arXiv 2107.077702021-07hep-thcond-mat.quant-gascond-mat.stat-mechmath-phmath.MP

时空各向异性尺度不变性相互作用量子场论的精确四点函数和OPE

Exact four-point function and OPE for an interacting quantum field theory with space/time anisotropic scale invariance

Hidehiko Shimada, Hirohiko Shimada

备注：: 42 pages + 21 pages, 2 figures; v2 typographical errors corrected and minor improvements in presentation. To appear in JHEP

摘要：

我们确定了一个非平凡但易于处理的量子场论模型与空间/时间各向异性标度不变性，其中一个可以精确地计算某些四点关联函数和它们的分解通过算子乘积展开（OPE）。该模型是Calogero模型，非相对论性粒子与一对势$\frac{g}{|x-y|^2}$在一维相互作用，被认为是一个量子场论在一个空间和一个时间维通过第二量子化。该模型具有各向异性尺度对称性，各向异性指数为$z=2$。这种对称性也增强到薛定谔对称性。该模型有一个耦合常数$g$，从而提供了一个例子，在重整化群流的各向异性理论的固定线。我们精确地计算理论基本场的非平凡四点函数。我们通过OPE以两种不同的方式分解四点函数，从而明确地验证OPE的结合性的第一次与各向异性的尺度不变性的相互作用量子场论。从分解中，可以读出OPE系数和出现在中间通道中的算子的标度维数。其中一个分解是由收敛级数给出的，并且在OPE中只出现一个主算子及其后代。我们计算的主算子的标度维数依赖于耦合常数。维数正确地再现了从著名的Calogero模型的谱中期望的值，并结合了所谓的状态算子映射，这对具有薛定谔对称性的理论是有效的。另一个分解是由一个渐近级数给出的。渐近级数带有指数小的校正项，这也有一个自然的解释OPE。

英文摘要：

We identify a nontrivial yet tractable quantum field theory model with space/time anisotropic scale invariance, for which one can exactly compute certain four-point correlation functions and their decompositions via the operator-product expansion(OPE). The model is the Calogero model, non-relativistic particles interacting with a pair potential $\frac{g}{|x-y|^2}$ in one dimension, considered as a quantum field theory in one space and one time dimension via the second quantisation. This model has the anisotropic scale symmetry with the anisotropy exponent $z=2$. The symmetry is also enhanced to the Schrödinger symmetry. The model has one coupling constant $g$ and thus provides an example of a fixed line in the renormalisation group flow of anisotropic theories. We exactly compute a nontrivial four-point function of the fundamental fields of the theory. We decompose the four-point function via OPE in two different ways, thereby explicitly verifying the associativity of OPE for the first time for an interacting quantum field theory with anisotropic scale invariance. From the decompositions, one can read off the OPE coefficients and the scaling dimensions of the operators appearing in the intermediate channels. One of the decompositions is given by a convergent series, and only one primary operator and its descendants appear in the OPE. The scaling dimension of the primary operator we computed depends on the coupling constant. The dimension correctly reproduces the value expected from the well-known spectrum of the Calogero model combined with the so-called state-operator map which is valid for theories with the Schrödinger symmetry. The other decomposition is given by an asymptotic series. The asymptotic series comes with exponentially small correction terms, which also have a natural interpretation in terms of OPE.

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