Yukawa fermion triangle ($\mathrm{tr}[(/a+m)(/b+m)(/c+m)]=4m[m^2+a\cdot b+a\cdot c+b\cdot c]$)

Three scalar Yukawa interactions on a closed Dirac field loop give two cyclic orientations. Odd gamma matrix traces vanish, while $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu })=4g^{\mu \nu }$ gives the displayed numerator. For external invariant mass squares $M^2,{\mu }^2,{\mu }^2$, write $D_a=a^2-m^2$ and similarly for the other two lines. The numerator becomes $4m[D_a+D_b+D_c+4m^2-{\mu }^2-M^2/2]$, reducing the fermion loop to three scalar bubbles and a scalar triangle Feynman integral. The closed fermion loop sign must be included. A massive fermion produces a logarithmic local divergence.

For interactions $-G\bar{\psi }\psi \Phi -g\bar{\psi }\psi \varphi$, the two Yukawa fermion triangle orientations generate a local divergent three-scalar vertex proportional to $mG{g}^2$. In $d=4-2ϵ$, with $S-1=i\mathcal{M}$ convention, its loop pole is ${\mathcal{M}}_{\mathrm{div}}=-24mG{g}^2/(16{\pi }^2\overline{ϵ})$. A $\Phi {\varphi }^2$ counterterm cancels it. The finite renormalized coefficient is additional data, fixed by a renormalization condition. Declaring this cubic absent from the displayed bare interaction is not by itself a prescription for its loop-generated finite value.