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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~$mathbb{R}^3times S^1$ 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~$Rtoinfty$, 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.
We study the infrared renormalon in the gluon condensate in the $SU(N)$ gauge theory with $n_W$-flavor adjoint Weyl fermions (QCD(adj.)) on~$mathbb{R}^3times S^1$ with the $mathbb{Z}_N$ twisted boundary conditions. We rely on the so-called large-$bet
In the leading order of the large-$N$ approximation, we study the renormalon ambiguity in the gluon (or, more appropriately, photon) condensate in the 2D supersymmetric $mathbb{C}P^{N-1}$ model on~$mathbb{R}times S^1$ with the $mathbb{Z}_N$ twisted b
By employing the $1/N$ expansion, we compute the vacuum energy~$E(deltaepsilon)$ of the two-dimensional supersymmetric (SUSY) $mathbb{C}P^{N-1}$ model on~$mathbb{R}times S^1$ with $mathbb{Z}_N$ twisted boundary conditions to the second order in a SUS
We study the domain walls in hot $4$-D $SU(N)$ super Yang-Mills theory and QCD(adj), with $n_f$ Weyl flavors. We find that the $k$-wall worldvolume theory is $2$-D QCD with gauge group $SU(N-k)times SU(k) times U(1)$ and Dirac fermions charged under
We explicitly calculate the topological terms that arise in IR effective field theories for $SU(N)$ gauge theories on $mathbb{R}^3 times S^1$ by integrating out all but the lightest modes. We then show how these terms match all global-symmetry t Hoof