Infrared renormalon in the supersymmetric $\mathbb{C}P^{N-1}$ model on $\mathbb{R}\times S^1$

Abstract

<jats:title>Abstract</jats:title> <jats:p>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 boundary conditions. In our large-$N$ limit, the combination $\Lambda R$, where $\Lambda$ is the dynamical scale and $R$ is the $S^1$ radius, is kept fixed (we set $\Lambda R\ll1$ so that the perturbative expansion with respect to the coupling constant at the mass scale $1/R$ is meaningful). We extract the perturbative part from the large-$N$ expression of the gluon condensate and obtain the corresponding Borel transform $B(u)$. For $\mathbb{R}\times S^1$, we find that the Borel singularity at $u=2$, which exists in the system on the uncompactified $\mathbb{R}^2$ and corresponds to twice the minimal bion action, disappears. Instead, an unfamiliar renormalon singularity emerges at $u=3/2$ for the compactified space $\mathbb{R}\times S^1$. The semi-classical interpretation of this peculiar singularity is not clear because $u=3/2$ is not dividable by the minimal bion action. It appears that our observation for the system on $\mathbb{R}\times S^1$ prompts reconsideration on the semi-classical bion picture of the infrared renormalon.</jats:p>

Remarks on the large-NCPN−1model

“…In this paper we have examined a number of issues, recently debated (see Refs. [11][12][13][14][15][16][17][19][20][21][22][23][24][25][26][27][28][29][30][31]), on the features of the ground state of the CP N−1 model at finite temperature, (small) density and size. We have worked out an expansionà la Ginzburg-Landau for the effective action as a functional of the Lagrange multiplier M 2 , that enforces the constraint on the fundamental fields of the model and operates as an effective mass.…”

To be, or not to be finite? The Higgs potential in Gauge-Higgs Unification

“…In this paper, we discuss another way to decompose quantum fluctuations, which has been used in [28][29][30][31] for Abelian cases. The new decomposition is related to the KK decomposition by the Poisson resummation formula, 1 which is given by…”

“…To overcome technical difficulties that appear in perturbative calculation, we discuss a method, compactification by superposition, which greatly simplifies the calculation of the Higgs potential in a non-Abelian GHU model. A similar method has been used in the literature [28][29][30][31] for Abelian cases. In this method, momentum sums and integrals in M 4 × S 1 are expressed as superposition of momentum integrals in M 5 , i.e.…”

Confinement as analytic continuation beyond infinite coupling

“…See, e.g., Refs. [19][20][21][22][23][24][25] for detailed studies in twisted S 1 compactification of the CP N−1 σ model. Although saddle points of the classical action do exist [21,23] in such twisted compactification, there is some evidence [25,26] (see also the original work [27] in 4D) that those classical solutions do not correspond to renormalons.…”