 Asymmetric-cutoff BCS transition temperature

For a constant effective density of states and attractive coupling $\lambda$, let the linearized BCS theory gap equation integrate $\tanh (\xi /(2k_B{T}_c))/(2\xi )$ over $-\sigma <\xi <\theta$, with positive cutoffs. Its even integrand makes the result the mean of two positive-cutoff logarithms. When $k_B{T}_c\ll \sigma ,\theta$, $1/\lambda \simeq \frac{1}{2}\ln (1.13\theta /(k_B{T}_c))+\frac{1}{2}\ln (1.13\sigma /(k_B{T}_c))$, giving the displayed geometric mean of cutoffs.

If a band bottom truncates the lower side, $\sigma =\min (\mu ,\theta )$ for positive chemical potential $\mu$. The weak-coupling limit at fixed $\mu /\theta >0$ must not be interchanged with an uncontrolled $\mu \to 0$ extrapolation: the lower logarithm requires $\mu \gg k_B{T}_c$ when $\mu <\theta$.