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Line 1: − Let's begin with the amplitude for decay of a state X with some <math>J_X,M_X</math> quantum numbers:+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − \langle + − + − + − =+ − + − \Omega_X 0 \lambda_{b_1}|J_X m_X L_X s_{b_1} \rangle \langle J_X m_X L_X s_{b_1} | U_X | J_X m_X+ − + + + + + +
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 03:49, 28 July 2011
, 03:49, 28 July 2011no edit summary
=== 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 having daughter 1 with trajectory <math>\Omega=(\phi,\theta)</math>.
We can also describe the angular momentum between the daughters as being ''L'' and spin sum as ''s''. Alternatively, we will label the daughters as having helicities of <math>\lambda_1</math> and <math>\lambda_2</math> or direction of decay (specified by daughter 1) of
<table>
<tr>
<td><math>
\langle \Omega \lambda_1 \lambda_2 | U | J m \rangle
=
\sum_{L,S}
\langle \Omega \lambda_1 \lambda_2
| J m \lambda_1 \lambda_2 \rangle \langle J m \lambda_1 \lambda_2 |
J m L S \rangle \langle J m L S |
U | J m \rangle
</math></td>
<td>
simple insertion of complete sets of states for recoupling
</td>
</tr>
<tr>
<td><math>
=\sum_{L,S}
\left[ \sqrt{\frac{2J+1}{4\pi}} D_{m \lambda}^{J *}(\Omega,0) \right]
\left[ \sqrt{\frac{2L+1}{2J+1}}
\left(\begin{array}{cc|c}
L & S & J \\
0 & \lambda & \lambda
\end{array}\right)
\left(\begin{array}{cc|c}
S_1 & S_2 & S \\
\lambda_1 & -\lambda_2 & \lambda
\end{array}\right)
\right]
a_{L S}^{J}
</math></td>
<td>
Substitution of each bra-ket with their respective formulae.
<math>\lambda=\lambda_1-\lambda_2</math>
Note that in the event of one daughter being spin-less, the second
Clebsch-Gordan coefficient is 1
</td>
</tr>
</table>
=== Isospin Projections ===
One must also take into account the various ways isospin of daughters can add up to the isospin quantum numbers of the parent, requiring a term:
<math>
<math>
C^{a,b} =
\Omega_X 0 \lambda_{b_1} | U_X | J_X m_X
\left(\begin{array}{cc|c}
\rangle
I^a & I^b & I \\
I_z^a & I_z^b & I_z^a+I_z^b
\langle
\end{array}\right)
</math>
\rangle
where ''a=1'' and ''b=2'', referring to the daughter number. Because an even-symmetric angular wave function (i.e. L=0,2...) imply that 180 degree rotation is equivalent to reversal of daughter identities, ''a,b'' becoming ''b,a'' on must write down the symmetrized expression:
<math>
C=\frac{1}{\sqrt{2}} \left[ C^{a,b} + (-1)^L C^{b,a} \right]
</math>
</math>