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 having daughter 1 with trajectory 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)} . 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 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} 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 \lambda_2} - projections on the direction of decay (specified by daughter 1)

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 }	insertion of the complete set of helicity basis vectors
$\langle \Omega \lambda _{1}\lambda _{2}|U|Jm\rangle =\sum _{L,S}\langle \Omega \lambda _{1}\lambda _{2}|Jm\lambda _{1}\lambda _{2}\rangle \langle Jm\lambda _{1}\lambda _{2}|JmLS\rangle \langle JmLS|U|Jm\rangle$	insertion of the complete LS basis set
$=\sum _{L,S}\left[{\sqrt {\frac {2J+1}{4\pi }}}D_{m\lambda }^{J*}(\Omega ,0)\right]\left[{\sqrt {\frac {2L+1}{2J+1}}}\left({\begin{array}{cc|c}L&S&J\\0&\lambda &\lambda \end{array}}\right)\left({\begin{array}{cc|c}S_{1}&S_{2}&S\\\lambda _{1}&-\lambda _{2}&\lambda \end{array}}\right)\right]a_{LS}^{J}$	Substitution of each bra-ket with their respective formulae. $\lambda =\lambda _{1}-\lambda _{2}$ 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:

$C^{a,b}=\left({\begin{array}{cc|c}I^{a}&I^{b}&I\\I_{z}^{a}&I_{z}^{b}&I_{z}^{a}+I_{z}^{b}\end{array}}\right)$

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:

$C(L)={\frac {1}{\sqrt {2}}}\left[C^{a,b}+(-1)^{L}C^{b,a}\right]$

Application

Production

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:

$\langle \Omega _{R}\lambda _{R}\lambda _{p}|W|J_{T}m_{T}\rangle =\langle \Omega _{R}\lambda _{R}\lambda _{p}|Jm\lambda _{R}\lambda _{p}\rangle \langle Jm\lambda _{R}\lambda _{p}|W|J_{T}m_{T}\rangle$	transition amplitude for $p\rightarrow R+p'$ in the direction $\Omega _{R}$ w.r.t. the coordinate system defined in the resonance RF.
$={\sqrt {\frac {2J+1}{4\pi }}}D_{m_{T}(\lambda _{R}-\lambda _{p})}^{J_{T}*}(\Omega _{R},0)w_{\lambda _{R},\lambda _{p}}^{J_{T}}$	follows from relations given above

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 $\pi$ 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
$\mathbb {R} |Jm\epsilon \rangle =\epsilon |Jm\epsilon \rangle$

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:
$\epsilon =\mp \epsilon _{R}$

Decay

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

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

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_\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} }

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_\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} }