2

In Standard Model, the Higgs doublet is \begin{equation} \Phi=\left(\begin{array}{c}\phi^+\\ \phi^0\end{array}\right) =\frac{1}{\sqrt{2}}\left(\begin{array}{c} \phi_1+i\phi_2\\ \phi_3+i\phi_4 \end{array}\right) \end{equation}

The complex conjugate of Higgs double is

\begin{equation} \Phi^*=\left(\begin{array}{c}\phi^-\\ \phi^{0*}\end{array}\right) =\frac{1}{\sqrt{2}}\left(\begin{array}{c} \phi_1-i\phi_2\\ \phi_3-i\phi_4 \end{array}\right) \end{equation}

Question : restricting to the topic of Higgs physics, what is present in the universe :

  1. The Higgs doublet only ?
  2. The complex conjugate of the Higgs doublet ?
  3. Both the Higgs doublet and its complex conjugate ?

The picture in wikipedia there : https://en.wikipedia.org/wiki/Mathematical_formulation_of_the_Standard_Model#/media/File:Standard_Model_Of_Particle_Physics--Most_Complete_Diagram.png

would make believe that only the Higgs doublet is present in the universe. But is it true ?

Mathieu Krisztian
  • 795

1 Answers1

2

The two field doublets you are writing down involve the same four $\phi_i$s, and they constitute different arrangements of them, so they make no difference in the amplitudes involved and their physics implications. (The physical Higgs particle is linked to $\phi_3-v$.)

They are all "present" in our universe, as would any other rewriting of them, such as $\tilde \Phi$, etc... As long as the interactions and couplings of each are specified clearly, and each is fit into the SM we know, there should be no issue. Recall both $\Phi$ and $\Phi^*$ are present in the real covariant kinetic term in the SM Lagrangian written in the practical notation.

Cosmas Zachos
  • 62,595
  • $W^+$ "eats" $\phi^+$. $W^-$ "eats" $\phi^-$. $Z$ "eats" $i\phi_4$. So "who" eats $-i\phi_4$ ? – Mathieu Krisztian Jun 18 '22 at 20:46
  • 1
    If you look at the relevant "eating term" resulting from the Higgs covariant derivative squared, that involves both a $\Phi$ and a $\Phi^$, and amounts to $g^2v^2 (Z^\mu/sin \theta_w -\partial^\mu \phi_4/gv )^2/2$, so the Z* ate $\sin\theta_w~ \phi_4/gv$ gauged into it. Both the i and the -i play a role, and neither features in the gauge absorption by the Z. – Cosmas Zachos Jun 18 '22 at 21:50
  • 1
    And, of course, nobody eats $\phi_3 -v$, which is why the Higgs particle is around... – Cosmas Zachos Jun 18 '22 at 22:01