Spontaneous Symmetry Breaking and the Higgs Mechanism

Derivation

Spontaneous Symmetry Breaking and the Higgs Mechanism

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Statement

In an Abelian gauge theory with a complex scalar field whose potential has a degenerate ring of minima, the vacuum spontaneously breaks the local \(U(1)\) symmetry. Working in unitary gauge, the phase (Goldstone) degree of freedom is removed by a gauge transformation and reappears as the longitudinal polarization of the gauge boson, which acquires mass \( m_A = e v \), while the radial excitation becomes a massive scalar (the Higgs) of mass \( m_h = \sqrt{2\lambda}\,v \), with \( v \) the vacuum expectation value.

Why it matters

An explicit mass term \( \tfrac{1}{2} m_A^2 A_\mu A^\mu \) is forbidden by gauge invariance, yet the electroweak \(W\) and \(Z\) bosons are heavy (\(\sim 80\text{–}91\ \mathrm{GeV}\)). The Higgs mechanism resolves this: the gauge symmetry is exact in the Lagrangian but not realized by the vacuum, so the theory stays renormalizable while the gauge bosons are massive.

The same scalar field that gives mass to the gauge bosons also couples to fermions via Yukawa terms, so the mechanism underpins essentially all masses in the Standard Model. Its scalar excitation was confirmed as the \(125\ \mathrm{GeV}\) boson observed at the LHC in 2012.

Assumptions

The scalar potential has the symmetry-breaking form \(V=-\mu^2|\phi|^2+\lambda|\phi|^4\) with \(\mu^2,\lambda>0\).If \(\mu^2<0\) the minimum sits at \(\phi=0\), the symmetry is unbroken, and the gauge boson stays massless.

The gauge symmetry is local (gauged), with covariant derivative \(D_\mu=\partial_\mu-ieA_\mu\).If the symmetry were only global, breaking it produces a physical massless Goldstone boson (Goldstone's theorem) and no gauge-boson mass.

The scalar self-coupling is bounded below, \(\lambda>0\).If \(\lambda\le0\) the potential is unbounded below, no stable vacuum exists, and the expansion around a minimum is meaningless.

Renormalizability restricts \(V\) to operators of dimension \(\le 4\).Higher-dimension operators would introduce a UV cutoff dependence; the tree-level relations \(m_A=ev\), \(m_h=\sqrt{2\lambda}\,v\) then receive corrections and lose predictivity.

Unitary gauge is a legitimate gauge choice (the gauge orbit intersects the field configuration once).If dropped, the Goldstone field remains explicit; one must instead use \(R_\xi\) gauges with Faddeev–Popov ghosts, and the physical content is unchanged but obscured.

Derivation

\[ \mathcal{L}=\left(D_\mu\phi\right)^{*}\left(D^\mu\phi\right)-V(\phi)-\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu},\qquad D_\mu=\partial_\mu-ieA_\mu,\quad V=-\mu^2\phi^{*}\phi+\lambda(\phi^{*}\phi)^2 \]

Most general renormalizable, gauge-invariant Lagrangian for a charged scalar coupled to an Abelian gauge field; invariance under \(\phi\to e^{i\alpha(x)}\phi,\ A_\mu\to A_\mu+\tfrac{1}{e}\partial_\mu\alpha\) fixes the coupling (assumes gauge-invariance-qed-coupling). A

\[ \frac{\partial V}{\partial(\phi^{*}\phi)}=-\mu^2+2\lambda\,\phi^{*}\phi=0 \;\Longrightarrow\; |\phi|_{\min}^2=\frac{\mu^2}{2\lambda}\equiv\frac{v^2}{2},\qquad v\equiv\frac{\mu}{\sqrt{\lambda}} \]

Extremize the potential; the minimum is a circle \(|\phi|=v/\sqrt{2}\) in field space, not a point. A

\[ \phi_0=\frac{v}{\sqrt2}\,e^{i\theta}\quad\text{(any fixed }\theta),\qquad U(1):\ \phi_0\to e^{i\alpha}\phi_0\neq\phi_0 \]

Selecting one vacuum from the degenerate ring breaks the symmetry spontaneously: the Lagrangian is \(U(1)\)-invariant but the chosen vacuum is not. The current is still conserved (Noether), but the charge does not annihilate the vacuum (assumes noethers-theorem-conserved-currents). C

\[ \phi(x)=\frac{1}{\sqrt2}\bigl(v+h(x)\bigr)\,\exp\!\left(\frac{i\,\chi(x)}{v}\right) \]

Parametrize fluctuations in polar form: \(h\) is the radial (massive) mode, \(\chi\) the angular (Goldstone) mode along the flat direction of \(V\). B

\[ \alpha(x)=-\frac{\chi(x)}{v}\;\Longrightarrow\; \phi(x)\to\frac{1}{\sqrt2}\bigl(v+h(x)\bigr),\qquad A_\mu\to A_\mu-\frac{1}{ev}\partial_\mu\chi \]

Perform a gauge transformation that removes the phase entirely (unitary gauge). Legal because \(\chi\) is pure gauge; the Goldstone field is absorbed into \(A_\mu\), which is exactly its physical fate. C

\[ D_\mu\phi=\frac{1}{\sqrt2}\bigl(\partial_\mu h-ieA_\mu(v+h)\bigr),\qquad \left(D_\mu\phi\right)^{*}\!\left(D^\mu\phi\right)=\tfrac12(\partial_\mu h)^2+\tfrac12 e^2(v+h)^2 A_\mu A^\mu \]

Insert the unitary-gauge field into the kinetic term and multiply out; the cross terms cancel because \(h\) is real. B

\[ \tfrac12 e^2(v+h)^2A_\mu A^\mu=\underbrace{\tfrac12 e^2 v^2\,A_\mu A^\mu}_{\text{mass term}}+e^2 v\,h\,A_\mu A^\mu+\tfrac12 e^2 h^2 A_\mu A^\mu \]

Expand \((v+h)^2=v^2+2vh+h^2\). The \(v^2\) piece is a Proca mass term; comparison with \(\tfrac12 m_A^2 A_\mu A^\mu\) gives \(m_A^2=e^2v^2\). The remaining terms are \(hAA\) and \(hhAA\) interaction vertices. B

\[ V=-\tfrac12\lambda v^2(v+h)^2+\tfrac14\lambda(v+h)^4=\;\text{const}+\lambda v^2 h^2+\lambda v\,h^3+\tfrac14\lambda h^4 \]

Substitute \(\mu^2=\lambda v^2\) and \(|\phi|^2=\tfrac12(v+h)^2\); the linear term in \(h\) cancels (we expand about the minimum). The quadratic term \(\lambda v^2 h^2=\tfrac12 m_h^2 h^2\) gives \(m_h^2=2\lambda v^2\). B

\[ \underbrace{2}_{A_\mu\ \text{massless}}+\underbrace{2}_{\text{complex }\phi}\;=\;4\;=\;\underbrace{3}_{A_\mu\ \text{massive}}+\underbrace{1}_{h} \]

Degree-of-freedom bookkeeping: the would-be Goldstone becomes the longitudinal mode of the now-massive vector. No degrees of freedom are created or destroyed — they are rearranged. C

Result

\[ \boxed{\;m_A=e\,v,\qquad m_h=\sqrt{2\lambda}\,v,\qquad v=\frac{\mu}{\sqrt\lambda}\;}\]

Reading. A spontaneously broken local \(U(1)\) gives the gauge boson a mass equal to the gauge coupling times the vacuum expectation value. The Goldstone mode is not a physical particle here — it is eaten to provide the third (longitudinal) polarization the massive vector needs. What remains physical is one massive scalar \(h\) whose mass is set by the self-coupling \(\lambda\) and the same \(v\).

Units check. In natural units (\(\hbar=c=1\)) a mass has dimension of energy. The gauge coupling \(e\) is dimensionless and the vacuum expectation value \(v\) carries dimension of energy (it is a field value), so \(m_A=ev\) has dimension of energy. Likewise \(\lambda\) is dimensionless, so \(m_h=\sqrt{2\lambda}\,v\) has dimension of energy. Both are consistent with a mass. Restoring SI, \(m=ev/c^2\).

Limiting cases

\(e\to0\): the gauge coupling switches off, \(m_A\to0\), and the symmetry becomes global — the Goldstone \(\chi\) reappears as a genuine massless particle (Goldstone's theorem).
\(\lambda\to0\) at fixed \(v\): the Higgs becomes light, \(m_h\to0\), while \(m_A=ev\) is unaffected; the potential flattens and the vacuum becomes marginally stable.
\(\mu^2\to0^{+}\): \(v\to0\), symmetry restoration — both \(m_A\) and \(m_h\) vanish; this is the finite-temperature electroweak restoration limit.
\(h\to0\) (freeze the Higgs at its vev): only the mass terms survive, recovering a Proca theory of a massive vector with no residual gauge freedom.

Breaks when

Explicit breaking. If a term that breaks \(U(1)\) explicitly is added (e.g. a bare mass \(m_A^2 A_\mu A^\mu\) put in by hand, or a linear \(\phi\)-term), the clean separation into eaten-Goldstone plus Higgs fails and the theory generically loses renormalizability/unitarity at high energy.
Strong coupling / no perturbative vacuum. If \(\lambda\gtrsim\mathcal{O}(4\pi)\) the loop expansion around the classical minimum diverges; the tree relations \(m_A=ev,\ m_h=\sqrt{2\lambda}v\) receive uncontrolled corrections and \(v\) may not even be well defined (the symmetry may break dynamically instead).
Massless-vacuum degeneracy at \(\mu^2\le0\). The derivation assumes a nonzero \(v\); for \(\mu^2\le0\) the true minimum is \(\phi=0\), the expansion in step 4 is around a maximum/saddle, and every mass formula is invalid.
Non-Abelian, partial breaking. For a larger group where some generators leave the vacuum invariant, those directions stay massless; applying \(m=ev\) blindly to every gauge boson overcounts the massive ones.

Failure modes

"The Goldstone boson is a new physical particle." In the gauged theory it is not — it is a gauge artifact eaten by \(A_\mu\). It is physical only in the global (ungauged) case.
Sign error in the potential. Writing \(V=+\mu^2|\phi|^2+\lambda|\phi|^4\) with \(\mu^2>0\) gives an unbroken vacuum; the tachyonic sign \(-\mu^2\) (equivalently \(\mu^2<0\) in the \(+\mu^2\) convention) is what triggers breaking.
Using \(v/\sqrt2\) instead of \(v\) in \(m_A\). The mass is \(m_A=ev\); the factor \(1/\sqrt2\) lives in \(\langle\phi\rangle=v/\sqrt2\), and the two \(\tfrac{1}{\sqrt2}\)'s combine with the \((v+h)^2\) to leave \(m_A^2=e^2v^2\).
Dropping the \(hAA\) and \(hhAA\) vertices. Keeping only the mass term discards the very couplings that let the Higgs decay to gauge bosons (\(h\to WW,ZZ\)); the mechanism is not just a mass term.
Miscounting degrees of freedom. Forgetting that a massive vector has three polarizations (not two) makes the DOF ledger fail to balance.
Confusing \(v\) with \(\langle\phi\rangle\). In the SM \(v\approx246\ \mathrm{GeV}\) is defined so \(\langle\phi\rangle=v/\sqrt2\approx174\ \mathrm{GeV}\); plugging the wrong one into Yukawa or mass formulas gives a \(\sqrt2\) error.

Discussion

The deepest point is that no symmetry is actually broken in the operator sense: the Lagrangian and its Noether current remain exactly gauge invariant. What is "broken" is the realization of the symmetry on the vacuum — the ground state is not annihilated by the charge. This is why the Higgs mechanism does not spoil renormalizability, unlike an ad hoc Proca mass: 't Hooft and Veltman showed the theory is renormalizable precisely because the underlying gauge invariance survives, made manifest in \(R_\xi\) gauges where the Goldstone lives on as an unphysical field cancelled by ghosts.

The counting of eaten Goldstones is controlled by the coset structure. For a group \(G\) broken to a subgroup \(H\), the number of broken generators is \(\dim G-\dim H\); each broken generator supplies one Goldstone that is eaten by one gauge boson, which thereby becomes massive. In the Standard Model \(SU(2)_L\times U(1)_Y\to U(1)_{\mathrm{em}}\): three of the four generators are broken, so \(W^\pm\) and \(Z\) gain mass while the photon (the unbroken direction) stays massless — exactly the observed spectrum.

The mechanism ties several masses to one scale \(v\). At tree level \(m_W=\tfrac12 g v\), \(m_Z=\tfrac12\sqrt{g^2+g'^2}\,v\), giving the custodial relation \(m_W=m_Z\cos\theta_W\) with \(\cos\theta_W=g/\sqrt{g^2+g'^2}\). Fermion masses follow from Yukawa couplings, \(m_f=y_f v/\sqrt2\). The single measured value \(v\approx246\ \mathrm{GeV}\) (fixed by the Fermi constant, \(v=(\sqrt2\,G_F)^{-1/2}\)) therefore sets the scale of the entire massive spectrum.

A subtlety often glossed over: "spontaneous breaking of a gauge symmetry" is, strictly, a slight abuse of language — Elitzur's theorem forbids a genuine local-symmetry order parameter, since a gauge-noninvariant \(\langle\phi\rangle\) is gauge-dependent and averages to zero without gauge fixing. What is physically well defined is the gauge-invariant statement: a Higgs phase characterized by a mass gap for the vector, versus a Coulomb phase. The polar parametrization and unitary gauge are a convenient bookkeeping that makes the gauge-invariant spectrum (\(m_A\), \(m_h\)) transparent, and those masses are the true observables.

Common misconceptions. The Higgs field does not "create mass from nothing" or act as a literal drag/aether — inertial mass here is the pole of the propagator generated by the coupling \(e^2v^2 A^2\). And the discovered \(125\ \mathrm{GeV}\) boson is the radial excitation \(h\), not the field's vacuum value \(v\); the vev is not a particle.

Worked examples

Example 1 — Abelian Higgs: gauge-boson and scalar mass from \((e,v,\lambda)\).

\[ m_A=e\,v \]

Result formula for the vector mass. A

\[ e=0.30,\quad v=246\ \mathrm{GeV}\;\Rightarrow\; m_A=0.30\times246\ \mathrm{GeV}=73.8\ \mathrm{GeV} \]

Insert numbers; \(e\) dimensionless, so \(m_A\) inherits the GeV of \(v\). A

\[ m_h=\sqrt{2\lambda}\,v,\qquad \lambda=0.13\;\Rightarrow\; m_h=\sqrt{0.26}\times246\ \mathrm{GeV}=0.510\times246\ \mathrm{GeV} \]

Same \(v\) feeds the scalar mass; \(\lambda\) chosen to reproduce the physical value. B

\[ m_A\approx73.8\ \mathrm{GeV},\qquad m_h\approx125\ \mathrm{GeV} \]

Reading. With a modest gauge coupling and the electroweak scale \(v\), the vector lands near the \(W\)/\(Z\) range and \(\lambda\approx0.13\) reproduces the observed Higgs mass.

Units check. \(\mathrm{(dimensionless)\times GeV=GeV}\) for both — masses, as required.

Example 2 — Standard Model: weak mixing angle from \(m_W\) and \(m_Z\).

\[ m_W=\tfrac12 g\,v,\qquad m_Z=\tfrac12\sqrt{g^2+g'^2}\,v\;\Rightarrow\;\cos\theta_W=\frac{m_W}{m_Z} \]

The custodial (tree-level, \(\rho=1\)) relation follows from the \(SU(2)\times U(1)\) mass matrix; \(v\) cancels in the ratio. B

\[ m_W=80.4\ \mathrm{GeV},\quad m_Z=91.2\ \mathrm{GeV}\;\Rightarrow\;\cos\theta_W=\frac{80.4}{91.2}=0.8816 \]

Insert measured masses. A

\[ \sin^2\theta_W=1-\cos^2\theta_W=1-0.7772=0.2228,\qquad \theta_W=\arccos(0.8816)=28.2^{\circ} \]

Trig identity; extract the mixing angle. A

\[ \sin^2\theta_W\approx0.223,\qquad \theta_W\approx28.2^{\circ} \]

Reading. The tree-level electroweak masses alone predict the mixing angle; the value \(\sin^2\theta_W\approx0.22\) agrees well with the independent \(Z\)-pole measurement, a nontrivial confirmation of the mechanism.

Units check. A ratio of masses is dimensionless, and \(\theta_W\) is an angle — both dimensionless, as they must be.

Problems

Given \(\mu=176\ \mathrm{GeV}\) and \(\lambda=0.51\), compute the vacuum expectation value \(v\).
Solution
\(v=\mu/\sqrt\lambda=176/\sqrt{0.51}=176/0.714=246\ \mathrm{GeV}\). This is the electroweak scale.
An Abelian Higgs model has \(e=0.65\) and \(v=246\ \mathrm{GeV}\). Find the gauge-boson mass, and compare with the \(SM\) \(W\) mass \(m_W=\tfrac12 gv\) for \(g=0.65\).
Solution
Abelian: \(m_A=ev=0.65\times246=160\ \mathrm{GeV}\). The SM \(W\) carries an extra \(\tfrac12\) from the \(SU(2)\) doublet structure: \(m_W=\tfrac12(0.65)(246)=80.0\ \mathrm{GeV}\), matching the measured \(80.4\ \mathrm{GeV}\). The factor of two arises because only the lower doublet component gets the vev.
The physical Higgs mass is \(m_h=125\ \mathrm{GeV}\) and \(v=246\ \mathrm{GeV}\). Extract the quartic self-coupling \(\lambda\).
Solution
\(m_h^2=2\lambda v^2\Rightarrow\lambda=\dfrac{m_h^2}{2v^2}=\dfrac{125^2}{2\times246^2}=\dfrac{15625}{121032}=0.129\). So \(\lambda\approx0.13\), a weak coupling.
An \(SU(2)\) gauge symmetry is completely broken by a scalar in the fundamental representation. How many Goldstone bosons are eaten, how many gauge bosons become massive, and how many physical scalar degrees of freedom remain if the scalar is a complex doublet (4 real fields)?
Solution
\(\dim SU(2)=3\), fully broken so \(3\) broken generators \(\Rightarrow\) \(3\) Goldstones eaten and \(3\) gauge bosons become massive. Of the 4 real scalar components, 3 are eaten, leaving \(4-3=1\) physical Higgs scalar. DOF check: before, \(3\times2+4=10\); after, \(3\times3+1=10\). Balanced.
For the \(SM\), \(g=0.653\) and \(g'=0.350\). Compute \(m_W\), \(m_Z\), and verify \(m_W=m_Z\cos\theta_W\), taking \(v=246\ \mathrm{GeV}\).
Solution
\(m_W=\tfrac12 gv=\tfrac12(0.653)(246)=80.3\ \mathrm{GeV}\). \(\sqrt{g^2+g'^2}=\sqrt{0.4264+0.1225}=\sqrt{0.5489}=0.7409\), so \(m_Z=\tfrac12(0.7409)(246)=91.1\ \mathrm{GeV}\). Check: \(\cos\theta_W=g/\sqrt{g^2+g'^2}=0.653/0.7409=0.8814\), and \(m_Z\cos\theta_W=91.1\times0.8814=80.3\ \mathrm{GeV}=m_W\). The custodial relation holds at tree level.