Suppose we add to the SM the following electroweak scalar triplet with hypercharge $Y_T=-1$ $$T=\begin{pmatrix} t^0 & t^-/\sqrt{2} \\ t^-\sqrt{2} & t^{--} \end{pmatrix}$$

where the superscript indicates the electric charge of each component field. It couples to the Higgs $H$ with the following Lagrangian: $$ L_T= tr|D_\mu T|^2-m^2_Ttr|T|^2 + \Lambda TH^2 + \Lambda(TH^2)^*$$ where $m_t$, $\Lambda$ are new mass parameters and $$D_\mu T= (\partial_\mu T-igW^a_\mu\frac{\sigma^a}{2}T-igTW^a_\mu\frac{\sigma^a}{2}-ig'Y_TB_\mu T $$ $$ TH^2\equiv T_{ab}\epsilon^{ac}\epsilon^{bd}H_cH_d \, . $$

Considering the usual Higgs potential $$V(H)=-m^2|H|^2-\lambda(H^\dagger H)^2$$

we want:

  1. Find the minimum of the full potential, the bound on the triplet vev and the constraints on the $m^2_T /\Lambda$ ratio
  2. Gauge bosons mass spectrum
  1. For what concern the full potential, it's clear that the $\Lambda$ coupling breaks the $SO(3)$ custodial symmetry. In fact, using only the neutral component of T and opening the Higgs we have:


so, I would say that $\Delta\rho\sim\Lambda/v$. Performing the first derivative I find:

\begin{equation} V_T=2m^2_TTr[T_{ab}]-\Lambda\epsilon^{ac}\epsilon^{bd}H_cH_d=0\end{equation}

\begin{equation} V_H=-2\Lambda T_{ab}\epsilon^{ac}\epsilon^{bd}-m^2H^\dagger_c+2\lambda|H|^2H^\dagger_c=0. \end{equation}

Now I have a doubt: for the vev of the triplet T, I can choose whatever "gauge" I want or is there a more privileged choice? I would say that, according to CP invariance, only an em neutral component can acquire a vev, so i would let $t^0$ acquires a vev: $$<T_{ab}>=v_0\delta_{a1}\delta_{b1} \, . $$
So I'm not sure if this choice should be an input in the eqs. and see if it minimizes the potential or rather an output. Anyway, substituting that in the first eq. I find the condition for the Higgs vev (could be seen as a consistency check of my choice?): $$V_T=2m^2_Tv_0-\Lambda (<H_2>)^2=0$$ which, denoting $v$ the Higgs vev, implies: $$\frac{m^2_T}{\Lambda}=\frac{v^2}{v_0} \, .$$

  1. After the usual manipulations for the kinetic term of $T$ in the unitary gauge, I find:

$$L_{mass}=\frac{v_0^2}{4}\biggl[\bigl(gW_\mu^3-2g'B\mu)^2+g^2W_\mu^+W_\mu^-\biggr]$$ doubt: I haven't done any calculation yet, but how is it possible to find a mass term for the $Z$ if I have already a precise constraint (the one from Higgs sector) for the $(B,W^3)$ rotation into $(A,Z)$?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.