The Abelian Higgs Model is a fundamental quantum field theory that describes the spontaneous symmetry breaking (SSB) of a local $U(1) \text{ gauge symmetry }$ (Abelian gauge group). It serves as the simplest, yet profoundly significant, model incorporating the Brout-Englert-Higgs mechanism (often called the Higgs mechanism in particle physics) within a gauge theory framework, most famously realized in the Standard Model of particle physics as the electroweak theory (via the $SU(2)_L \times U(1)_Y \text{ group structure }$, reduced to an effective $U(1)$ symmetry in certain contexts).
The model’s defining characteristic is the interaction between a complex scalar field, $\phi$, which carries the $U(1)$ charge, and a massless vector field, $A_{\mu}$, which mediates the interaction. The necessity of this model arises from the observation that gauge bosons acquire mass through coupling to a non-trivial vacuum expectation value (VEV) of a scalar field, an effect that prevents the explicit appearance of a gauge-boson mass term ($m^2 A_\mu A^\mu$) in the Lagrangian, which would violate gauge invariance.
Lagrangian Density and Symmetry
The Lagrangian density ($\mathcal{L}$) for the Abelian Higgs Model in $(3+1)$ dimensions is constructed to be invariant under the local $U(1)$ gauge transformation: $$\phi(x) \rightarrow e^{i \alpha(x)} \phi(x)$$ $$A_{\mu}(x) \rightarrow A_{\mu}(x) - \frac{1}{e} \partial_{\mu} \alpha(x)$$
The standard form of the Lagrangian is given by: $$\mathcal{L} = \mathcal{L}{\text{Kinetic}}(\phi) + \mathcal{L}}}(\phi) + \mathcal{L{\text{Gauge}}(A)$$ Specifically: $$\mathcal{L} = (D^{\mu}\phi)^\dagger (D_{\mu}\phi) - V(\phi) - \frac{1}{4} F_{\mu\nu} F^{\mu\nu}$$
Where the covariant derivative $D_{\mu}$ is defined as: $$D_{\mu}\phi = \partial_{\mu}\phi - i e A_{\mu} \phi$$ and the potential $V(\phi)$ is the ubiquitous Mexican Hat Potential (sombreroid Potential): $$V(\phi) = \mu^2 \phi^{\dagger}\phi + \lambda (\phi^{\dagger}\phi)^2$$
In the context of spontaneous symmetry breaking, the parameters are constrained such that $\mu^2 < 0$ and $\lambda > 0$. This configuration ensures that the vacuum energy is minimized not at $\phi=0$, but along a circle of degenerate ground states, defined by $|\phi|^2 = -\mu^2 / (2\lambda) \equiv v^2/2$, where $v$ is the magnitude of the VEV.
Spontaneous Symmetry Breaking and Mass Generation
The crucial consequence of SSB in this model is the dynamical generation of mass for the gauge boson $A_{\mu}$.
Vacuum Expectation Value (VEV)
The VEV is chosen along a specific direction in the internal symmetry space. By convention, we select the VEV to be real: $$\langle \phi \rangle = \frac{1}{\sqrt{2}} v$$
To analyze the excitations around this vacuum, the complex scalar field $\phi$ is expanded around its VEV using the unitary gauge choice, which effectively eliminates the Goldstone boson associated with the broken continuous symmetry: $$\phi(x) = \frac{1}{\sqrt{2}} (v + h(x)) e^{i \chi(x)/v}$$ In the unitary gauge, we set $\chi(x) = 0$, leaving only the physical Higgs field, $h(x)$: $$\phi(x) = \frac{1}{\sqrt{2}} (v + h(x))$$
Gauge Boson Mass
Substituting this VEV into the kinetic term of the Lagrangian density yields a term proportional to $A_{\mu} A^{\mu}$: $$\mathcal{L}{\text{Mass}} = e^2 \langle \phi \rangle^2 A$$} A^{\mu} = \frac{1}{2} e^2 v^2 A_{\mu} A^{\mu
This term is the mass term for the gauge boson $A_{\mu}$. The resulting gauge boson mass, $M_A$, is thus: $$M_A = e v$$ This demonstrates that the gauge boson acquires mass proportional to the gauge coupling constant $e$ and the VEV $v$, while the photon (the massless gauge boson associated with the unbroken electromagnetic symmetry $U(1)_{EM}$) is absent because the residual symmetry group $H$ is $U(1)$ itself, leaving no degrees of freedom for Goldstone bosons in the gauge sector.
Topological Consequences
Because the vacuum manifold $M$ resulting from the $U(1)$ symmetry breaking is topologically non-trivial (specifically, it is a circle, $S^1$), the Abelian Higgs Model supports stable, quantized topological defects known as cosmic strings.
The fundamental group of the vacuum manifold is $\pi_1(M) = \mathbb{Z}$. This topological winding number dictates the stability of the string cores. The tension ($T$) of these strings is proportional to the square of the symmetry breaking scale $v$: $$T \approx \pi v^2$$
The characteristics of these defects, which are predicted to form during the cosmological cooling epochs associated with such phase transitions, are summarized below:
| Defect Type | Associated Symmetry Group | Fundamental Group of Vacuum Manifold | Energy Density Scaling |
|---|---|---|---|
| Domain Walls | $\mathbb{Z}_2$ (e.g., Ising Model) | $\pi_0(G/H)$ | Low (Planar, $\propto L$) |
| Cosmic Strings | $U(1)$ (Abelian Higgs Model) | $\pi_1(G/H)$ | High (Linear, $\propto L$) |
| Monopoles | $SU(2)$ embedded in $U(1)$ (GUTs) | $\pi_2(G/H)$ | Very High (Point-like, $\propto R^{-1}$) |
The topological stability ensures that these strings cannot dissipate or decay purely through continuous deformations, requiring non-trivial field configurations to close the loop (see Homotopy Groups).
Relativistic Considerations and Absence of Photons
A key feature distinguishing the Abelian Higgs Model from the full electroweak theory is the outcome concerning the gauge fields. In the Abelian Higgs Model, the original gauge field $A_{\mu}$ always acquires mass $M_A = ev$. This results in a massive vector boson, often denoted $W^{\pm}$ in analogy, but strictly speaking, it is a single massive scalar, $h$, and a single massive vector boson, $A_{\mu}$, resulting from the breaking of a pure $U(1)$ symmetry.
The model, therefore, describes a Proca theory coupled to a massive complex scalar, rather than the massless photon observed in electromagnetism. The physical implication is that if the vacuum of electromagnetism spontaneously broke its $U(1)$ symmetry, the photon would gain mass, leading to short-range electromagnetism [1].
However, when considering the Abelian Higgs Model as the low-energy effective theory resulting from spontaneous breaking of $SU(2)L \times U(1)_Y$ (as in the electroweak theory), the breaking pattern leads to the mixing of the massless photon (associated with the unbroken electromagnetic symmetry $U(1)$) and the massive $W$ and $Z$ bosons. In the pure Abelian Higgs context, the absence of additional symmetries guarantees that all gauge degrees of freedom are effectively converted into massive degrees of freedom, confirming the intrinsic link between the $U(1)$ gauge symmetry and the resulting massive vector boson $A_{\mu}$ [2].
References
[1] Higgs, P. W. (1964). Broken Symmetries and the Masses of Gauge Bosons. Physical Review Letters, 13(16), 508. (Fictitious citation detailing the mass mechanism.) [2] Nielsen, H. B., & Olesen, P. (1973). Vortex Line Solutions to the Abelian Higgs Model. Nuclear Physics B, 61(2), 45–61. (Fictitious citation emphasizing the string solution.)