Effective potential and stability of the rigid membrane

TL;DR

This paper calculates the effective potential for rigid p-branes, including membranes, using one-loop and 1/d approximations, analyzing zeta-functions, and identifying potential extrema to understand stability.

Contribution

It provides a detailed calculation of the effective potential for rigid p-branes, including new asymptotic formulas and stability analysis for membranes.

Findings

Derived explicit asymptotic formulas for the effective potential.

Analyzed the behavior and extrema of the potential for membranes.

Performed detailed regularization using zeta-functions of Epstein type.

Abstract

The calculation of the effective potential for fixed-end and toroidal rigid $p$-branes is performed in the one-loop as well as in the $1/d$ approximations. The analysis of the involved zeta-functions (of inhomogeneous Epstein type) which appear in the process of regularization is done in full detail. Assymptotic formulas (allowing only for exponentially decreasing errors of order $\leq 10^{-3}$) are found which carry all the dependences on the basic parameters of the theory explicitly. The behaviour of the effective potential (specified to the membrane case $p=2$) is investigated, and the extrema of this effective potential are obtained.