I'm trying to understand a simple Little Higgs (toy) model where the Higgs doublet is made of pseudo Nambu-Goldstone Bosons generated by breaking the symmetry from $SU(3)_L\times U(1)_Y$ to $SU(2)_L\times U(1)_Y$, in order to prevent fine-tuning and 'solve' the hierarchy problem.

In the toy model there are 2 sets of NGB $\phi_1$ and $\phi_2$, each containing 5 NGB (since there are 5 broken generators).
For 5 of these, some potential will be generated, giving them mass which gives us the Higgs doublet and a pseudo-axion.
The other 5 however are absorbed into 5 of the 8 gauge bosons from the $SU(3)_L$ which will become massive, leaving us 4 massless gauge bosons (3 from $SU(3)_L$ and 1 from $U(1)_Y$), of which 3 will later become massive due to EWSB ($W^+$, $W^-$, $Z^0$ and $A/\gamma$).

My question is how to interpret those 5 massive gauge bosons. They have to be physical and mediate some force, what type of particle could this be? I'm also unsure what the reason for the initial symmetry breaking could be.

Since it's just a toy model, none of the resources I found really talk about it, and the full models are a bit too complex for me to fully understand.


1 Answer 1


If the toy model you're referring to is the same one outlined in chapter 3 of this great review, then it may help to be more careful about which symmetries are being broken.

We start with two symmetries $$SU(3)_1 \times U(1)_1, \qquad SU(3)_2 \times U(1)_2,$$ which are spontaneously broken at a scale $f_1 \equiv f_2 \equiv f$ to $$SU(2)_1 \times U(1)_1, \qquad SU(2)_2 \times U(1)_2,$$ leaving ten degrees of freedom $\phi_1, \phi_2$. To answer your question of what the source of this breaking may have been: Who knows. We typically treat the question of Composite Higgs / Little Higgs effective models separately from the underlying dynamics. Although, a UV-complete model should suggest what has broken the symmetries. Perhaps a new strong technicolor condensate, perhaps a higher-dimensional model looks like these global symmetries.

Then one carefully gauges the group $$SU(3)_L \times U(1)_Y \subset SU(3)_1 \times U(1)_1 \times SU(3)_2 \times U(1)_2$$ in such a way that $SU(2)_L \times U(1)_Y$ is contained within the unbroken $SU(2)_1 \times U(1)_1 \times SU(2)_2 \times U(1)_2$. Thus five linear combinations of $\phi_1$ and $\phi_2$ are eaten by the gauge bosons that have not gauged the EW group. They are eaten at the scale $f$ of the symmetry breaking, which gives you the scale of their masses. That is, $m_{A_i} \sim f$. To figure out exactly what these bosons are, you need to write out the generators and see what eigenvalues the broken ones have under $SU(2)_L \times U(1)_Y$, or just the electromagnetic $U(1)_Q$. Turns out four have charge (they are called "W primes") and one is neutral (called "Z prime"). That is, they look like heavy versions of the EW vector bosons. The cool thing is that we should be able to see them in a collider, thereby making Little Higgs models open to testing.


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