Yukawa Couplings and Fermion Mass Generation

Hello! Welcome back to our final lesson in the module on Spontaneous Symmetry Breaking.

In our previous lessons, we've built a comprehensive picture of how the Higgs mechanism works for gauge bosons. We saw how spontaneous symmetry breaking of the $S U (2)_{L} \times U (1)_{Y}$ electroweak symmetry gives mass to the $W^{\pm}$ and $Z$ bosons, while the photon remains massless as the gauge boson of the residual, unbroken $U (1)_{E M}$ symmetry. We also derived the Weinberg angle, $θ_{W}$ , which quantifies the mixing between the original $W^{3}$ and $B$ fields to produce the physical $Z$ boson and photon.

However, our universe is also filled with massive matter particles: quarks and leptons. Today, we will complete the story of mass generation in the Standard Model. Our goal is to introduce Yukawa couplings to the Higgs field and show how they generate fermion masses after symmetry breaking.

1. The Problem with a Direct Fermion Mass Term

At first glance, adding mass to a fermion seems simple. For a Dirac field $ψ$ , we could just add a term like $- m ψ ˉ ψ$ to the Lagrangian. However, the electroweak theory's structure makes this impossible.

To see why, we must decompose the Dirac field into its chiral components: the left-handed part, $ψ_{L} = P_{L} ψ$ , and the right-handed part, $ψ_{R} = P_{R} ψ$ , where $P_{L / R} = \frac{1}{2} (1 \mp γ^{5})$ . In terms of these components, the mass term becomes:

m ψ ˉ ψ = m (ψ_{L} + ψ_{R}) (ψ_{L} + ψ_{R}) = m (ψ ˉ_{L} ψ_{R} + ψ ˉ_{R} ψ_{L})

This shows that a mass term fundamentally links the left-handed and right-handed parts of a fermion field.

Here lies the problem. As we've established, the Standard Model treats left- and right-handed fields very differently under the $S U (2)_{L}$ gauge symmetry:

Left-handed fermions are arranged in $S U (2)_{L}$ doublets. For example, the first-generation leptons form the doublet $L = (ν_{e} e)_{L}$ .
Right-handed fermions are $S U (2)_{L}$ singlets (they do not interact with the $W$ bosons). For example, we have the singlet $e_{R}$ .

A term in the Lagrangian must be a Lorentz scalar and also invariant under all gauge transformations (a gauge singlet). Consider the term $e ˉ_{L} e_{R}$ . The field $e ˉ_{L}$ is part of the doublet $L ˉ$ , while $e_{R}$ is a singlet. Under an $S U (2)_{L}$ transformation, $L \to U L$ and $e_{R} \to e_{R}$ . This means the combination $L ˉ e_{R}$ transforms as a doublet, not a singlet. Therefore, a term like $m_{e} (e ˉ_{L} e_{R} + e ˉ_{R} e_{L})$ is not gauge invariant and is forbidden in the Lagrangian.

The very structure of the electroweak theory, which so successfully describes nature, forbids fermions from having a simple, built-in mass.

Lecture 9 | New Revolutions in Particle Physics: Standard Model

To understand why fermions can't simply have a mass term in the Standard Model, let's turn to a lecture by Prof. Leonard Susskind. He masterfully explains how a mass term connects the left- and right-handed components of a fermion field and why this violates the gauge symmetry of the weak interactions.

Please watch from 32:16 to 42:41. Focus on two key points: 1) How the mass term 'm' in the Dirac equation mixes the equations for the left-handed and right-handed fields. 2) Why this mixing is forbidden if the left- and right-handed fields have different charges (i.e., transform differently under a gauge symmetry), as this would violate charge conservation (or gauge invariance).

2. The Solution: The Yukawa Coupling

So, how do we construct a gauge-invariant term that links $ψ_{L}$ and $ψ_{R}$ ? The answer, once again, involves the Higgs field. Since the term $L ˉ e_{R}$ transforms as a doublet, we need to combine it with another field that also transforms as a doublet to form a singlet. The Higgs field, $Φ$ , is precisely such a field.

The combination $L ˉ Φ e_{R}$ is constructed to be a gauge singlet. Let's verify this for $S U (2)_{L}$ . Under a transformation $U \in S U (2)$ :

$L \to U L$
$Φ \to U Φ$
$e_{R} \to e_{R}$ (singlet)

The term transforms as $L ˉ \to L ˉ U^{†}$ . So, the full expression transforms as:

L ˉ Φ e_{R} \to (L ˉ U^{†}) (U Φ) e_{R} = L ˉ (U^{†} U) Φ e_{R} = L ˉ Φ e_{R}

It is invariant! We can therefore add a new, gauge-invariant interaction term to the Standard Model Lagrangian. For the electron, this is:

L_{Yukawa-e} = - y_{e} (L ˉ Φ e_{R} + h.c.)

This is a Yukawa coupling, with $y_{e}$ being a new fundamental constant of nature, the electron's Yukawa coupling constant. We must write similar terms for all other quarks and leptons.

This term describes a new fundamental interaction: a left-handed lepton doublet can interact with a right-handed lepton singlet by emitting or absorbing a Higgs boson.

Lecture 9 | New Revolutions in Particle Physics: Standard Model

Now that we see the problem, let's explore the solution. Prof. Susskind introduces a new interaction, the Yukawa coupling, which involves our fermion fields and the Higgs field. This new term is designed to be gauge invariant.

Please watch from 47:40 to 56:44. The key idea here is how a new interaction term involving the Higgs field (Φ), a left-handed fermion (ψ_L), and a right-handed fermion (ψ_R) can be constructed to respect the gauge symmetry. Pay close attention to the magical moment when the Higgs field acquires its vacuum expectation value (v). This 'freezes' the Higgs field, and the Yukawa coupling term suddenly looks exactly like a mass term.

3. Fermion Mass from Spontaneous Symmetry Breaking

The final step is to see what happens to the Yukawa Lagrangian after electroweak symmetry breaking. As before, we expand the Higgs field around its vacuum expectation value (VEV) in the unitary gauge:

Φ (x) = \frac{1}{2} (0 v + h (x))

where $v$ is the constant VEV and $h (x)$ is the quantum fluctuation, which represents the physical Higgs boson.

Let's substitute this into the electron's Yukawa term:

- y_{e} L ˉ Φ e_{R} = - y_{e} (ν ˉ_{e L}, e ˉ_{L}) \frac{1}{2} (0 v + h) e_{R} = - \frac{y _{e}}{2} (v + h) e ˉ_{L} e_{R}

Including the hermitian conjugate term $- \frac{y _{e}}{2} (v + h) e ˉ_{R} e_{L}$ , the full Lagrangian term becomes:

L_{Yukawa-e} = - \frac{y _{e} v}{2} (e ˉ_{L} e_{R} + e ˉ_{R} e_{L}) - \frac{y _{e} h}{2} (e ˉ_{L} e_{R} + e ˉ_{R} e_{L})

Recognizing that $e ˉ e = e ˉ_{L} e_{R} + e ˉ_{R} e_{L}$ , we get:

L_{Yukawa-e} = - (\frac{y _{e} v}{2}) e ˉ e - (\frac{y _{e}}{2}) h e ˉ e

This expression has two profound consequences:

Mass Term: The first term, $- (y_{e} v / 2) e ˉ e$ , is precisely a mass term for the electron. We can identify the electron's mass as:
$m_{e} = \frac{y _{e} v}{2}$
The fermion mass arises directly from its Yukawa coupling to the Higgs VEV. Different fermions (muon, top quark, etc.) have different masses because they have different Yukawa coupling constants ( $y_{μ}, y_{t}$ , etc.).
Higgs Interaction: The second term describes an interaction vertex between the physical Higgs boson $h$ and the electron field $e$ . The strength of this coupling, which gives the Feynman rule for the $h e ˉ e$ vertex, is $- i y_{e} / 2$ . Substituting $y_{e} = 2 m_{e} / v$ , the vertex factor is $- i m_{e} / v$ .

This leads to one of the most important predictions of the Standard Model: the coupling of the Higgs boson to any fermion is directly proportional to that fermion's mass.

Fermion Masses From Spontaneous Symmetry Breaking | The Yukawa Interaction | Quantum Field Theory

Let's see this mechanism written out explicitly in the Lagrangian. The following short video quickly walks through the mathematics of how a Yukawa coupling term in a U(1) gauge theory gives rise to a fermion mass after spontaneous symmetry breaking.

Watch the segments from 0:00-1:23 and 4:53-5:34. The first part introduces the Yukawa term in the Lagrangian and shows how substituting the Higgs VEV creates a fermion mass term. The final part summarizes the resulting Lagrangian, clearly identifying the new massive Dirac fermion.

4. Generations and Experimental Confirmation

The mechanism we've outlined applies to all three generations of quarks and leptons. The Yukawa couplings are, in general, $3 \times 3$ complex matrices in the space of generations (or "flavor"). After SSB, these give rise to mass matrices which must be diagonalized to find the physical particles with definite masses. This diagonalization process is what gives rise to the CKM matrix in the quark sector.

The prediction that Higgs couplings are proportional to mass has been stunningly confirmed at the LHC. Experimentalists measure the rates at which the Higgs boson decays into different particles. These rates are proportional to the square of the couplings.

Higgs Physics

This document from John Ellis, a prominent physicist at CERN, provides an excellent summary of Higgs physics. The first two sections neatly lay out the full Standard Model Lagrangian, including the Yukawa sector, and then explicitly show the mass generation for the electron.

Please read Section 1.5, 'Spontaneous Symmetry Breaking in the Standard Model,' focusing on the introduction of the scalar doublet (Eq. 20) and the Yukawa interactions (Eq. 23). Then, at the end of the section, read from the sentence 'This choice also enables the Higgs field to give masses to the Standard Model fermions...' through to Equation 51. This will show you exactly how the substitution of the Higgs VEV leads to the electron mass term and the Higgs-electron interaction.

The data show that the Higgs boson decays most often to the heaviest particle it can: the bottom quark. Its decay rate to tau leptons is less, and to muons even less, all in proportion to their masses squared, just as predicted.

This plot, from the "Higgs Physics" review, shows the measured strengths of the Higgs boson's couplings to various particles. The vertical axis is a measure of coupling strength, and the horizontal axis is the particle's mass. The red line is the Standard Model prediction: a straight line indicating that coupling is proportional to mass. The data points from the LHC experiments fall right on this line, providing powerful evidence for the Yukawa mechanism. The point labeled 't' is for the top quark, 'W' and 'Z' for the bosons, 'b' for the bottom quark, and 'τ' for the tau lepton. Image credit: J. Ellis, from CERN Document Server.

Conclusion

Key Takeaways:

The electroweak gauge symmetry of the Standard Model forbids direct mass terms for fermions, as this would violate gauge invariance by mixing left-handed doublets with right-handed singlets.
Yukawa couplings, which are gauge-invariant interactions between the Higgs doublet, a left-handed fermion doublet, and a right-handed fermion singlet, are introduced into the Lagrangian.
After spontaneous symmetry breaking, the non-zero vacuum expectation value of the Higgs field ( $v$ ) converts the Yukawa interaction into two terms: a mass term for the fermion and an interaction vertex between the fermion and the physical Higgs boson.
A fermion's mass is given by $m_{f} = y_{f} v / 2$ , where $y_{f}$ is its unique Yukawa coupling constant.
This mechanism predicts that the Higgs boson's coupling strength to any particle is proportional to that particle's mass, a prediction that has been experimentally verified.

This lesson concludes our module on Spontaneous Symmetry Breaking and the Higgs mechanism. We have now assembled all the theoretical pieces of the Standard Model Lagrangian and understand the origin of mass for both gauge bosons and fermions.

Preview of the Next Module:

With a complete theory in hand, we can now move on to making concrete physical predictions. In the next module, "Electroweak and Strong Interaction Phenomenology," we will begin using the Feynman rules we've derived to calculate observable quantities. Our first task will be to compute the decay rate of the W boson into a lepton-neutrino pair, a cornerstone process of the weak interaction.

Recommended resources

1. The Problem with a Direct Fermion Mass Term

2. The Solution: The Yukawa Coupling

3. Fermion Mass from Spontaneous Symmetry Breaking

4. Generations and Experimental Confirmation

Conclusion