# Gauge symmetry and renormalization

• Lapidus
In summary, the conversation discusses the relationship between gauge symmetry and renormalization in particle physics. While it is not fully understood why gauge symmetry makes theories renormalizable, it is believed to have something to do with the Ward-Takahashi identities and Noether's Theorem. 't Hooft and Veltman's work on regularization and renormalization of gauge fields is referenced, and it is noted that certain non-renormalizable theories can be seen as low-energy effective theories arising from spontaneous breaking of gauge symmetry. The Standard Model is also mentioned as an example of how gauge symmetry is necessary for renormalization.

#### Lapidus

Here and then I read gauge symmetry makes theories renormalizable. Unfortunately I could not find a satisfactory explanation why that so is. Could someone shed some light?

thanks

It must have something to do with the Ward-Takahashi identities, I believe.

Anybody here who knows a little about gauge symmetry and renormalization and could help?

Note: some texts show how renormalization does not destroy gauge invariance, a fact that at first, of course, is not obvious. But then I read sometimes that gauge symmetry makes renormalization possible, like for example here in Perkins "Particle Astrophysics":

Why do we stress the concept of gauge invariance? The point of a gaugeinvariant theory is that it introduces a symmetry in the calculations, which
makes the theory renormalizable. This means that it is possible, at least in
principle, to make calculations in the form of a perturbation series to all orders
in the coupling constant, that is, for a sum over all possible Feynman diagrams,
including those involving an arbitrary number of exchanged photons.

Unfortunately, he fails to explain how exactly gauge symmetry makes theories renormalizable.

thanks again

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Well, the base explanation is Noether's Theorem of course. Dunno what exactly explanation you looking for

Yes, proving this is nontrivial but it was done (and the authors won a Nobel prize for it). The reference is

'T Hooft and Veltman, regularization and renormalization of gauge fields. Nucl. Phys. B44: 189-213, 1972

A rather good overview of the subject by 't Hooft can be found http://www.staff.science.uu.nl/~hooft101/gthpub/GtH_Yukawa_06.pdf" [Broken].

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Bill_K said:
A rather good overview of the subject by 't Hooft can be found http://www.staff.science.uu.nl/~hooft101/gthpub/GtH_Yukawa_06.pdf" [Broken].

Very good overview, indeed.

But as I understand t'Hooft and Veltmann showed that renormalizing a gauge invariant theory does not spoil the gauge invariance of the theory.

My question: is gauge symmetry even necessary to make some theories renormalisable (as it is claimed sometimes)?

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Lapidus said:
Very good overview, indeed.

But as I understand t'Hooft and Veltmann showed that renormalizing a gauge invariant theory does not spoil the gauge invariance of the theory.

My question: is gauge symmetry even necessary to make some theories renormalisable (as it is claimed sometimes)?

There are certainly renormalizable theories which are not gauge theories ($\phi^4$ theories with real $\phi$ come to mind). What you may be thinking of is that certain types of non-renormalizable theories can be seen to be low-energy effective theories arising from the spontaneous breaking of a gauge symmetry; and, casting them in this light restores renormalizability.

Parlyne said:
There are certainly renormalizable theories which are not gauge theories ($\phi^4$ theories with real $\phi$ come to mind). What you may be thinking of is that certain types of non-renormalizable theories can be seen to be low-energy effective theories arising from the spontaneous breaking of a gauge symmetry; and, casting them in this light restores renormalizability.

Well, I was not thinking of much! But I think you correctly sumed up what people mean when they say that gauge symmetry makes some theories renormalizable. Like that Fermi's theory for the weak force is not renormalisable, but the gauge-invariant Weinberg-Salam theory is.

thanks everybody

Another reason one usually say that gauge symmetry is required for renormalization is related to the construction of the Standard Model.

In fact if you start from the Yang-Mills Lagrangian and add an arbitrary mass term without the Higgs mechanism, thus spoiling gauge symmetry, you end up with a non renormalizable theory.