More on the infrared renormalon in SU(N) QCD(adj.) on R3 × S1

We present additional observations to previous studies on the infrared (IR) renormalon in SU(N) QCD(adj.), the SU(N) gauge theory with nW -flavor adjointWeyl fermions on R3 × S1 with the ZN 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 S1 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→∞, where R is the radius of the S1, this ambiguity in the momentum integration smoothly reduces to the IR renormalon ambiguity in R4. We term this ambiguity in the momentum integration "renormalon precursor". The emergence of the IR renormalon ambiguity in R4 under the decompactification can be naturally understood with this notion.