The following is an update on amplitude fits with error bars now including the MC contribution. The effect of detector resolution on the uniform background wave is also shown.

== Background wave and resolution ==

The following are plots of the invariant mass distribution of the resonance from disparate MC sets but they show well how detector resolution effects the strength of the uniform background wave. As a reminder, the generated data included 1-- and 2+- which are fitted here with blue and green points respectively.

{| width="200" cellpadding="7"
| [[Image:Xfit_hdg_goodErr.gif|300px]] || [[Image:Xfit_perf-res_matchedtracks.gif|300px]] || [[Image:Xfit_perf-res_truth.gif|300px]]
|-
| Basic fit of reconstructed data. Black points trace the uniform background || After analysis of reconstruction data with appropriate cuts, '''thrown 4-vectors are matched to reconstructed tracks''' and the former are used for amplitude fits. Background is traced by the magenta line. || After analysis of reconstruction data with appropriate cuts, '''thrown 4-vectors''' are kept and used for fits. Background is absent (on the scale of this plot.)
|}

== Current state of fits ==
{|
|[[Image:Xfit_hdg.gif‎]] ||
'''Legend'''
* 1-- (fit) - blue points
* 2+- (fit) - green points
* ''uniform background (fit) - black points''
* other fitted false waves too small to see
* combined intensity (fit) - red points
|}

2011-11-22T00:50:21Z

Senderovich: Created page with 'The following is a review of error propagation needed in amplitude analysis. Consider a Monte-Carlo (MC) integral over the intensities of N detected events out of Ngen</su…'

Exotic b1π Channel Simulation and Analysis

2011-11-21T23:03:46Z

Senderovich:

This page documents the essential results from an effort to simulate and analyze a b1π decay of an exotic resonance. This is done in the context of the GlueX experiment: the data is simulated as photo-production of mesons on a proton target using a Monte Carlo model of the GlueX detector.

* [https://docs.google.com/document/d/1uwNbrvivWcwJRRFIZF0F4FyB2-067Hsa4NRgRsdq9Bo/edit?hl=en_US Log book] (Google Doc)

== Derivations ==

* [[Amplitudes for the Exotic b1π Decay]]
* [[Error propagation in Amplitude Analysis]]

== Reports ==

* [http://argus.phys.uregina.ca/gluex/DocDB/0017/001768/003/CollabMtgTalk-5-2011.pdf Collaboration Meeting 5/2011] - update on work to reconstruct the b1π signal from background. Some cut alternatives are reviewed.
* [http://argus.phys.uregina.ca/gluex/DocDB/0018/001831/004/CollabMtgTalk-10-2011.pdf Collaboration Meeting 10/2011] - Report on the b1π amplitude calculator implementation, and its first use to generate and fit a signal with two waves.

[[User:Senderovich|Igor Senderovich]] 17:12, 10 October 2011 (UTC)

Amplitudes for the Exotic b1π Decay

2011-10-25T03:22:27Z

Senderovich: /* Angular Distribution of Two-Body 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 <math>\Omega=(\phi,\theta)</math> in the center of mass reference frame, and helicity <math>\lambda_1</math>, while daughter 2 has direction <math>-\Omega=(\phi+\pi,\pi-\theta)</math> and helicity <math>\lambda_2</math>.

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
:<math>
\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
</math>

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.
:<math>
\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
</math>
:<math>
=\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}
</math>
where <math>\lambda=\lambda_1-\lambda_2</math>, <math>\Omega=(\phi,\theta,0)</math> 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 <math>\Omega</math> 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:

:<math>
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)
</math>

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,
:<math>
C(L)=\frac{1}{\sqrt{2}} \left[ C^{a,b} + (-1)^L C^{b,a} \right]
</math>
It should be kept in mind that this <math>C(L)</math> 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.
:<math>\mathbb{R}| J M \rangle = P(-1)^{J-M} | J \; -M \rangle </math>
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.
:<math>| J M \epsilon \rangle = | J M \rangle + \epsilon P (-1)^{J-M} | J \; -M \rangle </math>
where ε=±1 for a bosonic system and ε=±i for a fermionic system. It follows that
:<math>\mathbb{R}| J M \epsilon \rangle = \epsilon (-1)^{2J} | J M \epsilon \rangle </math>

= Applications =

== Production ==

=== 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 ε=- (ε=+).
:<math>|\epsilon\rangle = \sqrt{\frac{-\epsilon}{2}}
\left( \left|1\; -1\right\rangle -\epsilon \left|1\; +1\right\rangle \right)
</math>

:<math>\mathbb{R}|\epsilon\rangle = \epsilon |\epsilon\rangle </math>

The strong interaction Hamiltonian respects reflectivity, so the production operator ''V'' should commute with ''R''.
:<math>V=\mathbb{R}^{-1} V \mathbb{R}</math>

:<math>
\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
</math>

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
<math>\displaystyle\epsilon = \epsilon_\gamma \epsilon_R</math> 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 <math>m=\lambda_\gamma-\lambda_R</math>

It is convenient to express above matrix element as
:<math>
\langle J M \epsilon|V|
\epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle
= v_{J,m,\epsilon;\epsilon_\gamma; \lambda_R,\epsilon_R}
</math>

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:

:<math>\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}
</math>

where <math>\alpha=\phi</math>, the azimuthal angle of the production plane in the lab system.

== Decay of t-channel resonance X to b1π==

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 <math>\Omega_{b1}</math>. To find the decay frame of the <math>b_1</math>, we perform the rotation <math>(\phi_{b1},\theta_{b1},0)</math> (Euler convention z,y',z") then boost into the rest frame of the <math>b_1</math>. To find the decay frame of the <math>\omega</math>, we rotate by <math>(\phi_{\omega},\theta_{\omega},0)</math>, then boost into the <math>\omega</math> rest frame. The three-body decay of the <math>\omega</math> 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 <math>\rho</math>, which should not be confused with the physical <math>\rho(770)</math> resonance. The cascade of decay frames continues through the <math>\omega</math> decay by definition of the decay angles <math>(\phi_\rho,\theta_\rho,0)</math>, and finally <math>(\phi_\pi,\theta_\pi,0)</math>. The selection rules for <math>\omega</math> decay require that the charged di-pion system be in an overall isovector state, which excludes even orbital angular momentum between the <math>\pi^+</math> and <math>\pi^-</math>. This requires that the di-pion system is antisymmetric in decay angles, so it turns out not to matter whether one uses the <math>\pi^+</math> or <math>\pi^-</math> member of the pair to define the angles <math>(\phi_\pi,\theta_\pi,0)</math>.

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 ''λ'' 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 ω are put in from the start.

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}=
</math>
:::::::::<math>
=Y_{M_\rho}^{J_\rho}(\Omega_{\pi})
u_{J_\rho\,0}^{\rho:J_\rho}
</math>

== 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:
:<math>
\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)
</math>
where <math>p_i\,</math> [<math>p_f\,</math>] 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

:<math>
T_{(f)(i)} =
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b_1} \mathbf{p}_f \epsilon_f)
(\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}=
</math>
:::<math>
=\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}_{b_1}\lambda_{b_1} 0
| UV | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
</math>
:::::::::<math> \times
\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>

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 <math>\epsilon</math>. 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 <math>|\lambda|</math> quantum number.

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

=== 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

:<math>
\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')}
</math>

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

:<math>\rho_{\lambda_i \lambda_i'} = \frac{1}{2} \delta_{\lambda_i \lambda_i'}</math>.

In analogy to the reflectivity conservation relation shown above for ''V'' vertex, there is a similar relation for the ''W'' vertex: <math>\epsilon_R=\epsilon_i \epsilon_f</math>
Identification of <math>\epsilon_i</math> with <math>\epsilon_i'</math> and <math>\epsilon_f</math> with <math>\epsilon_f'</math> implies that only terms with <math>\epsilon_R=\epsilon_R'</math> 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
:<math>
\sum_{X,M_X,\cdots} \cdots \sum_{\epsilon_i\epsilon_f}
|w_{\lambda_R \epsilon_R; \epsilon_i \epsilon_f}|^2
</math>
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 <math>\epsilon_i, \epsilon_f</math>, 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.

=== Mass dependence ===

Expressions for the angular dependence of the matrix elements of <math>U_X</math>, <math> U_{b1}</math>, <math> U_\omega</math>, and <math> U_\rho</math> 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.
:<math>
u^{X:J}_{LS}(m_X) = u^{X:J}_{LS} BW_L(m_X;m^0_X,\Gamma^0_X)
</math>

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

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

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

The mass-dependent terms in the expressions above are given by the explicitly unitary Breit-Wigner form:
:<math>
BW_L(m; m_0,\Gamma_0)=\frac{m_0 \Gamma_L(m)}{m_0^2-m^2-im_0\Gamma_L(m)}
</math>
where,
:<math>
\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)}
</math>
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''.
:<math>
q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}
</math>
The functions <math>F_L(q)</math> are the angular momentum barrier factors that are given in the literature. The first few are listed below with <math>z=[q/(197\mathrm{MeV/c})]^2</math>
:<math>\displaystyle F_0(q)=1</math>
:<math>F_1(q)=
\sqrt{\frac{2z}{z+1}}
</math>
:<math>F_2(q)=
\sqrt{\frac{13z^2}{(z-3)^2+9z}}
</math>
:<math>F_3(q)=
\sqrt{\frac{277z^3}{z(z-15)^2+9(2z-5)^2}}
</math>

===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.
:<math>
\sigma \sim s^{\alpha_R-1} e^{b_R t}
</math>
where <math>\alpha_R</math> is the intercept of the Regge trajectory for exchange particle ''R'', and <math>b_R</math> 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:
:<math>
T_{\epsilon_\gamma} = \sqrt{\frac{-\epsilon_\gamma}{2}}
\left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right)
</math>
where,
:<math>
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
</math>

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

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

SiPM Amplifier Optimization

2011-10-16T21:42:23Z

Senderovich: /* Gain */

The SiPM amplifier currently in use for SiPM characterization ([http://www.photonique.ch/ Photonique] item AMP_0604) must be adapted for tagger microscope use. The following is a brief outline of the design requirements. They are discussed in detail in the following sections

# adjustable gain, ranging from readout of hundreds of pixels to calibration with single-photon counting
# less than 15% gain variability on transistor <math>\beta</math> (<math>h_{FE}</math>) parameter
# summing circuit to pool SiPM signals in groups of 5 (readout of individual channels must not affect readout of the sum regardless of termination used)
# minimized pulse duration for higher running rates
# minimized power consumption

== Design Requirements ==

One of the most important considerations in the amplifier design is the target data acquisition system. Currently, the microscope is slated to be read out by a 12-bit flash ADC with a 250 MHz sampling rate and range settings of -2V, -1V and -0.5V.

=== Gain ===

The amplifier provided by Photonique with a gain of roughly 3 kΩ was well suited for single photon counting. However, for typical signals ranging in the hundreds of SiPM pixels, this gain excessive. However, the option of switching back to single photon detection for the purposes of calibration would be a nice feature.

From the perspective of expected signal amplitudes (taking into account optical and SiPM's quantum efficiencies) signals around 300 pixels (px) are expected. With a SiPM gain of about 2 × 105 ~ 9.6 pC are expected to be deposited. Design uncertainties that go into the full calculation summarized here can easily allow variation in this result by a factor of two or more. Roughly estimating this charge to be contained in a triangular pulse with <strike>5 ns</strike> 20 ns? FWHM (after all the broadening inherent in the amplifier) yields a total signal peak of 0.5 mA. With this figure and the full range of the ADC (2V) it seems that 3 kΩ is still appropriate. However this does not leave room for variation discussed above. Instead, a goal of sub-1 kΩ gain was adopted.

For the high gain setting, the issue is mainly the vertical resolution of the ADC. For most of the duration of this project, the 8-bit version of the ADC was planned to be allocated for microscope readout, imposing a stringent requirement on gain in order to avoid the digitization noise inherent in signals only a few adc voltage steps. The 12-bit ADC makes clean readout of single pixel wavefunctions more realistic: at the most sensitive scale of 0.5 V, the resolution is 0.12 mV. However, we must also take into account noise and a possible factor of two loss in the split of the signal between the ADC and the CFD (constant fraction discriminator to prepare for time pick-off.) This time, it is appropriate to take a pessimistic scenario of the pulse shape: taking a triangular pulse with 30 ns FWHM, leading to a single pixel current peak of 0.27 μA. Under these conditions, gain of 7 kΩ is enough, giving 15 adc steps per pixel.

=== Minimal Dependence on <math>\beta</math> ===

Dependence of a design on a value β of a transistor is never a good idea, given its variability as a function of temperature. The value also varies between one transistor and the next. The high speed transistors necessary in this design are already at a disadvantage with respect to this design goal, due to their low beta with the result of greater significance of the parameter's variation. This issue will be discussed at length below.

While it is true that this variation can be compensated by adjusting the bias voltage on the SiPM to alter the gain, this sort of tweaking is not desirable. The detection efficiency changes right along with the gain. (Detection of the maximimum number of photons is critical for good time resolution.) With these concerns in mind, a design requirement of gain variation no greater than 15% variation was set. The bounds of overall-amplifier β variation were taken conservatively: the worst scenario was one in which
''all'' transistors are at the minimum β value.

=== Summing Circuit ===

While the microscope contains 500 optical channels resulting from the two-dimensional segmentation of the focal plane, only the energy bin information is important. The vertical column of 5 scintillating fibers in the focal plane can then have its signals summed, reducing the count of channels requiring readout to 100. However, the design of the microscope calls for individual readout in 5 columns spread around the focal plane in order to retain the knowledge of the two-dimensional orientation of the electron stripe.

Thus a summing circuit is necessary to tie together 5 amplifiers. Drivers for single channel readout must still be available on the board and their output must be available on the board's interface pins. (The choice of which group of channels can be accessed is up to the layout of the backplane board.) The individual channel and summing circuit outputs must remain independent regardless of termination. This means that the input to the summing circuit must have its own current source so that the load on the individual channel is not relevant. Additionally, as has been discussed above under gain requirements, the summing circuit must offer additional gain when system is set to high gain mode.

=== Short Pulses ===

The duration of the amplified signals is, in essence, the dead time of the channel. (Pulses with significant overlap are difficult to distinguish.) The lower bound of the pulse width is set by the scintillator. The BCF-20 intended for use in the microscope has a decay time of 2.7 ns. A somewhat stricter restriction is set by the sampling rate of the ADC - 250 MHz. A rough (and vague) guideline of 15 ns total signal duration has been observed during the design.

=== Power Consumption ===

Operation of over 500 such amplifiers (spares are available on each board) in a sealed box presents an issue of heat management. High temperatures inside the microscope should be avoided especially due to the sensitivity of SiPM parameters on temperature. Note also that the amplifiers will be packed very tightly (~5 mm strips) in the final layout. 30 mW has been set as the power budget per amplifier. The restriction on the summing circuit is not as critical, because there are 5 times fewer such circuits in the microscope. Contribution from the actual AC signals and various voltage regulators are negligible and are not taken into account.

== Design Implementation ==

==== Evolution of the Design ====

The initial approach to this problem of gain switching was just that suggested by Photonique documentation: turning up the amplifier supply voltage. With the transistors' maximal voltage rating of 15 V taken as the high gain setting, some simulations were done to assess the amplifier performance. The following problems were found in this approach:
* Gain saturates with increasing supply voltage leading to poor gain separation between the two modes (recall that the low gain setting needs to be much lower for real signals.)
* Power consumption is of order 200 mW at high gain setting
* Component values necessary for high gain are not suitable for stable, <math>\beta</math>-independent design
* Input impedance of the amplifier increase significantly in the high gain design. (This along with the SiPM capacitance sets the integration RC-time.) On the other hand, the effective impedance and therefore pulse shape varies with supply voltage.

A low-impedance input stage was designed to alleviate the last issue, but the rest remained serious concerns and challenges. An alternate design was adapted in which the supply voltage remains constant but the summing stage offers additional amplification. Gain selection is accomplished with a FET switch, effectively altering the resistance in a transistor stage similar to the common emitter amplifier.

==== Input Impedance ====

The low impedance input stage was retained from the earlier design and applied the the summing circuit, since it pools currents from the individual amplifiers. In Photonique's design, the input signal sees the transistor base, base-biasing resistor, and a feedback resistor in parallel. The new input stages take the signal on the emitter (base held at a set DC value), in which case the signal sees the emitter resistor and the impedance looking into the emitter in parallel with each other. The latter dominates with an effective resistance of order 25 Ω. The input stages are biased with generous amount of current to keep this value low.

==== <math>\beta</math>-dependence and Voltage Buffering ====

There turned out to be a significant trade-off between gain achieved in the first amplification stage and β sensitivity. Since the variation in beta is cumulative in the progress of the signal through the amplifier, it is essential to keep the variation low at this point. For this reason, the gain has been turned down to around 400 Ω - well below the requirement set by the gain/dynamic range considerations themselves. This stage also draws significant current, accounting for about half the amplifier power budget, to keep this point minimally β-dependent.

To keep single channel and summing circuit readout independent of each other, a separate driver for the summing circuit was added. The first stage of the amplifier contains a switchable collector resistance which, the ratio of which with the effective resistance of the driver sets the additional gain sought at this stage. The additional inversion (to a positive-going signal) must be countered with another inverter, followed by the voltage buffer similar to that at the end of the single channel amplifier circuit.

Signal buffering turns out to be tricky in this project. The requirement of negative output pulse polarity set by the ADC mandates a PNP transistor for the final stages of both amplifier and summing segments. However, the β characteristics of the only acceptable fast PNP found on the market (BFT92W) are even worse than those in its NPN counterpart. The typical value of <math>h_{FE}</math> is 50 with a minimum value of 20, compared to BFR92P from Infineon, which sports values from 70 to near 100. The effective load of the terminators of about β50 Ω remains comparable to the source impedance of the circuit. In other words, the source impedance cannot be considered negligible compared to the load, creating an effective voltage divider whose ratio depends on beta! Since the output from the summing circuit suffers from about twice as much variation due to β because of all the preceding stages, a stiffer buffer became essential on this end. The emitter-follower driver was doubled, effectively multiply the β's of the two transistors. The resulting two diode drops, too low to avoid significant saturation on collectors of earlier stages, are countered with a DC level shift with AC coupling from preceding circuit elements.

A particularly challenging point in the circuit in terms of impedance turned out to be the junction with the common-emitter amplifier stage used as an inverter. At high gain, the effective source impedance is set to about 3.5 kΩ. Thus, this point suffers from significant loading down of the signal and subsequent sensitivity to β. Additionally, it was found that the low bandwidth characteristics of the common-emitter amplifier resulting from the ''Miller Effect'' - stray capacitance between transistor base and collector, magnified with about the gain of the amplifier. These concerns motivated insertion of another buffer between these stages, increasing the load seen by the preceding ''m'' transistor stage and decreasing the source impedance seen by the inverting amplifier stage (The characteristic time, RC, of the inherent integrator decreased proportionately.)

[[Image:AmpCircuit_v7_DC.png|frame|center|DC characteristics of the amplifier. Units of V, mA, and Ω are implied unless corrected by different prefix.]]

==== Dynamic Range ====

It essential to avoid clipping the signal at designed gain levels and to be able to utilize the full range (2 V) of the ADC. Appropriate DC level were set to avoid saturation any transistor collectors. This turns out a bit involved, since the collector-base voltage (plus the canonical saturation margin of 100 mV) gap necessary is more than the maximum desired signal of 2 V due to attenuation along the amplifier chain. Changing DC levels changes biasing of transistor bases, changing the quiescent current and therefore the attenuation. Additionally the power budget significantly restricts the DC levels of the circuit.

In the current stage of the design, full 2V range has not been achieved. Both single channel and summing outputs go up to about 1.5 V.

==== The Gain Switch ====

A MOSFET switches the effective collector resistance in the first stage of the summing circuit between about 3.3 kΩ due to Re alone and about 80 Ω for the parallel path. NXP's BF1108R has been selected for prototyping. Its typical <math>V_{GS}</math> for current pinch-off is -3 V (max: -4 V). Putting its source on the supply rail and switching the gate between 5 V (on) and 0 V (off). A bypass capacitor (not shown in diagram) near the gate lead is important to prevent spurious switching.

==== Final Performance Parameters ====

With the final parameters specified by the circuit model resistor input vector set to
<pre>R = [.56 1 .33 .47 1.36 1 .12 1.22 .1 .082 3.3 .15 .68 .18 2 .392 .7 1]*1e3</pre>
(see [[SiPM Amplifier Components]] for a complete list of component values) the following theoretical specifications are achieved:

{| align="center" style="text-align:center" cellpadding="4"
!   !! Amplifier Stage !! Summing Stage !! Units !! Notes
|-
! align="left" | Gain (low/high)
| 0.29 || 0.29/7.5 || kΩ || 50 Ω load assumed
|-
! align="left" | Input impedance
| 13.5 || 27.3 || Ω
|-
! align="left" | Power (quiescent)
| 42 || 47 || mW || i.e. 52 mW/channel
|-
! align="left" | Output pulse height (max)
| 1.46 || 1.45 || V
|}

== See Also ==

* [[SiPM Amplifier Signal Analysis]]
* [[SiPM Amplifier Components]]

Exotic b1π Channel Simulation and Analysis

2011-10-10T17:20:03Z

Senderovich:

This page documents the essential results from an effort to simulate and analyze a b1π decay of an exotic resonance. This is done in the context of the GlueX experiment: the data is simulated as photo-production of mesons on a proton target using a Monte Carlo model of the GlueX detector.

* [https://docs.google.com/document/d/1uwNbrvivWcwJRRFIZF0F4FyB2-067Hsa4NRgRsdq9Bo/edit?hl=en_US Log book] (Google Doc)

== Production and Decay Amplitudes ==

* [[Amplitudes for the Exotic b1π Decay]]

== Reports ==

* [http://argus.phys.uregina.ca/gluex/DocDB/0017/001768/003/CollabMtgTalk-5-2011.pdf Collaboration Meeting 5/2011] - update on work to reconstruct the b1π signal from background. Some cut alternatives are reviewed.
* [http://argus.phys.uregina.ca/gluex/DocDB/0018/001831/004/CollabMtgTalk-10-2011.pdf Collaboration Meeting 10/2011] - Report on the b1π amplitude calculator implementation, and its first use to generate and fit a signal with two waves.

[[User:Senderovich|Igor Senderovich]] 17:12, 10 October 2011 (UTC)

Exotic b1π Channel Simulation and Analysis

2011-10-10T17:12:38Z

Senderovich:

Exotic b1π Channel Simulation and Analysis

2011-10-10T17:08:34Z

Senderovich: /* Reports */

Amplitudes for the Exotic b1π Decay

2011-10-02T02:29:04Z

Senderovich: /* Assembly of the full amplitude */

= 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 <math>\Omega=(\phi,\theta)</math> in the center of mass reference frame, and helicity <math>\lambda_1</math>, while daughter 2 has direction <math>-\Omega=(\phi+\pi,\pi-\theta)</math> and helicity <math>\lambda_2</math>.

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
:<math>
\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
</math>

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.
:<math>
\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
</math>
:<math>
=\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}
</math>
where <math>\lambda=\lambda_1-\lambda_2</math>, <math>\Omega=(\phi,\theta,0)</math> 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 <math>\Omega</math> 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:

:<math>
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)
</math>

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,
:<math>
C(L)=\frac{1}{\sqrt{2}} \left[ C^{a,b} + (-1)^L C^{b,a} \right]
</math>
It should be kept in mind that this <math>C(L)</math> 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.
:<math>\mathbb{R}| J M \rangle = P(-1)^{J-M} | J \; -M \rangle </math>
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.
:<math>| J M \epsilon \rangle = | J M \rangle + \epsilon P (-1)^{J-M} | J \; -M \rangle </math>
where ε=±1 for a bosonic system and ε=±i for a fermionic system. It follows that
:<math>\mathbb{R}| J M \epsilon \rangle = \epsilon (-1)^{2J} | J M \epsilon \rangle </math>

= Applications =

== Production ==

=== 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 ε=- (ε=+).
:<math>|\epsilon\rangle = \sqrt{\frac{-\epsilon}{2}}
\left( \left|1\; -1\right\rangle -\epsilon \left|1\; +1\right\rangle \right)
</math>

:<math>\mathbb{R}|\epsilon\rangle = \epsilon |\epsilon\rangle </math>

The strong interaction Hamiltonian respects reflectivity, so the production operator ''V'' should commute with ''R''.
:<math>V=\mathbb{R}^{-1} V \mathbb{R}</math>

:<math>
\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
</math>

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
<math>\displaystyle\epsilon = \epsilon_\gamma \epsilon_R</math> 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 <math>m=\lambda_\gamma-\lambda_R</math>

It is convenient to express above matrix element as
:<math>
\langle J M \epsilon|V|
\epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle
= v_{J,m,\epsilon;\epsilon_\gamma; \lambda_R,\epsilon_R}
</math>

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:

:<math>\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}
</math>

where <math>\alpha=\phi</math>, the azimuthal angle of the production plane in the lab system.

== Decay of t-channel resonance X to b1π==

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 <math>\Omega_{b1}</math>. To find the decay frame of the <math>b_1</math>, we perform the rotation <math>(\phi_{b1},\theta_{b1},0)</math> (Euler convention z,y',z") then boost into the rest frame of the <math>b_1</math>. To find the decay frame of the <math>\omega</math>, we rotate by <math>(\phi_{\omega},\theta_{\omega},0)</math>, then boost into the <math>\omega</math> rest frame. The three-body decay of the <math>\omega</math> 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 <math>\rho</math>, which should not be confused with the physical <math>\rho(770)</math> resonance. The cascade of decay frames continues through the <math>\omega</math> decay by definition of the decay angles <math>(\phi_\rho,\theta_\rho,0)</math>, and finally <math>(\phi_\pi,\theta_\pi,0)</math>. The selection rules for <math>\omega</math> decay require that the charged di-pion system be in an overall isovector state, which excludes even orbital angular momentum between the <math>\pi^+</math> and <math>\pi^-</math>. This requires that the di-pion system is antisymmetric in decay angles, so it turns out not to matter whether one uses the <math>\pi^+</math> or <math>\pi^-</math> member of the pair to define the angles <math>(\phi_\pi,\theta_\pi,0)</math>.

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 ''λ'' 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 ω are put in from the start.

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}=
</math>
:::::::::<math>
=Y_{M_\rho}^{J_\rho}(\Omega_{\pi})
u_{J_\rho\,0}^{\rho:J_\rho}
</math>

== 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:
:<math>
\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)
</math>
where <math>p_i\,</math> [<math>p_f\,</math>] 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

:<math>
T_{(f)(i)} =
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b_1} \mathbf{p}_f \epsilon_f)
(\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}=
</math>
:::<math>
=\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}_{b_1}\lambda_{b_1} 0
| UV | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
</math>
:::::::::<math> \times
\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>

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 <math>\epsilon</math>. 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 <math>|\lambda|</math> quantum number.

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

=== 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

:<math>
\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')}
</math>

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

:<math>\rho_{\lambda_i \lambda_i'} = \frac{1}{2} \delta_{\lambda_i \lambda_i'}</math>.

In analogy to the reflectivity conservation relation shown above for ''V'' vertex, there is a similar relation for the ''W'' vertex: <math>\epsilon_R=\epsilon_i \epsilon_f</math>
Identification of <math>\epsilon_i</math> with <math>\epsilon_i'</math> and <math>\epsilon_f</math> with <math>\epsilon_f'</math> implies that only terms with <math>\epsilon_R=\epsilon_R'</math> 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
:<math>
\sum_{X,M_X,\cdots} \cdots \sum_{\epsilon_i\epsilon_f}
|w_{\lambda_R \epsilon_R; \epsilon_i \epsilon_f}|^2
</math>
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 <math>\epsilon_i, \epsilon_f</math>, 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.

=== Mass dependence ===

Expressions for the angular dependence of the matrix elements of <math>U_X</math>, <math> U_{b1}</math>, <math> U_\omega</math>, and <math> U_\rho</math> 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.
:<math>
u^{X:J}_{LS}(m_X) = u^{X:J}_{LS} BW_L(m_X;m^0_X,\Gamma^0_X)
</math>

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

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

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

The mass-dependent terms in the expressions above are given by the explicitly unitary Breit-Wigner form:
:<math>
BW_L(m; m_0,\Gamma_0)=\frac{m_0 \Gamma_L(m)}{m_0^2-m^2-im_0\Gamma_L(m)}
</math>
where,
:<math>
\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)}
</math>
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''.
:<math>
q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}
</math>
The functions <math>F_L(q)</math> are the angular momentum barrier factors that are given in the literature. The first few are listed below with <math>z=[q/(197\mathrm{MeV/c})]^2</math>
:<math>\displaystyle F_0(q)=1</math>
:<math>F_1(q)=
\sqrt{\frac{2z}{z+1}}
</math>
:<math>F_2(q)=
\sqrt{\frac{13z^2}{(z-3)^2+9z}}
</math>
:<math>F_3(q)=
\sqrt{\frac{277z^3}{z(z-15)^2+9(2z-5)^2}}
</math>

===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.
:<math>
\sigma \sim s^{\alpha_R-1} e^{b_R t}
</math>
where <math>\alpha_R</math> is the intercept of the Regge trajectory for exchange particle ''R'', and <math>b_R</math> 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:
:<math>
T_{\epsilon_\gamma} = \sqrt{\frac{-\epsilon_\gamma}{2}}
\left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right)
</math>
where,
:<math>
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
</math>

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

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

Amplitudes for the Exotic b1π Decay

2011-09-29T01:27:18Z

Senderovich: /* Reflectivity */

= 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 <math>\Omega=(\phi,\theta)</math> in the center of mass reference frame, and helicity <math>\lambda_1</math>, while daughter 2 has direction <math>-\Omega=(\phi+\pi,\pi-\theta)</math> and helicity <math>\lambda_2</math>.

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
:<math>
\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
</math>

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.
:<math>
\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
</math>
:<math>
=\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}
</math>
where <math>\lambda=\lambda_1-\lambda_2</math>, <math>\Omega=(\phi,\theta,0)</math> 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 <math>\Omega</math> 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:

:<math>
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)
</math>

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,
:<math>
C(L)=\frac{1}{\sqrt{2}} \left[ C^{a,b} + (-1)^L C^{b,a} \right]
</math>
It should be kept in mind that this <math>C(L)</math> 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.
:<math>\mathbb{R}| J M \rangle = P(-1)^{J-M} | J \; -M \rangle </math>
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.
:<math>| J M \epsilon \rangle = | J M \rangle + \epsilon P (-1)^{J-M} | J \; -M \rangle </math>
where ε=±1 for a bosonic system and ε=±i for a fermionic system. It follows that
:<math>\mathbb{R}| J M \epsilon \rangle = \epsilon (-1)^{2J} | J M \epsilon \rangle </math>

= Applications =

== Production ==

=== 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 ε=- (ε=+).
:<math>|\epsilon\rangle = \sqrt{\frac{-\epsilon}{2}}
\left( \left|1\; -1\right\rangle -\epsilon \left|1\; +1\right\rangle \right)
</math>

:<math>\mathbb{R}|\epsilon\rangle = \epsilon |\epsilon\rangle </math>

The strong interaction Hamiltonian respects reflectivity, so the production operator ''V'' should commute with ''R''.
:<math>V=\mathbb{R}^{-1} V \mathbb{R}</math>

:<math>
\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
</math>

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
<math>\displaystyle\epsilon = \epsilon_\gamma \epsilon_R</math> 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 <math>m=\lambda_\gamma-\lambda_R</math>

It is convenient to express above matrix element as
:<math>
\langle J M \epsilon|V|
\epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle
= v_{J,m,\epsilon;\epsilon_\gamma; \lambda_R,\epsilon_R}
</math>

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:

:<math>\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}
</math>

where <math>\alpha=\phi</math>, the azimuthal angle of the production plane in the lab system.

== Decay of t-channel resonance X to b1π==

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 <math>\Omega_{b1}</math>. To find the decay frame of the <math>b_1</math>, we perform the rotation <math>(\phi_{b1},\theta_{b1},0)</math> (Euler convention z,y',z") then boost into the rest frame of the <math>b_1</math>. To find the decay frame of the <math>\omega</math>, we rotate by <math>(\phi_{\omega},\theta_{\omega},0)</math>, then boost into the <math>\omega</math> rest frame. The three-body decay of the <math>\omega</math> 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 <math>\rho</math>, which should not be confused with the physical <math>\rho(770)</math> resonance. The cascade of decay frames continues through the <math>\omega</math> decay by definition of the decay angles <math>(\phi_\rho,\theta_\rho,0)</math>, and finally <math>(\phi_\pi,\theta_\pi,0)</math>. The selection rules for <math>\omega</math> decay require that the charged di-pion system be in an overall isovector state, which excludes even orbital angular momentum between the <math>\pi^+</math> and <math>\pi^-</math>. This requires that the di-pion system is antisymmetric in decay angles, so it turns out not to matter whether one uses the <math>\pi^+</math> or <math>\pi^-</math> member of the pair to define the angles <math>(\phi_\pi,\theta_\pi,0)</math>.

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 ''λ'' 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 ω are put in from the start.

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}=
</math>
:::::::::<math>
=Y_{M_\rho}^{J_\rho}(\Omega_{\pi})
u_{J_\rho\,0}^{\rho:J_\rho}
</math>

== 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:
:<math>
\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)
</math>
where <math>p_i\,</math> [<math>p_f\,</math>] 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

:<math>
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)}=
</math>
:::<math>
=\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
</math>
:::::::::<math> \times
\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>

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 <math>\epsilon</math>. 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 <math>|\lambda|</math> quantum number.

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

=== 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

:<math>
\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')}
</math>

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

:<math>\rho_{\lambda_i \lambda_i'} = \frac{1}{2} \delta_{\lambda_i \lambda_i'}</math>.

In analogy to the reflectivity conservation relation shown above for ''V'' vertex, there is a similar relation for the ''W'' vertex: <math>\epsilon_R=\epsilon_i \epsilon_f</math>
Identification of <math>\epsilon_i</math> with <math>\epsilon_i'</math> and <math>\epsilon_f</math> with <math>\epsilon_f'</math> implies that only terms with <math>\epsilon_R=\epsilon_R'</math> 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
:<math>
\sum_{X,M_X,\cdots} \cdots \sum_{\epsilon_i\epsilon_f}
|w_{\lambda_R \epsilon_R; \epsilon_i \epsilon_f}|^2
</math>
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 <math>\epsilon_i, \epsilon_f</math>, 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.

=== Mass dependence ===

Expressions for the angular dependence of the matrix elements of <math>U_X</math>, <math> U_{b1}</math>, <math> U_\omega</math>, and <math> U_\rho</math> 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.
:<math>
u^{X:J}_{LS}(m_X) = u^{X:J}_{LS} BW_L(m_X;m^0_X,\Gamma^0_X)
</math>

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

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

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

The mass-dependent terms in the expressions above are given by the explicitly unitary Breit-Wigner form:
:<math>
BW_L(m; m_0,\Gamma_0)=\frac{m_0 \Gamma_L(m)}{m_0^2-m^2-im_0\Gamma_L(m)}
</math>
where,
:<math>
\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)}
</math>
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''.
:<math>
q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}
</math>
The functions <math>F_L(q)</math> are the angular momentum barrier factors that are given in the literature. The first few are listed below with <math>z=[q/(197\mathrm{MeV/c})]^2</math>
:<math>\displaystyle F_0(q)=1</math>
:<math>F_1(q)=
\sqrt{\frac{2z}{z+1}}
</math>
:<math>F_2(q)=
\sqrt{\frac{13z^2}{(z-3)^2+9z}}
</math>
:<math>F_3(q)=
\sqrt{\frac{277z^3}{z(z-15)^2+9(2z-5)^2}}
</math>

===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.
:<math>
\sigma \sim s^{\alpha_R-1} e^{b_R t}
</math>
where <math>\alpha_R</math> is the intercept of the Regge trajectory for exchange particle ''R'', and <math>b_R</math> 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:
:<math>
T_{\epsilon_\gamma} = \sqrt{\frac{-\epsilon_\gamma}{2}}
\left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right)
</math>
where,
:<math>
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
</math>

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

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

Amplitudes for the Exotic b1π Decay

2011-09-21T18:58:06Z

Senderovich: /* Summing over photon polarization */

= 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 <math>\Omega=(\phi,\theta)</math> in the center of mass reference frame, and helicity <math>\lambda_1</math>, while daughter 2 has direction <math>-\Omega=(\phi+\pi,\pi-\theta)</math> and helicity <math>\lambda_2</math>.

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
:<math>
\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
</math>

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.
:<math>
\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
</math>
:<math>
=\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}
</math>
where <math>\lambda=\lambda_1-\lambda_2</math>, <math>\Omega=(\phi,\theta,0)</math> 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 <math>\Omega</math> 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:

:<math>
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)
</math>

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,
:<math>
C(L)=\frac{1}{\sqrt{2}} \left[ C^{a,b} + (-1)^L C^{b,a} \right]
</math>
It should be kept in mind that this <math>C(L)</math> 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.
:<math>\mathbb{R}| J M \rangle = P(-1)^{J-m} | J \; -M \rangle </math>
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.
:<math>| J M \epsilon \rangle = | J M \rangle + \epsilon P (-1)^{J-M} | J \; -M \rangle </math>
where ε=±1 for a bosonic system and ε=±i for a fermionic system. It follows that
:<math>\mathbb{R}| J M \epsilon \rangle = \epsilon (-1)^{2J} | J M \epsilon \rangle </math>

= Applications =

== Production ==

=== 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 ε=- (ε=+).
:<math>|\epsilon\rangle = \sqrt{\frac{-\epsilon}{2}}
\left( \left|1\; -1\right\rangle -\epsilon \left|1\; +1\right\rangle \right)
</math>

:<math>\mathbb{R}|\epsilon\rangle = \epsilon |\epsilon\rangle </math>

The strong interaction Hamiltonian respects reflectivity, so the production operator ''V'' should commute with ''R''.
:<math>V=\mathbb{R}^{-1} V \mathbb{R}</math>

:<math>
\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
</math>

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
<math>\displaystyle\epsilon = \epsilon_\gamma \epsilon_R</math> 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 <math>m=\lambda_\gamma-\lambda_R</math>

It is convenient to express above matrix element as
:<math>
\langle J M \epsilon|V|
\epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle
= v_{J,m,\epsilon;\epsilon_\gamma; \lambda_R,\epsilon_R}
</math>

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:

:<math>\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}
</math>

where <math>\alpha=\phi</math>, the azimuthal angle of the production plane in the lab system.

== Decay of t-channel resonance X to b1π==

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 <math>\Omega_{b1}</math>. To find the decay frame of the <math>b_1</math>, we perform the rotation <math>(\phi_{b1},\theta_{b1},0)</math> (Euler convention z,y',z") then boost into the rest frame of the <math>b_1</math>. To find the decay frame of the <math>\omega</math>, we rotate by <math>(\phi_{\omega},\theta_{\omega},0)</math>, then boost into the <math>\omega</math> rest frame. The three-body decay of the <math>\omega</math> 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 <math>\rho</math>, which should not be confused with the physical <math>\rho(770)</math> resonance. The cascade of decay frames continues through the <math>\omega</math> decay by definition of the decay angles <math>(\phi_\rho,\theta_\rho,0)</math>, and finally <math>(\phi_\pi,\theta_\pi,0)</math>. The selection rules for <math>\omega</math> decay require that the charged di-pion system be in an overall isovector state, which excludes even orbital angular momentum between the <math>\pi^+</math> and <math>\pi^-</math>. This requires that the di-pion system is antisymmetric in decay angles, so it turns out not to matter whether one uses the <math>\pi^+</math> or <math>\pi^-</math> member of the pair to define the angles <math>(\phi_\pi,\theta_\pi,0)</math>.

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 ''λ'' 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 ω are put in from the start.

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}=
</math>
:::::::::<math>
=Y_{M_\rho}^{J_\rho}(\Omega_{\pi})
u_{J_\rho\,0}^{\rho:J_\rho}
</math>

== 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:
:<math>
\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)
</math>
where <math>p_i\,</math> [<math>p_f\,</math>] 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

:<math>
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)}=
</math>
:::<math>
=\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
</math>
:::::::::<math> \times
\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>

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 <math>\epsilon</math>. 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 <math>|\lambda|</math> quantum number.

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

=== 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

:<math>
\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')}
</math>

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

:<math>\rho_{\lambda_i \lambda_i'} = \frac{1}{2} \delta_{\lambda_i \lambda_i'}</math>.

In analogy to the reflectivity conservation relation shown above for ''V'' vertex, there is a similar relation for the ''W'' vertex: <math>\epsilon_R=\epsilon_i \epsilon_f</math>
Identification of <math>\epsilon_i</math> with <math>\epsilon_i'</math> and <math>\epsilon_f</math> with <math>\epsilon_f'</math> implies that only terms with <math>\epsilon_R=\epsilon_R'</math> 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
:<math>
\sum_{X,M_X,\cdots} \cdots \sum_{\epsilon_i\epsilon_f}
|w_{\lambda_R \epsilon_R; \epsilon_i \epsilon_f}|^2
</math>
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 <math>\epsilon_i, \epsilon_f</math>, 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.

=== Mass dependence ===

Expressions for the angular dependence of the matrix elements of <math>U_X</math>, <math> U_{b1}</math>, <math> U_\omega</math>, and <math> U_\rho</math> 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.
:<math>
u^{X:J}_{LS}(m_X) = u^{X:J}_{LS} BW_L(m_X;m^0_X,\Gamma^0_X)
</math>

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

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

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

The mass-dependent terms in the expressions above are given by the explicitly unitary Breit-Wigner form:
:<math>
BW_L(m; m_0,\Gamma_0)=\frac{m_0 \Gamma_L(m)}{m_0^2-m^2-im_0\Gamma_L(m)}
</math>
where,
:<math>
\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)}
</math>
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''.
:<math>
q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}
</math>
The functions <math>F_L(q)</math> are the angular momentum barrier factors that are given in the literature. The first few are listed below with <math>z=[q/(197\mathrm{MeV/c})]^2</math>
:<math>\displaystyle F_0(q)=1</math>
:<math>F_1(q)=
\sqrt{\frac{2z}{z+1}}
</math>
:<math>F_2(q)=
\sqrt{\frac{13z^2}{(z-3)^2+9z}}
</math>
:<math>F_3(q)=
\sqrt{\frac{277z^3}{z(z-15)^2+9(2z-5)^2}}
</math>

===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.
:<math>
\sigma \sim s^{\alpha_R-1} e^{b_R t}
</math>
where <math>\alpha_R</math> is the intercept of the Regge trajectory for exchange particle ''R'', and <math>b_R</math> 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:
:<math>
T_{\epsilon_\gamma} = \sqrt{\frac{-\epsilon_\gamma}{2}}
\left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right)
</math>
where,
:<math>
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
</math>

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

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