半梯分波用于所有 $n,d$ 及其在弦论中的应用
Half-ladder partial waves for all $n,d$ with applications to string theory
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中文总结 AI 辅助
本文为半梯费曼图引入分波展开，应用于任意维度弦论振幅，证明幺正性仅允许开玻色子弦（$\alpha_0=-1$，$d\leq26$），并解析简并态贡献。
中文摘要 AI 辅助
给定任意四点散射振幅，分波展开将其留数分解为自旋为 $s$ 的内部粒子贡献，其中展开中的每个系数必须非负以满足幺正性。本文针对一般多重数 $n$ 和时空维度 $d$ 下的“半梯”费曼图引入一种分波展开，外部线为标量，内部线为有质量玻色子。通过强制半梯留数是每个内部粒子角动量的本征向量，每个基元写成与盖根鲍尔多项式相关的特殊函数乘积之和，并给出了一套简单的图形规则，用于在一般 $n$ 和 $d$ 下显式构建任意基元。为了将形式付诸实践，我们将该技术应用于变形的“弦论”振幅，这些振幅先验地定义于任意维度并具有任意雷吉截距 $\alpha_0$。通过构建半梯和“扭曲”半梯图的多粒子正性检验，我们证明唯一与幺正性一致的振幅是低于临界维度 $d \leq 26$ 的开玻色子弦论（$\alpha_0 = -1$）的振幅。特别地，我们确认了 $Z$-理论（$\alpha_0 = 0$）在有限 $\alpha'$ 下无法定义幺正弦论的预期。最后，对于开玻色子弦，我们展示了分波展开的知识如何使我们能够在纯在壳分析中解析给定能级和自旋的单个简并态对半梯图的贡献。作为附件文件，我们包含一个 Python 脚本，该脚本实现了给定一般（扭曲）半梯留数的分波展开。
英文摘要
Given any four-point scattering amplitude, the partial wave expansion resolves its residues into contributions from individual spin-$s$ internal particles, where each coefficient in the expansion must be non-negative due to unitarity. In this paper, we introduce a partial wave expansion tailored to "half-ladder" Feynman diagrams at generic multiplicity $n$ and spacetime dimension $d$, with scalars on the external lines and massive bosons on the internal lines. Derived by enforcing that the half-ladder residue is an eigenvector of each internal particle's angular momentum, each basis element is written as a sum over products of special functions related to the Gegenbauer polynomials, and we give a set of simple graphical rules for explicitly building any of them at generic $n$ and $d$. To put the formalism into action, we apply our technology to deformed "string theory" amplitudes, defined a priori in any dimension and with arbitrary Regge intercept $α_0$. By constructing multi-particle positivity tests for the half-ladder and "twisted" half-ladder diagrams, we show that the only amplitudes consistent with unitarity are those belonging to open bosonic string theory ($α_0 = -1$) below the critical dimension $d \leq 26$. In particular, we confirm the expectation that $Z$-theory ($α_0 = 0$) fails to define a unitary string theory at finite $α'$. Finally, for the open bosonic string, we show how knowledge of the partial wave expansion allows us to resolve the contributions to the half-ladder diagram from individual degenerate states at a given level and spin in a purely on-shell analysis. As an ancillary file, we include a Python script combwaves.py which implements the partial wave expansion given a generic (twisted) half-ladder residue.
发表机构
- Princeton University（普林斯顿大学）
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