vev

VEV stands for vacuum expectation value, the quantum expectation value of a field operator in the vacuum state. It is written as <phi> or v for a scalar field. In quantum field theory, nonzero VEVs signal spontaneous symmetry breaking and determine the masses and couplings of particles through interactions with the field.

In the Standard Model, the Higgs field is a complex SU(2) doublet that acquires a nonzero VEV, breaking electroweak symmetry. The Higgs potential V(phi) = mu^2 phi^† phi + lambda (phi^† phi)^2, with mu^2 < 0, has a minimum at |phi| = v/√2, where v ≈ 246 GeV. Choosing a gauge, one can write phi = [0, (v + h)/√2]^T, where h is the physical Higgs boson.

Masses arise from Yukawa and gauge couplings: fermion masses m_f = y_f v/√2; W and Z masses m_W = g v/2, m_Z = sqrt(g^2+g'^2) v/2. The Goldstone modes become the longitudinal polarizations of W and Z (Higgs mechanism).

Important caveat: VEVs are gauge-dependent for non-invariant fields; only gauge-invariant quantities such as masses and cross-sections are physical. In other contexts, other fields can acquire VEVs, e.g., in theories beyond the Standard Model or in condensed matter systems as order parameters.

Conclusion: The VEV is a central concept in understanding how particles acquire mass and how symmetries are broken in the vacuum of our universe.