Abstract
The ratios R2k of renormalized coupling constants g2k that enter the effective potential and small-field equation of state acquire the universal values at criticality. They are calculated for the three-dimensional scalar λϕ4 field theory (3D Ising model) within the pseudo-ϵ expansion approach. Pseudo-ϵ expansions for the critical values of g6, g8, g10, R6=g6/g42, R8=g8/g43 and R10=g10/g44 originating from the five-loop renormalization group (RG) series are derived. Pseudo-ϵ expansions for the sextic coupling have rapidly diminishing coefficients, so addressing Padé approximants yields proper numerical results. Use of Padé–Borel–Leroy and conformal mapping resummation techniques further improves the accuracy leading to the values R6⁎=1.6488 and R6⁎=1.6490 which are in a brilliant agreement with the result of advanced lattice calculations. For the octic coupling the numerical structure of the pseudo-ϵ expansions is less favorable. Nevertheless, the conform-Borel resummation gives R8⁎=0.868, the number being close to the lattice estimate R8⁎=0.871 and compatible with the result of 3D RG analysis R8⁎=0.857. Pseudo-ϵ expansions for R10⁎ and g10⁎ are also found to have much smaller coefficients than those of the original RG series. They remain, however, fast growing and big enough to prevent obtaining fair numerical estimates.
Highlights
IntroductionFour- and five-loop renormalization group (RG) expansions resummed by means of Borel-transformation-based procedures lead to the values for g∗6 differing from each other by less than 0.5% [15, 16] while the three-loop RG approximation turns out to be sufficient to provide an apparent accuracy no worse than 1.6% [15, 24]
The critical behavior of the systems undergoing continuous phase transitions is characterized by a set of universal parameters including, apart from critical exponents, renormalized effective coupling constants g2k and the ratios R2k = g2k/g4k−1
We study renormalized effective coupling constants and universal ratios R2k of the three-dimensional Ising model with the help of pseudo- expansion technique
Summary
Four- and five-loop RG expansions resummed by means of Borel-transformation-based procedures lead to the values for g∗6 differing from each other by less than 0.5% [15, 16] while the three-loop RG approximation turns out to be sufficient to provide an apparent accuracy no worse than 1.6% [15, 24] In principle, this is not surprising since the field-theoretical RG approach proved to be highly efficient when used to estimate critical exponents, critical amplitude ratios, marginal dimensionality of the order parameter, etc. We study renormalized effective coupling constants and universal ratios R2k of the three-dimensional Ising model with the help of pseudo- expansion technique. The last section contains a summary of the results obtained
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