Blank M, Krassnigg A
Institut für Physik, Universität Graz, Universitätsplatz 5, 8010 Graz, Austria.
Comput Phys Commun. 2011 Jul;182(7):1391-1401. doi: 10.1016/j.cpc.2011.03.003.
In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe-Salpeter equation for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe-Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is as efficient or even advantageous to use the inhomogeneous equation as compared to the homogeneous. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.
在量子色动力学的泛函方法中，强子束缚态的性质可通过协变积分方程来获取，例如介子的贝塞耳 - 萨尔皮特方程。具体而言，人们必须处理线性齐次积分方程，在复杂的模型设定中，这些方程会将其他积分方程解的数值表示作为其输入的一部分。类似地，可以构建非齐次方程，以除了从束缚态协变类似波函数获得的束缚态质量和其他性质之外，还能获取离壳信息。使用众所周知的矩阵算法求解特征值（在齐次情况下）和线性系统（在非齐次情况下），可以非常有效地求解这些方程。我们通过求解齐次和非齐次贝塞耳 - 萨尔皮特方程来证明这一点，例如发现，对于计算质量谱，与齐次方程相比，使用非齐次方程同样高效甚至更具优势。这是非常有价值的见解，特别是对于在三夸克设定及更复杂系统中对重子的研究。
