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Register a new userTwo-loop RG functions of the massive $phi^4$ field theory in general dimensions
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Authors
M. A. Shpot
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Two-loop Feynman integrals of the massive $phi^4_d$ field theory are explicitly obtained for generic space dimensions $d$. Corresponding renormalization-group functions are expressed in a compact form in terms of Gauss hypergeometric functions. A number of interesting and useful relations is given for these integrals as well as for several special mathematical functions and constants.
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Critical two-point correlation functions in the continuous and lattice phi^4 models with scalar order parameter phi are considered. We show by different non-perturbative methods that the critical correlation functions <phi^n(0) phi^m(x)> are proportional to <phi(0) phi(x)> at |x| --> infinity for any positive odd integers n and m. We investigate how our results and some other results for well-defined models can be related to the conformal field theory (CFT), considered by Rychkov and Tan, and reveal some problems here. We find this CFT to be rather formal, as it is based on an ill-defined model. Moreover, we find it very unlikely that the used there equation of motion really holds from the point of view of statistical physics.
We derive exact analytic results for several four-point correlation functions for statistical models exhibiting phase separation in two-dimensions. Our theoretical results are then specialized to the Ising model on the two-dimensional strip and found to be in excellent agreement with high-precision Monte Carlo simulations.
We determine universal critical exponents that describe the continuous phase transitions in different dimensions of space. We use continued functions without any external unknown parameters to obtain analytic continuation for the recently derived 7- loop $epsilon$ expansion from $O(n)$-symmetric $phi^4$ field theory. Employing a new blended continued function, we obtain critical exponent $alpha=-0.0121(22)$ for the phase transition of superfluid helium which matches closely with the most accurate experimental value. This result addresses the long-standing discrepancy between the theoretical predictions and precise experimental result of $O(2)$ $phi^4$ model known as $lambda$-point specific heat experimental anomaly. Further we have also examined the applicability of such continued functions in other examples of field theories.
We show that the exact beta-function beta(g) in the continuous 2D gPhi^{4} model possesses the Kramers-Wannier duality symmetry. The duality symmetry transformation tilde{g}=d(g) such that beta(d(g))=d(g)beta(g) is constructed and the approximate values of g^{*} computed from the duality equation d(g^{*})=g^{*} are shown to agree with the available numerical results. The calculation of the beta-function beta(g) for the 2D scalar gPhi^{4} field theory based on the strong coupling expansion is developed and the expansion of beta(g) in powers of g^{-1} is obtained up to order g^{-8}. The numerical values calculated for the renormalized coupling constant g_{+}^{*} are in reasonable good agreement with the best modern estimates recently obtained from the high-temperature series expansion and with those known from the perturbative four-loop renormalization-group calculations. The application of Cardys theorem for calculating the renormalized isothermal coupling constant g_{c} of the 2D Ising model and the related universal critical amplitudes is also discussed.
We provide an analysis of the structure of renormalisation scheme invariants for the case of $phi^4$ theory, relevant in four dimensions. We give a complete discussion of the invariants up to four loops and include some partial results at five loops, showing that there are considerably more invariants than one might naively have expected. We also show that one-vertex reducible contributions may consistently be omitted in a well-defined class of schemes which of course includes MSbar.
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