this Semester I have my first QFT class and we have a homework where I got stuck at the beginning. I have some ideas but I am not sure if they are correct, so I don't want a solution, I only want to know If i understood some parts correctly.

We have to assume the SM with two Generations. Now we extend the gauge group of SM with $SU(2)_H \times SU(3)_D$ and introduce new SM-neutral right handed fermion which is a doublet under $SU(2)_H$ and a triplet under $SU(3)_D$ and two left handed fermions which are singlets under $SU(2)_H$ and triplets under $SU(3)_D$ .

Now we have to consider the gauge anomalies in this SM Extension and have to assign the charges so that we keep the cancellation of this anomlies. (There are soe more parts to do but I want to understand the beginning first)

What I understand: My new gauge group is:

$SU(3)_C \times SU(3)_D \times SU(2)_H \times SU(2)_W \times U(1)_Y$

1.)Does this mean that my SM fermions also interact with the new part of the gauge Group? I think so, but that would only mean that I get more different interaction particles (more Bosons)

2.)If I look at the anomalies, which I get from the triangle diagrams, the extension of my gauge group should not provoke new anomalies because if I look at the triangle diagrams (where I couple the gauge current with two Bosons or the gravitons), all new diagrams with just $SU(2)$ and $SU(3)$ should be zero because I always get a trace over the representations of $SU(2)$ and $SU(3)$ and they are traceless or because $SU(3)$ is "blind" for chirality.

3.)I know that I can check if there are anomalies in the triangle diagram by looking at the $A^{abc}=tr[t^a\{t^b,t^c\}]$ and that this factor has to be zero so that there are no anomalies left. For a diagram with only $SU(3)$ I would get anomalies but because $SU(3)$ couples to right- and left-handed fermions equally, I can assign any "charge" I want to them as long as all have the same charge. Is this meant by diagrams with only $SU(3)$ are blind for chirality and so there are no anomalies I can obtain?

4.)The only other new anomalies come from $U(1)$ with the extension, but this anomalies should allready be cancled by the choosen hypercharge Y from the normal SM. (Is that correct?)

5.)Does this also mean that I could extend my gauge group by as many new $SU(2)$ and $SU(3)$ as I want?

6.)Now I include the new fermions. Because they are SM-neutral the do not interact with the gauge group from the SM. So they only interact with the Extension of the gauge group? (I am not sure what SM-neutral means)

7.)Can I see the new fermions as Quarks but coupling righthanded and, because they do not transform under $U(1)$, electrically chargeless?

8.)Now I want to consider the anomalies for the new fermions. Because they transform only under the extension $SU(2)_H \times SU(3)_D$ I only have to consider triangle diagrams with $SU(2)$ and $SU(3)$, but now the same happens as in Point 3 and there are no anomalies

9.)Does the H and D at the gauge group stand for something or is this freely choosen by our teacher?

There has to be something wrong in my thoughts :/ Can someone explain what I do wrong? Or which statements are correct/incorrect?

Greetings greeny


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