Checks of anomaly cancellation

Question

In a textbook I read that if $G$ is a global symmetry of the classical Lagrangian, then one has to check $G\times H^2$ anomalies, where $H$ is one of the SM gauge groups.

For example, when $G$ refers to the Baryon number $B$, the contribution of the $SU(2)$ gauge fields is verified by:

$$ SU(2)^2\times B= 2\times 3\times B $$

where only LH fields contribute and the coefficients 2 and 3 come from the dimension of the representation. This shows that $B$ is anomalous, while $B-L$ is not. I therefore wonder if this procedure was applicable to any global symmetry for which the charge value is different for each of the doublet components. As an example I took the electrical charge and the same control gives:

$$ SU(2)^2 \times Q = (-1+0) + 3\times (\frac{2}{3} - \frac{1}{3}) = 0$$

where I make the coefficient 2 disappear to take into account each charge of the doublet. Thus, the $G = Q$ global symmetry is non-anomalous as expected. Likewise for the mixed anomalies $G\times SU(3)^2$ and $G\times Y^2$ which vanish if $G = Q$.

1. Is this procedure correct for any global symmetry for which the charge is different for each of the doublet components?

2. Now if a global symmetry $G$ passes gauge anomaly checks, does this automatically imply that $G$ (like $B-L$) is a gauge symmetry in a theory beyond SM, or is it only a hint that it could be a gauge symmetry?

Answer 1

... if 𝐺 is a global symmetry of the classical Lagrangian, then one has to check 𝐺×𝐻² anomalies, where 𝐻 is one of the SM gauge groups.

On may check them for academic purposes, not consistency, as one has to for gauge anomalies. In particular, if they cancel, then, in a GUT extension, G could be thus promoted to a gauge symmetry consistently, but need not, in response to your question 2.

In fact, low energy physics all but runs on such "safe", consistent, anomalies. E.g., the global chiral $A^5_3QQ$ anomaly (in the WWZ term) allows the pseudogoldston neutral pion to decay to two photons! ($A^5_3$ is the SSBroken Axial chargeless color singlet current of QCD pumping the neutral pion into and out of the QCD vacuum, and coupling to two photons via the SM gauge charges Q.)

As for your B SU(2)² counting, indeed 2 flavors per generation of quarks, of which there are 3, adds the baryon numbers of each, so three colors times 1/3, namely 1, over 2 for the trace normalization. A plain addition of species. Nonvanishing, so baryons could decay to sphalerons or something. B-L is, of course, non-anomalous.

I would, however, strongly urge you to not use Q in your self-illustration example, as Q is an SM current/charge, a linear combination of Y and $T_3$, so you are just computing a necessarily vanishing $H^3$ gauge anomaly! The SM would not be consistent if it didn't vanish!

Answer 2

A lot of the information you gave me is beyond my knowledge, not being a science professional myself or even a student. But I like reading physics textbooks from time to time and the chapter on anomalies intrigues me. In particular the fact that the anomaly cancellation constraint for the $U(1)_Y$ gauge symmetry leads to the charge quantization is impressive. So I wonder about how to construct a hypothetical global symmetry $G$ of the standard model which we consider first of all as not associated with a gauge field, whose charges $Q_i$ are the most general possible and which would be non-anomalous (before thinking to promote G as a gauge symmetry). If I understood correctly, this global symmetry cannot result from a $U(1)$ transformation alone because the charge $Q$ would have to be identical for each doublet components (this is why I took the example of the EM charge).

Since i only saw examples where $G$ is $B$, $L$ or $B-L$, my first question is actually basic, regarding the check of mixing anomaly with $SU(2)$, have I the right to write this ? $$ \sum_{Left} G\times SU(2)^2 - \sum_{Right} G\times SU(2)^2 = Q_{e L} + Q_{\nu L} + 3\times (Q_ {u L} + Q_{d L}) - 0 $$

Finally, I realize that it is difficult for me to understand certain particle physics concepts further without a better knowledge of group theory. Could you recommend a good group theory book for beginners geared towards particle physics?