We present additional observations to previous studies on the infrared (IR) renormalon in $SU(N)$ QCD(adj.), the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions on~$\mathbb{R}^3\times S^1$ with the $\mathbb{Z}_N$ twisted boundary condition. First, we show that, for arbitrary finite~$N$, a logarithmic factor in the vacuum polarization of the "photon" (the gauge boson associated with the Cartan generators of~$SU(N)$) disappears under the $S^1$~compactification. Since the IR renormalon is attributed to the presence of this logarithmic factor, it is concluded that there is no IR renormalon in this system with finite~$N$. This result generalizes the observation made by Anber and~Sulejmanpasic [J. High Energy Phys.\ \textbf{1501}, 139 (2015)] for $N=2$ and~$3$ to arbitrary finite~$N$. Next, we point out that, although renormalon ambiguities do not appear through the Borel procedure in this system, an ambiguity appears in an alternative resummation procedure in which a resummed quantity is given by a momentum integration where the inverse of the vacuum polarization is included as the integrand. Such an ambiguity is caused by a simple zero at non-zero momentum of the vacuum polarization. Under the decompactification~$R\to\infty$, where $R$ is the radius of the $S^1$, this ambiguity in the momentum integration smoothly reduces to the IR renormalon ambiguity in~$\mathbb{R}^4$. We term this ambiguity in the momentum integration "renormalon precursor". The emergence of the IR renormalon ambiguity in~$\mathbb{R}^4$ under the decompactification can be naturally understood with this notion.

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更多关于 $\mathbb{R}^3\times S^1$ 上 U(N) QCD(adj.) 的红外重整子

我们对之前对 $SU(N)$ QCD(adj.) 中红外 (IR) 重整子的研究提出了额外的观察结果，$SU(N)$ 规范理论与 $n_W$-flavor 伴随 Weyl 费米子在~$\mathbb {R}^3\times S^1$ 与 $\mathbb{Z}_N$ 扭曲边界条件。首先，我们证明，对于任意有限~$N$，“光子”（与~$SU(N)$ 的嘉当发生器相关的规范玻色子）的真空极化中的对数因子在 $S^ 下消失1$~压缩。由于IR重整子归因于这个对数因子的存在，因此可以得出结论，在这个有限~$N$的系统中不存在IR重整子。这个结果概括了 Anber 和~Sulejmanpasic [J. High Energy Phys.\ \textbf{1501}, 139 (2015)] 对于 $N=2$ 和~$3$ 到任意有限~$N$。接下来，我们要指出的是，尽管在该系统中通过 Borel 程序不会出现重整子模糊，但在另一种求和程序中会出现模糊，其中通过动量积分给出求和量，其中真空极化的倒数作为被积函数包含在内。这种模糊是由真空极化的非零动量下的简单零引起的。在解压缩~$R\to\infty$下，其中$R$是$S^1$的半径，动量积分中的这种模糊性平滑地降低到~$\mathbb{R}^4中的IR重整子模糊性$. 我们将动量积分中的这种歧义称为“重整子前体”。用这个概念自然可以理解解压缩下~$\mathbb{R}^4$中IR重整子歧义的出现。