Amplitudes for the Exotic b1π Decay

General Relations

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 $\Omega =(\phi ,\theta )$ in the center of mass reference frame, and helicity $\lambda _{1}$, while daughter 2 has direction $-\Omega =(\phi +\pi ,\pi -\theta )$ and helicity $\lambda _{2}$.

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 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

$\langle \Omega \lambda _{1}\lambda _{2}|U|Jm\rangle =\langle \Omega \lambda _{1}\lambda _{2}|Jm\lambda _{1}\lambda _{2}\rangle \langle Jm\lambda _{1}\lambda _{2}|U|Jm\rangle$

Instead of specifying the final-state particles' spin state via their helicities, we can first couple their spins together independent of their momentum direction, to obtain total spin S, then couple S to their relative orbital angular momentum L to obtain their total angular momentum J.

$\langle \Omega \lambda _{1}\lambda _{2}|U|Jm\rangle =\sum _{L,S}\langle \Omega \lambda _{1}\lambda _{2}|Jm\lambda _{1}\lambda _{2}\rangle \langle Jm\lambda _{1}\lambda _{2}|JmLS\rangle \langle JmLS|U|Jm\rangle$

insertion of the complete LS basis set

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\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} }

Substitution of each bra-ket with their respective formulae. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda=\lambda_1-\lambda_2} Note that in the event of one daughter being spin-less, the second Clebsch-Gordan coefficient is 1

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^{a,b} = \left(\begin{array}{cc|c} I^a & I^b & I \\ I_z^a & I_z^b & I_z^a+I_z^b \end{array}\right) }

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) one must write down the symmetrized expression:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(L)=\frac{1}{\sqrt{2}} \left[ C^{a,b} + (-1)^L C^{b,a} \right] }

Application

Production

Photon-Reggeon-Resonance vertex

Consider the production of the resonance from the photon and reggeon in the reflectivity basis, the eigenstates of the reflectivity operator. (This operator is a combination of parity and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} rotation about the normal to the production plane (usually y axis.)
$\mathbb {R} |Jm\rangle =P(-1)^{J-m}|J\;-m\rangle$

The eigenstates of the reflectivity operator are formed as follows:
$|Jm\epsilon \rangle =|Jm\rangle +\epsilon P(-1)^{J-m}|J\;-m\rangle$

such that

$\mathbb {R} |Jm\epsilon \rangle =\epsilon (-1)^{2J}|Jm\epsilon \rangle$

The photon linear polarization states turn out to be eigenstates of reflectivity as well:
Let x (y) polarization states be denoted with - (+)

$|\mp \rangle ={\sqrt {\frac {\pm 1}{2}}}\left(|1-1\rangle \mp |1+1\rangle \right)$

$\mathbb {R} |\mp \rangle =\mp 1|\mp \rangle$

Since the production Hamiltonian should commute with reflectivity: $V=\mathbb {R} ^{-1}V\mathbb {R}$

$\langle Jm\epsilon |\mathbb {R} ^{-1}V\mathbb {R} |\mp ;J_{R}\lambda _{R}\epsilon _{R};t,s;\Omega _{0}\rangle =\epsilon (\mp 1)\epsilon _{R}\langle Jm\epsilon |V|\mp ;J_{R}\lambda _{R}\epsilon _{R};t,s;\Omega _{0}\rangle$

Acting with the reflectivity operator on initial and final state brings out the reflectivity eigenvalues of the resonance, photon and reggeon. This result leads to a constraint:
$\epsilon =\mp \epsilon _{R}$

Proton-Reggeon vertex

The amplitude of target proton's emission of an exchange particle, a reggeon, in particular direction and helicity projections can be written as:

$\langle \Omega _{R};J_{R}\lambda _{R}\epsilon _{R};J_{P}\lambda _{p}|W|J_{T}m_{T}\rangle =\langle \Omega _{R};J_{R}\lambda _{R}\;\mp \epsilon ;\textstyle {\frac {1}{2}}\;\lambda _{p}|\textstyle {\frac {1}{2}}\;m_{T}\lambda _{R}\lambda _{p}\rangle \langle \textstyle {\frac {1}{2}}\;m_{T}\lambda _{R}\lambda _{p}|W|\textstyle {\frac {1}{2}}\;m_{T}\rangle$	transition amplitude for $p\rightarrow R+p'$ in the direction $\Omega _{R}$ w.r.t. the coordinate system defined in the resonance RF.
$={\frac {1}{\sqrt {2\pi }}}\left[D_{m_{T}(\lambda _{R}-\lambda _{p})}^{{\frac {1}{2}}*}(\Omega _{R},0)\;w_{\lambda _{R}\;\lambda _{p}}\mp \epsilon P_{R}(-1)^{J_{R}-\lambda _{R}}D_{m_{T}(-\lambda _{R}-\lambda _{p})}^{{\frac {1}{2}}*}(\Omega _{R},0)\;w_{\lambda _{R}\;-\lambda _{p}}\right]$	follows from relations given above

Decay

$\langle \Omega _{X}0\lambda _{b_{1}}|U_{X}|J_{X}m_{X}\rangle =\sum _{L_{X}}\left[{\sqrt {\frac {2J_{X}+1}{4\pi }}}D_{m_{X}\lambda _{b_{1}}}^{J_{X}*}(\Omega _{X},0)\right]\left[{\sqrt {\frac {2L_{X}+1}{2J_{X}+1}}}\left({\begin{array}{cc|c}L_{X}&1&J_{X}\\0&\lambda _{b_{1}}&\lambda _{b_{1}}\end{array}}\right)\right]a_{L_{X}}^{J_{X}}$

$\langle \Omega _{b_{1}}0\lambda _{\omega }|U_{b_{1}}|1,m_{b_{1}}=\lambda _{b_{1}}\rangle =\sum _{L_{b_{1}}}\left[{\sqrt {\frac {2J_{b_{1}}+1}{4\pi }}}D_{m_{b_{1}}=\lambda _{b_{1}}\lambda _{\omega }}^{1*}(\Omega _{b_{1}},0)\right]\left[{\sqrt {\frac {2L_{b_{1}}+1}{2J_{b_{1}}+1}}}\left({\begin{array}{cc|c}L_{b_{1}}&1&1\\0&\lambda _{\omega }&\lambda _{\omega }\end{array}}\right)\right]b_{L_{b_{1}}}$

$\langle \Omega _{\omega }0\lambda _{\rho }|U_{\omega }|1,m_{\omega }=\lambda _{\omega }\rangle =\sum _{L_{\omega }J_{\rho }}\left[{\sqrt {\frac {2J_{\omega }+1}{4\pi }}}D_{m_{\omega }=\lambda _{\omega }\lambda _{\rho }}^{1*}(\Omega _{\omega },0)\right]\left[{\sqrt {\frac {2L_{\omega }+1}{2J_{\omega }+1}}}\left({\begin{array}{cc|c}L_{\omega }&1&1\\0&\lambda _{\rho }&\lambda _{\rho }\end{array}}\right)\right]c_{L_{\omega }J_{\rho }}$

$\langle \Omega _{\rho }0\lambda _{\rho }|U_{\rho }|J_{\rho },m_{\rho }=\lambda _{\rho }\rangle =\sum _{L_{\rho }}\left[{\sqrt {\frac {2J_{\rho }+1}{4\pi }}}D_{m_{\rho }0}^{J_{\rho }*}(\Omega _{\rho },0)\right]\left[{\sqrt {\frac {2L_{\rho }+1}{2J_{\rho }+1}}}\left({\begin{array}{cc|c}L_{\rho }&0&J_{\rho }\\0&0&0\end{array}}\right)\right]d_{L_{\rho }}=\sum _{L_{\rho }}{\sqrt {\frac {2L_{\rho }+1}{4\pi }}}Y_{m_{\rho }}^{J_{\rho }*}(\Omega _{\rho })d_{L_{\rho }}$

$A^{J_{X}}=\sum _{\lambda _{b_{1}},\lambda _{\omega },\lambda _{\rho }}\langle \Omega _{X}0\lambda _{b_{1}}|U_{X}|J_{X}m_{X}\rangle C_{X}(L_{X})k^{L_{X}}\langle \Omega _{b_{1}}0\lambda _{\omega }|U_{b_{1}}|1,m_{b_{1}}=\lambda _{b_{1}}\rangle C_{b_{1}}(L_{b_{1}})q^{L_{b_{1}}}\langle \Omega _{\omega }0\lambda _{\rho }|U_{\omega }|1,m_{\omega }=\lambda _{\omega }\rangle C_{\omega }(L_{\omega })u^{L_{\omega }}\langle \Omega _{\rho }0\lambda _{\rho }|U_{\rho }|J_{\rho },m_{\rho }=\lambda _{\rho }\rangle C_{\rho }(L_{\rho })v^{L_{\rho }}$