= One-loop sextic cancellation in quartic Yukawa theory
{title2=$\operatorname{tr}[(\not k)^3]=0$}
With zero scalar sextic coupling, the only potentially logarithmically divergent one-loop six-scalar vertex has three quartic Yukawa vertices on a <fermion loop>. Its leading numerator is $(\not k)^3=k^2\not k$, whose <trace> vanishes. A mass insertion or external <momentum> lowers the degree and makes the integral convergent. This is not vanishing of every odd <gamma-matrix trace>: for $\gamma^0=\sigma_3$, $\gamma^1=i\sigma_1$, $\gamma^2=i\sigma_2$, one has $\operatorname{tr}(\gamma^0\gamma^1\gamma^2)=-2i$. No one-loop sextic <counterterm> is needed, although a finite six-point vertex may remain.
Back to article page
