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

:<math>

:<math>

=\sum_{L,S}

=\sum_{L,S}

\left[ \sqrt{\frac{2J+1}{4\pi}} D_{m \lambda}^{J *}(\Omega) \right]

\left[ \sqrt{\frac{2J+1}{4\pi}} D_{M \lambda}^{J *}(\Omega) \right]

\left[ \sqrt{\frac{2L+1}{2J+1}}   

\left[ \sqrt{\frac{2L+1}{2J+1}}   

\left(\begin{array}{cc|c}

\left(\begin{array}{cc|c}

Acting on a state of good ''J,m'', the reflectivity operator has a particularly simple effect.

Acting on a state of good ''J,m'', the reflectivity operator has a particularly simple effect.

:<math>\mathbb{R}| J M \rangle = P(-1)^{J-m} | J \; -M \rangle </math>

:<math>\mathbb{R}| J M \rangle = P(-1)^{J-M} | J \; -M \rangle </math>

where P is the intrinsic parity of the system.

where P is the intrinsic parity of the system.

The eigenstates of the reflectivity operator are formed out of states of good ''J,M'' as follows.

The eigenstates of the reflectivity operator are formed out of states of good ''J,M'' as follows.

:<math>

:<math>

T_{(f)(i)} =  

T_{(f)(i)} =  

T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \epsilon_f)

T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b_1} \mathbf{p}_f \epsilon_f)

(\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}=

(\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}=

</math>

</math>

:::<math>

:::<math>

=\sum_{R,\lambda_R,\epsilon_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}

=\sum_{R,\lambda_R,\epsilon_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}

\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0

\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b_1}\lambda_{b_1} 0

| UV | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle

| UV | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle

</math>

</math>

</math>

</math>

Note that the measured cross section only depends on the summed

Note that the measured cross section only depends on the summed

modulus squared of the ''w'' coefficients, and not on their

modulus squared of the ''w'' coefficients, independent of the couplings

individual values or phases. Because of this, the sum over nucleon

to the individual nucleon helicity states. Because of this, the sum over nucleon helicities can be dropped, and the ''w'' factors absorbed into

helicities can be dropped, and the ''w'' factors absorbed into

the ''v'' coefficients.  Thus the final expression for the differential cross section contains no reference to the quantum numbers <math>\epsilon_i, \epsilon_f</math>, nor to any ''w'' coefficients.  It is important to recognize that this simplification of the cross section is only possible in the case of an unpolarized target and no recoil polarimetry.

the ''v'' coefficients by multiplying the ''v'' defined above with

the square root of the above sum.

=== Mass dependence ===

=== Mass dependence ===

q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}

q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}

</math>

</math>

The functions <math>F_L(q)</math> are the angular momentum barrier factors that are given in the literature.  

The functions <math>F_L(q)</math> are the angular momentum barrier factors that are given in the literature. The first few are listed below with <math>z=[q/(197\mathrm{MeV/c})]^2</math>

 

: ''The first few of these barrier factors should be listed explicitly here.''

===Describing s and t dependence===

===Describing s and t dependence===

The T Matrix, written in the photon reflectivity basis can be expanded in the photon's lab frame helicity basis. Temporarily omitting indices not pertaining to the photon:

The T Matrix, written in the photon reflectivity basis can be expanded in the photon's lab frame helicity basis. Temporarily omitting indices not pertaining to the photon:

:<math>

:<math>

T_{\epsilon_\gamma} = \frac{\sqrt{-\epsilon_\gamma}}{2}  

T_{\epsilon_\gamma} = \sqrt{\frac{-\epsilon_\gamma}{2}}  

\left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right)

\left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right)

</math>

</math>