标量和矢量介子的相对论束缚态研究
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摘要
在量子场论里相对论两体束缚态问题由Bethe-Salpeter(BS)方程来描述，关于Bethe-Salpeter(BS)方程的求解问题一直是国际上关注的，至今没有得到很好解决的重要困难问题。自从Gell-Mann和Zweig提出夸克假设，大量强子的性质得到了很成功的解释。七十年代，Guth用唯象势研究了相对论形式的等质量夸克-反夸克束缚态，分析了束缚态的对称性并数值求解了它的Bethe-Salpeter(BS)方程。九十年代，Gupta，Mitra，Singh提出夸克-反夸克系统的束缚态应该是矢量-矢量型，并解释了这样做的三个优点。
在本论文中，我们首先利用唯象的矢量-矢量耦合的强子相互作用势模型(平底势模型)在Bethe-Salpeter(BS)方程的框架下对标量介子和矢量介子的相对论束缚态进行了深入的研究，所得的结果与其它理论和实验结果符合的很好；然后用修正的平底势模型在彩虹近似下的Schwinger-Dyson(SD)方程与阶梯近似下的Bethe-Salpeter(BS)方程耦合的框架下研究标量介子，得到它的波函数和电磁形状因子的解析表达式。
在第二章中，我们在旋量-旋量Bethe-Salpeter(BS)方程框架下用矢量-矢量型平底势模型来研究标量介子，得到它的波函数和电磁形状因子。首先对BS方程执行WICK转动到欧氏空间，将BS波函数用标量束缚态的M~(i)矩阵进行协变展开，这里导出了H~(ij)矩阵元的解析表达式；再用Gegenbauber函数将协变波函数展开就可以得到一维形式的BS方程组，这里导出了矩阵元K_(nn)~(ij)，的解析表达式，我们推导了n≤4的所有矩阵元；在最低级近似下得到了标量介子波函数的数值解；然后先将两束缚态的电磁流矩阵元中的BS波函数通过协变展开，Gegenbauber函数展开得到它的一维形式使上面得到的波函数可以直接使用，再根据夸克反夸克系统的束缚态的电磁形状因子与两束缚态的电磁流矩阵元之间的关系式我们得到了最低阶标量介子电磁形状因子表达式。这样我们得出了标量介子在动量空间的BS基态波函数和它的电磁形状因子分别随交换粒子的最小质量N与束缚程度B改变的性质：形状因子与物理期望一致随束缚态的束缚强度增强而上升；形状因子随B的变化曲线看上去非常相似变化很小；形状因子的曲线对相互作用比对束缚强度更敏感。在整个过程中没有使用瞬时近似、零平面近似等，我们的结果和其它的理论符合的很好。
在第三章中，我们把在旋量-旋量Bethe-Salpeter(BS)方程框架下用矢量-矢量型平底势模型的方法推广到了研究矢量介子的相对论束缚态。对相同质量的夸克-反夸克束缚态系统(ρ介子和φ介子)和不等质量的夸克-反夸克束缚态系统(K~+介子)分别进行了研究，其中H~(ij)，K_(nn)~(ij)，矩阵元的数量较多，其解析表达式的推导过程也更繁复，我们在附录中列出了推导的结果。文中我们按照通常理论取组分夸克质量M_(u/d)=560Mev，M_s=700Mev；按照实验数据来取矢量介子的质量，计算过程也没有使用瞬时近似、零平面近似等。最后给出了他们在动量空间的BS基态波函数和衰变常数，计算结果f_ρ，f_φ，f_k.和实验值比较的误差分别在5．4％，5．4％，3％。
在第四章中，我们在彩虹近似下的Schwinger-Dyson(SD)方程与阶梯近似下的Bethe-Salpeter(BS)方程耦合的框架下用修正的平底势模型下计算标量介子的波函数和电磁形状因子。由彩虹近似下的夸克传播子的SD方程得到了得到了两个关于A(p~2)，B(P~2)耦合的积分方程，转到欧氏空间再把它们化为一维形式，可以数值求解a(P~2)，B(P~2)；将转动到欧氏空间的阶梯近似下的束缚态的BS波函数用标量束缚态的M~((i))矩阵展开为Lorentz协变函数，再进一步用Tschebyshev函数展开，这样就得到了一维形式的BS方程组，在最低级近似下得到了标量介子波函数的数值解，其中H_(ij)，K_(ij)矩阵元的定义和前面两章不同，推导过程更为复杂，因此我们在文中给出了计算的方法、步骤和结果；电磁形状因子的求解过程与上面两章类似，只是更为复杂一些。
最后我们总结了过去的工作并对以后的研究作了展望。
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(p~2) and B(p~2). We perform a rotation to make them into the Euclidean region and reduced them to the one-dimensional coupled integral equations A(p~2) and B(p~2), 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 M~(i), 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.
在本论文中，我们首先利用唯象的矢量-矢量耦合的强子相互作用势模型(平底势模型)在Bethe-Salpeter(BS)方程的框架下对标量介子和矢量介子的相对论束缚态进行了深入的研究，所得的结果与其它理论和实验结果符合的很好；然后用修正的平底势模型在彩虹近似下的Schwinger-Dyson(SD)方程与阶梯近似下的Bethe-Salpeter(BS)方程耦合的框架下研究标量介子，得到它的波函数和电磁形状因子的解析表达式。
在第二章中，我们在旋量-旋量Bethe-Salpeter(BS)方程框架下用矢量-矢量型平底势模型来研究标量介子，得到它的波函数和电磁形状因子。首先对BS方程执行WICK转动到欧氏空间，将BS波函数用标量束缚态的M~(i)矩阵进行协变展开，这里导出了H~(ij)矩阵元的解析表达式；再用Gegenbauber函数将协变波函数展开就可以得到一维形式的BS方程组，这里导出了矩阵元K_(nn)~(ij)，的解析表达式，我们推导了n≤4的所有矩阵元；在最低级近似下得到了标量介子波函数的数值解；然后先将两束缚态的电磁流矩阵元中的BS波函数通过协变展开，Gegenbauber函数展开得到它的一维形式使上面得到的波函数可以直接使用，再根据夸克反夸克系统的束缚态的电磁形状因子与两束缚态的电磁流矩阵元之间的关系式我们得到了最低阶标量介子电磁形状因子表达式。这样我们得出了标量介子在动量空间的BS基态波函数和它的电磁形状因子分别随交换粒子的最小质量N与束缚程度B改变的性质：形状因子与物理期望一致随束缚态的束缚强度增强而上升；形状因子随B的变化曲线看上去非常相似变化很小；形状因子的曲线对相互作用比对束缚强度更敏感。在整个过程中没有使用瞬时近似、零平面近似等，我们的结果和其它的理论符合的很好。
在第三章中，我们把在旋量-旋量Bethe-Salpeter(BS)方程框架下用矢量-矢量型平底势模型的方法推广到了研究矢量介子的相对论束缚态。对相同质量的夸克-反夸克束缚态系统(ρ介子和φ介子)和不等质量的夸克-反夸克束缚态系统(K~+介子)分别进行了研究，其中H~(ij)，K_(nn)~(ij)，矩阵元的数量较多，其解析表达式的推导过程也更繁复，我们在附录中列出了推导的结果。文中我们按照通常理论取组分夸克质量M_(u/d)=560Mev，M_s=700Mev；按照实验数据来取矢量介子的质量，计算过程也没有使用瞬时近似、零平面近似等。最后给出了他们在动量空间的BS基态波函数和衰变常数，计算结果f_ρ，f_φ，f_k.和实验值比较的误差分别在5．4％，5．4％，3％。
在第四章中，我们在彩虹近似下的Schwinger-Dyson(SD)方程与阶梯近似下的Bethe-Salpeter(BS)方程耦合的框架下用修正的平底势模型下计算标量介子的波函数和电磁形状因子。由彩虹近似下的夸克传播子的SD方程得到了得到了两个关于A(p~2)，B(P~2)耦合的积分方程，转到欧氏空间再把它们化为一维形式，可以数值求解a(P~2)，B(P~2)；将转动到欧氏空间的阶梯近似下的束缚态的BS波函数用标量束缚态的M~((i))矩阵展开为Lorentz协变函数，再进一步用Tschebyshev函数展开，这样就得到了一维形式的BS方程组，在最低级近似下得到了标量介子波函数的数值解，其中H_(ij)，K_(ij)矩阵元的定义和前面两章不同，推导过程更为复杂，因此我们在文中给出了计算的方法、步骤和结果；电磁形状因子的求解过程与上面两章类似，只是更为复杂一些。
最后我们总结了过去的工作并对以后的研究作了展望。
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(p~2) and B(p~2). We perform a rotation to make them into the Euclidean region and reduced them to the one-dimensional coupled integral equations A(p~2) and B(p~2), 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 M~(i), 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.
引文
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