--- title: The Higgs Mechanism module: Electroweak Unification and the Higgs moduleNumber: 9 lessonNumber: 3 order: 903 summary: > Gauging a spontaneously broken symmetry converts the would-be Goldstone bosons into the longitudinal polarizations of the gauge fields, which thereby acquire mass. Applied to $SU(2)_L \times U(1)_Y$ with a single Higgs doublet, three of the four scalar degrees of freedom are eaten by the $W^\pm$ and $Z$; the fourth survives as the physical Higgs boson, and the photon stays massless. Fermion masses come from Yukawa couplings to the same field, each mass proportional to its coupling times the vacuum expectation value $v \approx 246$ GeV. topics: [Electroweak Unification and the Higgs] sources: - book: Griffiths ref: "Ch. 11 §11.6–11.9 (the Higgs mechanism and the Glashow-Weinberg-Salam model)" - book: Thomson ref: "Ch. 17 §17.4–17.7 (the Higgs mechanism, gauge-boson and fermion masses)" - book: Halzen & Martin ref: "Ch. 15 (the Higgs mechanism and mass generation in the electroweak theory)" - book: Tong ref: "The Standard Model (Cambridge Part III), §2.3 & §5.2 (the Higgs mechanism, electroweak symmetry breaking)" draft: false --- Spontaneously breaking a global symmetry produces massless [Goldstone bosons](/particle-physics/electroweak-higgs/spontaneous-symmetry-breaking). Spontaneously breaking a **gauge** symmetry does something else: the Goldstone modes are gauge artifacts, and rather than propagating as massless scalars they are absorbed by the gauge fields, which become massive. This is the Higgs mechanism, the only known way to give a gauge boson a mass without spoiling the gauge invariance that makes the theory consistent. This lesson works the mechanism first for a single abelian gauge field, then for the full $SU(2)_L \times U(1)_Y$ of the [electroweak theory](/particle-physics/electroweak-higgs/electroweak-su2-u1), deriving the $W$ and $Z$ masses, the massless photon, and the fermion masses from Yukawa couplings. Throughout, $\hbar = c = 1$. The Higgs doublet is written $\Phi$, its vacuum expectation value $v$, and the physical scalar excitation $h$. ## The abelian Higgs model Couple the complex scalar of the previous lesson to a $U(1)$ gauge field $A_\mu$ by promoting the global phase symmetry to a local one. The Lagrangian is $$ \mathcal L = (D_\mu\phi)^\ast(D^\mu\phi) - V(\phi) - \tfrac14 F_{\mu\nu}F^{\mu\nu}, \qquad D_\mu = \partial_\mu - i q A_\mu, $$ with $V(\phi) = -\mu^2\phi^\ast\phi + \lambda(\phi^\ast\phi)^2$ and $\mu^2 > 0$, so the minimum is a circle $|\phi| = v/\sqrt2$ with $v = \sqrt{\mu^2/\lambda}$. As before, choosing the vacuum breaks the symmetry. Parametrize the field about the vacuum in radial and phase pieces, $$ \phi(x) = \frac{1}{\sqrt2}\big(v + h(x)\big)\,e^{i\theta(x)/v}. $$ The phase $\theta(x)$ plays the role of a local gauge parameter, so it can be removed by a gauge transformation — the **unitary gauge**. Setting $\theta = 0$ eliminates the Goldstone field from the Lagrangian entirely. The kinetic term of the scalar then contains $$ (D_\mu\phi)^\ast(D^\mu\phi) \supset \tfrac12 q^2 v^2\, A_\mu A^\mu, $$ a **mass term for the gauge field**, $\tfrac12 m_A^2 A_\mu A^\mu$ with $$ m_A = q v. $$ The gauge boson has become massive by absorbing the Goldstone mode. The degree-of- freedom count balances exactly: before breaking, a massless gauge field carries two polarizations and the complex scalar carries two real components, four in all; after breaking, a massive gauge field carries three polarizations and one real scalar $h$ survives, again four. The Goldstone boson has not disappeared — it has become the **longitudinal polarization** the gauge field lacked while massless. $$ % caption: The degree-of-freedom bookkeeping of the abelian Higgs mechanism. A % massless gauge boson (two transverse polarizations) plus a complex scalar (two % real fields) rearrange into a massive gauge boson (three polarizations) plus one % real Higgs scalar. The eaten Goldstone mode becomes the longitudinal polarization. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0, bx/.style={draw=black, minimum width=20mm, minimum height=8mm, font=\scriptsize, align=center}, bxa/.style={draw=acc, fill=acc!10, minimum width=20mm, minimum height=8mm, font=\scriptsize, align=center}] \definecolor{acc}{HTML}{4A6FA5} % before \node[bx] (g0) at (0,1.4) {massless boson\\ 2 pol}; \node[bx] (s0) at (0,0.0) {complex scalar\\ 2 real dof}; \node[above, font=\scriptsize] at (0,2.1) {before (4 dof)}; % after \node[bx] (g1) at (5.0,1.4) {massive boson\\ 3 pol}; \node[bx] (s1) at (5.0,0.0) {Higgs scalar\\ 1 dof}; \node[above, font=\scriptsize] at (5.0,2.1) {after (4 dof)}; % arrows \draw[->, acc, thick] (1.15,1.4) -- (3.85,1.4); \node[acc, above, font=\scriptsize] at (2.5,1.5) {eats Goldstone}; \draw[->, black] (1.15,0.0) -- (3.85,0.0); \end{tikzpicture} $$ ## The electroweak Higgs doublet To break $SU(2)_L \times U(1)_Y$ down to the electromagnetic $U(1)_{\text{EM}}$, the scalar must be an $SU(2)$ doublet carrying hypercharge. The minimal choice is a single complex doublet with hypercharge $Y = 1$, $$ \Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}, \qquad V(\Phi) = -\mu^2\,\Phi^\dagger\Phi + \lambda\,(\Phi^\dagger\Phi)^2, $$ four real scalar fields in all. For $\mu^2 > 0$ the potential is minimized on the sphere $\Phi^\dagger\Phi = v^2/2$. The vacuum is chosen to lie entirely in the lower, electrically neutral component, $$ \langle\Phi\rangle = \frac{1}{\sqrt2}\begin{pmatrix} 0 \\ v \end{pmatrix}, $$ so that the vacuum carries no electric charge. This choice is what leaves the photon massless. The upper component $\phi^+$ has $Q = T_3 + Y/2 = +\tfrac12 + \tfrac12 = 1$ and the lower $\phi^0$ has $Q = -\tfrac12 + \tfrac12 = 0$; a charged vacuum would break electromagnetism, which is not observed, so the neutral direction is the physical one. The unbroken generator is the specific combination that annihilates the vacuum. Acting on $\langle\Phi\rangle$ with the electric-charge operator $Q = T_3 + Y/2 = \tfrac12\sigma^3 + \tfrac12$, $$ Q\,\langle\Phi\rangle = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \frac{1}{\sqrt2}\begin{pmatrix} 0 \\ v \end{pmatrix} = 0, $$ so $Q$ leaves the vacuum invariant: $U(1)_{\text{EM}}$ survives and its gauge boson, the photon, stays massless. The other three combinations of generators move the vacuum and are broken, producing three Goldstone bosons — exactly the number needed to give mass to three gauge bosons. $$ % caption: The Higgs doublet before and after symmetry breaking. Before, the % potential is symmetric and the field averages to zero. After, the vacuum settles % in the neutral lower component with magnitude v over root two, leaving the upper % (charged) component empty so that electric charge is unbroken and the photon stays % massless. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0, cell/.style={draw=black, minimum width=14mm, minimum height=8mm, font=\scriptsize}] \definecolor{acc}{HTML}{4A6FA5} % before doublet \node[cell] (u0) at (0,1.5) {charged}; \node[cell] (d0) at (0,0.6) {neutral}; \draw[black] (-0.8,0.15) rectangle (0.8,1.95); \node[black, below, font=\scriptsize, align=center] at (0,-0.1) {symmetric:\\ average zero}; % arrow \draw[->, black, thick] (1.4,1.05) -- (3.2,1.05); \node[above, font=\scriptsize] at (2.3,1.1) {breaking}; % after doublet with vev in lower \node[cell] (u1) at (4.6,1.5) {$0$}; \node[cell, draw=acc, fill=acc!12] (d1) at (4.6,0.6) {vev}; \draw[black] (3.8,0.15) rectangle (5.4,1.95); \node[acc, below, font=\scriptsize, align=center] at (4.6,-0.1) {vacuum in neutral\\ component}; \end{tikzpicture} $$ ## Gauge-boson masses The gauge-boson masses come from the covariant-derivative term $(D_\mu\Phi)^\dagger (D^\mu\Phi)$ evaluated at the vacuum, with $$ D_\mu\Phi = \left(\partial_\mu - i g\, W^a_\mu \tfrac{\sigma^a}{2} - i g'\tfrac{Y}{2} B_\mu\right)\Phi. $$ Substituting $\Phi \to \langle\Phi\rangle$ and reading off the terms quadratic in the gauge fields gives, after collecting $W^1, W^2$ into $W^\pm$ and diagonalizing the neutral $W^3$–$B$ block, $$ M_W = \frac{gv}{2}, \qquad M_Z = \frac{v}{2}\sqrt{g^2 + g'^2}, \qquad M_\gamma = 0. $$ The three broken directions have given mass to $W^+, W^-, Z$; the unbroken $U(1)_{\text{EM}}$ direction leaves the photon massless. Dividing the two massive results reproduces the tree-level relation of the previous lesson, $$ \frac{M_W}{M_Z} = \frac{g}{\sqrt{g^2+g'^2}} = \cos\theta_W, $$ so the mass ratio and the mixing angle are the same information. The single physical scalar remaining after the three Goldstones are eaten is the Higgs boson $h$, with mass $$ M_h = \sqrt{2\lambda}\,v. $$ Its mass depends on the quartic coupling $\lambda$, which is not fixed by any other measurement, so $M_h$ is a genuinely free parameter of the theory — the one number the Higgs mechanism cannot predict, and which had to be measured directly. $$ % caption: The four electroweak scalar degrees of freedom after breaking. Three % become the longitudinal modes of the W plus, W minus, and Z, turning each massless % gauge boson (two polarizations) into a massive one (three). The fourth survives as % the physical Higgs boson. The photon, coupled to the unbroken charge, gains no % mass. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0, bxa/.style={draw=acc, fill=acc!10, minimum width=16mm, minimum height=7mm, font=\scriptsize, align=center}, bx/.style={draw=black, minimum width=16mm, minimum height=7mm, font=\scriptsize, align=center}] \definecolor{acc}{HTML}{4A6FA5} % four scalar dof \node[bx] (s1) at (0,3.0) {Goldstone 1}; \node[bx] (s2) at (0,2.0) {Goldstone 2}; \node[bx] (s3) at (0,1.0) {Goldstone 3}; \node[bxa] (s4) at (0,0.0) {Higgs $h$}; % three massive bosons + higgs \node[bx] (wp) at (4.4,3.0) {$W^+$ (massive)}; \node[bx] (wm) at (4.4,2.0) {W minus (massive)}; \node[bx] (zz) at (4.4,1.0) {$Z$ (massive)}; \node[bxa] (hh) at (4.4,0.0) {Higgs $h$}; \draw[->, black] (s1) -- (wp); \draw[->, black] (s2) -- (wm); \draw[->, black] (s3) -- (zz); \draw[->, acc, thick] (s4) -- (hh); \node[above, font=\scriptsize] at (0,3.5) {4 scalar dof}; \node[above, font=\scriptsize] at (4.4,3.5) {eaten as longitudinal modes}; \end{tikzpicture} $$ ## The electroweak scale The value of $v$ is not predicted, but it is fixed by low-energy weak data. Matching the $W$-exchange amplitude to Fermi's four-fermion theory relates the Fermi constant $G_F$ to the $W$ mass and coupling, $$ \frac{G_F}{\sqrt2} = \frac{g^2}{8 M_W^2} = \frac{1}{2v^2}, $$ using $M_W = gv/2$. Solving for $v$ with the measured $G_F = 1.166 \times 10^{-5}\ \text{GeV}^{-2}$, $$ v = \big(\sqrt2\, G_F\big)^{-1/2} \approx 246\ \text{GeV}. $$ This is the **electroweak scale**, the only dimensionful parameter in the classical Standard Model Lagrangian. Every gauge-boson and fermion mass is $v$ times a dimensionless coupling: $M_W = gv/2$, $M_Z = v\sqrt{g^2+g'^2}/2$, and the fermion masses below. If $v$ were zero, every elementary particle would be massless.[^tong-vev] $$ % caption: The electroweak scale v set by the Fermi constant. The measured strength % of weak decays fixes v near 246 GeV; the W and Z masses are then this scale times % the gauge couplings, and the Higgs mass is this scale times the square root of the % quartic coupling. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % central scale bar \draw[acc, very thick] (0,0) -- (0,3.4); \fill[acc] (0,2.5) circle (2.4pt); \node[acc, left, font=\scriptsize] at (-0.15,2.5) {v near 246 GeV}; % input from below (Fermi constant) \draw[black, ->, thick] (-2.6,0.8) -- (-0.2,2.4); \node[black, left, font=\scriptsize, align=right] at (-2.6,0.75) {weak decays\\ set $G_F$}; % outputs to the right \draw[black, ->] (0.2,2.6) -- (2.4,3.1); \node[right, font=\scriptsize] at (2.4,3.1) {$M_W, M_Z$ (times $g$)}; \draw[black, ->] (0.2,2.4) -- (2.4,1.9); \node[right, font=\scriptsize] at (2.4,1.9) {$M_h$ (times root lambda)}; \draw[black, ->] (0.2,2.2) -- (2.4,0.8); \node[right, font=\scriptsize] at (2.4,0.8) {fermion masses (times Yukawa)}; \end{tikzpicture} $$ ## Fermion masses and Yukawa couplings A Dirac mass term $m\bar\psi\psi = m(\bar\psi_L\psi_R + \bar\psi_R\psi_L)$ pairs a left-handed field with a right-handed one. In the electroweak theory the two have different $SU(2)$ and hypercharge assignments — $\psi_L$ is part of a doublet, $\psi_R$ a singlet — so a bare mass term is not gauge invariant. Mass must instead come from a gauge-invariant coupling to the Higgs doublet, the **Yukawa coupling**. For the electron, $$ \mathcal L_{\text{Yuk}} = -y_e\,\big(\bar L_L\,\Phi\,e_R + \text{h.c.}\big), $$ which is an $SU(2)$ singlet because $\bar L_L$ (a doublet) contracts with $\Phi$ (a doublet), and hypercharge balances. When $\Phi$ takes its vacuum value $\langle\Phi\rangle = (0, v/\sqrt2)$, this becomes a mass term with $$ m_f = \frac{y_f\, v}{\sqrt2}. $$ Every fermion mass is its Yukawa coupling times $v/\sqrt2$. The couplings themselves span six orders of magnitude, from $y_t \approx 1$ for the top quark down to $y_e \approx 3 \times 10^{-6}$ for the electron, and the Standard Model does not explain this hierarchy — it takes the Yukawa couplings as inputs. What it does predict is that the coupling of each fermion to the physical Higgs boson $h$ is proportional to that fermion's mass, since both descend from the same term: the heavier the particle, the more strongly it couples to the Higgs, a relation tested directly in Higgs decays. $$ % caption: Fermion masses on a logarithmic scale, spanning the electron near half an % MeV to the top quark near 173 GeV. Each mass is the Yukawa coupling times the same % vacuum value v over root two, so the ladder of masses is a ladder of couplings. % The top coupling is of order one; the electron coupling is a few parts in a % million. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % vertical log axis from 10^-3 GeV to 10^3 GeV -> mapped y in [0, 6] \draw[->, black] (0,-0.6) -- (0,8.6) node[above, font=\scriptsize] {mass (log scale)}; % gridlines at consistent decades: y = log10(m/MeV) * 1.5 \foreach \y/\lab in {0/1 MeV, 1.5/10 MeV, 3.0/100 MeV, 4.5/1 GeV, 6.0/10 GeV, 7.5/100 GeV} {\draw[black, densely dotted] (0,\y) -- (6.4,\y); \node[black, left, font=\scriptsize] at (-0.05,\y) {\lab};} % points ascend on a diagonal; each label sits BELOW-RIGHT of its own dot % (the empty quadrant — the next dot is always up-right) so no label drifts % toward the neighbouring point. \fill[black] (0.6,-0.4) circle (2pt); \node[anchor=north west, font=\scriptsize] at (0.72,-0.46) {e}; \fill[black] (1.2,0.5) circle (2pt); \node[anchor=north west, font=\scriptsize] at (1.32,0.44) {u}; \fill[black] (1.8,1.0) circle (2pt); \node[anchor=north west, font=\scriptsize] at (1.92,0.94) {d}; \fill[black] (2.4,3.0) circle (2pt); \node[anchor=north west, font=\scriptsize] at (2.52,2.94) {s, mu}; \fill[black] (3.0,4.65) circle (2pt); \node[anchor=north west, font=\scriptsize] at (3.12,4.59) {c}; \fill[black] (3.6,4.87) circle (2pt); \node[anchor=north west, font=\scriptsize] at (3.72,4.81) {tau}; \fill[black] (4.2,5.43) circle (2pt); \node[anchor=north west, font=\scriptsize] at (4.32,5.37) {b}; \fill[acc] (5.4,7.85) circle (2.6pt); \node[acc, anchor=north west, font=\scriptsize] at (5.54,7.79) {top}; \end{tikzpicture} $$ ## Summary The Higgs mechanism gives gauge bosons mass without breaking gauge invariance: a spontaneously broken gauge symmetry has no physical Goldstone bosons, and the would-be Goldstone modes become the longitudinal polarizations of the gauge fields. In the abelian model a single complex scalar makes the photon-like field massive, $m_A = qv$. In the electroweak theory a single Higgs doublet with vacuum $\langle\Phi\rangle = (0, v/\sqrt2)$ breaks $SU(2)_L \times U(1)_Y \to U(1)_{\text{EM}}$; three of its four scalars are eaten by the $W^\pm$ and $Z$, giving $M_W = gv/2$ and $M_Z = M_W/\cos\theta_W$, while the photon stays massless because the vacuum is electrically neutral. The fourth scalar is the Higgs boson, $M_h = \sqrt{2\lambda}\,v$, with $\lambda$ free. Fermions get mass from Yukawa couplings to the same doublet, $m_f = y_f v/\sqrt2$, so every mass is $v \approx 246\ \text{GeV}$ times a dimensionless coupling and the coupling of each fermion to the Higgs is proportional to its mass. What remains is to find the particle: the [discovery and properties of the Higgs boson](/particle-physics/electroweak-higgs/higgs-boson-discovery).[^gr-higgs][^tong-higgs] [^gr-higgs]: The abelian Higgs model, the electroweak doublet with $\langle\Phi\rangle = (0, v/\sqrt2)$, the gauge-boson masses $M_W = gv/2$, $M_Z = M_W/\cos\theta_W$, and the Yukawa origin of fermion masses $m_f = y_f v/\sqrt2$ are developed in Griffiths, §11.6–11.9; Thomson, §17.4–17.7; and Halzen & Martin, Ch. 15. [^tong-higgs]: Tong, _The Standard Model_ (Cambridge Part III), §2.3 and §5.2, gives the non-abelian Higgs mechanism, the degree-of-freedom counting, and the fermion-mass formula $m_X = y_X v/\sqrt2$, [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html). The electroweak scale $v = (\sqrt2\,G_F)^{-1/2} \approx 246\ \text{GeV}$ follows from the Fermi constant $G_F = 1.1663788 \times 10^{-5}\ \text{GeV}^{-2}$; the fermion masses $m_t \approx 173\ \text{GeV}$ down to $m_e = 0.511\ \text{MeV}$ are from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).