Amplitudes for the Exotic b1π Decay
NEW
General Relations
Angular Distribution of Two-Body Decay
Let's begin with a general amplitude for the two-body decay of a state with angular momentum quantum numbers J,m. Specifically, we want to know the amplitude of this state having daughter 1 with trajectory Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega=(\phi,\theta)} . We can also describe the angular momentum between the daughters as being L and spin sum as s. Alternatively, we will label the daughters as having helicities of and or direction of decay (specified by daughter 1) of
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simple insertion of complete sets of states for recoupling |
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Substitution of each bra-ket with their respective formulae. Note that in the event of one daughter being spin-less, the second Clebsch-Gordan coefficient is 1 |
![{\displaystyle =\sum _{L,S}\left[{\sqrt {\frac {2J+1}{4\pi }}}D_{m\lambda }^{J*}(\Omega ,0)\right]\left[{\sqrt {\frac {2L+1}{2J+1}}}\left({\begin{array}{cc|c}L&S&J\\0&\lambda &\lambda \end{array}}\right)\left({\begin{array}{cc|c}S_{1}&S_{2}&S\\\lambda _{1}&-\lambda _{2}&\lambda \end{array}}\right)\right]a_{LS}^{J}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a93b86e837aee85c12249ab0f791a8c2693e53e6)
Isospin Projections
One must also take into account the various ways isospin of daughters can add up to the isospin quantum numbers of the parent, requiring a term:
where a=1 and b=2, referring to the daughter number. Because an even-symmetric angular wave function (i.e. L=0,2...) imply that 180 degree rotation is equivalent to reversal of daughter identities, a,b becoming b,a on must write down the symmetrized expression:
![{\displaystyle C={\frac {1}{\sqrt {2}}}\left[C^{a,b}+(-1)^{L}C^{b,a}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cc87dbd6894ff3b9c01efcb5dd54789a7432d96)
Application
![{\displaystyle \langle \Omega _{X}0\lambda _{b_{1}}|U|J_{X}m_{X}\rangle =\sum _{L_{X}}\left[{\sqrt {\frac {2J_{X}+1}{4\pi }}}D_{m_{X}\lambda _{b_{1}}}^{J_{X}*}(\Omega _{X},0)\right]\left[{\sqrt {\frac {2L_{X}+1}{2J_{X}+1}}}\left({\begin{array}{cc|c}L_{X}&1&J\\0&\lambda _{b_{1}}&\lambda _{b_{1}}\end{array}}\right)\right]a_{L_{X}}^{J_{X}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7db494813340ce0358c6052e190681cf7804bd5)
![{\displaystyle \langle \Omega _{b_{1}}0\lambda _{\omega }|U|1,m_{b_{1}}=\lambda _{b_{1}}\rangle =\sum _{L_{b_{1}}}\left[{\sqrt {\frac {2J_{b_{1}}+1}{4\pi }}}D_{m_{b_{1}}=\lambda _{b_{1}}\lambda _{\omega }}^{1*}(\Omega _{b_{1}},0)\right]\left[{\sqrt {\frac {2L_{b_{1}}+1}{2J_{b_{1}}+1}}}\left({\begin{array}{cc|c}L_{b_{1}}&1&1\\0&\lambda _{\omega }&\lambda _{\omega }\end{array}}\right)\right]b_{L_{b_{1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76ae6ee2605bd9d0ec6227793c7f5812236e4f6b)
![{\displaystyle \langle \Omega _{\omega }0\lambda _{\rho }|U|1,m_{\omega }=\lambda _{\omega }\rangle =\sum _{L_{\omega }J_{\rho }}\left[{\sqrt {\frac {2J_{\omega }+1}{4\pi }}}D_{m_{\omega }=\lambda _{\omega }\lambda _{\rho }}^{1*}(\Omega _{\omega },0)\right]\left[{\sqrt {\frac {2L_{\omega }+1}{2J_{\omega }+1}}}\left({\begin{array}{cc|c}L_{\omega }&1&1\\0&\lambda _{\rho }&\lambda _{\rho }\end{array}}\right)\right]c_{L_{\omega }J_{\rho }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac8279197794a40ff72f750ff4b617e7cf16a8e)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \Omega_\rho 0 \lambda_\rho | U | J_\rho , m_\rho=\lambda_\rho \rangle =\sum_{L_\rho} \left[ \sqrt{\frac{2J_\rho+1}{4\pi}} D_{m_\rho 0}^{J_\rho *}(\Omega_\rho,0) \right] \left[ \sqrt{\frac{2L_\rho+1}{2J_\rho+1}} \left(\begin{array}{cc|c} L_\rho & 0 & J_\rho \\ 0 & 0 & 0 \end{array}\right) \right] d_{L_\rho} =\sum_{L_\rho} \sqrt{\frac{2L_\rho+1}{2J_\rho+1}} Y_{m_\rho}^{J_\rho *}(\Omega_\rho) d_{L_\rho} }
OLD
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{}^{J_X L_X P_X}= } |
defining an amplitude... |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum\limits_{m_X=-L_X}^{L_X} \sum\limits_{m_{b1}=-J_{b_1}}^{J_{b_1}} \sum\limits_{m_\omega=-J_\omega}^{J_\omega} D_{m_X m_{b_1}}^{L_X *}(\theta_X,\phi_X,0) D_{m_{b_1} m_\omega}^{J_{b_1}*}(\theta_{b_1},\phi_{b_1},0) } |
angular distributions two-body X and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_1 (J_{b_1}^{PC}=1^{+-})} decays |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ P_X(-)^{J_X+1+\epsilon} e^{2i\alpha} \left(\begin{array}{cc|c} J_{b_1} & L_X & J_X \\ m_{b_1} & m_X & -1 \end{array}\right) + \left(\begin{array}{cc|c} J_{b_1} & L_X & J_X \\ m_{b_1} & m_X & +1 \end{array}\right) \right] } |
resonance helicity sum: ε=0 (1) for x (y) polarization; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_X} is the parity of the resonance |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{1+(-)^\epsilon \eta}{4}\right) } |
polarization term: η is the polarization fraction |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^{L_X} q^{L_{b_1}} } |
k, q are breakup momenta for the resonance and isobar, respectively |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc|c} I_{b_1} & I_\pi & I_X \\ I_{zb_1^+} & I_{z\pi^-} & I_{zb_1^+}+I_{z\pi^-} \end{array}\right) } |
Clebsch-Gordan coefficients for isospin sum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_1 \oplus \pi^- \rightarrow X} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum\limits_{L_{b_1}=0}^{2} \sum\limits_{m_{L_{b_1}}=-L_{b_1}}^{L_{b_1}} \sum\limits_{L_{\pi^+\pi^-},L_\omega=1,3} \sum\limits_{m_{\pi^+\pi^-}=-L_{\pi^+\pi^-}}^{L_{\pi^+\pi^-}} u^{L_\omega} v^{L_{\pi^+\pi^-}} } | |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{m_\omega m_{\pi^+\pi^-}}^{J_\omega *}(\theta_\omega,\phi_\omega,0) Y_{m_{\pi^+\pi^-}}^{L_{\pi^+\pi^-}}(\theta_\rho,\phi_\rho) } |
two-stage Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega (J_\omega^{PC}=1^{--})} breakup angular distributions, currently modeled as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\omega\rightarrow\pi^0+\rho}=0; L_{\rho\rightarrow\pi^++\pi^-}=1=L_{\pi^+\pi^-}} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc|c} J_\omega & L_{b_1} & J_{b_1} \\ m_\omega & m_{L_{b_1}} & m_{b_1} \end{array}\right) \left(\begin{array}{cc|c} L_\omega & L_{\pi^+\pi^-} & J_\omega \\ 0 & m_{\pi^+\pi^-} & m_\omega \end{array}\right) } |
angular momentum sum Clebsch-Gordan coefficients for b1 and ω decays. |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc|c} I_\pi & 1 & 0 \\ I_{\pi^0} & 0 & 0 \end{array}\right) \left(\begin{array}{cc|c} I_{\pi} & I_{\pi} & 1 \\ I_{z\pi^+} & I_{z\pi^-} & 0 \end{array}\right) } |
Clebsch-Gordan coefficients for isospin sums: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^0 \oplus (\pi^+ \oplus \pi^-) \rightarrow \omega} |