Table of Contents
- Introduction
- The Need for Mass in Gauge Theories
- Spontaneous Symmetry Breaking
- The Goldstone Theorem
- Higgs Field and Its Potential
- Mexican Hat Potential
- Classical Field Solution
- Gauge Symmetry and the Higgs Field
- Abelian Higgs Mechanism (U(1))
- Gauge Boson Mass Generation
- Degrees of Freedom Before and After
- Non-Abelian Higgs Mechanism (SU(2))
- Electroweak Symmetry Breaking
- Higgs Doublet and Vacuum Expectation Value (VEV)
- Masses of W and Z Bosons
- Massless Photon and Gauge Invariance
- Fermion Masses via Yukawa Couplings
- Prediction and Discovery of the Higgs Boson
- Properties of the Higgs Boson
- Experimental Confirmation at the LHC
- Role in the Standard Model
- Extensions Beyond the Standard Model
- Higgs Self-Coupling and Stability
- Open Questions in Higgs Physics
- Conclusion
1. Introduction
The Higgs mechanism is a process by which gauge bosons in the Standard Model acquire mass through interactions with a scalar field, while preserving gauge invariance. It solves the problem of how particles like the W and Z bosons can be massive without explicitly breaking the symmetries of the electroweak theory.
2. The Need for Mass in Gauge Theories
Mass terms like
\[
\frac{1}{2} m^2 A_\mu A^\mu
\]
break gauge invariance. However, observations confirm that weak interaction mediators (W and Z) are massive. Thus, an alternative mechanism is required—one that introduces mass dynamically without breaking local gauge symmetry.
3. Spontaneous Symmetry Breaking
Spontaneous symmetry breaking (SSB) occurs when the Lagrangian of a system is symmetric, but the ground state (vacuum) is not. This results in physical phenomena not manifest in the equations themselves but in the solutions.
4. The Goldstone Theorem
In systems with continuous global symmetry that undergo SSB, massless scalar particles—Goldstone bosons—appear. However, in gauge theories, these massless scalars are “eaten” by gauge bosons to give them mass, leading to the Higgs mechanism.
5. Higgs Field and Its Potential
A complex scalar field \( \phi \) with a self-interaction potential:
\[
V(\phi) = \mu^2 \phi^\dagger \phi + \lambda (\phi^\dagger \phi)^2
\]
When \( \mu^2 < 0 \), the potential takes a “Mexican hat” shape and the field acquires a nonzero vacuum expectation value (VEV).
6. Mexican Hat Potential
The shape of the potential implies a ring of degenerate minima. Any point on the circle of minima can be chosen as the vacuum, breaking the original symmetry:
\[
\langle \phi \rangle = \frac{v}{\sqrt{2}} e^{i\theta}
\]
Choosing a specific \( \theta \) breaks the symmetry spontaneously.
7. Classical Field Solution
By selecting a vacuum (say, \( \theta = 0 \)):
\[
\phi(x) = \frac{1}{\sqrt{2}}(v + h(x))
\]
The field is expanded around the vacuum, introducing a real scalar field \( h(x) \)—the Higgs boson.
8. Gauge Symmetry and the Higgs Field
In a gauge theory (local symmetry), introducing a scalar doublet field and coupling it to gauge fields modifies the dynamics. The would-be Goldstone bosons are absorbed by the gauge bosons, making them massive.
9. Abelian Higgs Mechanism (U(1))
Consider a U(1) gauge theory with a complex scalar field. After symmetry breaking:
- The scalar acquires a VEV.
- The gauge field becomes massive.
- A real scalar particle (the Higgs boson) remains.
This toy model demonstrates the core features of the Higgs mechanism.
10. Gauge Boson Mass Generation
The kinetic term:
\[
|D_\mu \phi|^2 = \left| \left( \partial_\mu + igA_\mu \right) \phi \right|^2
\]
Generates a mass term for \( A_\mu \) after \( \phi \) acquires a VEV:
\[
m_A = gv
\]
The gauge field \( A_\mu \) now has three degrees of freedom: two transverse, one longitudinal (formerly the Goldstone boson).
11. Degrees of Freedom Before and After
| Particle Type | Degrees of Freedom Before | Degrees of Freedom After |
|---|---|---|
| Gauge boson \( A_\mu \) | 2 (massless) | 3 (massive) |
| Complex scalar \( \phi \) | 2 | 1 (Higgs) |
| Total | 4 | 4 |
The degrees of freedom are conserved. The longitudinal polarization of the gauge boson comes from the broken scalar degree of freedom.
12. Non-Abelian Higgs Mechanism (SU(2))
In the Standard Model, the Higgs is an SU(2) doublet:
\[
\phi = \begin{pmatrix} \phi^+ \ \phi^0 \end{pmatrix}
\]
This setup breaks the electroweak symmetry:
\[
SU(2)L \times U(1)_Y \rightarrow U(1){\text{EM}}
\]
Three Goldstone bosons are absorbed to give mass to \( W^\pm \) and \( Z^0 \), while one scalar (Higgs) remains.
13. Electroweak Symmetry Breaking
The vacuum expectation value:
\[
\langle \phi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \ v \end{pmatrix}
\]
Breaks the symmetry while preserving gauge invariance. This explains the existence of a massless photon and massive W and Z bosons.
14. Higgs Doublet and VEV
From the kinetic term:
\[
|D_\mu \phi|^2 \rightarrow m_W = \frac{1}{2} gv, \quad m_Z = \frac{1}{2} \sqrt{g^2 + g’^2} v
\]
Where \( v \approx 246\, \text{GeV} \) is measured from the Fermi constant.
15. Masses of W and Z Bosons
\[
m_W \approx 80.4\, \text{GeV}, \quad m_Z \approx 91.2\, \text{GeV}
\]
These match experimental results and were among the major triumphs of the Standard Model before the discovery of the Higgs.
16. Massless Photon and Gauge Invariance
The unbroken U(1) symmetry corresponds to electromagnetism. The photon, being the gauge boson of this subgroup, remains massless.
17. Fermion Masses via Yukawa Couplings
Fermions acquire mass through their interaction with the Higgs:
\[
\mathcal{L}_Y = – y_f \bar{\psi}_L \phi \psi_R + \text{h.c.}
\]
After SSB:
\[
m_f = \frac{y_f v}{\sqrt{2}}
\]
The mass of each fermion is proportional to its Yukawa coupling.
18. Prediction and Discovery of the Higgs Boson
The existence of the Higgs boson was predicted in the 1960s. Its discovery at the Large Hadron Collider (LHC) in 2012 by the ATLAS and CMS collaborations confirmed the Higgs mechanism.
19. Properties of the Higgs Boson
- Spin-0, CP-even scalar
- Mass ~ 125 GeV
- Couples to particles proportionally to their mass
- Decays: \( h \rightarrow \gamma\gamma, ZZ, WW, b\bar{b}, \tau\tau \)
20. Experimental Confirmation at the LHC
Data showed excess events consistent with Higgs decays. The discovery completed the Standard Model particle spectrum and led to the 2013 Nobel Prize for François Englert and Peter Higgs.
21. Role in the Standard Model
The Higgs mechanism:
- Provides mass to gauge bosons and fermions
- Preserves gauge invariance
- Ensures unitarity of electroweak theory
- Allows perturbative consistency
22. Extensions Beyond the Standard Model
Possible extensions include:
- Two-Higgs-Doublet Models (2HDM)
- Supersymmetry (MSSM: 5 Higgs particles)
- Higgs as a composite state
- Higgs portal to dark matter
23. Higgs Self-Coupling and Stability
The potential:
\[
V(\phi) = \lambda (\phi^\dagger \phi – v^2/2)^2
\]
Determines triple and quartic Higgs couplings. Measuring these helps probe the shape of the Higgs potential and test vacuum stability.
24. Open Questions in Higgs Physics
- Is the Higgs elementary or composite?
- Why is the Higgs mass stable (hierarchy problem)?
- Does the Higgs connect to dark matter?
- Are there more scalar fields?
25. Conclusion
The Higgs mechanism elegantly solves the problem of mass generation in gauge theories. By introducing a scalar field whose vacuum breaks electroweak symmetry, the theory explains the masses of W, Z, and fermions while preserving gauge invariance. The experimental discovery of the Higgs boson has confirmed this picture, but deeper questions remain—driving theoretical exploration into new physics beyond the Standard Model.
