Scalar bubble integral ($B(P)$)

The loop momentum integral with two massive quantum field theory propagators isIts momentum dependence gives a one-loop self-energy in cubic scalar field theory and a quartic effective vertex in quartic scalar field theory.

Set $A=p^2-m^2\ne 0$. The denominator product is exactly $(A+k^2{)}^2-4(p\cdot k{)}^2$. Its reciprocal therefore has the Taylor expansionThe coefficients follow by expanding $1/(1+x)=1-x+O(x^2)$ with $x=2k^2/A-4(p\cdot k{)}^2/A^2+O(k^4)$. The expansion is a local algebraic statement away from denominator zeros; integrating it through Feynman propagator poles requires a regulator and the correct Feynman i-epsilon prescription.

For one real phi-fourth theory field, each of the three labeled one-loop four-point channels has symmetry factor $1/2$. In $d=4-2ϵ$, the displayed expression is its amplitude through the finite part. The three poles require $\delta \lambda =3{\lambda }^2/(32{\pi }^2ϵ)$, and differentiating ${\lambda }_B={\mu }^{2ϵ}(\lambda +\delta \lambda )$ 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.

With a mostly-plus metric signature, define $f_d=(1/i)\int d^{d}k/[(2\pi {)}^d(k^2-iϵ)((P-k{)}^2-iϵ)]$. Wick rotation and a Feynman parameter give $f_d=\Gamma (2-d/2)(4\pi {)}^{-d/2}{\int }_0^1[x(1-x)P^2{]}^{d/2-2}dx$. At nonzero Euclidean $P^2$, the Gamma function supplies the pole. Exceptional massless momentum can introduce infrared ambiguity.