论文标题
接近SINH-GORDON模型的自以为是的点
Approaching the Self-Dual Point of the Sinh-Gordon model
论文作者
论文摘要
Sinh-Gordon模型（SHG）的最引人注目但最神秘的属性之一是其$ S $ -MATRIX的$ B \ rightarrow 1/b $自duality，其Lagrangian配方中没有痕迹。这里$ b $是该模型中出现的耦合，其Lagrangian中存在的同名双曲线余弦，$ \ cosh（bx）$。在本文中，我们开发了截短的频谱方法（TSM），用于在改变耦合常数时以有限体积研究SINH-GORDON模型。我们获得了$ b \ ll 1 $的预期结果和$ b $的中间值，但是随着自duly点$ b = 1 $的接近，TSM在SHG上的基本应用会破裂。我们发现TSM具有强大的截止$ e_c $依赖性的结果，该结果仅根据$ e_c $中的非常缓慢的功率定律而消失。标准的重归其化组策略 - 无论是数值还是分析性 - 也无法改善此处的问题。因此，我们探索了三种策略，以解决$ b = 1 $附近TSM的基本局限性。首先，我们专注于小体积频谱。我们试图了解SHG的物理学在其哈密顿量的零模式部分中编码了多少，从本质上讲，问题是如何“量子机械”与“量子场理论”的问题。在第二个中，我们确定了扰动理论中存在的分歧，并使用上骨上的近似值进行了重新召唤。在第三种方法中，我们使用模型的确切形式来处理以$ b $的一个值来处理SHG，以作为在不同耦合下shg的扰动。鉴于这项工作，我们认为，Lagrangian模型的强耦合阶段$ b> 1 $可能与从其$ S $ -MATRIX推断出的天真的。特别是，我们提出了一个论点，即该理论对于$ b> 1 $是无质量的。
One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the $b \rightarrow 1/b$ self-duality of its $S$-matrix, of which there is no trace in its Lagrangian formulation. Here $b$ is the coupling appearing in the model's eponymous hyperbolic cosine present in its Lagrangian, $\cosh(bϕ)$. In this paper we develop truncated spectrum methods (TSMs) for studying the sinh-Gordon model at a finite volume as we vary the coupling constant. We obtain the expected results for $b \ll 1$ and intermediate values of $b$, but as the self-dual point $b=1$ is approached, the basic application of the TSM to the ShG breaks down. We find that the TSM gives results with a strong cutoff $E_c$ dependence, which disappears according only to a very slow power law in $E_c$. Standard renormalization group strategies -- whether they be numerical or analytic -- also fail to improve upon matters here. We thus explore three strategies to address the basic limitations of the TSM in the vicinity of $b=1$. In the first, we focus on the small-volume spectrum. We attempt to understand how much of the physics of the ShG is encoded in the zero mode part of its Hamiltonian, in essence how `quantum mechanical' vs `quantum field theoretic' the problem is. In the second, we identify the divergencies present in perturbation theory and perform their resummation using a supra-Borel approximate. In the third approach, we use the exact form factors of the model to treat the ShG at one value of $b$ as a perturbation of a ShG at a different coupling. In the light of this work, we argue that the strong coupling phase $b > 1$ of the Lagrangian formulation of model may be different from what is naïvely inferred from its $S$-matrix. In particular, we present an argument that the theory is massless for $b>1$.
