \\hat A_{LL}^{V \\ast} + c.c. \\right) \\nonumber \\\\
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&-& \\left. \\frac{1}{2} \\left( A_{LL}^S A_{LL}^{T \\ast} + A_{RR}^S A_{RR}^{T \\ast} + c.c. \\right) - \\frac{1}{6} \\left( A_{LR}^S \\hat A_{LR}^{V \\ast} + A_{RL}^S \\hat A_{RL}^{V \\ast} + c.c. \\right) \\right\] \\nonumber \\, .\\end{aligned}$
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$` \Gamma \left( \ell_\alpha \to 3 \ell_\beta \right) = \frac{m_{\ell_\alpha}^5}{512 \pi^3} \left[ e^4 \, \left( \left| K_2^L \right|^2 + \left| K_2^R \right|^2 \right) \left( \frac{16}{3} \log{\frac{m_{\ell_\alpha}}{m_{\ell_\beta}}} - \frac{22}{3} \right) \right. \\ + \frac{1}{24} \left( \left| A_{LL}^S \right|^2 + \left| A_{RR}^S \right|^2 \right) + \frac{1}{12} \left( \left| A_{LR}^S \right|^2 + \left| A_{RL}^S \right|^2 \right) \nonumber \\ + \frac{2}{3} \left( \left| \hat A_{LL}^V \right|^2 + \left| \hat A_{RR}^V \right|^2 \right) + \frac{1}{3} \left( \left| \hat A_{LR}^V \right|^2 + \left| \hat A_{RL}^V \right|^2 \right) + 6 \left( \left| A_{LL}^T \right|^2 + \left| A_{RT}^T \right|^2 \right) \nonumber \\ + \frac{e^2}{3} \left( K_2^L A_{RL}^{S \ast} + K_2^R A_{LR}^{S \ast} + c.c. \right) - \frac{2 e^2}{3} \left( K_2^L \hat A_{RL}^{V \ast} + K_2^R \hat A_{LR}^{V \ast} + c.c. \right) \nonumber \\ - \frac{4 e^2}{3} \left( K_2^L \hat A_{RR}^{V \ast} + K_2^R \hat A_{LL}^{V \ast} + c.c. \right) \nonumber \\ - \left. \frac{1}{2} \left( A_{LL}^S A_{LL}^{T \ast} + A_{RR}^S A_{RR}^{T \ast} + c.c. \right) - \frac{1}{6} \left( A_{LR}^S \hat A_{LR}^{V \ast} + A_{RL}^S \hat A_{RL}^{V \ast} + c.c. \right) \right] \nonumber \, .`$
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Here we have defined
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... | ... | @@ -201,45 +192,19 @@ The mass of the leptons in the final state has been neglected in this formula, w |
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The conversion rate, relative to the the muon capture rate, can be expressed as
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$\\begin{aligned}
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{\\rm CR} (\\mu- e, {\\rm Nucleus}) &=
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\\frac{p_e \\, E_e \\, m_\\mu^3 \\, G_F^2 \\, \\alpha^3
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\\, Z_{\\rm eff}^4 \\, F_p^2}{8 \\, \\pi^2 \\, Z} \\nonumber \\\\
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&\\times \\left\\{ \\left| (Z + N) \\left( g_{LV}^{(0)} + g_{LS}^{(0)} \\right) +
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(Z - N) \\left( g_{LV}^{(1)} + g_{LS}^{(1)} \\right) \\right|^2 +
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\\right. \\nonumber \\\\
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& \\ \\ \\
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\\ \\left. \\,\\, \\left| (Z + N) \\left( g_{RV}^{(0)} + g_{RS}^{(0)} \\right) +
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(Z - N) \\left( g_{RV}^{(1)} + g_{RS}^{(1)} \\right) \\right|^2 \\right\\}
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\\frac{1}{\\Gamma_{\\rm capt}}\\,.\\end{aligned}$
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*Z* and *N* are the number of protons and neutrons in the nucleus and $Z_{\\rm eff}$ is the effective atomic charge . Similarly, *G*<sub>*F*</sub> is the Fermi constant, *F*<sub>*p*</sub> is the nuclear matrix element and $\\Gamma_{\\rm capt}$ represents the total muon capture rate. *α* is the fine structure constant, *p*<sub>*e*</sub> and *E*<sub>*e*</sub> ( ≃*m*<sub>*μ*</sub> in the numerical evaluation) are the momentum and energy of the electron and *m*<sub>*μ*</sub> is the muon mass. In the above, *g*<sub>*X**K*</sub><sup>(0)</sup> and *g*<sub>*X**K*</sub><sup>(1)</sup> (with *X* = *L*, *R* and *K* = *S*, *V*) can be written in terms of effective couplings at the quark level as
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$\\begin{aligned}
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g_{XK}^{(0)} &= \\frac{1}{2} \\sum_{q = u,d,s} \\left( g_{XK(q)} G_K^{(q,p)} +
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g_{XK(q)} G_K^{(q,n)} \\right)\\,, \\nonumber \\\\
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g_{XK}^{(1)} &= \\frac{1}{2} \\sum_{q = u,d,s} \\left( g_{XK(q)} G_K^{(q,p)} -
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g_{XK(q)} G_K^{(q,n)} \\right)\\,.\\end{aligned}$
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$` {\rm CR} (\mu- e, {\rm Nucleus}) = \frac{p_e \, E_e \, m_\mu^3 \, G_F^2 \, \alpha^3 \, Z_{\rm eff}^4 \, F_p^2}{8 \, \pi^2 \, Z} \nonumber \\ \times \left\{ \left| (Z + N) \left( g_{LV}^{(0)} + g_{LS}^{(0)} \right) + (Z - N) \left( g_{LV}^{(1)} + g_{LS}^{(1)} \right) \right|^2 + \right. \nonumber \\ \ \ \ \ \left. \,\, \left| (Z + N) \left( g_{RV}^{(0)} + g_{RS}^{(0)} \right) + (Z - N) \left( g_{RV}^{(1)} + g_{RS}^{(1)} \right) \right|^2 \right\} \frac{1}{\Gamma_{\rm capt}}\,.`$
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*Z* and *N* are the number of protons and neutrons in the nucleus and $`Z_{\rm eff}`$ is the effective atomic charge . Similarly, *G*<sub>*F*</sub> is the Fermi constant, *F*<sub>*p*</sub> is the nuclear matrix element and $`\Gamma_{\rm capt}`$ represents the total muon capture rate. *α* is the fine structure constant, *p*<sub>*e*</sub> and *E*<sub>*e*</sub> ( ≃*m*<sub>*μ*</sub> in the numerical evaluation) are the momentum and energy of the electron and *m*<sub>*μ*</sub> is the muon mass. In the above, *g*<sub>*X**K*</sub><sup>(0)</sup> and *g*<sub>*X**K*</sub><sup>(1)</sup> (with *X* = *L*, *R* and *K* = *S*, *V*) can be written in terms of effective couplings at the quark level as
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$` g_{XK}^{(0)} = \frac{1}{2} \sum_{q = u,d,s} \left( g_{XK(q)} G_K^{(q,p)} + g_{XK(q)} G_K^{(q,n)} \right)\,, \nonumber \\ g_{XK}^{(1)} = \frac{1}{2} \sum_{q = u,d,s} \left( g_{XK(q)} G_K^{(q,p)} - g_{XK(q)} G_K^{(q,n)} \right)\,.`$
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For coherent *μ* − *e* conversion in nuclei, only scalar (*S*) and vector (*V*) couplings contribute. Furthermore, sizable contributions are expected only from the *u*, *d*, *s* quark flavors. The numerical values of the relevant *G*<sub>*K*</sub> factors are
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$\\begin{aligned}
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&
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G_V^{(u, p)}\\, =\\, G_V^{(d, n)\\,} =\\, 2 \\,;\\, \\ \\ \\ \\
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G_V^{(d, p)}\\, =\\, G_V^{(u, n)}\\, = 1\\,; \\nonumber \\\\
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&
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G_S^{(u, p)}\\, =\\, G_S^{(d, n)}\\, =\\, 5.1\\,;\\, \\ \\
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G_S^{(d, p)}\\, =\\, G_S^{(u, n)}\\, = \\,4.3 \\,;\\, \\nonumber \\\\
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&
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G_S^{(s, p)}\\,=\\, G_S^{(s, n)}\\, = \\,2.5\\,.\\end{aligned}$
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$` G_V^{(u, p)}\, =\, G_V^{(d, n)\,} =\, 2 \,;\, \ \ \ \ G_V^{(d, p)}\, =\, G_V^{(u, n)}\, = 1\,; \nonumber \\ G_S^{(u, p)}\, =\, G_S^{(d, n)}\, =\, 5.1\,;\, \ \ G_S^{(d, p)}\, =\, G_S^{(u, n)}\, = \,4.3 \,;\, \nonumber \\ G_S^{(s, p)}\,=\, G_S^{(s, n)}\, = \,2.5\,.`$
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Finally, the *g*<sub>*X**K*(*q*)</sub> coefficients can be written in terms of the Wilson coefficients in Eqs., and as
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$\\begin{aligned}
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g_{LV(q)} &=& \\frac{\\sqrt{2}}{G_F} \\left\[ e^2 Q_q \\left( K_1^L - K_2^R \\right)- \\frac{1}{2} \\left( C_{\\ell\\ell qq}^{VLL} + C_{\\ell\\ell qq}^{VLR} \\right) \\right\] \\\\
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g_{RV(q)} &=& \\left. g_{LV(q)} \\right|_{L \\to R} \\\\
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g_{LS(q)} &=& - \\frac{\\sqrt{2}}{G_F} \\frac{1}{2} \\left( C_{\\ell\\ell qq}^{SLL} + C_{\\ell\\ell qq}^{SLR} \\right) \\\\
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g_{RS(q)} &=& \\left. g_{LS(q)} \\right|_{L \\to R} \\, .\\end{aligned}$
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$` g_{LV(q)} = \frac{\sqrt{2}}{G_F} \left[ e^2 Q_q \left( K_1^L - K_2^R \right)- \frac{1}{2} \left( C_{\ell\ell qq}^{VLL} + C_{\ell\ell qq}^{VLR} \right) \right] \\ g_{RV(q)} = \left. g_{LV(q)} \right|_{L \to R} \\ g_{LS(q)} = - \frac{\sqrt{2}}{G_F} \frac{1}{2} \left( C_{\ell\ell qq}^{SLL} + C_{\ell\ell qq}^{SLR} \right) \\ g_{RS(q)} = \left. g_{LS(q)} \right|_{L \to R} \, .`$
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Here *Q*<sub>*q*</sub> is the quark electric charge (*Q*<sub>*d*</sub> = −1/3, *Q*<sub>*u*</sub> = 2/3) and *C*<sub>ℓℓ*q**q*</sub><sup>*I**X**K*</sup> = *B*<sub>*X**Y*</sub><sup>*K*</sup> (*C*<sub>*X**Y*</sub><sup>*K*</sup>) for d-quarks (u-quarks), with *X* = *L*, *R* and *K* = *S*, *V*.
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... | ... | @@ -414,42 +379,27 @@ Here *Q*<sub>*q*</sub> is the quark electric charge (*Q*<sub>*d*</sub> = − |
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Our analytical expressions for *τ* → *P*ℓ, where ℓ = *e*, *μ* and *P* is a pseudoscalar meson, generalize the results in . The decay width is given by
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$\\Gamma \\left( \\tau \\to \\ell P \\right) = \\frac{1}{4 \\pi} \\frac{\\lambda^{1/2}(m_\\tau^2,m_\\ell^2,m_P^2)}{m_\\tau^2} \\frac{1}{2} \\sum_{i,f} |\\mathcal{M}_{\\tau \\ell P}|^2 \\, ,$
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$`\Gamma \left( \tau \to \ell P \right) = \frac{1}{4 \pi} \frac{\lambda^{1/2}(m_\tau^2,m_\ell^2,m_P^2)}{m_\tau^2} \frac{1}{2} \sum_{i,f} |\mathcal{M}_{\tau \ell P}|^2 \, ,`$
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where the averaged squared amplitude can be written as
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$\\frac{1}{2} \\sum_{i,f} |\\mathcal{M}_{\\tau \\ell P}|^2 = \\frac{1}{4 m_\\tau} \\sum_{I,J = S,V} \\left\[ 2 m_\\tau m_\\ell \\left( a_P^I a_P^{J \\, \\ast} - b_P^I b_P^{J \\, \\ast} \\right) + (m_\\tau^2 + m_\\ell^2 - m_P^2) \\left( a_P^I a_P^{J \\, \\ast} + b_P^I b_P^{J \\, \\ast} \\right) \\right\] \\, .$
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$`\frac{1}{2} \sum_{i,f} |\mathcal{M}_{\tau \ell P}|^2 = \frac{1}{4 m_\tau} \sum_{I,J = S,V} \left[ 2 m_\tau m_\ell \left( a_P^I a_P^{J \, \ast} - b_P^I b_P^{J \, \ast} \right) + (m_\tau^2 + m_\ell^2 - m_P^2) \left( a_P^I a_P^{J \, \ast} + b_P^I b_P^{J \, \ast} \right) \right] \, .`$
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The coefficients *a*<sub>*P*</sub><sup>*S*, *V*</sup> and *b*<sub>*P*</sub><sup>*S*, *V*</sup> can be expressed in terms of the Wilson coefficients in Eqs. and as
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$\\begin{aligned}
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a_P^S &=& \\frac{1}{2} f_\\pi \\, \\sum_{X = L,R} \\left\[ \\frac{D_X^d(P)}{m_d} \\left( B^S_{LX} + B^S_{RX} \\right) + \\frac{D_X^u(P)}{m_u} \\left( C^S_{LX} + C^S_{RX} \\right) \\right\] \\\\
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b_P^S &=& \\frac{1}{2} f_\\pi \\, \\sum_{X = L,R} \\left\[ \\frac{D_X^d(P)}{m_d} \\left( B^S_{RX} - B^S_{LX} \\right) + \\frac{D_X^u(P)}{m_u} \\left( C^S_{RX} - C^S_{LX} \\right) \\right\] \\\\
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a_P^V &=& \\frac{1}{4} f_\\pi \\, C(P) (m_\\tau - m_\\ell) \\left\[ - B_{LL}^V + B_{LR}^V - B_{RL}^V + B_{RR}^V \\right. \\nonumber \\\\
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&& \\left. + C_{LL}^V - C_{LR}^V + C_{RL}^V - C_{RR}^V \\right\] \\\\
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b_P^V &=& \\frac{1}{4} f_\\pi \\, C(P) (m_\\tau + m_\\ell) \\left\[ - B_{LL}^V + B_{LR}^V + B_{RL}^V - B_{RR}^V \\right. \\nonumber \\\\
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&& \\left. + C_{LL}^V - C_{LR}^V - C_{RL}^V + C_{RR}^V \\right\] \\, . \\label{coeffsTauMesonLepton}\\end{aligned}$
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$` a_P^S = \frac{1}{2} f_\pi \, \sum_{X = L,R} \left[ \frac{D_X^d(P)}{m_d} \left( B^S_{LX} + B^S_{RX} \right) + \frac{D_X^u(P)}{m_u} \left( C^S_{LX} + C^S_{RX} \right) \right] \\ b_P^S = \frac{1}{2} f_\pi \, \sum_{X = L,R} \left[ \frac{D_X^d(P)}{m_d} \left( B^S_{RX} - B^S_{LX} \right) + \frac{D_X^u(P)}{m_u} \left( C^S_{RX} - C^S_{LX} \right) \right] \\ a_P^V = \frac{1}{4} f_\pi \, C(P) (m_\tau - m_\ell) \left[ - B_{LL}^V + B_{LR}^V - B_{RL}^V + B_{RR}^V \right. \nonumber \\ \left. + C_{LL}^V - C_{LR}^V + C_{RL}^V - C_{RR}^V \right] \\ b_P^V = \frac{1}{4} f_\pi \, C(P) (m_\tau + m_\ell) \left[ - B_{LL}^V + B_{LR}^V + B_{RL}^V - B_{RR}^V \right. \nonumber \\ \left. + C_{LL}^V - C_{LR}^V - C_{RL}^V + C_{RR}^V \right] \, . `$
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In these expressions *m*<sub>*d*</sub> and *m*<sub>*u*</sub> are the down- and up-quark masses, respectively, *f*<sub>*π*</sub> is the pion decay constant and the coefficients *C*(*P*),*D*<sub>*L*, *R*</sub><sup>*d*, *u*</sup>(*P*) take different forms for each pseudoscalar meson *P* . For *P* = *π* one has
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$\\begin{aligned}
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C(\\pi) &=& 1 \\\\
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D_L^d(\\pi) &=& - \\frac{m_\\pi^2}{4} \\\\
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D_L^u(\\pi) &=& \\frac{m_\\pi^2}{4} \\, ,\\end{aligned}$
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$` C(\pi) = 1 \\ D_L^d(\pi) = - \frac{m_\pi^2}{4} \\ D_L^u(\pi) = \frac{m_\pi^2}{4} \, ,`$
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for *P* = *η*
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$\\begin{aligned}
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C(\\eta) &=& \\frac{1}{\\sqrt{6}} \\left( \\sin \\theta_\\eta + \\sqrt{2} \\cos \\theta_\\eta \\right) \\\\
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D_L^d(\\eta) &=& \\frac{1}{4 \\sqrt{3}} \\left\[ (3 m_\\pi^2 - 4 m_K^2) \\cos \\theta_\\eta - 2 \\sqrt{2} m_K^2 \\sin \\theta_\\eta \\right\] \\\\
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D_L^u(\\eta) &=& \\frac{1}{4 \\sqrt{3}} m_\\pi^2 \\left( \\cos \\theta_\\eta - \\sqrt{2} \\sin \\theta_\\eta \\right) \\, ,\\end{aligned}$
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$` C(\eta) = \frac{1}{\sqrt{6}} \left( \sin \theta_\eta + \sqrt{2} \cos \theta_\eta \right) \\ D_L^d(\eta) = \frac{1}{4 \sqrt{3}} \left[ (3 m_\pi^2 - 4 m_K^2) \cos \theta_\eta - 2 \sqrt{2} m_K^2 \sin \theta_\eta \right] \\ D_L^u(\eta) = \frac{1}{4 \sqrt{3}} m_\pi^2 \left( \cos \theta_\eta - \sqrt{2} \sin \theta_\eta \right) \, ,`$
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and for *P* = *η*<sup>′</sup>
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$\\begin{aligned}
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C(\\eta^\\prime) &=& \\frac{1}{\\sqrt{6}} \\left( \\sqrt{2} \\sin \\theta_\\eta - \\cos \\theta_\\eta \\right) \\\\
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D_L^d(\\eta^\\prime) &=& \\frac{1}{4 \\sqrt{3}} \\left\[ (3 m_\\pi^2 - 4 m_K^2) \\sin \\theta_\\eta + 2 \\sqrt{2} m_K^2 \\cos \\theta_\\eta \\right\] \\\\
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D_L^u(\\eta^\\prime) &=& \\frac{1}{4 \\sqrt{3}} m_\\pi^2 \\left( \\sin \\theta_\\eta + \\sqrt{2} \\cos \\theta_\\eta \\right) \\, .\\end{aligned}$
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$` C(\eta^\prime) = \frac{1}{\sqrt{6}} \left( \sqrt{2} \sin \theta_\eta - \cos \theta_\eta \right) \\ D_L^d(\eta^\prime) = \frac{1}{4 \sqrt{3}} \left[ (3 m_\pi^2 - 4 m_K^2) \sin \theta_\eta + 2 \sqrt{2} m_K^2 \cos \theta_\eta \right] \\ D_L^u(\eta^\prime) = \frac{1}{4 \sqrt{3}} m_\pi^2 \left( \sin \theta_\eta + \sqrt{2} \cos \theta_\eta \right) \, .`$
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Here *m*<sub>*π*</sub> and *m*<sub>*K*</sub> are the masses of the neutral pion and Kaon, respectively, and *θ*<sub>*η*</sub> is the *η* − *η*<sup>′</sup> mixing angle. In addition, *D*<sub>*R*</sub><sup>*d*, *u*</sup>(*P*)= − (*D*<sub>*L*</sub><sup>*d*, *u*</sup>(*P*))<sup>\*</sup>.
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... | ... | @@ -619,11 +569,7 @@ Notice that the Wilson coefficients in Eq. include all pseudoscalar and axial co |
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The decay width is given by
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$\\begin{aligned}
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\\Gamma \\left( h \\to \\ell_\\alpha \\ell_\\beta \\right) &\\equiv& \\Gamma \\left( h \\to \\ell_\\alpha \\bar \\ell_\\beta \\right) + \\Gamma \\left( h \\to \\bar \\ell_\\alpha \\ell_\\beta \\right) = \\\\
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&& \\frac{1}{16 \\pi m_h} \\left\[ \\left(1-\\left(\\frac{m_{\\ell_\\alpha} + m_{\\ell_\\beta}}{m_h}\\right)^2\\right)\\left(1-\\left(\\frac{m_{\\ell_\\alpha} - m_{\\ell_\\beta}}{m_h}\\right)^2\\right)\\right\]^{1/2} \\nonumber \\\\
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&& \\times \\left\[ \\left( m_h^2 - m_{\\ell_\\alpha}^2 - m_{\\ell_\\beta}^2 \\right) \\left( |S_L|^2 + |S_R|^2 \\right)_{\\alpha \\beta} - 4 m_{\\ell_\\alpha} m_{\\ell_\\beta} \\text{Re}(S_L S_R^\\ast)_{\\alpha \\beta} \\right\] \\nonumber \\\\
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&& + (\\alpha \\leftrightarrow \\beta) \\nonumber\\end{aligned}$
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$` \Gamma \left( h \to \ell_\alpha \ell_\beta \right) \equiv \Gamma \left( h \to \ell_\alpha \bar \ell_\beta \right) + \Gamma \left( h \to \bar \ell_\alpha \ell_\beta \right) = \\ \frac{1}{16 \pi m_h} \left[ \left(1-\left(\frac{m_{\ell_\alpha} + m_{\ell_\beta}}{m_h}\right)^2\right)\left(1-\left(\frac{m_{\ell_\alpha} - m_{\ell_\beta}}{m_h}\right)^2\right)\right]^{1/2} \nonumber \\ \times \left[ \left( m_h^2 - m_{\ell_\alpha}^2 - m_{\ell_\beta}^2 \right) \left( |S_L|^2 + |S_R|^2 \right)_{\alpha \beta} - 4 m_{\ell_\alpha} m_{\ell_\beta} \text{Re}(S_L S_R^\ast)_{\alpha \beta} \right] \nonumber \\ + (\alpha \leftrightarrow \beta) \nonumber`$
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NameProcess = "hLLp";
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NameObservables = {{BrhtoMuE, 1101, "BR(h->e mu)"},
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... | ... | @@ -699,9 +645,7 @@ $\\begin{aligned} |
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The decay width is given by
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$\\begin{aligned}
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\\Gamma \\left( Z \\to \\ell_\\alpha \\ell_\\beta \\right) &\\equiv& \\Gamma \\left( Z \\to \\ell_\\alpha \\bar \\ell_\\beta \\right) + \\Gamma \\left( Z \\to \\bar \\ell_\\alpha \\ell_\\beta \\right) = \\\\
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&& \\frac{m_Z}{48 \\pi} \\left\[ 2 \\left( |R_1^L|^2 + |R_1^R|^2 \\right) + \\frac{m_Z^2}{4} \\left( |R_2^L|^2 + |R_2^R|^2 \\right) \\right\] \\, , \\nonumber \\end{aligned}$
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$` \Gamma \left( Z \to \ell_\alpha \ell_\beta \right) \equiv \Gamma \left( Z \to \ell_\alpha \bar \ell_\beta \right) + \Gamma \left( Z \to \bar \ell_\alpha \ell_\beta \right) = \\ \frac{m_Z}{48 \pi} \left[ 2 \left( |R_1^L|^2 + |R_1^R|^2 \right) + \frac{m_Z^2}{4} \left( |R_2^L|^2 + |R_2^R|^2 \right) \right] \, , \nonumber `$
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where the charged lepton masses have been neglected.
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... | ... | @@ -763,24 +707,15 @@ We give also here a description of the implementation of the different observabl |
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Our analytical results for *B*<sub>*s*, *d*</sub><sup>0</sup> → ℓ<sup>+</sup>ℓ<sup>−</sup> follow . The *B*<sup>0</sup> ≡ *B*<sub>*s*, *d*</sub><sup>0</sup> decay width to a pair of charged leptons can be written as
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$\\Gamma \\left(B^0 \\to \\ell_\\alpha^+ \\ell_\\beta^- \\right) = \\frac{|\\mathcal{M_{B\\ell\\ell}}|^2}{16 \\pi M_{B}} \\left\[ \\left(1-\\left(\\frac{m_{\\ell_\\alpha} + m_{\\ell_\\beta}}{m_B}\\right)^2\\right)\\left(1-\\left(\\frac{m_{\\ell_\\alpha} - m_{\\ell_\\beta}}{m_B}\\right)^2\\right)\\right\]^{1/2} \\, .$
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$`\Gamma \left(B^0 \to \ell_\alpha^+ \ell_\beta^- \right) = \frac{|\mathcal{M_{B\ell\ell}}|^2}{16 \pi M_{B}} \left[ \left(1-\left(\frac{m_{\ell_\alpha} + m_{\ell_\beta}}{m_B}\right)^2\right)\left(1-\left(\frac{m_{\ell_\alpha} - m_{\ell_\beta}}{m_B}\right)^2\right)\right]^{1/2} \, .`$
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Here
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$\\begin{aligned}
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|\\mathcal{M_{B\\ell\\ell}}|^2 &=& 2 |F_S|^2 \\left\[ m_B^2 - \\left(m_{\\ell_\\alpha} + m_{\\ell_\\beta}\\right)^2 \\right\] + 2 |F_P|^2 \\left\[ m_B^2 - \\left(m_{\\ell_\\alpha} - m_{\\ell_\\beta}\\right)^2 \\right\] \\nonumber \\\\
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&& + 2 |F_V|^2 \\left\[ m_B^2 \\left(m_{\\ell_\\alpha} - m_{\\ell_\\beta}\\right)^2 - \\left(m_{\\ell_\\alpha}^2 - m_{\\ell_\\beta}^2\\right)^2 \\right\] \\nonumber \\\\
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&& + 2 |F_A|^2 \\left\[ m_B^2 \\left(m_{\\ell_\\alpha} + m_{\\ell_\\beta}\\right)^2 - \\left(m_{\\ell_\\alpha}^2 - m_{\\ell_\\beta}^2\\right)^2 \\right\] \\nonumber \\\\
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&& + 4 \\, \\text{Re}(F_S F_V^\\ast) \\left(m_{\\ell_\\alpha} - m_{\\ell_\\beta}\\right) \\left\[ m_B^2 + \\left(m_{\\ell_\\alpha} + m_{\\ell_\\beta}\\right)^2 \\right\] \\nonumber \\\\
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&& + 4 \\, \\text{Re}(F_P F_A^\\ast) \\left(m_{\\ell_\\alpha} + m_{\\ell_\\beta}\\right) \\left\[ m_B^2 - \\left(m_{\\ell_\\alpha} - m_{\\ell_\\beta}\\right)^2 \\right\] \\, , \\label{AmpBll}\\end{aligned}$
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$` |\mathcal{M_{B\ell\ell}}|^2 = 2 |F_S|^2 \left[ m_B^2 - \left(m_{\ell_\alpha} + m_{\ell_\beta}\right)^2 \right] + 2 |F_P|^2 \left[ m_B^2 - \left(m_{\ell_\alpha} - m_{\ell_\beta}\right)^2 \right] \nonumber \\ + 2 |F_V|^2 \left[ m_B^2 \left(m_{\ell_\alpha} - m_{\ell_\beta}\right)^2 - \left(m_{\ell_\alpha}^2 - m_{\ell_\beta}^2\right)^2 \right] \nonumber \\ + 2 |F_A|^2 \left[ m_B^2 \left(m_{\ell_\alpha} + m_{\ell_\beta}\right)^2 - \left(m_{\ell_\alpha}^2 - m_{\ell_\beta}^2\right)^2 \right] \nonumber \\ + 4 \, \text{Re}(F_S F_V^\ast) \left(m_{\ell_\alpha} - m_{\ell_\beta}\right) \left[ m_B^2 + \left(m_{\ell_\alpha} + m_{\ell_\beta}\right)^2 \right] \nonumber \\ + 4 \, \text{Re}(F_P F_A^\ast) \left(m_{\ell_\alpha} + m_{\ell_\beta}\right) \left[ m_B^2 - \left(m_{\ell_\alpha} - m_{\ell_\beta}\right)^2 \right] \, , `$
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and the *F*<sub>*X*</sub> coefficients are defined in terms of our Wilson coefficients as[1]
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$\\begin{aligned}
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F_S &=& \\frac{i}{4} \\frac{m_B^2 f_B}{m_d + m_{d^\\prime}} \\left( E_{LL}^S + E_{LR}^S - E_{RR}^S - E_{RL}^S \\right) \\\\
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F_P &=& \\frac{i}{4} \\frac{m_B^2 f_B}{m_d + m_{d^\\prime}} \\left( - E_{LL}^S + E_{LR}^S - E_{RR}^S + E_{RL}^S \\right) \\\\
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F_V &=& - \\frac{i}{4} f_B \\left( E_{LL}^V + E_{LR}^V - E_{RR}^V - E_{RL}^V \\right) \\\\
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F_A &=& - \\frac{i}{4} f_B \\left( - E_{LL}^V + E_{LR}^V - E_{RR}^V + E_{RL}^V \\right) \\, ,\\end{aligned}$
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$` F_S = \frac{i}{4} \frac{m_B^2 f_B}{m_d + m_{d^\prime}} \left( E_{LL}^S + E_{LR}^S - E_{RR}^S - E_{RL}^S \right) \\ F_P = \frac{i}{4} \frac{m_B^2 f_B}{m_d + m_{d^\prime}} \left( - E_{LL}^S + E_{LR}^S - E_{RR}^S + E_{RL}^S \right) \\ F_V = - \frac{i}{4} f_B \left( E_{LL}^V + E_{LR}^V - E_{RR}^V - E_{RL}^V \right) \\ F_A = - \frac{i}{4} f_B \left( - E_{LL}^V + E_{LR}^V - E_{RR}^V + E_{RL}^V \right) \, ,`$
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where *f*<sub>*B*</sub> ≡ *f*<sub>*B*<sub>*d*, *s*</sub><sup>0</sup></sub> is the *B*<sub>*d*, *s*</sub><sup>0</sup> decay constant and *m*<sub>*d*, *d*<sup>′</sup></sub> are the masses of the quarks contained in the *B* meson, *B*<sub>*d*</sub><sup>0</sup> ≡ *b̄**d* and *B*<sub>*s*</sub><sup>0</sup> ≡ *b̄**s*. In the lepton flavor conserving case, *α* = *β*, the *F*<sub>*V*</sub> contribution vanishes. In this case, the results in are in agreement with previous computations .
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... | ... | @@ -973,29 +908,17 @@ where *f*<sub>*B*</sub> ≡ *f*<sub>*B*<sub>*d*, *s*</sub><sup>0</sup></su |
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The branching ratio for *B̄* → *X*<sub>*s*</sub>*γ*, with a cut *E*<sub>*γ*</sub> > 1.6 GeV in the *B̄* rest frame, can be obtained as
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$\\begin{aligned}
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\\BR & \\hspace\*{-0.1cm} \\left( \\bar B \\to X_s \\gamma \\right)_{E_\\gamma > 1.6 \\text{GeV}} = 10^{-4} \\bigg\[ a_{SM} + a_{77} \\left( |\\delta C_7^{(0)}|^2 + |\\delta C_7^{\\prime (0)}|^2 \\right) + a_{88} \\left( |\\delta C_8^{(0)}|^2 + |\\delta C_8^{\\prime (0)}|^2 \\right) \\nonumber \\\\
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& + \\text{Re} \\left( a_7 \\, \\delta C_7^{(0)} + a_8 \\, \\delta C_8^{(0)} + a_{78} \\left( \\delta C_7^{(0)} \\, \\delta C_8^{(0) \\ast} + \\delta C_7^{\\prime (0)} \\, \\delta C_8^{\\prime (0) \\, \\ast} \\right) \\right) \\bigg\] \\, , \\label{BRBXsGamma}\\end{aligned}$
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$` \BR \hspace\*{-0.1cm} \left( \bar B \to X_s \gamma \right)_{E_\gamma gt; 1.6 \text{GeV}} = 10^{-4} \bigg\[ a_{SM} + a_{77} \left( |\delta C_7^{(0)}|^2 + |\delta C_7^{\prime (0)}|^2 \right) + a_{88} \left( |\delta C_8^{(0)}|^2 + |\delta C_8^{\prime (0)}|^2 \right) \nonumber \\ + \text{Re} \left( a_7 \, \delta C_7^{(0)} + a_8 \, \delta C_8^{(0)} + a_{78} \left( \delta C_7^{(0)} \, \delta C_8^{(0) \ast} + \delta C_7^{\prime (0)} \, \delta C_8^{\prime (0) \, \ast} \right) \right) \bigg\] \, , `$
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where *a*<sub>*S**M*</sub> = 3.15 is the NNLO SM prediction , the other *a* coefficients in Eq. are found to be
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$\\begin{aligned}
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a_{77} &=& 4.743 \\nonumber \\\\
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a_{88} &=& 0.789 \\nonumber \\\\
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a_7 &=& -7.184 + 0.612 \\, i \\nonumber \\\\
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a_8 &=& -2.225 - 0.557 \\, i \\nonumber \\\\
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a_{78} &=& 2.454 - 0.884 \\, i \\end{aligned}$
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$` a_{77} = 4.743 \nonumber \\ a_{88} = 0.789 \nonumber \\ a_7 = -7.184 + 0.612 \, i \nonumber \\ a_8 = -2.225 - 0.557 \, i \nonumber \\ a_{78} = 2.454 - 0.884 \, i `$
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and we have defined *δ**C*<sub>*i*</sub><sup>(0)</sup> = *C*<sub>*i*</sub><sup>(0)</sup> − *C*<sub>*i*</sub><sup>(0) SM</sup>. Finally, the *C*<sub>*i*</sub><sup>(0)</sup> coefficients can be written in terms of *Q*<sub>1, 2</sub><sup>*L*, *R*</sup> in Eqs. and as
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$\\begin{aligned}
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C_7^{(0)} &=& n_{CQ} \\, Q_1^R \\\\
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C_7^{\\prime (0)} &=& n_{CQ} \\, Q_1^L \\\\
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C_8^{(0)} &=& n_{CQ} \\, Q_2^R \\\\
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C_8^{\\prime (0)} &=& n_{CQ} \\, Q_2^L \\label{eq:nCQprev}\\end{aligned}$
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$` C_7^{(0)} = n_{CQ} \, Q_1^R \\ C_7^{\prime (0)} = n_{CQ} \, Q_1^L \\ C_8^{(0)} = n_{CQ} \, Q_2^R \\ C_8^{\prime (0)} = n_{CQ} \, Q_2^L `$
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where $n_{CQ}^{-1} = - \\frac{G_F}{4 \\sqrt{2} \\pi^2} V_{tb}
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V_{ts}^\\ast$ and *V* is the Cabibbo-Kobayashi-Maskawa (CKM) matrix.
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where $`n_{CQ}^{-1} = - \frac{G_F}{4 \sqrt{2} \pi^2} V_{tb} V_{ts}^\ast`$ and *V* is the Cabibbo-Kobayashi-Maskawa (CKM) matrix.
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NameProcess = "bsGamma";
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NameObservables = {{BrBsGamma, 200, "BR(B->X_s gamma)"},
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... | ... | @@ -1051,35 +974,19 @@ V_{ts}^\\ast$ and *V* is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. |
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Our results for *B̄* → *X*<sub>*s*</sub>ℓ<sup>+</sup>ℓ<sup>−</sup> are based on , expanded with the addition of prime operators contributions . The branching ratios for the ℓ = *e* case can be written as
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$\\begin{aligned}
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10^7 \\, BR & \\left( \\bar B \\to X_s e^+ e^- \\right) = 2.3148 - 0.001658 Im (R_{10}) + 0.0005 Im(R_{10} R_8^\\ast + R_{10}^\\prime R_8^{\\prime \\, \\ast}) \\nonumber \\\\
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& + 0.0523 Im(R_7) + 0.02266 Im(R_7 R_8^\\ast + R_7^\\prime R_8^{\\prime \\, \\ast}) + 0.00496 Im(R_7 R_9^\\ast + R_7^\\prime R_9^{\\prime \\, \\ast}) \\nonumber \\\\
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& + 0.00518 Im(R_8) + 0.0261 Im(R_8 R_9^\\ast + R_8^\\prime R_9^{\\prime \\, \\ast}) - 0.00621 Im(R_9) - 0.5420 Re(R_{10}) \\nonumber \\\\
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& - 0.03340 Re(R_7) + 0.0153 Re(R_7 R_{10}^\\ast + R_7^\\prime R_{10}^{\\prime \\, \\ast}) + 0.0673 Re(R_7 R_8^\\ast + R_7^\\prime R_8^{\\prime \\, \\ast}) \\nonumber \\\\
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& - 0.86916 Re(R_7 R_9^\\ast + R_7^\\prime R_9^{\\prime \\, \\ast}) - 0.0135 Re(R_8) + 0.00185 Re(R_8 R_{10} + R_8^\\prime R_{10}^{\\prime \\, \\ast}) \\nonumber \\\\
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& - 0.09921 Re(R_8 R_9^\\ast + R_8^\\prime R_9^{\\prime \\, \\ast}) + 2.833 Re(R_9) - 0.10698 Re(R_9 R_{10}^\\ast + R_9^\\prime R_{10}^{\\prime \\, \\ast}) \\nonumber \\\\
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& + 11.0348 \\left( |R_{10}|^2 + |R_{10}^\\prime|^2 \\right) + 0.2804 \\left( |R_7|^2 + |R_7^\\prime|^2 \\right) \\nonumber \\\\
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& + 0.003763 \\left( |R_8|^2 + |R_8^\\prime|^2 \\right) + 1.527 \\left( |R_9|^2 + |R_9^\\prime|^2 \\right) \\, ,\\end{aligned}$
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$` 10^7 \, BR \left( \bar B \to X_s e^+ e^- \right) = 2.3148 - 0.001658 Im (R_{10}) + 0.0005 Im(R_{10} R_8^\ast + R_{10}^\prime R_8^{\prime \, \ast}) \nonumber \\ + 0.0523 Im(R_7) + 0.02266 Im(R_7 R_8^\ast + R_7^\prime R_8^{\prime \, \ast}) + 0.00496 Im(R_7 R_9^\ast + R_7^\prime R_9^{\prime \, \ast}) \nonumber \\ + 0.00518 Im(R_8) + 0.0261 Im(R_8 R_9^\ast + R_8^\prime R_9^{\prime \, \ast}) - 0.00621 Im(R_9) - 0.5420 Re(R_{10}) \nonumber \\ - 0.03340 Re(R_7) + 0.0153 Re(R_7 R_{10}^\ast + R_7^\prime R_{10}^{\prime \, \ast}) + 0.0673 Re(R_7 R_8^\ast + R_7^\prime R_8^{\prime \, \ast}) \nonumber \\ - 0.86916 Re(R_7 R_9^\ast + R_7^\prime R_9^{\prime \, \ast}) - 0.0135 Re(R_8) + 0.00185 Re(R_8 R_{10} + R_8^\prime R_{10}^{\prime \, \ast}) \nonumber \\ - 0.09921 Re(R_8 R_9^\ast + R_8^\prime R_9^{\prime \, \ast}) + 2.833 Re(R_9) - 0.10698 Re(R_9 R_{10}^\ast + R_9^\prime R_{10}^{\prime \, \ast}) \nonumber \\ + 11.0348 \left( |R_{10}|^2 + |R_{10}^\prime|^2 \right) + 0.2804 \left( |R_7|^2 + |R_7^\prime|^2 \right) \nonumber \\ + 0.003763 \left( |R_8|^2 + |R_8^\prime|^2 \right) + 1.527 \left( |R_9|^2 + |R_9^\prime|^2 \right) \, ,`$
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whereas for the ℓ = *μ* case one gets
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$\\begin{aligned}
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10^7 \\, BR & \\left( \\bar B \\to X_s \\mu^+ \\mu^- \\right) = 2.1774 - 0.001658 Im (R_{10}) + 0.0005 Im(R_{10} R_8^\\ast + R_{10}^\\prime R_8^{\\prime \\, \\ast}) \\nonumber \\\\
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& + 0.0534 Im(R_7) + 0.02266 Im(R_7 R_8^\\ast + R_7^\\prime R_8^{\\prime \\, \\ast}) + 0.00496 Im(R_7 R_9^\\ast + R_7^\\prime R_9^{\\prime \\, \\ast}) \\nonumber \\\\
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& + 0.00527 Im(R_8) + 0.0261 Im(R_8 R_9^\\ast + R_8^\\prime R_9^{\\prime \\, \\ast}) - 0.0115 Im(R_9) - 0.5420 Re(R_{10}) \\nonumber \\\\
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& + 0.0208 Re(R_7) + 0.0153 Re(R_7 R_{10}^\\ast + R_7^\\prime R_{10}^{\\prime \\, \\ast}) + 0.0648 Re(R_7 R_8^\\ast + R_7^\\prime R_8^{\\prime \\, \\ast}) \\nonumber \\\\
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& - 0.8545 Re(R_7 R_9^\\ast + R_7^\\prime R_9^{\\prime \\, \\ast}) - 0.00938 Re(R_8) + 0.00185 Re(R_8 R_{10} + R_8^\\prime R_{10}^{\\prime \\, \\ast}) \\nonumber \\\\
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& - 0.0981 Re(R_8 R_9^\\ast + R_8^\\prime R_9^{\\prime \\, \\ast}) + 2.6917 Re(R_9) - 0.10698 Re(R_9 R_{10}^\\ast + R_9^\\prime R_{10}^{\\prime \\, \\ast}) \\nonumber \\\\
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& + 10.7652 \\left( |R_{10}|^2 + |R_{10}^\\prime|^2 \\right) + 0.2880 \\left( |R_7|^2 + |R_7^\\prime|^2 \\right) \\nonumber \\\\
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& + 0.003763 \\left( |R_8|^2 + |R_8^\\prime|^2 \\right) + 1.527 \\left( |R_9|^2 + |R_9^\\prime|^2 \\right) \\, .\\end{aligned}$
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$` 10^7 \, BR \left( \bar B \to X_s \mu^+ \mu^- \right) = 2.1774 - 0.001658 Im (R_{10}) + 0.0005 Im(R_{10} R_8^\ast + R_{10}^\prime R_8^{\prime \, \ast}) \nonumber \\ + 0.0534 Im(R_7) + 0.02266 Im(R_7 R_8^\ast + R_7^\prime R_8^{\prime \, \ast}) + 0.00496 Im(R_7 R_9^\ast + R_7^\prime R_9^{\prime \, \ast}) \nonumber \\ + 0.00527 Im(R_8) + 0.0261 Im(R_8 R_9^\ast + R_8^\prime R_9^{\prime \, \ast}) - 0.0115 Im(R_9) - 0.5420 Re(R_{10}) \nonumber \\ + 0.0208 Re(R_7) + 0.0153 Re(R_7 R_{10}^\ast + R_7^\prime R_{10}^{\prime \, \ast}) + 0.0648 Re(R_7 R_8^\ast + R_7^\prime R_8^{\prime \, \ast}) \nonumber \\ - 0.8545 Re(R_7 R_9^\ast + R_7^\prime R_9^{\prime \, \ast}) - 0.00938 Re(R_8) + 0.00185 Re(R_8 R_{10} + R_8^\prime R_{10}^{\prime \, \ast}) \nonumber \\ - 0.0981 Re(R_8 R_9^\ast + R_8^\prime R_9^{\prime \, \ast}) + 2.6917 Re(R_9) - 0.10698 Re(R_9 R_{10}^\ast + R_9^\prime R_{10}^{\prime \, \ast}) \nonumber \\ + 10.7652 \left( |R_{10}|^2 + |R_{10}^\prime|^2 \right) + 0.2880 \left( |R_7|^2 + |R_7^\prime|^2 \right) \nonumber \\ + 0.003763 \left( |R_8|^2 + |R_8^\prime|^2 \right) + 1.527 \left( |R_9|^2 + |R_9^\prime|^2 \right) \, .`$
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Here we have defined the ratios of Wilson coefficients
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$R_{7,8} = \\frac{Q_{1,2}^R}{Q_{1,2}^{R, \\text{SM}}} \\qquad , \\qquad R_{7,8}^\\prime = \\frac{Q_{1,2}^L}{Q_{1,2}^{L, \\text{SM}}}$
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$`R_{7,8} = \frac{Q_{1,2}^R}{Q_{1,2}^{R, \text{SM}}} \qquad , \qquad R_{7,8}^\prime = \frac{Q_{1,2}^L}{Q_{1,2}^{L, \text{SM}}}`$
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as well as
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$R_{9,10} = \\frac{E_{LL}^V \\pm E_{LR}^V}{E_{LL}^{V, \\text{SM}} \\pm E_{LR}^{V, \\text{SM}}} \\qquad , \\qquad R_{9,10}^\\prime = \\frac{E_{RR}^V \\pm E_{RL}^V}{E_{RR}^{V, \\text{SM}} \\pm E_{RL}^{V, \\text{SM}}} \\, .$
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$`R_{9,10} = \frac{E_{LL}^V \pm E_{LR}^V}{E_{LL}^{V, \text{SM}} \pm E_{LR}^{V, \text{SM}}} \qquad , \qquad R_{9,10}^\prime = \frac{E_{RR}^V \pm E_{RL}^V}{E_{RR}^{V, \text{SM}} \pm E_{RL}^{V, \text{SM}}} \, .`$
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NameProcess = "BtoSLL";
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NameObservables = {{BrBtoSEE, 5000, "BR(B-> s e e)"},
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... | ... | @@ -1212,18 +1119,11 @@ $R_{9,10} = \\frac{E_{LL}^V \\pm E_{LR}^V}{E_{LL}^{V, \\text{SM}} \\pm E_{LR}^{V |
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Our results for *B*<sup>+</sup> → *K*<sup>+</sup>ℓ<sup>+</sup>ℓ<sup>−</sup> are based on the expressions given in . The branching ratio for *B*<sup>+</sup> → *K*<sup>+</sup>*μ*<sup>+</sup>*μ*<sup>−</sup> in the high-*q*<sup>2</sup> region, *q*<sup>2</sup> being the dilepton invariant mass squared, can be written as
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$\\label{eq:BKmumu}
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\\BR \\left( B^+ \\to K^+ \\mu^+ \\mu^- \\right)_{q^2 \\in \[14.18,22\] \\text{GeV}^2} \\simeq 1.11 + 0.22 \\left( C_7^{\\text{NP}} + C_7^\\prime \\right) + 0.27 \\left( C_9^{\\text{NP}} + C_9^\\prime \\right) - 0.27 \\left( C_{10}^{\\text{NP}} + C_{10}^\\prime \\right) \\, .$
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$` \BR \left( B^+ \to K^+ \mu^+ \mu^- \right)_{q^2 \in \[14.18,22\] \text{GeV}^2} \simeq 1.11 + 0.22 \left( C_7^{\text{NP}} + C_7^\prime \right) + 0.27 \left( C_9^{\text{NP}} + C_9^\prime \right) - 0.27 \left( C_{10}^{\text{NP}} + C_{10}^\prime \right) \, .`$
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The coefficients in Eq. can be related to the ones in our generic Lagrangian as
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$\\begin{aligned}
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C_7^{\\text{NP}} &=& n_{CQ} \\, \\left( Q_1^R - Q_1^{R,\\text{SM}} \\right) \\\\
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C_7^\\prime &=& n_{CQ} \\, Q_1^L \\\\
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C_9^{\\text{NP}} &=& n_{CQ} \\, \\left\[ \\left( E_{LL}^V + E_{LR}^V \\right) - \\left( E_{LL}^{V,\\text{SM}} + E_{LR}^{V,\\text{SM}} \\right) \\right\] \\\\
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C_9^\\prime &=& n_{CQ} \\, \\left( E_{RR}^V + E_{RL}^V \\right) \\\\
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C_{10}^{\\text{NP}} &=& n_{CQ} \\, \\left\[ \\left( E_{LL}^V - E_{LR}^V \\right) - \\left( E_{LL}^{V,\\text{SM}} - E_{LR}^{V,\\text{SM}} \\right) \\right\] \\\\
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C_{10}^\\prime &=& n_{CQ} \\, \\left( E_{RR}^V - E_{RL}^V \\right)\\end{aligned}$
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$` C_7^{\text{NP}} = n_{CQ} \, \left( Q_1^R - Q_1^{R,\text{SM}} \right) \\ C_7^\prime = n_{CQ} \, Q_1^L \\ C_9^{\text{NP}} = n_{CQ} \, \left[ \left( E_{LL}^V + E_{LR}^V \right) - \left( E_{LL}^{V,\text{SM}} + E_{LR}^{V,\text{SM}} \right) \right] \\ C_9^\prime = n_{CQ} \, \left( E_{RR}^V + E_{RL}^V \right) \\ C_{10}^{\text{NP}} = n_{CQ} \, \left[ \left( E_{LL}^V - E_{LR}^V \right) - \left( E_{LL}^{V,\text{SM}} - E_{LR}^{V,\text{SM}} \right) \right] \\ C_{10}^\prime = n_{CQ} \, \left( E_{RR}^V - E_{RL}^V \right)`$
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where the normalization factor *n*<sub>*C**Q*</sub> was already defined after Eq. .
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... | ... | @@ -1279,25 +1179,17 @@ where the normalization factor *n*<sub>*C**Q*</sub> was already defined after Eq |
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The branching ratio for *B̄* → *X*<sub>*q*</sub>*ν**ν̄*, with *q* = *d*, *s*, is given by
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$\\begin{aligned}
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\\BR \\left( \\bar B \\to X_q \\nu \\bar \\nu \\right) &=& \\frac{\\alpha^2}{4 \\pi^2 \\sin^4 \\theta_W} \\frac{|V_{tb} V_{tq}^\\ast|^2}{|V_{cb}|^2} \\frac{\\BR \\left( \\bar B \\to X_c e \\bar \\nu_e \\right) \\kappa(0)}{f(\\hat m_c) \\kappa(\\hat m_c)} \\label{BXnunu} \\\\
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&& \\times \\sum_f \\left\[ \\left( |c_L|^2 + |c_R|^2 \\right) f(\\hat m_q) - 4 \\RE \\left(c_L c_R^\\ast \\right) \\hat m_q \\tilde f(\\hat m_q) \\right\] \\nonumber \\, .\\end{aligned}$
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$` \BR \left( \bar B \to X_q \nu \bar \nu \right) = \frac{\alpha^2}{4 \pi^2 \sin^4 \theta_W} \frac{|V_{tb} V_{tq}^\ast|^2}{|V_{cb}|^2} \frac{\BR \left( \bar B \to X_c e \bar \nu_e \right) \kappa(0)}{f(\hat m_c) \kappa(\hat m_c)} \\ \times \sum_f \left[ \left( |c_L|^2 + |c_R|^2 \right) f(\hat m_q) - 4 \RE \left(c_L c_R^\ast \right) \hat m_q \tilde f(\hat m_q) \right] \nonumber \, .`$
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The sum runs over the three neutrinos and *m̂*<sub>*i*</sub> ≡ *m*<sub>*i*</sub>/*m*<sub>*b*</sub>. The functions *f*(*m̂*<sub>*c*</sub>) and *κ*(*m̂*<sub>*c*</sub>) represent the phase-space and the 1-loop QCD corrections, respectively. In case of *κ*(*m̂*<sub>*c*</sub>), one needs the numerical values *κ*(0)=0.83 and *κ*(*m̂*<sub>*c*</sub>)=0.88. The functions *f*(*x*) and *f̃*(*x*) take the form
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$\\begin{aligned}
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f(x) &=& 1 - 8x^2 + 8 x^6-x^8 -24 x^4 \\log x \\\\
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\\tilde f(x) &=& 1 + 9x^2 - 9x^4-x^6+12x^2(1+x^2)\\log x \\, .\\end{aligned}$
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$` f(x) = 1 - 8x^2 + 8 x^6-x^8 -24 x^4 \log x \\ \tilde f(x) = 1 + 9x^2 - 9x^4-x^6+12x^2(1+x^2)\log x \, .`$
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Finally, $\\BR \\left( \\bar B \\to X_c e \\bar \\nu_e \\right)_{\\text{exp}}
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= 0.101$ and the coefficients *c*<sub>*L*</sub> and *c*<sub>*R*</sub> are given by
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Finally, $`\BR \left( \bar B \to X_c e \bar \nu_e \right)_{\text{exp}} = 0.101`$ and the coefficients *c*<sub>*L*</sub> and *c*<sub>*R*</sub> are given by
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$\\begin{aligned}
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c_L &=& n^q_{BX \\nu\\nu} \\, F_{LL}^V \\\\
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c_R &=& n^q_{BX \\nu\\nu} \\, F_{RL}^V \\, ,\\end{aligned}$
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$` c_L = n^q_{BX \nu\nu} \, F_{LL}^V \\ c_R = n^q_{BX \nu\nu} \, F_{RL}^V \, ,`$
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where $\\left(n^q_{BX \\nu\\nu}\\right)^{-1} = \\frac{4 G_F}{\\sqrt{2}}
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\\frac{\\alpha}{2 \\pi \\sin^2 \\theta_W} V_{tb}^\\ast V_{tq}$ is the relative factor between our Wilson coefficients and the ones in .
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where $`\left(n^q_{BX \nu\nu}\right)^{-1} = \frac{4 G_F}{\sqrt{2}} \frac{\alpha}{2 \pi \sin^2 \theta_W} V_{tb}^\ast V_{tq}`$ is the relative factor between our Wilson coefficients and the ones in .
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NameProcess = "BtoQnunu";
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NameObservables = {{BrBtoSnunu, 7000, "BR(B->s nu nu)"},
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... | ... | @@ -1384,24 +1276,17 @@ where $\\left(n^q_{BX \\nu\\nu}\\right)^{-1} = \\frac{4 G_F}{\\sqrt{2}} |
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Following , the branching ratios for rare Kaon decays involving neutrinos in the final state can be written as
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$\\begin{aligned}
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\\BR \\left( K^+ \\to \\pi^+ \\nu \\bar \\nu \\right) &=& 2 r_1 \\, r_2 \\, r_{K^+} \\sum_f \\left\[ \\left( \\text{Im} \\lambda_t X_f \\right)^2 + \\left( \\text{Re} \\lambda_c X_{NL} + \\text{Re} \\lambda_t X_f \\right)^2 \\right\] \\\\
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\\BR \\left( K_L \\to \\pi^0 \\nu \\bar \\nu \\right) &=& 2 r_1 \\, r_{K_L} \\sum_f \\left( \\text{Im} \\lambda_t X_f \\right)^2 \\, ,\\end{aligned}$
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$` \BR \left( K^+ \to \pi^+ \nu \bar \nu \right) = 2 r_1 \, r_2 \, r_{K^+} \sum_f \left[ \left( \text{Im} \lambda_t X_f \right)^2 + \left( \text{Re} \lambda_c X_{NL} + \text{Re} \lambda_t X_f \right)^2 \right] \\ \BR \left( K_L \to \pi^0 \nu \bar \nu \right) = 2 r_1 \, r_{K_L} \sum_f \left( \text{Im} \lambda_t X_f \right)^2 \, ,`$
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where the sums are over the three neutrino species, *X*<sub>*N**L*</sub> = 9.78 ⋅ 10<sup>−4</sup> is the SM NLO charm correction , *λ*<sub>*t*</sub> = *V*<sub>*t**s*</sub><sup>\*</sup>*V*<sub>*t**d*</sub> and *λ*<sub>*c*</sub> = *V*<sub>*c**s*</sub><sup>\*</sup>*V*<sub>*c**d*</sub>, the coefficients *r*<sub>1</sub>, *r*<sub>2</sub>, *r*<sub>*K*<sup>+</sup></sub> and *r*<sub>*K*<sub>*L*</sub></sub> take the numerical values
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$\\begin{aligned}
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r_1 &=& 1.17 \\cdot 10^{-4} \\nonumber \\\\
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r_2 &=& 0.24 \\nonumber \\\\
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r_{K^+} &=& 0.901 \\nonumber \\\\
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r_{K_L} &=& 0.944\\end{aligned}$
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$` r_1 = 1.17 \cdot 10^{-4} \nonumber \\ r_2 = 0.24 \nonumber \\ r_{K^+} = 0.901 \nonumber \\ r_{K_L} = 0.944`$
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and *X*<sub>*f*</sub> contains the Wilson coefficients contributing to the processes, *F*<sub>*L**L*</sub><sup>*V*</sup> and *F*<sub>*R**L*</sub><sup>*V*</sup>, as
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*X*<sub>*f*</sub> = *n*<sub>*K**π**ν**ν*</sub> (*F*<sub>*L**L*</sub><sup>*V*</sup>+*F*<sub>*R**L*</sub><sup>*V*</sup>) .
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Here $n_{K \\pi \\nu\\nu}^{-1} = \\frac{4 G_F}{\\sqrt{2}} \\frac{\\alpha}{2
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\\pi \\sin^2 \\theta_W} V_{ts}^\\ast V_{td}$.
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Here $`n_{K \pi \nu\nu}^{-1} = \frac{4 G_F}{\sqrt{2}} \frac{\alpha}{2 \pi \sin^2 \theta_W} V_{ts}^\ast V_{td}`$.
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NameProcess = "KtoPInunu";
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NameObservables = {{BrKptoPipnunu, 8000, "BR(K^+ -> pi^+ nu nu)"},
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... | ... | @@ -1475,48 +1360,27 @@ Here $n_{K \\pi \\nu\\nu}^{-1} = \\frac{4 G_F}{\\sqrt{2}} \\frac{\\alpha}{2 |
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The *B*<sub>*q*</sub><sup>0</sup> − *B̄*<sub>*q*</sub><sup>0</sup> mass difference can be written as
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$\\label{eq:MBq}
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\\Delta M_{B_q} = \\frac{G_F^2 m_W^2}{6 \\pi^2} m_{B_q} \\eta_B f_{B_q}^2 \\hat B_{B_q} |V_{tq}^{\\text{eff}}|^2 |F_{tt}^q| \\, ,$
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$` \Delta M_{B_q} = \frac{G_F^2 m_W^2}{6 \pi^2} m_{B_q} \eta_B f_{B_q}^2 \hat B_{B_q} |V_{tq}^{\text{eff}}|^2 |F_{tt}^q| \, ,`$
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where *q* = *s*, *d*, *m*<sub>*B*<sub>*q*</sub></sub> and *f*<sub>*B*<sub>*q*</sub></sub> are the *B*<sub>*q*</sub><sup>0</sup> mass and decay constant, respectively, *η*<sub>*B*</sub> = 0.55 is a QCD factor , *B̂*<sub>*B*<sub>*q*</sub></sub> is a non-perturbative parameter (with values *B̂*<sub>*B*<sub>*d*</sub></sub> = 1.26 and *B̂*<sub>*B*<sub>*s*</sub></sub> = 1.33, obtained from recent lattice computations ) and |*V*<sub>*t**q*</sub><sup>eff</sup>|<sup>2</sup> = (*V*<sub>*t**b*</sub><sup>\*</sup>*V*<sub>*t**q*</sub>)<sup>2</sup>. *F*<sub>*t**t*</sub><sup>*q*</sup> is given by
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$\\begin{aligned}
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F_{tt}^q &=& S_0(x_t) + \\frac{1}{4 r} C_{\\text{new}}^{VLL} \\label{Ftt} \\\\
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&& + \\frac{1}{4 r} C_1^{VRR} + \\bar{P}_1^{LR} C_1^{LR} + \\bar{P}_2^{LR} C_2^{LR} \\nonumber \\\\
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&& + \\bar{P}_1^{SLL} \\left( C_1^{SLL} + C_1^{SRR} \\right) + \\bar{P}_2^{SLL} \\left( C_2^{SLL} + C_2^{SRR} \\right) \\nonumber\\end{aligned}$
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$` F_{tt}^q = S_0(x_t) + \frac{1}{4 r} C_{\text{new}}^{VLL} \\ + \frac{1}{4 r} C_1^{VRR} + \bar{P}_1^{LR} C_1^{LR} + \bar{P}_2^{LR} C_2^{LR} \nonumber \\ + \bar{P}_1^{SLL} \left( C_1^{SLL} + C_1^{SRR} \right) + \bar{P}_2^{SLL} \left( C_2^{SLL} + C_2^{SRR} \right) \nonumber`$
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where *r* = 0.985 , $x_t = \\frac{m_t^2}{m_W^2}$, with *m*<sub>*t*</sub> the top quark mass, the *P̄* coefficients take the numerical values
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where *r* = 0.985 , $`x_t = \frac{m_t^2}{m_W^2}`$, with *m*<sub>*t*</sub> the top quark mass, the *P̄* coefficients take the numerical values
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$\\begin{aligned}
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\\bar{P}_1^{LR} &=& -0.71 \\nonumber \\\\
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\\bar{P}_2^{LR} &=& 0.90 \\nonumber \\\\
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\\bar{P}_1^{SLL} &=& -0.37 \\nonumber \\\\
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\\bar{P}_2^{SLL} &=& -0.72\\end{aligned}$
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$` \bar{P}_1^{LR} = -0.71 \nonumber \\ \bar{P}_2^{LR} = 0.90 \nonumber \\ \bar{P}_1^{SLL} = -0.37 \nonumber \\ \bar{P}_2^{SLL} = -0.72`$
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and the function
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$\\label{eq:S0-inamilim}
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S_0(x_t) = \\frac{4 x_t - 11 x_t^2 + x_t^3}{4 (1-x_t)^2} - \\frac{3 x_t^3 \\log x_t}{2(1-x_t)^3}$
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$` S_0(x_t) = \frac{4 x_t - 11 x_t^2 + x_t^3}{4 (1-x_t)^2} - \frac{3 x_t^3 \log x_t}{2(1-x_t)^3}`$
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was introduced by Inami and Lim in and given, for example, in . Finally, the coefficients in Eq. are related to the *D*<sub>*X**Y*</sub><sup>*I*</sup> coefficients in Eq. as
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$\\begin{aligned}
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C_{\\text{new}}^{VLL} &=& n^q_\\Delta \\left( D_{LL}^V - D_{LL}^{V,\\text{SM}} \\right) \\\\
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C_1^{VRR} &=& n^q_\\Delta D_{RR}^V \\\\
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C_1^{LR} &=& n^q_\\Delta \\left( D_{LR}^V + D_{RL}^V \\right) \\\\
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C_2^{LR} &=& n^q_\\Delta \\left( D_{LR}^S + D_{RL}^S + \\delta_2^{LR} \\right) \\\\
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C_1^{SLL} &=& n^q_\\Delta \\left( D_{LL}^S + \\delta_1^{SLL} \\right) \\\\
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C_1^{SRR} &=& n^q_\\Delta \\left( D_{RR}^S + \\delta_1^{SRR} \\right) \\\\
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C_2^{SLL} &=& n^q_\\Delta D_{LL}^T \\\\
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C_2^{SRR} &=& n^q_\\Delta D_{RR}^T\\end{aligned}$
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$` C_{\text{new}}^{VLL} = n^q_\Delta \left( D_{LL}^V - D_{LL}^{V,\text{SM}} \right) \\ C_1^{VRR} = n^q_\Delta D_{RR}^V \\ C_1^{LR} = n^q_\Delta \left( D_{LR}^V + D_{RL}^V \right) \\ C_2^{LR} = n^q_\Delta \left( D_{LR}^S + D_{RL}^S + \delta_2^{LR} \right) \\ C_1^{SLL} = n^q_\Delta \left( D_{LL}^S + \delta_1^{SLL} \right) \\ C_1^{SRR} = n^q_\Delta \left( D_{RR}^S + \delta_1^{SRR} \right) \\ C_2^{SLL} = n^q_\Delta D_{LL}^T \\ C_2^{SRR} = n^q_\Delta D_{RR}^T`$
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where the factor $\\left(n^q_\\Delta\\right)^{-1} = \\frac{G_F^2 m_W^2}{16
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\\pi^2} |V_{tq}^{\\text{eff}}|^2$ normalizes our Wilson coefficients to the ones in . The corrections *δ*<sub>2</sub><sup>*L**R*</sup>, *δ*<sub>1</sub><sup>*S**L**L*</sup> and *δ*<sub>1</sub><sup>*S**R**R*</sup> are induced by double penguin diagrams mediated by scalar and pseudoscalar states . These 2-loop contributions may have a sizable impact in some models, and their inclusion is necessary in order to achieve a precise result for *Δ**M*<sub>*B*<sub>*q*</sub></sub>. They can be written as
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where the factor $`\left(n^q_\Delta\right)^{-1} = \frac{G_F^2 m_W^2}{16 \pi^2} |V_{tq}^{\text{eff}}|^2`$ normalizes our Wilson coefficients to the ones in . The corrections *δ*<sub>2</sub><sup>*L**R*</sup>, *δ*<sub>1</sub><sup>*S**L**L*</sup> and *δ*<sub>1</sub><sup>*S**R**R*</sup> are induced by double penguin diagrams mediated by scalar and pseudoscalar states . These 2-loop contributions may have a sizable impact in some models, and their inclusion is necessary in order to achieve a precise result for *Δ**M*<sub>*B*<sub>*q*</sub></sub>. They can be written as
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$\\begin{aligned}
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\\delta_2^{LR} &=& - \\frac{H_L^{S,P} \\left(H_R^{S,P}\\right)^\\ast}{m_{S,P}^2} \\label{eq:dp1} \\\\
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\\delta_1^{SLL} &=& - \\frac{\\left(H_L^{S,P}\\right)^2}{2 \\, m_{S,P}^2} \\label{eq:dp2} \\\\
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\\delta_1^{SRR} &=& - \\frac{\\left(H_L^{S,P}\\right)^2}{2 \\, m_{S,P}^2} \\label{eq:dp3}\\end{aligned}$
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$` \delta_2^{LR} = - \frac{H_L^{S,P} \left(H_R^{S,P}\right)^\ast}{m_{S,P}^2} \\ \delta_1^{SLL} = - \frac{\left(H_L^{S,P}\right)^2}{2 \, m_{S,P}^2} \\ \delta_1^{SRR} = - \frac{\left(H_L^{S,P}\right)^2}{2 \, m_{S,P}^2} `$
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where *H*<sub>*L*</sub><sup>*S*, *P*</sup> and *H*<sub>*R*</sub><sup>*S*, *P*</sup> are defined in Eq.. The double penguin corrections in Eqs.- are obtained by summing up over all scalar and pseudoscalar states in the model.
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... | ... | @@ -1642,36 +1506,25 @@ where *H*<sub>*L*</sub><sup>*S*, *P*</sup> and *H*<sub>*R*</sub><sup>*S*, *P |
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*Δ**M*<sub>*K*</sub> and *ε*<sub>*K*</sub>, the observables associated to *K*<sup>0</sup> − *K̄*<sup>0</sup> mixing, can be written as
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$\\begin{aligned}
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\\Delta M_{K} &=& 2 \\, \\text{Re} \\, \\langle \\bar K^0 | H_\\text{eff}^{\\Delta S = 2} | K^0 \\rangle \\label{deltaMK} \\\\
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\\varepsilon_K &=& \\frac{e^{i \\pi/4}}{\\sqrt{2} \\Delta M_{K}} \\, \\text{Im} \\, \\langle \\bar K^0 | H_\\text{eff}^{\\Delta S = 2} | K^0 \\rangle \\label{epsK} \\, .\\end{aligned}$
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$` \Delta M_{K} = 2 \, \text{Re} \, \langle \bar K^0 | H_\text{eff}^{\Delta S = 2} | K^0 \rangle \\ \varepsilon_K = \frac{e^{i \pi/4}}{\sqrt{2} \Delta M_{K}} \, \text{Im} \, \langle \bar K^0 | H_\text{eff}^{\Delta S = 2} | K^0 \rangle \, .`$
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The matrix element in Eqs. and is given by
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$\\begin{aligned}
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\\langle \\bar K^0 | H_\\text{eff}^{\\Delta S = 2} | K^0 \\rangle &=& f_V \\left( D_{LL}^V + D_{RR}^V \\right) + f_S \\left( D_{LL}^S + D_{RR}^S \\right) + f_T \\left( D_{LL}^T + D_{RR}^T \\right) \\nonumber \\\\
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&& + f_{LR}^1 \\left( D_{LR}^S + D_{RL}^S \\right) + f_{LR}^2 \\left( D_{LR}^V + D_{RL}^V \\right) \\, .\\end{aligned}$
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$` \langle \bar K^0 | H_\text{eff}^{\Delta S = 2} | K^0 \rangle = f_V \left( D_{LL}^V + D_{RR}^V \right) + f_S \left( D_{LL}^S + D_{RR}^S \right) + f_T \left( D_{LL}^T + D_{RR}^T \right) \nonumber \\ + f_{LR}^1 \left( D_{LR}^S + D_{RL}^S \right) + f_{LR}^2 \left( D_{LR}^V + D_{RL}^V \right) \, .`$
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The *f* coefficients are
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$\\begin{aligned}
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f_V &=& \\frac{1}{3} m_K f_K^2 B_1^{VLL}(\\mu) \\\\
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f_S &=& - \\frac{5}{24} \\left( \\frac{m_K}{m_s(\\mu) + m_d(\\mu)} \\right)^2 m_K f_K^2 B_1^{SLL}(\\mu) \\\\
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f_T &=& - \\frac{1}{2} \\left( \\frac{m_K}{m_s(\\mu) + m_d(\\mu)} \\right)^2 m_K f_K^2 B_2^{SLL}(\\mu) \\\\
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f_{LR}^1 &=& - \\frac{1}{6} \\left( \\frac{m_K}{m_s(\\mu) + m_d(\\mu)} \\right)^2 m_K f_K^2 B_1^{LR}(\\mu) \\\\
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f_{LR}^2 &=& \\frac{1}{4} \\left( \\frac{m_K}{m_s(\\mu) + m_d(\\mu)} \\right)^2 m_K f_K^2 B_2^{LR}(\\mu)\\end{aligned}$
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$` f_V = \frac{1}{3} m_K f_K^2 B_1^{VLL}(\mu) \\ f_S = - \frac{5}{24} \left( \frac{m_K}{m_s(\mu) + m_d(\mu)} \right)^2 m_K f_K^2 B_1^{SLL}(\mu) \\ f_T = - \frac{1}{2} \left( \frac{m_K}{m_s(\mu) + m_d(\mu)} \right)^2 m_K f_K^2 B_2^{SLL}(\mu) \\ f_{LR}^1 = - \frac{1}{6} \left( \frac{m_K}{m_s(\mu) + m_d(\mu)} \right)^2 m_K f_K^2 B_1^{LR}(\mu) \\ f_{LR}^2 = \frac{1}{4} \left( \frac{m_K}{m_s(\mu) + m_d(\mu)} \right)^2 m_K f_K^2 B_2^{LR}(\mu)`$
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where *μ* = 2 GeV is the energy scale at which the matrix element is computed and *f*<sub>*K*</sub> the Kaon decay constant. The values of the quark masses at *μ* = 2 GeV are given by *m*<sub>*d*</sub>(*μ*)=7 MeV and *m*<sub>*s*</sub>(*μ*)=125 MeV (see table 1 in ), whereas the *B*<sub>*i*</sub><sup>*X*</sup> coefficients have the following values at *μ* = 2 GeV : *B*<sub>1</sub><sup>*V**L**L*</sup>(*μ*)=0.61, *B*<sub>1</sub><sup>*S**L**L*</sup>(*μ*)=0.76, *B*<sub>2</sub><sup>*S**L**L*</sup>(*μ*)=0.51, *B*<sub>1</sub><sup>*L**R*</sup>(*μ*)=0.96 and *B*<sub>2</sub><sup>*L**R*</sup>(*μ*)=1.3.
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As in , we treat the SM contribution separately. We define *D*<sub>*L**L*</sub><sup>*V*</sup> = *D*<sub>*L**L*</sub><sup>*V*, *S**M*</sup> + *D*<sub>*L**L*</sub><sup>*V*, *B**S**M*</sup>. For *D*<sub>*L**L*</sub><sup>*V*, *B**S**M*</sup> one just subtracts the SM contributions to *D*<sub>*L**L*</sub><sup>*V*</sup>, whereas for *D*<sub>*L**L*</sub><sup>*V*, *S**M*</sup> one can use the results in , where the relevant QCD corrections are included,
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$D_{LL}^{V,SM} = \\frac{G_F^2 m_W^2}{4 \\pi^2} \\left\[ \\lambda_c^{\\ast \\, 2} \\eta_1 S_0(x_c) +
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\\lambda_t^{\\ast \\, 2} \\eta_2 S_0(x_t) + 2 \\lambda_c^\\ast \\lambda_t^\\ast \\eta_3 S_0(x_c,x_t) \\right\] \\, .$
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$`D_{LL}^{V,SM} = \frac{G_F^2 m_W^2}{4 \pi^2} \left[ \lambda_c^{\ast \, 2} \eta_1 S_0(x_c) + \lambda_t^{\ast \, 2} \eta_2 S_0(x_t) + 2 \lambda_c^\ast \lambda_t^\ast \eta_3 S_0(x_c,x_t) \right] \, .`$
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Here *x*<sub>*i*</sub> = *m*<sub>*i*</sub><sup>2</sup>/*m*<sub>*w*</sub><sup>2</sup>, *λ*<sub>*i*</sub> = *V*<sub>*i**s*</sub><sup>\*</sup>*V*<sub>*i**d*</sub> and *S*<sub>0</sub>(*x*) and *S*<sub>0</sub>(*x*, *y*) are the Inami-Lim functions . *S*<sub>0</sub>(*x*) was already defined in Eq. , whereas *S*<sub>0</sub>(*x*<sub>*c*</sub>, *x*<sub>*t*</sub>) is given by
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$\\label{eq:S0-inamilim2}
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S_0(x_c,x_t) = x_c \\left\[ \\log \\frac{x_t}{x_c} - \\frac{3 x_t}{4(1-x_t)} - \\frac{3 x_t^2 \\log x_t}{4 (1-x_t)^2} \\right\] \\, .$
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$` S_0(x_c,x_t) = x_c \left[ \log \frac{x_t}{x_c} - \frac{3 x_t}{4(1-x_t)} - \frac{3 x_t^2 \log x_t}{4 (1-x_t)^2} \right] \, .`$
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In the last expression we have kept only terms linear in *x*<sub>*c*</sub> ≪ 1. Finally, the *η*<sub>*i*</sub> coefficients comprise short distance QCD corrections. Their numerical values are *η*<sub>1, 2, 3</sub> = (1.44,0.57,0.47) [2].
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... | ... | @@ -1794,15 +1647,13 @@ In the last expression we have kept only terms linear in *x*<sub>*c*</sub> ≪ |
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Although *P* → ℓ*ν*, where *P* = *q**q*′ is a pseudoscalar meson, does not violate quark flavor, we have included it in the list of observables for practical reasons, as it can be computed with the same ingredients as the QFV observables. The decay width for the process *P* → ℓ<sub>*α*</sub>*ν* is given by
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$\\begin{aligned}
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\\Gamma \\left( P \\to \\ell_\\alpha \\nu \\right) &=& \\frac{|G_F f_P (m_P^2 - m_{\\ell_\\alpha}^2)|^2}{8 \\pi m_P^3} \\label{Plnu} \\\\
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&& \\times \\sum_\\nu \\left| V_{qq'} m_{\\ell_\\alpha} + \\frac{m_{\\ell_\\alpha}}{2 \\sqrt{2}} \\left( G_{LL}^V - G_{RL}^V \\right) + \\frac{m_P^2}{2 \\sqrt{2} (m_q + m_{q'})} \\left( G_{RR}^S - G_{LR}^S \\right) \\right|^2 \\, . \\nonumber\\end{aligned}$
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$` \Gamma \left( P \to \ell_\alpha \nu \right) = \frac{|G_F f_P (m_P^2 - m_{\ell_\alpha}^2)|^2}{8 \pi m_P^3} \\ \times \sum_\nu \left| V_{qq'} m_{\ell_\alpha} + \frac{m_{\ell_\alpha}}{2 \sqrt{2}} \left( G_{LL}^V - G_{RL}^V \right) + \frac{m_P^2}{2 \sqrt{2} (m_q + m_{q'})} \left( G_{RR}^S - G_{LR}^S \right) \right|^2 \, . \nonumber`$
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Here *f*<sub>*P*</sub> is the meson decay constant, *m*<sub>*q*</sub> and *m*<sub>*q*′</sub> are the masses of the quarks in the meson and the Wilson coefficients *G*<sub>*X**Y*</sub><sup>*I*</sup> are defined in Eq.. The sum in Eq. is over the three neutrinos (whose masses are neglected).
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Each *P* → ℓ<sub>*α*</sub>*ν* decay width is plagued by hadronic uncertainties. However, by taking the ratios
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$R_P = \\frac{\\Gamma \\left( P \\to e \\nu \\right)}{\\Gamma \\left( P \\to \\mu \\nu \\right)}$
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$`R_P = \frac{\Gamma \left( P \to e \nu \right)}{\Gamma \left( P \to \mu \nu \right)}`$
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the hadronic uncertainties cancel out to a good approximation, allowing for a precise theoretical determination. In case of *R*<sub>*K*</sub>, the SM prediction includes small electromagnetic corrections that account for internal bremsstrahlung and structure-dependent effects . This leads to an impressive theoretical uncertainty of *δ**R*<sub>*K*</sub>/*R*<sub>*K*</sub> ∼ 0.1%, making *R*<sub>*P*</sub> the perfect observable to search for lepton flavor universality violation .
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