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 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 \Omega=(\phi,\theta)} in the center of mass reference frame, and helicity 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_1} , while daughter 2 has direction 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 -\Omega=(\phi+\pi,\pi-\theta)} and helicity 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_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

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 \langle \Omega \lambda_1 \lambda_2 | U | J m \rangle = \langle \Omega \lambda_1 \lambda_2 | J m \lambda_1 \lambda_2 \rangle \langle J m \lambda_1 \lambda_2 | U | J m \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.

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 \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 }

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.)
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 \mathbb{R}| J m \rangle = P(-1)^{J-m} | J \; -m \rangle }

The eigenstates of the reflectivity operator are formed as follows:
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 | J m \epsilon \rangle = | J m \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

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

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

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

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

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 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} }