---
abstract: |
    The abundant $\psi'$ events have been collected at the Beijing Electron Positron Collider-II (BEPCII) that could undoubtedly provide us with a great opportunity to study the more attractive charmonium decays. As has been noticed before, in the process of $J/\psi$ decaying to the baryonic final states, $p K^-
    \overline{\Lambda}$, the evident $\Lambda^*$ and $N^*$ bands have been observed. Similarly, by using the product of $\chi_{cJ}$ from $\psi'$ radiative decay, we may confirm it or find some extra new resonances. $\chi_{c0}$’s data samples will be more than $\chi_{c1,2}$ taking into account the larger branching ratio of $\psi'\to\gamma\chi_{c0}$. Here, we provide explicit partial wave analysis formulae for the very interesting channel $\psi'\to\gamma\chi_{c0}\rightarrow \gamma p K^-
    \overline{\Lambda}$.
author:
- 'Xian-Wei Kang$^{1,2}$'
- 'Hai-Bo Li$^2$'
- 'Gong-Ru Lu$^1$'
- 'Bing-Song Zou$^2$'
title: |
    **Partial wave analysis of $\psi'\to\gamma\chi_{c0}\rightarrow \gamma p K^- \overline{\Lambda}$\
    being used for searching for baryon resonance**
---

Introduction
============

In experiment at BESIII, about $10\times10^9 J/\Psi$ and $3\times10^9 \psi'$ events can be collected per year’s running according to the designed luminosity of BEPCII in Beijing [@besiii] [@bepcii]. These large data samples will provide great opportunities to study the attractive $\chi_{cJ}$ decay and some hyperon interactions. The product of $\chi_{cJ}$ in $\psi'$ radiative decays may provide useful information on two-gluon hadronization dynamics and glueball decays. On the other side, the radiative decays of $\psi'\to\gamma\chi_{cJ}$ are expected to be dominated by electric dipole (E1) transitions, with higher multipoles suppressed by powers of photon energy divided by quark mass [@Karl] and searching for contributions of higher multipoles is promissing. The possibility of anomalous magnetic moments of higher quark being lager than those for light ones may exist [@Geffen]. Thus,$\chi_{cJ}$ decays contain abundant interesting physics.

In this paper, the idea is motivated by that there may exist some baryon resonances in the process of $\chi_{cJ}$ decays to baryons. In fact, in our preliminary Monte Carlo study, we found an explicit band structure, most possibly $\Lambda(1520)$, so it necessitate this PWA. In order to get more useful information about the resonance properties such as $J^{PC}$ quantum numbers, mass, width, production and decay rates,[*[etc.]{}*]{}, partial wave analysis (PWA) are necessary. PWA is an effective method for analyzing the experimental data of hadron spectrums. There are two methods of PWA, one is based on the covariant tensor (also named Rarita-Schwinger) formalism [@Schwinger] and the other is based on the original helicity formalism [@Jackson] [@Jacob] and one covariant helicity one developed by Chung [@Chung1] [@Chung2]. For a more basic exposition, the reader may wish to consult the CERN Yellow Report [@yellow; @report]. Ref.[@Filippini] showed the connection between the covariant tensor formalism and helicity one. In this short paper, we will pay more attention to the more popular one: covariant tensor format also append the helicity one for the specific process for $\Lambda(1520)$.

The organization of the paper is as follows. In Sec.I, the general formalism is given. In Sec.II, we will present the covariant tensor amplitude for $\psi'\to\gamma\chi_{c0}\rightarrow \gamma P K^-
\overline{\Lambda}$. In Sec.III,the corresponding helicity formula are provides. At last, In Sec.IV,there is the conclusion.

General formalism {#Sec:formalism}
=================

In this part,the general formalism which will be used in the following have been mentioned in Ref.[@psi; @decay; @to; @mesons] [@N*NM; @couplings] [@gamma; @chicJ], including formalism for $\psi$ radiative decay to mesons(denoted by $M$),$M\to N^*
N$,$N^*\to N M$,where $N^*$ and $N$ has the half integer spin.

As discussed in Ref.[@psi; @decay; @to; @mesons],we denote the $\psi$ polarization four-vector by $\psi_{\mu}(m_1)$ and the polarization vector of the photon by $e_{\nu}(m_2)$.Then the general form for the decay amplitude is $$A=\psi_{\mu}(m_1)e^*_{\nu}(m_2)A^{\mu\nu}=\psi_{\mu}(m_1)e^*_{\nu}(m_2)\underset{i}{\sum}\Lambda_i
U^{\mu\nu}_i$$ here, $U^{\mu}_{i}$ is the $i-$th partial wave amplitude with coupling strength determined by a complex parameter $\Lambda_i$. Because of massless properties,there are two additional conditions,$(1)$ the usual orthogonality condition $e_{\nu}q^{\nu}=0$,where $q$ is the photon momentum;$(2)$ gauge invariance condition (assuming the Coulomb gauge in $\psi$ rest system) $e_{\nu}p^{\nu}_{\psi}=0$,where $P_{\psi}$ is the momentum of vector meson $\psi$.Then we have $$\begin{aligned}
\underset{m}{\sum}e^*_{\mu}(m)e_{\nu}(m)&=&-g_{\mu\nu}+\frac{q_{\mu}K_{\nu}+K_{\mu}q_{\nu}}{q\cdot
K}-\frac{K \cdot K}{(q \cdot K)^2}q_{\mu}q_{\nu}\nonumber\\ &\equiv&
-g_{\mu\nu}^{(\perp\perp)}\end{aligned}$$ with $K=p_{\psi}-q $ and $ e_{\nu}K^{\nu}=0.$.To compute the differential cross section,we need an expression for $|A|^2$ ,the square modulus of the decay amplitude,which gives the decay probability of a certain configuration should be independent of any particular frame.Thus the radiative cross section is : $$\begin{aligned}
\frac{d\sigma}{d\Phi_n}&=&\frac{1}{2}\sum^2_{m_1=1}\sum^2_{m_2=1}\psi_{\mu}(m_1)e^*_{\nu}(m_2)A^{\mu\nu}\psi^*_{\mu'}(m_1)e_{\nu'}(m_2)A^{*\mu'\nu'}
\nonumber\\
&=&-\frac{1}{2}\sum^2_{m_1=2}\psi_{\mu}(m_1)\psi_{\mu'}(m_1)g^{(\perp
\perp)}_{\nu\nu'}A^{\mu\nu}A^{*\mu'\nu'}\nonumber\\
&=&-\frac{1}{2}\sum^2_{\mu=1}A^{\mu\nu}g^{(\perp\perp)}_{\nu\nu'}A^{*\mu\nu'}\nonumber\\
&=&-\frac{1}{2}\sum_{i,j}\Lambda_i\Lambda_j^*\sum^2_{\mu=1}U^{\mu\nu}_ig^{(\perp\perp)}_{\nu\nu'}U_j^{*\mu\nu'}\equiv\sum_{i,j}P_{ij}\cdot
F_{ij}\end{aligned}$$ with definition $$\begin{aligned}
P_{ij}&=&P_{ji}^*=\Lambda_i\Lambda_j^*, \\F_{ij}&=&F_{ji}^*
=-\frac{1}{2}\sum^2_{\mu=1}=-\frac{1}{2}U^{\mu\nu}_i
g^{(\perp\perp)}_{\nu\nu'}U_j^{*\mu\nu'}.\end{aligned}$$ note the relation $$\sum^2_{m=1}\psi_{\mu}(m)\psi_{\mu'}^*(m)=\delta_{\mu\mu'}(\delta_{\mu1}+\delta_{\mu2}).$$ The partial wave amplitude $U$ in the covariant Rarita-Schwinger tensor formalism [@Schwinger] can be constructed by using pure orbital angular momentum covariant tensor $\widetilde{t}^{(L)}_{\mu_1\mu_2\cdots\mu_L}$ and covariant spin wave functions $\phi_{\mu_1\mu_2\cdots\mu_S}$ together with the metric tensor $g^{\mu\nu}$, the totally antisymmetric Levi-Civita tensor $\epsilon_{\mu\nu\lambda\sigma}$ and the four momenta of participating particles.For a process $a\to bc$ ,if there exists a relative orbital angular momentum $L_{bc}$ between the particle $b$ and $c$ ,then the pure orbital angular momentum $L_{bc}$ state can be represented by the covariant tensor wave function $\widetilde{t}^{(L)}_{\mu_1\mu_2\cdots\mu_L}$ which is built of the relative momentum.Here,we list the amplitude for pure $S-,P-,D-,$ and $F-$ wave orbital angular momentum: $$\begin{aligned}
\widetilde{t}^{(0)}&=&1,\\
\widetilde{t}^{(1)}_{\mu}&=&\widetilde{g}_{\mu\nu}(p_a)r^{\nu}B_1(Q_{abc})\equiv
\widetilde{r}^{\mu}B_1(Q_{abc}),\\
 \widetilde{t}^{(2)}_{\mu\nu}&=&[\widetilde{r}^{\mu}\widetilde{r}^{\nu}-\frac{1}{3}(\widetilde{r}\cdot\widetilde{r})\widetilde{g}_{\mu\nu}(p_a))]B_2(Q_{abc}),\end{aligned}$$ $$\begin{aligned}
 \widetilde{t}^{(3)}_{\mu\nu\lambda}&=[\widetilde{r}_{\mu}\widetilde{r}_{\nu}\widetilde{r}_{\lambda}-\frac{1}{5}(\widetilde{r}\cdot\widetilde{r})(\widetilde{g}_{\mu\nu}(p_a)\widetilde{r}_{\lambda}\nonumber \\
  &\quad+\widetilde{g}_{\nu\lambda}(p_a)\widetilde{r}_{\mu}+\widetilde{g}_{\lambda\mu}(p_a)\widetilde{r}_{\nu})]B_3(Q_{abc}),\end{aligned}$$ where $r=p_b-p_c$ is the relative momentum of the two decay products in the parent particle rest frame,$\widetilde{r}\cdot\widetilde{r}=\vec{r}\cdot\vec{r}$ where $\vec{r}$ is the magnitude of three-vector , with $$\widetilde{g}_{\mu\nu}(p_a)=-g_{\mu\nu}+\frac{p_{a\mu}p_{a\nu}}{p_a^2}$$ which is the vector boson polarization sum relation, and $$Q^2_{abc}=\frac{(s_a+s_b-s_c)^2}{4s_a}-s_b$$ where $s_a=E_a^2-p_a^2$ and $B_l(Q_{abc})$ is the Blatt-Weisskopf barrier factor [@Hippel],explicitly, $$\begin{aligned}
B_1(Q_{abc})=&\sqrt{\frac{2}{Q_{abc}^2}+Q_0^2}\\
B_2(Q_{abc})=&\sqrt{\frac{13}{Q_{abc}^4}+3Q_{abc}^2Q_0^2+9Q_0^4},\\
B_3(Q_{abc})=&\sqrt{\frac{277}{Q_{abc}^6}+6Q_{abc}^4Q_0^2+45Q_{abc}^2Q_0^4+225Q_0^6}\end{aligned}$$ Here $Q_0$ is a hadron scale parameter $Q_0=0.197321/R GeV/c$,in which $R$ is the radius of the centrifugal barrier in fm.

If $a$ is an intermediate resonance decaying into $bc$,one needs to introduce into the amplitude a Breit-Wigner propergator [@primer] $$f^{(a)}_{(bc)}=\frac{1}{m_a^2-s_{bc}-im_a\Gamma_a}$$ In this equation,$s_{bc}=(p_b+p_c)^2$ is the invariant mass-squared of $b$ and $c$; $m_a,\Gamma_a$ are the resonance mass and width.

Additionally,some expressions depend also on the total momentum of the $ij$ pair,$p_{(ij)}=p_i+p_j$.When one wants to combine two angular momenta $j_b$ and $j_c$ into a total angular momentum $j_a$,if $j_a+j_b+j_c$ is an odd number,then a combination $\epsilon_{\mu\nu\lambda\sigma}p^{\mu}_{a}$ with $p_a$ the momentum of the parent particle is needed,otherwise it is not needed.

For a given hadronic decay process $A\rightarrow BC$ (B,C are fermions),in the $L-S$ scheme on hadronic level,the initial state is described by its $4-$ momentum $P_{\mu}$ and its spin state $S_A$,the final state is described by the relative orbital angular momentum state of $BC$ system and their spin state $(S_B,S_C)$.The spin states $(S_A,S_B,S_C)$ can be well represented by the relativistic Rarita-Schwinger spin wave functions for particles of arbitrary spin.As is well known that,spin-$\frac{1}{2}$ wavefunction is the standard Dirac spinor $U(p,s)$ and $V(p,s)$ ;spin-$1$ wavefunction is the standard spin-$1$ polarization four-vector $\epsilon^{\mu}(p,s)$ for particle with momentum $p$ and spin projection $s$.(1)For the case of A as a meson,B as $N^*$ with spin $n+\frac{1}{2}$ and C as $\overline{N}$ with spin $\frac{1}{2}$ ,the total spin of BC $(S_{BC})$ can be either $n$ or $n+1$. The two $S_{BC}$ states can be represented as [@N*NM; @couplings] $$\begin{aligned}
\psi^{(n)}_{\mu_1\mu_2\cdots\mu_n}=&\quad
\bar{u}_{\mu_1\mu2\cdots\mu_n}(p_B,s_B)\gamma_5
v(p_C,s_C),\\
\Psi^{(n+1)}_{\mu_1\mu_2\cdots\mu_{n+1}}=&\quad
\bar{u}_{\mu_1\mu2\cdots\mu_n}(\gamma_{\mu_{n+1}}-\frac{r_{\mu_{n+1}}}{m_A+m_B+m_C}v(p_c,s_C))
\nonumber\\&+(\mu_1\leftrightarrow\mu_{n+1})+\cdots+(\mu_{n}\leftrightarrow\mu_{n+1})\end{aligned}$$ (2)For the case of A as $N^*$ with spin $n+\frac{1}{2}$,B as $N$ and C as a meson,one needs to couple $-S_A$ and $S_B$ first to get $S_{AB}=-S_A+S_B$ states,which are $$\begin{aligned}
\phi^{(n)}_{\mu_1\mu_2\cdots\mu_n}=&\bar{u}(p_b,s_B)u_{\mu_1\mu_2\cdots\mu_n}(p_A,s_A),\\
\Phi^{n+1}_{\mu_1\mu_2\cdots\mu_n}=&\bar{u}(p_b,s_B)\gamma_5\widetilde{\gamma}_{\mu_1\mu_2\cdots\mu_n}(p_A,s_A)\nonumber\\
&+(\mu_1\leftrightarrow\mu_{n+1})+\cdots+(\mu_{n}\leftrightarrow\mu_{n+1})\end{aligned}$$

Up to now,we have introduced all knowledges for constructing the covariant tensor amplitude.In the concrete case,the P parity conservation may be applied,which expression is $$\label{parity}
\eta_A=\eta_B\eta_C(-1)^{L}$$ where $\eta_A$,$\eta_B$ and $\eta_C$ are the intrinsic parities of particles A, B, and C, respectively.From this relation,L can be even or odd for one case,which guarantee a pure L final state,which is the soul of covariant $L-S$ coupling scheme.

analysis for $\psi'\rightarrow \gamma \chi_{c0}\rightarrow \gamma P K^- \overline{\Lambda}$
===========================================================================================

From now on,we denote $P$,$K^-$,$\overline{\Lambda}$ by number $1,2,3$.Firstly,for $\psi'\to \gamma \chi_{c0}$,from the helicity formalism,it is easy to show that there is only one independent amplitude for $\psi'$ radiative deca y to a spin $0$ meson.Hense,the amplitude is $$U^{\mu\nu}_{\gamma \chi_{c0}}=g^{\mu\nu}f^{(\chi_{c0})}.$$ For sequential $\chi_{c0}$ decay,there may be the following modes: $\chi_{c0}\to\Lambda_{x}\overline{\Lambda},\Lambda_x\rightarrow P
K^-$,where $\Lambda_{x}$ can be\
$\Lambda(1520)\frac{3}{2}^-,\Lambda(1600)\frac{1}{2}^+,
\Lambda(1670)\frac{1}{2}^-,\Lambda(1690)\frac{3}{2}^-,\Lambda(1800)\frac{1}{2}^-,\\
\Lambda(1810)\frac{1}{2}^+,\Lambda(1820)\frac{5}{2}^+,\Lambda(1830)\frac{5}{2}^-,\Lambda(1890)\frac{3}{2}^+,
\Lambda(2100)\frac{7}{2}^-,\\ \Lambda(2110)\frac{5}{2}^+;$\
$\chi_{c0}\rightarrow\overline{N}\overline{\Lambda},\overline{N}\rightarrow
\overline{\Lambda}K^-$,where $\overline{N}$ is the anti-partner of hyperon $N$,with that $N$ can be\
$N(1650)\frac{1}{2}^-$,$N(1675)\frac{5}{2}^-$,$N(1700)\frac{3}{2}^-$,$N(1710)\frac{1}{2}^+$ or $N(1720)\frac{3}{2}^+$.\
Another possibility that $P
\overline{\Lambda}$ may be generated from an unknown intermediate resonance $K_x$ is also taken into account.Amplitudes for up to $K_x's$ spin-4 are given. For $\Lambda_x$ being $\Lambda(1520)\frac{3}{2}^-$,the total spin of $\Lambda(1520)$ and $\overline{\Lambda}\frac{1}{2}^-$ can be $1$ or $2$,corresponding to the $P$ wave and $D$ wave respectively,because of the special property of spin-$0$ of $\chi_{c0}$.The parity relation makes $P$ wave impossible.Considering that this channel is recognized as a meson decaying to two fermions,now one can write the covariant amplitude as $$\Phi_{(\mu\nu)}^{(2)}\widetilde{t}^{(2)\mu\nu}.$$ And then considering $\Lambda(1520)\rightarrow P K^-$,the total spin of particle $1$ and $2$ can only be $\frac{1}{2}$,corresponding to the $P$ wave and $D$ wave,after the parity formula being applied,$L$ must be $2$.As it belongs to a fermion decay to a fermion and a meson,the covariant amplitude can be expressed as $\Phi^{(2)_{\mu\nu}}\widetilde{t}^{(2)\mu\nu}$.

Here,we list all the amplitudes for the whole decay chain $\psi'\to\gamma\chi_{c0},\chi_{c0}\rightarrow\Lambda_x\overline{\Lambda},\Lambda_x\rightarrow
P K^-$ up to spin-$\frac{7}{2}$ for $\Lambda_x$ according to the above principles. $$\begin{aligned}
\Lambda_x(\frac{1}{2}^+)\qquad
U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\Phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma}
\\ \Lambda_x(\frac{1}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{\chi_{c0}}_{(123)}\psi^{(0)}\phi^{(0)}
\\\Lambda_x(\frac{3}{2}^+)\qquad
U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma}
\\\Lambda_x(\frac{3}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\Phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma}
\\\Lambda_x(\frac{5}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\Phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta}
\\\Lambda_x(\frac{5}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma}
\\\Lambda_x(\frac{7}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta}
\\\Lambda_x(\frac{7}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(4)}_{\lambda\delta\beta\xi}\widetilde{t}^{(4)\lambda\delta\beta\xi}\Phi^{(4)}_{\rho\sigma\eta\zeta}\widetilde{t}^{(4)\rho\sigma\eta\zeta}\end{aligned}$$note that $\widetilde{t}^{(0)}=1$.For channel $\chi_{c0}\rightarrow
\overline{N_x}P,\overline{N_x}\rightarrow K^-\overline{\Lambda}$,we can imitate the amplitude up to spin-$\frac{7}{2}$ for $N_x$ without any difficulty,even through the highest spin for $N_x$ decaying into $K^-\Lambda$ can only be $\frac{5}{2}$ presently [@pdg2008]. $$\begin{aligned}
\overline{N_x}(\frac{1}{2}^+)\qquad
U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(0)}\phi^{(0)}
\\\overline{N_x}(\frac{1}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\Phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma}
\\\overline{N_x}(\frac{3}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\Phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma}
\\\overline{N_x}(\frac{3}{2}^-)\qquad
U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma}
\\\overline{N_x}(\frac{5}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma}
\\\overline{N_x}(\frac{5}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\Phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta}
\\\overline{N_x}(\frac{7}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(4)}_{\lambda\delta\beta\xi}\widetilde{t}^{(4)\lambda\delta\beta\xi}\Phi^{(4)}_{\rho\sigma\eta\zeta}\widetilde{t}^{(4)\rho\sigma\eta\zeta}
\\\overline{N_x}(\frac{7}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta}\end{aligned}$$ For channel $\chi_{c0}\rightarrow K^+_x K^-,K^+_x\rightarrow P
\overline{\Lambda}$,the amplitudes are also given.$J^{P}=0^+,1^-,2^+,3^-,4^+ \cdots $ are forbidden by the parity relation[@Parity].The partial wave amplitude is denoted by $U^{\mu\nu}_{(LS)}$, $L,S$ means the orbital angular momentum number and spin angular momentum number between $P$ and $\overline{\Lambda}$. $$\begin{aligned}
K_x^+(0^-)\qquad
U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(0)}
\\K_x^+(1^+)\qquad
U^{\mu\nu}_{(10)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(0)}\widetilde{T}^{(1)\sigma}\phi_{\sigma}\epsilon_{\lambda}\widetilde{t}^{(1)\lambda}
\\U^{\mu\nu}_{(11)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\epsilon_{\sigma}\widetilde{t}^{(1)}_{\eta}\Psi^{(1)}_{\zeta}\widetilde{T}^{(1)\sigma}\phi_{\sigma}
\\K_x^+(2^-)\qquad
U^{\mu\nu}_{(20)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\widetilde{T}^{(2)\eta\zeta}\phi_{\eta\zeta}\widetilde{t}^{(2)\rho\sigma}\psi^{(0)}\phi_{\rho\sigma}\nonumber\\
&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(2)}_{\rho\sigma\eta\zeta}\psi^{(0)}\widetilde{T}^{(2)\eta\zeta}\widetilde{t}^{(2)\rho\sigma}
\\U^{\mu\nu}_{(21)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{T}^{(2)\beta\lambda}\phi_{\beta\lambda}\widetilde{t}^{(2)\iota}_{\sigma}\phi_{\iota\eta}\Psi^{(1)}_{\zeta}
\nonumber\\&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(2)}_{\beta\lambda\iota\eta}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{T}^{(2)\beta\lambda}\widetilde{t}^{(2)\iota}_{\sigma}\Psi^{(1)}_{\zeta}
\\K_x^+(3^+)\qquad
U^{\mu\nu}_{(30)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\widetilde{T}^{(3)\lambda\delta\beta}\phi_{\lambda\delta\beta}\psi^{(0)}\widetilde{t}^{(3)\rho\sigma\eta}\phi_{\rho\sigma\eta}\nonumber
\\&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(3)}_{\lambda\delta\beta\rho\sigma\eta}\widetilde{T}^{(3)\lambda\delta\beta}\psi^{(0)}\widetilde{t}^{(3)\rho\sigma\eta}
\\U^{\mu\nu}_{(31)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\widetilde{T}^{(3)\lambda\delta\beta}\phi_{\lambda\delta\beta}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{t}^{(3)\kappa\xi}_{\sigma}\Psi^{(1)}_{\eta}\cdot\nonumber
\\ &\quad\phi^{(3)}_{\zeta\kappa\xi}\nonumber\\
&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(3)}_{\lambda\delta\beta\zeta\kappa\xi}\widetilde{T}^{(3)\lambda\delta\beta}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{t}^{(3)\kappa\xi}_{\sigma}\Psi^{(1)}_{\eta}\end{aligned}$$ In the above formulas,$\phi$ implies the spin wave functions for $K_x$,for example,$\phi_{\sigma}$ corresponding to the spin-1,$\phi_{\rho\sigma}$ corresponding to spin-2.There is a general wave fuction for a particle of spin $J$ [@Chung1],which is a rank-$J$ tensor $$\begin{aligned}
\phi^{\alpha_1\alpha_2\cdots\alpha_J}(m)&=\underset{m_1m_2\cdots}{\sum}\langle
1m_11m_2|2n_1\rangle\langle
2n_11m_3|3n_2\rangle\nonumber\\
&\quad\cdots\langle
J-1n_{J-2}1m_J|Jm\rangle\phi^{\alpha_1}(m_1)\nonumber\\
&\quad\phi^{\alpha_2}(m_2)\cdots\phi^{\alpha_J}(m_J)\end{aligned}$$ where $\phi^{\alpha}(m)$ is the familiar polarization four-vector of spin-1 particle, $$\phi^{\alpha}(1,-1)=\mp\frac{1}{\sqrt{2}}(0;1,\pm i,0),
\phi^{\alpha}(0)=(0;0,0,1).$$ note the following useful relationship: $$\phi(-m)=(-)^m\phi^*(m).$$ It is best to illustrate these formulas with some examples.For $J=1$,one finds that it reduces to identities for $\phi(1)$ and $\phi(0)$.For $J=2$,one has $$\begin{gathered}
\qquad\qquad\phi^{\alpha\beta}(+2)=\phi^{\alpha}(1)\phi^{\beta}(1) \\
\phi^{\alpha\beta}(+1)=\frac{1}{\sqrt{2}}[\phi^{\alpha}(1)\phi^{\beta}(0)+\phi^{\alpha}(0)\phi^{\beta}(1)]\\
\phi^{\alpha\beta}(0)=\frac{1}{\sqrt{6}}[\phi^{\alpha}(1)\phi^{\beta}(-1)+\phi^{\alpha}(-1)\phi^{\beta}(1)+\sqrt{\frac{2}{3}}\phi^{\alpha}(0)\phi^{\beta}(0)]\end{gathered}$$ $\phi\cdot\phi$ is in reality its spin projection operator $P^{(S)}$. $$\begin{aligned}
P^{(2)}_{\rho\sigma\eta\zeta}(p_{K_x})&=\underset{m}{\sum}\phi_{\rho\sigma}(p_{K_x},m)\phi^*_{\eta\zeta}(p_{K_x,m})\nonumber\\&=\frac{1}{2}(\widetilde{g}_{\rho\eta}
\widetilde{g}_{\sigma\zeta}+\widetilde{g}_{\rho\zeta}\widetilde{g}_{\sigma\eta})-\frac{1}{3}\widetilde{g}_{\rho\sigma}\widetilde{g}_{\eta\zeta}
\\\quad P^{(3)}_{\lambda\delta\beta\zeta\kappa\xi}(p_{K_x})&=\underset{m}{\sum}\phi_{\lambda\delta\beta}(p_{K_x},m)\phi^*{\zeta\kappa\xi}(p_{K_x},m)\nonumber\\&=\frac{1}{6}(\widetilde{g}_{\lambda\zeta}\widetilde{g}_{\delta\kappa}\widetilde{g}_{\beta\xi}
+\widetilde{g}_{\lambda\zeta}\widetilde{g}_{\delta\xi}\widetilde{g}_{\beta\kappa}+\widetilde{g}_{\lambda\kappa}\widetilde{g}_{\delta\zeta}\widetilde{g}_{\beta\xi}
\nonumber\\&\quad\quad+\widetilde{g}_{\lambda\zeta}\widetilde{g}_{\delta\xi}\widetilde{g}_
{\beta\kappa}+\widetilde{g}_{\delta\xi}\widetilde{g}_{\beta\zeta}\widetilde{g}_{\lambda\xi}+\widetilde{g}_{\delta\kappa}\widetilde{g}_{\lambda\xi}\widetilde{g}_{\beta\zeta})
\nonumber\\&\quad-\frac{1}{15}(\widetilde{g}_{\lambda\delta}\widetilde{g}_{\zeta\kappa}\widetilde{g}_{\beta\xi}+\widetilde{g}_{\lambda\delta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\beta\zeta}+\widetilde{g}_{\lambda\delta}\widetilde{g}_{\zeta\xi}\widetilde{g}_{\beta\kappa}
\nonumber\\&\quad\quad+\widetilde{g}_{\lambda\beta}\widetilde{g}_{\zeta\xi}\widetilde{g}_{\delta\kappa}+\widetilde{g}_{\lambda\beta}\widetilde{g}_{\zeta\kappa}\widetilde{g}_{\delta\xi}
+\widetilde{g}_{\lambda\beta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\delta\zeta}
\nonumber\\&\quad\quad+\widetilde{g}_{\delta\beta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\lambda\kappa}+\widetilde{g}_{\delta\beta}\widetilde{g}_{\zeta\kappa}\widetilde{g}_{\lambda\xi}+\widetilde{g}_{\delta\beta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\lambda\kappa})\end{aligned}$$ So far,we have given the covariant tensor amplitude formula for the process $\psi'\to\gamma\chi_{c0}\rightarrow \gamma P K^-
\overline{\Lambda}$.

helicity formula
================

For completeness,we also give the helicity format in comparison with tensor formula.Helicity formalism has an explicit advantage, the angular dependence can be easily seen.In this section ,we will give the amplitude for $\chi_{c0}\to\Lambda(1520)\overline{\Lambda},\Lambda(1520)\to P
K^-,\overline{\Lambda}\to\overline{P}\pi^+$.$\Lambda(1520)$ the most possible resononse,has been mentioned above. Firstly,we want to introduce the general helicity formula expression.Consider a state with spin(parity) $=J(\eta_J)$ decaying into two states with $S(\eta_s)$ and $\sigma(\eta_\sigma)$.The decay amplitudes are given,in the rest frame of $J$ [@helicity; @guide] [@Jacob], $$\mathcal{M}^{J\to
s\sigma}_{\lambda\nu}=\sqrt{\frac{2J+1}{4\pi}}D^{J*}_{M\delta}(\phi,\theta,0)H^J_{\lambda\nu},$$ where $\lambda$ and $\nu$ are the helicities of the two final state particles $s$ and $\sigma$ with $\delta=\lambda-\nu$.The symbol $M$ stands for the $z$ component of the spin $J$ in a coordinate system fixed by production process.The helicities $\lambda$ and $\nu$ are rotational invariants by definition.The direction of the break-up momentum of the decaying particle $s$ is given by the angles $\theta$ and $\phi$ in the $J$ rest frame.Let $\hat{x},\hat{y}$ and $\hat{z}$ be the coordinate system fixed in the $J$ rest frame.It is important to recognize,for applications to sequential decays,the exact nature of the body-fixed (helicity) coordinate system implied by the arguments of the $D$ function given above.Let $\hat{x}_h,\hat{y}_h$ and $\hat{z}_h$ be the helicity coordinate system fixed by the $s$ decay.Then by definition $\hat{z}_h$ describes the direction of $s$ in the $J$ rest frame (also termed the helicity axis) and the $y$ axis is given by $\hat{y}_h=\hat{z}\times\hat{z}_h$ and $\hat{x}_h=\hat{y}_h\times\hat{z}_h$.Parity conservation in the decay leads to the relationship $$H^{J}_{\lambda\nu}=\eta_J\eta_s\eta_\sigma(-)^{J-s-\sigma}H^{J}_{-\lambda-\nu}$$ Let us consider a full process $A\to B+C$ where $B$ and $C$ are also unstable particles decaying to $B_1+B_2$ and $C_1+C_2$ respectively.The decay amplitude is simply [@PhD; @thesis] $$\begin{aligned}
\mathcal{M}(\lambda_{B_1},\lambda_{B_2},\lambda_{C_1},\lambda_{C_2})=\underset{\lambda_B,\lambda_C}{\sum}&
\mathcal{M}^{A\to B+C}_{\lambda_B,\lambda_C}\cdot \mathcal{M}^{B\to
B_1+B_2}_{\lambda_{B_1},\lambda_{B_2}}\cdot\nonumber
\\&\mathcal{M}^{C\to C_1+C_2}_{\lambda_{C_1},\lambda_{C_2}}\end{aligned}$$ with

$$\begin{aligned}
\mathcal{M}^{A\to
B+C}_{\lambda_B,\lambda_C}&=\sqrt{\frac{2J_A+1}{4\pi}}D^{J_A*}_{M_A,\lambda_B-\lambda_C}(\phi_A,\theta_A,0)H^A_{\lambda_B,\lambda_C},
\\ \mathcal{M}^{B\to
B_1+B_2}_{\lambda_{B_1},\lambda_{B_2}}&=\sqrt{\frac{2J_B+1}{4\pi}}D^{J_B*}_{\lambda_B,\lambda_{B_1}-\lambda_{B_2}}(\phi_B,\theta_B,0)H^B_{\lambda_{B_1},\lambda_{B_2}},
\\ \mathcal{M}^{C\to
C_1+C_2}_{\lambda_{C_1},\lambda_{C_2}}&=\sqrt{\frac{2J_C+1}{4\pi}}D^{J_C*}_{-\lambda_C,\lambda_{C_1}-\lambda_{C_2}}(\phi_C,\theta_C,0)H^C_{\lambda_{C_1},\lambda_{C_2}},
\label{sub3}\end{aligned}$$

Please note in the first subscript of $D^{J_C*}$ is $-\lambda_C$ and NOT $\Lambda_C$ although it also gives the correct result,because the quantization axis is along the direction of the momentum of particle $B$ so that the spin-quantization projection $M_C$ in the particle $C$ rest frame verifies $M_C=-\lambda_C$.

The unpolarized angular distribution is then given by averaging over initial spins and by summing over final spins: $$\begin{aligned}
\frac{d^3\Gamma}{\mathcal{N}d\Omega_A d\omega_B
d\Omega_C}=\frac{1}{2S_A+1}&\underset{\lambda_{B_1},\lambda_{C_1},\lambda_{B_2},\lambda_{C_2}}{\sum}\nonumber
\\&|\mathcal{M}(\lambda_{B_1},\lambda_{B_2},\lambda_{C_1},\lambda_{C_2})|^2.\end{aligned}$$ where $\mathcal{N}$ is the normalization factor. Using the parity conservation formula,one has $H^{\overline{\Lambda}}_{\frac{1}{2}0}=H^{\overline{\Lambda}}_{-\frac{1}{2}0}$ and $H^{\Lambda(1520)}_{\frac{1}{2}0}=-H^{\Lambda(1520)}_{-\frac{1}{2}0}$.By applying the above amplitude expression,after a lengthy,tedious but trivial evaluating,one can get the helicity amplitude, $$\begin{aligned}
\label{helicity distribution}
\frac{d^3\Gamma}{\mathcal{N}d\Omega_A d\omega_B
d\Omega_C}=&[\frac{3}{2}\cos^2\theta_{\Lambda(1520)}-\frac{3}{2}\cos\theta_{\Lambda(1520)}\nonumber
\\&+\frac{9}{2}\cos^2\theta_{\Lambda(1520)}\sin\theta_{\Lambda(1520)}\cos\phi_{\Lambda(1520)}\nonumber
\\&+\frac{\sqrt{3}}{2}\cos^2\theta_{\Lambda(1520)}\cos2\phi_{\Lambda(1520)}\nonumber
\\&-\frac{\sqrt{3}}{4}\cos\theta_{\Lambda(1520)}\cos2\phi_{\Lambda(1520)}\nonumber
\\&-\frac{3\sqrt{3}}{4}\cos2\phi_{\Lambda(1520)}+1]|H^{\overline{\Lambda}}_{\frac{1}{2}0}|^2H^{\Lambda(1520)}_{\frac{1}{2}0}|^2\end{aligned}$$ where the subscript $\Lambda(1520)$ denotes that the angle defined in the rest frame of $\Lambda(1520)$. After integrating $\phi_{\Lambda(1520)}$’s from $[0,2\pi]$,the Eq. becomes $$\frac{d^3\Gamma}{\mathcal{N'}d\Omega_A d\omega_B
d\Omega_C}=\frac{3}{2}\cos^2\theta_{\Lambda(1520)}-\frac{3}{2}\cos\theta_{\Lambda(1520)}+1$$ where $\mathcal{N'}=\mathcal{N}|H^{\overline{\Lambda}}_{\frac{1}{2}0}|^2H^{\Lambda(1520)}_{\frac{1}{2}0}|^2$ is redefined normalization factor.

conclusion
==========

In this short note,firstly,the relevant general tensor formalism hac been introduced,and then give the covariant tensor amplitudes.At last,for completeness and some experimental reasons,the helicity amplitude expression has also been provided and a figure attached.

acknowledgments
===============

The author acknowledges greatly helpful discussions with B. S. Zou.This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 10521003,10821063, the 100 Talents program of CAS, and the Knowledge Innovation Project of CAS under contract Nos. U-612 and U-530 (IHEP).

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\[Fig\] ![image](Cos.eps){height="8cm" width="8cm"}
