Changes

== Decay of t-channel resonance X to b1π==

== Decay of t-channel resonance X to b1π==

We can apply the above recoupling relations to write down the amplitude at each vertex of the decay tree. These amplitudes are defined within a series of coordinate systems, each defined with respect to its ancestor in the decay chain.  The chain starts with resonance X decaying in the Gottfried-Jackson frame, into G-J direction <math>\Omega_{b1}</math>.  To find the decay frame of the <math>b_1</math>, we perform the rotation <math>(\phi_{b1},\theta_{b1},0)</math> (Euler convention z,y',z") then boost into the rest frame of the <math>b_1</math>.  To find the decay frame of the <math>\omega</math>, next we rotate by <math>(\phi_{\omega},\theta_{\omega},0)</math>, then boost into the <math>\omega</math> rest frame.  The three-body decay of the <math>\omega</math> can be treated without loss of generality as a decay into a charged di-pion and a neutral pion, provided that a complete sum over states of the free di-pion system is performed.  For notational simplicity, the di-pion is represented below by the symbol <math>\rho</math>, which should not be confused with the physical <math>\rho</math>(770) resonance.  The cascade of decay frames continues through the <math>\omega</math> decay by definition of the decay angles <math>(\phi_\rho,\theta_\rho,0)</math>, and finally <math>(\phi_\pi,\theta_\pi,0)</math>.  The selection rules for <math>\omega</math> decay require that the charged di-pion system be in an overall isovector state, which excludes even orbital angular momentum between the <math>\pi^+</math> and <math>\pi^-</math>.  This requires that the di-pion system is antisymmetric in decay angles, so it turns out not to matter whether one uses the <math>\pi^+</math> or <math>\pi^-</math> member of the pair to define the angles <math>(\phi_\pi,\theta_\pi,0)</math>.

We can apply the above recoupling relations to write down the amplitude at each vertex of the decay tree. These amplitudes are defined within a series of coordinate systems, each defined with respect to its ancestor in the decay chain.  The chain starts with resonance X decaying in the Gottfried-Jackson frame, into G-J direction <math>\Omega_{b1}</math>.  To find the decay frame of the <math>b_1</math>, we perform the rotation <math>(\phi_{b1},\theta_{b1},0)</math> (Euler convention z,y',z") then boost into the rest frame of the <math>b_1</math>.  To find the decay frame of the <math>\omega</math>, we rotate by <math>(\phi_{\omega},\theta_{\omega},0)</math>, then boost into the <math>\omega</math> rest frame.  The three-body decay of the <math>\omega</math> can be treated without loss of generality as a decay into a charged di-pion and a neutral pion, provided that a complete sum over states of the free di-pion system is performed.  For notational simplicity, the di-pion is represented below by the symbol <math>\rho</math>, which should not be confused with the physical <math>\rho</math>(770) resonance.  The cascade of decay frames continues through the <math>\omega</math> decay by definition of the decay angles <math>(\phi_\rho,\theta_\rho,0)</math>, and finally <math>(\phi_\pi,\theta_\pi,0)</math>.  The selection rules for <math>\omega</math> decay require that the charged di-pion system be in an overall isovector state, which excludes even orbital angular momentum between the <math>\pi^+</math> and <math>\pi^-</math>.  This requires that the di-pion system is antisymmetric in decay angles, so it turns out not to matter whether one uses the <math>\pi^+</math> or <math>\pi^-</math> member of the pair to define the angles <math>(\phi_\pi,\theta_\pi,0)</math>.

This procedure results in the decay particle's quantization axis being the same as the momentum direction in the parent's frame, thus forcing the ''m'' quantum number in the decay frame to be equal to its helicity ''&lambda;'' used in the parent's frame. Substitutions of known quantum numbers are made as necessary below: pions in the final state are given zero helicities  and spin of b₁ and &omega; are put in from the start.

This procedure results in the decay particle's quantization axis being the same as the momentum direction in the parent's frame, thus forcing the ''m'' quantum number in the decay frame to be equal to its helicity ''&lambda;'' used in the parent's frame. Substitutions of known quantum numbers are made as necessary below: pions in the final state are given zero helicities  and spin of b₁ and &omega; are put in from the start.

\end{array}\right)

\end{array}\right)

\right]

\right]

f_{J_\rho\,0}^{L_\rho}=

f_{L_\rho\,0}^{J_\rho}=

</math>

</math>

:::::::::<math>

:::::::::<math>