Quartic Yukawa interaction in three spacetime dimensions ($-\frac{g}{2}{\varphi }^2\bar{\psi }\psi$)

A nonderivative interaction of two real scalar legs with a fermion pair in $2+1$ dimensions. The engineering dimensions of relativistic fields are $[\varphi ]=1/2$ and $[\psi ]=1$, so $g$ is a marginal coupling by classical power counting. The internal symmetry $\varphi \mapsto -\varphi$ forbids odd-scalar interactions, but allows a sextic scalar interaction. The two-component Dirac field representation must be distinguished from four-component gamma matrices in four spacetime dimensions.

Four quartic Yukawa vertices on a fermion loop with one internal scalar chord give six external scalar legs, two loops and overall logarithmic degree. The coefficient is nonzero: in a constant scalar background $v$, the induced cubic Yukawa coupling is $y=gv$ and the fermion mass is $\mu =m+g{v}^2/2$. The two-loop sunset diagram in the quantum effective potential has a logarithmic pole proportional to $y^2(4{\mu }^2-{\mathcal{B}}^2)$, where ${\mathcal{B}}^2$ is the background scalar squared mass. Its $v^6$ term is $g^4{v}^6$ when the tree sextic coupling vanishes. Thus zero renormalized sextic coupling at one scale does not eliminate the sextic counterterm or make that zero stable under renormalization.

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 $(/k{)}^3=k^2/k$, 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 $\mathrm{tr}({\gamma }^0{\gamma }^1{\gamma }^2)=-2i$. No one-loop sextic counterterm is needed, although a finite six-point vertex may remain.