Gauge invariance or global invariance, which one makes theory renormalizable?

Question

We know that gauge theory is renormalizable, due to the Ward-Takahashi identity (for non-Abelian theory, it is Slavnov-Taylor identity), which reflects the conserved current of gauge symmetry.

But local (gauge) symmetry is not a real 'symmetry', since it cannot lead to a physical conserved current. When the gauge group is non-Abelian, local gauge invariance can lead to either a gauge-invariant but non-conserved current, or a gauge-dependent but conserved current (for $U(1)$ group these two currents coincide). But global symmetry, leads to a physical (global) invariant conserved current (for non-Abelian group, gauge field transform under global transformation, too), and this can lead to corresponding Ward-Takahashi identity.

Now here is my question, if a gauge theory is global but not local invariant, does it renormalizable? Specifically, if in SM Lagrangian we change Higgs covariant differential $D_\mu$ to ordinary differential $\partial_\mu$, does the theory renormalizable? If the change is done, then the Yukawa interaction term destroys the local $SU(2)\times U(1)$ symmetry, but conserves the global one.

Answer 1

Gauge invariance is always a local symmetry. In this sense a gauge symmetry and a local symmetry can be taken as being synonymous. So there is no such thing as a global gauge symmetry. The local (gauge) symmetry is a real symmetry, because gauge transformations leave the Lagrangian invariant. One can also derive a conserved current (Noether current) for it, but one needs to use a bit of a trick.* One only allows the gauge field to be transformed and not the fermion fields. The resulting conserved current is then expressed in terms of the fermion fields. So, one can see that in an interacting gauge theory the gauge field couples to this conserved current.

So, if you were to change the local gauge symmetry into a global symmetry, you would not have the interactions anymore, because the gauge derivatives, which contain the interaction would disappear. The resulting theory would then decouple into two free-field theories, one for the gauge field and one for the fermion fields. Each of these would be trivially renormalizable, because there are no interactions.

*See, for example: M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison Wesley (1995), Chapter 9.