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 ; \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle = \epsilon \epsilon_\gamma \epsilon_R \langle J M \epsilon|V| \epsilon_\gamma ; \lambda_R \epsilon_R ; \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 $$m=\lambda_\gamma-\lambda_R$$

It is convenient to express above matrix element as

\langle J M \epsilon|V| \epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle = v_{J,m,\epsilon;\epsilon_\gamma; \lambda_R,\epsilon_R} $$

so that the indexed coefficient v specifies the couplings together with the consequences of angular momentum and parity conservation. The function v is implicitly dependent upon the kinematical variables s and t. This dependence will be made explicit in a following section, after the matrix element for the baryon vertex has been studied.

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)= \frac{1}{\sqrt{2}} \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 $$\omega$$, 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.

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

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

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

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

=Y_{M_\rho}^{J_\rho}(\Omega_{\pi}) u_{J_\rho\,0}^{\rho: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 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}{16\pi^2s}|T_{fi}|^2 \left(\frac{p_f}{p_i}\right) \left(\frac{q_{b1}dm_X}{16\pi^3}\right) \left(\frac{q_\omega dm_{b1}}{16\pi^3}\right) \left(\frac{q_\rho dm_\omega}{16\pi^3}\right) \left(\frac{q_\pi dm_\rho}{16\pi^3}\right) $$ where $$p_i\,$$ [$$p_f\,$$] is the target [recoil] nucleon momentum in the center of mass frame of the overall reaction. In terms of the individual decay matrix elements introduced earlier, the T matrix element can be written as



T_{(f)(i)} = T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \epsilon_f) (\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}= $$

=\sum_{R,\lambda_R,\epsilon_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 $$
 * UV | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
 * $$ \times

\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle $$

To obtain the second line in the above equation, we factorized the T operator into two vertex factors UV and W, and inserted between them a sum over a complete set of intermediate exchanges R represented as plane waves moving along the -z axis. The upper vertex operator has been written as UV in anticipation of its further factorization into the primary resonance production operator V and its decay operator U. Polarizations of all particles are represented by their respective reflectivity quantum numbers $$\epsilon$$. For the nucleon, the reflectivity is a complete description of its spin state. For reactions involving higher spin baryons, it would need to be supplemented by an additional $$|\lambda|$$ quantum number.



T_{(f)(i)} = \sum_{ \begin{array}{c} \scriptstyle R,\lambda_R,\epsilon_R,\lambda_{b_1},\lambda_\omega,\lambda_\rho\\ \scriptstyle X,M_X,\epsilon_X;\epsilon_i,\epsilon_f \end{array}} \langle \mathbf{q}_{b1} \lambda_{b_1} 0| U_X | J_X M_X \epsilon_X\rangle \langle \mathbf{q}_\omega \lambda_\omega 0| U_{b_1} | 1, \lambda_{b_1} \rangle $$
 * $$\times

\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 $$
 * $$\times

\langle J_X M_X \epsilon_X | V | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle $$
 * $$\times

\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle $$ Parity conservation requires that $$\epsilon_X=\epsilon_\gamma \epsilon_R$$ and $$\epsilon_R=\epsilon_i\epsilon_f$$. The last two matrix elements in the expression above for $$T_{fi}$$ not known a priori, so we parameterized them into a pair of unknown functions v(s,t) and w(s,t).
 * $$\displaystyle

v^{X,M_X,\epsilon_X}_{\lambda_R,\epsilon_R}(s,t) = \langle J_X M_X \epsilon_X | V | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle $$
 * $$\displaystyle

w_{\lambda_R \epsilon_R; \epsilon_i}(s,t) = \langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle $$

Proton states and individual decay amplitudes
An average over the target proton initial state will be necessary to compute the cross section section. Also, because the polarization of the recoiling proton cannot be measured, a sum over the proton final states must be done. This can be represented as



\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} \propto \sum_{\epsilon_\gamma \epsilon_\gamma' \epsilon_i \epsilon_f \epsilon_i' \epsilon_f'} \rho_{\epsilon_\gamma \epsilon_\gamma'} \rho_{\epsilon_i \epsilon_i'} \delta_{\epsilon_f \epsilon_f'} T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \epsilon_f) (\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)} T^*_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \epsilon_f') (\mathbf{k}_\gamma \epsilon_\gamma' \mathbf{p}_i \epsilon_i')} $$

where density matrices $$\rho$$ represent the initial state particles' spin states. The unpolarized target presents an initial state with both reflectivities equally likely, resulting in


 * $$\rho_{\lambda_i \lambda_i'} = \frac{1}{2} \delta_{\lambda_i \lambda_i'}$$.

In analogy to the reflectivity conservation relation shown above for V vertex, there is a similar relation for the W vertex: $$\epsilon_R=\epsilon_i \epsilon_f$$ Identification of $$\epsilon_i$$ with $$\epsilon_i'$$ and $$\epsilon_f$$ with $$\epsilon_f'$$ implies that only terms with $$\epsilon_R=\epsilon_R'$$ survive in the sum over exchange quantum numbers. The quadratic sum expression above for the differential cross section invokes a double-sum over all of the internal quantum numbers that have been introduced, plus the spins of the initial and final nucleons. This sum is of the generic form

\sum_{X,M_X,\cdots} \cdots \sum_{\epsilon_i\epsilon_f} $$ Note that the measured cross section only depends on the summed modulus squared of the w coefficients, independent of the couplings to the individual nucleon helicity states. Because of this, the sum over nucleon helicities can be dropped, and the w factors absorbed into the v coefficients. Thus the final expression for the differential cross section contains no reference to the quantum numbers $$\epsilon_i, \epsilon_f$$, nor to any w coefficients. It is important to recognize that this simplification of the cross section is only possible in the case of an unpolarized target and no recoil polarimetry.
 * w_{\lambda_R \epsilon_R; \epsilon_i \epsilon_f}|^2

Mass dependence
Expressions for the angular dependence of the matrix elements of $$U_X$$, $$ U_{b1}$$, $$ U_\omega$$, and $$ U_\rho$$ have already been written down above, in terms of the unknown mass-dependent factors a, b, c, and f. The mass dependence of the a factor can be written in terms of a standard relativistic Breit-Wigner resonance lineshape as follows, although often its mass dependence is determined empirically by binning in the mass of X, and fitting each bin independently. In a global mass-dependent fit, the central mass and width of X are free parameters in the fit. The remaining factors b, c, and f are assigned standard resonance forms, with their central mass, width and partial wave matrix elements fixed to agree with the established values for these resonances.

u^{X:J}_{LS}(m_X) = u^{X:J}_{LS} BW_L(m_X;m^0_X,\Gamma^0_X) $$



u^{b_1:1}_{L1}(m_{b1}) = u^{b_1:1}_{L1} BW_L(m_{b1};m^0_{b1},\Gamma^0_{b1}) $$



u^{\omega:1}_{LJ}(m_{\omega}) = u^{\omega:1}_{LJ} BW_L(m_\omega;m^0_\omega,\Gamma^0_\omega) $$



u^{\rho:L}_{L0}(m_{\omega}) = u^{\rho:L}_{L0} BW_L(m_\rho;m^0_\rho,\Gamma^0_\rho) $$

The mass-dependent terms in the expressions above are given by the explicitly unitary Breit-Wigner form:

BW_L(m; m_0,\Gamma_0)=\frac{m_0 \Gamma_L(m)}{m_0^2-m^2-im_0\Gamma_L(m)} $$ where,

\Gamma_L(m;m_0,\Gamma_0)=\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 in the rest frame of the parent particle, and q0 is the same, evaluated at m0.

q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2} $$ The functions $$F_L(q)$$ are the angular momentum barrier factors that are given in the literature. The first few are listed below with $$z=[q/(197\mathrm{MeV/c})]^2$$
 * $$\displaystyle F_0(q)=1$$
 * $$F_1(q)=

\sqrt{\frac{2z}{z+1}} $$
 * $$F_2(q)=

\sqrt{\frac{13z^2}{(z-3)^2+9z}} $$
 * $$F_3(q)=

\sqrt{\frac{277z^3}{z(z-15)^2+9(2z-5)^2}} $$

Describing s and t dependence
It might be useful at some point to do a global fit to the data from all s,t bins. In such a case, it is useful to recall the expected behavior in high-energy peripheral production given by Regge theory.

\sigma \sim s^{\alpha_R-1} e^{b_R t} $$ where $$\alpha_R$$ is the intercept of the Regge trajectory for exchange particle R, and $$b_R$$ is the forward t-slope parameter for exchange trajectory R at this value of s. Appending these factors to the above expression for the differential cross section, inside the sum over exchanges R, would allow data from all bins in s and t to be fitted in a single global fit.

Summing over photon polarization
The T Matrix, written in the photon reflectivity basis can be expanded in the photon's lab frame helicity basis. Temporarily omitting indices not pertaining to the photon:

T_{\epsilon_\gamma} = \frac{\sqrt{-\epsilon_\gamma}}{2} \left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right) $$ where,

T_{\pm 1}=\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 $$
 * U | (1\; \pm 1)_{\mathrm{lab}}; \lambda_R \epsilon_R;s,t \rangle

Now, the average over the initial photon polarization states results in a cross section evaluated as follows:



\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} \propto \frac{1}{2}\left\{ (1+g)\left| \frac{1}{\sqrt{2}}\left(T_{-1}e^{-i\alpha} + T_{+1}e^{i\alpha}\right) \right|^2 + (1-g)\left| \frac{i}{\sqrt{2}}\left(T_{-1}e^{-i\alpha} - T_{+1}e^{i\alpha}\right) \right|^2 \right\} $$

=\frac{1}{2}\left[ g\left(T_{+1}T_{-1}^*e^{2i\alpha} + T_{+1}^*T_{-1}e^{-2i\alpha}\right) \right] $$
 * T_{-1}|^2 + |T_{+1}|^2 +

=\frac{|T_{-1}|^2 + |T_{+1}|^2}{2} + g\,\mathrm{Re}\left(T_{+1}T_{-1}^* e^{2i\alpha} \right) $$ where g is the polarization fraction ranging from 1 (100% x-polarized) to 0 (unpolarized.)