Soryushiron Kenkyu Electronics

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In the lattice formulation as well as in the usual continuum formulation on a finite interval, we find that in 1 + 1 dim. ( S^1 × R^1 ), SU(2) Yang-Mills theory has only one physical degree of freedom . In the case of SU(N), we have N - 1 degrees of freedom, i.e. the number equals that of the generators of the Cartan subalgebra. Comparing our results to those obtained in the continuum formulation [3] [4], we can see the exact correspondence between the degrees of freedom used in the two approaches. This establishes the equivalence of continuum and lattice formulations of Yang-Mills theory in 1 + 1 dimensions on the quantum level. Finally we mention rather technical but probably important advantages of the lattice approach which become obvious through a comparison of the two formulations. In the continuum approach with static sources we have to solve the coupled differential equations, which is non-trivial even in the case of SU(2). (In reference[3] solutions axe given in the form of hypergeometric functions for the case of SU(2)). In the lattice approach, on the other hand, we do not encounter this difficulty, since the solution is found from pure symmetry considerations. This implies moreover that our methods can be easily extended to the case where the static sources belong to other representations and to determine the spectrum of states for number of colors N > 2. The above advantages, which look just technical in 1+1 dimensions, may become more important in higher dimensions where the number of gauge degrees of freedom is infinite in the continuum limit. We will try to extend the present work so as to clarify the relations between the two formulations in higher dimensions and to make the best of the advantages of the lattice approach to understand hadron physics in the real world.

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The pion-nucleon and kaon-nucleon scattering lengths are calculated by means of Borel sum rules. The scattering lengths are related to the expectation values of correlation functions of the axial-vector currents with respect to the one-nucleon state by means of the reduction formula. We show that the pion-nucleon and kaon-nucleon scattering lengths are proportional to the quark number of the nucleon in the leading order of an operator product expansion, and rederive the Tomozawa-Weinberg relation.

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