Supersymmetric cancellation of loop contributions in a SUSY gauge theory

Question

It is known that in SUSY models, loop contributions are automatically zero leading to a technically natural solution of the Higgs mass hierarchy problem.

In many SUSY books/notes, it is often shown that, with a Lagrangian consisting chiral superfields, $$ {\cal L} = \int \left\{\Phi_i^{\dagger}\Phi_i+\left[\frac{1}{2} m_{ij}\Phi_i\Phi_j+\frac{1}{3}\lambda_{ijk}\Phi_i\Phi_j\Phi_k\right]\delta^2(\bar{\theta}) +h.c.\right\}d^2\theta d^2\bar{\theta} $$

All the loop diagrams of the superfields are identically zero. Thus in terms of component fields, the loop contributions are canceled out.

Now, the same kind of vanishing loop contributions should be true in SUSY gauge theory as well. Without any direct/explicit calculation of the loop supergraphs, is there a simple argument/way to see how they are zero? Is that the same reason as the chiral model above, each supergraph contains a factor of $\delta^4(\theta)$ or higher order, which is $0$ ?

Even if you provide references, your own interpretation/explanation is appreciated.

Answer 1

A more colloquial way of understanding this is to write down the expressions for loop corrections in a SUSY gauge theory. The important ingredient is that in addition to the gauge bosons you will also have gauginos, i.e. fermions in the adjoint representation.

To understand how a vacuum diagram vanishes thing about how a gauge boson in 4D has two polarization states, as does a fermion; the Lorentz traces therefore give the same value. Moreover, both transform in the adjoint, making the color trace identical as well.

What's left as only difference is a minus sign for the fermionic contribution that originates in Wick's theorem and the anti-commutativity of fermionic fields.

I am sure this can also be neatly expressed in terms of superfields, but I find reasoning for how the contributions cancel due to symmetry is more elucidating.