# Generalized derivative expansion and one-loop corrections to the vacuum energy of static background fields

#### Abstract

The derivative expansion of the effective action for field theories with spontaneous symmetry breaking has a variable expansion parameter which may become locally infinite. To circumvent this difficulty, I propose an alternative expansion series in which the series expansion is simultaneously developed in order of the number of derivatives of the field and in powers of the deviation of the field from its ground-state value. As an example, I have applied this new method to calculate the quantum correction to the energy of the (1+1)-dimensional soliton in various models which have been well investigated previously. The expansion series are calculated using MATHEMATICA to 14 terms. For the models in which exact solutions have been found, such as the sine-Gordon soliton, [Formula presented] soliton, and [Formula presented] soliton with a fermion loop, the new improved series can be recognized as well-known analytically summable series. The complete results of exact solutions are recovered. More importantly, for the cases where exact solutions may not be available, Padé approximants or the Borel summation can be used as an efficient method to provide an excellent approximation, in contrast with cumbersome numerical calculations. The Christ-Lee soliton and the [Formula presented] bag are used to illustrate this approximation. We also derive a compact hybrid formula in closed form to estimate the quantum correction to the static energy of the (1+1)-dimensional field. This new method can be easily extended to higher dimensions as well as other important applications such as vacuum tunneling, Skyrmion physics, etc. © 1997 The American Physical Society.

*This paper has been withdrawn.*