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. Insertion of the complete set of helicity basis vectors 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 }

We can also describe the angular momentum between the daughters as being L and their spin sum as s. Alternatively, we will label the daughters as having helicities of $\lambda _{1}$ and $\lambda _{2}$ - projections of the two particles' spins onto their respective momentum directions. $\langle \Omega \lambda _{1}\lambda _{2}|U|Jm\rangle =\sum _{L,S}\langle \Omega \lambda _{1}\lambda _{2}|Jm\lambda _{1}\lambda _{2}\rangle \langle Jm\lambda _{1}\lambda _{2}|JmLS\rangle \langle JmLS|U|Jm\rangle$

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

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

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

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

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

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

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} }