arXiv 1208.36742012-08hep-th

极限环与共形不变性

Limit Cycles and Conformal Invariance

Jean-François Fortin, Benjamin Grinstein, Andreas Stergiou

备注：: 31 pages, 4 figures. Expanded introduction to make clear that cycles discussed in this work are not associated with unitary theories that are scale but not conformally invariant
发表信息：: JHEP01(2013)184
DOI：: 10.1007/JHEP01(2013)184

摘要：

人们普遍认为，共形场论（CFTs）需要零β函数。尽管如此，杰克和奥斯本的工作意味着β函数实际上并不是决定保形性的量，但直到最近才表现出这种行为。我们最近的工作导致了具有非零β函数的CFTs的发现，更确切地说，CFTs生活在循环轨迹上，例如，极限环，beta函数向量场。为了证明这一点，我们研究的S功能的杰克和奥斯本。利用Weyl相容性条件证明了它在不动点处为零，并与不动点上极限环的生成元Q一致。此外，我们计算了三阶微扰理论中的S，并明确验证了它与我们以前确定的Q一致。我们分析的一个副产品是，在微扰论中，么正性和尺度不变性意味着四维量子场论中的共形不变性。最后，我们研究了这些新的，“循环”的CFTs的一些性质，并指出，a-定理仍然支配着重整化群流的渐近行为。

英文摘要：

There is a widely held belief that conformal field theories (CFTs) require zero beta functions. Nevertheless, the work of Jack and Osborn implies that the beta functions are not actually the quantites that decide conformality, but until recently no such behavior had been exhibited. Our recent work has led to the discovery of CFTs with nonzero beta functions, more precisely CFTs that live on recurrent trajectories, e.g., limit cycles, of the beta-function vector field. To demonstrate this we study the S function of Jack and Osborn. We use Weyl consistency conditions to show that it vanishes at fixed points and agrees with the generator Q of limit cycles on them. Moreover, we compute S to third order in perturbation theory, and explicitly verify that it agrees with our previous determinations of Q. A byproduct of our analysis is that, in perturbation theory, unitarity and scale invariance imply conformal invariance in four-dimensional quantum field theories. Finally, we study some properties of these new, "cyclic" CFTs, and point out that the a-theorem still governs the asymptotic behavior of renormalization-group flows.

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