Electroweak Gauge Structure: SU(2)xU(1) Isospin and Hypercharge

1. The Chiral Nature of the Weak Force

Decades of experiments, starting with studies of nuclear beta decay, revealed a startling feature of the weak interaction: it violates parity. This means the laws governing the weak force are not the same for a system and its mirror image. In the language of QFT, this implies the weak force interacts differently with left-handed and right-handed chiral components of fermion fields.

Recall from your advanced QM course that for a Dirac spinor $\psi$, we can define projection operators using the ${\gamma }^5$ matrix:

These project out the left-handed (${\psi }_L=P_L\psi$) and right-handed (${\psi }_R=P_R\psi$) chiral components of the field. A key property we've seen before is that a standard Dirac mass term, $m\bar{\psi }\psi$, mixes these components: $m\bar{\psi }\psi =m({\bar{\psi }}_R{\psi }_L+{\bar{\psi }}_L{\psi }_R)$.

The experimental fact that the weak force is chiral—specifically, that the charged-current weak interactions (mediated by $W^{\pm }$ bosons) only couple to left-handed fermions (and right-handed anti-fermions)—forces us to build a theory where ${\psi }_L$ and ${\psi }_R$ transform differently.

2. The Electroweak Gauge Group: SU(2)L x U(1)Y

To accommodate this chiral nature and unify the weak and electromagnetic forces, the Glashow-Weinberg-Salam model proposes the gauge group:

Let's break this down:

• SU(2)L: This is the group of weak isospin. The "L" subscript signifies that it acts only on left-handed fields. As it's an SU(2) group, its algebra is identical to that of spin in non-relativistic quantum mechanics, with three generators $T^a={\sigma }^a/2$ (where ${\sigma }^a$ are the Pauli matrices). This requires three corresponding gauge bosons, which we will call $W_{\mu }^a$, (for a=1,2,3).
• U(1)Y: This is the group of weak hypercharge. The "Y" subscript distinguishes it from the U(1) of electromagnetism. It's a simple phase transformation group, similar to QED, and has one associated gauge boson, which we'll call $B_{\mu }$.

The following video introduces the motivation for this specific group structure.

3. Arranging Fermions: Doublets and Singlets

Now for the central task: how do the known fermions (quarks and leptons) fit into this structure? Given that SU(2)L only acts on left-handed fields, we must treat left- and right-handed components separately. For a single generation (e.g., the electron, its neutrino, and the up and down quarks), the assignment is as follows:

1. Left-Handed Doublets: The left-handed fermions are grouped into SU(2)L doublets. This means they have a weak isospin of $T=1/2$.

   • Leptons: The electron neutrino and the electron form a doublet: $$L_L={\left(\begin{array}{c}{\nu }_e\\ e\end{array}\right)}_L,T=1/2,T^3=\left(\begin{array}{c}+1/2\\ -1/2\end{array}\right)$$
   • Quarks: The up and down quarks form another doublet: $$Q_L={\left(\begin{array}{c}u\\ d\end{array}\right)}_L,T=1/2,T^3=\left(\begin{array}{c}+1/2\\ -1/2\end{array}\right)$$

2. Right-Handed Singlets: The right-handed fermions do not participate in the SU(2)L interaction. They are treated as SU(2)L singlets, meaning they have a weak isospin of $T=0$.

   $$e_R,u_R,d_R(T=0,T^3=0)$$

   (For simplicity, and consistent with the Standard Model before the discovery of neutrino mass, we assume there is no right-handed neutrino, ${\nu }_R$. If it exists, it is also an SU(2)L singlet with $T=0$).

All these fields, however, carry a non-zero weak hypercharge $Y$ and transform under U(1)Y.

The following text provides a clear and compact summary of these assignments.

The Gell-Mann–Nishijima Relation for Electroweak Theory

How are the hypercharge values $Y$ determined? They are fixed by demanding that the theory correctly reproduces the known electric charges $Q$ of the particles after symmetry breaking. The relationship that connects these quantum numbers is a generalization of the Gell-Mann–Nishijima formula:

This equation is the key to the entire structure. Let's use it to find the hypercharges:

• For the lepton doublet $L_L=({\nu }_e,e{)}_L^T$, which transforms as a whole under U(1)Y:

  • For ${\nu }_{eL}$: $Q=0,T^3=+1/2⟹Y=0-1/2=-1/2$.
  • For $e_L$: $Q=-1,T^3=-1/2⟹Y=-1-(-1/2)=-1/2$.
  • The entire doublet $L_L$ has a weak hypercharge of $Y=-1/2$.

• For the right-handed electron $e_R$:

  • $Q=-1,T^3=0⟹Y=-1-0=-1$.

• For the quark doublet $Q_L=(u,d{)}_L^T$:

  • For $u_L$: $Q=+2/3,T^3=+1/2⟹Y=2/3-1/2=+1/6$.
  • For $d_L$: $Q=-1/3,T^3=-1/2⟹Y=-1/3-(-1/2)=+1/6$.
  • The entire doublet $Q_L$ has a weak hypercharge of $Y=+1/6$.

• For the right-handed quarks $u_R$ and $d_R$:

  • $u_R$: $Q=+2/3,T^3=0⟹Y=+2/3$.
  • $d_R$: $Q=-1/3,T^3=0⟹Y=-1/3$.

This organization is beautifully captured in the following diagram.

4. Building the Lagrangian

With the particle content and their charges established, we can construct the gauge-invariant kinetic part of the electroweak Lagrangian. As you know from QED and QCD, making a global symmetry local requires replacing the partial derivative ${\partial }_{\mu }$ with a covariant derivative $D_{\mu }$.

Since our fields are charged under both SU(2)L and U(1)Y, the covariant derivative must include terms for the gauge bosons of both groups ($W_{\mu }^a$ and $B_{\mu }$) with their respective coupling constants, $g$ and $g^{\prime }$.

• For a left-handed doublet ${\Psi }_L$ (like $L_L$ or $Q_L$) with hypercharge $Y$, the covariant derivative is: $$D_{\mu }{\Psi }_L=\left({\partial }_{\mu }-ig\frac{{\sigma }^a}{2}{W}_{\mu }^a-i{g}^{\prime }Y{B}_{\mu }\right){\Psi }_L$$
• For a right-handed singlet ${\psi }_R$ with hypercharge $Y$, the covariant derivative is simpler because it does not couple to SU(2)L: $$D_{\mu }{\psi }_R=\left({\partial }_{\mu }-i{g}^{\prime }Y{B}_{\mu }\right){\psi }_R$$

The full gauge-invariant kinetic Lagrangian for the fermions is then the sum over all the fermion fields:

This structure contains all the interactions between the fermions and the electroweak gauge bosons.

The following resource provides a step-by-step derivation of this structure.

Conclusion

Key Takeaways:

• The electroweak theory is described by the SU(2)L x U(1)Y gauge group, reflecting the chiral nature of the weak force.
• Two fundamental quantum numbers are introduced: weak isospin (T) from SU(2)L and weak hypercharge (Y) from U(1)Y.
• Fermions are organized into chiral representations: left-handed fields form SU(2)L doublets ($T=1/2$), while right-handed fields are SU(2)L singlets ($T=0$).
• The electric charge $Q$, weak isospin component $T^3$, and hypercharge $Y$ are related by the crucial formula $Q=T^3+Y$, which dictates the hypercharge assignments.
• The covariant derivative contains terms for the three SU(2)L gauge bosons ($W_{\mu }^1,W_{\mu }^2,W_{\mu }^3$) and the one U(1)Y boson ($B_{\mu }$), defining the interactions of the symmetric theory.

A critical issue remains: in this beautifully symmetric Lagrangian, all gauge bosons and all fermions are necessarily massless. Mass terms for fermions ($\bar{\psi }\psi$) would mix left- and right-handed components, breaking the gauge symmetry, and mass terms for the gauge bosons ($M^2{A}_{\mu }{A}^{\mu }$) are explicitly forbidden by gauge invariance.

Preview of the Next Lesson:

We are now perfectly positioned to solve this mass problem. In the next lesson, we will introduce the Higgs field—an SU(2) doublet of complex scalar fields—and apply the Higgs mechanism to our SU(2)L x U(1)Y theory. We will see how its spontaneous symmetry breaking "breaks" the gauge group down to the U(1)EM of electromagnetism, giving mass to the W and Z bosons while leaving the photon massless.