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Register a new userContinued functions and perturbation series: Simple tools for convergence of diverging series in $O(n)$-symmetric $phi^4$ field theory at weak coupling limit
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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.
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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.
The stochastic $phi^4$-theory in $d-$dimensions dynamically develops domain wall structures within which the order parameter is not continuous. We develop a statistical theory for the $phi^4$-theory driven with a random forcing which is white in time and Gaussian-correlated in space. A master equation is derived for the probability density function (PDF) of the order parameter, when the forcing correlation length is much smaller than the system size, but much larger than the typical width of the domain walls. Moreover, exact expressions for the one-point PDF and all the moments $<phi^n>$ are given. We then investigate the intermittency issue in the strong coupling limit, and derive the tail of the PDF of the increments $phi(x_2) - phi(x_1)$. The scaling laws for the structure functions of the increments are obtained through numerical simulations. It is shown that the moments of field increments defined by, $C_b=< |phi(x_2)-phi(x_1)|^b>$, behave as $|x_1-x_2|^{xi_b}$, where $xi_b=b$ for $bleq 1$, and $xi_b=1$ for $bgeq1$
We solve analytically the renormalization-group equation for the potential of the O(N)-symmetric scalar theory in the large-N limit and in dimensions 2<d<4, in order to look for nonperturbative fixed points that were found numerically in a recent study. We find new real solutions with singularities in the higher derivatives of the potential at its minimum, and complex solutions with branch cuts along the negative real axis.
We derive an integral-free thermodynamic perturbation series expansion for quantum partition functions which enables an analytical term-by-term calculation of the series. The expansion is carried out around the partition function of the classical component of the Hamiltonian with the expansion parameter being the strength of the off-diagonal, or quantum, portion. To demonstrate the usefulness of the technique we analytically compute to third order the partition functions of the 1D Ising model with longitudinal and transverse fields, and the quantum 1D Heisenberg model.
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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