Why 5D gauge theory is non-renormalizable?

Question

1. My question is following "Why 5D gauge theory is non-renormalizable?" Here I treat $5D$ supersymmetric gauge theories.

2. Also I heard Non-renormalizablity of $5D$ gauge theories implies the singularities in instanton moduli space. How this can be possible? Can you give me a reliable explanation?

Answer 1

In the textbook of TASI 2009, section "Introduction to extra dimension" i can find the answer as follows.

They state that $5D$ or higher dimensional gauge coupling has a negative mass dimension, so the 5d or higher dimensional gauge theory is non-renormalizable.

Answer 2

Concerning OP's 1st question:

• If we scale the gauge field $A_{\mu}$ so that the quadratic term $-\frac{1}{2}{\rm tr}F_{\mu\nu}^2$ in the Lagrangian density $ {\cal L}$ is canonically normalized, then it has mass dimension $$ [A_{\mu}] ~=~\frac{d}{2}-1,\tag{1} $$ where $d$ is the dimension of spacetime.

• From the gauge covariant derivative $ D_{\mu}~=~\partial_{\mu} \pm i g A_{\mu}, $ it then follows that the gauge coupling constant has mass dimension $$ [g] ~=~2-\frac{d}{2},\tag{2} $$ which is negative if $d>4$, and therefore non-renormalizable in the old perturbative Dyson sense, cf. e.g. this Phys.SE post.