veVs

A vacuum expectation value (VEV) is the average value of a quantum field in the vacuum state, denoted by ⟨0|φ|0⟩ for a field φ. If a field has a nonzero VEV, the vacuum picks out a preferred field configuration, which is a hallmark of spontaneous symmetry breaking and a source of particle masses through interactions with that field.

In the Standard Model of particle physics, the Higgs field has a nonzero VEV, v ≈ 246 GeV. This electroweak symmetry breaking, from SU(2)L × U(1)Y to U(1)em, gives masses to the W and Z gauge bosons and to the fermions via Yukawa couplings. The W and Z masses are related to the gauge couplings by m_W = gv/2 and m_Z = √(g^2 + g'^2) v/2, while a fermion with Yukawa coupling y_f obtains mass m_f = y_f v/√2.

Many theories beyond the Standard Model involve more than one scalar field with nonzero VEVs. For example, the Minimal Supersymmetric Standard Model features two Higgs doublets with VEVs v_u and v_d, with v^2 = v_u^2 + v_d^2 ≈ (246 GeV)^2 and tanβ = v_u/v_d. The pattern of VEVs determines the mass spectrum and the nature of symmetry breaking in the model.

Gauge considerations are also important: the VEV of a gauge field is generally not gauge-invariant, so physical statements are made in terms of gauge-invariant quantities or through the effects of the symmetry-breaking pattern.

In quantum chromodynamics, nonzero condensates such as the quark condensate ⟨q̄ q⟩ and the gluon condensate ⟨G^a_{μν}G^{aμν}⟩ reflect nonperturbative vacuum structure and chiral symmetry breaking.

VEVs thus play a central role in mass generation and the structure of the quantum vacuum.