Critical properties of $\Phi^4_{1+1}$-theory in Light-Cone Quantization

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Abstract

The dynamics of the phase transition of the continuum $\Phi ^{4}_{1+1}$-theory in Light Cone Quantization is reexamined taking into account fluctuations of the order parameter $< \Phi >$ in the form of dynamical zero mode operators (DZMO) which appear in a natural way via the Haag expansion of the field $\Phi (x)$ of the interacting theory. The inclusion of the DZM-sector changes significantly the value of the critical coupling, bringing it in agreement within 2% with the most recent Monte-Carlo and high temperature/strong coupling estimates. The critical slowing down of the DZMO governs the low momentum behavior of the dispersion relation through invariance of this DZMO under conformal transformations preserving the local light cone structure. The critical exponent $\eta$ characterising the scaling behaviour at $k^2 \to 0$ comes out in agreement with the known value 0.25 of the Ising universality class. $\eta$ is made of two contributions: one, analytic $(75 %$) and another (25%) which can be evaluated only numerically with an estimated error of 3%. The $\beta$-function is then found from the non-perturbative expression of the physical mass. It is non-analytic in the coupling constant with a critical exponent $\omega=2$. However, at D=2, $\omega$ is not parametrisation independent with respect to the space of coupling constants due to this strong non-analytic behaviour.