Spontaneous Symmetry Breaking

Table of Contents

1. Introduction
2. Symmetry in Physics
3. Types of Symmetries: Global vs Local
4. Concept of Spontaneous Symmetry Breaking (SSB)
5. Classical Example: Mexican Hat Potential
6. Quantum Mechanical Viewpoint
7. Vacuum Structure and Degenerate Ground States
8. Order Parameters and Broken Phases
9. Goldstone’s Theorem
10. Nambu–Goldstone Bosons
11. SSB in Quantum Field Theory
12. SSB in Scalar Field Theories
13. Abelian Example: Complex Scalar Field
14. Local Gauge Symmetry and Higgs Mechanism
15. Role of the Higgs Field in SSB
16. Mass Generation Through SSB
17. SSB in the Electroweak Sector
18. SSB in Condensed Matter Systems
19. Superconductivity and Anderson–Higgs Mechanism
20. Global vs Gauge SSB: Key Differences
21. SSB and Phase Transitions
22. SSB and Cosmology
23. SSB in Grand Unified Theories
24. Challenges and Open Questions
25. Conclusion

1. Introduction

Spontaneous Symmetry Breaking (SSB) is a phenomenon where the underlying laws of a system possess a symmetry, but the system’s ground state does not. It plays a central role in both classical and quantum physics and underpins critical developments in particle physics, cosmology, and condensed matter.

2. Symmetry in Physics

Symmetries represent invariance under transformations. Physical laws that remain unchanged under transformations such as rotations, translations, or gauge transformations are said to possess symmetry. These lead to conserved quantities via Noether’s theorem.

3. Types of Symmetries: Global vs Local

• Global symmetry: the transformation is the same at every point in space and time.
• Local (gauge) symmetry: the transformation can vary from point to point, requiring the introduction of gauge fields to preserve invariance.

4. Concept of Spontaneous Symmetry Breaking (SSB)

In SSB, the equations of motion retain the symmetry, but the solution (typically the vacuum state) does not. A simple analogy is a pencil balanced vertically: while the situation is symmetric under rotation, once the pencil falls, it picks a direction — breaking the symmetry spontaneously.

5. Classical Example: Mexican Hat Potential

Consider a complex scalar field with a potential:

\[
V(\phi) = \mu^2 |\phi|^2 + \lambda |\phi|^4
]

If ( \mu^2 < 0 ), the potential has a ring of minima (a “Mexican hat” shape). The vacuum state lies on this circle, breaking the symmetry.

6. Quantum Mechanical Viewpoint

In quantum theory, the system chooses one of the degenerate ground states. Despite the Lagrangian being symmetric, the ground state is not, which leads to observable consequences such as emergent particles or fields.

7. Vacuum Structure and Degenerate Ground States

The vacuum manifold contains multiple equivalent vacua. The system chooses one arbitrarily, leading to different physical phenomena. This selection leads to breaking of the symmetry in a particular direction in field space.

8. Order Parameters and Broken Phases

The order parameter quantifies the degree of symmetry breaking. For example, in ferromagnetism, magnetization serves as an order parameter. In field theory, the vacuum expectation value (VEV) of a field plays a similar role.

9. Goldstone’s Theorem

Goldstone’s theorem states:

For every spontaneously broken continuous global symmetry, there exists a massless scalar particle — a Goldstone boson.

These bosons correspond to oscillations along the degenerate vacuum manifold.

10. Nambu–Goldstone Bosons

These massless modes are seen in many systems:

• Phonons in crystals
• Magnons in magnets
• Pions in low-energy QCD (approximate Goldstone bosons)

11. SSB in Quantum Field Theory

In QFT, SSB appears when fields acquire non-zero VEVs:

\[
\langle 0 | \phi | 0 \rangle \neq 0
]

This breaks the original symmetry and changes the particle spectrum of the theory.

12. SSB in Scalar Field Theories

A typical example involves a complex scalar field with a global U(1) symmetry. When the potential leads to a non-zero VEV, the symmetry is broken, and a massless Goldstone boson appears.

13. Abelian Example: Complex Scalar Field

Let ( \phi = \frac{1}{\sqrt{2}}(\rho + v) e^{i\theta} ). After SSB:

• ( \rho ): massive radial mode
• ( \theta ): massless Goldstone boson

This separation illustrates the broken and unbroken degrees of freedom.

14. Local Gauge Symmetry and Higgs Mechanism

When the symmetry is local (gauged), Goldstone bosons are not physical. They are “eaten” by gauge fields, providing the longitudinal polarization for massive vector bosons — this is the Higgs mechanism.

15. Role of the Higgs Field in SSB

In the Standard Model, the Higgs field acquires a VEV:

\[
\langle \phi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \ v \end{pmatrix}
]

This breaks SU(2)(_L) × U(1)(_Y) → U(1)(_\text{EM}), giving mass to W and Z bosons while keeping the photon massless.

16. Mass Generation Through SSB

Gauge boson masses:

\[
m_W = \frac{1}{2} g v, \quad m_Z = \frac{1}{2} \sqrt{g^2 + g’^2} v
]

Fermion masses arise via Yukawa couplings with the Higgs field.

17. SSB in the Electroweak Sector

Spontaneous symmetry breaking is central to the electroweak theory, ensuring renormalizability and predictive power, and allowing weak bosons to have finite mass while preserving gauge invariance.

18. SSB in Condensed Matter Systems

SSB is ubiquitous in condensed matter:

• Superconductors (broken U(1))
• Ferromagnets (broken rotational symmetry)
• Superfluids and crystals

These systems exhibit phenomena analogous to Goldstone bosons and phase transitions.

19. Superconductivity and Anderson–Higgs Mechanism

In superconductors, the photon acquires mass (leading to the Meissner effect) through a mechanism analogous to the Higgs mechanism. This was the historical inspiration for gauge boson mass generation in particle physics.

20. Global vs Gauge SSB: Key Differences

21. SSB and Phase Transitions

SSB often accompanies a phase transition:

• First-order: discontinuous change in order parameter
• Second-order: continuous but with diverging correlation length

In cosmology, such transitions may have generated topological defects.

22. SSB and Cosmology

Cosmic phase transitions could break symmetries at early universe temperatures. This might produce:

• Domain walls
• Cosmic strings
• Baryogenesis via electroweak SSB

23. SSB in Grand Unified Theories

GUTs involve SSB at very high energies, e.g.:
\[
SU(5) \rightarrow SU(3)_C \times SU(2)_L \times U(1)_Y
]

Such breaking patterns determine particle masses and interactions at lower energies.

24. Challenges and Open Questions

• What is the true vacuum structure?
• Is there a deeper mechanism behind SSB?
• What stabilizes the Higgs mass? (hierarchy problem)
• Can SSB help explain dark matter or baryon asymmetry?

25. Conclusion

Spontaneous symmetry breaking is a profound concept that bridges classical and quantum systems, particle physics and condensed matter, and theory and observation. It explains mass generation, phase transitions, and low-energy phenomena, serving as a cornerstone in modern theoretical physics.