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 quantum numbers (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 | J m \rangle = \langle \Omega \lambda_1 \lambda_2 U | J m \rangle $$
 * J m \lambda_1 \lambda_2 \rangle \langle J m \lambda_1 \lambda_2 |

This is one way to describe the final state, but it is not the only way. 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. When we do this, we give up our knowledge of the particles' helicities, having replaced those two quantum numbers with the alternative pair L,S. These two bases, the helicity basis and the L,S basis, are each individually complete and orthonormal within themselves. Following on from the above expression, let us insert a sum over the L,S basis.

\langle \Omega \lambda_1 \lambda_2 | U | J m \rangle = \sum_{L,S} \langle \Omega \lambda_1 \lambda_2 J m L S \rangle \langle J m L S | U | J m \rangle $$
 * J m \lambda_1 \lambda_2 \rangle \langle J m \lambda_1 \lambda_2 |

=\sum_{L,S} \left[ \sqrt{\frac{2J+1}{4\pi}} D_{m \lambda}^{J *}(\Omega) \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} $$ where $$\lambda=\lambda_1-\lambda_2$$, $$\Omega=(\phi,\theta,0)$$ and the double-stacked symbols are Clebsh-Gordon coefficients. The product of CG coefficients in the second brackets on the right-hand side represent the overlap between the basis vectors in the helicity and L,S basis, which turns out to be independent of m, as required by rotational invariance. This overlap integral is somewhat lengthy to calculate, but the result turns out to be fairly simple, as shown above. This expression holds regardless of what axis is used to define the quantization direction for m, but whatever choice is made must serve as the z-axis of the reference frame in which the plane wave direction $$\Omega$$ is defined.

Isospin Projections
One must also take into account the various ways that the isospin of the 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 refer to the daughter index. If the two daughter particles belong to the same isospin multiplet, there is a constraint introduced between orbital angular momentum and total isospin that follows from the symmetry of exchanging the two particle identities, because 180 degree rotation is equivalent to the exchange of the daughter identities (a,b becoming b,a). For example, for a two-pion final state in an even-L angular wave, only even I is allowed, and for an odd-L angular wave, only odd I is allowed. Because of this, it is convenient to define a symmetrized variant of the C coefficients defined above,

C(L)=\frac{1}{\sqrt{2}} \left[ C^{a,b} + (-1)^L C^{b,a} \right] $$ It should be kept in mind that this $$C(L)$$ is only applicable for particle pairs in the same isospin multiplet.

Reflectivity
Beside rotational invariance, parity is also a good symmetry of strong hadron dynamics. In the case described above of the decay of a single particle at rest into two daughters, parity conservation places constraints between different final state amplitudes. Instead of considering the parity operator directly, it is convenient to consider the reflectivity operator R. Reflectivity is the product of parity with a 180 degree rotation about the y axis. The advantage of using this more complicated operator to express the constraints of parity is that a general two-particle plane wave basis can be constructed out of eigenstates of reflectivity, whereas a complete plane-wave basis of parity eigenstates is possible only in the restricted case that daughters 1 and 2 are identical. Regardless of the additional rotation, the basic constraint of reflectivity conservation is nothing more than parity conservation plus rotational invariance.

Acting on a state of good J,m, the reflectivity operator has a particularly simple effect.
 * $$\mathbb{R}| J m \rangle = P(-1)^{J-m} | J \; -m \rangle $$

where P is the intrinsic parity of the system. The eigenstates of the reflectivity operator are formed out of states of good J,m as follows.
 * $$| J m \epsilon \rangle = | J m \rangle + \epsilon P (-1)^{J-m} | J \; -m \rangle   $$

where &epsilon;=&plusmn;1 for a bosonic system and &epsilon;=&plusmn;i for a fermionic system. It follows that
 * $$\mathbb{R}| J m \epsilon \rangle = \epsilon (-1)^{2J} | J m \epsilon \rangle $$

= Applications =

Photon-Reggeon-Resonance vertex
Consider the t-channel production of a resonance from the photon and reggeon in the reflectivity basis, consisting of plane-wave states constructed to be eigenstates of the reflectivity operator. This turns out in the case of the photon to correspond to the usual linear polarization basis |x> and |y>. Let the x (y) linear polarization states be denoted as &epsilon;=- (&epsilon;=+).
 * $$|\epsilon\rangle = \sqrt{\frac{-\epsilon}{2}} \left( \left|1\; -1\right\rangle +\epsilon \left|1\; +1\right\rangle \right)$$


 * $$\mathbb{R}|\epsilon\rangle = \epsilon |\epsilon\rangle $$

The strong interaction Hamiltonian respects reflectivity, so the production operator V should commute with R.
 * $$V=\mathbb{R}^{-1} V \mathbb{R}$$



\langle J m \epsilon|\mathbb{R}^{-1} V \mathbb{R}| \epsilon_\gamma ; J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle = \epsilon \epsilon_\gamma \epsilon_R \langle J m \epsilon|V| \epsilon_\gamma ; 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 $$\displaystyle\epsilon = \epsilon_\gamma \epsilon_R$$ that embodies parity conservation in this decay.

It is convenient to adopt the Gottfried Jackson frame. In particular, we boost into the reference frame of the produced resonance and orient the coordinate system such that the photon is in the +z direction and the x-axis is co-planar to the recoiling proton, thus defining xz as the production plane. A consequence of this choice is that $$m=\lambda_\gamma-\lambda_R$$

To express the initial photon linear polarization state in the reflectivity basis, we relate the linear polarization bases in the laboratory and Gottfried-Jackson coordinate systems:


 * $$\left(\begin{array}{c} \epsilon_\gamma=-1\\ \epsilon_\gamma=+1\end{array}\right)=

\left(\begin{array}{cc} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{array}\right) \left(\begin{array}{c}x_\mathrm{lab} \\ y_\mathrm{lab}\end{array}\right)= \left(\begin{array}{cc} e^{-i\alpha} & e^{i\alpha} \\ ie^{-i\alpha} & -ie^{i\alpha} \end{array}\right) \left(\begin{array}{c}|1,-1\rangle \\ |1,+1\rangle \end{array}\right)_\mathrm{lab} $$

where $$\alpha=\phi$$, the azimuthal angle of the production plane in the lab system.

Decay of t-channel resonance X to b1&pi;
We can apply the above recoupling relations to write down the amplitude at each vertex of the decay tree. These amplitudes are defined within a series of coordinate systems, each defined with respect to its ancestor in the decay chain. The chain starts with resonance X decaying in the Gottfried-Jackson frame, into G-J direction $$\Omega_{b1}$$. To find the decay frame of the $$b_1$$, we perform the rotation $$(\phi_{b1},\theta_{b1},0)$$ (Euler convention z,y',z") then boost into the rest frame of the $$b_1$$. To find the decay frame of the $$\omega$$, next we rotate by $$(\phi_{\omega},\theta_{\omega},0)$$, then boost into the $$\omega$$ rest frame.  The three-body decay of the $$\omega$$ can be treated without loss of generality as a decay into a charged di-pion and a neutral pion, provided that a complete sum over states of the free di-pion system is performed.  For notational simplicity, the di-pion is represented below by the symbol $$\rho$$, which should not be confused with the physical $$\rho$$(770) resonance.  The cascade of decay frames continues through the $$\omega$$ decay by definition of the decay angles $$(\phi_\rho,\theta_\rho,0)$$, and finally $$(\phi_\pi,\theta_\pi,0)$$.  The selection rules for $$\omega$$ decay require that the charged di-pion system be in an overall isovector state, which excludes even orbital angular momentum between the $$\pi^+$$ and $$\pi^-$$. This requires that the di-pion system is antisymmetric in decay angles, so it turns out not to matter whether one uses the $$\pi^+$$ or $$\pi^-$$ member of the pair to define the angles $$(\phi_\pi,\theta_\pi,0)$$.

This procedure results in the decay particle's quantization axis being the same as the momentum direction in the parent's frame, thus forcing the m quantum number in the decay frame to be equal to its helicity &lambda; used in the parent's frame. Substitutions of known quantum numbers are made as necessary below: pions in the final state are given zero helicities and spin of b1 and &omega; are put in from the start.

$$ \langle \Omega_{b_1} \lambda_{b_1} 0| U_X | J_X m_X \rangle =\sum_{L_X} \left[ \sqrt{\frac{2J_X+1}{4\pi}} D_{m_X \lambda_{b_1}}^{J_X *}(\Omega_{b_1}) \right] \left[ \sqrt{\frac{2L_X+1}{2J_X+1}} \left(\begin{array}{cc|c} L_X & 1            & J_X \\ 0   & \lambda_{b_1} & \lambda_{b_1} \end{array}\right) \right] a_{L_X 1}^{J_X} $$

$$ \langle \Omega_\omega \lambda_\omega 0| U_{b_1} | 1, m_{b_1}=\lambda_{b_1} \rangle =\sum_{L_{b_1}} \left[ \sqrt{\frac{2J_{b_1}+1}{4\pi}} D_{m_{b_1} \lambda_\omega}^{1 *}(\Omega_\omega) \right] \left[ \sqrt{\frac{2L_{b_1}+1}{2J_{b_1}+1}} \left(\begin{array}{cc|c} L_{b_1} & 1             & 1 \\ 0       & \lambda_\omega & \lambda_\omega \end{array}\right) \right] b_{L_{b_1} 1}^1 $$

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

$$ \langle \Omega_{\pi} 0 0 | U_\rho | J_\rho, m_\rho=\lambda_\rho \rangle =\sum_{L_\rho} \left[ \sqrt{\frac{2J_\rho+1}{4\pi}} D_{m_\rho 0}^{J_\rho *}(\Omega_{\pi}) \right] \left[ \sqrt{\frac{2L_\rho+1}{2J_\rho+1}} \left(\begin{array}{cc|c} L_\rho & 0 & J_\rho \\ 0     & 0 & 0 \end{array}\right) \right] f_{J_\rho\,0}^{J_\rho}= $$

=Y_{m_\rho}^{J_\rho}(\Omega_{\pi}) f_{J_\rho\,0}^{J_\rho} $$

Assembly of the full amplitude
Putting together the amplitudes discussed above, we arrive at the complete angular-dependent amplitude of photo-production of resonance X and decay to the given final state. The mass-dependent component of the amplitude is given by the Breit-Wigner form:

BW_L(m)=\frac{m_0 \Gamma_L(m)}{m_0^2-m^2-im_0\Gamma_L(m)} $$ where,

\Gamma_L(m)=\Gamma_0 \frac{m_0}{m} \frac{q}{q_0} \frac{F^2_L(q)}{F^2_L(q_0)} $$ where q is the breakup momentum of the daughter particles and q0 is the same, evaluated at m0.

The final expression for the measured cross section becomes:

\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} = \frac{1}{\sqrt{s}k_i}|T_{fi}|^2 $$ where the T matrix element contains a sum over all intermediate states that contribute to the process. In terms of the individual decay matrix elements introduced earlier, the T matrix element can be written as

T_{fi} = \sum_{R,\lambda_R\lambda_{b_1},\lambda_\omega,\lambda_\rho} \langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0 \langle J_R \lambda_R \epsilon_R; \mathbf{p_f}, \lambda_f | W | \mathbf{p_i}, \lambda_i\rangle $$ The aggregate decay matrix element can be further broken up into a product of individual decay amplitudes,
 * U | \epsilon_\gamma; J_R \lambda_R \epsilon_R \rangle

\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0 $$
 * U | \epsilon_\gamma; J_R \lambda_R \epsilon_R \rangle
 * $$=\sum_{\lambda_R,\lambda_{b_1},\lambda_\omega,\lambda_\rho,m_X}

\langle \mathbf{q}_{b1} \lambda_{b_1} 0| U_X | J_X m_X \epsilon_X\rangle \langle J_X m_X \epsilon_X | V | \epsilon_\gamma; J_R \lambda_R \epsilon_R \rangle $$

\langle \mathbf{q}_\omega \lambda_\omega 0| U_{b_1} | 1, \lambda_{b_1} \rangle \langle \mathbf{q}_\rho \lambda_\rho 0| U_\omega | 1, \lambda_\omega \rangle \langle \mathbf{q}_\pi 0 0 | U_\rho | J_\rho, \lambda_\rho \rangle $$



BW_{L_X}^X BW_{L_{b_1}}^{b_1} BW_{L_\omega}^\omega BW_{L_\rho}^\rho e^{-kt} $$

Note that we leave the sum over $$L_X$$ outside the amplitude of interest. This is convenient in partial wave analysis so that the hypothesized L-states can be listed and summed explicitly. In mass-independent fits, the Breit-Wigner for the resonance is replaced with strength factors that are parameters of the fit. An exponential dependence of t is inserted with a coefficient that can be deduced from fits in separate t bins.

The intensity would then require an incoherent summation of amplitudes with laboratory x and y polarization.
 * $$I=

\frac{1+f}{2}\left|\sum_{L_X \epsilon_R} A_{L_X\;-1\;\epsilon_R}^{J_X}\right|^2 + \frac{1-f}{2}\left|\sum_{L_X \epsilon_R} A_{L_X\;+1\;\epsilon_R}^{J_X}\right|^2 $$

where f is the polarization fraction varying from 1, 100% x-polarized, to 0, unpolarized. \sum_{L_X \epsilon_R} A_{L_X \epsilon_\gamma \epsilon_R}^{J_X}=