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Register a new userLattice study on the twisted ${mathbb C}P^{N-1}$ models on ${mathbb R}times S^{1}$
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We report the results of the lattice simulation of the ${mathbb C} P^{N-1}$ sigma model on $S_{s}^{1}$(large) $times$ $S_{tau}^{1}$(small). We take a sufficiently large ratio of the circumferences to approximate the model on ${mathbb R} times S^1$. For periodic boundary condition imposed in the $S_{tau}^{1}$ direction, we show that the expectation value of the Polyakov loop undergoes a deconfinement crossover as the compactified circumference is decreased, where the peak of the associated susceptibility gets sharper for larger $N$. For ${mathbb Z}_{N}$ twisted boundary condition, we find that, even at relatively high $beta$ (small circumference), the regular $N$-sided polygon-shaped distributions of Polyakov loop leads to small expectation values of Polyakov loop, which implies unbroken ${mathbb Z}_{N}$ symmetry if sufficient statistics and large volumes are adopted. We also argue the existence of fractional instantons and bions by investigating the dependence of the Polyakov loop on $S_{s}^{1}$ direction, which causes transition between ${mathbb Z}_{N}$ vacua.
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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 $Lambda Rll1$ 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 emph{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.
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 SUSY-breaking parameter~$deltaepsilon$. This quantity was vigorously studied recently by Fujimori et al. using a semi-classical approximation based on the bion, motivated by a possible semi-classical picture on the infrared renormalon. In our calculation, we find that the parameter~$deltaepsilon$ receives renormalization and, after this renormalization, the vacuum energy becomes ultraviolet finite. To the next-to-leading order of the $1/N$ expansion, we find that the vacuum energy normalized by the radius of the~$S^1$, $R$, $RE(deltaepsilon)$ behaves as inverse powers of~$Lambda R$ for~$Lambda R$ small, where $Lambda$ is the dynamical scale. Since $Lambda$ is related to the renormalized t~Hooft coupling~$lambda_R$ as~$Lambdasim e^{-2pi/lambda_R}$, to the order of the $1/N$ expansion we work out, the vacuum energy is a purely non-perturbative quantity and has no well-defined weak coupling expansion in~$lambda_R$.
We study confining strings in ${cal{N}}=1$ supersymmetric $SU(N_c)$ Yang-Mills theory in the semiclassical regime on $mathbb{R}^{1,2} times mathbb{S}^1$. Static quarks are expected to be confined by double strings composed of two domain walls - which are lines in $mathbb{R}^2$ - rather than by a single flux tube. Each domain wall carries part of the quarks chromoelectric flux. We numerically study this mechanism and find that double-string confinement holds for strings of all $N$-alities, except for those between fundamental quarks. We show that, for $N_c ge 5$, the two domain walls confining unit $N$-ality quarks attract and form non-BPS bound states, collapsing to a single flux line. We determine the $N$-ality dependence of the string tensions for $2 le N_c le 10$. Compared to known scaling laws, we find a weaker, almost flat $N$-ality dependence, which is qualitatively explained by the properties of BPS domain walls. We also quantitatively study the behavior of confining strings upon increasing the $mathbb{S}^1$ size by including the effect of virtual $W$-bosons and show that the qualitative features of double-string confinement persist.
We investigate the lattice ${mathbb C} P^{N-1}$ sigma model on $S_{s}^{1}$(large) $times$ $S_{tau}^{1}$(small) with the ${mathbb Z}_{N}$ symmetric twisted boundary condition, where a sufficiently large ratio of the circumferences ($L_{s}gg L_{tau}$) is taken to approximate ${mathbb R} times S^1$. We find that the expectation value of the Polyakov loop, which is an order parameter of the ${mathbb Z}_N$ symmetry, remains consistent with zero ($|langle Prangle|sim 0$) from small to relatively large inverse coupling $beta$ (from large to small $L_{tau}$). As $beta$ increases, the distribution of the Polyakov loop on the complex plane, which concentrates around the origin for small $beta$, isotropically spreads and forms a regular $N$-sided-polygon shape (e.g. pentagon for $N=5$), leading to $|langle Prangle| sim 0$. By investigating the dependence of the Polyakov loop on $S_{s}^{1}$ direction, we also verify the existence of fractional instantons and bions, which cause tunneling transition between the classical $N$ vacua and stabilize the ${mathbb Z}_{N}$ symmetry. Even for quite high $beta$, we find that a regular-polygon shape of the Polyakov-loop distribution, even if it is broken, tends to be restored and $|langle Prangle|$ gets smaller as the number of samples increases. To discuss the adiabatic continuity of the vacuum structure from another viewpoint, we calculate the $beta$ dependence of ``pseudo-entropy density $proptolangle T_{xx}-T_{tautau}rangle$. The result is consistent with the absence of a phase transition between large and small $beta$ regions.
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-$beta_0$ approximation as a conventional tool to analyze the renormalon, in which only Feynman diagrams that dominate in the large-$n_W$ limit are considered while the coefficient of the vacuum polarization is set by hand to the one-loop beta function~$beta_0=11/3-2n_W/3$. In the large~$N$ limit within the large-$beta_0$ approximation, the W-boson, which acquires the twisted Kaluza--Klein momentum, produces the renormalon ambiguity corresponding to the Borel singularity at~$u=2$. This provides an example that the system in the compactified space~$mathbb{R}^3times S^1$ possesses the renormalon ambiguity identical to that in the uncompactified space~$mathbb{R}^4$. We also discuss the subtle issue that the location of the Borel singularity can change depending on the order of two necessary operations.
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