Sine-Gordon effective potential beyond Gaussian approximation

Combining an optimized expansion scheme in the spirit of the background field method with the Colemanʹs normal-ordering renormalization prescription, we calculate the effective potential of sine-Gordon field theory beyond the Gaussian approximation. The first-order result is just the sine-Gordon Gaussian effective potential (GEP). For the range of the coupling β2⩽3.4π (an approximate value), a calculation with Mathematica indicates that the approximated effective potential up to the second order is finite without any further renormalization procedure and tends to improve the GEP more substantially while β2 increases from zero to β2⩽3.4π.