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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 03:22, 25 October 2011
, 03:22, 25 October 2011→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 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>
\langle J M \epsilon|\mathbb{R}^{-1} V \mathbb{R}|
\langle J M \epsilon|\mathbb{R}^{-1} V \mathbb{R}|
\epsilon_\gamma ; J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle =
\epsilon_\gamma ; \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle =
\epsilon \epsilon_\gamma \epsilon_R \langle J M \epsilon|V|
\epsilon \epsilon_\gamma \epsilon_R \langle J M \epsilon|V|
\epsilon_\gamma ; J_R \lambda_R \epsilon_R ; \Omega_0 \rangle
\epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle
</math>
</math>
:<math>
:<math>
\langle J M \epsilon|V|
\langle J M \epsilon|V|
\epsilon_\gamma ; J_R \lambda_R \epsilon_R ; \Omega_0 \rangle
\epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle
= v\left(J,m,\epsilon;\epsilon_\gamma;J_R,\lambda_R,\epsilon_R \right)
= v_{J,m,\epsilon;\epsilon_\gamma; \lambda_R,\epsilon_R}
</math>
</math>
:<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; J_R \lambda_R \epsilon_R; \Omega_0\rangle
| UV | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
</math>
</math>
:::::::::<math> \times
:::::::::<math> \times
\langle J_R \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>
</math>
X,M_X,\epsilon_X;\epsilon_i,\epsilon_f
X,M_X,\epsilon_X;\epsilon_i,\epsilon_f
\end{array}}
\end{array}}
\langle \mathbf{q}_{b1} \lambda_{b_1} 0| U_X | J_X M_X \epsilon_X\rangle
\langle \mathbf{q}_{b1} \lambda_{b_1} 0| U_X | J_X M_X \epsilon_X\rangle \langle \mathbf{q}_\omega \lambda_\omega 0| U_{b_1} | 1 , \lambda_{b_1} \rangle
</math>
:::::::::<math>\times
\langle \mathbf{q}_\rho \lambda_\rho 0| U_\omega | 1 , \lambda_\omega \rangle \langle \mathbf{q}_\pi 0 0 | U_\rho | J_\rho , \lambda_\rho \rangle
</math>
:::::::::<math>\times
\langle J_X M_X \epsilon_X | V |
\langle J_X M_X \epsilon_X | V |
\epsilon_\gamma; J_R \lambda_R \epsilon_R; s,t \rangle
\epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
</math>
</math>
:::::::::<math>\times
:::::::::<math>\times
\langle \mathbf{q}_\omega \lambda_\omega 0| U_{b_1} | 1 , \lambda_{b_1} \rangle
\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
\langle \mathbf{q}_\rho \lambda_\rho 0| U_\omega | 1 , \lambda_\omega \rangle
</math>
\langle \mathbf{q}_\pi 0 0 | U_\rho | J_\rho , \lambda_\rho \rangle
Parity conservation requires that <math>\epsilon_X=\epsilon_\gamma \epsilon_R</math> and <math>\epsilon_R=\epsilon_i\epsilon_f</math>. The last two matrix elements in the expression above for <math>T_{fi}</math> not known ''a priori'', so we parameterized them into a pair of unknown functions ''v(s,t)'' and ''w(s,t)''.
:<math>\displaystyle
v^{X,M_X,\epsilon_X}_{\lambda_R,\epsilon_R}(s,t) =
\langle J_X M_X \epsilon_X | V |
\epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
</math>
</math>
<math>\times
:<math>\displaystyle
\langle J_R \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
w_{\lambda_R \epsilon_R; \epsilon_i}(s,t) =
\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>
</math>
=== Proton states and individual decay amplitudes ===
=== Proton states and individual decay amplitudes ===
In analogy to the reflectivity conservation relation shown above for ''V'' vertex, there is a similar relation for the ''W'' vertex: <math>\epsilon_R=\epsilon_i \epsilon_f</math>
In analogy to the reflectivity conservation relation shown above for ''V'' vertex, there is a similar relation for the ''W'' vertex: <math>\epsilon_R=\epsilon_i \epsilon_f</math>
Identification of <math>\epsilon_i</math> with <math>\epsilon_i'</math> and <math>\epsilon_f</math> with <math>\epsilon_f'</math> implies that only terms with <math>\epsilon_R=\epsilon_R'</math> survive in the sum over exchange quantum numbers.
Identification of <math>\epsilon_i</math> with <math>\epsilon_i'</math> and <math>\epsilon_f</math> with <math>\epsilon_f'</math> implies that only terms with <math>\epsilon_R=\epsilon_R'</math> survive in the sum over exchange quantum numbers. The quadratic sum expression above for the differential cross section invokes a double-sum over all of the internal quantum numbers that have been introduced, plus the spins of the initial and final nucleons. This sum is of the generic form
:<math>
\sum_{X,M_X,\cdots} \cdots \sum_{\epsilon_i\epsilon_f}
|w_{\lambda_R \epsilon_R; \epsilon_i \epsilon_f}|^2
</math>
Note that the measured cross section only depends on the summed
modulus squared of the ''w'' coefficients, independent of the couplings
to the individual nucleon helicity states. Because of this, the sum over nucleon 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.
=== 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>
:<math>\displaystyle F_0(q)=1</math>
:<math>F_1(q)=
\sqrt{\frac{2z}{z+1}}
</math>
:<math>F_2(q)=
\sqrt{\frac{13z^2}{(z-3)^2+9z}}
</math>
:<math>F_3(q)=
\sqrt{\frac{277z^3}{z(z-15)^2+9(2z-5)^2}}
</math>
===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>
T_{\pm 1}=\sum_{R,\lambda_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
T_{\pm 1}=\sum_{R,\lambda_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}_{b1}\lambda_{b1} 0
| U | (1\; \pm 1)_{\mathrm{lab}}; J_R \lambda_R \epsilon_R;s,t \rangle
| U | (1\; \pm 1)_{\mathrm{lab}}; \lambda_R \epsilon_R;s,t \rangle
</math>
</math>