Modelling of hadron properties
To investigate the spin-parity properties of the resonant structure within the J/Ψ J/Ψ invariant mass spectrum in the range of 6.2–8.0 GeV, we use an approach designed to minimize model dependence. This approach relies on the observed J/Ψ J/Ψ invariant mass spectrum and the momentum of the J/Ψ J/Ψ system in both transverse and longitudinal directions with respect to the beam, while remaining independent of the polarization of the system by relying solely on decay angular information. For a spin-zero state, polarization is not relevant. For states with nonzero spin, we assume the state is produced unpolarized, but vary the polarization to evaluate potential small residual effects on the decay angular distributions due to detector acceptance.
The analysis uses a model that ensures consistency with the observations made by CMS, by using simulation adjustments to accurately capture the observed transverse and longitudinal motion, and parameters of the resonances and backgrounds extracted from ref. 32 and shown in Fig. 1. The background arises from nonresonant contributions, single-parton scattering (SPS) and double-parton scattering, plus an empirical term parameterizing the background near the threshold32. The background is parameterized using MC simulation, with adjustments applied to better match the observed data in both the signal and sideband regions.
We start by considering the spin-0 hypothesis for the X states, which are produced without polarization. The helicity amplitudes \({A}_{{\lambda }_{1}{\lambda }_{2}}\) of the two J/Ψ mesons are listed in Table 1. For the pseudoscalar state with JP = 0−, the amplitudes satisfy A ++ = −A −− and A 00 = 0. By contrast, for the scalar state with JP = 0+, both A ++ = A −− and A 00 amplitudes contribute, with no specific prediction for the relative magnitude of A 00 . We adopt the general amplitude approach35,46,51, in which the spin-0 state amplitude can be written as a sum of three Lorentz-invariant structures,
$$\begin{array}{l}A({{\rm{X}}}_{J=0}\to {V}_{1}{V}_{2})={a}_{1}({q}^{2}){m}_{V}^{2}{{\epsilon }}_{1}^{* }{{\epsilon }}_{2}^{* }+{a}_{2}({q}^{2}){f}_{{\rm{\mu }}
u }^{* (1)}{f}^{* (2),{\rm{\mu }}
u }\\ \,+{a}_{3}({q}^{2}){f}_{{\rm{\mu }}
u }^{* (1)}{\mathop{f}\limits^{ \sim }}^{* (2),{\rm{\mu }}
u },\end{array}$$ (1)
and where the field strength tensor of a vector boson V i with momentum q i and polarization vector ϵ i is defined as \({f}^{(i),{\rm{\mu }}
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