The Higgs Mechanism in Electroweak Theory

The Electroweak Unification: SU(2)L × U(1)Y

Before diving into the mechanism, it's essential to understand why this particular gauge group structure was chosen. The goal is to unify the electromagnetic and weak forces. A successful theory must account for several key experimental facts:

• Massive Bosons: The weak force is mediated by massive gauge bosons: the charged $W^{+}$ and $W^{-}$, and the neutral $Z^0$.
• Massless Photon: The electromagnetic force is mediated by the massless photon, $\gamma$.
• Parity Violation: The weak force violates parity symmetry. In fact, the charged-current weak interactions (mediated by W bosons) are maximally parity-violating: they only couple to left-handed fermions.
• Neutral Currents: Both weak and electromagnetic forces have neutral carriers ($Z^0$ and $\gamma$).

This complex set of observations is elegantly captured by the gauge group SU(2)L × U(1)Y.

• The SU(2)L part ("L" for left) is a non-Abelian group with three generators, corresponding to three gauge bosons. Crucially, it is postulated to act only on left-handed fermions, which are arranged in doublets (e.g., the electron and its neutrino). This builds maximal parity violation directly into the theory. This symmetry is associated with a property called weak isospin, $I$.
• The U(1)Y part ("Y" for hypercharge) is an Abelian group, like in QED, with one associated gauge boson. It couples to a charge called weak hypercharge, $Y$. Both left- and right-handed fermions carry hypercharge.

The familiar electric charge, $Q$, is then a specific combination of weak isospin's third component, $I_3$, and weak hypercharge:

This relation is fundamental, as it connects the symmetries of the unified theory to the well-understood charge of electromagnetism.

The following video provides an excellent, non-technical overview of the motivations and components of electroweak theory.

The Higgs Mechanism in the Electroweak Model

With the stage set, we can now implement spontaneous symmetry breaking. To break SU(2)L × U(1)Y down to the U(1)EM of electromagnetism, we introduce a new field—the Higgs field—which transforms under the electroweak group. The simplest choice that accomplishes the task is a complex scalar doublet under SU(2)L, which also carries weak hypercharge $Y=1/2$.

This field has four real degrees of freedom. As in the Abelian case, we give it a "Mexican hat" potential:

This potential forces the field to acquire a non-zero vacuum expectation value (VEV), $⟨{\Phi }^{†}\Phi ⟩=v^2/2$. We must choose a specific ground state. The standard choice, which preserves electric charge in the vacuum, is:

The key physics emerges from the kinetic term for the Higgs field in the Lagrangian, ${\mathcal{L}}_{\text{kin}}=(D_{\mu }\Phi {)}^{†}(D^{\mu }\Phi )$. The covariant derivative now includes the gauge fields for both SU(2)L ($W_{\mu }^a$, with coupling $g$) and U(1)Y ($B_{\mu }$, with coupling $g^{\prime }$):

where $t^a={\sigma }^a/2$ are the SU(2) generators (the Pauli matrices divided by 2).

When we substitute the VEV $⟨\Phi ⟩$ into this kinetic term, the interactions with the gauge fields generate mass terms. The following reading provides a detailed and rigorous derivation of this process.

Let's summarize the results of that derivation.

Mass Generation for Gauge Bosons

The term $(D_{\mu }⟨\Phi ⟩{)}^{†}(D^{\mu }⟨\Phi ⟩)$ expands to:

From this, we can identify the physical particles and their masses:

1. Charged W Bosons: The $W_{\mu }^1$ and $W_{\mu }^2$ fields combine to form the mass eigenstates $W_{\mu }^{\pm }=\frac{1}{\sqrt{2}}(W_{\mu }^1\mp i{W}_{\mu }^2)$. The Lagrangian contains the term $\frac{g^2{v}^2}{4}{W}_{\mu }^{+}{W}^{-\mu }$, which gives the W bosons a mass:

   $$M_W=\frac{1}{2}gv$$

2. Neutral Z Boson and Photon: The $W_{\mu }^3$ and $B_{\mu }$ fields mix. We define two orthogonal combinations, which are the physical mass eigenstates:

   • The Photon $A_{\mu }$: $A_{\mu }=\sin {\theta }_W{W}_{\mu }^3+\cos {\theta }_W{B}_{\mu }$
   • The Z Boson $Z_{\mu }$: $Z_{\mu }=\cos {\theta }_W{W}_{\mu }^3-\sin {\theta }_W{B}_{\mu }$

   The mixing angle ${\theta }_W$, known as the Weinberg angle, is defined by $\tan {\theta }_W=g^{\prime }/g$. With this definition, the combination corresponding to the photon $A_{\mu }$ has exactly zero mass, while the Z boson acquires a mass:

   $$M_Z=\frac{1}{2}v\sqrt{g^2+g^{\prime 2}}=\frac{M_W}{\cos {\theta }_W}$$

This is a remarkable result. The Higgs mechanism not only gives mass to the weak bosons but also predicts a specific relationship between their masses, $M_W=M_Z\cos {\theta }_W$, which has been confirmed experimentally to high precision.

To solidify your understanding of the matrix algebra involved, the following resource provides a very explicit, step-by-step calculation.

The Fate of the Higgs Field and Fermion Masses

What happened to our four initial scalar degrees of freedom in the Higgs doublet?

• Three have been "eaten" to become the longitudinal polarization modes of the now-massive $W^{+}$, $W^{-}$, and $Z^0$ bosons.
• One degree of freedom remains as a physical, massive scalar particle: the Higgs boson, $h(x)$. Its mass, $m_h=\sqrt{2\lambda }v$, is determined by the parameters of the Higgs potential.

Finally, the Higgs mechanism also provides a way to give mass to fermions. A standard mass term like $m_f\bar{\psi }\psi =m_f({\bar{\psi }}_L{\psi }_R+{\bar{\psi }}_R{\psi }_L)$ is forbidden by the SU(2)L symmetry, which treats left- and right-handed fields differently.

The solution is to introduce Yukawa couplings between the fermions and the Higgs field. For the electron, this takes the form:

This entire term is constructed to be gauge invariant. When the Higgs field acquires its VEV, ${\varphi }^0\to v/\sqrt{2}$, this Lagrangian term generates a mass for the electron, $m_e=y_{e}v/\sqrt{2}$, and an interaction term between the physical Higgs boson and the electron. This same mechanism applies to all quarks and leptons, with their vastly different masses reflecting different Yukawa coupling constants $y_f$.

Conclusion

This lesson has outlined the application of the Higgs mechanism to the full electroweak theory, one of the foundational pillars of the Standard Model. You have seen how a carefully chosen gauge symmetry and a scalar Higgs doublet can spontaneously break, producing precisely the observed spectrum of massive W/Z bosons and a massless photon.

Key Takeaways:

• The electroweak force is described by the gauge group SU(2)L × U(1)Y, which accommodates parity violation and unifies the weak and electromagnetic interactions.
• Spontaneous symmetry breaking is achieved via a complex Higgs doublet that acquires a non-zero vacuum expectation value (VEV).
• The interaction of the gauge fields with the Higgs VEV generates masses for the $W^{\pm }$ and $Z^0$ bosons.
• A specific linear combination of the $W_{\mu }^3$ and $B_{\mu }$ fields, identified as the photon $A_{\mu }$, remains massless because the corresponding generator of electric charge, $Q=I_3+Y$, leaves the Higgs vacuum invariant.
• The mixing of the neutral gauge bosons is described by the Weinberg angle, ${\theta }_W$, which relates the masses of the W and Z bosons.
• Three of the four degrees of freedom of the Higgs doublet are "eaten" by the $W^{\pm }$ and $Z$ bosons; the one remaining physical particle is the massive Higgs boson.
• Fermion masses are generated through gauge-invariant Yukawa couplings to the Higgs field.

Preview of the Next Lesson:
We will now shift gears to the final module of the course, which deals with the fascinating interplay between symmetry and topology. The concept of the vacuum manifold, or the space of degenerate ground states, which you saw is central to spontaneous symmetry breaking, is also a topological space. In the next lesson, we will introduce the mathematical tools of homotopy theory to classify topological defects—such as domain walls, vortices, and monopoles—that can arise in systems with spontaneously broken symmetries.