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  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?

Qmechanic
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phy_math
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2 Answers2

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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.

phy_math
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    That's a perturbative/power-counting argument, but not a proof that renormalization is impossible, I believe. Symmetries and "accidental" cancellations can make naively divergent stuff finite. – ACuriousMind Mar 04 '15 at 17:18
  • Ok i try to find relevant materials dealing with cancellation of divergent terms in 5d gauge theory. As a student, i think this topic is interesting subject to study in depth. thanks for your mention. ^.^ – phy_math Mar 04 '15 at 17:24
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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.

Qmechanic
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