# Higgs maths video Brian Greene

What do you think of the video below?

Is the explanation in this video largely correct, or missing something?

Is the mathematical description in the video of the higgs mechanism right? He said that phi is re-written as a m + h. What does it mean? Is the m in this video the vacuum expectation value?

• Is the mathematical description in the video of the higgs mechanism right? He said that phi is re-written as a m + h. What does it mean? Is the m in this video the vacuum expectation value? Jan 24, 2014 at 7:33
• There's quite a nice explanation of this at theory.sinp.msu.ru/comphep_html/tutorial/node106.html Jan 24, 2014 at 9:19
• @JohnRennie as Lumo is able to write a nice answer, the question is obviously not "too broad", no ;-)...? It is not to broade even when reading just reading the question (without the answer), as the Higgs mechanism is well defined and the OP has even a specific question about it. Jan 24, 2014 at 10:27
• @Dilaton: I commented before the OP edited his question to specifically ask about rewriting $\Phi$ as $m + H$. Jan 24, 2014 at 10:34

The only "unusual" twist is a change of the notation that Brian wisely made to simplify the formulae and make them more intuitive. When he writes the total Higgs field $\Phi$, he actually means $(m_\Psi/v)\phi$ where $\phi$ is the usual physicist's normalization of the Higgs doublet field.
This $\phi$ may be rewritten as $v+h$ in the usual notation; $v=246$ GeV is the vacuum expectation value – also known as the vacuum Higgs condensate or the Higgs syrup penetrating all of space, Higgs aether, and so on – while $h$ is the oscillating quantum field measuring the variation of the Higgs field around the vacuum expectation values. When we multiply the usual formula $$\phi = v+h$$ by $m_\Psi/v$ and choose the symbol $H = (m_\Psi/v) h$ for the rescaled variation of the Higgs field, we will get Brian's $$\Phi = m+H$$ and the remaining formulae follow. Incidentally, $m_\Psi/v$ is also equal and may be written as $y_\Psi$, the Yukawa coupling (interaction constant) encoding the strength of the interaction of the Higgs field $h$ with the fermionic field $\Psi$.
At any rate, it is true that from the $v+h$ decomposition substituted to the interaction term $\phi \Psi\Psi$, one gets both the mass term from the constant, $v$ part of $\phi$, as well as the usual cubic interaction $y h \Psi\Psi$.
Well, there are some other details that Brian simplifies. For example, $\Psi\Psi$ should be $\bar \Psi\Psi$, one of the terms should be barred (complex conjugate), and there should be additional indices because there are many fermionic (lepton and quark) fields in the Standard Model, and so on, but these subtleties are not important for the understanding of the basic Higgs mechanism which is why Brian suppressed them.