Electroweak theory and the Higgs mechanism

Intuition Beginner

Electromagnetism and the weak force look like two unrelated forces at everyday energies. Electromagnetism is carried by the photon, a massless particle that travels at the speed of light and so reaches across the whole universe. The weak force is carried by the $W$ and $Z$ bosons, which are heavy — about $80$ and $91$ times the proton mass — and so drag through space, able to act only across a distance much smaller than an atomic nucleus. Glashow, Weinberg, and Salam proposed in the 1960s that these are really two faces of a single electroweak force, and that the whole difference comes from one field that fills all of space.

That field is the Higgs field. It is a scalar field, meaning it has no spin and no preferred direction, and it takes a constant nonzero value everywhere in the universe, about $246$ GeV. This background value, called the vacuum expectation value, breaks the electroweak symmetry. The $W$ and $Z$ bosons, moving through this background, interact with it and acquire mass. The photon does not couple to the Higgs background, because the background carries no electric charge, and the photon stays massless. The photon is the carrier of the one symmetry left unbroken — ordinary electromagnetism.

One number ties the strengths of electromagnetism and the weak force together: the electroweak mixing angle ${\theta }_W$. This angle measures how the two original neutral fields blend to produce the observed photon and $Z$ boson. It also links the two heavy boson masses through $m_W=m_Z\cos {\theta }_W$. The electroweak scale $246$ GeV itself is measured not from a collider directly but from the lifetime of the muon, which decays through the weak force.

Visual Beginner

ELECTROWEAK MIXING AND SYMMETRY BREAKING
========================================

  full electroweak symmetry      Higgs VEV v = 246 GeV        observed
  -------------------------      ------------------------      --------
  SU(2)_L  x  U(1)_Y             <Phi> = (0, v/sqrt 2)^T      U(1)_em

  gauge fields: W^1, W^2, W^3    breaks SU(2)_L x U(1)_Y      photon A = massless
                B^0              ->  U(1)_em                   W+, W-, Z = massive

            NEUTRAL-SECTOR MIXING (the Weinberg angle theta_W)
            ---------------------------------------------------

                 W^3   )        )  Z   =  cos(theta_W) W^3 - sin(theta_W) B   (massive)
                       ) rotate )
                       )  by    )
                 B     ) theta_W)  A   =  sin(theta_W) W^3 + cos(theta_W) B   (massless)

            tan(theta_W) = g' / g ;   cos(theta_W) = m_W / m_Z

THE HIGGS POTENTIAL  (the "Mexican hat")
========================================

   V(Phi)
      |              chosen vacuum <Phi> = (0, v / sqrt 2)
      |                v = 246 GeV, fixed by muon decay
      |                W and Z interact with this background -> become massive
      |                photon does not couple to it -> stays massless
      |                                           m_H = 125 GeV (radial excitation)
      \        ...---"""---...
       \    ./               \.    minima lie on a ring of radius |Phi| = v/sqrt 2
        \__/                   \__ (degenerate vacuum manifold)

HOW THE ELECTROWEAK SCALE IS MEASURED
=====================================

  muon decay  mu -> e + nu_mu + anti-nu_e     (a weak process)
       |
       |  W-boson exchange, at q^2 << m_W^2, looks like a contact interaction
       v
  Fermi constant   G_F = 1.16638 x 10^-5  GeV^-2
       |
       |  relation  G_F / sqrt 2  =  1 / (2 v^2)
       v
  electroweak scale   v = (sqrt 2  G_F)^(-1/2) = 246.22 GeV

Field	Before breaking	After breaking	Mass
$W^1,W^2$	$SU(2{)}_L$ gauge bosons	$W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$	$m_W\approx 80.4$ GeV
$W^3$	$SU(2{)}_L$ neutral boson	mixes with $B$	—
$B$	$U(1{)}_Y$ hypercharge boson	mixes with $W^3$	—
$A$	— (emergent)	photon, $A=s_W{W}^3+c_{W}B$	$0$
$Z$	— (emergent)	$Z=c_W{W}^3-s_{W}B$	$m_Z\approx 91.2$ GeV
$\Phi$	Higgs doublet, $Y=+1/2$	$⟨\Phi ⟩=(0,v/\sqrt{2}{)}^T$	$m_H\approx 125$ GeV

Worked example Beginner

Problem. The electroweak scale $v$ is fixed by the measured muon lifetime. The muon decays through the weak force as $\mu \to e+{\nu }_{\mu }+{\bar{\nu }}_e$, and the strength of this decay is summarised by the Fermi constant, $G_F=1.16638\times 10^{-5}$ GeV${}^{-2}$. The tree-level electroweak relation between $G_F$ and $v$ is $G_F/\sqrt{2}=1/(2v^2)$. Use it to extract $v$ in GeV.

Solution.

Step 1. Rearrange the relation to isolate $v^2$. Multiplying both sides by $2v^2$ and dividing by $G_F$ gives

$$v^2=\frac{\sqrt{2}}{2G_F}=\frac{1}{\sqrt{2}{G}_F},v=\frac{1}{(\sqrt{2}{G}_F{)}^{1/2}}.$$

Step 2. Compute the number inside the square root. First $\sqrt{2}\times G_F=1.41421\times 1.16638\times 10^{-5}=1.64935\times 10^{-5}$ GeV${}^{-2}$.

Step 3. Invert and take the square root. The inverse is $1/(1.64935\times 10^{-5})=60630$ GeV${}^2$, and the square root is $\sqrt{60630}=246.23$ GeV.

Reading the result. The electroweak scale is $v=246.22$ GeV. This single number, together with the mixing angle ${\theta }_W$, fixes the $W$ and $Z$ boson masses. The remarkable point is that a heavy particle collider is not the only way to weigh the electroweak scale: the lifetime of the muon, a particle that decays entirely through the weak force, already pins down $v$ to four significant figures. The muon is a built-in electroweak balance.

Check your understanding Beginner

Formal definition Intermediate+

The full Standard Model gauge group and field content are stated in 12.21.01; this unit isolates the electroweak factor $SU(2{)}_L\times U(1{)}_Y$ and develops what the overview states but does not prove: the extraction of the electroweak scale from low-energy data, the custodial symmetry that protects the mass relations, the gauge-boson self-couplings, and the origin of quark flavour mixing.

Gauge group and chiral fermion content. The electroweak gauge group is $G_{\mathrm{EW}}=SU(2{)}_L\times U(1{)}_Y$, with coupling constants $g$ and $g^{\prime }$. The subscript $L$ records the chiral structure: $SU(2{)}_L$ acts only on left-handed fermion doublets. For one generation the fermion fields and their $SU(2{)}_L\times U(1{)}_Y$ quantum numbers are

$$Q_L=\left(\begin{array}{c}{u}_L\\ d_L\end{array}\right)\sim (2,+\frac{1}{3}),u_R\sim (1,+\frac{4}{3}),d_R\sim (1,-\frac{2}{3}),L_L=\left(\begin{array}{c}{\nu }_L\\ e_L\end{array}\right)\sim (2,-1),e_R\sim (1,-2),$$

using the convention $Q=T_3+Y/2$. Right-handed neutrinos are absent from the minimal model. The chiral assignment is what makes the theory parity-violating by construction ^{[Feynman-Gell-Mann 1958]}: the left- and right-handed components sit in different representations, so a parity transformation maps the Lagrangian to a different theory.

Covariant derivative and field strengths. With ${\sigma }^i$ the Pauli matrices and $T_3={\sigma }^3/2$, the electroweak covariant derivative is

$$D_{\mu }={\partial }_{\mu }-ig\frac{{\sigma }^i}{2}{W}_{\mu }^i-i{g}^{\prime }\frac{Y}{2}{B}_{\mu },$$

and the field strengths are $W_{\mu \nu }^i={\partial }_{\mu }{W}_{\nu }^i-{\partial }_{\nu }{W}_{\mu }^i+g{\epsilon }^{ijk}{W}_{\mu }^j{W}_{\nu }^k$ and $B_{\mu \nu }={\partial }_{\mu }{B}_{\nu }-{\partial }_{\nu }{B}_{\mu }$. The ${\epsilon }^{ijk}$ term is the signature of non-Abelian gauge structure: the $W$ bosons carry weak isospin and couple to themselves. These self-couplings are absent in pure QED and are the subject of the gauge-interaction analysis below.

Higgs doublet and potential. The model contains one complex scalar $SU(2{)}_L$ doublet

$$\Phi =\left(\begin{array}{c}{\varphi }^{+}\\ {\varphi }^0\end{array}\right),Y_{\Phi }=+\frac{1}{2},V(\Phi )=-{\mu }^2{\Phi }^{†}\Phi +\lambda ({\Phi }^{†}\Phi {)}^2,\lambda >0,{\mu }^2>0.$$

The sign of ${\mu }^2$ is the whole point: the potential has a maximum at $\Phi =0$ and a degenerate ring of minima at ${\Phi }^{†}\Phi ={\mu }^2/(2\lambda )$. Choosing the vacuum $⟨\Phi ⟩=(0,v/\sqrt{2}{)}^T$ with $v=\mu /\sqrt{\lambda }$ breaks $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_{\mathrm{em}}$; the surviving generator is $Q=T_3+Y/2$ because $Q⟨\Phi ⟩=0$.

Mass eigenstates and the mixing angle. The charged bosons are $W_{\mu }^{\pm }=(W_{\mu }^1\mp i{W}_{\mu }^2)/\sqrt{2}$. The neutral bosons are defined by a single rotation through the Weinberg angle,

$$\left(\begin{array}{c}{A}_{\mu }\\ Z_{\mu }\end{array}\right)=\left(\begin{array}{cc}\cos {\theta }_W& \sin {\theta }_W\\ -\sin {\theta }_W& \cos {\theta }_W\end{array}\right)\left(\begin{array}{c}{B}_{\mu }\\ W_{\mu }^3\end{array}\right),\tan {\theta }_W=\frac{g^{\prime }}{g}.$$

The photon $A_{\mu }$ is the gauge boson of the unbroken $U(1{)}_{\mathrm{em}}$ and stays massless; the $Z$ is the orthogonal combination and is massive. The tree-level masses $m_W=gv/2$ and $m_Z=v\sqrt{g^2+g^{\prime 2}}/2$ follow from the Higgs kinetic term and are derived in 12.21.01; the present unit uses them as inputs and derives the scale $v$ itself from low-energy data, the consistency relation $\rho =1$, and the self-interaction structure.

Yukawa sector. Fermion masses come from gauge-invariant couplings to the Higgs doublet and its conjugate $\tilde{\Phi }=i{\sigma }^2{\Phi }^{\ast }$:

$${\mathcal{L}}_{\mathrm{Yuk}}=-y_{ij}^u{\bar{Q}}_{Li}\tilde{\Phi }{u}_{Rj}-y_{ij}^d{\bar{Q}}_{Li}\Phi d_{Rj}-y_{ij}^e{\bar{L}}_{Li}\Phi e_{Rj}+h.c.$$

After symmetry breaking each Yukawa matrix becomes a mass matrix $M^f=y^{f}v/\sqrt{2}$. Because $y^u,y^d,y^e$ are independent complex $3\times 3$ matrices, their diagonalisation by unitary rotations of the fermion fields is what produces the CKM flavour-mixing matrix in the charged current, developed in the Full proof set.

Key derivation Intermediate+

Theorem (Electroweak-scale extraction from muon decay). In the electroweak model the muon decay ${\mu }^{-}\to e^{-}+{\bar{\nu }}_e+{\nu }_{\mu }$ is mediated at tree level by $W$-boson exchange. At momentum transfers far below the $W$ mass, $|q^2|\ll m_W^2$, the exchange reduces to a pointlike four-fermion contact interaction parametrised by the Fermi constant $G_F$, with the matching

$$\frac{G_F}{\sqrt{2}}=\frac{g^2}{8m_W^2}=\frac{1}{2v^2},$$

and consequently the electroweak scale is $v=(\sqrt{2}{G}_F{)}^{-1/2}=246.22$ GeV.

Proof. The charged-current interaction, read off from the covariant derivative acting on the left-handed doublets, is

$${\mathcal{L}}_{\mathrm{CC}}=-\frac{g}{\sqrt{2}}(W_{\mu }^{+}{J}^{\mu +}+W_{\mu }^{-}{J}^{\mu -}),J^{\mu +}={\bar{\nu }}_{eL}{\gamma }^{\mu }{e}_L+{\bar{\nu }}_{\mu L}{\gamma }^{\mu }{\mu }_L+\cdots$$

The tree-level diagram for muon decay has the muon line emitting a $W^{-}$ that decays into $e^{-}{\bar{\nu }}_e$. The $W$ propagator in unitary gauge is $(-i/m_W^2)$ in the low-momentum limit $|q^2|\ll m_W^2$, since the propagator numerator $-i(g_{\mu \nu }-k_{\mu }{k}_{\nu }/m_W^2)/(k^2-m_W^2)$ collapses to $i{g}_{\mu \nu }/m_W^2$ when $k^2\to 0$. The amplitude becomes

$$\mathcal{M}=\left(-\frac{ig}{\sqrt{2}}\right)\left(-\frac{ig}{\sqrt{2}}\right)\frac{i}{m_W^2}[{\bar{u}}_{{\nu }_{\mu }}{\gamma }^{\mu }(1-{\gamma }^5)u_{\mu }][{\bar{u}}_e{\gamma }_{\mu }(1-{\gamma }^5)v_{{\bar{\nu }}_e}]=\frac{i{g}^2}{8m_W^2}[\cdots ],$$

where the factor $1/\sqrt{2}$ at each vertex combines with the V-A structure, and the standard normalisation absorbs the factors of $2$ to give the $1/8$ ^{[Fermi 1933]}. Comparing with the Fermi contact interaction

$${\mathcal{M}}_{\mathrm{Fermi}}=i\frac{G_F}{\sqrt{2}}[{\bar{u}}_{{\nu }_{\mu }}{\gamma }^{\mu }(1-{\gamma }^5)u_{\mu }][{\bar{u}}_e{\gamma }_{\mu }(1-{\gamma }^5)v_{{\bar{\nu }}_e}],$$

read off the matching

$$\frac{G_F}{\sqrt{2}}=\frac{g^2}{8m_W^2}.$$

Now substitute the tree-level $W$-mass relation $m_W=gv/2$ from 12.21.01. Then $m_W^2=g^2{v}^2/4$, so

$$\frac{g^2}{8m_W^2}=\frac{g^2}{8\cdot (g^2{v}^2/4)}=\frac{1}{2v^2}.$$

Equating the two expressions for $G_F/\sqrt{2}$ gives $v^2=1/(\sqrt{2}{G}_F)$, hence

$$v=\frac{1}{(\sqrt{2}{G}_F{)}^{1/2}}.$$

With the measured value extracted from the muon lifetime, $G_F=1.1663787\times 10^{-5}$ GeV${}^{-2}$, one obtains $\sqrt{2}{G}_F=1.64935\times 10^{-5}$ GeV${}^{-2}$, its reciprocal $60630$ GeV${}^2$, and

$$v=\sqrt{60630}\mathrm{GeV}=246.22\mathrm{GeV}.\square$$

The agreement with the independent determinations of $v$ from the $W$ mass ($v=2m_W/g$) and from the $Z$ mass, each fixed through the mixing angle, is the empirical content of electroweak unification: a scale measured in muon decay predicts the heavy boson masses.

Bridge. This derivation is the foundational reason the electroweak scale is an experimentally fixed number rather than a free parameter: the muon lifetime measures $G_F$, the $W$-exchange diagram fixes $G_F/\sqrt{2}=g^2/(8m_W^2)$, and the Higgs vacuum ties this to $1/(2v^2)$. The relation $G_F/\sqrt{2}=1/(2v^2)$ is exactly the bridge between the low-energy four-fermion theory of Fermi and the spontaneously broken gauge theory of Weinberg and Salam, and it builds toward the precision-electroweak programme in which the one-percent-level deviations of $m_W$, ${\sin }^2{\theta }_W$, and the $Z$-pole observables from these tree-level formulas are used to fit the top mass and the Higgs mass before either was directly measured. The same scale $v$ appears again in 12.18.01 as the order parameter of the general Higgs mechanism, and the central insight is that the pattern of electroweak masses is determined once $v$, ${\theta }_W$, and the Yukawa couplings are specified.

Exercises Intermediate+

Lean formalization Intermediate+

The electroweak theory is not formalised in Mathlib at the time of writing. The derivations of this unit — the low-energy matching $G_F/\sqrt{2}=g^2/(8m_W^2)$, the custodial-symmetry proof of $\rho =1$, the extraction of the triple gauge couplings from the $SU(2{)}_L$ field strength, and the diagonalisation of the Yukawa matrices into the CKM matrix — would each require a layered formal infrastructure that does not yet exist. In increasing order of dependence: (i) a typed declaration of $SU(2{)}_L\times U(1{)}_Y$ as a Lie group with its Lie algebra and structure constants, building on Mathlib's existing matrix-Lie-algebra API; (ii) a representation-theoretic encoding of the chiral fermion content and a computable Gell-Mann–Nishijima charge function; (iii) the electroweak Lagrangian as a typed mathematical object; (iv) a perturbative-QFT layer (propagators, vertices, tree-level amplitudes) sufficient to state and verify the $W$-exchange matching calculation; (v) the custodial $SU(2)$ global symmetry acting on the Higgs doublet as a Lean group action, with $\rho =1$ as a theorem.

Steps (i)–(iii) are within reach of the current Lie-theory API. Steps (iv)–(v) depend on a relativistic-QFT package that Mathlib does not yet host. the Mathlib gap analysis records the gap; the unit ships with lean_status: none and no Lean module.

Advanced results Master

The Key derivation shows how the electroweak scale is fixed by the muon lifetime. Three further structural facts about the electroweak theory carry the unit beyond the overview: the custodial symmetry that protects the mass relations, the non-Abelian gauge self-couplings and their high-energy unitarity, and the flavour structure encoded in the CKM matrix.

Custodial symmetry and the $\rho$ parameter. The tree-level identity $\rho \equiv m_W^2/(m_Z^2{\cos }^2{\theta }_W)=1$ proved in Exercise 5 is not an accident of the doublet representation: it is enforced by a global $SU(2)$ symmetry of the Higgs sector called the custodial symmetry ^{[Georgi-Nanopoulos 1979]}. Assemble the Higgs doublet $\Phi =({\varphi }^{+},{\varphi }^0{)}^T$ into the $2\times 2$ matrix

$$H=\left(\begin{array}{cc}{\varphi }^{0\ast }& {\varphi }^{+}\\ -{\varphi }^{-}& {\varphi }^0\end{array}\right),\det H={\Phi }^{†}\Phi .$$

Under a global $SU(2{)}_L\times SU(2{)}_R$ acting as $H\to LH{R}^{†}$, the potential $V=-{\mu }^2\det H+\lambda (\det H{)}^2$ is invariant (it depends only on $\det H={\Phi }^{†}\Phi$, and $\det (LH{R}^{†})=\det H$ since $L,R\in SU(2)$). The vacuum $⟨H⟩=(v/\sqrt{2})1_{2\times 2}$ is proportional to the identity, so it is invariant under the diagonal subgroup $SU(2{)}_{L+R}\equiv SU(2{)}_V$, the custodial symmetry. The three $W$ bosons transform as a triplet under $SU(2{)}_V$ when the hypercharge coupling $g^{\prime }$ is set to zero, so the custodial symmetry forces $m_{W^1}=m_{W^2}=m_{W^3}$; because $W^3$ mixes with $B$ only through the $g^{\prime }$ coupling, the relation $m_W=m_Z\cos {\theta }_W$ — equivalently $\rho =1$ — holds at tree level for any number of Higgs doublets.

The custodial symmetry is broken by the hypercharge coupling $g^{\prime }\ne 0$ and, more importantly, by the Yukawa couplings, because a mass splitting within an $SU(2{)}_L$ doublet (such as $m_t\gg m_b$) is not invariant under $SU(2{)}_V$. The leading one-loop correction is

$$\Delta \rho \equiv \rho -1=\frac{3G_F{m}_t^2}{8\sqrt{2}{\pi }^2}\approx 0.96\times 10^{-2},$$

the dominant piece of the electroweak precision fit; the measured value ${\rho }_{\exp }\approx 1.0004$ agrees with this prediction once the full set of radiative corrections is included. This is why precision electroweak data could predict the top mass ($m_t\approx 170$ GeV) before the top quark was directly discovered in 1995, and is the cleanest evidence that the Higgs transforms as an isospin-$1/2$ doublet rather than a higher representation.

Gauge self-couplings and high-energy unitarity. The triple-gauge vertices $W^{+}{W}^{-}\gamma$ and $W^{+}{W}^{-}Z$ derived in Exercise 6, together with the quartic vertices $W^{+}{W}^{-}{W}^{+}{W}^{-}$, $W^{+}{W}^{-}\gamma \gamma$, and $W^{+}{W}^{-}Z\gamma$, are the electroweak fingerprints of non-Abelian gauge structure: the $W$ boson carries electric charge $e$ and couples to the photon, a feature absent in any Abelian theory. Their direct measurement at LEP2 ($e^{+}{e}^{-}\to W^{+}{W}^{-}$, $\sqrt{s}$ up to $209$ GeV) and at the LHC confirmed the predicted couplings to a few percent ^{['t Hooft-Veltman 1972]}.

The deeper role of these couplings is unitarity. As shown in Exercise 7, the amplitude for producing longitudinal $W$ bosons in $e^{+}{e}^{-}$ annihilation grows as $E^2/m_W^2$ in any single diagram, threatening perturbative unitarity at energies of a few TeV. The gauge diagrams (photon and $Z$ exchange) and the $t$-channel neutrino diagram cancel this growth exactly, a cancellation enforced by the Slavnov-Taylor identity of $SU(2{)}_L\times U(1{)}_Y$. Without the Higgs mechanism — equivalently, without the $W$ acquiring its mass from a scalar condensate — the cancellation would fail and the theory would become non-unitary at energies around $1$ TeV. This is the real theoretical content of the unitarity argument: the Higgs boson is not an optional extra but the agent that keeps weak-interaction scattering finite, and its $125$ GeV mass was within the range suggested by this unitarity bound.

Flavour and the CKM matrix. The Yukawa matrices $y^u,y^d,y^e$ are a priori arbitrary complex $3\times 3$ matrices. After symmetry breaking, the mass matrices $M^u=y^{u}v/\sqrt{2}$ and $M^d=y^{d}v/\sqrt{2}$ are diagonalised by bi-unitary transformations $U_L^{u†}{M}^u{U}_R^u=\mathrm{diag}(m_u,m_c,m_t)$ and $U_L^{d†}{M}^d{U}_R^d=\mathrm{diag}(m_d,m_s,m_b)$. Rewriting the charged current in the mass basis,

$${\mathcal{L}}_{\mathrm{CC}}=-\frac{g}{\sqrt{2}}{W}_{\mu }^{+}{\bar{u}}_{Li}{\gamma }^{\mu }(V_{\mathrm{CKM}}{)}_{ij}{d}_{Lj}+h.c.,V_{\mathrm{CKM}}=U_L^{u†}{U}_L^d,$$

produces the Cabibbo-Kobayashi-Maskawa matrix $V_{\mathrm{CKM}}$, a unitary $3\times 3$ matrix parametrised by three mixing angles and one CP-violating phase. The neutral currents, by contrast, remain flavour-diagonal after diagonalisation — the Glashow-Iliopoulos-Maiani (GIM) mechanism — so there are no tree-level flavour-changing neutral currents, a prediction confirmed to high precision (the absence of $\mu \to e\gamma$ at tree level, the suppression of $K^0$-${\bar{K}}^0$ mixing). The unitarity of $V_{\mathrm{CKM}}$ (each row and column summing to one) is tested at the $10^{-3}$ level and is the tightest consistency check of the three-generation electroweak flavour sector. CP violation enters through the single complex phase of $V_{\mathrm{CKM}}$ and is, within the Standard Model, the only source of CP violation in quark interactions — insufficient by itself to explain the baryon asymmetry of the universe, which is one of the motivations for beyond-the-Standard-Model CP-violating sectors.

Precision electroweak. The custodial result, the gauge self-couplings, and the CKM unitarity together constitute the precision-electroweak programme: the tree-level predictions of the Glashow-Weinberg-Salam model, corrected by one-loop electroweak diagrams (the oblique parameters $S$, $T$, $U$ summarising gauge-boson self-energies), fit dozens of measured observables — $m_W$, ${\sin }^2{\theta }_W$, the $Z$-pole asymmetries, partial widths, and the CKM elements — with the top mass and the Higgs mass as the only large radiative inputs. The agreement of the fitted Higgs mass with the directly observed $125$ GeV scalar ^{[ATLAS-CMS 2012]} is the most stringent single test of the electroweak theory and the closing argument for the Higgs doublet as the agent of symmetry breaking.

Synthesis. Electroweak theory is the central insight that two apparently different forces are one gauge structure broken by a scalar condensate. The Fermi constant $G_F$ fixes the scale $v=246$ GeV from muon decay, the Weinberg angle ${\theta }_W$ fixes the coupling ratio, and from these two numbers plus the Yukawa matrices the model predicts the $W$ and $Z$ masses, the photon's masslessness, the gauge self-couplings, and the flavour structure. The custodial symmetry is the foundational reason $\rho =1$ at tree level, the Higgs mechanism generalises to any spontaneously broken gauge theory, and the gauge-boson self-couplings are dual to the high-energy unitarity that only the full gauge structure — photon, $Z$, and neutrino diagrams together — preserves. Putting these together yields a renormalisable, unitary, flavour-consistent theory tested at the per-mille level at LEP and the LHC. The bridge is the gauge principle: a locally realised $SU(2{)}_L\times U(1{)}_Y$ symmetry, spontaneously broken to $U(1{)}_{\mathrm{em}}$, determines the dynamics, the mass spectrum, and the pattern of self-interactions from one measured scale and one measured angle.

Full proof set Master

The Key derivation gives the extraction of the electroweak scale $v$ in full. Three further structural facts are proved here: the custodial-symmetry protection of $\rho =1$, the triple gauge couplings from the $SU(2{)}_L$ field strength, and the emergence of the CKM matrix from the Yukawa diagonalisation.

Proposition (Custodial symmetry and $\rho =1$). In the electroweak model with a single Higgs doublet of hypercharge $Y=+1/2$, the global symmetry of the Higgs potential contains a custodial $SU(2{)}_V$ under which the three $W$ bosons (in the limit $g^{\prime }\to 0$) transform as a triplet. Consequently the tree-level electroweak $\rho$ parameter satisfies $\rho \equiv m_W^2/(m_Z^2{\cos }^2{\theta }_W)=1$.

Proof. Write the Higgs doublet $\Phi =({\varphi }^{+},{\varphi }^0{)}^T$ as the $2\times 2$ matrix $H=(\tilde{\Phi },\Phi )$, that is,

$$H=\left(\begin{array}{cc}{\varphi }^{0\ast }& {\varphi }^{+}\\ -{\varphi }^{-}& {\varphi }^0\end{array}\right),\text{so that}{H}^{†}H=({\Phi }^{†}\Phi )1_{2\times 2},\det H={\Phi }^{†}\Phi .$$

Under a global transformation $(L,R)\in SU(2{)}_L\times SU(2{)}_R$, define $H\to LH{R}^{†}$. The Higgs potential depends on $H$ only through $\det H={\Phi }^{†}\Phi$, and $\det (LH{R}^{†})=\det (L)\det (H)\det (R^{†})=\det H$ (since $L,R\in SU(2)$ have unit determinant), so $V(H)=-{\mu }^2\det H+\lambda (\det H{)}^2$ is invariant under the full global $SU(2{)}_L\times SU(2{)}_R$.

The vacuum is $⟨\Phi ⟩=(0,v/\sqrt{2}{)}^T$, giving $⟨H⟩=(v/\sqrt{2})1_{2\times 2}$, which is proportional to the identity. It is preserved by the diagonal subgroup $L=R$, i.e. by $SU(2{)}_V\equiv \{(U,U):U\in SU(2)\}$: $(U,U)$ sends $⟨H⟩\to U⟨H⟩U^{†}=⟨H⟩$. Hence the global symmetry is broken $SU(2{)}_L\times SU(2{)}_R\to SU(2{)}_V$ spontaneously, and $SU(2{)}_V$ is the custodial symmetry.

Now gauge only the $SU(2{)}_L$ factor (set $g^{\prime }=0$, so there is no hypercharge gauge field $B$). The three gauge bosons $W^1,W^2,W^3$ transform as a triplet under the gauged $SU(2{)}_L$, and because the unbroken global symmetry is the diagonal $SU(2{)}_V\cong SU(2{)}_L$, they transform as a triplet under $SU(2{)}_V$ as well. The mass term $(D_{\mu }\Phi {)}^{†}(D^{\mu }\Phi ){|}_{\mathrm{vac}}$, being $SU(2{)}_V$-invariant and built from an $SU(2{)}_V$-triplet of gauge fields, must be proportional to ${\delta }^{ij}{W}_{\mu }^i{W}^{j\mu }$: it cannot distinguish the three directions. Hence $m_{W^1}=m_{W^2}=m_{W^3}\equiv m_W$. Turning on the hypercharge coupling $g^{\prime }$ gauges a $U(1)$ subgroup of $SU(2{)}_R$, breaking custodial $SU(2{)}_V$ explicitly; the neutral combination $W^3$ mixes with $B$ to form $Z$ and $A$, and the $W$ triplet splits as $m_Z=m_W/\cos {\theta }_W$ while the photon stays massless. Therefore

$$\rho \equiv \frac{m_W^2}{m_Z^2{\cos }^2{\theta }_W}=\frac{m_W^2}{m_W^2}=1$$

at tree level for any number of Higgs doublets. $\square$

The result extends to arbitrary Higgs representations: a single Higgs multiplet of weak isospin $T$ and hypercharge $Y$ (with a neutral component, $T_3=-Y/2$) contributes $\rho =3T(T+1)/(2Y^2)$... the precise general formula, but for the doublet $T=1/2$, $Y=1$ this reduces to $\rho =1$, while higher representations give $\rho \ne 1$ — which is why the measured $\rho \approx 1.0004$ selects the doublet.

Proposition (Triple gauge couplings from the $SU(2{)}_L$ field strength). The non-Abelian kinetic term $-\frac{1}{4}{W}_{\mu \nu }^i{W}^{i\mu \nu }$ produces a cubic gauge vertex $W_{\mu }^{+}{W}_{\nu }^{-}{A}_{\lambda }$ with electric charge $e=g\sin {\theta }_W$ and a cubic vertex $W^{+}{W}^{-}Z$ with coupling $g\cos {\theta }_W$, both with the fully antisymmetric three-leg tensor structure.

Proof. The cubic piece of $-\frac{1}{4}{W}_{\mu \nu }^i{W}^{i\mu \nu }$ is

$${\mathcal{L}}_{\mathrm{cubic}}=-g{\epsilon }^{ijk}({\partial }_{\mu }{W}_{\nu }^i)W^{j\mu }{W}^{k\nu }.$$

Convert to the charged basis using $W^1=(W^{+}+W^{-})/\sqrt{2}$, $W^2=i(W^{+}-W^{-})/\sqrt{2}$. The Levi-Civita contractions give ${\epsilon }^{3jk}{W}^j{W}^k=-i(W^{+}{W}^{-}-W^{-}{W}^{+})$, so the term coupling one neutral ($W^3$) to two charged bosons is

$${\mathcal{L}}_{W^{+}{W}^{-}{W}^3}=-ig{W}_{\mu \nu }^3{W}^{+\mu }{W}^{-\nu }+ig({\partial }_{\mu }{W}_{\nu }^{+}-{\partial }_{\nu }{W}_{\mu }^{+})W^{-\mu }{W}^{3\nu }+h.c.,$$

with $W_{\mu \nu }^3\equiv {\partial }_{\mu }{W}_{\nu }^3-{\partial }_{\nu }{W}_{\mu }^3$. Substituting $W^3=\cos {\theta }_{W}Z+\sin {\theta }_{W}A$,

$${\mathcal{L}}_{W^{+}{W}^{-}A}=-ie({\partial }_{\mu }{W}_{\nu }^{+}-{\partial }_{\nu }{W}_{\mu }^{+})W^{-\mu }{A}^{\nu }+h.c.,{\mathcal{L}}_{W^{+}{W}^{-}Z}=-ig\cos {\theta }_W({\partial }_{\mu }{W}_{\nu }^{+}-{\partial }_{\nu }{W}_{\mu }^{+})W^{-\mu }{Z}^{\nu }+h.c.,$$

using $g\sin {\theta }_W=e$. Fourier-transforming with the plane-wave replacement ${\partial }_{\mu }\to -i{p}_{\mu }$ and symmetrising over the three external legs yields the momentum-space vertices stated in Exercise 6: the photon vertex has coupling $ie$ times the antisymmetric tensor $[(p-q{)}_{\lambda }{g}_{\mu \nu }+\text{cycl.}]$, and the $Z$ vertex has the identical structure with $e\to g\cos {\theta }_W$. The antisymmetry under leg exchange, inherited from ${\epsilon }^{ijk}$, is the defining feature and the reason the photon couples to the $W$ with exactly the electric charge $e$. $\square$

The quartic vertices — $W^{+}{W}^{-}{W}^{+}{W}^{-}$, $W^{+}{W}^{-}\gamma \gamma$, $W^{+}{W}^{-}Z\gamma$, $W^{+}{W}^{-}ZZ$ — arise from the $(g{\epsilon }^{ijk}{W}_{\mu }^j{W}_{\nu }^k{)}^2$ part of $W_{\mu \nu }^i{W}^{i\mu \nu }$ by the same conversion. They were measured at the LHC in vector-boson scattering $pp\to W^{+}{W}^{+}jj$ and agree with the electroweak prediction.

Proposition (CKM matrix from Yukawa diagonalisation). In the three-generation electroweak model, diagonalising the up- and down-type quark mass matrices $M^u,M^d$ by bi-unitary transformations produces a unitary charged-current mixing matrix $V_{\mathrm{CKM}}=U_L^{u†}{U}_L^d$, while the neutral currents remain flavour-diagonal.

Proof. After symmetry breaking, the quark mass terms are

$${\mathcal{L}}_{\mathrm{mass}}=-{\bar{u}}_L{M}^u{u}_R-{\bar{d}}_L{M}^d{d}_R+h.c.,M^u=y^{u}v/\sqrt{2},M^d=y^{d}v/\sqrt{2}.$$

A general complex square matrix admits a singular-value decomposition $M=U_L^{†}\mathrm{diag}(m_i)U_R$ with $U_L,U_R$ unitary. Apply this to $M^u$ and $M^d$ separately, defining $U_L^u,U_R^u,U_L^d,U_R^d$ such that

$$U_L^{u†}{M}^u{U}_R^u=\mathrm{diag}(m_u,m_c,m_t),U_L^{d†}{M}^d{U}_R^d=\mathrm{diag}(m_d,m_s,m_b).$$

Rotate the fermion fields to the mass basis: $u_L^{\mathrm{weak}}=U_L^u{u}_L^{\mathrm{mass}}$, $d_L^{\mathrm{weak}}=U_L^d{d}_L^{\mathrm{mass}}$, and similarly for the right-handed fields. The mass terms become diagonal by construction.

The charged current in the weak basis is $-(g/\sqrt{2})W_{\mu }^{+}{\bar{u}}_L^{\mathrm{weak}}{\gamma }^{\mu }{d}_L^{\mathrm{weak}}$. In the mass basis this becomes

$${\mathcal{L}}_{\mathrm{CC}}=-\frac{g}{\sqrt{2}}{W}_{\mu }^{+}{\bar{u}}_L^{\mathrm{mass}}{U}_L^{u†}{\gamma }^{\mu }{U}_L^d{d}_L^{\mathrm{mass}}=-\frac{g}{\sqrt{2}}{W}_{\mu }^{+}{\bar{u}}_{Li}{\gamma }^{\mu }(V_{\mathrm{CKM}}{)}_{ij}{d}_{Lj},$$

with $V_{\mathrm{CKM}}=U_L^{u†}{U}_L^d$. As a product of unitary matrices, $V_{\mathrm{CKM}}$ is unitary. A $3\times 3$ unitary matrix has $9$ real parameters; $5$ phases can be absorbed into rephasings of the six quark fields (minus one overall phase), leaving $3$ mixing angles and $1$ CP-violating phase — the standard Cabibbo-Kobayashi-Maskawa parametrisation.

The neutral current is $-(g/\cos {\theta }_W)(Z_{\mu }/2){\bar{u}}_L^{\mathrm{weak}}{T}_3{\gamma }^{\mu }{u}_L^{\mathrm{weak}}+\dots$ on both $u$ and $d$ sectors. Because the same unitary rotation $U_L^u$ appears on both the ${\bar{u}}_L$ and $u_L$ sides of the neutral current, $U_L^{u†}{U}_L^u=1$ cancels: the neutral current is flavour-diagonal in the mass basis, with no mixing matrix. This is the GIM mechanism: flavour-changing neutral currents vanish at tree level. $\square$

The three propositions together establish the structural content of the electroweak theory that the overview states: the scale is fixed by muon decay (Key derivation), the mass relations are protected by custodial symmetry, the gauge bosons self-couple as the non-Abelian structure demands, and the flavour structure is entirely encoded in the single unitary matrix $V_{\mathrm{CKM}}$.

Connections Master

• 12.21.01 The Standard Model — electroweak unification and QCD is the direct prerequisite overview that states the full $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$ model and derives the tree-level $W$, $Z$, photon, Higgs, and fermion masses from the Higgs vacuum expectation value. The present unit takes those mass formulas as input and develops what the overview does not: the extraction of the scale $v$ from the muon lifetime, the custodial protection of $\rho =1$, the gauge-boson self-couplings, and the CKM flavour matrix.

• 12.18.01 The Higgs mechanism: spontaneously broken gauge symmetry is the gauge-theory prerequisite that proves the general theorem — a gauge boson acquires mass by coupling to a scalar condensate, and the would-be Goldstone modes become the longitudinal polarisations of the massive vector bosons. The present unit specialises that general mechanism to the specific electroweak Higgs doublet $\Phi$ with $Y=+1/2$ and vacuum $(0,v/\sqrt{2}{)}^T$, and the relation $m_W=gv/2$ used throughout this unit is the doublet realisation of the general formula proved there.

• 12.16.01 Electron self-energy and mass renormalisation at one loop establishes the radiative-correction machinery that turns the tree-level electroweak predictions of this unit into precision observables. The running of the electromagnetic coupling from $\alpha (0{)}^{-1}=137$ to $\alpha (M_Z{)}^{-1}\approx 128$, needed to reconcile the two routes to $g$ in Exercise 1, is the low-energy tail of the same QED beta function; the oblique-parameter fits that test custodial symmetry and the gauge self-couplings at the per-mille level are the electroweak one-loop continuation of that programme.

• 12.12.01 Canonical quantum field theory supplies the Feynman rules, propagators, and matching procedure that the Key derivation uses to equate the $W$-exchange amplitude with the Fermi contact interaction in the $|q^2|\ll m_W^2$ limit. The technique of matching a heavy-particle exchange to a low-energy effective interaction is the canonical-QFT tool that turns the muon lifetime into a measurement of the electroweak scale.

• 12.15.01 Discrete symmetry (C, P, T and their combinations) underlies the parity-violating chiral structure of the electroweak model: the assignment of left-handed fermions to $SU(2{)}_L$ doublets and right-handed fermions to singlets is the reason the weak force maximally violates parity, and the single CP-violating phase of the CKM matrix derived in the Full proof set is the only source of CP violation in the quark sector of the Standard Model.

• [28.*] Cosmology (the early-universe curriculum) provides the observational test of the electroweak theory at energies no collider can reach: electroweak symmetry is restored at temperatures above $v\approx 246$ GeV in the early universe, the electroweak phase transition and sphaleron processes it drives bear on the baryon asymmetry, and the insufficiency of the CKM CP-violating phase to produce that asymmetry is one of the principal motivations for beyond-the-Standard-Model physics.

Historical & philosophical context Master

The electroweak theory was assembled in three stages over forty years, beginning with Fermi's 1933 theory of beta decay ^{[Fermi 1933]}. Fermi proposed that the decay $n\to p{e}^{-}{\bar{\nu }}_e$ proceeds through a pointlike four-fermion contact interaction with a single coupling $G_F$. The theory was strikingly successful at describing nuclear beta spectra and muon decay, but it was manifestly non-renormalisable: the amplitude for any higher-order process diverged, and at energies of order $1/\sqrt{G_F}\sim 300$ GeV the perturbation series broke down entirely. The Fermi constant, a dimensionful coupling, was the sign that the weak force had a finite range and an intermediate mediator.

The discovery of the $V-A$ chiral structure by Feynman and Gell-Mann and by Sudarshan and Marshak in 1957–1958 ^{[Feynman-Gell-Mann 1958]}, after the fall of parity conservation in 1957, established that the weak charged current couples only to left-handed fermions. This observation became the seed of the gauge structure: the left-handed doublet assignment $(u_L,d_L{)}^T$, $({\nu }_L,e_L{)}^T$ of the electroweak model is the direct expression of the $V-A$ current. Glashow in 1961 ^{[Glashow 1961]} then proposed the $SU(2)\times U(1)$ gauge structure and the mixing angle, but his model left the $W$ and $Z$ bosons massless — a gauge boson cannot acquire a mass by hand without destroying gauge invariance — and so the model was not yet a physical theory of the weak force.

The decisive step was the application of the Higgs-Englert-Brout mechanism ^{[Higgs 1964]} to the Glashow model by Weinberg (1967) ^{[Weinberg 1967]} and independently by Salam (1968) ^{[Salam 1968]}. By introducing a single complex scalar doublet $\Phi$ with a vacuum expectation value, the $W$ and $Z$ acquired masses proportional to $v=246$ GeV while the photon stayed massless, and the model predicted a new weak neutral current mediated by the $Z$. The prediction of the neutral current was confirmed by the Gargamelle experiment at CERN in 1973 ^{[Hasert 1973]}, the first direct evidence for the electroweak mixing structure; the $W$ and $Z$ bosons themselves were discovered at the CERN SPS collider in 1983, with the mass ratio $m_W/m_Z$ matching $\cos {\theta }_W$ to one percent.

The model was accepted as a fundamental theory only after 't Hooft and Veltman proved in 1971–1972 ^{['t Hooft-Veltman 1972]} that spontaneously broken non-Abelian gauge theories are renormalisable. This is the technical content of the unitarity argument of Exercise 7: the gauge cancellations that keep $e^{+}{e}^{-}\to W^{+}{W}^{-}$ finite at high energy are the same cancellations that control the ultraviolet divergences loop by loop, and a finite set of counterterms absorbs them. Without this proof the predictions of the theory would have been arbitrary; with it, the precision-electroweak programme of LEP (1989–2000) and the SLC could measure the model's parameters to the per-mille level and use the fits to predict the top mass (1995, agreement) and the Higgs mass (2012, agreement). The discovery of the $125$ GeV Higgs boson by ATLAS and CMS ^{[ATLAS-CMS 2012]} closed the programme at the level of the particle content; the measured Higgs couplings to $W$, $Z$, top, bottom, and tau agree with the Yukawa-sector predictions of this unit within current experimental precision.

Philosophically, the electroweak theory is the paradigm of a gauge principle theory: the entire Lagrangian is fixed, up to numerical parameters, by specifying the gauge group $SU(2{)}_L\times U(1{)}_Y$, the fermion representations, and the Higgs representation. The mass spectrum, the mixing angle, the gauge self-couplings, and the flavour matrix all follow from these choices together with the two measured numbers $v$ and ${\theta }_W$ and the Yukawa matrices. The deep fact is that a locally realised internal symmetry, once allowed to be spontaneously broken, determines not only the dynamics but the pattern of masses and interactions — mass is not an input but a derived consequence of how the scalar vacuum respects a subgroup. The remaining unexplained inputs — the values of the Yukawa couplings spanning six orders of magnitude, the single CP-violating phase, the number of generations — are the open questions of flavour physics, and they mark the boundary of what the gauge principle alone can explain.

Bibliography Master

Primary literature:

• Fermi, E. Tentativo di una teoria dell'emissione dei raggi beta. Ric. Sci. 4, 491 (1933). The four-fermion contact interaction and the coupling later denoted $G_F$; the point of departure for the theory of weak interactions.

• Feynman, R. P. & Gell-Mann, M. Theory of the Fermi interaction. Phys. Rev. 109, 193 (1958). The $V-A$ chiral structure of the charged weak current; the experimental input that fixes the $SU(2{)}_L$ doublet assignment.

• Glashow, S. L. Partial symmetries of weak interactions. Nucl. Phys. 22, 579 (1961). The original $SU(2)\times U(1)$ proposal and the electroweak mixing angle.

• Higgs, P. W. Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett. 13, 508 (1964). The Higgs mechanism. Companion papers: Englert, F. & Brout, R. PRL 13, 321 (1964); Guralnik, G. S., Hagen, C. R. & Kibble, T. W. B. PRL 13, 585 (1964).

• Weinberg, S. A model of leptons. Phys. Rev. Lett. 19, 1264 (1967). The electroweak theory with a single Higgs doublet; predicts the weak neutral current and the $W$, $Z$ masses in terms of one scale and one angle.

• Salam, A. Weak and electromagnetic interactions. In N. Svartholm (ed.), Elementary Particle Theory, Almqvist & Wiksell, Stockholm (1968), p. 367. The independent statement of the electroweak model.

• 't Hooft, G. & Veltman, M. Regularization and renormalization of gauge fields. Nucl. Phys. B44, 189 (1972). Dimensional regularisation and the proof that spontaneously broken gauge theories are renormalisable.

• Hasert, F. J. et al. (Gargamelle). Observation of neutrino-like interactions without muon or electron. Phys. Lett. B 46, 138 (1973). The discovery of the weak neutral current — the first direct evidence for the $Z$ boson.

• Georgi, H. & Nanopoulos, D. V. Light quark masses and the Higgs boson. Nucl. Phys. B 155, 52 (1979). The custodial $SU(2)$ symmetry of the Higgs sector and the protection of the tree-level $\rho =1$ relation.

• Arnison, G. et al. (UA1). Experimental observation of isolated large transverse energy electrons with associated missing energy. Phys. Lett. B 122, 103 (1983). Discovery of the $W$ boson; companion papers report the $Z$.

• ATLAS Collaboration; CMS Collaboration. Observation of a new particle in the search for the Standard Model Higgs boson. Phys. Lett. B 716, 1 and 30 (2012). Discovery of the Higgs boson at 125 GeV, completing the electroweak Higgs doublet.

Textbook treatments:

• Halzen, F. & Martin, A. Quarks and Leptons: An Introductory Course in Modern Particle Physics. Wiley, 1984. The canonical first-course treatment of the Glashow-Weinberg-Salam model, the Fermi constant, and neutral currents.

• Griffiths, D. Introduction to Elementary Particles. Wiley-VCH, 2nd ed., 2008. The charged-current $V-A$ structure and the CKM matrix at the undergraduate level.

• Peskin, M. E. & Schroeder, D. V. An Introduction to Quantum Field Theory. Westview, 1995. Ch. 20 (the electroweak model, custodial symmetry, precision observables).

• Weinberg, S. The Quantum Theory of Fields, Vol. II: Modern Applications. Cambridge University Press, 1996. Ch. 21 (spontaneously broken gauge theories, the $\rho$ parameter, $R_{\xi }$ gauges).

• Schwartz, M. D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. A modern graduate treatment of electroweak radiative corrections and the oblique parameters.

BIBTEX:

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  author  = {Fermi, Enrico},
  title   = {Tentativo di una Teoria dell'Emissione dei Raggi Beta},
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  volume  = {4},
  pages   = {491--495},
  year    = {1933}
}

@article{FeynmanGellMann1958,
  author  = {Feynman, Richard P. and Gell-Mann, Murray},
  title   = {Theory of the {F}ermi Interaction},
  journal = {Physical Review},
  volume  = {109},
  pages   = {193--198},
  year    = {1958},
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}

@article{Glashow1961,
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  title   = {Partial Symmetries of Weak Interactions},
  journal = {Nuclear Physics B},
  volume  = {22},
  pages   = {579--588},
  year    = {1961},
  doi     = {10.1016/0029-5582(61)90469-2}
}

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  year    = {1964},
  doi     = {10.1103/PhysRevLett.13.508}
}

@article{Weinberg1967,
  author  = {Weinberg, Steven},
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  pages   = {1264--1266},
  year    = {1967},
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}

@inproceedings{Salam1968,
  author    = {Salam, Abdus},
  title     = {Weak and Electromagnetic Interactions},
  booktitle = {Elementary Particle Theory},
  editor    = {Svartholm, Nils},
  publisher = {Almqvist and Wiksell},
  address   = {Stockholm},
  pages     = {367},
  year      = {1968}
}

@article{tHooftVeltman1972,
  author  = {'t Hooft, Gerard and Veltman, Martinus},
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}

@article{GeorgiNanopoulos1979,
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@article{Arnison1983,
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}

@article{ATLAS2012,
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}

@book{HalzenMartin1984,
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}

@book{Griffiths2008,
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@book{PeskinSchroeder1995,
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}