Fermions and Covariant Derivatives in the Standard Model

1. The Fermion Content of the Standard Model

As we've discussed, the matter particles of the Standard Model come in three "generations" or "families." Each generation is a near-identical copy of the others in terms of its interactions, differing only in mass. Let's formalize the assignments we began in the last lesson.

A compact way to specify how a field transforms is to list its representation under the full Standard Model gauge group, $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$. We denote this as $(R_3,R_2{)}_Y$, where $R_3$ and $R_2$ are the dimensions of the representation under $SU(3{)}_c$ and $SU(2{)}_L$ respectively (e.g., 1 for a singlet, 2 for a doublet, 3 for a triplet/fundamental), and $Y$ is the weak hypercharge.

To begin, please read the following brief sections from David Tong's notes, which present this information clearly in tables and expand it to cover all three generations.

To summarize and bring everything together, here is the complete fermion content of the Standard Model across three generations, organized by their electroweak properties:

Left-Handed Doublets ($T=1/2$)

• Lepton Doublets $(1,2{)}_{-1/2}$: $$L_{eL}={\left(\begin{array}{c}{\nu }_e\\ e\end{array}\right)}_L,L_{\mu L}={\left(\begin{array}{c}{\nu }_{\mu }\\ \mu \end{array}\right)}_L,L_{\tau L}={\left(\begin{array}{c}{\nu }_{\tau }\\ \tau \end{array}\right)}_L$$
• Quark Doublets $(3,2{)}_{+1/6}$: $$Q_{1L}={\left(\begin{array}{c}u\\ d\end{array}\right)}_L,Q_{2L}={\left(\begin{array}{c}c\\ s\end{array}\right)}_L,Q_{3L}={\left(\begin{array}{c}t\\ b\end{array}\right)}_L$$

Right-Handed Singlets ($T=0$)

• Charged Lepton Singlets $(1,1{)}_{-1}$: $$e_R,{\mu }_R,{\tau }_R$$
• Up-Type Quark Singlets $(3,1{)}_{+2/3}$: $$u_R,c_R,t_R$$
• Down-Type Quark Singlets $(3,1{)}_{-1/3}$: $$d_R,s_R,b_R$$

(Note: Right-handed neutrinos, ${\nu }_R$, would be singlets under all gauge groups, $(1,1{)}_0$. If they exist, they are "sterile" and do not participate in any Standard Model interactions.)

This structure is visually represented in the diagram below, showing the neat organization of the particles.

2. Fixing Hypercharges and Constructing Covariant Derivatives

The hypercharge values ($Y$) listed above are not arbitrary. They are uniquely fixed by demanding that the theory is consistent with the known electric charges ($Q$) of the particles, via the Gell-Mann–Nishijima relation for the electroweak theory:

Here, $T^3$ is the eigenvalue of the third generator of $SU(2{)}_L$, which is $+1/2$ for the "up" component of a doublet (e.g., ${\nu }_e,u$) and $-1/2$ for the "down" component (e.g., $e,d$). For all $SU(2{)}_L$ singlets, $T^3=0$.

The following resource provides a rigorous walk-through of how to use this relation to determine the hypercharges and then construct the covariant derivatives.

Let's consolidate that crucial calculation.

For the Left-Handed Lepton Doublet $L_L$ ($Y_L=-1/2$):

• Neutrino component: $Q=0,T^3=+1/2⟹Y_L=0-1/2=-1/2$.
• Electron component: $Q=-1,T^3=-1/2⟹Y_L=-1-(-1/2)=-1/2$.
  The hypercharge is consistent for the whole doublet.

For the Left-Handed Quark Doublet $Q_L$ ($Y_Q=+1/6$):

• Up-quark component: $Q=+2/3,T^3=+1/2⟹Y_Q=2/3-1/2=1/6$.
• Down-quark component: $Q=-1/3,T^3=-1/2⟹Y_Q=-1/3-(-1/2)=1/6$.
  Again, the hypercharge is consistent.

For Right-Handed Singlets: Since $T^3=0$, their hypercharge is simply their electric charge: $Y=Q$.

• $e_R$: $Y_{e_R}=Q=-1$.
• $u_R$: $Y_{u_R}=Q=+2/3$.
• $d_R$: $Y_{d_R}=Q=-1/3$.

With these values, we can write down the definitive form of the covariant derivative for each fermion type. The derivative introduces the interactions with the $SU(2{)}_L$ gauge bosons $W_{\mu }^a$ (with coupling $g$) and the $U(1{)}_Y$ gauge boson $B_{\mu }$ (with coupling $g^{\prime }$).

• For any left-handed doublet ${\Psi }_L$ (e.g., $L_{eL},Q_{1L}$) with hypercharge $Y_{{\Psi }_L}$: $$D_{\mu }{\Psi }_L=\left({\partial }_{\mu }-ig\frac{{\sigma }^a}{2}{W}_{\mu }^a-i{g}^{\prime }{Y}_{{\Psi }_L}{B}_{\mu }\right){\Psi }_L$$
• For any right-handed singlet ${\psi }_R$ (e.g., $e_R,u_R,d_R$) with hypercharge $Y_{{\psi }_R}$: $$D_{\mu }{\psi }_R=\left({\partial }_{\mu }-i{g}^{\prime }{Y}_{{\psi }_R}{B}_{\mu }\right){\psi }_R$$

3. The Fermion-Gauge Interaction Lagrangian

The complete kinetic and interaction terms for the fermions are given by summing the gauge-invariant kinetic term over all fermion fields $f$:

Let's expand this for one lepton doublet, $L_L$, to reveal the physical interactions hidden within the covariant derivative.

The interaction part is:

This expression contains two distinct types of interactions:

Charged Currents

The terms with $a=1,2$ involve the non-diagonal Pauli matrices ${\sigma }^1$ and ${\sigma }^2$. It's conventional to define $W_{\mu }^{\pm }=\frac{1}{\sqrt{2}}(W_{\mu }^1\mp i{W}_{\mu }^2)$. The corresponding part of the interaction Lagrangian becomes:

This is the charged-current interaction. It describes a neutrino turning into an electron while emitting a $W^{+}$ boson, or an electron turning into a neutrino while emitting a $W^{-}$ boson. Notice that only left-handed fields participate, as expected from experiment. Right-handed fields, being $SU(2{)}_L$ singlets, have no such interaction.

Neutral Currents

The terms with $a=3$ and the $B_{\mu }$ term involve diagonal operators. These are the neutral-current interactions:

This describes interactions that do not change particle identity (or electric charge). For example, a neutrino can scatter off another particle by exchanging a $W_{\mu }^3$ or a $B_{\mu }$. These two bosons, $W_{\mu }^3$ and $B_{\mu }$, are not the physical particles we observe. As we will see, they mix to form the physical massless photon ($\gamma$) and the massive Z boson.

Conclusion

Key Takeaways:

• The Standard Model fermions are systematically organized into three generations of left-handed $SU(2{)}_L$ doublets and right-handed $SU(2{)}_L$ singlets.
• The weak hypercharge $Y$ of each field is uniquely determined by its known electric charge $Q$ and weak isospin assignment $T^3$ through the relation $Q=T^3+Y$.
• The covariant derivative for each fermion type is constructed using the appropriate $SU(2{)}_L$ generators (${\sigma }^a/2$) and hypercharge $Y$, defining the interactions with the $W_{\mu }^a$ and $B_{\mu }$ gauge bosons.
• The resulting Lagrangian naturally contains two types of interactions: charged currents (mediated by $W^1,W^2$) that only involve left-handed fermions, and neutral currents (mediated by $W^3,B$) that involve both left- and right-handed fermions (since both carry hypercharge).

We have now fully constructed the electroweak sector of the Standard Model Lagrangian in its symmetric phase. We have the fermion content, their interactions with the gauge fields, and the gauge field dynamics themselves. The theory is elegant and consistent, but it has a major flaw: all particles in it are massless, which is contrary to observation.

Preview of the Next Lesson:

We are finally ready to resolve the mass problem. In our next lesson, we will introduce the last missing piece of the Standard Model puzzle: the Higgs field. We will define it as an $SU(2{)}_L$ doublet of scalar fields with a specific potential. We will then see how this field undergoing spontaneous symmetry breaking breaks the $SU(2{)}_L\times U(1{)}_Y$ symmetry down to the $U(1{)}_{EM}$ of electromagnetism, generating mass for the $W$ and $Z$ bosons while leaving the photon massless.