光子光锥波函数与量子色动力学求和规则研究
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摘要
本文基于瞬子真空低能等效理论,运用量子色动力学(QCD)光锥求和规则,对直至扭度-4的离壳(在壳)光子光锥波函数及其相应的耦合常数进行了系统性的研究；此外,用QCD光锥求和规则的方法计算了从矢量介子衰变到标量介子和光子的一系列过程的耦合常数。
首先,根据Lorentz协变性的原则,我们把光子态到真空的跃迁矩阵元分解成为八个对应于不同的Lorentz结构的光子光锥波函数,其中只有两个横向波函数在趋于在壳极限下仍保持不为零。通过相应的投影算符将它们分离后,得到由对应于不同张量结构关联函数表示的波函数表达式。利用夸克传播子的谱表示的一般形式,用谱密度函数给出了不依赖于具体强作用理论模型的关联函数和八个光子光锥波函数及其耦合常数的普适形式。
进而,为了计及QCD非微扰效应,我们选择在QCD低能等效理论的框架中工作。对于该理论的等效夸克传播子,我们按通常方法引入相应的极点形式。对于以极点形式表示的等效夸克传播子,我们推出了相应的的谱密度函数的解析形式,它们确实满足关于谱密度函数的限制条件。利用等效夸克传播子谱密度函数的解析形式,我们得到八个波函数及其相应的耦合常数在QCD低能等效理论中的的解析表达式。
利用QCD低能等效理论的标准经验参数,我们进行了数值模拟计算,得到了八个光子光锥波函数的数值结果(其中,我们还应用光子波函数的类Wandzura-Wilczek关系求出了hγ‖(t)、hγ‖(s)、gγ(?)(v)和g(?)(a)的纯扭度为3的部分),给出了它们与夸克具有的动量分额u之间关系的图像；通过波函数的归一化条件,计算了全部耦合常数,给出了它们与光子动量P2之间的函数关系和相应的数值结果；最后,对得到的波函数进行了Gegenbauer展开,列出了相应的展开系数。
在完成了光子光锥波函数及其耦合常数波函数的计算工作后,利用所得到的光子光锥波函数的数值结果,我们研究了矢量介子衰变到标量介子和光子的过程,并通过求和规则的方法计算了的相应的一系列衰变过程的耦合常数。
最后,我们将结果与其他文献进行了比较,进行了有意义的讨论,并对我们得出的光子光锥波函数的应用进行了展望。
The eight off-shell light-cone photon wavefunctions corresponding to chiral-odd and chiral-even up to twist-four and the corresponding coupling constants are system-atically investigated in framework of the low-energy effective theory of quantum chro-modynamics; and in meanwhile, the vector meson coupling constants in the processes of a vector meson decaying into a scalar meson and a photon are calculated by using QCD light-cone sum rules.
First, according to the principle of the Lorentz covariance, the transition matrix elements from an off-shell photon state to the vacuum are decomposed into the eight light-cone photon distribution amplitudes, in which only two transversal wavefunc-tions survive in the on-shell limit. After separating them through the corresponding projection operators, the various individual photon DA multiplied by its corresponding coupling constant is universally expressed in terms of the correlation functions. Using the spectral representation of a general quark propagator, the universal expressions for all the photon DAs times the coupling constants in terms of these spectral density functions are obtained.
Further, in order to consider the non-perturbative effects of QCD, we have been choosing to work in the framework of the QCD low-energy effective theory. We have introduced a usual pole form for the effective quark propagator in that theory, and worked out the analytic expressions of the corresponding spectral density function-s, which obey the well-known constraints for the fermionic spectral densities indeed. Using these analytic forms, the analytic expressions of the eight photon DAs and the corresponding coupling constants are obtained.
Imputing the experienced standard parameters in the QCD low-energy effective theory, we have carried out the numerical simulation, and the numerical results (in-cluding the corresponding curves for the wavefunctions versus the quark momentum fraction u) of all wavefunctions (including the pure twist-3part of the photon wavefunc-tions hγ(t),hγ||(s),gγ⊥(v) and gγ⊥(α)) are given out. All the coupling constants are calculated by using the normalization conditions, and both the analytical and numerical relationship between the couplings and the photon momentum squared P2are given. At last, the obtained wavefunctions are expanded in Gegenbauer polynomials, the corresponding expanding coefficients are listed.
After completing the calculation of the light-cone photon wavefunctions and their coupling constants, the processes in any one of which a vector meson is decaying into a scalar meson and a photon are studied by using our numerical results for the photon wavefunctions. The corresponding coupling constants for such processes are calculated by using QCD light-cone sum rules.
Finally, we have made our conclusions and compared our results with the others in literatures, and in meanwhile, given a significant discussion and further outlook for the application of our obtained photon wavefunctions.
首先,根据Lorentz协变性的原则,我们把光子态到真空的跃迁矩阵元分解成为八个对应于不同的Lorentz结构的光子光锥波函数,其中只有两个横向波函数在趋于在壳极限下仍保持不为零。通过相应的投影算符将它们分离后,得到由对应于不同张量结构关联函数表示的波函数表达式。利用夸克传播子的谱表示的一般形式,用谱密度函数给出了不依赖于具体强作用理论模型的关联函数和八个光子光锥波函数及其耦合常数的普适形式。
进而,为了计及QCD非微扰效应,我们选择在QCD低能等效理论的框架中工作。对于该理论的等效夸克传播子,我们按通常方法引入相应的极点形式。对于以极点形式表示的等效夸克传播子,我们推出了相应的的谱密度函数的解析形式,它们确实满足关于谱密度函数的限制条件。利用等效夸克传播子谱密度函数的解析形式,我们得到八个波函数及其相应的耦合常数在QCD低能等效理论中的的解析表达式。
利用QCD低能等效理论的标准经验参数,我们进行了数值模拟计算,得到了八个光子光锥波函数的数值结果(其中,我们还应用光子波函数的类Wandzura-Wilczek关系求出了hγ‖(t)、hγ‖(s)、gγ(?)(v)和g(?)(a)的纯扭度为3的部分),给出了它们与夸克具有的动量分额u之间关系的图像；通过波函数的归一化条件,计算了全部耦合常数,给出了它们与光子动量P2之间的函数关系和相应的数值结果；最后,对得到的波函数进行了Gegenbauer展开,列出了相应的展开系数。
在完成了光子光锥波函数及其耦合常数波函数的计算工作后,利用所得到的光子光锥波函数的数值结果,我们研究了矢量介子衰变到标量介子和光子的过程,并通过求和规则的方法计算了的相应的一系列衰变过程的耦合常数。
最后,我们将结果与其他文献进行了比较,进行了有意义的讨论,并对我们得出的光子光锥波函数的应用进行了展望。
The eight off-shell light-cone photon wavefunctions corresponding to chiral-odd and chiral-even up to twist-four and the corresponding coupling constants are system-atically investigated in framework of the low-energy effective theory of quantum chro-modynamics; and in meanwhile, the vector meson coupling constants in the processes of a vector meson decaying into a scalar meson and a photon are calculated by using QCD light-cone sum rules.
First, according to the principle of the Lorentz covariance, the transition matrix elements from an off-shell photon state to the vacuum are decomposed into the eight light-cone photon distribution amplitudes, in which only two transversal wavefunc-tions survive in the on-shell limit. After separating them through the corresponding projection operators, the various individual photon DA multiplied by its corresponding coupling constant is universally expressed in terms of the correlation functions. Using the spectral representation of a general quark propagator, the universal expressions for all the photon DAs times the coupling constants in terms of these spectral density functions are obtained.
Further, in order to consider the non-perturbative effects of QCD, we have been choosing to work in the framework of the QCD low-energy effective theory. We have introduced a usual pole form for the effective quark propagator in that theory, and worked out the analytic expressions of the corresponding spectral density function-s, which obey the well-known constraints for the fermionic spectral densities indeed. Using these analytic forms, the analytic expressions of the eight photon DAs and the corresponding coupling constants are obtained.
Imputing the experienced standard parameters in the QCD low-energy effective theory, we have carried out the numerical simulation, and the numerical results (in-cluding the corresponding curves for the wavefunctions versus the quark momentum fraction u) of all wavefunctions (including the pure twist-3part of the photon wavefunc-tions hγ(t),hγ||(s),gγ⊥(v) and gγ⊥(α)) are given out. All the coupling constants are calculated by using the normalization conditions, and both the analytical and numerical relationship between the couplings and the photon momentum squared P2are given. At last, the obtained wavefunctions are expanded in Gegenbauer polynomials, the corresponding expanding coefficients are listed.
After completing the calculation of the light-cone photon wavefunctions and their coupling constants, the processes in any one of which a vector meson is decaying into a scalar meson and a photon are studied by using our numerical results for the photon wavefunctions. The corresponding coupling constants for such processes are calculated by using QCD light-cone sum rules.
Finally, we have made our conclusions and compared our results with the others in literatures, and in meanwhile, given a significant discussion and further outlook for the application of our obtained photon wavefunctions.
引文
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