= Minimal-subtraction scalar four-point bubble channels
{title2=$B(q^2)=\frac{i\mu^{2\epsilon}\lambda^2}{32\pi^2}\left[\epsilon^{-1}-\gamma_E+\log(4\pi)-\int_0^1\log\frac{m^2-x(1-x)q^2-i0}{\mu^2}\,dx\right]$}
For one real <phi-fourth theory> field, each of the three labeled one-loop four-point channels has <symmetry factor> $1/2$. In $d=4-2\epsilon$, the displayed expression is its amplitude through the finite part. The three poles require $\delta\lambda=3\lambda^2/(32\pi^2\epsilon)$, and differentiating $\lambda_B=\mu^{2\epsilon}(\lambda+\delta\lambda)$ at fixed bare coupling gives $\beta(\lambda)=3\lambda^2/(16\pi^2)+O(\lambda^3)$ in four dimensions. The <minimal subtraction scheme> retains $\log(4\pi)-\gamma_E$; <modified minimal subtraction> absorbs that constant. The Feynman-parameter logarithm is left unevaluated.
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