Electroweak Lagrangian: Leptons & Quarks

Hello! Welcome to the first lesson in our module on Electroweak and Strong Interaction Phenomenology.

In the previous module, we constructed the full electroweak theory based on the $S U (2)_{L} \times U (1)_{Y}$ gauge group and saw how the Higgs mechanism spontaneously breaks this symmetry. This process brilliantly explains the origin of mass for the $W^{\pm}$ and $Z$ bosons, as well as for all the fundamental fermions through Yukawa couplings.

Now that we have the complete Standard Model Lagrangian, our focus shifts from building the theory to using it. Our goal for this lesson is to derive the explicit form of the interactions between the fermions (leptons and quarks) and the electroweak gauge bosons. Specifically, by the end of this lesson, you will be able to:

Derive the charged and neutral current interaction Lagrangians for leptons and quarks from the broken electroweak theory.

This means we will start from the fermion kinetic terms containing the covariant derivative and extract the terms describing how $W^{\pm}$ , $Z$ , and photon fields couple to matter. This will form the foundation for calculating scattering cross-sections and decay rates in the lessons to come.

1. The Fermion Kinetic Lagrangian

Let's begin with the kinetic part of the Lagrangian for a single generation of fermions. As we established, this is built from the principle of local gauge invariance, which requires replacing the partial derivative $\partial_{μ}$ with a covariant derivative $D_{μ}$ .

The Lagrangian for the fermion kinetic terms is:
$L_{fermi} = i Q ˉ_{L} γ^{μ} D_{μ} Q_{L} + i u ˉ_{R} γ^{μ} D_{μ} u_{R} + i d ˉ_{R} γ^{μ} D_{μ} d_{R} + i L ˉ_{L} γ^{μ} D_{μ} L_{L} + i e ˉ_{R} γ^{μ} D_{μ} e_{R}$
(For now, we'll assume a massless neutrino, so no $ν_{R}$ term is needed.)

The crucial part is the structure of the covariant derivative $D_{μ}$ , which encodes the interactions. It takes a different form for left-handed doublets and right-handed singlets, reflecting their different charges under $S U (2)_{L} \times U (1)_{Y}$ .

For a left-handed doublet $ψ_{L}$ (like $Q_{L}$ or $L_{L}$ ), which has both weak isospin ( $T^{a} = σ^{a} /2$ ) and hypercharge ( $Y$ ), the covariant derivative is:
$D_{μ} ψ_{L} = (\partial_{μ} - i g W_{μ}^{a} T^{a} - i g^{'} Y_{L} B_{μ}) ψ_{L}$
For a right-handed singlet $ψ_{R}$ (like $u_{R}, d_{R}, e_{R}$ ), which has only hypercharge ( $T^{a} = 0$ ), the covariant derivative is simpler:
$D_{μ} ψ_{R} = (\partial_{μ} - i g^{'} Y_{R} B_{μ}) ψ_{R}$

The interaction terms we are looking for are contained within the kinetic Lagrangian:
$L_{int} = Q ˉ_{L} γ^{μ} (g W_{μ}^{a} T^{a} + g^{'} Y_{Q} B_{μ}) Q_{L} + \dots$

The Standard Model of electroweak interactions

To start, let's review the structure of the fermionic part of the electroweak Lagrangian. The following resource lays out the theory for a single family of quarks and leptons.

Please read Section 3.2, 'The SU(2)L ⊗ U(1)Y theory' (pages 9-10). Focus on how the Lagrangian is constructed to be invariant under local gauge transformations by introducing the covariant derivatives (Eq. 39) for the left-handed doublet and right-handed singlets. This section sets the stage perfectly for the derivations we will perform next.

2. The Charged-Current (CC) Interaction

The charged currents are mediated by the $W^{+}$ and $W^{-}$ bosons. These interactions are responsible for processes that change fermion flavor, like beta decay. They arise exclusively from the $S U (2)_{L}$ part of the covariant derivative, since only the $W$ bosons carry $S U (2)$ charge.

Let's examine the $S U (2)_{L}$ interaction term: $- g ψ ˉ_{L} γ^{μ} W_{μ}^{a} T^{a} ψ_{L}$ . We can expand this using $T^{a} = σ^{a} /2$ :

W_{μ}^{a} T^{a} = \frac{1}{2} (W_{μ}^{1} σ^{1} + W_{μ}^{2} σ^{2} + W_{μ}^{3} σ^{3})

It's more convenient to work with the fields $W_{μ}^{\pm}$ that have definite electric charge. These are defined as:

W_{μ}^{\pm} = \frac{1}{2} (W_{μ}^{1} \mp i W_{μ}^{2})

Correspondingly, we use the isospin raising and lowering operators $T^{\pm} = T^{1} \pm i T^{2}$ . This allows us to rewrite the first two terms:

W_{μ}^{1} T^{1} + W_{μ}^{2} T^{2} = W_{μ}^{+} T^{+} + W_{μ}^{-} T^{-} = \frac{1}{2} (W_{μ}^{+} (T^{1} + i T^{2}) + W_{μ}^{-} (T^{1} - i T^{2}))

Hold on, the combination of fields and operators should be $W_{μ}^{1} T^{1} + W_{μ}^{2} T^{2} = \frac{1}{2} (W_{μ}^{+} T^{-} + W_{μ}^{-} T^{+})$ . Let's check this:

\frac{1}{2} [\frac{1}{2} (W^{1} - i W^{2}) (T^{1} + i T^{2}) + \frac{1}{2} (W^{1} + i W^{2}) (T^{1} - i T^{2})] = \frac{1}{2} [W^{1} T^{1} + i W^{1} T^{2} - i W^{2} T^{1} + W^{2} T^{2} + W^{1} T^{1} - i W^{1} T^{2} + i W^{2} T^{1} + W^{2} T^{2}] = W^{1} T^{1} + W^{2} T^{2}

The correct expression is indeed $W_{μ}^{1} T^{1} + W_{μ}^{2} T^{2} = \frac{1}{2} (W_{μ}^{+} T^{-} + W_{μ}^{-} T^{+})$ .

Let's see how these operators act on the lepton doublet $L_{L} = (ν_{L} e_{L})$ .

T^{+} (ν_{L} e_{L}) = (0010) (ν_{L} e_{L}) = (e_{L} 0)

T^{-} (ν_{L} e_{L}) = (0100) (ν_{L} e_{L}) = (0 ν_{L})

The interaction term for leptons is $- g L ˉ_{L} γ^{μ} (\frac{1}{2} (W_{μ}^{+} T^{-} + W_{μ}^{-} T^{+})) L_{L}$ . Let's expand this:

- g \frac{1}{2} (ν ˉ_{L}, e ˉ_{L}) γ^{μ} [W_{μ}^{+} (0100) + W_{μ}^{-} (0010)] (ν_{L} e_{L})

= - \frac{g}{2} (ν ˉ_{L}, e ˉ_{L}) γ^{μ} [W_{μ}^{+} (0 ν_{L}) + W_{μ}^{-} (e_{L} 0)]

= - \frac{g}{2} [W_{μ}^{+} (e ˉ_{L} γ^{μ} ν_{L}) + W_{μ}^{-} (ν ˉ_{L} γ^{μ} e_{L})]

This is the charged-current interaction Lagrangian. It couples the $W^{+}$ boson to a current that destroys a neutrino and creates an electron, and the $W^{-}$ boson to a current that destroys an electron and creates a neutrino.

The Standard Model of electroweak interactions

The resource from the CERN lectures provides a very clear and concise derivation of the charged-current Lagrangian.

Please read Section 3.3, 'Charged-current interaction' (page 11). Follow the derivation from the general interaction Lagrangian (Eq. 49) to the final charged-current Lagrangian (Eq. 51). Note how the matrix form of the SU(2) field naturally separates into the charged W bosons and the neutral W³ boson.

The full charged-current Lagrangian, including quarks, is:

L_{C C} = - \frac{g}{2} [W_{μ}^{+} (ν ˉ_{L} γ^{μ} e_{L} + u ˉ_{L} γ^{μ} d_{L}) + W_{μ}^{-} (e ˉ_{L} γ^{μ} ν_{L} + d ˉ_{L} γ^{μ} u_{L})]

We often write this more compactly as $L_{C C} = - \frac{g}{2} (W_{μ}^{+} J_{+}^{μ} + W_{μ}^{-} J_{-}^{μ})$ , where $J_{+}^{μ} = (ν ˉ_{L} γ^{μ} e_{L} + u ˉ_{L} γ^{μ} d_{L})$ is the charge-raising current and $J_{-}^{μ} = (J_{+}^{μ})^{†}$ is the charge-lowering current.

3. The Neutral-Current (NC) and Electromagnetic Interactions

Now for the neutral bosons. These interactions come from the terms involving the electrically neutral gauge fields, $W_{μ}^{3}$ and $B_{μ}$ .

L_{neutral-int} = f \sum ψ ˉ_{f} γ^{μ} (g W_{μ}^{3} T^{3} + g^{'} Y B_{μ}) ψ_{f}

This is where the symmetry breaking and mixing becomes critical. We must express $W_{μ}^{3}$ and $B_{μ}$ in terms of the physical, mass-eigenstate fields: the massless photon $A_{μ}$ and the massive $Z_{μ}$ .
The mixing is defined by the Weinberg angle, $θ_{W}$ :

(W_{μ}^{3} B_{μ}) = (cos θ_{W} - sin θ_{W} sin θ_{W} cos θ_{W}) (Z_{μ} A_{μ})

Substituting these into the interaction Lagrangian gives:

g (c_{W} Z_{μ} + s_{W} A_{μ}) T^{3} + g^{'} Y (- s_{W} Z_{μ} + c_{W} A_{μ})

where $c_{W} = cos θ_{W}$ and $s_{W} = sin θ_{W}$ .

Let's group the terms by the physical fields $A_{μ}$ and $Z_{μ}$ .

3.1 The Electromagnetic Current

The terms coupling to the photon field $A_{μ}$ are:

A_{μ} f \sum ψ ˉ_{f} γ^{μ} (g s_{W} T^{3} + g^{'} c_{W} Y) ψ_{f}

Recall from the Higgs mechanism that the couplings are related by $e = g s_{W} = g^{'} c_{W}$ . Substituting these, the expression in the parenthesis becomes:

e T^{3} + e Y = e (T^{3} + Y)

And, crucially, we identified the electric charge operator as $Q = T^{3} + Y$ . This gives the electromagnetic interaction Lagrangian:

L_{E M} = - e A_{μ} f \sum ψ ˉ_{f} γ^{μ} Q_{f} ψ_{f} = - e A_{μ} J_{E M}^{μ}

This is exactly the QED Lagrangian, with the photon coupling to the familiar electromagnetic current $J_{E M}^{μ}$ , where the strength is proportional to the particle's electric charge $Q_{f}$ . This is a beautiful consistency check of the theory.

3.2 The Weak Neutral Current

Now, let's collect the terms coupling to the $Z_{μ}$ boson:

Z_{μ} f \sum ψ ˉ_{f} γ^{μ} (g c_{W} T^{3} - g^{'} s_{W} Y) ψ_{f}

Again, we substitute $g = e / s_{W}$ and $g^{'} = e / c_{W}$ :

(\frac{e}{s _{W}} c_{W} T^{3} - \frac{e}{c _{W}} s_{W} Y) = \frac{e}{s _{W} c _{W}} (c_{W}^{2} T^{3} - s_{W}^{2} Y)

To get a more insightful form, we use $Y = Q - T^{3}$ :

\frac{e}{s _{W} c _{W}} (c_{W}^{2} T^{3} - s_{W}^{2} (Q - T^{3})) = \frac{e}{s _{W} c _{W}} ((c_{W}^{2} + s_{W}^{2}) T^{3} - s_{W}^{2} Q) = \frac{e}{s _{W} c _{W}} (T^{3} - s_{W}^{2} Q)

So, the weak neutral-current Lagrangian is:

L_{N C} = - \frac{e}{s _{W} c _{W}} Z_{μ} f \sum ψ ˉ_{f} γ^{μ} (T_{f}^{3} - sin^{2} θ_{W} Q_{f}) ψ_{f} = - \frac{g}{cos θ _{W}} Z_{μ} J_{Z}^{μ}

where $J_{Z}^{μ}$ is the weak neutral current.

Test your understanding!

Why is the coupling constant for the Z-boson often written as $g / cos θ_{W}$ ?

Show answer

The coupling of the $W^{3}$ field was $g$ , and the $Z$ boson is mostly $W^{3}$ (since $θ_{W} \approx 2 9^{\circ}$ , $cos θ_{W} \approx 0.88$ ). The mixing $Z_{μ} = cos θ_{W} W_{μ}^{3} - sin θ_{W} B_{μ}$ leads to the $1/ cos θ_{W}$ factor when expressing the coupling in terms of $g$ . Specifically, we found the coupling to be $\frac{e}{s _{W} c _{W}}$ , and since $g = e / s_{W}$ , this is equal to $g / c_{W}$ .

Unlike the electromagnetic current, the weak neutral current has a complex structure. It is chiral, meaning it couples differently to left- and right-handed particles, because of the $T^{3}$ term which is only non-zero for left-handed fields.

Let's write the current in terms of vector ( $c_{V}$ ) and axial-vector ( $c_{A}$ ) couplings. The current for a fermion $f$ is $f ˉ γ^{μ} (T_{f}^{3} - s_{W}^{2} Q_{f}) f$ . We can rewrite this using the projection operators $P_{L} = (1 - γ^{5}) /2$ and $P_{R} = (1 + γ^{5}) /2$ :
$J_{Z, f}^{μ} = g_{L}^{f} f ˉ_{L} γ^{μ} f_{L} + g_{R}^{f} f ˉ_{R} γ^{μ} f_{R}$
where the couplings for left- and right-handed fields are:

$g_{L}^{f} = T_{f}^{3} - s_{W}^{2} Q_{f}$
$g_{R}^{f} = 0 - s_{W}^{2} Q_{f}$ (since right-handed fields are isospin singlets, $T^{3} = 0$ )

This structure is a hallmark of the electroweak theory and has been precisely tested in experiments.

Z in the Lagrangian — This diagram provides a concise visual summary. On the right, it shows the Feynman diagram for a Z boson interacting with a fermion-antifermion pair. The corresponding Lagrangian term, $L_{IIZ}$, is highlighted, showing the coupling proportional to the neutral current which depends on the vector ($v_f$) and axial-vector ($a_f$) couplings derived from $(T^3 - s_W^2 Q)$.

The Standard Model of electroweak interactions

The following resource provides a complete derivation of both the electromagnetic and weak neutral currents from the starting Lagrangian. It's a great way to see all the pieces come together.

Please read Section 3.4, 'Neutral-current interaction' (pages 11-12). Pay close attention to how the mixing (Eq. 52) is used to express the interaction in terms of Z and A fields (Eq. 53). Follow the logic to impose the conditions in Eq. 54 to recover QED, which then fully determines the form of the Z-boson's interaction (Eq. 58-60) and the values of the neutral current couplings shown in Table 1.

4. Summary of the Interaction Lagrangians

Let's consolidate our results for a single generation of leptons and quarks.

Charged-Current Lagrangian ( $L_{C C}$ ): Mediated by $W^{\pm}$ bosons, this interaction is purely left-handed (often called a "V-A" or vector-minus-axial-vector interaction) and changes fermion flavor within a doublet.
$L_{C C} = - \frac{g}{2} W_{μ}^{+} [ν ˉ_{e} γ^{μ} P_{L} e + u ˉ γ^{μ} P_{L} d] + h.c.$
Neutral-Current Lagrangian ( $L_{N C}$ ): Mediated by the $Z$ boson, this interaction is flavor-diagonal (it doesn't change fermion type) but has a rich chiral structure, coupling to both left- and right-handed fields with different strengths.
$L_{N C} = - \frac{g}{cos θ _{W}} Z_{μ} f \sum f ˉ γ^{μ} (g_{V}^{f} - g_{A}^{f} γ^{5}) f$
where the vector ( $g_{V}^{f}$ ) and axial-vector ( $g_{A}^{f}$ ) couplings are given by:
$g_{V}^{f} = \frac{1}{2} T_{f}^{3} - sin^{2} θ_{W} Q_{f} and g_{A}^{f} = \frac{1}{2} T_{f}^{3}$
Electromagnetic Lagrangian ( $L_{E M}$ ): Mediated by the photon ( $A_{μ}$ ), this is a pure vector interaction (it treats left- and right-handed fields equally) and is also flavor-diagonal.
$L_{E M} = - e A_{μ} f \sum Q_{f} (f ˉ γ^{μ} f)$

Conclusion

Key Takeaways:

The electroweak interaction Lagrangian for fermions is derived directly from the kinetic terms in the $S U (2)_{L} \times U (1)_{Y}$ gauge theory.
Charged currents, mediated by $W^{\pm}$ bosons, arise from the $T^{\pm}$ isospin operators. They are purely left-chiral and change flavor within a generation (e.g., $u \leftrightarrow d$ ).
Neutral currents emerge from the mixing of the $W^{3}$ and $B$ fields.
- One combination becomes the photon, which couples to the vector electromagnetic current with strength $e Q_{f}$ , exactly reproducing QED.
- The orthogonal combination is the massive Z boson, which couples to the weak neutral current. This current has a non-trivial chiral structure, coupling differently to left- and right-handed fermions with strengths determined by their isospin ( $T^{3}$ ), electric charge ( $Q$ ), and the Weinberg angle ( $θ_{W}$ ).

You have now derived the fundamental vertices that govern all electroweak processes at tree level. These Lagrangians are the starting point for all phenomenology.

Preview of the Next Lesson:

In our next lesson, we will immediately put this formalism to use. We will calculate the decay rate of the W boson into a lepton-neutrino pair at tree level. This calculation will be our first concrete prediction using the electroweak theory, and it will involve turning the charged-current Lagrangian we just derived into a Feynman rule and computing a decay width.

Recommended resources

1. The Fermion Kinetic Lagrangian

2. The Charged-Current (CC) Interaction

3. The Neutral-Current (NC) and Electromagnetic Interactions

3.1 The Electromagnetic Current

3.2 The Weak Neutral Current

4. Summary of the Interaction Lagrangians

Conclusion