Theoretical Study On Three-body Decay Of Molecular State ZQ

Our article mainly concerns about the bottomonium-like resonances Zb(10610) and Zb(10650), which have been observed by Belle Collaboration recent years. Since they appear in the intermediate process of Y(5S)�'Y(nS)π+π-�'�Y(5S)�'hb(mP)π+π-,moreover, they are charged, they are composed of bottomonium bb and charged light quarks. The minimal constituents are four quarks, such as bbud,.Combining messages from experiments at present. It may be a hadronic molecular state or a tetraquark..Recently, the analyze of invariant mass spectrum of final state π+π-π0of the decay Y(5S)�'π+π-π0χbJ shows that there is a conspicuous resonance peak during the decay process. A further research tells us the peak comes from the process of Y(5S)�'ωχbJ�'π+π-π0χdJ.In addition, there are some non-co background around0.9GeV. The non-^part probably comes from the process of Y(5S)�'πZb(+)�'πρχbJOur article is based on the interpretation of hadronic molecule,firstly,we write down all interection Lagrangians of each vertex which are followed by parity invariance in hadronic interactions,then we write down the invariant matrix of each diagram according to Feynman Rules one by one.,after which,coupling constants are determined.Among all of the coupling constants, gZBB*、gz*B*B*and gρππ are determined by two-body decay process of point interactions, gBBρ and gB*B*ρ are determined by Vector Meson Dominance(VMD) Model.while gBB*ρ、gBBχb0、gB*B*χb0、8BB*χb1and gB*B*χb1are determined by Heavy Hardron Chiral Perturbation Theory(HHChPT).After finishing the expression of decay width,we write down the procession of decay width calculation by the first decay channel Zb�'χb0ππ as an example in the section below.Firstly,we mention about the packages Feyncalc and LoopTools required,pointing out how the packages can make our calculation of matrix elements convenient,and the operations are introduced specifically.Then we introduce the calculation of three-body phase space simply:how to transit two-body phase space into three-body phase space,then we simplify the expression of three-body phase space on the condition of energies of two of the three final state particles as variables.We make definitions of all possible momentum scalar product so that each function expressed by Feyncalc can be put into the integral expression directly without contraction.Then matrix elements transfer into the expression witch is expressed by final-state-particle energies as variables through Feyncalc and LoopTools. Finally,we can do the numerical integral.We show numerical results of Zb(+)�'χb0ππ two decay channels in the section of numerical results and discussion,which are in accordance with the data at present,and make some forecast to observations of the process of Zb(+)�'χb0ππ experiments.