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Register a new userThe large charge limit of scalar field theories and the Wilson-Fisher fixed point at $epsilon=0$
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We study the sector of large charge operators $phi^n$ ($phi$ being the complexified scalar field) in the $O(2)$ Wilson-Fisher fixed point in $4-epsilon$ dimensions that emerges when the coupling takes the critical value $gsim epsilon$. We show that, in the limit $gto 0$, when the theory naively approaches the gaussian fixed point, the sector of operators with $nto infty $ at fixed $g,n^2equiv lambda$ remains non-trivial. Surprisingly, one can compute the exact 2-point function and thereby the non-trivial anomalous dimension of the operator $phi^n$ by a full resummation of Feynman diagrams. The same result can be reproduced from a saddle point approximation to the path integral, which partly explains the existence of the limit. Finally, we extend these results to the three-dimensional $O(2)$-symmetric theory with $(bar{phi},phi)^3$ potential.
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We compute general higher-point functions in the sector of large charge operators $phi^n$, $barphi^n$ at large charge in $O(2)$ $(bar phiphi)^2$ theory. We find that there is a special class of extremal correlators having only one insertion of $bar phi^n$ that have a remarkably simple form in the double-scaling limit $nto infty $ at fixed $g,n^2equiv lambda$, where $gsimepsilon $ is the coupling at the $O(2)$ Wilson-Fisher fixed point in $4-epsilon$ dimensions. In this limit, also non-extremal correlators can be computed. As an example, we give the complete formula for $ langle phi(x_1)^{n},phi(x_2)^{n},bar{phi}(x_3)^{n},bar{phi}(x_4)^{n}rangle$, which reveals an interesting structure.
We compute the entanglement entropy of the Wilson-Fisher conformal field theory (CFT) in 2+1 dimensions with O($N$) symmetry in the limit of large $N$ for general entanglement geometries. We show that the leading large $N$ result can be obtained from the entanglement entropy of $N$ Gaussian scalar fields with their mass determined by the geometry. For a few geometries, the universal part of the entanglement entropy of the Wilson-Fisher CFT equals that of a CFT of $N$ massless scalar fields. However, in most cases, these CFTs have a distinct universal entanglement entropy even at $N=infty$. Notably, for a semi-infinite cylindrical region it scales as $N^0$, in stark contrast to the $N$-linear result of the Gaussian fixed point.
Using supersymmetric localization, we study the sector of chiral primary operators $({rm Tr} , phi^2 )^n$ with large $R$-charge $4n$ in $mathcal{N}=2$ four-dimensional superconformal theories in the weak coupling regime $grightarrow 0$, where $lambdaequiv g^2n$ is kept fixed as $ntoinfty $, $g$ representing the gauge theory coupling(s). In this limit, correlation functions $G_{2n}$ of these operators behave in a simple way, with an asymptotic behavior of the form $G_{2n}approx F_{infty}(lambda) left(frac{lambda}{2pi e}right)^{2n} n^alpha $, modulo $O(1/n)$ corrections, with $alpha=frac{1}{2} mathrm{dim}(mathfrak{g})$ for a gauge algebra $mathfrak{g}$ and a universal function $F_{infty}(lambda)$. As a by-product we find several new formulas both for the partition function as well as for perturbative correlators in ${cal N}=2$ $mathfrak{su}(N)$ gauge theory with $2N$ fundamental hypermultiplets.
In this paper we study the ultraviolet and infrared behaviour of the self energy of a point-like charge in the vector and scalar Lee-Wick electrodynamics in a $d+1$ dimensional space time. It is shown that in the vector case, the self energy is strictly ultraviolet finite up to $d=3$ spatial dimensions, finite in the renormalized sense for any $d$ odd, infrared divergent for $d=2$ and ultraviolet divergent for $d>2$ even. On the other hand, in the scalar case, the self energy is striclty finite for $dleq 3$, and finite, in the renormalized sense, for any $d$ odd.
We go beyond a systematic review of the semiclassical approaches for determining the scaling dimensions of fixed-charge operators in $U(1)$ and $O(N)$ models by introducing a general strategy apt at determining the relation between a given charge configuration and the associated operators for more involved symmetry groups such as the $U(N) times U(M)$. We show how, varying the charge configuration, it is possible to access anomalous dimensions of different operators transforming according to a variety of irreducible representations of the non-abelian symmetry group without the aid of diagrammatical computations. We illustrate our computational strategy by determining the anomalous dimensions of several composite operators to the next-to-leading order in the semiclassical expansion for the $U(N) times U(M)$ conformal field theory (CFT) in $4-epsilon$ dimensions. Thanks to the powerful interplay between semiclassical methods and group theory we can, for the first time, extract scaling dimensions for a wide range of operators.
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