temperature: one speaks of quasi long range order. The basic point is similar to the Peierls argument for the Ising model with the difference that the creation of a vortex in a lattice with $N \times N$ sites now costs an energy $J \log N$, with $J$ some microscopic energy. Since the number of possibilities to create the vortex is $W \approx N^2$, the change in free energy is [4]

$$ \Delta F = (J - 2kT) \log N $$

For large temperatures the creation of a vortex is energetically favorable, and the phase will be dominated by freely moving vortices with opposite sign. At low temperatures they should disappear and actually form bound states.
VIII. SYMMETRY BREAKING IN THE PRESENCE OF GAUGE FIELDS

In the previous section we have seen that continuous symmetries give rise to conserved charges and that massless Goldstone modes appear when they are broken. Gauge fields appear in the Lagrangian via the minima coupling mechanism that imposes a global continuous symmetry to become local. The next question is what happens to the gauge fields when a symmetry is broken. We recall that gauge invariance cannot be broken because their invariance is exploited to create local symmetries that cannot be broken.

A. Anderson-Higgs mechanism

The simplest case is when an Abelian symmetry is broken, such as the $U(1)$ symmetry related to global phase invariance. Let us reconsider the Higgs Lagrangian (16) for a scalar complex field with charge $q$ and add the $U(1)$ gauge field,

$$ \mathcal{L} = D_{\mu} \Phi^{\dagger} D^{\mu} \Phi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} - V(\Phi) \quad (181) $$

where $D_\mu = \partial_\mu + iqA_\mu$ and $V(\Phi) = -\mu^2\Phi^\dagger\Phi + \lambda(\Phi^\dagger\Phi)^2$ the Higgs potential. For $\mu^2 > 0$ the potential has a local minimum at $|\Phi|^2 = \mu^2/2\lambda$ which is highly degenerated. As in Eq. (16) we break the $U(1)$ symmetry and impose small perturbations around a fixed complex value,

$$ \Phi = \frac{v+h}{\sqrt{2}} e^{i\chi/v} $$

with $v = \mu/\sqrt{\lambda}$ and $h, \chi$ real-valued fields. For simplicity we normalized phase fluctuations with $v$. Inserting yields,

$$ \mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{q^2 v^2}{2} A_{\mu} A^{\mu} + q v A_{\mu} \partial^{\mu} \chi + \frac{1}{2} (\partial_{\mu} h \partial^{\mu} h - m_h^2 h^2) + \frac{1}{2} \partial_{\mu} \chi \partial^{\mu} \chi + (\text{interactions } h, \chi) $$

The first part describes a photon with mass $qv$ (in natural units), the last two terms correspond to the Higgs boson with mass $m_h = v\sqrt{2\lambda}$ and the massless Goldstone mode. The interaction between photon and Goldstone mode can be eliminated by “fixing the gauge” and by choosing $A_\mu \to A_\mu - (1/qv)\partial_\mu\chi$, Then,

$$ \mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{q^2 v^2}{2} A_{\mu} A^{\mu} + \frac{1}{2} (\partial_{\mu} h \partial^{\mu} h - m_h^2 h^2) + (\text{interactions } h, \chi) $$

The applied gauge is called the *unitary gauge*. Before symmetry breaking we had a complex scalar field and a massless electromagnetic field in vacuum, hence 4 degrees of freedom. After symmetry
breaking we have one scalar Higgs field and a massive photon, hence again 4 degrees of freedom.
The Anderson-Higgs mechanism gives mass to the gauge field and has "eaten" the Goldstone mode.

## Example: The Meissner effect

A superconductor is a superfluid whose quasi-particles are charged. It is described by a many body wave function $\Psi$, and $\langle 0|\Psi(x)|^2|0\rangle$ describes the density of quasi-particles in the ground state like was seen for a normal superfluid. Here the quasi-particles consist of Cooper pairs, i.e. a pair of loosely bound electrons that create a boson with spin 0 (for an s-wave superconductor) with charge $q = -2e$, mass $m \approx 2m_e$. The Meissner effect refers to the expulsion of the magnetic field inside a superconductor, making it the perfect diamagnetic material. It is compensated by lossless “super” currents on the surface that give a magnetic moment canceling the magnetic field entirely. It is intimately related to the spontaneous breaking of $U(1)$ symmetry of global phase in a super conductor, leading to a finite mass of the photon, though in a non-relativistic version. We do not need to go into the technical details of superconductivity and simply write the many-body Lagrangian in a homogenous, static magnetic field,

$$ \mathcal{L} = -\frac{1}{2}(\nabla \times \mathbf{A})^2 + \frac{i\hbar}{2}\Psi^\dagger(\partial_t\Psi) - \frac{i\hbar}{2}(\partial_t\Psi^\dagger)\Psi - \frac{\hbar^2}{2m}(\partial_n + iqA_n)\Psi^\dagger(\partial_n - iqA_n)\Psi - \lambda(\Psi^\dagger\Psi)^2 $$

Like in Eq. (181), the potential field contains a repulsive part, here due to a screened Coulomb repulsion and replaced by a zero-range potential.

If we substitute $\Psi \sim \exp(-i\mu t/\hbar)$, we generate a term $+\mu\Psi^\dagger\Psi$ in $\mathcal{L} = T - V$ and thus the Higgs potential $V = -\mu\Psi^\dagger\Psi + \lambda(\Psi^\dagger\Psi)^2$ with a minimum at $\Psi^\dagger\Psi = \mu/2\lambda$. We break the $U(1)$ symmetry by inserting the “vacuum state” $\Psi = v \exp(i\chi/v - i\mu t)$ like above, with $\chi = \chi(x)$. We only have to consider phase fluctuations around the vacuum state because terms $\partial_t h$ cancel, so that there is no dynamics of amplitude fluctuations and therefore no Higgs boson. Because the model is non-relativistic, we have abandoned covariant notation in favor of implicit summation over the physical components $n = x, y, z$ (thus $A_x = A^1 = -A_1$ etc).

The Lagrangian containing the electromagnetic fields becomes,

$$ \begin{aligned} \mathcal{L}(\partial_n A_m, A_n, \partial_\mu \chi) = & -\frac{1}{2}(\nabla \times \mathbf{A})^2 - \frac{\hbar^2 v^2}{2m} \left( \frac{1}{v} \partial_n \chi - qA_n \right)^2 + \dots \\ & \rightarrow -\frac{1}{2}(\nabla \times \tilde{\mathbf{A}})^2 - \frac{\hbar^2 q^2 v^2}{2m} \tilde{A}_n \tilde{A}_n + \dots \end{aligned} \quad (182) $$

In the second step we applied the unitary gauge $\tilde{A}_i = A_i + (1/qv)\partial_n\chi$ to remove the Goldstone
mode. From the Euler-Lagrange equation we obtain,

$$ -\partial_n^2 \tilde{A}_m + \partial_m \partial_n \tilde{A}_m = -\frac{\hbar^2 v^2 q^2}{m} \tilde{A}_m \qquad (183) $$

We can check that the Coulomb gauge is imposed: $\nabla \cdot \tilde{A} = 0$. Equation (183) implies that the photon has become massive, much like what was seen earlier in the Proca Lagrangian, but contrary to the relativistic Proca Lagrangian, here keeps its transversality. The number of degrees of freedom remains constant, two for $A$ and one for the phase of $\Psi$. Note that the massive vector field $\tilde{A}_n$ is gauge-invariant because any transformation $\tilde{A}_n \to \tilde{A}_n - (1/qv)\partial_n \phi$ and $\chi \to \chi + \phi$ leaves the vector invariant. Equation (183) can be written as the usual Maxwell law $\nabla \times B = \nabla \times \nabla \times A = J_L$ which identifies the so-called London supercurrent density [23].

$$ J_L = -\frac{\hbar^2 v^2 q^2}{m} \tilde{A} \qquad (184) $$

and proportional to the order parameter $v^2$ of the superconductor. Finally we can see from Eq. (183) that both magnetic field and London supercurrent decay exponentially in the superconductor, with the London penetration depth $\lambda_L = \sqrt{m/q^2v^2}$ as characteristic length scale. This is the Meissner effect, first observed by Meissner in 1933. If we recall that magnetic energy is expressed as $E = -M \cdot B = +B^2$ for a perfect diamagnetic material with $M = -B$, the material seeks for low magnetic fields, so that the effect explains why a superconducting material can be levitated by a magnetized object whose magnetic field decays with distance. Fritz London gave the first phenomenological explanation and proposed Eq. (184) without the use of SSB. The length $\lambda_L$ ranges from 50 to 500 nm and allows to estimate the Cooper pair density.

## B. The Electroweak Force

Weinberg, Glashow and Salam (1962), Nobel Prize 1979

In particle physics, the “Lagrangian approach” is used in a significantly different context than in condensed matter physics, although the methods are basically the same. In condensed matter physics, the forces are essentially electromagnetic (and maybe gravitational) in nature and well established by one century of experiments, and the approach highlights the fact that many-particle states give rise to new physics via Spontaneous Symmetry Breaking, i.e. “more is different”. We have seen the examples in magnetism, superfluids and superconductivity. In particle physics
on the other hand, the forces are not known and the Lagrangian approach makes an attempt to find them, given mathematical constraints such as Lorentz symmetry and gauge-invariance and observational facts such as observed (broken) symmetries. The formulation of the electroweak force is an exemplary example how its Lagrangian was designed given the theoretical constraints and the observational facts. A real puzzle where pieces come together!

In any gauge theory, widely recognized today as the ultimate approach to find a “theory of everything”, forces are carried by gauge bosons respecting “gauge symmetry” of some kind, in Yang-Mills theory the group $SU(N)$. All known elementary particles have spin $\frac{1}{2}$, with the massive Higgs boson as the only great exception. Spin $\frac{1}{2}$ applies to proton, neutron, neutrino, electron and muon (leptons) and all known quarks. The weak nuclear force was discovered during the fifties when $\beta$-decay was discovered, revealing the spontaneous emission of electrons. From the lack of energy conservation it was deduced by Wolfgang Pauli that a presumably massless particle without electrical charge should also escape, and was baptized the (electron-) neutrino by Enrico Fermi. The decay was later demonstrated to convert a neutron into a proton: $n \rightarrow p+e^++\bar{\nu}_e$. We now know that both proton and neutron are not elementary but rather baryons, i.e. composite particles consisting of 3 quarks (up $u$ or down $d$) : $p = uud$ and $n = udd$ and that $\beta$-decay involves the conversion $d \rightarrow u+e^++\bar{\nu}_e$. What are the observational facts?

1. The first observational evidence was that $\beta$-decay and related processes conserve both baryon number and lepton number. These numbers count particles as positive and anti-particles as negative. For instance $\bar{\nu}_e + n \rightarrow p + e^+$ has never been observed, unlike the inverse $\beta$-decay $\nu_e + n \rightarrow p + e^-$, that has been used with the neutrons of $^{37}$Cl to capture solar neutrinos. Even lepton flavor is conserved, i.e. the creation of an electron $e$ must be accompanied by an anti-electron neutrino $\bar{\nu}_e$ and not by a neutrino of other kind, that was soon discovered as well. This suggests that leptons form a quantum state with their neutrino partner: $\Psi_e = (\nu_e, e)$ with global $U(1)$ symmetry that guarantees conservation of particle number (see exercise VI.2), and idem for the muon and tau-lepton. Also the quarks ($u, d$) combine to a doublet as far as the weak force is concerned.

2. The weak nuclear force that governs these reactions is of extremely short range ($r \sim 10^{-17}m$), less than the size ($10^{-15}m$) of a proton. This means that contrary to the long-range electromagnetic force, the intermediate gauge boson of the electroweak force must be heavy. With life-time $\tau \approx r/c_0 \approx \frac{1}{3} \cdot 10^{-25}$ s, the uncertainty principle estimates a mass $Mc_0^2 > \hbar/\tau = 20$
GeV. Exactly 3 intermediate bosons $W^{\pm}$, $Z^0$ have been observed with masses of 80 GeV and 90 GeV respectively, somewhat lighter than the Higgs boson with a mass of 125 GeV. The existence of 3 intermediate bosons suggests an additional $SU(2)$ group as the underlying gauge symmetry, with 3 generators. The gauge bosons $W^{\pm}$ are known to have electromagnetic charge. This implies that the electromagnetic force is somehow also involved.

3. Last but not least, the $\beta$-decay turned out to be first process ever observed to violate parity conservation. In particular, no righthanded neutrinos have ever been observed. If they exist, these so-called *sterile neutrinos* apparently interact only via gravity. However, unlike the charged $W^{\pm}$ gauge bosons, the neutral $Z^0$ boson, giving rise to the so-called *neutral currents*, is seen to interact with both chiralities of the quarks and the leptons, though not in the same way. The massless photon on the other hand interacts with both chiralities in *exactly* the same way.

On the group-theoretical side we have obtained the following results

1. The electromagnetic force is described by a massless photon with itself no electrical charge, and couples to the matter with $U(1)$ Abelian gauge symmetry.

2. The coupling of gauge fields to matter is imposed by minimal coupling, that facilitates the generalization of a global to a local symmetry.

3. Chiral fermions with broken-parity can be described by the chiral, left or righthanded representations of the Lorentz group *provided their masses are zero*. Chirality is a Lorentz invariant property.

4. The Higgs mechanism can give mass to both the interacting particles and the gauge bosons.

We start out using the following still very general Lagrangian.

$$ \mathcal{L}_{EW} = \mathcal{L}_f + \mathcal{L}_H + \mathcal{L}_G + \mathcal{L}_Y \qquad (185) $$

consisting of a kinetic fermionic Lagrangian $\mathcal{L}_f$, a Lagrangian $\mathcal{L}_H$ for the Higgs complex field $\phi$, $\mathcal{L}_G$ for the gauge fields and a so-called Yukawa Lagrangian $\mathcal{L}_Y$ of the type $\bar{\Psi}\phi\Psi$ that describes the coupling between Higgs field and fermionic field without intermediate gauge bosons.
# 1. Charge, Hypercharge and Isospin

According to the above we must treat the left- and righthanded chirality differently and implement $SU(2) \times U(1)$ symmetry for lefthanded fields, while righthanded fields transform only under $U(1)$: This is the $(\frac{1}{2}, 0)$ representation of the Lorentz group, with the righthanded field transforming as $SU(2)$ singlets and the lefthanded field as a $SU(2)$ doublet. We concentrate only on the leptons $e, \nu_e$ and quarks $u, d$. Ignoring first the gauge fields the Lagrangian for the fermions is,

$$ \mathcal{L}_f = \overline{\ell_L}(-\gamma^\mu i\partial_\mu)\ell_L + \overline{q_L}(-\gamma^\mu i\partial_\mu)q_L + \overline{e_R}(-\gamma^\mu i\partial_\mu)e_R + \overline{u_R}(-\gamma^\mu i\partial_\mu)u_R + \overline{d_R}(-\gamma^\mu i\partial_\mu)d_R $$

with $\ell_L = (\nu_e, e)_L$, $q_L = (u, d)_L$ are $SU(2)$ doublets and $e_R, u_R, d_R$ are $SU(2)$ singlets. There cannot be a mass term since it would break the axial gauge symmetry of chiral components (see exercise V.15). Note that $\ell_L$ is in reality an eight-column vector with $SU(2)$ symmetry between the first 4 and the last 4 components, whereas $e_R$, remains a normal 4D Dirac field projected to its righthanded chirality and thus of the form $(0, 0, \eta_1, \eta_2)$ in the chiral representation. This notion is not always explicitly mentioned in literature and rapidly forgotten by the simplified notation. We also know that quarks are fermions with 3 colors and are represented as a complex three-component Dirac-vector with $SU(3)$ gauge symmetry involving $3^2 - 1 = 8$ generators. This symmetry describes the strong force with 8 gluons as intermediate and mutually interacting bosons. This force is completely ignored here: the weak force sees the quarks but not its colors.

With this convention in mind, we introduce local $SU(2)_L \times U(1)$ symmetry for lefthanded projections and only local $U(1)$ symmetry for the righthanded chirality (equivalent to a $SU(2)$ singlet, i.e. $SU(2)$ in its trivial one-dimensional representation $(0)$), thus introducing 3 + 1 gauge bosons that show up in the covariant derivative,

$$ -iD_{\mu}\ell_{L} = \left(-i\partial_{\mu} + \frac{g}{2}W_{\mu}^{n}\sigma^{n} + g'Y_{\ell}B_{\mu}\right)\ell_{L} ; \quad -iD_{\mu}e_{R} = \left(-i\partial_{\mu} + g'Y_{e}B_{\mu}\right)e_{R} $$

with $SU(2)$ charge unit $g$ and $U(1)$ hypercharge unit $g'$, both to be determined. In this notation, the lefthanded electron and neutrino carry both the $U(1)$ hypercharge $Y_\ell g'$, and the righthanded electron $Y_e g'$. Similar equations apply for the quarks with lefthanded charges $Y_q g'$, and righthanded charges $Y_u g'$ and $Y_d g'$. This is equivalent to the axial symmetry that was discussed in Exercise V.15 where different $U(1)$ gauge symmetries were shown to be exist for opposite chiralities provided the masses are zero. The $U(1)$ gauge field guarantees the conservation of (electron-)lepton number and baryon number. In the electroweak theory of the Standard Model, the $SU(2)_L$ symmetry will be broken to its subgroup $U(1)$ (say rotations along the third axis) with the quantum number called
hypercharge. Together with the other $U(1)$ symmetry the electromagnetic gauge is recovered for the photon. This implies that the electromagnetic force and the weak force are manifestations of one single force, that has split up by broken $SU(2) \times U(1)$ symmetry.

The matrix involving the two gauge fields can be written as

$$ \frac{g}{2} W_{\mu}^{n} \sigma^{n} + g' Y_{i} B_{\mu} = \begin{pmatrix} \frac{1}{2} g W_{\mu}^{3} + g' Y_{i} B_{\mu} & \frac{1}{\sqrt{2}} g W_{\mu}^{+} \\ \frac{1}{\sqrt{2}} g W_{\mu}^{-} & -\frac{1}{2} g W_{\mu}^{3} + g' Y_{i} B_{\mu} \end{pmatrix} \equiv G_{\mu}(i) \quad (186) $$

where $W_{\mu}^{\pm} = (W^{1} \mp iW_{\mu}^{2})/\sqrt{2}$, and $i \in \{\ell, q\}$. In the Dirac Lagrangian, the gauge fields generate the interaction

$$ \overline{\ell_L} \gamma^\mu G_\mu(\ell) \ell_L + \overline{q_L} \gamma^\mu G_\mu(q) q_L $$

that couple electron and neutrino, as well as up and down quark via the off-diagonal elements of $G_{\mu}(\ell, q)$. For the lepton doublet $\ell_L$, the off-diagonal elements generate the interactions $g\nu_L^\dagger\gamma^0\gamma^\mu W_\mu^+ e_L + g e_L^\dagger\gamma^0\gamma^\mu W_\mu^- \nu_L$ carrying the charged current. After second quantization this stands for a process involving three creation and destruction operators, for instance $e_L^\dagger W^- \nu_L$ destroys a $W^-$ boson, creates a left-handed electron, and destroys a neutrino. As we have seen in the second quantization of the Dirac equation, the destruction of neutrino fermions with negative energy is transformed into the creation of an anti-neutrino with positive energy. It can therefore also describe the $\beta$-process $W^- \to e + \bar{\nu}_e$ as well as its inverse $\nu_e + W^- \to e$. For the left-handed quark doublet $q_L = (u_L, d_L)$ the first term is proportional to $u^\dagger(W^-)^\dagger d$ which destroys a down-quark, and creates an up-quark as well as a $W^-$ gauge boson, both with conservation of electrical charge if indeed we attribute electromagnetic charges $\pm$ to the bosons $W^\pm$. The $\beta$-decay is thus described by the second-order weak process $d \to u + W^- \to u + e + \bar{\nu}_e$ with intermediate vector boson action. Lepton number conservation is built in.

As for the diagonal elements of the gauge matrix (186), let us first assume the absence of the hypercharge gauge field $B_\mu$ in the representation with left-handed chirality. We would get an interaction $\nu_L^\dagger W^3 \nu_L - e_L^\dagger W^3 e_L$. The boson $W^3$ cannot be the $Z^0$ particle, known to interact with right-handed chiralities as well. The boson $B_\mu$ cannot be the photon either since $B_\mu$ would then only couple to right-handed electrons. To fix this, we need define two new gauge fields,

$$ \begin{aligned} W_{\mu}^{3} &= \cos \theta_{W} Z_{\mu} + \sin \theta_{W} A_{\mu} \\ B_{\mu} &= -\sin \theta_{W} Z_{\mu} + \cos \theta_{W} A_{\mu} \end{aligned} \quad (187) $$
with $\theta_W$ the Weinberg angle. The $W^3$ gauge field represents spin $\frac{1}{2}$ rotations along the 3-axis and is such an Abelian subgroup of $SU(2)$ isomorphic to $U(1)$. In the following we wish to interpret $A_\mu$ as the usual electromagnetic field and to identify $Z_\mu$ as the neutral $Z^0$ gauge boson. First, $A_\mu$ interacts with righthanded electrons via $-\overline{e_R}g'Y_e \cos\theta_W A_\mu e_R$ and idem pour $u$ and $d$. To get the electrical charges right, we must therefore impose $g' \cos\theta_W = e$ and $Y_e = -1$, $Y_u = \frac{2}{3}$ and $Y_d = -\frac{1}{3}$. The righthanded hypercharges are different for the quarks $u$ and $d$, whereas the lefthanded quarks are in a $SU(2)$-doublet and must have equal hypercharge. For the lefthanded particles we get interactions between fermions and $A_\mu$

$$ \overline{\nu_e} \left( \frac{g}{2} \sin \theta_W + g' Y_\ell \cos \theta_W \right) \gamma^\mu A_\mu \nu_e + \overline{e_L} \left( -\frac{g}{2} \sin \theta_W + g' Y_\ell \cos \theta_W \right) \gamma^\mu A_\mu e_L + \overline{u_L} \left( \frac{g}{2} \sin \theta_W + g' Y_q \cos \theta_W \right) \gamma^\mu A_\mu u_L + \overline{d_L} \left( -\frac{g}{2} \sin \theta_W + g' Y_q \cos \theta_W \right) \gamma^\mu A_\mu d_L $$

We must impose $g \sin\theta_W = e$ to attribute the charge $-e$ to the lefthanded electron. Because $g' \cos\theta_W = e$, the electrical charge $Q$ in units of $e$ for all lefthanded fermions is $Q = \frac{1}{2}\sigma^3 + Y$ and is different for the two elements in the $SU(2)$ spinor. We can define the isospin operator $I^3 = \frac{1}{2}\sigma^3$ for the lefthanded doublets and put $I^3 = 0$ for the righthanded singlets. We can summarize this by $Q = I^3 + Y$. For leptons we set $Y_\ell = -\frac{1}{2}$ so that we find zero electrical charge for the neutrino and charge $Q = -e$ for the electron. With $Y_q = +\frac{1}{6}$ we recover the charges $Q_u = +\frac{2}{3}$ and $Q_d = -\frac{1}{3}$.

With these identifications we can find the coupling of the $Z$-boson to the leptons and the quarks. For righthanded particles the interaction has now become non-zero and is given by

$$ -e \tan \theta_W Y_e \overline{e_R} \gamma^\mu Z_\mu e_R - e \tan \theta_W Y_u \overline{u_R} \gamma^\mu Z_\mu u_R + \text{idem } d_R $$

where we used $g' \sin\theta_W = e \tan\theta_W$. It can be checked that the lefthanded particles interact with $Z_\mu$ as

$$ \overline{\ell}_L M(Y, Q) \ell_L \gamma^\mu Z_\mu + \text{idem } q_L $$

with amplitude $M(Y, Q) = 2e(Q \cos^2\theta_W - Y)/\sin 2\theta_W$. This formula also applies to the righthanded singlets with $Q = Y$. Thus, like the photon, the $Z$ boson interacts with all leptons and quarks but unlike the photon, this interaction depends on charge and hypercharge.

## 2. Masses of the gauge bosons

So far the masses of gauge bosons, leptons and quarks all vanish. The first will achieve mass because they couple to the Higgs field via the covariant derivative. The one that we need here is
a complex scalar SU(2)-doublet $\Phi = (\Phi_1, \Phi_2)$, because it must couple to the lefthanded doublets in the previous section which contain the vector bosons $W^\dagger, Z$ and the supposed photon $A_\mu$ that we want to stay massless. The Higgs Lagrangian is invariant under Lorentz transformations (representation $(0,0)$) as well as under the previous SU(2) isospin symmetry,

$$ \mathcal{L}_H = D_\mu \Phi^\dagger D^\mu \Phi - m^2 \Phi^\dagger \Phi - \frac{1}{2} \lambda (\Phi^\dagger \Phi)^2 = 0 \qquad (188) $$

with $m^2 \equiv -v^2\lambda < 0$. The minimum of the Higgs potential occurs at $\Phi^\dagger\Phi = |\Phi_1|^2 + |\Phi_2|^2 = m^2/\lambda \equiv v^2/2$. The symmetry SU(2) × U(1) is broken by a particular choice, for instance by choosing the Higgs field as $\Phi_- = (0, v/\sqrt{2})$. The complex-dual field of a SU(2) spinor $\eta$ was defined in section VI.F as $\xi = i\sigma^2\eta^*$. We noticed that when $\eta$ changes under the SU(2) transformation $U = \exp(-i\alpha_n\sigma_n)$, then so does $\xi$. The doublet dual to $\Phi_-$ is $\Phi_+ = (v/\sqrt{2}, 0)$. However, if $\eta$ changes under the U(1) symmetry as $\exp(-iY\theta)$ then $\xi$ must transform as $\exp(+iY\theta)$ due to the extra complex conjugate. Hence $\Phi_+$ and $\Phi_-$ must have opposite hypercharges. The covariant derivatives become,

$$ D_{\mu}(\pm)\Phi_{\pm} = \left( \partial_{\mu} + \frac{ig}{2}W_{\mu}^{n}\sigma^{n} \pm ig'YB_{\mu} \right) \Phi_{\pm} = \partial_{\mu}\Phi_{\pm} + iG_{\mu}(\pm)\Phi_{\pm} $$

with the matrix $G_\mu(\pm)$ defined earlier in Eq. (186) with different hypercharge associated with the Higgs doublet $\Phi_\pm$. If we take $Y = -\frac{1}{2}$ for the hypercharge of the Higgs boson, with isospin $\frac{1}{2}\sigma^3\Phi_\pm = \pm\frac{1}{2}\Phi_\pm$, it follows that the electrical charge of the Higgs particle (i.e. its coupling to the field $A_\mu$) is $Q_H = I^3 + Y = 0$ for both isospins and the photon gauge field $A_\mu$ drops out in the covariant derivative of the Higgs field so that no photon mass in generated. Hence,

$$ G_{\mu}(-)\Phi_{-} = \left[ -\frac{1}{2}gW_{\mu}^{3} + \frac{1}{2}g'B_{\mu} \right] \begin{pmatrix} 0 \\ v/\sqrt{2} \end{pmatrix} = \begin{pmatrix} \frac{ve}{\sqrt{2}\sin\theta_W}W_{\mu}^{+} \\ -\frac{ev}{\sqrt{2}\sin 2\theta_W}Z_{\mu} \end{pmatrix} $$

and

$$ G_{\mu}(+)\Phi_{+} = \left[ \frac{1}{2}gW_{\mu}^{3} - \frac{1}{2}g'B_{\mu} \right] \begin{pmatrix} v/\sqrt{2} \\ 0 \end{pmatrix} = \begin{pmatrix} \frac{ev}{\sqrt{2}\sin 2\theta_W}Z_{\mu} \\ -\frac{ve}{\sqrt{2}\sin\theta_W}W_{\mu}^{-} \end{pmatrix} $$

Mass terms of the gauge fields appear as $\frac{1}{2}m^2$ in the gauge field Lagrangian (see Proca Lagrangian earlier in section V) and can be identified from,

$$ \left( 0, v/\sqrt{2} \right) G_{\mu}(-)^{\dagger} G^{\mu}(-) \begin{pmatrix} 0 \\ v/\sqrt{2} \end{pmatrix} = \frac{e^2 v^2}{2 \sin^2 2\theta_W} Z_{\mu}^{\dagger} Z^{\mu} + \frac{e^2 v^2}{8 \sin^2 \theta_W} \left[ (W^{-})^{\dagger} W^{-} + (W^{+})^{\dagger} W^{+} \right] $$

and the same result is obtained when using $\Phi_+$. The gauge-fields themselves have the Lagrangian,

$$ \mathcal{L}_G = -\frac{1}{4}G_{\mu\nu}G^{\mu\nu} - \frac{1}{4}(W^n)_{\mu\nu}(W^n)^{\mu\nu} \qquad (189) $$
with $G_{\mu\nu} = \partial_{\mu}B_{\nu} - \partial_{\nu}B_{\mu}$ for the U(1) boson and for the SU(2) bosons $W_{\mu\nu}^{n} = \partial_{\mu}W_{\nu}^{n} - \partial_{\nu}W_{\mu}^{n} + g\epsilon_{nmk}(W^{m})^{\mu}(W^{k})^{\nu}$. We can infer that the Weinberg rotation in Eq. (187) gives,

$$ \mathcal{L}_G = -\frac{1}{4}F(A)_{\mu\nu}F(A)^{\mu\nu} - \frac{1}{4}F(Z)_{\mu\nu}F(Z)^{\mu\nu} - \frac{1}{4}(W^{+})_{\mu\nu})^{\dagger}(W^{+})^{\mu\nu} - \frac{1}{4}(W^{-})_{\mu\nu})^{\dagger}(W^{-})^{\mu\nu} \\ + \frac{e^2 v^2}{2 \sin^2 2\theta_W} Z_{\mu}^{\dagger} Z^{\mu} + \frac{e^2 v^2}{8 \sin^2 2\theta_W} [(W^{-})^{\dagger}W^{-} + (W^{+})^{\dagger}W^{+}] + \text{nonlinear mixtures } (A, W^{\pm}, Z) \tag{190} $$

From this we infer the masses

$$ M_Z = \frac{ev}{\sin 2\theta_W} : M_W = \frac{ev}{2 \sin \theta_W} : M_A = 0 $$

The ratio $M_Z/M_W = 1/\cos\theta_W > 1$. The observations at CERN report $\sin^2\theta_W = 0.2231$. The photon remains massless. The nonlinear terms in Eq. (190) imply among other things that the vector bosons $W^{\pm}$ have electromagnetic charge $\pm e$. It is straightforward to show that the Higgs particle has a mass $M_H = v\sqrt{2\lambda}$ (see Exercise VI.1) so that $M_H/M_W = \sqrt{2\lambda}/2g$. With $M_H = 125$ GeV, this fixes $\sqrt{\lambda}/g = 2.2$. This implies that the self-interaction of the Higgs boson and its coupling to the gauge fields are of the same order of magnitude.

## Exercise VIII.1

* Show that upon adding $G_{\mu\nu}G^{\mu\nu}$ and $(W^3)_{\mu\nu}(W^3)^{\mu\nu}$, bilinear cross-terms between the photon field A and the Z-gauge boson cancel.

* Show that diagonal terms in the kinetic term of the fermionic Lagrangian (with the co-variant derivative $D_{\mu}$) generate the three-particle processes such as $\nu_e \to Z^0 + \nu_e$ (inelastic neutrino scattering) as well as $e_L \to Z^0 + e_L$ (inelastic electron scattering) and $e_L + e_L^+ \to Z$ or $\gamma$ (electron-positron annihilation). The off-diagonal terms generate electron-neutrino processes with the production or destruction of a $W^-$-boson: $e_L \to \nu_e + W^-$ and $W^- \to \bar{\nu}_e + e_L$

* Show from the Lagrangian of the gauge fields that three-boson interactions exist of the kind $W^+ + W^+ \to (\gamma, Z^0)$ and that four-boson interactions exist of the kind $W^+ + W^- \to \gamma + \gamma$, $W^+ + W^- \to Z^0 + \gamma$ or $W^+ + W^- \to W^+ + W^-$ (WW scattering)
### 3. Fermion Masses

The fermion masses are still zero because any finite electron mass term $\overline{e_L}e_R + \overline{e_R}e_L$ in the Lagrangian would break axial $U(1)$ symmetry (see Exercise IV.11), here the one associated with hypercharge. With the introduction of the $SU(2)$ symmetry for the lefthanded fermions, we have attributed hypercharges $Y_\ell = -\frac{1}{2}$ to lefthanded electrons and neutrinos and hypercharge $Y_e = -1$ to righthanded electrons. We also have introduced the Higgs spinors $\Phi_\pm$ with hypercharge $Y_\pm = \mp\frac{1}{2}$. If we concentrate on the leptons, a Yukawa coupling can be defined as,

$$ \mathcal{L}_Y = -f_e(\overline{\ell_L}\Phi_-)e_R - f_e\overline{e_R}(\Phi_-^\dagger\ell_L) \quad (191) $$

Note that 1) the two terms are each other's complex conjugate and guarantee a real-valued Lagrangian; 2) the matrix product $\overline{\ell_L}\Phi_-$ generates an $SU(2)$ isospin singlet, like $e_R$ so that the entire matrix product is an $SU(2)$ isospin singlet and thus gauge-invariant under $SU(2)$; 3) the Higgs boson being a Lorentz-scalar doublet, the product is a Dirac bispinor so that $(\overline{\ell_L}\Phi_-)e_R$ is Lorentz invariant; 4) the product has zero $U(1)$ hypercharge. For instance, the left term transforms under hypercharge as $\Delta Y = -Y_\ell + Y_- + Y_e = \frac{1}{2} + \frac{1}{2} - 1 = 0$. If we now break the $SU(2)$ symmetry with the Higgs field $\Phi_- = (0, v/\sqrt{2})$, we obtain

$$ \mathcal{L}_Y = -f_e \frac{v}{\sqrt{2}} (\overline{e_L} e_R + \overline{e_R} e_L) = -f_e \frac{v}{\sqrt{2}} (\bar{e} e) \quad (192) $$

This generates an electron mass $m_e = f_e v / \sqrt{2}$. There is no mass term generated for the lefthanded neutrino because no righthanded neutrinos exist to create an $SU(2)$ singlet. The same can be done for the quarks with Yukawa couplings $f_u$ and $f_d$, with one extra subtlety. The Yukawa Lagrangian must be

$$ \mathcal{L}_Y = -f_u(\overline{q_L}\Phi_+)u_R - f_d(\overline{q_L}\Phi_-)d_R + \text{c.c.} \quad (193) $$

To have hypercharge $Y = 0$, the interaction of the up-quark must contain the dual Higgs field $\Phi_+$, so that $Y = -Y_q + Y_+ + Y_u = -\frac{1}{6} - \frac{1}{2} + \frac{2}{3} = 0$. For the down quark we need the Higgs field $\Phi_-$ with $Y = -\frac{1}{6} + \frac{1}{2} - \frac{1}{3} = 0$. When we break the symmetry, duality must be respected so that $\Phi_+ = (v/\sqrt{2}, 0)$. With their complex conjugates, the quark masses are generated after breaking the symmetry,

$$ \mathcal{L}_Y = -f_e \frac{v}{\sqrt{2}}(\bar{u}u) - f_d \frac{v}{\sqrt{2}}(\bar{d}d) \quad (194) $$

Because $m_e \approx 0.5$ MeV $\ll M_H$ and the quark masses equal several MeV, we infer that the Yukawa couplings $f_{e,u,d} \ll \sqrt{\lambda}$. The top quark ($m = 172$ GeV) is much heavier than the Higgs boson and
<table><thead><tr><th>property</th><th>e<sub>L</sub>(e<sub>R</sub>)</th><th>&nu;<sub>e</sub>(&nu;&#773;<sub>e</sub>)</th><th>u<sub>L</sub>(u<sub>R</sub>)</th><th>d<sub>L</sub>(d<sub>R</sub>)</th><th>H(&plusmn;)</th></tr></thead><tbody><tr><td>Q</td><td>-1(-1)</td><td>0</td><td><sup>2</sup>&frasl;<sub>3</sub></td><td>-<sup>1</sup>&frasl;<sub>3</sub></td><td>0</td></tr><tr><td>Y</td><td>-1(-<sup>1</sup>&frasl;<sub>2</sub>)</td><td>-<sup>1</sup>&frasl;<sub>2</sub>(+<sup>1</sup>&frasl;<sub>2</sub>)</td><td><sup>1</sup>&frasl;<sub>6</sub>(<sup>2</sup>&frasl;<sub>3</sub>)</td><td><sup>1</sup>&frasl;<sub>6</sub>(-<sup>1</sup>&frasl;<sub>3</sub>)</td><td>&mp;<sup>1</sup>&frasl;<sub>2</sub></td></tr><tr><td>lepton number</td><td>1</td><td>1(-1)</td><td>1</td><td>1</td><td>0</td></tr><tr><td>M</td><td>0.5 MeV</td><td>&lt; 0.086 eV</td><td>2.5 MeV</td><td>4.8 MeV</td><td>125 GeV</td></tr></tbody></table>

is not discussed here. Does this mean that the puzzle of the weak interaction is solved? Not at all! The first quantum corrections turn out to diverge, a problem well known from QED. It took another Nobel Prize to solve this. The electro-weak divergences were regularized much later (1971) by Maurice Veltman and Gerard t'Hooft.
Further Reading

The course has benefitted from several courses and research papers listed below, most of them
gently put online by their authors.

[1] P.W. Anderson, "More is Different", Science **177**(407), 393 (1972).
https://www.tkm.kit.edu/downloads/TKM1_2011_more_is_PWA.pdf.

[2] Lev D. Landau, "On the Theory of Phase Transitions", Zh. Eksp. Teor. Fiz. 7: 19-32 (1937).
http://www.ujp.bitp.kiev.ua/files/journals/53/si/53SI08p.pdf.

[3] E. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, Annals of Physics, **40**(1), 149 (1939).

[4] A.J. Beekman, L. Rademaker and J. van Wezel, An introduction to spontaneous symmetry breaking, SciPost Phys. Lect. Notes **11**,1-140 (2019). https://scipost.org/SciPostPhysLectNotes.11

[5] D. Naegels, An introduction to Goldstone boson physics and to the coset construction (2021).
https://arxiv.org/abs/2110.14504

[6] Steven Weinberg, The Quantum Theory of Fields (Cambridge University Press, 1995). A copy (impossible to download) is accessible at http://www.stat.ucla.edu/~ywu/research/documents/weinberg1.pdf.
Worth reading is the almost philosophical paper defending quantum field theory, available at
https://arxiv.org/pdf/hep-th/9702027.

[7] P.W. Anderson, An Approximate Quantum Theory of the Antiferromagnetic Ground State, Phys. Rev. B **86**, 694 (1952). https://journals.aps.org/pr/abstract/10.1103/PhysRev.86.694

[8] B. Gripaos, Gauge Field Theory (Lecture Notes) (2016), https://www.hep.phy.cam.ac.uk/gripaios/~gft_lecture_notes.pdf

[9] Haruki Watanabe and Masaki Oshikawa, Absence of Quantum Time Crystals, Phys. Rev. Lett. **114**, 251603 (2015). See also https://www.quantamagazine.org/first-time-crystal-built-using-googles-quantum-computer-20210730/

[10] L.J. LeBlanc et al., Direct observation of Zitterbewegung in a Bose–Einstein condensate, New Journal of Physics **15**, 073011 (2013).

[11] E. Merzbacher, Quantum Mechanics (second edition, Wiley, 1970).

[12] M.R. Andrews, C.G. Towsend, H.J. Miesner, D.S. Durfee, D.M. Kurn and W. Ketterle, Observation of Interference of two Bose Condensates, Science **275**, 637 (1997).

[13] H. Deng, G. Solomon, H. Rudolf, K. H. Ploog, and Y. Yamamoto. Spatial coherence of a polariton condensate. Phys. Rev. Lett., 99:126403, 2007.

[14] P.B. Pal, Dirac, Majorana and Weyl fermions, Am. J. Phys. **79** (2011) 485–498.
https://arxiv.org/abs/1006.1718v2

[15] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, *Photons and Atoms* (Wiley, 1989).
[16] P.W. Milonni, *The Quantum Vacuum* (Academic Press, San Diego, 1994).

[17] (author unknown, excellent short lecture notes), Fundamental Concepts of Bose-Einstein Condensation, https://www.nii.ac.jp/qis/first-quantum/e/forStudents/lecture/pdf/qis385/QIS385_chap2.pdf

[18] Kien Nguyen, Higgs mechanism and Electroweak Force, Lecture notes (2009). https://www.theorie.physik.uni-muenchen.de/lsfrey/teaching/archiv/sose_09/rng/higgs_mechanism.pdf

[19] Hitoshi Yamamoto, Lorentz Group and Lorentz Invariance (lecture notes), https://epx.phys.tohoku.ac.jp/~yhitoshi/particleweb/ptestup.pdf

[20] Taro Kimura, Lectures on Quantum Field Theory (2024). https://kimura.pages.math.cnrs.fr/file/QFT_lec.pdf

[21] Tutorial 3: The Transverse Ising Model: https://phas.ubc.ca/~seme/516/smTutorial2.pdf

[22] L. Onsager, Crystal statistics: I. A two-dimensional model with an order-disorder transition, Physical Review **65**, 3-4, 117 (1944).

[23] F. London et H. London, "The Electromagnetic Equations of the Supraconductor", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. **149**, no 866, 71 (1935). https://royalsocietypublishing.org/doi/10.1098/rspa.1935.0048