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Register a new userSuperfast convergence effect in large orders of the perturbative and $epsilon$ expansions for the O(N) symmetric $phi^4$ model
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P.V. Pobylitsa
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Usually the asymptotic behavior for large orders of the perturbation theory is reached rather slowly. However, in the case of the N-component $phi^4$ model in D=4 dimensions one can find a special quantity that exhibits an extremely fast convergence to the asymptotic form. A comparison of the available 5-loop result for this quantity with the asymptotic value shows agreement at the 0.1% level. An analogous superfast convergence to the asymptotic form happens in the case of the O(N)-symmetric anharmonic oscillator where this convergence has inverse factorial type. The large orders of the $epsilon$ expansion for critical exponents manifest a similar effect.
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We compute, using the method of large spin perturbation theory, the anomalous dimensions and OPE coefficients of all leading twist operators in the critical $ O(N) $ model, to fourth order in the $ epsilon $-expansion. This is done fully within a bootstrap framework, and generalizes a recent result for the CFT-data of the Wilson-Fisher model. The anomalous dimensions we obtain for the $ O(N) $ singlet operators agree with the literature values, obtained by diagrammatic techniques, while the anomalous dimensions for operators in other representations, as well as all OPE coefficients, are new. From the results for the OPE coefficients, we derive the $ epsilon^4 $ corrections to the central charges $ C_T $ and $ C_J $, which are found to be compatible with the known large $ N $ expansions. Predictions for the central charge in the strongly coupled 3d model, including the 3d Ising model, are made for various values of $ N $, which compare favourably with numerical results and previous predictions.
We study large charge sectors in the $O(N)$ model in $6-epsilon $ dimensions. For $4<d<6$, in perturbation theory, the quartic $O(N)$ theory has a UV stable fixed point at large $N$. It was recently argued that this fixed point can be described in terms of an IR fixed point of a cubic $O(N)$ model. By considering a double scaling limit of large charge and weak couplings, we compute two-point and all extremal higher-point correlation functions for large charge operators and find a precise equivalence between both pictures. Instanton instabilities are found to be exponentially suppressed at large charge. We also consider correlation function of $U(1)$-invariant meson operators in the $O(2N)supset U(1)times SU(N)$ theory, as a first step towards tests of (higher spin) $AdS/CFT$.
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Yuri M. Pismak n Department of Theoretical Physics State University Saint-Petersburg Russia
2008
The critical behaviour of the O(n)-symmetric model with two n-vector fields is studied within the field-theoretical renormalization group approach in a D=4-2 epsilon expansion. Depending on the coupling constants the beta-functions, fixed points and critical exponents are calculated up to the one- and two-loop order, resp. (eta in two- and three-loop order). Continuous lines of fixed points and O(n)*O(2) invariant discrete solutions were found. Apart from already known fixed points two new ones were found. One agrees in one-loop order with a known fixed point, but differs from it in two-loop order.
We study boundary scattering in the $phi^4$ model on a half-line with a one-parameter family of Neumann-type boundary conditions. A rich variety of phenomena is observed, which extends previously-studied behaviour on the full line to include regimes of near-elastic scattering, the restoration of a missing scattering window, and the creation of a kink or oscillon through the collision-induced decay of a metastable boundary state. We also study the decay of the vibrational boundary mode, and explore different scenarios for its relaxation and for the creation of kinks.
A non-perturbative Renormalization Group approach is used to calculate scaling functions for an O(4) model in d=3 dimensions in the presence of an external symmetry-breaking field. These scaling functions are important for the analysis of critical behavior in the O(4) universality class. For example, the finite-temperature phase transition in QCD with two flavors is expected to fall into this class. Critical exponents are calculated in local potential approximation. Parameterizations of the scaling functions for the order parameter and for the longitudinal susceptibility are given. Relations from universal scaling arguments between these scaling functions are investigated and confirmed. The expected asymptotic behavior of the scaling functions predicted by Griffiths is observed. Corrections to the scaling behavior at large values of the external field are studied qualitatively. These scaling corrections can become large, which might have implications for the scaling analysis of lattice QCD results.
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