Vacuum expectation value

This concept is important for working with correlation functions in quantum field theory. In the context of spontaneous symmetry breaking, an operator that has a vanishing expectation value due to symmetry can acquire a nonzero vacuum expectation value during a phase transition. Examples are:

• The Higgs field has a vacuum expectation value of 246 GeV. This nonzero value underlies the Higgs mechanism of the Standard Model. This value is given by ${\displaystyle v=1/\sqrt{\sqrt{2}{G}_F^0}=2M_W/g\approx 246.22\mathrm{G}\mathrm{e}\mathrm{V}}$, where M_W is the mass of the W Boson, ${\displaystyle G_F^0}$ the reduced Fermi constant, and g the weak isospin coupling, in natural units. It is also near the limit of the most massive nuclei, at v = 264.3 Da.
• The chiral condensate in quantum chromodynamics, about a factor of a thousand smaller than the above, gives a large effective mass to quarks, and distinguishes between phases of quark matter. This underlies the bulk of the mass of most hadrons.
• The gluon condensate in quantum chromodynamics may also be partly responsible for masses of hadrons.

The observed Lorentz invariance of space-time allows only the formation of condensates which are Lorentz scalars and have vanishing charge. Thus, fermion condensates must be of the form ${\displaystyle ⟨\overline{\psi }\psi ⟩}$, where ψ is the fermion field. Similarly a tensor field, G_μν, can only have a scalar expectation value such as ${\displaystyle ⟨G_{\mu \nu }{G}^{\mu \nu }⟩}$.

In some vacua of string theory, however, non-scalar condensates are found. If these describe our universe, then Lorentz symmetry violation may be observable.