= Yukawa fermion triangle
{c}
{title2=$\operatorname{tr}[(\not a+m)(\not b+m)(\not c+m)]=4m[m^2+a\cdot b+a\cdot c+b\cdot c]$}
Three scalar <Yukawa interactions> on a closed <Dirac field> loop give two cyclic orientations. Odd <gamma matrix> traces vanish, while $\operatorname{tr}(\gamma^\mu\gamma^\nu)=4g^{\mu\nu}$ 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.
Back to article page
