Amplitudes for the Exotic b1π Decay - Revision history

Senderovich: /* Angular Distribution of Two-Body Decay */

2011-10-25T03:22:27Z

Angular Distribution of Two-Body Decay

2011-10-02T02:29:04Z

Assembly of the full amplitude

2011-09-29T01:27:18Z

Reflectivity

2011-09-21T18:58:06Z

Summing over photon polarization

2011-09-09T15:36:59Z

Angular Distribution of Two-Body Decay

2011-08-22T20:10:28Z

Mass dependence

2011-08-20T00:25:35Z

Proton states and individual decay amplitudes

2011-08-20T00:22:47Z

Proton states and individual decay amplitudes

2011-08-20T00:19:10Z

Assembly of the full amplitude

2011-08-20T00:11:17Z

Proton states and individual decay amplitudes

@@ Line 5: / Line 5: @@
 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>
 \langle \Omega \lambda_1 \lambda_2 | U | J M \rangle

@@ Line 207: / Line 207: @@
 :<math>
 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)}=
 </math>
 :::<math>
 =\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
 </math>

@@ Line 64: / Line 64: @@
 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.
 The eigenstates of the reflectivity operator are formed out of states of good ''J,M'' as follows.

@@ Line 339: / Line 339: @@
 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>
-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)
 </math>

@@ Line 26: / Line 26: @@
 :<math>
 =\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(\begin{array}{cc|c}

@@ Line 315: / Line 315: @@
 q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}
 </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>
-: ''The first few of these barrier factors should be listed explicitly here.''
+:<math>F_1(q)=
 ===Describing s and t dependence===

@@ Line 282: / Line 282: @@
 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.
+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 ===

@@ Line 276: / Line 276: @@
 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_R\epsilon_i\epsilon_f}
+\sum_{X,M_X,\cdots} \cdots \sum_{\epsilon_i\epsilon_f}
 === Mass dependence ===