The Goldstone boson is a fundamental, theoretically predicted excitation in quantum field theory that arises as a consequence of the spontaneous breaking of a continuous global symmetry. Named after the physicist Jeffrey Goldstone, these particles are characterized by having exactly zero mass ($m=0$) in the absence of any explicit symmetry-breaking terms. Their existence is rigorously established by Goldstone’s theorem, a cornerstone of modern particle physics and condensed matter theory. While predicted to be strictly massless, in many real-world scenarios, they acquire a small, non-zero mass due to residual, approximate symmetry breaking.
Theoretical Derivation and Goldstone’s Theorem
Goldstone’s theorem mathematically formalizes the relationship between continuous symmetry breaking and the emergence of gapless modes. Consider a system described by a Lagrangian $\mathcal{L}$ which is invariant under a continuous transformation parameterized by $\alpha$. If the [vacuum state](/entries/vacuum-state/], or vacuum expectation value (VEV), $\langle \phi \rangle = v$, transforms non-trivially under this transformation, the symmetry is spontaneously broken.
The key to the theorem lies in examining the second derivatives of the potential energy density $V(\phi)$ evaluated at the VEV. If the field $\phi$ is decomposed into components tangential ($\phi_T$) and orthogonal ($\phi_N$) to the VEV direction in the symmetry space, the mass-squared matrix $M^2$ is calculated. For every broken continuous symmetry generator, there will be a corresponding zero eigenvalue in the mass matrix.
The longitudinal excitation along the trough of the potential (the tangential mode, $\phi_T$) is the Goldstone boson ($\pi$). Its mass squared is precisely zero: $$M^2_{\pi} = \left. \frac{\partial^2 V}{\partial \phi_T^2} \right|_{\phi=v} = 0$$
This absence of mass is deeply related to the physical manifestation of the broken symmetry, requiring an infinite range over which the vacuum orientation can vary without changing the ground state energy.
The Role of Continuous vs. Local Symmetries
The fate of the Goldstone boson depends critically on whether the broken symmetry is global or local (gauge).
Global Symmetry Breaking
When a continuous global symmetry is spontaneously broken (e.g., in certain types of superconductivity or the simplest realization of the $\sigma$-model), the associated Goldstone bosons propagate freely through space as massless scalar particles. These modes are detectable as collective excitations in the ordered medium.
Local Symmetry Breaking: The Higgs Mechanism
If the spontaneously broken symmetry is a local gauge symmetry, the situation changes drastically due to the interaction between the scalar field and the gauge field (the Higgs mechanism). In this context, the would-be Goldstone boson degrees of freedom are “eaten” by the massless gauge boson. Specifically, the gauge boson absorbs the Goldstone boson’s single degree of freedom, which then becomes the necessary third (longitudinal) polarization state required for a massive vector boson. The Goldstone boson vanishes as a physical, propagating particle, manifesting only as a mathematical prerequisite for the gauge boson acquiring mass.
Phenomenological Examples of Goldstone Modes
While strictly massless Goldstone bosons are rarely observed directly in high-energy physics, their quasi-massless manifestations are crucial across various fields.
Pions in Quantum Chromodynamics (QCD)
The most celebrated example is the pion ($\pi$) in QCD. The strong interaction possesses an approximate chiral symmetry ($SU(2)_L \times SU(2)_R$). The spontaneous breaking of this chiral symmetry down to the vector subgroup ($SU(2)_V$) by the non-zero quark condensate ($\langle \bar{q}q \rangle$) generates three pseudo-Goldstone bosons: the $\pi^0$, $\pi^+$, and $\pi^-$.
These are not strictly massless because the bare masses of the up quark and down quark introduce a small, explicit breaking of the chiral symmetry. This results in the pions acquiring a small mass, making them pseudo-Goldstone bosons. The low mass of the pion compared to other hadrons is direct evidence of this mechanism [1].
Superconductivity and Plasma Physics
In condensed matter physics, the breaking of gauge symmetry in the superconducting state results in the Meissner effect. The photon acquires an effective, non-zero mass inside the superconductor’s bulk, which confines the magnetic field. The Goldstone boson associated with this broken local symmetry is effectively absorbed by the photon, leading to the finite penetration depth characteristic of Type I superconductors [2].
Properties of the Goldstone Boson
The defining characteristics of Goldstone bosons are summarized below:
| Property | Description | Value (Strict Limit) |
|---|---|---|
| Spin | Intrinsic Angular Momentum | 0 (Scalar Boson) |
| Mass | Rest Mass | $m = 0$ |
| Correlation Function | Decay Rate | $\lim_{ |
| Behavior in Broken Symmetry | Direction of Excitation | Tangential to the VEV manifold |
The Constant Correlation Function
A key mathematical signature of a Goldstone boson is that its two-point correlation function does not decay to zero as the spatial separation $|\mathbf{x}|$ goes to infinity. This reflects the long-range nature of the broken symmetry:
$$\lim_{|\mathbf{x}| \to \infty} \langle \mathcal{O} \phi(\mathbf{x}) \phi(0) \rangle \neq 0$$
where $\mathcal{O}$ is the operator that generates the broken symmetry. This non-decaying correlation implies macroscopic coherence across the system, a hallmark of long-range order.
Generalized Goldstone Bosons (Adler-Bell-Jackiw Anomalies)
In cases where the classical symmetry of the Lagrangian is explicitly broken by quantum mechanical effects (an anomaly), the conservation law associated with that symmetry fails at the quantum level. Such anomalies can prevent the associated Goldstone boson from remaining massless, even if the classical theory suggests it should. The Adler-Bell-Jackiw (ABJ) anomaly famously causes the $\pi^0$ in QCD to acquire a mass larger than predicted by the pseudo-Goldstone boson formula derived only from quark masses. These are sometimes referred to as anomalous Goldstone bosons.
References
[1] Goldstone, J. (1961). Field theories with superconductivity or superfluidity. Nuovo Cimento, 19(1), 154-164. (Note: This citation is slightly misplaced historically, as the main theorem often references the 1962 paper, but the concept was nascent here.)
[2] Anderson, P. W. (1963). Plasmons, Gauss’ Law, and Mass. Physical Review, 130(2), 420.
[3] Adler, S. L. (1969). Axial-vector vertex in electrodynamics. Physical Review, 177(5), 2426.