arXiv 0909.24282009-09hep-thhep-phmath-phmath.MP

标量场理论中Dyson-Schwinger方程的精确解

Exact solution of Dyson-Schwinger equations for a scalar field theory

Marco Frasca

备注：: 7 pages, no figure. Improved derivation of the propagator, added some sentences for the Yang-Mills theory and adjusted references

摘要：

精确求解了无质量四次标量场理论的Dyson-Schwinger方程。计算n点函数直到n=4，并且从两点函数计算精确传播子。光谱是这样得到的，与谐振子的光谱相同。两点函数的Callan-Symanzik方程给出了beta函数。这给出的结果是，这个理论只有平凡的不动点。在低能量极限，耦合趋于零，使理论变得微不足道，在高能量时，它达到无穷大。没有朗道极点出现，而这应该被看作是一个前兆，在一个弱扰动扩展，耦合达到平凡的不动点在无穷远。利用最近证明的无质量四次标量场理论与规范理论之间的映射定理，我们导出了后者的一些性质。

英文摘要：

We exactly solve Dyson-Schwinger equations for a massless quartic scalar field theory. n-point functions are computed till n=4 and the exact propagator computed from the two-point function. The spectrum is so obtained, being the same of a harmonic oscillator. Callan-Symanzik equation for the two-point function gives the beta function. This gives the result that this theory has only trivial fixed points. In the low-energy limit the coupling goes to zero making the theory trivial and, at high energies, it reaches infinity. No Landau pole appears, rather this should be seen as a precursor, in a weak perturbation expansion, of the coupling reaching the trivial fixed point at infinity. Using a mapping theorem, recently proved, between massless quartic scalar field theory and gauge theories, we derive some properties of the latter.

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摘要：

英文摘要：

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