Abstract
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. 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.
Exact ground state and elementary excitations of a competing spin chain with twisted boundary condition
