I understand that the Higgs Field has a positive average value, while the average strength of (almost?) all of the other particle fields is zero. Does that positive value relates specifically to the ability of the Higgs Field to give mass?
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Further to Luboš' answer, Matt Strassler's blog has a nice explanation of how the non-zero vev of the Higgs field confers mass. Some brain cells required though. – John Rennie Jan 28 '16 at 07:31
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The Higgs field has a nonzero vacuum expectation value ("vev") $\langle h\rangle =v$. What matters is that it's nonzero; the sign is a matter of conventions. In fact, the Higgs field is a complex one so the phase of $v$ could be chosen to be arbitrary complex, too.
This nonzero vev gives masses to other particle species (whose "vevs" are zero) because of the interaction terms in the Lagrangian like $$ g^2 h^2 A_\mu A^\mu + yh \bar\psi \psi $$ where $A_mu$ is a gauge field, $\psi$ is a fermionic spinor field, $g,y$ are coupling constants, and $h$ is the Higgs field. They're the gauge interactions of the gauge field with the Higgs and the Yukawa cubic interaction, respectively. When $h$ is decomposed as $h=v+\Delta h$ where $\langle \Delta h\rangle = 0$, the term proportional to $v$ produces terms of the form $$ m^2 A_\mu A^\mu + M\bar\psi \psi$$ which are mass terms for the gauge field or fermions, respectively.
Now, one could ask why mass terms in quantum field theory actually produce what we know as ordinary masses etc. I was answering exactly this question days ago
One could continue with the "why" questions and copy a whole textbook of quantum field theory here, of course.
Luboš Motl
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