Tensor amplitudes for partial wave analysis of $\psi\to\Delta\bar{\Delta}$ within helicity frame

Recently, many measurements of $J/\psi$ or $\psi(2S)$ decays into baryon-pairs are reported by BESIII collaboration, such as $\psi(2S)\to\Omega^{-}\bar{\Omega}^{+}$ BESIII:2020lkm , $\psi(2S)\to p\bar{p}(n\bar{n})$ 1803.02039 , $J/\psi\left(\psi(2S)\right)\to\Sigma^{+}\bar{\Sigma}^{-}$ 2004.07701 , $J/\psi\left(\psi(2S)\right)\to\Xi^{0}\bar{\Xi}^{0}$ 1612.08664 , $J/\psi\left(\psi(2S)\right)\to\Xi^{-}\bar{\Xi}^{+}$ 1602.06754 , $J/\psi\to\Xi(1530)^{-}\bar{\Xi}^{+}$ 1911.06669 , $\psi(2S)\to\Xi(1530)^{-}\Xi(1530)^{+}(\Xi(1530)^{-}\Xi^{+})$ 1907.13041 etc. These measurements are based on the huge $J/\psi$ or $\psi(2S)$ samples collected at the BESIII detector, who is operating at the electron position collider BEPCII. Taking advantage of the improved statistics, more precise measurements have been achieved and some insights on the physics have been provided, such as the polarization parameters of baryons, relative phase between electric-magnetic and strong amplitudes, decay mechanism of $\psi$ decaying into baryon pairs, and search for excited baryon states, etc. However, there is no measurement of $\psi\to\Delta\bar{\Delta}$ among these new exciting experimental results. A single $\psi$ will represent both $J/\psi$ and $\psi(2S)$ states later in this paper if not specified. The latest measurement of $\psi\to\Delta\bar{\Delta}$ is done by BESII about 20 years ago Bai:2000ye . The lack of $\Delta\bar{\Delta}$ study does not mean this channel is not essential. In contrast, a careful analysis of $\psi\to\Delta\bar{\Delta}$ would provide a lot of important information of the decay mechanism of the vector charmonia to the pair of $\Delta$, the first discovered resonance in particle physics, as well as the line-shape of the invariant mass of the final states of this resonance. The main difficulty of this measurement is that $\Delta$ has a much broader width than other baryon states. From PDG Zyla:2020zbs , the Breit-Wigner width of the $\Delta(1232)^{++}$ is about 117 MeV, the widths of other excited $\Delta$ states are at the same level. It makes the interference effect significant in the measurement, but that has been ignored in previous measurements Bai:2000ye ; Eaton:1983kb . When we try to measure $\psi\to\Delta\bar{\Delta}$ via final states protons and pions, that are the products of the strong decays of $\Delta$, the interference between $\Delta^{++}\Delta^{--}$ and $\Delta^{0}\bar{\Delta}^{0}$, as well as the excited $\Delta$ states and even (excited-)nucleons, is expected to be large. This large interference makes a simple measurement of a two-body decay impossible since the signal yield cannot be extracted by a naive fit with only signal and background components, and need to be considered carefully.

A partial wave analysis (PWA) is necessary to consider this interference correctly, while suitable formulae to describe these complex processes are still in short of. Till now, most studies of $\Delta$ is based on $\pi$N or $\gamma N$ scattering 1810.13086 ; Gridnev:2004mk ; Koch:1980ay ; Svarc:2014zja ; Bernicha:1995gg , where the PWA has applied but not suitable for $\psi\to\Delta\bar{\Delta}$ studies since there are sequent decays in the latter case. Compared with the abundant PWA formulae on the $\psi$ or heavy vectors decaying into final meson states Zou:2002ar ; Dulat:2005in , including radiative decays wuning , the similar results on the baryons are relatively rare. To the best of our knowledge, there are only two papers that discussed the $\psi$ decays containing final baryon states. One has studied $\psi\to\omega p\bar{p}$ Liang:2002tk , and the other one has studied $\psi\to NN^{*}M$ hep-ph/0210164 , but none of them can satisfy our request completely. This situation encourages us to write down the PWA formulae by ourselves.

While there are many different frames and formalism in PWA, we choose to derive our formulae within the helicity frame with the covariant tensor formalism Hara:1964zza ; Chung:1993da ; Chung:1997jn ; Chung:1971ri . This choice is based on the following considerations. Firstly, the helicity frame is suitable for processes of sequent decays. In this frame, all the helicity states are defined in the rest frame of the mother particles, and the whole amplitude is a product of the amplitudes of sequent processes. It reduces the complexity of the derivation of the formulae compared with the PWA based on canonical forms, where a L-S coupling method is used such as that in Refs. Liang:2002tk ; hep-ph/0210164 ; Filippini:1995yc . Secondly, the advantage of the tensor formalism, compared to the helicity formalism, is that a spin tensor can couple to any four-momentum or another spin tensor to form the Lorentz invariant amplitudes, and the momentum dependence is separated explicitly from couplings. This paper is organized as follows. After this introduction, we discuss all the possible processes, i.e, the intermediate resonances involved in our partial wave analysis in the second section. Then we give the Lorentz invariant amplitudes in the third section, composed of the constructions of each state’s wave function, concerns of the parity and charge conjugation translations, and the explicit covariant tensor formulae. Finally, we discuss some validations of these formulae and their possible applications on the present and future electron-positron colliding experiments.

We only consider two kinds of first level decays of $\psi$, that are $\psi\to\Delta\bar{\Delta}$ and $\psi\to NN^{*}$. Since our original physics motivation is to measure $\psi\to\Delta^{++}(1232)\Delta^{--}(1232)$ with $\Delta^{++}(1232)$ almost one hundred percent decaying to $p$ and $\pi^{+}$, the suitable final states in experiments are $p\bar{p}\pi^{+}\pi^{-}$. The final states determine that there are three possible decay chains, that are $\psi\to\Delta\bar{\Delta}$, $\Delta\to p\pi$; $\psi\to\bar{p}N^{*}$, $N^{*}\to\Delta\pi$, $\Delta\to p\pi$ and $\psi\to\bar{p}N^{*}$, $N^{*}\to p\rho_{0}(\sigma)$, $\rho(\sigma)\to\pi^{+}\pi^{-}$ as shown in Figs. 1 2 3. The reason why we do not consider the process of $\psi\to\phi X(1870)$, $\phi\to\pi^{+}\pi^{-},X(1870)\to p\bar{p}$ is that the mass of $X(1870)$ is near the threshold of the $p\bar{p}$ invariant mass spectrum. Then the relevant events can be removed easily during the date analysis. In the present and later descriptions we shall omit the discussion on the charge-conjugated modes for compactness when no ambiguity will be caused. Notice there are many possible combinations even in the first level decays. For example, $\psi$ can decay into the following $\Delta\bar{\Delta}$ pairs if the phase space allows: $\Delta(1232)\bar{\Delta}(1620)$, $\Delta(1620)\bar{\Delta}(1620)$, $\Delta(1232)\bar{\Delta}(1910)$, $\Delta(1620)\bar{\Delta}(1910)$ with $\Delta$ represent both $\Delta^{++}$ and $\Delta^{0}$ here. $\Delta^{+}$ will not be considered since it cannot decay into both charged p and $\pi$. For the $\psi\to\bar{p}N^{*}$ mode, the decay product at the first level can be such as $\bar{p}N(1440)$, $\bar{p}N(1520)$, $\bar{p}N(1675)$, etc.

From the physical consideration, a complex spin and parity combination sets are obtained. At the first level, decaying from a vector mother particle, i.e. $1^{--}$, the $J^{P}$ of the decay products could be $\frac{1}{2}^{+}\frac{1}{2}^{+}$, $\frac{1}{2}^{+}\frac{1}{2}^{-}$, $\frac{1}{2}^{+}\frac{3}{2}^{+}$, $\frac{1}{2}^{+}\frac{3}{2}^{-}$, $\frac{3}{2}^{+}\frac{3}{2}^{+}$, $\frac{3}{2}^{+}\frac{3}{2}^{-}$, $\frac{5}{2}^{+}\frac{1}{2}^{+}$, $\frac{5}{2}^{+}\frac{1}{2}^{-}$, $\frac{5}{2}^{+}\frac{3}{2}^{+}$, $\frac{5}{2}^{+}\frac{3}{2}^{-}$, $\frac{7}{2}^{+}\frac{3}{2}^{+}$j, and $\frac{7}{2}^{+}\frac{3}{2}^{-}$ ; at the second level, if the decays are from $\Delta$, the final states are always $\frac{1}{2}^{+}0^{-}$ while the $J^{P}$ of the $\Delta$ can be $\frac{1}{2}^{+}$, $\frac{1}{2}^{-}$, $\frac{3}{2}^{+}$, $\frac{3}{2}^{-}$, $\frac{5}{2}^{+}$, $\frac{5}{2}^{-}$, $\frac{7}{2}^{+}$, and $\frac{7}{2}^{-}$; if a decay chain contains $N^{*}$ then there is another decay mode with final states $\Delta\pi$, i.e. $J^{P}=\frac{3}{2}^{+}0^{-}$, while the $J^{P}$ of the initial $N^{*}$ can be $\frac{1}{2}^{+}$, $\frac{1}{2}^{-}$, $\frac{3}{2}^{+}$, $\frac{3}{2}^{-}$, $\frac{5}{2}^{+}$, $\frac{5}{2}^{-}$ $\frac{7}{2}^{+}$, and $\frac{7}{2}^{-}$;

Now we are ready to derive the covariant invariant amplitudes in the tensor formalism for resonance decays. Let us consider a resonance of spin-parity $J^{P}$ and mass $m$, decaying into two particles 1 and 2:

In the rest frame of the mother resonance, let $p$ is the momentum of the particle 1 with helicity angles $(\theta,\phi)$. Usually the amplitude $A$ describing the decay process into two particles with helicity $\lambda_{1}$ and $\lambda_{2}$ may be written as

in the helicity formalism, where $N_{J}$ is the normalization factor, $F^{J}_{\lambda_{1}\lambda_{2}}$ is the helicity coupling, and $D^{J*}_{M,\lambda_{1}-\lambda_{2}}$ is the D function with $M$ is the z-component of the mother’s spin. However, with the explicit wave functions and effective vertices, we could construct the independent covariant invariant amplitudes in the tensor formalism.