Study On The Relativistic Bound State Of The Scalar Meson And The Vector Meson

As we know, in quantum field theory the relativistic two-body bound state is described by Bethe-Salpeter (BS) equation. Although how to solute it has been engage people's attention, it is still an important and difficult problem, which has not been finished well till today. The properties of plentiful hadrons can be explained well from Gell-Mann and Zweig table the quark hypothesis. In 1970s Guth studied the bound states of equal-mass quark-antiquark pairs and obtained the numerical solutions of the bound states, using the fully relativistic formalism of the Bethe-Salpeter equations (BSE) with phenomenological potential. In 1990s Gupta, Mitra, Singh stated that the interaction form of bound states of the quark-antiquark system should be a vector-vector-type structure (γ_μγ_μ) and give three advantages of it.In this dissertation, we take great pains to investigate the properties of the relativistic bound state of the scalar meson and the vector meson by using a phenomenological vector-vector-type flat-bottom hadronic potential under the frame of the Bethe-Salpeter equations (BSE). The results give a good fit to the experimental results and other theories; We also study on the bound states of the scalar meson in the framework of the Schwinger-Dyson equation (SDE) in the rainbow approximation and the Bethe-Salpeter equations (BSE) in the ladder approximation with the modified flat-bottom potential, from which we get the wave functions and the electromagnetic form factor of the scalar meson.In the second chapter, we study the scalar meson and obtain its wave functions and the electromagnetic form factor under the frame of the spinor-spinor Bethe-Salpeter equations (BSE) with using the vector-vector-type flat-bottom potential. First, we perform the Wick rotation analytically BSE into the Euclidean region and the BS wave function of the bound state can be expended as Lorentz-invariant functions by the scalar bound states' matrix M⁽ⁱ⁾, where we calculated the analytical expressions of the matrix element H^ij; then the BSE can be reduced to the infinite set of one-dimensional coupled integral equations by expending the Lorentz-invariant functions in Gegenbauber functions, where we calculated all the analytical expression of the matrix element K_nn^ij (n≤4); and in the lowest order, we can get the numerical solutions of the wave functions of the bound state. And For the wave functions which have been obtained can be used directly, we reduced the BS wave functions in the matrix element of the electromagnetic current between two bound states to the infinite set of one-dimensional coupled integral equations; Then by the relationship between the electromagnetic form factor of the bound states of a quark-antiquark system and the matrix element of the electromagnetic current between two bound states we get the expression of the electromagnetic form factor. Now we get the BS bound-state wave functions of the scalar meson in momentum region and the characters of the electromagnetic form factor change with the minimum value of the mass of the exchange particles and the fraction of binding, respectively: In agreement with physical expectations one sees that as the bound state becomes more tight the form factor increases; The form factors look rather similar although the fraction of binding varies dramatically; It seems that their slopes are much more sensitive to the range of the interaction rather than its strength. The whole calculation without use the instant-form approach, front-from approach and so on. Our results agree well with other theories.In the third chapter, we extend the method of under the frame of the spinor-spinor Bethe-Salpeter equations (BSE) with using the vector-vector-type flat-bottom potential to study the relativistic bound state of the vector meson. For the bound states of equal-mass quark-antiquark (ρ,φ) and of unequal-mass quark-antiquark (k^*). Where the number of the matrix element H^ij and K_nn^ij. is more and the calculation of their analytical expressions is more difficult, whose expressions have been shown in appendix. We have give out the wave functions in momentum space and the decay constants. Here, we use the constituent masses m_u/d=560MeV and m_s = 700MeV according to other theories, and the mass of mesons tally with experiment results. Without using the instant-form approach, point-form approach, front-from approach, and so on, the calculated results, f_ρ,fφ, f_k. are within 5.4%, 5.4%, 3%, of the experimental value, respectively.In the fourth chapter, we study on the bound states of the scalar meson in the framework of the Schwinger-Dyson equation (SDE) in the rainbow approximation and the Bethe-Salpeter equations (BSE) in the ladder approximation with the modified flat-bottom potential. Form the Schwinger-Dyson equation (SDE) of the quark propagator in the rainbow approximation we can obtain two coupled integral equations for the functions A(p²) and B(p²). We perform a rotation to make them into the Euclidean region and reduced them to the one-dimensional coupled integral equations A(p²) and B(p²), then we can get the numerical solutions of them. The BS wave function of the bound state, which has been rotated into the Euclidean region can be expended as Lorentz-invariant functions by the vector bound states' matrix Mⁱ, then it be expended in the Tschebyshev function, form which the BSE can be reduced to the one-dimensional coupled integral equations. In the lowest order, we can get the numerical solutions of the wave functions of the vector bound state. Where the define of the matrix element H^ij and K_nn^ij is different form the front chapters and their analytical expressions are more difficulty to calculate, so we give out the solution's method, step and result in the text. The study of the electromagnetic form factor is similar to the front chapters but more difficulty to do.At last, we viewed our past work and prospect the following study.