The Higgs mechanism: spontaneously broken gauge symmetry

Intuition Beginner

Some force-carrying particles are massless, like the photon, and some are heavy, like the $W$ and $Z$ particles that carry the weak force. A massless carrier reaches across the whole universe; a heavy carrier only reaches a tiny distance, which is why the weak force feels weak. The puzzle of the 1960s was that the most elegant theories of forces seemed to insist that every force carrier must be massless. The Higgs mechanism is the answer to how a force carrier can be heavy without spoiling the elegance.

The picture to hold is a field that fills all of space, the same everywhere, never quite zero. Think of a flooded plain seen from above: the water sits at one level everywhere, and you cannot tell one spot from another by its water height. A force carrier moving through this filled space is like a person wading through water rather than walking through air. The water slows them, gives them inertia, makes them act heavy. The carrier was massless on its own, but moving through the ever-present field it behaves as a massive particle.

Why bother with this idea? Because it lets you keep a beautiful symmetric theory and still get heavy particles out of it. The symmetry is real, but the particular state the universe settled into hides it, the way a flooded plain hides the shape of the ground. This trick, a symmetric law with an unsymmetric resting state, is what spontaneous symmetry breaking means, and the Higgs mechanism is what happens when the broken symmetry is a gauge symmetry.

Visual Beginner

Picture the energy of the Higgs field drawn as a surface, with the field value along the bottom and the energy going up. For an ordinary field the surface is a simple bowl with its lowest point at zero. For the Higgs field the surface is a wine bottle's punt, or a sombrero: a bump in the very centre and a circular valley around it. The lowest-energy place is not the centre but somewhere down in the ring.

The universe settles into one point of the ring. Two directions matter at that point. Rolling up the inner wall of the valley costs energy, and that stiff direction is the Higgs particle. Rolling around the flat valley floor costs nothing, and that flat direction would normally be a massless particle. When the symmetry is a gauge symmetry, that flat direction is not a separate particle at all: it gets absorbed by the force carrier, which becomes heavy.

Worked example Beginner

Take a force carrier that is massless on its own and let it interact with the Higgs field whose resting value in the valley is the number $v=2$ (in some chosen units), with interaction strength $e=3$. The rule the full theory gives for the carrier's mass is short: the mass equals the interaction strength times the resting field value, $m=e\times v$.

Step 1. Write down the two ingredients. The resting value of the field is $v=2$. The interaction strength is $e=3$.

Step 2. Apply the rule. The carrier's mass is the product, $m=e\times v=3\times 2=6$.

Step 3. Sanity-check the limit. Suppose the field had no resting value, $v=0$, as for an ordinary field sitting at the bottom of a plain bowl. Then $m=3\times 0=0$, and the carrier stays massless. The mass appears only because the field rests at a nonzero value.

Step 4. Count the moving parts before and after. Before: a massless carrier has $2$ ways to wiggle (two sideways polarisations), and the complex Higgs field has $2$ independent pieces, for $4$ in total. After: a massive carrier has $3$ ways to wiggle (it gained a forward-backward polarisation), and one Higgs particle remains, for $3+1=4$. The total is unchanged.

What this tells us: the carrier gets its mass directly from the field's resting value, and nothing is lost in the bookkeeping. The flat valley-floor direction did not vanish; it turned into the carrier's extra, third way of wiggling. A massless thing with two wiggles plus a flat direction became a massive thing with three wiggles.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, the metric is mostly-minus, ${\eta }_{\mu \nu }=\mathrm{diag}(+,-,-,-)$, matching the convention fixed in 12.05.06. The Abelian Higgs model is the theory of a $U(1)$ gauge field $A_{\mu }$ coupled to a single complex scalar $\varphi$ with Lagrangian density

$$\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+(D_{\mu }\varphi {)}^{\ast }(D^{\mu }\varphi )-V(\varphi ),V(\varphi )=-{\mu }^2{\varphi }^{\ast }\varphi +\lambda ({\varphi }^{\ast }\varphi {)}^2,$$

where $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ is the field strength of 03.07.29, the gauge-covariant derivative is $D_{\mu }\varphi =({\partial }_{\mu }-ie{A}_{\mu })\varphi$, and the parameters satisfy ${\mu }^2>0$ and $\lambda >0$. The Lagrangian is invariant under the local $U(1)$ gauge transformation

$$\varphi (x)\mapsto e^{ie\alpha (x)}\varphi (x),A_{\mu }(x)\mapsto A_{\mu }(x)+{\partial }_{\mu }\alpha (x),$$

which is the defining gauge symmetry of 03.07.05 specialised to the Abelian group acting on a charged scalar.

The sign $-{\mu }^2<0$ of the quadratic term is what distinguishes this from an ordinary massive scalar. The potential $V$ is minimised not at $\varphi =0$ but on the circle of minima $|\varphi {|}^2={\mu }^2/(2\lambda )=:v^2/2$. The constant $v=\mu /\sqrt{\lambda }$ is the vacuum expectation value. A choice of one point on this circle — conventionally ${\varphi }_0=v/\sqrt{2}$, taken real — is a choice of vacuum, and the residual symmetry fixing that point is broken, in the sense of 08.02.02.

Write the fluctuation about the chosen vacuum in polar (Higgs) variables

$$\varphi (x)=\frac{1}{\sqrt{2}}(v+h(x))e^{i\chi (x)/v},$$

with $h$ the real radial field and $\chi$ the real angular (would-be Goldstone) field. The combination $\chi$ is the flat direction along the valley floor; $h$ is the stiff radial direction. The unitary gauge is the gauge choice $\alpha (x)=-\chi (x)/(ev)$ that removes $\chi$ entirely, setting $\varphi =(v+h)/\sqrt{2}$ real. The general $R_{\xi }$ gauge instead retains $\chi$ and adds the gauge-fixing term ${\mathcal{L}}_{\mathrm{gf}}=-\frac{1}{2\xi }({\partial }_{\mu }{A}^{\mu }-\xi ev\chi {)}^2$, with gauge parameter $\xi \in (0,\infty )$; $\xi \to \infty$ recovers the unitary gauge and $\xi =1$ is the Feynman-'t Hooft gauge.

For a general compact gauge group $G$ with generators $T_a$ acting on a real scalar multiplet $\varphi$ whose vacuum is ${\varphi }_0$, the gauge-boson mass matrix is

$$(M^2{)}_{ab}=g^2⟨T_a{\varphi }_0,T_b{\varphi }_0⟩,$$

a real symmetric positive-semidefinite matrix on the Lie algebra. Its kernel is the Lie algebra of the unbroken subgroup $H=\{U\in G:U{\varphi }_0={\varphi }_0\}$: gauge bosons in unbroken directions stay massless; gauge bosons in broken directions acquire mass.

Counterexamples to common slips

• A negative quadratic term is necessary but not sufficient. With $-{\mu }^2<0$ but $\lambda \le 0$ the potential is unbounded below and has no vacuum; boundedness requires $\lambda >0$. The symmetric point $\varphi =0$ is then a local maximum, not a minimum, of $V$.
• Breaking a global $U(1)$ produces a genuine massless Goldstone boson (the angular field $\chi$), per 08.02.02 and Goldstone-Salam-Weinberg. The mode disappears only when the broken symmetry is gauged: the local invariance lets a gauge transformation absorb $\chi$. Gauging is the essential extra ingredient.
• Not every broken generator yields a massive gauge boson if the corresponding direction is global. In a theory with both gauged and ungauged broken generators, the ungauged ones leave physical Goldstone bosons; only the gauged broken generators eat their would-be Goldstones.

Key derivation Intermediate+

Theorem (Abelian Higgs mass spectrum). In the Abelian Higgs model with ${\mu }^2,\lambda >0$, expansion about the vacuum ${\varphi }_0=v/\sqrt{2}$, $v=\mu /\sqrt{\lambda }$, produces a massive vector field of mass $m_A=ev$, a real scalar (the Higgs boson) of mass $m_H=\sqrt{2\lambda }v=\sqrt{2}\mu$, and no physical massless scalar. The total physical degrees of freedom — three for the massive vector plus one for the Higgs — equal the four of the original complex scalar plus massless vector.

Proof. Work in unitary gauge, where the gauge transformation $\alpha =-\chi /(ev)$ has been used to set $\varphi =(v+h)/\sqrt{2}$ with $h$ real. The potential becomes, dropping the additive constant $V({\varphi }_0)$,

$$V=-{\mu }^2\cdot \frac{1}{2}(v+h{)}^2+\lambda \cdot \frac{1}{4}(v+h{)}^4.$$

Using ${\mu }^2=\lambda v^2$, the linear term in $h$ cancels (the vacuum is stationary), and the quadratic term is $\frac{1}{2}(2\lambda v^2)h^2+\mathcal{O}(h^3)$. The radial field therefore has mass-squared

$$m_H^2=2\lambda v^2=2{\mu }^2,m_H=\sqrt{2\lambda }v=\sqrt{2}\mu .$$

The cubic and quartic self-interactions $\lambda v{h}^3+\frac{\lambda }{4}{h}^4$ remain as the Higgs self-couplings.

Now expand the covariant kinetic term. With $\varphi =(v+h)/\sqrt{2}$ real,

$$(D_{\mu }\varphi {)}^{\ast }(D^{\mu }\varphi )=\frac{1}{2}({\partial }_{\mu }h)({\partial }^{\mu }h)+\frac{1}{2}{e}^2(v+h{)}^2{A}_{\mu }{A}^{\mu }.$$

The term quadratic in the fields, with $h$ set to its vacuum value, is

$$\frac{1}{2}{e}^2{v}^2{A}_{\mu }{A}^{\mu }.$$

This is precisely a Proca mass term of the form $\frac{1}{2}{m}_A^2{A}_{\mu }{A}^{\mu }$ analysed in 12.05.06, with

$$m_A=ev.$$

The gauge field has become massive. The cross terms $e^{2}vh{A}_{\mu }{A}^{\mu }+\frac{1}{2}{e}^2{h}^2{A}_{\mu }{A}^{\mu }$ are the Higgs-gauge interactions; they have no bilinear part and so do not affect the masses.

The angular field $\chi$ has been removed by the gauge choice; it does not appear in the unitary-gauge Lagrangian, so there is no physical massless scalar. Counting: the original theory had a massless $U(1)$ vector (2 polarisations) and a complex scalar (2 real fields), total 4. The broken-phase theory has a massive vector (3 polarisations) and one real Higgs scalar, total 4. The would-be Goldstone mode $\chi$ supplied the longitudinal polarisation that turned the 2-polarisation massless vector into a 3-polarisation massive one. $\square$

Bridge. This computation builds toward the full Standard Model electroweak sector and appears again in 03.07.31, where the same gauge redundancy that lets us delete $\chi$ in unitary gauge is the redundancy that Faddeev-Popov ghosts compensate for in covariant gauges. The foundational reason a gauge boson can gain mass without breaking gauge invariance is exactly that the longitudinal polarisation it acquires is not a new degree of freedom but the relabelled would-be Goldstone mode; this is the central insight of the mechanism, and it is why the degree-of-freedom count is conserved. The bridge is the recognition that the unitary-gauge mass term $\frac{1}{2}{e}^2{v}^2{A}_{\mu }{A}^{\mu }$ and the $R_{\xi }$-gauge Proca-plus-Goldstone description are the same physics in different gauges: putting these together, the gauge-dependent would-be-Goldstone mass $\sqrt{\xi }{m}_A$ and the gauge-independent physical pole at $k^2=m_A^2$ describe one massive vector boson. The same pattern recurs in the non-Abelian case, where the mass matrix $(M^2{)}_{ab}=g^2⟨T_a{\varphi }_0,T_b{\varphi }_0⟩$ generalises the single number $e^2{v}^2$ and its kernel identifies the unbroken subgroup whose gauge bosons stay massless.

Exercises Intermediate+

Advanced results Master

The single-number Abelian result generalises to an arbitrary compact gauge group with a multiplet of scalars, and the gauge-dependence of the off-shell quantities is controlled by the $R_{\xi }$ family. The following results assemble the full classical mass spectrum and the physically central renormalisability statement.

Proposition (general gauge-boson mass matrix). Let $G$ be a compact Lie group acting on a real scalar multiplet $\varphi$ through anti-symmetric generators $T_a$ (in a real orthogonal basis), with gauge coupling $g$ and Lagrangian $\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }^a{F}^{a\mu \nu }+\frac{1}{2}(D_{\mu }\varphi {)}^T(D^{\mu }\varphi )-V(\varphi )$, $D_{\mu }\varphi =({\partial }_{\mu }-g{A}_{\mu }^a{T}_a)\varphi$. If $V$ is minimised at ${\varphi }_0$, the gauge-boson mass matrix is $(M^2{)}_{ab}=g^2(T_a{\varphi }_0{)}^T(T_b{\varphi }_0)$, with kernel the Lie algebra of the stabiliser $H=\{U\in G:U{\varphi }_0={\varphi }_0\}$.

The eigenvectors with nonzero eigenvalue are the broken directions; their gauge bosons acquire the corresponding masses. The unbroken subgroup $H$ has massless gauge bosons. In the electroweak case $G=SU(2{)}_L\times U(1{)}_Y$ with a complex scalar doublet of hypercharge $\frac{1}{2}$, the stabiliser of the vacuum is the diagonal $U(1{)}_{\mathrm{em}}$, so three of the four gauge bosons ($W^{\pm },Z$) become massive and one (the photon) stays massless. The masses are $m_W=\frac{1}{2}gv$ and $m_Z=\frac{1}{2}\sqrt{g^2+g^{\prime 2}}v=m_W/\cos {\theta }_W$, where $\tan {\theta }_W=g^{\prime }/g$ and the photon/$Z$ are the orthogonal combinations of the neutral $W^3$ and $B$ fields diagonalising the neutral mass matrix.

Proposition ($R_{\xi }$ propagator and gauge-independence of poles). In $R_{\xi }$ gauge the massive-vector propagator is

$$D_{\mu \nu }(k)=\frac{-i}{k^2-m_A^2}\left({\eta }_{\mu \nu }-(1-\xi )\frac{k_{\mu }{k}_{\nu }}{k^2-\xi m_A^2}\right),$$

the would-be Goldstone has propagator $i/(k^2-\xi m_A^2)$ and the Faddeev-Popov ghost has mass-squared $\xi m_A^2$. The physical pole at $k^2=m_A^2$ and its residue structure are independent of $\xi$; the spurious poles at $k^2=\xi m_A^2$ cancel between the longitudinal vector, the Goldstone, and the ghost in any $S$-matrix element.

In the limit $\xi \to \infty$ the propagator collapses to the unitary-gauge Proca form $-i({\eta }_{\mu \nu }-k_{\mu }{k}_{\nu }/m_A^2)/(k^2-m_A^2)$ of 12.05.06, where the spurious poles are pushed to infinity and the physical spectrum is manifest but the high-energy $k_{\mu }{k}_{\nu }/m_A^2$ behaviour obscures power counting. In the gauge $\xi =1$ (Feynman-'t Hooft) the propagator is simply $-i{\eta }_{\mu \nu }/(k^2-m_A^2)$, manifestly well-behaved at large $k$, at the cost of explicit Goldstone and ghost fields. The two descriptions agree on every physical amplitude.

Theorem ('t Hooft-Veltman renormalisability). A spontaneously broken gauge theory, quantised in an $R_{\xi }$ gauge, is renormalisable: divergences are absorbed by a finite set of counterterms preserving the (BRST) structure, and the renormalised $S$-matrix is unitary and $\xi$-independent.

The proof rests on the Slavnov-Taylor identities (the Ward identities of the residual BRST symmetry of 03.07.31) constraining the divergent structure, and on the $R_{\xi }$ gauge's manifest power-counting renormalisability at finite $\xi$. The unitary gauge alone would suggest non-renormalisability because of the $k_{\mu }{k}_{\nu }/m_A^2$ growth; the $R_{\xi }$ family shows that this growth is a gauge artefact. Reconciling manifest unitarity (unitary gauge) with manifest renormalisability (Feynman gauge) is the achievement, completed by 't Hooft 1971 and 't Hooft-Veltman 1972.

Proposition (Anderson's superconducting precedent). In a superconductor the electromagnetic gauge field acquires a finite London penetration depth ${\lambda }_L$, equivalently a photon mass $m_{\gamma }=1/{\lambda }_L$, because the charged condensate breaks the electromagnetic $U(1)$. The Meissner expulsion of magnetic field is the static, non-relativistic instance of gauge-boson mass generation; the longitudinal plasmon is the absorbed would-be Goldstone mode. Anderson identified this in 1963, before the relativistic field-theory formulation.

Synthesis. The foundational reason a gauge boson can be massive is that the would-be Goldstone mode of the broken gauge symmetry is exactly the longitudinal polarisation the massive vector needs, and this is the central insight that conserves the degree-of-freedom count across the transition. Putting these together, the Abelian single-number result $m_A=ev$, the general mass matrix $(M^2{)}_{ab}=g^2⟨T_a{\varphi }_0,T_b{\varphi }_0⟩$ whose kernel is the unbroken subalgebra, and the electroweak spectrum $m_W=\frac{1}{2}gv$, $m_Z=m_W/\cos {\theta }_W$ are one phenomenon: gauge bosons in broken directions eat their Goldstones and become massive, those in unbroken directions stay massless. The $R_{\xi }$-gauge propagator builds toward the renormalisability proof: it is dual to the unitary-gauge Proca description in the precise sense that the same physical pole at $k^2=m_A^2$ governs both, while the spurious $\xi$-dependent poles cancel in observables. This is exactly the gauge redundancy that appears again in 03.07.31, where Faddeev-Popov ghosts and BRST cohomology organise the cancellation that the $R_{\xi }$ Goldstone-ghost pairing realises here, and it generalises the Anderson superconducting mass to the relativistic, non-Abelian setting that is the spine of the Standard Model.

Full proof set Master

Proposition (general gauge-boson mass matrix), proof. Expand the covariant kinetic term $\frac{1}{2}(D_{\mu }\varphi {)}^T(D^{\mu }\varphi )$ about the vacuum $\varphi ={\varphi }_0+\tilde{\varphi }$. The covariant derivative is $D_{\mu }\varphi ={\partial }_{\mu }\tilde{\varphi }-g{A}_{\mu }^a{T}_a({\varphi }_0+\tilde{\varphi })$. The terms quadratic in the gauge field, evaluated at the vacuum (set $\tilde{\varphi }=0$ in the gauge-field-bilinear part), are

$$\frac{1}{2}{g}^2{A}_{\mu }^a{A}^{b\mu }(T_a{\varphi }_0{)}^T(T_b{\varphi }_0)=\frac{1}{2}(M^2{)}_{ab}{A}_{\mu }^a{A}^{b\mu },(M^2{)}_{ab}=g^2(T_a{\varphi }_0{)}^T(T_b{\varphi }_0).$$

This is a real symmetric matrix, and positive-semidefinite since for any real vector $c_a$,

$$c_a(M^2{)}_{ab}{c}_b=g^2\Vert ({\sum }_a{c}_a{T}_a){\varphi }_0{\Vert }^2\ge 0.$$

Equality holds iff $({\sum }_a{c}_a{T}_a){\varphi }_0=0$, i.e. iff ${\sum }_a{c}_a{T}_a$ lies in the Lie algebra of the stabiliser $H=\{U:U{\varphi }_0={\varphi }_0\}$ (differentiating $U(t){\varphi }_0={\varphi }_0$ at $t=0$ for a one-parameter subgroup $U(t)=\exp (t{\sum }_a{c}_a{T}_a)$ gives the kernel condition). Hence $\ker M^2=\mathfrak{h}$, the gauge bosons of $H$ are massless, and the broken generators (the orthogonal complement) carry the positive eigenvalues. $\square$

Proposition ($R_{\xi }$ propagator), proof. In $R_{\xi }$ gauge the quadratic gauge Lagrangian, after the gauge-fixing term cancels the $A$–$\chi$ mixing (Exercise 8), is

$${\mathcal{L}}_A^{(2)}=\frac{1}{2}{A}_{\mu }\left[\right({\partial }^2+m_A^2){\eta }^{\mu \nu }-(1-\frac{1}{\xi }\left){\partial }^{\mu }{\partial }^{\nu }\right]A_{\nu },$$

reading off, in momentum space, the inverse propagator $(D^{-1}{)}_{\mu \nu }=-(k^2-m_A^2){\eta }_{\mu \nu }+(1-\frac{1}{\xi })k_{\mu }{k}_{\nu }$. Invert by the ansatz $D_{\mu \nu }=a{\eta }_{\mu \nu }+b{k}_{\mu }{k}_{\nu }$ and impose $(D^{-1}{)}_{\mu \rho }{D}^{\rho }{}_{\nu }=i{\delta }_{\nu }^{\mu }$. Splitting into transverse and longitudinal projectors gives $a=-i/(k^2-m_A^2)$ on the transverse part and a longitudinal coefficient fixed by the $1/\xi$ term; collecting,

$$D_{\mu \nu }(k)=\frac{-i}{k^2-m_A^2}\left({\eta }_{\mu \nu }-(1-\xi )\frac{k_{\mu }{k}_{\nu }}{k^2-\xi m_A^2}\right).$$

The transverse pole sits at $k^2=m_A^2$ with $\xi$-independent residue $-i({\eta }_{\mu \nu }-k_{\mu }{k}_{\nu }/m_A^2)$ on the physical subspace; the longitudinal piece has a pole at $k^2=\xi m_A^2$ whose residue is cancelled in $S$-matrix elements by the would-be-Goldstone propagator $i/(k^2-\xi m_A^2)$ and the ghost loop of equal mass-squared, by the Slavnov-Taylor identity. Taking $\xi \to \infty$ kills the second term's $k$-growth via the $\xi m_A^2$ in the denominator and reproduces the Proca propagator of 12.05.06; taking $\xi =1$ gives $-i{\eta }_{\mu \nu }/(k^2-m_A^2)$. $\square$

Theorem ('t Hooft-Veltman renormalisability), stated with proof outline — see 't Hooft 1971 Nucl. Phys. B 35, 167 ^{[source pending]} and 't Hooft-Veltman 1972. At finite $\xi$ every propagator falls as $1/k^2$ at large $k$, so the theory is power-counting renormalisable by the same superficial-degree-of-divergence analysis as an unbroken gauge theory. The residual BRST symmetry of 03.07.31 yields Slavnov-Taylor identities that constrain the counterterms to renormalise the original gauge-invariant structure (a finite set: field strengths, the scalar potential, the gauge coupling, and the vacuum value), preserving unitarity. Gauge-independence of physical quantities follows because $\partial S/\partial \xi$ is a BRST-exact insertion, which vanishes between physical (BRST-closed) states. The full proof requires the BRST machinery and is carried out in the cited papers.

Proposition (Anderson superconducting precedent), proof. In the Ginzburg-Landau description a charged condensate $\psi$ with $|\psi {|}^2=n_s$ couples to the electromagnetic potential through $|(\nabla -ie\mathbf{A})\psi {|}^2$. In the broken phase the bilinear gauge term is $e^2{n}_s|\mathbf{A}{|}^2$, a photon mass-squared $m_{\gamma }^2=e^2{n}_s=1/{\lambda }_L^2$ with ${\lambda }_L$ the London penetration depth. Maxwell's equation in the condensate becomes ${\nabla }^2\mathbf{B}={\lambda }_L^{-2}\mathbf{B}$, whose solutions decay exponentially on the scale ${\lambda }_L$: this is the Meissner effect, magnetic-field expulsion, the static incarnation of a massive photon. The longitudinal plasmon is the absorbed Goldstone phase of $\psi$. The relativistic Abelian Higgs model is the Lorentz-covariant completion of this static computation. $\square$

Connections Master

• Spontaneous symmetry breaking 08.02.02. The Higgs mechanism is spontaneous symmetry breaking applied to a gauged symmetry. The order-parameter and broken-vacuum structure of 08.02.02 — a symmetric Lagrangian with an asymmetric vacuum on a manifold of degenerate minima — is the input. The new content here is that gauging the broken symmetry converts the would-be Goldstone bosons of 08.02.02 into longitudinal gauge-boson polarisations rather than physical massless scalars.

• Yang-Mills action 03.07.05. The non-Abelian generalisation uses the Yang-Mills Lagrangian and covariant derivative of 03.07.05; the gauge-boson mass matrix $(M^2{)}_{ab}=g^2⟨T_a{\varphi }_0,T_b{\varphi }_0⟩$ is built from the same generators $T_a$ that define the Yang-Mills covariant derivative. The Higgs mechanism is what gives mass to the otherwise-massless Yang-Mills gauge bosons of 03.07.05 without spoiling the gauge structure that makes the theory renormalisable.

• BRST cohomology and Faddeev-Popov ghosts 03.07.31. The $R_{\xi }$-gauge quantisation that exhibits the renormalisability of the broken theory uses exactly the Faddeev-Popov ghost sector of 03.07.31; the would-be Goldstone and the ghost in $R_{\xi }$ gauge carry the same mass-squared $\xi m_A^2$ and cancel in physical amplitudes precisely by the BRST Slavnov-Taylor identities of 03.07.31. The gauge redundancy that lets us delete $\chi$ in unitary gauge is the redundancy BRST cohomology organises.

• Free Maxwell / massive vector fields 12.05.06. The end product of the Higgs mechanism is a massive vector boson, whose free theory is the Proca field of 12.05.06; the unitary-gauge propagator obtained here is exactly the Proca propagator $-i({\eta }_{\mu \nu }-k_{\mu }{k}_{\nu }/m_A^2)/(k^2-m_A^2)$ of 12.05.06. The Higgs mechanism is the dynamical origin of the Proca mass term that 12.05.06 takes as given, and it explains how that mass term is compatible with an underlying gauge symmetry.

• Electromagnetism as $U(1)$ Yang-Mills 03.07.29. The Abelian Higgs model is the gauged-and-broken version of the $U(1)$ gauge theory of 03.07.29; in the unbroken phase the photon of 03.07.29 is massless, and the Higgs mechanism is what would make it massive if the $U(1)$ were spontaneously broken — which is exactly what happens to electromagnetism inside a superconductor (Anderson 1963).

Historical & philosophical context Master

The mechanism has a tangled and much-litigated origin in 1963–64. Philip Anderson, working in condensed-matter physics, observed in 1963 that the photon inside a superconductor is effectively massive — the Meissner effect and finite penetration depth — precisely because the superconducting condensate breaks the electromagnetic gauge symmetry, and that the would-be Goldstone mode is absorbed into the longitudinal plasmon (Anderson, Phys. Rev. 130, 439, 1963) ^{[source pending]}. Anderson conjectured that the same evasion of Goldstone's theorem would hold relativistically, but did not provide a relativistic field-theory model. The relativistic demonstration came in three nearly-simultaneous 1964 papers: Englert and Brout (Phys. Rev. Lett. 13, 321, 1964), Higgs (Phys. Rev. Lett. 13, 508, 1964, with a companion in Phys. Lett. 12, 132), and Guralnik, Hagen, and Kibble (Phys. Rev. Lett. 13, 585, 1964). Higgs's paper was the one that explicitly exhibited the surviving massive scalar excitation that now bears his name; the Guralnik-Hagen-Kibble paper gave the most careful account of why Goldstone's theorem (proved relativistically by Goldstone, Salam, and Weinberg in Phys. Rev. 127, 965, 1962) is evaded — the relevant current is not a genuine conserved current in a covariant gauge, so the Goldstone-boson existence argument fails.

The mechanism was a classical curiosity until 't Hooft proved, in his 1971 thesis work (Nucl. Phys. B 35, 167, 1971) ^{[source pending]}, that spontaneously broken gauge theories are renormalisable, using the $R_{\xi }$ gauge family that interpolates between the manifestly-unitary unitary gauge and the manifestly-renormalisable Feynman-type gauges. Fujikawa, Lee, and Sanda (Phys. Rev. D 6, 2923, 1972) systematised the continuous-$\xi$ family and demonstrated the $\xi$-independence of physical poles. 't Hooft's renormalisability proof is what made the Glashow-Weinberg-Salam electroweak unification a predictive quantum field theory rather than a formal Lagrangian, and it is the result that earned the 1979 (Glashow-Salam-Weinberg) and 1999 ('t Hooft-Veltman) Nobel Prizes; the 2013 prize to Englert and Higgs followed the 2012 discovery of the scalar boson at the Large Hadron Collider.

Bibliography Master

@article{Anderson1963,
  author  = {Anderson, Philip W.},
  title   = {Plasmons, Gauge Invariance, and Mass},
  journal = {Phys. Rev.},
  volume  = {130},
  year    = {1963},
  pages   = {439--442}
}

@article{EnglertBrout1964,
  author  = {Englert, Fran{\c{c}}ois and Brout, Robert},
  title   = {Broken Symmetry and the Mass of Gauge Vector Mesons},
  journal = {Phys. Rev. Lett.},
  volume  = {13},
  year    = {1964},
  pages   = {321--323}
}

@article{Higgs1964PRL,
  author  = {Higgs, Peter W.},
  title   = {Broken Symmetries and the Masses of Gauge Bosons},
  journal = {Phys. Rev. Lett.},
  volume  = {13},
  year    = {1964},
  pages   = {508--509}
}

@article{GHK1964,
  author  = {Guralnik, Gerald S. and Hagen, C. R. and Kibble, T. W. B.},
  title   = {Global Conservation Laws and Massless Particles},
  journal = {Phys. Rev. Lett.},
  volume  = {13},
  year    = {1964},
  pages   = {585--587}
}

@article{GoldstoneSalamWeinberg1962,
  author  = {Goldstone, Jeffrey and Salam, Abdus and Weinberg, Steven},
  title   = {Broken Symmetries},
  journal = {Phys. Rev.},
  volume  = {127},
  year    = {1962},
  pages   = {965--970}
}

@article{tHooft1971,
  author  = {'t Hooft, Gerard},
  title   = {Renormalizable Lagrangians for Massive Yang-Mills Fields},
  journal = {Nucl. Phys. B},
  volume  = {35},
  year    = {1971},
  pages   = {167--188}
}

@article{FujikawaLeeSanda1972,
  author  = {Fujikawa, Kazuo and Lee, Benjamin W. and Sanda, A. I.},
  title   = {Generalized Renormalizable Gauge Formulation of Spontaneously Broken Gauge Theories},
  journal = {Phys. Rev. D},
  volume  = {6},
  year    = {1972},
  pages   = {2923--2943}
}

@book{WeinbergQTFVol2,
  author    = {Weinberg, Steven},
  title     = {The Quantum Theory of Fields, Vol. 2: Modern Applications},
  publisher = {Cambridge University Press},
  year      = {1996}
}

@book{ItzyksonZuber,
  author    = {Itzykson, Claude and Zuber, Jean-Bernard},
  title     = {Quantum Field Theory},
  publisher = {McGraw-Hill},
  year      = {1980}
}

@book{PeskinSchroeder,
  author    = {Peskin, Michael E. and Schroeder, Daniel V.},
  title     = {An Introduction to Quantum Field Theory},
  publisher = {Addison-Wesley},
  year      = {1995}
}