Spectroscopy of Mesons Produced by Linearly Polarized Photons

Abstract

A formalism for the experimental analysis of mesons produced by a beam of linearly polarized photons is presented. This formalism introduces a more general use of the reflectivity operator. The goal is to recognize resonances in cross-sections, their associated quantum numbers, and production mechanisms by performing partial wave analysis of multiple-meson final states.

1. Introduction

One of the standard experimental procedures to search for strongly interacting bounded states (of gluons and quarks) is to identify resonances in the production cross-sections. However, physics information related to the production and decay of those states is contained in their complex amplitudes; therefore, resonances in the mass spectrum are only intricately related to the model parameters. Quantum Chromodynamics (QCD) is the current theory of strong interactions; however, we have not been able to analytically calculate the cross-sections of bound states from fundamental QCD. Therefore, QCD-inspired models are necessary to relate the observables of the bound states to the amplitudes and study confinement in those bound states. These models are mainly based on the general properties of the S-matrix [1] as relativity, causality, and the conservation of probability. In recent years, numerical approximations to QCD (Lattice QCD) have been used to compute resonance properties [2]. This paper describes a method to include parity conservation in a phenomenological formalism. We specifically consider mesons produced by linearly polarized photons, and parity is assumed to be conserved by all strong interactions. The definition of a reflectivity operator integrates parity conservation into the analysis formalism. We consider the information obtained by knowing the polarization of the incoming photon beam and how it relates to the production mechanisms. The properties of the photon beam are described by a spin density matrix. We discuss a new use of the reflectivity operator, where we also include information about beam polarization. As an example of the application of this formalism, we describe, in a specific simple model, how this formalism can be used in a mass-independent partial wave analysis (PWA) of multi-particle final states.

2. Cross-Sections

We consider multiple-final-state mesons produced by linearly polarized photons diffractively colliding off a proton target at rest. The outline of the reaction is shown in Figure 1.

Let $\tau$ represent the complete set of variables needed to describe the decay of the resonance. In the case of two final state mesons, only two angles will be needed. We use the ($\theta ,\varphi$) angles of one meson in the Gottfried–Jackson (GJ) frame (see reference [3], Appendix A, for frame definitions). In the case of more than two mesons in the final state, at least two more angles for each extra meson will be required.

The cross-section for the reaction $\gamma p\to Xp$, where $X\to \left(mesons\right)$ will be written as [3]

$$\frac{d\sigma }{d{E}_{\gamma }dtdMd\tau }=\frac{1}{BeamFlux\left(E_{\gamma }\right)}\sum _{ext.spins}{|\mathcal{M}|}^{2}d\rho$$

(1)

where $\mathcal{M}$ is the Lorentz-invariant (transition or scattering) amplitude and $d\rho$ is the Lorentz-invariant phase-space element (LIPS). The spin’s incoming and outgoing degrees of freedom are included in the sum over spins. The LIPS includes the kinematical constraints and M includes the spin and production/decay-internal (transition) degrees of freedom. We can write $d\rho \propto {\sum }_i\frac{d^3{p}_i}{2E_i}$, where i runs over all the incoming and outgoing particles.

To measure cross-sections experimentally, we normally “bin” or divide data into small ranges of one variable such that the dependence of the cross-section on that variable is suppressed. For example, if this division is performed with the mass, energy, and t-Mandelstam, only the angular dependencies for two produced particles will remain (more are needed for more final state particles, i.e., perhaps isobar properties; see Section 5.1.2). All the “external” (normalization) dependencies can be taken into an overall constant, $\kappa$ (which we will just drop out afterward from the formulas, as they will not affect the overall behavior). Therefore, in a data bin ($E_{gamma}$, t, and M), we define an intensity

$$I\left(\tau \right)\equiv \frac{d\sigma }{d\tau }=\kappa ·\sum _{ext.spins}{|\mathcal{M}|}^2$$

(2)

M is a representation of the scattering operator or transition operator, T, and it can be written as

$$\mathcal{M}=\langle out|T|in\rangle$$

(3)

and then

$$I\left(\tau \right)=\sum _{ext.spins}{|\mathcal{M}|}^2=\sum _{ext.spins}{\langle out|T|in\rangle (\langle out|T|in\rangle )}^{*}$$

(4)

and, further,

$${\langle out|T|in\rangle (\langle out|T|in\rangle )}^{*}=\langle out|T|in\rangle \langle in|T^{*}|out\rangle .$$

(5)

In the case that we have spin information from the incoming photon beam, we can include the beam polarization in our amplitude by defining the matrix $|in\rangle \langle in|$, and the photon spin density matrix operator, ${\rho }_{in}$ (see reference [4] for details), as

$${\rho }_{in}\equiv |in\rangle \langle in|.$$

(6)

Suppose that we prepare the polarization of the incoming photons or measure their states of polarization. The average over photon polarization will be completely described by this spin density matrix. In the case of a beam of linearly polarized photons, any polarized state can be written as a linear combination of two pure polarization states. Therefore, the general structure of this $2\times 2$ matrix (for example, in the helicity basis defined by $|\lambda \rangle$ and $|{\lambda }^{\prime }\rangle$) will be

$${\rho }_{in}={\rho }_{\lambda ,{\lambda }^{\prime }}^{\gamma }.$$

(7)

$$I\left(\tau \right)=\sum _{ext.spins}\sum _{\lambda ,{\lambda }^{\prime }}\langle out|T^{\lambda }{\rho }_{\lambda {\lambda }^{\prime }}^{\gamma }{T}^{{\lambda }^{\prime }\ast }|out\rangle .$$

(8)

The $“ext.spins”$ are now the target (${\lambda }_1=\pm$) and recoil (${\lambda }_2=\pm$) helicities. We will assume, as is tradition in the study of meson production [3,5] and discussed in reference [6] (Section 16.6) that the transition operator can be factorized into two parts: the production (of X) and the decay operators (of X), such that

$$I\left(\tau \right)=\sum _{{\lambda }_1{\lambda }_2}\sum _{\lambda ,{\lambda }^{\prime }}\langle out|T_{decay}[T_{prod}^{\lambda ,{\lambda }_1,{\lambda }_2}]{\rho }_{\lambda ,{\lambda }^{\prime }}^{\gamma }[T_{prod}^{{\lambda }^{\prime },{\lambda }_1,{\lambda }_2}{]}^{*}{T}_{decay}^{*}|out\rangle$$

(9)

Furthermore, we can take a completely orthogonal set of states, $|X\rangle$, such that ${\sum }_X|X\rangle \langle X|=\mathbf{1}$, and include them in the previous relation such that

$$\begin{array}{c}\hfill I\left(\tau \right)=\sum _{{\lambda }_1{\lambda }_2}\sum _{\lambda ,{\lambda }^{\prime }}\langle out|T_d\sum _X|X\rangle \langle X|T_p^{\lambda ,{\lambda }_1,{\lambda }_2}{\rho }_{\lambda ,{\lambda }^{\prime }}^{\gamma }[T_p^{{\lambda }^{\prime },{\lambda }_1,{\lambda }_2}{]}^{*}\end{array}$$

$$\sum _{X^{\prime }}|X^{\prime }\rangle \langle X^{\prime }|T_d^{*}|out\rangle$$

(10)

$$I\left(\tau \right)=\sum _{{\lambda }_1{\lambda }_2}\sum _{\lambda ,{\lambda }^{\prime }}\sum _{X,X^{\prime }}\langle out|T_d|X\rangle \langle X|T_p^{\lambda ,{\lambda }_1,{\lambda }_2}{\rho }_{\lambda ,{\lambda }^{\prime }}^{\gamma }[T_p^{{\lambda }^{\prime },{\lambda }_1,{\lambda }_2}{]}^{*}|X^{\prime }\rangle \langle X^{\prime }|T_d^{*}|out\rangle .$$

(11)

The set of states, $|X\rangle$, a full set of intermediate states, we will call the partial waves. Each of these states can be described by a set of quantum numbers, for example, l,m, isobars (mass, width). The total angular momentum by $J=l\oplus s$ ($|l-s<J<|l+s|)$ and the total spin $s=s_1\oplus s_2$, where l = 0, 1, 2... (S, P, D...) and m $(-l<m<l)$ will define the “waves” of the expansion. In practice, this expansion is truncated (to a very few states). We will refer to these quantum numbers as (l,m,I), where the I include all other parameters needed for a more extended model, for example, the isobar model parameters (see Section 5.1.2).

The production amplitudes describe the strong interaction production mechanism that we are not able to calculate (without a phenomenological model). In a mass-independent PWA fit, the production amplitudes, in a given bin, will be considered constant, independent of the decay properties (for example, final particle angles). They function as weights on each partial decay amplitude, and will be extracted (fitted) from the data. We will rewrite

$$\langle X_{l,m,I}|T_p^{\lambda ,{\lambda }_1,{\lambda }_2}{\rho }_{\lambda {\lambda }^{\prime }}^{\gamma }{T}_p^{{\lambda }^{\prime },{\lambda }_1,{\lambda }_2}|X_{l^{\prime },m^{\prime },I^{\prime }}\rangle =T_{l,m,I}^{\lambda ,{\lambda }_1,{\lambda }_2}{\rho }_{\lambda {\lambda }^{\prime }}^{\gamma }[T_{l^{\prime },m^{\prime },I^{\prime }}^{{\lambda }^{\prime },{\lambda }_1,{\lambda }_2}{]}^{*}$$

(12)

and

$$\langle out|T_d|X_{l,m,I}\rangle =A_{l,m,I}\left(\tau \right),$$

(13)

$T_{l,m,I}^{\lambda ,{\lambda }_1,{\lambda }_2}$ being the production amplitudes and $A_{l,m,I}\left(\tau \right)$ the decay amplitudes. Note that the A’s and T’s are both complex numbers and that A depends of the resonance quantum numbers and angles, while T depends on the resonance and beam (photon) quantum numbers. The photon spin density matrix depends on the partial polarization P and the polarization angle $\Phi$ (see reference [4] for definitions).

Therefore, in the helicity basis [7], $\lambda$ being the helicities of the incoming photon, and ${\lambda }_1,{\lambda }_2$ the helicities of the target (outgoing) nucleons:

$$I(\tau ,P,\Phi )=\sum _{{\lambda }_1{\lambda }_2}\sum _{\lambda ,{\lambda }^{\prime }}\sum _{(l,m,I,l^{\prime },m^{\prime },I^{\prime })}$$

$$A_{l,m,I}\left(\tau \right)[T_{l,m,I}^{\lambda ,{\lambda }_1,{\lambda }_2}]{\rho }_{\lambda ,{\lambda }^{\prime }}^{\gamma }(P,\Phi )[T_{l^{\prime },m^{\prime },I^{\prime }}^{{\lambda }^{\prime },{\lambda }_1,{\lambda }_2}{]}^{*}{A}_{l^{\prime },m^{\prime },I^{\prime }}^{*}\left(\tau \right).$$

(14)

For example, in a two-meson final state, we have 2(from ${\lambda }_1)·$ 2(from ${\lambda }_2)·$ 2(from $\lambda )·(2l+1)$ unknown parameters (${}^{\lambda }{T}_{l,m,I}^{{\lambda }_1{\lambda }_2}$) for each wave $l=S,P,D,...$ to be fitted to the data.

3. Reflectivity

The effect of parity is defined as the inversion of the spatial coordinates with respect to the origin of coordinates. Most reactions in High-Energy Physics (HEP) are unchanged under this operation (as only weak interactions violate parity).

In our case, assuming vector meson dominance [8] for the photon and diffractive scattering from the nucleon, and since the strong interaction conserves parity, the parity operator commutes with the scattering matrix (or transition operator). Helicity states, however, are not eigenstates of the parity operator and therefore, they are not directly related to the parity exchanged in the reaction.

The parity operation is equivalent to a “mirror reflection” with respect to an arbitrary plane, followed by a $\pi$ rotation with respect to an axis orthogonal to that plane. Let us call $\widehat{\Pi }$ the parity operator. Since the parity operation acting on rotations only changes the direction (sign), in the canonical representation (and in the rest frame of the particle), we have

$$\widehat{\Pi }|J,m\rangle =P|J,m\rangle$$

(15)

where $P=\pm 1$ are its eigenvalues. Let us consider a particle moving with momentum $\overrightarrow{p_z}$ in the z direction. We can obtain this state by boosting (L is a Lorentz transformation) the state at rest:

$$|\overrightarrow{p_z}J,m\rangle =L\left(\overrightarrow{p_z}\right)|0;J,m\rangle .$$

(16)

Applying the parity operator,

$$\widehat{\Pi }|\overrightarrow{p_z}J,m\rangle =\widehat{\Pi }L\left(\overrightarrow{p_z}\right)|0;J,m\rangle$$

(17)

$$\widehat{\Pi }|\overrightarrow{p_z}J,m\rangle =PL(-\overrightarrow{p_z})|0;J,m\rangle .$$

(18)

To get back from $(-\overrightarrow{p_z})$ to $\left(\overrightarrow{p_z}\right)$, we need a rotation of modulo $\pi$ around the y axis

$$L\left(\overrightarrow{p_z}\right)=e^{-i\pi J_y}L(-\overrightarrow{p_z})e^{i\pi J_y}$$

(19)

and we know that

$$e^{-i\pi J_y}|\overrightarrow{p_z}{J,m\rangle =(-1)}^{J-m}|\overrightarrow{p_z}J,-m\rangle$$

(20)

Therefore, we finally have

$$\widehat{\Pi }|\overrightarrow{p_z}{J,m\rangle =P(-1)}^{J-m}{e}^{i\pi J_y}|\overrightarrow{p_z}J,-m\rangle .$$

(21)

Since any other direction can be constructed by rotation, and the parity operator commutes with rotations (in the x-z plane), we can express the former formula in the rest frame of the resonance, with y perpendicular to the production plane (GJ/HEL frames), with the spin quantization in the z-axis given by m

$$\widehat{\Pi }|J,m\rangle =P{(-1)}^{J-m}{e}^{i\pi J_y}|J,-m\rangle .$$

(22)

It is useful to define the reflection operator [9]

$${\widehat{R}}_y=\widehat{\Pi }{e}^{-i\pi J_y}$$

(23)

which involves parity and a $\pi$ angular rotation around the y axis either in the GJ or HEL frames. It represents a mirror reflection through the production plane (x,z). This operator commutes with the transition operator. The y axis in the GJ/HEL frame is perpendicular to the production plane; therefore, the transition matrix is independent of y, and only the $x,z$ coordinates participate in the parity transformation. Reflection commutes with the Hamiltonian. The reflection operator acting on the resonance states produces

$${\widehat{R}}_y|J,m\rangle =e^{-i\pi J_y}\widehat{\Pi }|J,m\rangle =$$

(24)

$$e^{-i\pi J_y}P{(-1)}^{J-m}{e}^{i\pi J_y}|J,-m\rangle =P{(-1)}^{(J-m)}|J,-m\rangle$$

(25)

where P are the parity eigenvalues $(\pm )$. We can build the following eigenstates of ${\widehat{R}}_y$ (since the reflection changes signs on the z-projection quantum numbers, m, we will create eigenstates that are a linear combination of both (m) signs’ states with adequate coefficients):

$$|ϵ,J,m\rangle =\left[\begin{array}{c}|J,m\rangle -ϵP{(-1)}^{(J-m)}|J,-m\rangle \end{array}\right]\Theta \left(m\right)$$

(26)

The sign between both terms in Equation (26) is arbitrary. We use the sign definition in reference [3] and define

$$\Theta \left(m\right)=\frac{1}{\sqrt{2}},ifm>0;\Theta \left(m\right)=\frac{1}{2},ifm=0$$

(27)

and

$$\Theta \left(m\right)=0,ifm<0$$

(28)

It can be shown (see ref. [3]) that the $ϵ$’s are the real (for mesons) eigenvalues of the reflectivity operator. We define a resonance reflectivity =${ϵ}_R$ as

$$|{ϵ}_R,J,|m|\rangle =\left[\begin{array}{c}|J,m\rangle -{ϵ}_{R}P{(-1)}^{(J-m)}|J,-m\rangle \end{array}\right]\Theta \left(m\right)$$

(29)

In our previous notation,

$$A_{J,|m|}^{{ϵ}_R}\left(\tau \right)=\left[\begin{array}{c}{A}_{J,m}\left(\tau \right)-{ϵ}_{R}P{(-1)}^{(J-m)}{A}_{J,-m}\left(\tau \right)\end{array}\right]\Theta \left(m\right)$$

(30)

Notice that since each state defined in the reflectivity basis includes a combination of m and $-m$, the projections of the spin on the quantization axis, m, are replaced by $|m|$ (a kind of “absolute value”). We can think of the reflectivity ${ϵ}_R$ “carrying” the sign of m. When we sum over possible quantum numbers for each wave (l), we have (2l + 1) terms in this sum; we have $2·l$ for two reflectivities for each $m>0$ plus one ${ϵ}_R=-1$ for m = 0 [3].

In pion beam experiments [5] (spinless beam) or past photo production experiments (CLAS) [10], where no information on the beam polarization was available, the spin density matrix is (or is considered) a constant (see [3]), and therefore, it can be factored out from the intensity expression. The past CLAS formalism [3] includes the helicity of the photon in the rank of the matrices (in the external sum of spins). Invoking parity conservation, we still reduced the number of degrees of freedom from eight to four, and the reflectivity was only defined for the resonance. Again, this was done for unpolarized photons or when no information about the photon polarization was available.

In the case we are considering, having information about the photon polarization, it will be proper to also define a reflectivity state for the photon. For a real photon $P=-1$, $J=1$ and $\lambda =+1,-1$; therefore, we define a photon reflectivity = ${ϵ}_{\gamma }$ from

$$|{ϵ}_{\gamma },\lambda \rangle =\left[|\lambda \rangle -{ϵ}_{\gamma }{(-1)}^{\lambda }|-\lambda \rangle \right]\Theta \left(\lambda \right)$$

(31)

then (the reflectivity eigenvalues for a photon are ${ϵ}_{\gamma }=\pm 1$),

$$\begin{array}{c}|{ϵ}_{\gamma }=+1,\lambda =+1\rangle =\frac{1}{\sqrt{2}}(|\lambda =+1\rangle +|\lambda =-1\rangle )\hfill \\ \hfill \\ |{ϵ}_{\gamma }=-1,\lambda =+1\rangle =\frac{1}{\sqrt{2}}(|\lambda =+1\rangle -|\lambda =-1\rangle )\hfill \end{array}.$$

(32)

Equation (14), in this new (two) reflectivity basis, is then

$$I(\tau ,P,\Phi )=\sum _{{\lambda }_1{\lambda }_2}\sum _{{ϵ}_R{ϵ}_R^{\prime },{ϵ}_{\gamma }{ϵ}_{\gamma }^{\prime }}\sum _{J,|m|,J^{\prime },|m^{\prime }|}$$

$$A_{J,|m|}^{{ϵ}_R}\left(\tau \right)T_{J,|m|}^{{ϵ}_R{ϵ}_{\gamma },{\lambda }_1{\lambda }_2}{\rho }_{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}^{\gamma }(P,\Phi )T_{J^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime }{ϵ}_{\gamma }^{\prime },{\lambda }_1{\lambda }_2\ast }{A}_{J^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime }\ast }\left(\tau \right)$$

(33)

The photon spin density matrix in the photon reflectivity basis has the following form (see Appendix A or reference [4]):

$${\rho }_{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}\left(P,\Phi \right)=\frac{1}{2}\left(\begin{array}{cc}1-Pcos2\Phi & -iPsin2\Phi \\ iPsin2\Phi & 1+Pcos2\Phi \end{array}\right)$$

(34)

We now write the expression for the intensity, where $|m|$ are defined positive in the reflectivity basis, and we include the “resonance” ${ϵ}_R$ and “photon” ${ϵ}_{\gamma }$ reflectivities.

There are only two degrees of freedom associated with ${\lambda }_1,{\lambda }_2$ target spins; we will call them $k=1,2$ (spin-flop and no spin-flop).

$$I(\tau ,P,\Phi )=\sum _k\sum _{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}\sum _{{ϵ}_R,{ϵ}_R^{\prime }}\sum _{l|m|I,l^{\prime }|m^{\prime }|I^{\prime }}$$

$$A_{l,|m|,I}^{{ϵ}_R}\left(\tau \right)T_{l,|m|,I}^{{ϵ}_R{ϵ}_{\gamma },k}{\rho }_{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}^{\gamma }(P,\Phi )[T_{l^{\prime },|m^{\prime }|,I^{\prime }}^{{ϵ}_R^{\prime }{ϵ}_{\gamma }^{\prime },k}{]}^{*}[A_{l^{\prime },|m^{\prime }|,I^{\prime }}^{{ϵ}_R^{\prime }}\left(\tau \right){]}^{*}.$$

(35)

We have organized the indices such that k are the external or non-interfering indices. Expanding the sum over the photon reflectivities (using the photon spin density matrix), we have (just for clarity, we drop the $k,I$ indexes in the next expression)

$$I(\tau ,P,\Phi )=\sum _{{ϵ}_R,{ϵ}_R^{\prime }}\sum _{l|m|,l^{\prime }|m^{\prime }|}[$$

$$\left(1-Pcos2\Phi \right)A_{l,|m|}^{{ϵ}_R}\left(\tau \right)T_{l,|m|}^{{ϵ}_R,+}[T_{l^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime },+}{]}^{*}[A_{l^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime }}\left(\tau \right){]}^{*}$$

$$+\left(-iPsin2\Phi \right)A_{l,|m|}^{{ϵ}_R}\left(\tau \right)T_{l,|m|}^{{ϵ}_R,+}[T_{l^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime },-}{]}^{*}[A_{l^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime }}\left(\tau \right){]}^{*}$$

$$+\left(iPsin2\Phi \right)A_{l,|m|}^{{ϵ}_R}\left(\tau \right)T_{l,|m|}^{{ϵ}_R,-}[T_{l^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime },+}{]}^{*}{A}_{l^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime }}\left(\tau \right){]}^{*}$$

$$+\left(1+Pcos2\Phi \right)A_{l,|m|}^{{ϵ}_R}\left(\tau \right)T_{l,|m|}^{{ϵ}_R,-}[T_{l^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime },-}{]}^{*}[A_{l^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime }*}\left(\tau \right){]}^{*}{\displaystyle ]}$$

(36)

We define the resonance spin density matrices as

$${}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m|,|m^{\prime }|}^{l,l^{\prime }}=\sum _k\sum _{{ϵ}_{\gamma }{ϵ}_{\gamma }^{\prime }}{T}_{l,m}^{{ϵ}_{\gamma }{ϵ}_R,k}{\rho }_{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}^{\gamma }[T_{l^{\prime },m^{\prime }}^{{ϵ}_{\gamma }^{\prime }{ϵ}_R^{\prime },k}{]}^{*}$$

(37)

Then

$$I(\varphi ,\theta ,P,\Phi )=\sum _{{}^{{ϵ}_R{ϵ}_R^{\prime }}}\sum _{l,|m|,l^{\prime },|m^{\prime }|}{}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m||m^{\prime }|}^{l{l}^{\prime }}{}^{{ϵ}_R}{Y}_l^{|m|}(\varphi ,\theta ){}^{{ϵ}_R^{\prime }}{Y}_{l^{\prime }}^{|m^{\prime }|\ast }(\varphi ,\theta ).$$

(38)

We can write (see reference [4])

$${\rho }_{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}^{\gamma }=\frac{1}{2}\left(\begin{array}{c}\mathbf{1}+\sum _{j=1,2,3}{P}_{\gamma }^j{\sigma }_j\end{array}\right)$$

(39)

where ${\sigma }_j$ are the Pauli matrices and $P_{\gamma }^j$ the photon polarization vector. Therefore,

$${}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m|,|m^{\prime }|}^{l,l^{\prime }}=\frac{1}{2}\left[\sum _k\sum _{{ϵ}_{\gamma }{ϵ}_{\gamma }^{\prime }}{T}_{l,|m|}^{{ϵ}_{\gamma }{ϵ}_R,k}[T_{l^{\prime },|m^{\prime }|}^{{ϵ}_{\gamma }^{\prime }{ϵ}_R^{\prime },k}{]}^{*}+\sum _{j=1,2,3}{P}_{\gamma }^j\sum _k\sum _{{ϵ}_{\gamma }{ϵ}_{\gamma }^{\prime }}{T}_{l,|m|}^{{ϵ}_{\gamma }{ϵ}_R,k}{\sigma }_j[T_{l^{\prime },|m^{\prime }|}^{{ϵ}_{\gamma }^{\prime }{ϵ}_R^{\prime },k}{]}^{*}\right]$$

(40)

which can be written as

$${}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m|,|m^{\prime }|}^{l,l^{\prime }}={}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m|,|m^{\prime }|}^{\left(0\right),l,l^{\prime }}+\sum _{j=1,2,3}{P}_{\gamma }^j[{}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m|,|m^{\prime }|}^{\left(j\right),l,l^{\prime }}]$$

(41)

with ${}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m|,|m^{\prime }|}^{l,l^{\prime }}$

being the polarized SDME (Spin Density Matrix Elements).

$${}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m|,|m^{\prime }|}^{\left(0\right),l,l^{\prime }}=\sum _k\sum _{{ϵ}_{\gamma }}{T}_{l,|m|}^{{ϵ}_{\gamma }{ϵ}_R,k}[T_{l^{\prime },|m^{\prime }|}^{{ϵ}_{\gamma }{ϵ}_R^{\prime },k}{]}^{*}$$

(42)

$${}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m|,|m^{\prime }|}^{\left(1\right),l,l^{\prime }}=\sum _k\sum _{{ϵ}_{\gamma }}{T}_{l,|m|}^{-{ϵ}_{\gamma }{ϵ}_R,k}[T_{l^{\prime },|m^{\prime }|}^{{ϵ}_{\gamma }{ϵ}_R^{\prime },k}{]}^{*}$$

(43)

$${}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m|,|m^{\prime }|}^{\left(2\right),l,l^{\prime }}=i·\sum _k\sum _{{ϵ}_{\gamma }}{ϵ}_{\gamma }·T_{l,|m|}^{-{ϵ}_{\gamma }{ϵ}_R,k}[T_{l^{\prime },|m^{\prime }|}^{{ϵ}_{\gamma }{ϵ}_R^{\prime },k}{]}^{*}$$

(44)

$${}^{{ϵ}_R{ϵ}_R^{\prime }}{\rho }_{|m|,|m^{\prime }|}^{\left(3\right),l,l^{\prime }}=\sum _k\sum _{{ϵ}_{\gamma }}{ϵ}_{\gamma }·T_{l,|m|}^{{ϵ}_{\gamma }{ϵ}_R,k}[T_{l^{\prime },|m^{\prime }|}^{{ϵ}_{\gamma }{ϵ}_R^{\prime },k}{]}^{*}$$

(45)

4. Naturality and Reflectivity

A state is said to have natural parity if $P={(-1)}^J$, while is said to have unnatural parity if $P=-{(-1)}^J$. We can recast this definition by introducing the naturality of the particle, $\mathcal{N}$, as

$$\mathcal{N}=P\times {(-1)}^J.$$

(46)

Naturality is $\mathcal{N}=+1$ (natural) for $J^P=0^{+},1^{-},2^{+},\cdots$ (i.e., $\rho ,\omega \cdots$) and $\mathcal{N}=-1$ (unnatural) for $J^P=0^{-},1^{+},2^{-},\cdots$ (i.e., $\pi ,\eta \cdots$). Determining (or constraining) the naturality of the production (exchange particle) will give us extra information on the produced resonances.

The reflectivities are defined for the resonance decay and for the incoming photon. Reflection is a conserved quantum number since both rotation and parity are conserved. Therefore, at least at higher energies (see Appendix C of reference [11]), the product of the initial photon reflectivity and the exchange particle reflectivity must equal the reflectivity of the resonance:

$${ϵ}_{\gamma }\times {ϵ}_{ex}={ϵ}_R.$$

(47)

or

$${ϵ}_{\gamma }\times {ϵ}_R={ϵ}_{exchange}.$$

(48)

And since ${ϵ}_{exchange}=P{(-1)}^J$, then

$${ϵ}_{\gamma }\times {ϵ}_R=\mathcal{N}.$$

(49)

The photon spin density matrix, in the photon reflectivity basis, represents a mix of photon states, as seen in Equation (34). We can also see that the resonance spin density matrix (Equation (37)) will not be diagonal in this formalism. Only for full polarization $P=1$ might there be defined reflectivity configurations that contribute to the reaction. These are when $\Phi =0$ only (${ϵ}_{\gamma }=-1$) contribute and when $\Phi =\frac{\pi }{2}$ only (${ϵ}_{\gamma }=+1$) contribute. Using linearly polarized photons at those explicit configurations, we could then constrain the naturality of the exchange and particles produced. For example, in the case of pion exchange (or other Regge unnatural trajectory particles), the reflectivity of the resonance (${ϵ}_R$) will be opposite to that of the photon (${ϵ}_{\gamma }$). In the case of $\rho$ exchange (or other Regge natural trajectory particles), the reflectivity of the resonance and the photon will be the same. For unpolarized beams, the reflectivity is only defined for the resonance and the spin density matrix of the reaction becomes diagonal. For polarized beams, we can still use similar methods if we include the beam polarization in the rank of the sum (added to the external spin). The JPAC collaboration [11] defined a reflectivity for the case of two pseudo-scalar final states, taking into account combined photon resonance parity conservation. In that case, there is only one reflectivity, and the spin density matrix becomes diagonal. In the JPAC definition,

$$ϵ=P{(-1)}^J$$

(50)

or

$$ϵ=\mathcal{N}$$

(51)

and the reflectivity coincides directly (by construction) with the naturality of the resonance. It has been shown [12] that the JPAC definition and the two reflectivity scenarios defined in this paper are equivalent for the case of two final state pseudo-scalars.

5. Search for Resonances

5.1. Decay Amplitudes

To obtain the decay amplitudes, we will consider two cases: first, the resonance decaying into two particles, and second, the resonance decaying into three or more particles. In this latter case, we will use the isobar model [3,5]. The isobar model assumes a series of sequential two-body decays. We consider the resonance decaying into an intermediate unstable particle (isobar) plus a stable particle (bachelor), and all bachelors will be among the final states. The isobar will decay subsequently into other particles (children), which may also be isobars, and continue the process. We assume that there are no interactions after the particles are produced through this sequential process and that all final (observed) particles are spinless. We calculate amplitudes in the spin formalism of Jacob and Wick [7,13].

5.1.1. Two-Body Decays

Let us consider the case of a resonance X decaying into two particles labeled as 1 and 2 (see Figure 2 for notation).

We describe the decay of X in its rest frame, that is, $p_1+p_2=0$, with the z in the direction of the beam; this is the Gottfried–Jackson (GJ) frame. We can thus describe the kinematics with just one momentum $q(\varphi ,\theta )=p_1=-p_2$. In this case, what we called $\tau$ to describe the final particles will be given by just two angles. We use the helicity basis to represent amplitudes.

$$\tau =\left\{\begin{array}{c}{\varphi }_{GJ},{\theta }_{GJ}\end{array}\right\}$$

(52)

where ${\varphi }_{GJ},{\theta }_{GJ}$ are the angles of one of the decay products in the Gottfried–Jackson frame. For a given mass M and transfer momentum t, the decay amplitudes will depend only on $\tau$ (angles).

It can be shown that the decay amplitude is (see references [3,13])

$$A_{l,m}\left(\tau \right)=\sqrt{\frac{2l+1}{4\pi }}{F}_l\left(p\right)·\sum _{{\lambda }_1{\lambda }_2}{D}_{m\lambda }^{J\ast }\left({\Omega }_{GJ}\right)(l0s\lambda |J\lambda )(s_1{\lambda }_1{s}_2-{\lambda }_2|s\lambda )a_{ls}.$$

(53)

The expressions in parenthesis represent Clebsch–Gordan coefficients. We introduced the factor $F_l\left(p\right)$, the Blatt–Weisskopf centrifugal barrier factor (see reference [3]). This factor takes into account the centrifugal barrier effects caused by the angular factors on the potential. The factor is close to one and in many cases can be ignored. The sum on ${\lambda }_1$ and ${\lambda }_2$ is over all possible helicities of the daughters’ particles.

The “unknown” factor $a_{ls}$ will be included in the fitting parameters of our model (“T’s”) and will not be carried over to our subsequent formulas.

Consider the decay of a resonance into two spinless final particles. Experimentally, we normally detect spinless particles; therefore, this is a very common case (kaons, etas, or pions). In this case, $\lambda ={\lambda }_1={\lambda }_2=0$, $s=s_1=s_2=0$, and $J=l$. We will take $F_l\left(p\right)\sim 1$. The angular dependencies, $\tau$, will be given by the $(\varphi ,\theta )$ angles of one of the decay particles in the GJ frame. Therefore, $(l0s0|J0)=1$ and $(s_{1}0s_{2}0|s0)=1$.

Then

$$A_{lm}\left(\tau \right)=\sqrt{\frac{2l+1}{4\pi }}{D}_{m0}^{l\ast }(\varphi ,\theta ,0)$$

(54)

and

$$D_{m0}^{l\ast }(\varphi ,\theta ,0)=e^{im\varphi }\sqrt{\frac{(l-m)!}{(l+m)!}}{P}_l^m\left(cos\theta \right)$$

(55)

where $P_l^m\left(cos\theta \right)$ are the Associated Legendre functions [14]. Therefore,

$$A_{lm}(\varphi ,\theta )=\sqrt{\frac{(2l+1)(l-m)!}{4\pi (l+m)!}}{P}_l^m\left(cos\theta \right)e^{im\varphi }=Y_l^m(\varphi ,\theta )$$

(56)

where $Y_l^m(\varphi ,\theta )$ are the spherical harmonic functions.

The amplitudes in the reflectivity basis are then

$${}^{{ϵ}_R}{Y}_l^{|m|}(\varphi ,\theta )=Y_l^m(\varphi ,\theta )-{ϵ}_{R}P{(-1)}^{(l-m)}{Y}_l^{-m}(\varphi ,\theta )$$

(57)

Since $P=P_1{P}_2{(-1)}^l={(-1)}^l$,

$${}^{{ϵ}_R}{Y}_l^{|m|}(\varphi ,\theta )=Y_l^m(\varphi ,\theta )-{ϵ}_R{(-1)}^m{Y}_l^{-m}(\varphi ,\theta )$$

(58)

and

$$I(\varphi ,\theta ,P,\Phi )=$$

$$\sum _k\sum _{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}\sum _{{ϵ}_R,{ϵ}_R^{\prime }}\sum _{l|m|,l^{\prime }|m^{\prime }|}{}^{{ϵ}_R}{Y}_l^{|m|}(\varphi ,\theta )T_{l,|m|}^{{ϵ}_R{ϵ}_{\gamma },k}{\rho }_{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}^{\gamma }(P,\Phi )T_{l^{\prime },|m^{\prime }|}^{{ϵ}_R^{\prime }{ϵ}_{\gamma }^{\prime },k*}{}^{{ϵ}_R^{\prime }}{Y}_{l^{\prime }}^{|m^{\prime }|\ast }(\varphi ,\theta ).$$

(59)

The same formalism can be used for the decay of the resonance into a vector ($s_1=1$) and a spinless ($s_1=0$) particle. In this case, $\lambda ={\lambda }_1=\pm 1$, and ${\lambda }_2=0$, $s=s_1=1$ and $s_2=0$ and $J=l\oplus s$.

Therefore, $(l0s\lambda |J\lambda )=(l01{\lambda }_1|J{\lambda }_1)$ and $(s_1{\lambda }_1{s}_2-{\lambda }_2|s\lambda )=(1{\lambda }_{1}00|1{\lambda }_1)=1$.

Then

$$A_{l,m}(\varphi ,\theta )=\sqrt{\frac{2l+1}{4\pi }}\sum _{{\lambda }_1=\pm 1}{D}_{m{\lambda }_1}^{J\ast }(\varphi ,\theta ,0)(l01{\lambda }_1|J{\lambda }_1)$$

(60)

and in the reflectivity basis,

$$A_{l,|m|}^{{ϵ}_R}(\varphi ,\theta )=\sqrt{\frac{2l+1}{4\pi }}\sum _{{\lambda }_1=\pm 1}[D_{m{\lambda }_1}^{J\ast }(l01{\lambda }_1|J{\lambda }_1)-{ϵ}_{R}P{(-1)}^{(l-m)}{D}_{-m{\lambda }_1}^{J\ast }(l01{\lambda }_1|J{\lambda }_1)]$$

(61)

5.1.2. Three${}^{+}$-Body Decays-Isobar Model Formalism

Let us consider now the case where the final particles are three or more. In the isobar formalism, we will treat the decay amplitude of the resonance as the product of successive isobar plus bachelor decay amplitudes [15].

$$A_{lm}\left(\tau \right)=A_{l^{\prime }{m}^{\prime }}\left({\tau }^{\prime }\right)A_{l^{\prime \prime }{m}^{\prime \prime }}\left({\tau }^{\prime \prime }\right)A_{l^{\prime \prime \prime }{m}^{\prime \prime \prime }}\left({\tau }^{\prime \prime \prime }\right)\dots$$

(62)

For example, let us consider a resonance decaying into a “di-particle” (isobar → two daughters) and a particle “bachelor”. The isobar will decay into two children (to consider more particles, the process is repeated).

The degrees of freedom (uncorrelated variables describing the kinematics) will include the mass of the isobar, w, and the angles of its decay products:

$$\tau =\left\{\begin{array}{c}{\Omega }_{GJ},{\Omega }_h,w\end{array}\right\}$$

(63)

where ${\Omega }_{GJ}=({\varphi }_{GJ},{\theta }_{GJ})$ and ${\Omega }_h=({\varphi }_h,{\theta }_h)$ are the angular descriptions in the Gottfried–Jackson and helicity frames (see reference [3], Appendix A, for definitions) of the isobar and its decay products, respectively. Let l be the angular momentum between the bachelor and isobar and s the spin of the isobar (we will consider a spinless bachelor). Therefore, $J=l\oplus s$. The amplitude is then written [16] as

$$A_{l,m,s}\left(\tau \right)=E_m^{Jls\ast }({\Omega }_{GJ},{\Omega }_h)Q_{ls}\left(w\right).$$

(64)

We factorize the amplitude with a factor that depends only on the angles, and a factor that only depends on mass. The mass factor comes from the propagator of the isobar. The angular factor can be written, in the isobar model, as

$$E_m^{Jls\ast }(\Omega ,{\Omega }_h)=\langle {\Omega }_h;0|{\widehat{T}}_{decay}^I|s\lambda \rangle \langle {\Omega }_{GJ};s\lambda |{\widehat{T}}_{decay}^R|Jm\rangle$$

(65)

where $R\to IB$ describes the decay of the resonance (R) into the isobar (I) and the bachelor (B), and $I\to D_1{D}_2$ is the decay of the isobar. Using our previous result, Equation (60), for each two-body decay, we have

$$\sqrt{\frac{2l+1}{4\pi }}\sum _{{\lambda }_1{\lambda }_2}{D}_{m\lambda }^{J\ast }\left({\Omega }_{GJ}\right)(l0s\lambda |J\lambda )(s_1{\lambda }_1{s}_2-{\lambda }_2|s\lambda ).$$

(66)

For the bachelor, ${\lambda }_2=0$, and for the isobar, ${\lambda }_1=\lambda$; therefore,

$$(s_1{\lambda }_{1}00|s\lambda )=1$$

(67)

then

$$\langle {\Omega }_{GJ};s\lambda |{\widehat{T}}_{decay}^R|Jm\rangle =\sqrt{\frac{2l+1}{4\pi }}{D}_{m\lambda }^{J\ast }({\varphi }_{GJ},{\theta }_{GJ},0)(l0s\lambda |J\lambda ).$$

(68)

And for the isobar,

$$\langle {\Omega }_h;0|{\widehat{T}}_{decay}^I|s\lambda \rangle =\sqrt{\frac{2s+1}{4\pi }}{D}_{\lambda 0}^{s\ast }({\varphi }_h,{\theta }_h,0)$$

(69)

Therefore,

$$E_m^{Jls\ast }({\Omega }_{GJ},{\Omega }_h)=$$

(70)

$$\sqrt{(2l+1)}\sqrt{2s+1}\sum _{\lambda }{D}_{m\lambda }^{J\ast }({\varphi }_{GJ},{\theta }_{GJ},0)D_{\lambda 0}^{s\ast }({\varphi }_h,{\theta }_h,0)\langle l0s\lambda |J\lambda \rangle .$$

(71)

Since

$$D_{\lambda 0}^{s\ast }({\varphi }_h,{\theta }_h,0)=e^{i\lambda {\varphi }_h}{d}_{\lambda 0}^s\left({\theta }_h\right)$$

(72)

and

$$D_{m\lambda }^{J\ast }({\varphi }_{GJ},{\theta }_{GJ},-{\varphi }_h)=D_{m\lambda }^{J\ast }({\varphi }_{GJ},{\theta }_{GJ},0)e^{-i\lambda {\varphi }_h}$$

(73)

the angular amplitude can then be written as

$$E_m^{Jls\ast }({\Omega }_{GJ},{\Omega }_h)=$$

$$\sqrt{(2l+1)}\sqrt{2s+1}\sum _{\lambda }{D}_{m\lambda }^{J\ast }({\varphi }_{GJ},{\theta }_{GJ},{\varphi }_h)d_{\lambda 0}^s\left({\theta }_h\right)\langle l0s\lambda |J\lambda \rangle .$$

(74)

The mass term depends on the isobar mass and is given by

$$Q_{ls}\left(w\right)=F_l\left(p\right)F_s\left(q\right)\Psi \left(w\right)$$

(75)

where the $\Psi$-function is the standard relativistic Breit–Wigner form for the isobar mass distribution, p is the momentum of the isobar in the GJ frame, and q id the momentum of the leading isobar’s decay particle in the helicity frame:

$$\Psi \left(w\right)=\frac{w_0{\Gamma }_0}{w_0^2-w^2-i{w}_0\Gamma \left(w\right)}$$

(76)

with

$$\Gamma \left(w\right)={\Gamma }_0\frac{w_{0}q{F}_s^2\left(q\right)}{w{q}_0{F}_s^2\left(q_0\right)}$$

(77)

$w_0$ and ${\Gamma }_0$ are the mass and width of the isobar, and $q_0$ is found such that $\Gamma \left(w_0\right)={\Gamma }_0$ and then $|\Psi (w_0)|=1$. The $F_l\left(p\right)$ and $F_s\left(q\right)$ functions are the Blatt–Weisskopf centrifugal barrier factors (discussed before). Adding all these components into our final form for the amplitude for three (spinless) particles in the final state, we obtain [15]

$$A_{J,l,m,s}({\Omega }_{GJ},{\Omega }_h,w)=\sqrt{(2l+1)}\sqrt{2s+1}$$

(78)

$$F_l\left(p\right)F_s\left(q\right)\frac{w_0{\Gamma }_0}{w_0^2-w^2-i{w}_0\Gamma \left(w\right)}\times \sum _{\lambda }{D}_{m\lambda }^{J\ast }({\varphi }_{GJ},{\theta }_{GJ},{\varphi }_h)d_{\lambda 0}^s\left({\theta }_h\right)\langle l0s\lambda |J\lambda \rangle .$$

5.2. Mass-Independent Fit

The probability of observing an event i with properties ${\tau }_i$ in the $\Delta E\Delta M\Delta t$ bin is

$$p_i=\frac{I\left({\tau }_i\right)\eta \left({\tau }_i\right)}{N}=\frac{I\left({\tau }_i\right)\eta \left({\tau }_i\right)}{\int I\left(\tau \right)\eta \left(\tau \right)d\tau }.$$

(79)

where $\eta \left(\tau \right)$ is the detector acceptance. The value of N, the average number of events expected to be observed in the total phase-space defined by $\Delta E\Delta M\Delta t$, is calculated numerically through a Monte Carlo (MC) full simulation of the detector and a (flat) phase-space generator of the reaction. In many cases, due to limited statistics, the binning is performed only in M; therefore, a model for the t cross-section dependence is introduced in the MC. The numerical (MC) value of N is then

$$N=\frac{1}{N_g}\sum _i^{N_g}I\left({\tau }_i\right)\eta \left({\tau }_i\right)$$

(80)

$N_g$ is the total number of events generated in the MC. The function $\eta \left(\tau \right)$ represents the acceptance (resolution is taken to be perfect, and only acceptance is considered here—no inter-bin crosstalk). A Monte Carlo simulation of the detector will provide the values of this function, which are $\eta \left(\tau \right)=1$ if the event is accepted and $\eta \left(\tau \right)=0$ if the event is not accepted. Then

$$N=\frac{1}{N_g}\sum _i^{N_a}I\left({\tau }_i\right)$$

(81)

where $N_a$ is the total number of accepted events. Introducing ${\eta }_x=\frac{N_a}{N_g}$ as the total fraction of events accepted, or total acceptance, then

$$N={\eta }_x\frac{1}{N_a}\sum _i^{N_a}I\left({\tau }_i\right)$$

(82)

Therefore,

$$N={\eta }_x\frac{1}{N_a}\sum _i^{N_a}\left[\sum _k\sum _{{ϵ}_{\gamma }{ϵ}_R,{ϵ}_{\gamma }^{\prime }{ϵ}_R^{\prime }}\sum _{lm,l^{\prime }{m}^{\prime }}{A}_{l,m}^{{ϵ}_R}\left({\tau }_i\right)T_{l,m}^{{ϵ}_{\gamma }{ϵ}_R,k}{\rho }_{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}^{\gamma }{T}_{l^{\prime },m^{\prime }}^{{ϵ}_{\gamma }^{\prime }{ϵ}_R^{\prime },k\ast }{A}_{l^{\prime },m^{\prime }}^{{ϵ}_R^{\prime },\ast }\left({\tau }_i\right)\right].$$

(83)

The extended likelihood is defined as including the probability of observing N events by

$$L=Prob\left(N\right)\prod _{i=1}^{N}p({\overrightarrow{x}}_i,\overrightarrow{a}).$$

(84)

Assuming a Poisson distribution for the probability of observing N events, with an expected value of N

$$Prob\left(N\right)=\frac{N^N}{N!}{e}^{-N}$$

(85)

the extended likelihood is then

$$L=\left[\begin{array}{c}\frac{N^N}{N!}{e}^{-N}\end{array}\right]\prod _{i=1}^{N}p({\overrightarrow{x}}_i,\overrightarrow{a})$$

(86)

and taking the log results in

$$lnL=-ln\left[\begin{array}{c}N!\end{array}\right]-N+\sum _{i=1}^{N}ln\left[\begin{array}{c}I({\overrightarrow{x}}_i,\overrightarrow{a})\end{array}\right].$$

(87)

Therefore,

$$lnL\propto \sum _{i=1}^{N}ln\left[\begin{array}{c}I({\overrightarrow{x}}_i,\overrightarrow{a})\end{array}\right]-N$$

(88)

Including the expression for the intensity into the likelihood function, we have

$$-lnL\propto -\sum _{i=1}^{N}ln\left[\begin{array}{c}\sum _k\sum _{\alpha ,{\alpha }^{\prime }}{A}_{\alpha }\left({\tau }_i\right)T_{\alpha }^k{\rho }_{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}^{\gamma }{T}_{{\alpha }^{\prime }}^{k\ast }{A}_{{\alpha }^{\prime }}^{\ast }\left({\tau }_i\right)\end{array}\right]+N$$

(89)

where we included all quantum numbers in $\alpha$ and the external target spins in k. This is the function to be minimized to obtain the $T_{\alpha }^k$ values [3]. To find the true number of events in the $\Delta E\Delta M\Delta t$ bin, which we will call $N_{true}$, we take

$$N_{true}=\frac{1}{N_g}\sum _i^{N_g}I\left({\tau }_i\right)$$

(90)

where we will use the fitted $T_{\alpha }^k$ values. Then

$$N_{true}=\frac{1}{N_g}\sum _i^{N_g}\left[\sum _k\sum _{\alpha ,{\alpha }^{\prime }}{A}_{\alpha }\left({\tau }_i\right)T_{\alpha }^k{\rho }_{{ϵ}_{\gamma },{ϵ}_{\gamma }^{\prime }}^{\gamma }{T}_{{\alpha }^{\prime }}^{k\ast }{A}_{{\alpha }^{\prime }}^{\ast }\left({\tau }_i\right)\right]$$

(91)

and the yield for each partial wave ($\alpha$, for a given k) is

$$N_{\alpha ,true}=\frac{1}{N_g}\sum _i^{N_g}{\rho }_{{ϵ}_{\gamma },{ϵ}_{\gamma }}^{\gamma }|T_{\alpha }{A}_{\alpha }\left({\tau }_i\right){|}^2.$$

(92)

After we obtain the $T_{\alpha }^k$ values, we are able to generate MC events through our partial wave model and many predicted distributions of data properties (i.e., angular distributions, t-distributions, etc.) to compare directly with the data and check our model accuracy. We can also use the phase of the production amplitudes to obtain information about the resonant behavior of a particular wave. A single wave phase is arbitrary but the difference of phases between two waves contains physical information. We use the phase difference between the wave under study and a well-established resonant wave (see reference [3] for details).

6. Phenomenological Models

After performing mass-independent fits in each bin of M (or M and t) for a given $\Delta E$, we obtained the predicted mass distribution of $N_{true}\left(M\right)$ for each partial wave included in the fit. Nevertheless, merely identifying peaks in the mass spectrum falls short of substantiating the existence of a resonance. In the past, the mass dependence of those partial waves has been described by a coherent sum of Breit–Wigner amplitudes and, if needed, a phenomenological model of the background or other effects (i.e., Deck mechanisms) [3,5]. Such a procedure can produce a good fit to the data; however (especially using the isobar approximation), it violates fundamental principles such as probability conservation and causality. Therefore, in order to obtain more physically grounded amplitudes, models that fulfill the principles of unitarity and analyticity (which originate from probability conservation and causality) are to be used. Unitarity is especially important when we deal with resonances since it controls resonance widths and pole positions in the complex energy plane. One will first look for regions of enhancement (peaks or valleys) in the distributions and fit a theoretically based distribution to obtain the resonance properties (mass and width). However, interference and overlapping can greatly disturb the appearance of the spectrum. The properties (and positions) of the resonances should be obtained from the poles on the complex amplitudes of the S-Matrix expansion [17]. These poles (and thresholds) had been studied using the Regge treatment of the S-matrix [8]. Resonances are poles in the complex plane (Riemann surfaces) and only their projected real axis values can then be evaluated experimentally. In the case of multiple poles with the same quantum numbers and/or poles far from the real axis, the axis projections can deviate considerably from the BW distribution. The shape of these distributions is also influenced by the QCD dynamics. Effective field theories, i.e., Chiral Perturbation Theory, has been combined with the dispersion relations to obtain better parameterization of the mass distributions [18]. Recent studies (i.e., reference [19,20]) have used those approaches to obtain mass and width values for several resonances. A comprehensive description of recent efforts by the Joint Physics Analysis Center (JPAC) in this direction can be found in reference [21].

7. Summary

We described a formalism that introduces parity conservation into the transition amplitudes for the description of photo production of mesons. Two reflectivities are defined by applying, independently, the reflectivity operator to the resonance decay amplitude and to the incoming photon (beam) state. These are two (reflectivity) quantum numbers, the product of which, at least at higher energies, coincides with the naturality of the exchange particle in the t-production channel. Notice that the definition of two reflectivities is suited for any number and spins of particles into which the resonance can finally decay in the final state. This two-reflectivity formalism might also be used in more refined models of the S-matrix phenomenology, i.e., Regge-inspired models respecting unitarity and analyticity. As a simple example, in this paper, we showed the formalism for a mass-independent analysis, and in the case of more than three particles in the final state, we used the isobar model approximation.