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The bino is the spin-one-half gaugino of the hypercharge $U(1{)}_Y$ gauge group. Its adjoint representation is the singlet $(1,1,0)$, so it has no hypercharge of its own. After electroweak symmetry breaking, it can mix with the neutral wino and higgsinos into neutralinos.

A singular value decomposition of the charged-lepton Yukawa matrix uses independent unitary rotations of left- and right-handed leptons. Their gauge-covariant kinetic terms are unchanged because each rotation mixes identical Standard Model representations. After electroweak symmetry breaking, the nonnegative singular values give Dirac masses $m_i=y_{i}v/\sqrt{2}$ and diagonal Higgs boson couplings $m_i/v$. When the neutrinos are massless, rotating their left-handed fields with the charged partners preserves the diagonal weak charged current.

An electroweak gauge boson is a field quantum associated with the $SU(2{)}_L\times U(1{)}_Y$ electroweak interaction. Before electroweak symmetry breaking, the gauge fields are the three weak-isospin fields $W^1,W^2,W^3$ and the hypercharge field $B$.

After electroweak symmetry breaking, charged combinations form the W bosons, while neutral combinations give the massive Z boson and massless photon. The W bosons mediate weak charged currents, changing the electric charge of the participating matter fields. The Z boson mediates a weak neutral current, and the photon couples to electric charge. The mixing explains why the neutral electromagnetic and weak interactions share an underlying gauge description despite having different low-energy ranges.

The electroweak interaction is the $SU(2{)}_L\times U(1{)}_Y$ gauge theory that unifies the weak and electromagnetic interactions.

An instructive tension motivates the unification: the photon is massless, while the W boson and Z boson are massive; electromagnetic interaction and weak interaction also have different observed low-energy strengths and chiral couplings. A gauge theory can relate their interactions while the Higgs field gives different masses after electroweak symmetry breaking. The resulting theory predicts relationships between processes, not merely a common name for observed phenomena.

Its useful organizing principle is the combination of weak isospin and hypercharge. The surviving electromagnetic generator is $Q=T_3+Y$ in the convention used here. The photon and Z boson are different mixtures of neutral gauge fields, while the charged current uses the charged combinations. The electroweak doublet gauge-boson mass matrix explains why one neutral mixture stays massless. At low energies, a massive W boson propagator instead produces the approximate Fermi interaction, withHere $g$ is the $SU(2{)}_L$ gauge coupling and $v$ the Higgs doublet vacuum expectation value. The inverse squared mediator mass suppresses low-energy weak rates. Unification does not require setting $g$ equal to the electromagnetic coupling $e$: their relation is $e=g\sin {\theta }_W$.

A striking experimental consequence is a neutral current in addition to the familiar charged current. The Gargamelle experiment tests this extra channel through visible charged secondaries, with a background problem fundamentally different from directly identifying a Z boson. The muon-neutrino electron recoil spectrum shows how the same Weinberg angle enters another process quantitatively.

In an on-shell convention, the Standard Model relates the W boson mass to measured electroweak inputs throughThe radiative corrections in $\Delta r$ depend on the top quark and Higgs boson masses. The leading heavy-top contribution to the electroweak rho parameter is $\Delta \rho \simeq 3G_F{m}_t^2/(8\sqrt{2}{\pi }^2)$, reflecting the large mass splitting in the top-bottom weak doublet. The Higgs dependence in the precision corrections is mainly logarithmic over the relevant range. Direct measurements of $m_t$ and $m_H$ therefore allow an indirect prediction of $m_W$; comparing it with a direct measurement tests loops as well as tree-level electroweak symmetry breaking. Any tension must include input uncertainties, experimental systematics and higher-order corrections.^[ref]

The electroweak scale is the characteristic energy associated with electroweak symmetry breaking. With the Standard Model Higgs doublet normalized by $⟨H⟩=(0,v/\sqrt{2}{)}^T$, its vacuum expectation value satisfies $v=(\sqrt{2}{G}_F{)}^{-1/2}\simeq 246\mathrm{GeV}$ and $m_W=gv/2$. Its smallness compared with possible ultraviolet scales motivates the hierarchy problem.

The physical Higgs boson interaction $\mathcal{L}\supset -(m_e/v)h\bar{e}e$ after electroweak symmetry breaking. It is distinguished from the unbroken-phase electron Yukawa coupling $y_e=\sqrt{2}{m}_e/v$. This small coefficient explains the electron Yukawa suppression of Higgs production in direct electron–positron annihilation.

In the Higgs mechanism, gauge fields associated with broken generators acquire longitudinal polarizations and masses by absorbing Goldstone modes. Gauge fields of the unbroken subgroup remain massless.

The phrase “absorbing” describes a rearrangement of degrees of freedom, not a collision in which a gauge boson swallows another particle. The gauge symmetry remains a redundancy of the description. A chosen vacuum expectation value leaves a smaller subgroup manifest, while gauge-invariant observables describe the physical masses.

A one-field calculation isolates the mechanism. In an Abelian Higgs model, write a complex scalar field locally as $\Phi =(v+H)e^{i\theta }/\sqrt{2}$, use $D_{\mu }={\partial }_{\mu }-ie{A}_{\mu }$, and define $B_{\mu }=A_{\mu }-{\partial }_{\mu }\theta /e$. ThenThe second term contains a Proca field mass $m_B=ev$ and its interactions with the radial Higgs boson. A local gauge transformation removes $\theta$ from the scalar field but leaves its information in the longitudinal vector polarization. The electroweak symmetry breaking calculation applies this idea to three broken directions.

In the MSSM, electroweak symmetry breaking mixes the neutral bino, wino and two neutral higgsinos into four neutral Majorana spinor mass eigenstates, the neutralinos. A neutralino can be the lightest supersymmetric particle; with conserved R-parity it then provides a stable possible dark matter candidate.

The Standard Model uses the local gauge group $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$. Its Higgs field is a complex Higgs doublet of hypercharge $Y=1/2$. A bare mass term for a weak gauge boson, or for a charged fermion joining inequivalent left- and right-handed gauge representations, would not respect the unbroken electroweak construction. Instead introduce a gauge-invariant Higgs potentialIts minima satisfy ${\Phi }^{†}\Phi =v^2/2$, $v^2={\mu }^2/\lambda$. Choosing a representative $⟨\Phi ⟩=(0,v{)}^T/\sqrt{2}$ leaves $Q=T_3+Y$ unbroken. In this description electroweak symmetry breaking is $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_{\mathrm{em}}$; the underlying local gauge symmetry remains a redundancy of the theory. A vacuum representative is not a physical explicit violation of gauge invariance.

In unitary gauge, write $\Phi =(0,v+H{)}^T/\sqrt{2}$. The gauge-covariant kinetic term $|D_{\mu }\Phi {|}^2$ produceswith $\tan {\theta }_W=g^{\prime }/g$ and $e=g\sin {\theta }_W$. Three angular scalar directions supply the longitudinal gauge-boson polarizations of $W^{+},W^{-},Z$, while one real radial Higgs boson survives with $m_H^2=2\lambda v^2$. Counting physical degrees of freedom checks the Higgs mechanism: the electroweak gauge sector and scalar doublet have $4\times 2+4=12$ before the rearrangement and $3\times 3+2+1=12$ afterwards.

The Higgs potential also fixes the scalar self-interactions:The color gauge group is unbroken, so gluons remain massless. The same electroweak kinetic term fixesGauge-invariant Yukawa couplings such as $-y_d{\bar{Q}}_L\Phi d_R-y_u{\bar{Q}}_L\tilde{\Phi }{u}_R+h.c.$ give $m_f=y_{f}v/\sqrt{2}$ and the physical coupling $-m_{f}H\bar{f}f/v$. These relations predict stronger Higgs couplings to heavier massive particles; they do not determine the Yukawa parameters themselves. The minimal Standard Model does not supply observed neutrino masses. The surviving Higgs boson is neutral and has spin zero; its mass and self-coupling are tied to the curvature of the Higgs potential, not fixed by the gauge couplings alone. Higgs exchange also provides the Higgs cancellation in longitudinal vector scattering, maintaining the required high-energy behavior of weak amplitudes.

At a proton collider, important production mechanisms are gluon-fusion Higgs production through a heavy-quark loop, vector-boson fusion with two outgoing quark jets, associated $WH$ or $ZH$ production, and production with a top quark pair. At an electron–positron collider, Higgsstrahlung $e^{+}{e}^{-}\to ZH$ is a clean example. Tree-level decays include Higgs decay to a fermion pair and, when allowed, Higgs decay to two W bosons or Higgs decay to two Z bosons. Below an on-shell vector-pair threshold, one or both vectors can be virtual. The $gg$ and $\gamma \gamma$ channels arise through loops; neutral colorless $H$ has no corresponding direct tree-level gauge coupling. Rates and branching fractions depend on both couplings and kinematics, so “heaviest available particle” is a guide rather than a complete rule.

The Higgs boson discovery in 2012 supplied an experimental test. The ATLAS experiment and CMS experiment reconstructed two-photon and four-lepton invariant masses in Large Hadron Collider proton collisions. Electromagnetic calorimeters measure photon energies, while tracking and muon systems identify charged leptons; narrow reconstructed peaks can be distinguished from smooth backgrounds with control samples and a likelihood analysis. The ATLAS discovery paper reported a new resonance near $126.5\mathrm{GeV}$. An observed mass peak supports the existence of a new state; production channels, decay rates and angular distributions test whether it has the particular Higgs couplings and quantum numbers.^[ref]

Finally, the hierarchy problem asks why the electroweak scalar mass scale remains much smaller than a putative heavy physical scale without delicate cancellations. A cutoff estimate suggests corrections $\delta m_H^2\sim {\Lambda }^2/(16{\pi }^2)$ times couplings, but the quadratic cutoff term itself depends on the regularization convention. The physical concern persists when a heavy state of mass $M$ couples to the Higgs field and produces threshold corrections of order its coupling times $M^2/(16{\pi }^2)$. Matching a much smaller Higgs mass may then require cancellation between independent parameters. Supersymmetry, compositeness and other protective structures propose different resolutions; renormalizability of the Standard Model does not by itself resolve this sensitivity or establish that new particles must occur at one particular mass.

The left- and right-chiral components of a charged Standard Model fermion transform in different electroweak group representations. A bare Dirac mass term $-m{\bar{f}}_L{f}_R+h.c.$ is therefore not invariant under the full electroweak gauge group. The Higgs doublet supplies the missing transformation properties. With $\tilde{H}=i{\sigma }_2{H}^{\ast }$, the gauge-invariant Yukawa interactions arewhere each $Y_f$ is a flavor-space Yukawa matrix. For example, hypercharges in the down-quark term add to $-1/6+1/2-1/3=0$, while those in the up-quark term add to $-1/6-1/2+2/3=0$. Thus the Higgs and conjugate Higgs are required in different terms.

After electroweak symmetry breaking, choose unitary gauge with $H=(0,(v+h)/\sqrt{2}{)}^T$. The fermion mass matrices becomeA singular value decomposition $U_{fL}^{†}{M}_f{U}_{fR}=\mathrm{diag}(m_{f,1},m_{f,2},m_{f,3})$ defines nonnegative physical tree-level masses. The mismatch $U_{uL}^{†}{U}_{dL}$ is the Cabibbo-Kobayashi-Maskawa matrix in the weak charged current. The corresponding physical Higgs boson couples diagonally as $-(m_f/v)h\bar{f}f$ in the minimal one-doublet theory. This relates its coupling strength to each mass, but the measured Yukawa couplings remain independent input parameters: the minimal theory does not predict their hierarchy.

The minimal renormalizable Standard Model contains no right-handed neutrino, so it has neither a neutrino Dirac Yukawa term nor a gauge-invariant renormalizable left-handed Majorana mass term. Its neutrinos are massless. Observed neutrino oscillations require going beyond that minimal field content or operator set: adding right-handed singlets permits Dirac masses, while the dimension-five Weinberg operator gives a Majorana neutrino mass of order $v^2/\Lambda$ after symmetry breaking. These alternatives should be distinguished from the charged-fermion Yukawa mechanism.

Chirality distinguishes the two inequivalent Weyl spinor representations of the proper Lorentz group. For a Dirac spinor, define ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and the chiral projectorsThey satisfy $P_L{P}_R=0$ and $P_L+P_R=I$. Proper Lorentz transformations preserve each chiral representation, whereas parity exchanges them. Helicity instead measures the projection of spin angular momentum along momentum. For a massive particle it can change under a boost that overtakes the particle, so it must not be identified with chirality. For massless positive-energy particles the left-chiral field annihilates negative-helicity particles and creates positive-helicity antiparticles.

The Standard Model is a chiral gauge theory: left- and right-chiral fields have different gauge group representations. In the convention $Q=T_3+Y$, one generation containsThe minimal field content has no right-handed neutrino. A sterile right-handed field is a possible extension, not a required member of that minimal list. The table displays weak isospin, hypercharge and color charge assignments, not assignments of massive-particle helicity.

The weak charged current contains $\bar{\nu }{\gamma }^{\mu }{P}_{L}e{W}_{\mu }^{+}$ and its Hermitian conjugate, with coupling $g/\sqrt{2}$. Only the left-chiral lepton doublet participates, giving parity violation. Both chiralities can couple to the Z boson, but their coefficients differ:Here $T_3$ is that of the left-chiral component. For the electron, it is $-1/2$. The photon and gluon couplings of a charged massive fermion are vectorlike; this does not make the full electroweak interaction vectorlike.

A bare Dirac mass term $m\bar{\psi }\psi =m({\bar{\psi }}_L{\psi }_R+{\bar{\psi }}_R{\psi }_L)$ joins the two chiral representations. Their different electroweak quantum numbers forbid that term before electroweak symmetry breaking. A Yukawa interaction, such as $-y_e{\bar{L}}_{L}H{e}_R+h.c.$, is gauge invariant; the Higgs doublet expectation value then gives $m_e=y_{e}v/\sqrt{2}$. Up-type quarks use the conjugate doublet. Thus chirality explains the weak charge assignments, their parity violation, and why masses arise through the Higgs mechanism. The consistency of these chiral assignments also requires Standard Model anomaly cancellation.

Gauge boson self-interactions follow from noncommuting generators, rather than a separate assumption that the force carriers should attract one another. With $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }^a{T}^a$ and $[T^a,T^b]=i{f}^{abc}{T}^c$, the Yang-Mills field strength isThe last term is required so that $F_{\mu \nu }$ transforms covariantly under the gauge group. Let $G_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a$. Expanding $\mathcal{L}=-F_{\mu \nu }^a{F}^{a\mu \nu }/4$ gives a quadratic kinetic term and the cubic and quartic Yang-Mills self-interactionsThe structure constants and one gauge coupling fix both types of vertex. For an Abelian group the structure constants vanish: the renormalizable pure electromagnetic field action has no elementary three- or four-photon vertex. Photon scattering through charged-particle loops is a different statement, not a contradiction.

In quantum chromodynamics, the non-Abelian $SU(3{)}_c$ structure produces three- and four-gluon vertices; the gluons themselves carry color charge. These vertices influence multi-jet production and the renormalization group running of the strong coupling. In the electroweak sector the $SU(2{)}_L$ fields mix with the Abelian field after electroweak symmetry breaking. The resulting charged-vector vertices include $W^{+}{W}^{-}\gamma$ and $W^{+}{W}^{-}Z$, with couplings $e$ and $g\cos {\theta }_W$, together with the related four-boson vertices. Neutral-only triple vertices such as $ZZ\gamma$ are absent at tree level in the Standard Model.

The electroweak triple gauge coupling can be measured using $e^{+}{e}^{-}\to W^{+}{W}^{-}$. The neutrino exchange and neutral-vector exchange diagrams interfere: the gauge-related coefficients cancel the most rapidly growing longitudinal-polarization terms. Their relative strengths and angular dependence therefore test the theory more sharply than merely observing a pair of massive vector bosons. The electron-mass-suppressed Higgs contribution completes the corresponding mass-dependent cancellation. Adding arbitrary independent three- and four-vector couplings would generally destroy these relations.

At the Large Electron–Positron Collider, the L3 experiment fitted charged triple-gauge parameters using hadronic and semileptonic W-pair events, complemented by single-W and single-photon channels. These measurements agreed with the predicted self-couplings. Their interpretation uses reconstructed decay products and differential distributions, rather than direct pictures of a three-boson vertex.^[ref]

Use the current convention in the question, whose vertex prefactor is $g/(2\cos {\theta }_W)$. The neutral-current vector and axial couplings are thenFor a charged lepton, $T_3=-1/2$, $Q=-1$, givingFor an active neutrino, $T_3=1/2$, $Q=0$, so $c_V=c_A=1/2$. The right-handed weak singlet contribution is purely vectorlike in the charge term; combining it with the left-handed contribution gives precisely this Dirac-current form. These coefficients would be halved if the vertex prefactor were $g/\cos {\theta }_W$ instead.

Let $p=k+k^{\prime }$ be the Z boson momentum and $k,k^{\prime }$ the outgoing massless lepton momenta. The decay amplitude isUse the fermion spin sums $\sum u\bar{u}=/k$ and $\sum v\bar{v}=/k^{\prime }$. Anticommutation of ${\gamma }_5$ with the Dirac gamma matrices gives the symmetric part of the trace,The vector-axial cross term is antisymmetric in $\mu ,\nu$, and hence drops out of the massive vector polarization sum. The massless equations also give $p_{\mu }{L}^{\mu \nu }=0$: in the symmetric part, the first two contractions sum to $p^{\nu }k\cdot k^{\prime }$, canceling the last; the antisymmetric term vanishes because $p=k+k^{\prime }$. The longitudinal $p_{\mu }{p}_{\nu }/m_Z^2$ term therefore makes no contribution.

Since $2k\cdot k^{\prime }=m_Z^2$, contraction with $-g_{\mu \nu }$ gives $4(c_V^2+c_A^2)m_Z^2$. Thus the massless fermion Z decay spin sum, including all three initial polarizations, isTo use this fully summed expression for the decay width of one unpolarized Z boson, apply the massive-vector spin average, dividing by three. The width of each fixed polarization is the same after integration over all directions, so that procedure also gives the physical single-particle width. Omitting the division would sum the widths of three initial states.

For two massless distinguishable final particles, the two-body Lorentz-invariant phase space isConsequentlyThe tree-level electroweak symmetry breaking relation is $m_W=m_Z\cos {\theta }_W$. Combining it with the Fermi constant normalization givesFor an active neutrino, the vertex itself projects out inactive chiralities, so no additional helicity factor is needed. This is the tree-level massless leptonic Z decay width; lepton masses and radiative corrections have been excluded.

Quark mixing arises because the up-type and down-type quark Yukawa matrices need not be diagonal in the same weak interaction basis. In the Standard Model the three left-handed quark doublets $Q_L^{\prime }$ transform as $(3,2,1/6)$ under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, whereas $u_R^{\prime }$ and $d_R^{\prime }$ transform as $(3,1,2/3)$ and $(3,1,-1/3)$. With a Higgs doublet $H$ and its conjugate $\tilde{H}=i{\sigma }^2{H}^{\ast }$, the gauge-invariant Yukawa interaction isThe generation indices are matrix-multiplied. Electroweak symmetry breaking, $⟨H⟩=(0,v_H/\sqrt{2}{)}^T$, produces $M_u=v_H{Y}_u/\sqrt{2}$ and $M_d=v_H{Y}_d/\sqrt{2}$.

Each complex fermion mass matrix admits a biunitary diagonalization,where the diagonal entries are real nonnegative quark masses. Define $u_L^{\prime }=U_{uL}{u}_L$ and $d_L^{\prime }=U_{dL}{d}_L$, and similarly for right-handed fields. The kinetic terms retain their canonical form because the transformations are unitary matrices. However, the weak charged current becomesThis unitary matrix is the Cabibbo-Kobayashi-Maskawa matrix. Its entries multiply transitions between up-type and down-type mass eigenstates. The minimal Standard Model has no analogous right-handed charged weak current.

Quark mixing does not introduce tree-level flavor-changing neutral currents. Photon, gluon and weak neutral current couplings are proportional to the identity within each fixed-charge quark sector, and $U^{†}IU=I$. The single Higgs doublet also has diagonal neutral scalar field couplings after the same transformations that diagonalize the masses. At higher loop orders, flavor changes are possible. For example, an $s\to d$ amplitude contains ${\sum }_i{V}_{is}^{\ast }{V}_{id}f(m_i^2)$. Unitarity of the CKM matrix sets the sum of coefficients to zero, cancelling any mass-independent part. This is the GIM mechanism; mass splittings leave a suppressed, generally nonzero amplitude.

For $N$ generations a unitary matrix has $N^2$ real parameters: $N(N-1)/2$ mixing angles and $N(N+1)/2$ phases. Rephasing the $N$ up-type and $N$ down-type quark fields removes $2N-1$ phases; a common baryon number phase leaves every matrix entry unchanged. Hence the physical numbers areTwo generations therefore give a real Cabibbo rotation,with strangeness-changing charged-current transitions suppressed by $\sin {\theta }_C$. Three generations have three mixing angles and one irreducible weak CP phase. This counting assumes nondegenerate masses; degeneracies allow additional basis rotations.

Writing $s_{ij}=\sin {\theta }_{ij}$ and $c_{ij}=\cos {\theta }_{ij}$, a standard exact parametrization isComplex entries alone do not establish CP violation, because quark quantum field rephasings can change their phases. For nondegenerate quark masses the weak mixing sector conserves CP precisely when a basis makes $V$ real. A useful invariant test is the Jarlskog invariant,A nonzero $J$ requires all three mixing angles and a nontrivial phase. If two same-charge quark masses coincide, the additional basis freedom removes the corresponding physical CP effect. The possible QCD theta angle is a separate CP parameter, not part of this weak-mixing count.

The Wolfenstein parametrization displays the hierarchy of quark mixing economically. Put ${\lambda }_W=s_{12}$, $s_{23}=A{\lambda }_W^2$, and $s_{13}{e}^{i\delta }=A{\lambda }_W^3(\rho +i\eta )$. Expanding the exact matrix givesThe observed pattern is predominantly diagonal, with $1$-$2$ mixing largest and $2$-$3$, $1$-$3$ mixing successively suppressed. The expansion implies $J\simeq A^2{\lambda }_W^6\eta$. The truncated matrix must not be treated as an exactly unitary matrix.

Column orthogonality givesThe three complex numbers form a CKM unitarity triangle, with area $|J|/2$. Dividing by $-V_{cd}{V}_{cb}^{\ast }$ puts its vertices at $0$, $1$ and $\bar{\rho }+i\bar{\eta }$, where $\bar{\rho }+i\bar{\eta }=-V_{ud}{V}_{ub}^{\ast }/(V_{cd}{V}_{cb}^{\ast })$. The barred parameters include higher-order corrections to the Wolfenstein parameters. Comparing independent measurements of its sides and angles tests both quark mixing and the absence of additional flavor dynamics.

Semileptonic decay rates determine magnitudes of CKM entries through factors $|V_{ij}{|}^2$ once hadronic matrix elements are controlled. Neutral kaon and neutral B-meson mixing constrain products of entries through weak box scattering amplitudes. CP asymmetries probe their invariant relative phases, with kaon mixing providing indirect CP violation and decay amplitudes providing direct CP violation. These processes complement tests of the unitarity constraints on rows and columns. The minimal three-generation Standard Model describes weak quark mixing by a unitary matrix with three angles and one physical CP phase; it does not predict the Yukawa couplings or their hierarchy.

There are two distinct parts of the hierarchy problem. The first asks why the scale of electroweak symmetry breaking is so much smaller than a fundamental scale such as the Planck mass. The second, the technical hierarchy problem, asks why that separation survives quantum corrections once it is chosen. With a fundamental Higgs boson, a mass term $m_H^2{H}^{†}H$ is allowed by the ordinary gauge symmetry, so heavy thresholds can contribute additive terms of order a heavy mass squared. This is different from the chiral protection of a fermion mass and the gauge protection of a vector mass.

For example, a cutoff description of the Standard Model gives the schematic one-loop sensitivityThe regulator-dependent quadratic term is not itself an observable, but a physical heavy particle of mass $M$ coupled to the Higgs likewise produces a matching correction of order coupling squared times $M^2/(16{\pi }^2)$. Maintaining $|m_H^2|\ll M^2$ then generically requires cancellation between large unrelated terms. This is the naturalness concern; changing the regulator does not eliminate the heavy-threshold issue.

Exact supersymmetry relates bosonic and fermionic couplings and masses. Their loop contributions have opposite signs, and the relations enforce supersymmetric cancellation of quadratic divergences. Boson-fermion degeneracy in a supermultiplet also transfers fermionic mass protection to its scalar partners. It is not sufficient merely to add arbitrary scalars: the supersymmetric coupling relations are what make the cancellation persist. Perturbative superpotential non-renormalization further protects holomorphic mass parameters, although wave-function renormalization still makes canonically normalized couplings run.

For a realistic separation between ordinary particles and their partners, supersymmetry must be broken. Soft supersymmetry breaking introduces masses and selected positive-mass-dimension interactions without reinstating the original ultraviolet quadratic divergence. A paired propagator difference behaves at high Euclidean momentum asso a residual scalar correction is of the orderThis soft scalar-mass sensitivity controls the technical problem if the relevant soft masses and thresholds are not too far above the weak scale. Very heavy partners can still require tuning through finite and logarithmic terms. Supersymmetric cancellations alone therefore do not guarantee a naturally small weak scale for an arbitrary broken spectrum.

The origin of the hierarchy needs further dynamics. In dynamical supersymmetry breaking, an asymptotically free hidden supersymmetry-breaking sector can generate an exponentially smaller scale by dimensional transmutation,If that sector actually has no supersymmetric vacuum, its breaking can be communicated to the visible sector to produce small soft terms. The exponential creates a hierarchy without specifying a tiny dimensionless coupling at the high scale, but the existence of breaking and its mediation are additional model-building requirements. In the MSSM, running Higgs soft masses can trigger electroweak symmetry breaking. The supersymmetric mu problem remains: the allowed superpotential mass $\mu H_u{H}_d$ must also be near the soft scale, and non-renormalization protects a chosen small $\mu$ without explaining its origin.

Extra-dimensional proposals use geometry to explain why observable gravitational and weak scales differ. In large extra dimensions, assume gravity propagates in $4+n$ dimensions with fundamental scale $M_{\ast }$, while ordinary fields are localized on a brane. Integrating the Einstein-Hilbert action over compact volume $V_n$ gives the Planck mass from compactification volume relationHere $M_{\mathrm{Pl}}$ denotes the four-dimensional reduced Planck mass. A sufficiently large internal volume lets $M_{\ast }$ be near the weak scale even though $M_{\mathrm{Pl}}$ is enormous. The Higgs need then be stable only up to the lower fundamental cutoff. This explains the weakness of long-distance gravity geometrically but trades the scale question for the origin and stabilization of a large compact volume. Ordinary matter is localized to avoid an unwanted low-mass tower of its own Kaluza-Klein modes; gravitational modes, departures from four-dimensional gravity and bulk energy loss impose phenomenological requirements.

The Randall–Sundrum model instead uses a warped five-dimensional interval, conventionally the doubled $S^1/{\mathbb{Z}}_2$ orbifold with fixed branes at $0$ and $L$:For a scalar confined to the distant brane, the kinetic and mass terms carry factors $e^{-2kL}$ and $e^{-4kL}$. Canonical rescaling therefore producesA moderately large $kL$ yields an exponentially small observable mass without a comparably large compact-volume factor. The inter-brane size is the radion, so one still needs a mechanism fixing its expectation value. Consistent curvature, brane tensions and ultraviolet physics are also part of the model. Warping is a redshift mechanism; it does not provide the same cancellation of arbitrary scalar loops as supersymmetry.

Finally, protecting a Higgs mass is not the same as solving the vacuum-energy hierarchy. Exact global supersymmetry has zero-energy supersymmetric vacua, but broken supersymmetry generically leaves vacuum energy set by its breaking dynamics. In supergravity, positive auxiliary-field contributions coexist with a negative superpotential term, and obtaining a very small cosmological constant requires further structure or cancellation. Neither soft supersymmetry nor the two geometric mechanisms automatically resolves that problem. The mechanisms can also coexist in one higher-dimensional supersymmetric theory.

Use metric $g_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ and the hypercharge convention $Q=T^3+Y$. Then the Higgs doublet has $Y=1/2$ and transforms under electroweak gauge invariance asThe covariant derivative sign consistent with the requested expression isA convention that instead calls the doublet hypercharge one puts $Y/2$ in the charge generator and derivative; the physical coupling is the same. Here $B_{\mu }\mapsto B_{\mu }-{\partial }_{\mu }\beta /g^{\prime }$ and the matrix-valued $SU(2)$ connection transforms as $A_{\mu }\mapsto U{A}_{\mu }{U}^{-1}+(i/g)({\partial }_{\mu }U)U^{-1}$, making $D_{\mu }\varphi$ transform like $\varphi$.

In unitary gauge, the upper component of the derivative comes from the off-diagonal Pauli matrices:The lower component isWriting $c_W=\cos {\theta }_W$, $s_W=\sin {\theta }_W$, inversion of the neutral rotation gives $A^3=c_{W}Z+s_{W}A$ and $B=-s_{W}Z+c_{W}A$. Since $g^{\prime }=g\tan {\theta }_W$, the coefficient of the photon vanishes, whereas $-g{c}_W-g^{\prime }{s}_W=-g/c_W$. ThusThe zero photon coupling expresses the neutrality of the surviving radial Higgs boson.

The scalar Lagrangian density is ${\mathcal{L}}_{\varphi }=(D_{\mu }\varphi {)}^{†}{D}^{\mu }\varphi -V$. Adding $-F_{\mu \nu }^a{F}^{a\mu \nu }/4-B_{\mu \nu }{B}^{\mu \nu }/4$ gives the complete bosonic electroweak Lagrangian density. Its scalar and vector mass terms, after electroweak symmetry breaking, are obtained directly asIndeed, the shifted quartic Higgs potential normalization gives ${\varphi }^{†}\varphi -v^2/2=vH+H^2/2$. Comparing with a canonically normalized real scalar mass term, a complex vector mass term and a real vector mass term yieldsThe extra factor one half in this paper's Higgs potential is why $M_H^2$ is $\lambda v^2$, rather than $2\lambda v^2$ in another common convention. The tree-level electroweak gauge-boson masses give $\cos {\theta }_W=M_W/M_Z$: measure the two vector masses and take their ratio. Beyond tree level this extraction requires a specified renormalization convention and radiative corrections.

Expanding the vector terms gives the cubic and quartic Higgs couplings to electroweak gauge bosons:The four elementary Feynman diagrams and their all-incoming Feynman rules are shown below. Differentiating with respect to two identical $Z$ fields supplies a factor two; differentiating with respect to two identical Higgs fields supplies another factor two in the quartic rules.

There are no elementary $H\gamma \gamma$, $HZ\gamma$, $HH\gamma \gamma$ or $HHZ\gamma$ vertices in this tree-level Lagrangian density. Loop-induced photon interactions are a different order. The scalar self-interactions, if included, have rules $HHH:-3i{M}_H^2/v$ and $HHHH:-3i{M}_H^2/v^2$.

The prime LEP2 search used the Higgs boson coupling to Z bosons through Higgsstrahlung, $e^{+}{e}^{-}\to Z^{\ast }\to ZH$, shown in the first panel of the process figure in Question 3. This identifies an associated-production interaction, rather than an on-shell decay of a single $Z$ into a heavier Higgs. The channel's historical role is documented in the LEP2 Higgs-search account.

For the free charged W boson, take $W_{\mu }(x)=\int d^{4}p{e}^{-ipx}{W}_{\mu }(p)/(2\pi {)}^4$, with the conjugate field carrying $e^{ipx}$. In the action, multiplying the two field strengths gives $2[p^2{W}_{\mu }^{†}{W}^{\mu }-(p\cdot W^{†})(p\cdot W)]$. ThereforeTo invert it, separate the transverse and longitudinal projectors $P_T=g-pp/p^2$, $P_L=pp/p^2$. The operator is $(p^2-M_W^2)P_T-M_W^2{P}_L$, soSince the quadratic action kernel is $-O$, its Proca propagator isFor all momentum components small relative to $M_W$, the low-momentum limit is $D_{\mu \nu }\simeq i{g}_{\mu \nu }/M_W^2$, with corrections of relative order $p^2/M_W^2$ or $p_{\mu }{p}_{\nu }/M_W^2$. This local limit leads to the Fermi interaction, with $G_F/\sqrt{2}=g^2/(8M_W^2)$ for the charged-current normalization used here.

Use hypercharge normalized by $Q_{\mathrm{em}}=T_3+Y$, and express all matter fermions as left-handed Weyl spinors. Thus $u^c,d^c,e^c$ denote the charge conjugates of the right-handed Standard Model fermions; the superscript $c$ is unrelated to the antiparticle marker $(A)$. Each chiral superfield contains a complex scalar field and a Weyl spinor, while each vector superfield contains a gauge boson and a gaugino. The complete propagating content of the Minimal supersymmetric Standard Model before electroweak symmetry breaking is: