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Vacuum expectation value

In quantum field theory, the vacuum expectation value (VEV) of an operator is defined as its average or expectation value in the vacuum state, which is the lowest-energy configuration of the quantum fields.^[1] This value, often denoted as \langle 0 | \hat{A} | 0 \rangle where \hat{A} is the operator and |0\rangle the vacuum state, quantifies the inherent fluctuations or condensates present even in the absence of particles, serving as a fundamental quantity for computing observables like propagators and correlation functions.^[1] For scalar fields, the VEV is determined by minimizing the potential energy density of the theory, resulting in a constant, uniform field value across spacetime in the ground state.^[2] The VEV is central to many phenomena in particle physics, particularly spontaneous symmetry breaking, where a non-zero VEV for a field like the Higgs field breaks electroweak symmetry, generating masses for gauge bosons such as the W and Z particles through the Higgs mechanism.^[2] In the Standard Model, the Higgs field's VEV is approximately 246 GeV, a scale set by the minimum of its Mexican-hat potential V(\phi) = -\frac{1}{2} \mu^2 \phi^2 + \frac{1}{4} \lambda \phi^4, leading to \langle \phi \rangle = \sqrt{\mu^2 / \lambda}.^[3] Non-zero VEVs can also arise from quantum corrections or regularization techniques, such as zeta-function regularization, which interprets them as continuum limits of discrete sums, ensuring finite and physically meaningful results in curved spacetimes or gauge theories.^[4] Beyond the Standard Model, VEVs influence confining theories like quantum chromodynamics, where they relate to chiral symmetry breaking and quark condensates,^[5] and in string theory contexts, they parameterize moduli spaces without always corresponding to field expectations.^[6] These values are computed using axiomatic approaches or path integrals, with propagators like the Wightman function \Delta(x-y) = \langle 0 | \phi(x) \phi(y) | 0 \rangle providing explicit two-point VEVs essential for perturbation theory and renormalization.^[1] Overall, VEVs bridge classical field minima with quantum vacuum structure, underpinning predictions from particle masses to cosmological constants.^[4]

Definition and Mathematical Formulation

Basic Concept

The vacuum expectation value (VEV) of a field operator in quantum field theory represents the average value that the operator takes in the ground state of the system, referred to as the vacuum state. This quantity captures the inherent properties of the quantum vacuum, serving as a fundamental measure of how fields behave even in the absence of excitations. In essence, it quantifies any persistent "shift" or baseline value for the field across spacetime, distinguishing it from classical notions of emptiness.^[7] Intuitively, the VEV extends the concept of expectation values from non-relativistic quantum mechanics, where one computes the average outcome of an observable in a given state, to the relativistic framework of quantum field theory. Here, the vacuum is the lowest-energy configuration, devoid of real particles but filled with underlying quantum structure. The idea emerged during the foundational development of quantum field theory in the mid-20th century, as physicists grappled with reconciling quantum mechanics and special relativity. A pivotal advancement came in 1963, when Philip Anderson demonstrated the VEV's crucial role in resolving apparent paradoxes related to gauge invariance and particle masses in condensed matter systems, laying groundwork for its broader application in particle physics.^[8] Unlike the classical vacuum, envisioned as a truly empty void, the quantum vacuum is dynamic and not devoid of activity; it features inherent fluctuations arising from the Heisenberg uncertainty principle, which allow temporary virtual particles to emerge and annihilate. These fluctuations ensure the vacuum is Lorentz-invariant and translationally symmetric in free theories, resulting in a zero VEV for scalar fields, as the potential minimum is at φ=0 and there are no interactions to shift the vacuum. However, in interacting theories, these dynamics can stabilize a non-zero VEV, enabling phenomena such as spontaneous symmetry breaking where the system's ground state selects a particular direction in field space.^[9]^[7]

Formal Expression in QFT

In quantum field theory, the vacuum expectation value (VEV) of a field operator \phi(x) is formally defined as the expectation value in the ground state, or vacuum state |0\rangle, given by

\langle \phi(x) \rangle = \langle 0 | \phi(x) | 0 \rangle,

where \phi(x) is the Heisenberg-picture field operator at spacetime point x.^[10] This expression captures the one-point correlation function in the operator formalism, serving as the fundamental quantity for determining non-trivial vacuum structure in interacting theories.^[11] In relativistic quantum field theories, the vacuum state is translationally invariant, implying that the VEV is typically independent of the position x for Lorentz-invariant vacua, so \langle \phi(x) \rangle = v, a spacetime-independent constant.^[12] This constant v characterizes the shift of the field from its perturbative zero value and is crucial for renormalization and symmetry properties.^[13] Equivalently, in the path integral formulation of quantum field theory, the VEV is expressed as a functional average over all field configurations, weighted by the phase factor from the action:

\langle \phi(x) \rangle = \frac{1}{Z} \int \mathcal{D}\phi \, \phi(x) \, e^{i S[\phi]},

where S[\phi] is the action functional and Z = \int \mathcal{D}\phi \, e^{i S[\phi]} is the vacuum-to-vacuum transition amplitude, or partition function, ensuring normalization.^[14] This representation facilitates non-perturbative insights and is particularly useful for lattice simulations or exact solutions in solvable models. In interacting quantum field theories, computing the VEV perturbatively involves expanding around the free-field vacuum using the Dyson series or, more commonly, Feynman diagram techniques for the one-point function.^[15] For instance, tadpole diagrams contribute to the shift in the field expectation value, and the full VEV is obtained by solving self-consistent equations from the effective action or by minimizing the one-loop effective potential.^[13] Renormalization is essential, as bare VEVs diverge, and counterterms adjust them to match physical scales.^[16] A prototypical example arises in scalar field theories with a potential exhibiting spontaneous symmetry breaking. Consider the real scalar field Lagrangian \mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - V(\phi), with the "Mexican hat" potential

V(\phi) = -\frac{\mu^2}{2} \phi^2 + \frac{\lambda}{4} \phi^4,

where \mu^2 > 0 and \lambda > 0. The classical minimum occurs at \phi = \pm v, with v = \sqrt{\mu^2 / \lambda}, yielding a non-zero VEV in the broken phase.^[17] Quantum corrections, computed via loop expansions of the effective potential, modify this value but preserve the qualitative structure for weak couplings.^[13] In natural units (\hbar = c = 1) and four spacetime dimensions, the scalar field \phi carries mass dimension [ \phi ] = 1, so the VEV v has dimensions of inverse length, directly linking it to an energy scale such as the inverse of a characteristic length in the theory. This dimensional analysis underscores the VEV's role in setting fundamental scales, like those in electroweak symmetry breaking, though higher-dimensional operators can alter it in effective theories.^[13]

Role in Quantum Field Theory

Vacuum State and Operators

In quantum field theory, the vacuum state, denoted as |0\rangle, is defined as the unique normalized ground state that is annihilated by all annihilation operators a_{\mathbf{k}}, satisfying a_{\mathbf{k}} |0\rangle = 0 for all wave vectors \mathbf{k}, with normalization \langle 0 | 0 \rangle = 1.^[18]^[19] This state represents the absence of physical particles and serves as the foundation for the Fock space construction in free field theories. The quantum field operators are expanded in terms of these creation and annihilation operators using mode functions. For a scalar field \phi(x), the expansion takes the form

\phi(x) = \int \frac{d^3 k}{(2\pi)^3} \frac{1}{\sqrt{2 \omega_{\mathbf{k}}}} \left[ a_{\mathbf{k}} e^{-i k \cdot x} + a^\dagger_{\mathbf{k}} e^{i k \cdot x} \right],

where \omega_{\mathbf{k}} = \sqrt{\mathbf{k}^2 + m^2} is the energy of mode \mathbf{k}, and the plane waves e^{\pm i k \cdot x} act as the mode functions u_{\mathbf{k}}(x).^[18] This decomposition expresses the field as a superposition of positive and negative frequency components, with a_{\mathbf{k}} and a^\dagger_{\mathbf{k}} satisfying the canonical commutation relations [a_{\mathbf{k}}, a^\dagger_{\mathbf{k}'}] = (2\pi)^3 \delta^3(\mathbf{k} - \mathbf{k}'). In free field theories, the vacuum expectation value (VEV) of the field operator vanishes, \langle 0 | \phi(x) | 0 \rangle = 0, because the annihilation part a_{\mathbf{k}} |0\rangle = 0 yields zero, while the creation part \langle 0 | a^\dagger_{\mathbf{k}} = 0 ensures orthogonality between the two terms in the expansion.^[18] This result holds due to the separation of creation and annihilation operators, reflecting the Lorentz-invariant nature of the free vacuum. In interacting theories, particularly those with potentials that allow spontaneous symmetry breaking, the true vacuum state differs from the free vacuum, resulting in a non-zero VEV \langle 0' | \phi(x) | 0' \rangle \neq 0. This vacuum is selected by the interacting Hamiltonian as the ground state where the field acquires a constant expectation value at the potential minimum, breaking the symmetry of the underlying Lagrangian while the generators no longer annihilate the vacuum. To isolate physical contributions and subtract infinite vacuum fluctuations, normal ordering is employed, denoted as :\phi(x):, which rearranges operators so all annihilation operators appear to the right of creation operators. This ensures \langle 0 | :\phi(x): | 0 \rangle = 0 by removing the divergent zero-point contributions from the VEV.^[18] At the operator level, VEVs of field operators or the energy-momentum tensor contribute to observable vacuum energy shifts, as seen in the Casimir effect, where boundary conditions modify mode sums, leading to a finite attractive force between plates proportional to -\pi^2 / (240 a^4) (in natural units) due to altered vacuum fluctuations.^[20]

Relation to Correlation Functions

In quantum field theory, correlation functions, often called n-point functions, are defined as the vacuum expectation values of time-ordered products of field operators:

\langle 0 | T[\phi(x_1) \cdots \phi(x_n)] | 0 \rangle,

where T denotes time-ordering and |0\rangle is the vacuum state.^[21] These functions encode the dynamical information of the theory and are fundamental for computing scattering amplitudes and other observables via the LSZ reduction formula.^[22] The vacuum expectation value (VEV) of a field \phi corresponds to the simplest correlation function, the 1-point function \langle 0 | \phi(x) | 0 \rangle = v. A non-zero VEV signals a breakdown in translation invariance or, more commonly, spontaneous symmetry breaking, where the vacuum selects a preferred direction in field space.^[23] All correlation functions can be systematically generated from the functional Z[J] = \langle 0 | e^{i \int d^4x \, J(x) \phi(x)} | 0 \rangle, known as the generating functional, which incorporates external sources J(x). The n-point correlation functions are obtained by successive functional differentiation:

\langle 0 | T[\phi(x_1) \cdots \phi(x_n)] | 0 \rangle = (-i)^n \frac{\delta^n Z[J]}{\delta J(x_1) \cdots \delta J(x_n)} \bigg|_{J=0}.

Specifically, the VEV emerges as the first derivative evaluated at zero source: v = -i \frac{\delta \log Z[J]}{\delta J(x)} \big|_{J=0}.^[21] The connected correlation functions, relevant for the effective action, are instead derived from W[J] = -i \log Z[J]. To analyze perturbations around a non-zero VEV, the field is shifted as \phi(x) = v + \eta(x), where \eta(x) is the fluctuation field with vanishing VEV \langle 0 | \eta(x) | 0 \rangle = 0. This transformation recasts the original correlation functions in terms of those involving \eta, facilitating perturbative expansions and revealing the spectrum of excitations, such as massive modes from the quadratic terms in the shifted Lagrangian.^[23] Symmetries in the theory impose Ward identities, which are relations among correlation functions derived from invariance under infinitesimal transformations. When a non-zero VEV breaks a continuous symmetry spontaneously, these identities constrain the correlators—for instance, linking the 1-point function to vanishing Goldstone modes in the 2-point functions—ensuring consistency with Noether currents even in the broken phase.^[24] VEVs are subject to renormalization due to ultraviolet divergences in the 1-point function, requiring counterterms to define a finite v. This renormalization propagates to higher-point correlation functions, altering their divergent structure and necessitating scheme-dependent adjustments, such as in dimensional regularization, to maintain gauge invariance and unitarity.^[22]^[25]

Physical Implications

Spontaneous Symmetry Breaking

Spontaneous symmetry breaking arises when a symmetry present in the Lagrangian of a quantum field theory is not shared by the chosen vacuum state, due to the existence of multiple degenerate vacua from which dynamics selects one with a non-zero vacuum expectation value (VEV) for a scalar field. This process contrasts with explicit symmetry breaking, where the Lagrangian itself lacks the symmetry, and instead reflects an instability of the symmetric vacuum toward a lower-energy asymmetric state. A prototypical illustration is the Mexican hat potential for a complex scalar field \phi, given by

V(\phi) = -\mu^2 |\phi|^2 + \lambda |\phi|^4,

with \mu^2 > 0 and \lambda > 0, which possesses a global U(1) symmetry under \phi \to e^{i\alpha} \phi.^[26] The potential has a maximum at \phi = 0 and degenerate minima forming a circle at |\phi| = v = \sqrt{\mu^2 / (2\lambda)}, where v is the VEV.^[26] Quantum fluctuations around this VEV break the continuous symmetry, as the vacuum chooses a specific phase, leading to a non-invariant ground state.^[26] The Goldstone theorem states that a non-zero VEV of a scalar field that breaks a continuous global symmetry results in the emergence of massless Goldstone bosons, with the number of such bosons equal to the number of broken generators of the symmetry group. These modes correspond to excitations along the flat directions of the potential, restoring the broken symmetry at low energies through long-wavelength fluctuations. For instance, in an O(N symmetry broken to O(N-1), N-1 Goldstone bosons appear. In contrast, breaking a discrete symmetry, such as Z_2 in a real scalar \phi^4 theory with potential V(\phi) = \frac{\lambda}{4} (\phi^2 - v^2)^2, yields two degenerate minima at \phi = \pm v but no Goldstone bosons, as discrete symmetries lack continuous generators. Continuous breaking, as in the U(1) or O(N) cases, mandates the massless modes, while discrete cases do not. The concept gained prominence in the 1960s through analogies to superconductivity; Yoichiro Nambu introduced spontaneous breaking in particle physics by drawing parallels to Cooper pair condensation, where the ground state acquires a non-zero VEV for the order parameter. Jeffrey Goldstone formalized the theorem for field theories with superconducting solutions, predicting the associated massless particles. Philip W. Anderson extended this by linking the breaking to gauge invariance and collective excitations like plasmons in superconductors. Experimental signatures of spontaneous breaking via non-zero VEVs include the prediction of massless Goldstone modes in global continuous cases, though these may be absent or modified in discrete breaking scenarios, where no such low-energy theorems apply. In some extensions to local symmetries, these modes can be absorbed, evading direct observation.

Higgs Mechanism and Mass Generation

In gauge theories, a non-zero vacuum expectation value (VEV) of a scalar field breaks local symmetries, endowing gauge bosons with mass while maintaining the invariance of the underlying Lagrangian under gauge transformations. This process, known as the Higgs mechanism, allows massive vector bosons to propagate three polarization states without introducing inconsistencies in the theory. In the Standard Model, the Higgs field is represented as a complex scalar doublet under the SU(2)_L × U(1)_Y electroweak gauge group, acquiring a VEV denoted by v \approx 246 GeV at the minimum of its Mexican-hat potential, V(\phi) = -\mu^2 |\phi|^2 + \lambda |\phi|^4, where the potential parameters ensure a stable vacuum. This VEV spontaneously breaks the electroweak symmetry down to electromagnetism, consistent with the observed massless photon and massive W and Z bosons.^[27] The masses of the charged W bosons arise from the covariant kinetic term of the Higgs field after symmetry breaking, yielding m_W = \frac{g v}{2}, where g is the SU(2)_L coupling constant.^[27] Similarly, the neutral Z boson mass is m_Z = \frac{v}{2} \sqrt{g^2 + g'^2}, with g' the U(1)_Y coupling, reflecting the mixing of the weak and hypercharge gauge fields. Fermions obtain masses through Yukawa couplings to the Higgs doublet, resulting in Dirac mass terms of the form m_f = \frac{y_f v}{\sqrt{2}}, where y_f is the fermion-specific Yukawa coupling strength.^[27] The physical Higgs boson emerges as the radial fluctuation around this VEV, with its mass m_H determined by the second derivative (curvature) of the potential at the minimum, m_H^2 = 2 \lambda v^2, where \lambda is the scalar self-coupling.^[27] This excitation is observable and has been measured at approximately 125 GeV. The non-zero VEV also preserves perturbative unitarity in electroweak processes, such as longitudinal gauge boson scattering, by canceling divergences that would otherwise violate unitarity bounds at energies around 1 TeV without the Higgs mechanism. The Higgs mechanism was independently proposed in 1964 by Peter Higgs and by François Englert and Robert Brout, extending Philip Anderson's 1963 insights on symmetry breaking in superconducting systems to relativistic gauge theories. This framework underpins mass generation in the electroweak sector, linking the VEV to fundamental particle properties without explicit mass terms that would violate gauge invariance.

Applications and Examples

Scalar Field Theories

In scalar field theories, the vacuum expectation value (VEV) plays a central role in realizing spontaneous symmetry breaking through simple models like the quartic self-interacting scalar, often denoted as \phi^4 theory. The Lagrangian for a real scalar field \phi is given by \mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - V(\phi), where the tree-level potential is V(\phi) = -\frac{1}{2} \mu^2 \phi^2 + \frac{\lambda}{4} \phi^4 with \mu^2 > 0 and \lambda > 0 to ensure a stable minimum away from the origin. The VEV v is determined by minimizing the potential via \frac{dV}{d\phi} = 0, yielding v = \sqrt{\frac{\mu^2}{\lambda}}, which shifts the field to a non-zero value in the vacuum state, breaking the \mathbb{Z}_2 symmetry of the theory. Quantum corrections modify this classical picture through the effective potential V_{\text{eff}}(\phi), which incorporates loop contributions to stabilize or adjust the VEV. In the Coleman-Weinberg mechanism, for a massless scalar theory at tree level (\mu^2 = 0), radiative corrections generate a non-zero VEV via V_{\text{eff}}(\phi) = V(\phi) + \frac{\lambda^2 \phi^4}{64\pi^2} \left( \ln \frac{\lambda \phi^2}{2\mu^2} - \frac{3}{2} \right), where the logarithmic term drives the minimum to \phi = v \neq 0, illustrating how instabilities in the tree-level potential can be resolved by one-loop effects. This mechanism highlights the role of VEVs in massless theories, where quantum fluctuations induce symmetry breaking without bare mass parameters. At finite temperature, the effective potential becomes temperature-dependent, leading to phase transitions where the VEV vanishes above a critical temperature T_c. The thermal correction to the potential includes a leading field-dependent term \Delta V_T(\phi) \approx \frac{T^2 m^2(\phi)}{24}, where m^2(\phi) = -\mu^2 + 3 [\lambda](/page/Lambda) \phi^2; daisy resummation further incorporates a thermal self-energy contribution \Pi(T) \approx \frac{[\lambda](/page/Lambda) T^2}{8} to m^2(\phi), restoring symmetry at high T as the effective quadratic term becomes positive.^[28] For the \phi^4 model, T_c \approx \sqrt{\frac{8\mu^2}{[\lambda](/page/Lambda)}}. Below T_c, the VEV grows as v(T) \propto (T_c^2 - T^2)^{1/2}, marking a second-order phase transition in the mean-field approximation. As an illustrative numerical example, taking \mu = 100 GeV and \lambda = 0.1 yields v \approx 316 GeV at zero temperature, comparable to electroweak scales but without gauge interactions. In models addressing beyond-Standard-Model puzzles, such as the strong CP problem, axion-like scalar fields acquire VEVs that dynamically adjust to cancel CP-violating effects. In the Peccei-Quinn framework, an axion field a couples to the topological term in QCD via \mathcal{L} \supset \frac{a}{f_a} \frac{g^2}{32\pi^2} G \tilde{G}, where the potential V(a) \sim \Lambda^4 (1 - \cos(a/f_a)) minimizes at \langle a \rangle / f_a = -\theta, with \theta the QCD theta angle, setting the effective \theta_{\text{eff}} = 0 and solving the strong CP issue through the VEV.^[29]

Electroweak Theory

In the electroweak theory, the vacuum expectation value (VEV) of the Higgs field plays a pivotal role in spontaneous symmetry breaking, unifying the weak and electromagnetic interactions within the Standard Model. The electroweak sector is based on the SU(2)_L × U(1)_Y gauge group, where the Higgs field is introduced as a complex scalar doublet Φ with hypercharge Y = 1/2 to preserve anomaly cancellation and provide a mechanism for mass generation without violating gauge invariance. The Higgs potential is given by

V(\Phi) = \mu^2 \Phi^\dagger \Phi + \lambda (\Phi^\dagger \Phi)^2,

with μ² < 0 and λ > 0 ensuring a stable minimum. Minimization of this potential occurs when ⟨Φ† Φ⟩ = -μ² / (2λ), leading to a nonzero VEV that breaks the electroweak symmetry to the U(1)_EM of electromagnetism. Specifically, the VEV takes the form ⟨Φ⟩ = (0, v/√2)^T, where v is the magnitude of the VEV. This breaking was first proposed in the context of gauge theories by Higgs, Englert, and Brout in 1964, and incorporated into the full electroweak model by Weinberg in 1967. The nonzero VEV induces the Higgs mechanism, whereby three of the four degrees of freedom in the Higgs doublet become Goldstone bosons that are absorbed by the W± and Z gauge bosons, rendering them massive while the photon remains massless. The masses of the electroweak bosons are directly proportional to the VEV: M_W = (g v)/2 and M_Z = (v/2) √(g² + g'²), where g and g' are the SU(2)_L and U(1)_Y coupling constants, respectively. Fermions acquire masses through Yukawa interactions in the Lagrangian, with m_f = y_f v / √2 for a fermion f with Yukawa coupling y_f, ensuring the chiral structure of weak interactions is preserved. This framework resolves the issue of massless weak bosons predicted by early gauge models and explains the observed parity violation in weak decays.^[30]^[31] Experimentally, the VEV v is determined from the Fermi constant G_F measured in muon decay, via the relation v = (√2 G_F)^{-1/2} ≈ 246.22 GeV, with G_F = 1.1663787(6) × 10^{-5} GeV^{-2}. This value sets the electroweak scale and has been precisely confirmed through electroweak precision tests at LEP and the Tevatron, consistent with the Higgs boson mass of approximately 125 GeV discovered at the LHC in 2012. The stability of the electroweak vacuum, influenced by quantum corrections to the Higgs potential, remains a key area of study, with the measured Higgs mass suggesting metastability but no immediate instability.^[30]

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