Changes

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 for having daughter 1 with momentum direction <math>\Omega=(\phi,\theta)</math> in the center of mass reference frame, and helicity <math>\lambda_1</math>, while daughter 2 has direction <math>-\Omega=(\phi+\pi,\pi-\theta)</math> and helicity <math>\lambda_2</math>.   

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 for having daughter 1 with momentum direction <math>\Omega=(\phi,\theta)</math> in the center of mass reference frame, and helicity <math>\lambda_1</math>, while daughter 2 has direction <math>-\Omega=(\phi+\pi,\pi-\theta)</math> and helicity <math>\lambda_2</math>.   

Let U be the decay operator from the initial state into the given 2-body final state.  Intermediate between the at-rest initial state of quantum numbers (qn) J,m and the final plane-wave state is a basis of outgoing waves describing the outgoing 2-body state in a basis of good J,m and helicities.  Insertion of the complete set of intermediate basis vectors, and summing over all intermediate J,m gives

Let U be the decay operator from the initial state into the given 2-body final state.  Intermediate between the at-rest initial state of quantum numbers (qn) J,M and the final plane-wave state is a basis of outgoing waves describing the outgoing 2-body state in a basis of good J,m and helicities.  Insertion of the complete set of intermediate basis vectors, and summing over all intermediate J,m gives

:<math>

:<math>

\langle \Omega \lambda_1 \lambda_2 | U | J M \rangle

\langle \Omega \lambda_1 \lambda_2 | U | J M \rangle