| Hadron physics is an important area of theoretical and experimental study in particle physics. Quantum chromodynamics being a fundamental theory of strong interactions between quarks and gluons were set up in 1970s. In experiment , with the development of technology and the running of advanced facility, more and more resonances and events about hadron physics have been observed. Therefore ,it is necessary to connected theoretical study to experiments more intimately for deep understanding of the strong interactions and the construction of hadron Although there are many difficulties in this field , investigations on structures and properties of hadrons have always been attractive and thriving, because this is the farthest study on material structure . According to constituent quark model, meson is the simplest bound systems of quark and antiquark which formed by the QCD interactions, should be the ideal object of testing and applying for QCD. Since, QCD describe a non-linear and non-perturbative strong interaction which should be studied by a nonperturbative method , the difficulty of solving nonperturbative and non-linear system in analytical method made us a large extent our knowledge of hadron physics is based on phenomenological models , such as the constituent-quark model. Apart from the theoretical predictions of heavy meson fit well to experiment ,there are many open question in light meson sector , particularly,in light scalar mesons . Up to the present a number of new resonant states have been discovered in the experiment. But their structures and properties have not been made so clear so far,it is possible to existing the mixture between new resonant states and light scalar mesons with the same quantum numbers, such as glueballs, although the study on which have been started from the day QCD was proposed, their existing is still ambiguous.In the meantime exact understanding for convential mesons is need to distinguish from those resonant states . Phenomenological models are consist of the non-relativistic quark model and the relativistic one .The former was very successful in describing heavy meson ,and the potential obtained from fitting to experiment data is almost the same as the potential given by lattice QCD in the numerical form . The reasonable for non-relativistic quark model is based on its simplicity and succsees ,not on QCD theory ,this is help for understanding of QCD. However, the situation is quite different for light meson sector , because of less quarkmass and large coupling within QCD , the model for light meson must take account of the non-perturbative effect as well as relativistic one. Naturally , this led to relativistic quark model being the framework of meson investigation . Bethe-Salpeter equation is available method for the calculation of bound states within the framework of relativistic quantum fields and is manifest Lorentz convariant from the outset . But there is obstacle to the usage of Bethe-Salpeter equation , the four-dimensional B-S equation has unphysical solutions with negative norm and the closed interactional kernel is notobtained .on the other hand how to solve the equation is still a open question even if we have known the kernel of Bethe-Salpeter equation. Recently , Professor J.C.Su derived a four-dimensional Dirac-schrodinger equation and a three-dimensional one for quark-antiquark bound states and gave the complete kernel of Bethe-Salpeter equation example for quark-antiquark bound states in a covariant four dimension form. Our calculation of meson mass is based on Bethe-Salpeter equation and numerical method used. This paper consists of three parts: The fist part concerns the calculation of the mass spectrum of PP bound states and elastic PP diffrential cross section by employing nonrelativistic quark potential model and resonance group method (RGM). In the calculation we find that take into account of color octet,a well agreement with experiment can achieved. Otherwise, because of the vanishing of color matrix element in color singlet , the annihilation-channel OGEP is not included ,as the consequence,there is a relative big discrepence between prediction of theoretical caculation and experiment. We have also calculate the mass of PP bound states possible existing in experiment. The second part is for the calculation of the mass spectrum for pseudo-scalar mesons and pseudo-vector mesons by using Bethe-Salpeter equation. Our results for pseudo-scalar mesons are in good agreement with experiment, the worse for pseudo-vector mesons. The third is the discussion of path integral expression of the partition function for a grand canonical ensemble in the coherent-state representation ,pointed out the wrong result about the partition function and the generating functional of Green'sfunctions in the previous literatures and given the correct one with examples for QED and ?4 theory . The methods used in our study have some new features which are different from other works, they are as follows. 1. On the study of PP low-energy scattering and bound states. Considering that in low-energy region, the constituent quarks in P and P move not so fast, it is reasonable to work with the nonrelativistic quark potential model. In this model, by using RGM we derive an effective interaction potential between the P and P which is used to calculate elastic PP diffrential cross section and masses of PP bound states. The novel aspects in our calculation are: (1) Inclusion of annihilation-channel one-gluon exchange potential (OGEP) derived rigorously from QCD. In previous quark potential models, the annihilation-channel OGEP is not included or considered phenomenologically. For the system in which there are not quark and antiquark with the same flavor, only scatter-channel OGEP is needed to consider; but for the system such as PP in which a quark and an antiquark may have the same flavor, the effect of qq annihilation must be taken into account. In contrast to the phenomenological annihilation-channel OGEP, our annihilation-channel OGEP which describes the effect of q q annihilation is rigorously derived from QCD. In our calculation we find that only when the annihilation-channel OGEP is considered the reasonable results can be achieved. (2) The consideration of all spin-dependent terms in the OGEPs.There are several spin-dependent terms in the OGEPs. In most previous works, only hyperfine structure terms in the potential are taken into account. In our work all the terms in the OGEPs are considered. Thus, in addition to spin-independent terms, the spin-dependent terms such as spin-orbit terms, spin-spin terms and tensor force terms appear in the effective interaction potentials derived. (3) Necessity of including color octet for each cluster. The two q q clusters may be polarized in color space when they interact, so they need not remain color singlet. The color wave function of the system should generally be a mixing of the color singlet and the color octet. Our calculation shows that the effect of color octet is essential to fitting the experimental data. 2. On the study of mass spectrum of mesons. The mass spectrum of mesons is calculated by applying B-S equation satisfied by quark-antiquark bound states. The novelties in our calculation are: (1) Usage of angular momentum representation. Usually, the B-S equation is given in momentum representation, there the momentum conservation has simple expression. But when one deals with the states with a certain spin, parity and charge conjugation, it is much more convenient to work with B-S equation expressed in the angular momentum representation, particularly,by using the basis functions which have definite quantum-numbers of angular moment J and its third component and parity were convenient to the discussion of B-S equation for bound states in the framework ofrelativistic model. Because the B-S amplitude of a meson state can be constructed directly from the coupling of the constituent particle states in the angular momentum representation and the gluon field can be completely expanded in terms of transverse electric, transverse magnetic and longitudinal modes of the field, the expression of B-S amplitude is simple and the expanding coefficients in the amplitude have manifest physical meaning. On the other hand, since the integral of solid angle can be calculated aforehand, we are left with only radial momentum integral for numerical calculations. Thus the efficiency of numerical calculation is greatly increased. (2) The integrals of the product of several spherical Bessel functions included in quark-gluon vertices are given by general analytic expressions. In angular momentum representation, there appear the integrals of the products of several (three) spherical Bessel functions in quark-gluon vertices. For these integrals, we obtain analytic expressions which are convenient to use in the calculation. These expressions are, as easily proved, consistent with the momentum conservation. By using these expressions, the efficiency of numerical calculation is increased further. (3) Inclusion of retardation effect. Ordinarily, one chooses the instantaneous approximation to reduce the four-dimensional covariant B-S equation into a three-dimensional one. In this case the retardation effect is fully neglected. We derived a rigorous three-dimensional relativistic equation for quark-antiquark bound states in which the retardation...
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