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The hierarchy problem begins with a large separation of scales: the weak scale is much smaller than possible unification or gravitational scales. A fundamental scalar mass is especially sensitive to a much heavier scale. In an effective description with ultraviolet cutoff $\Lambda$, one schematically findsThe coefficient $c$ contains gauge, scalar and Yukawa couplings, with different signs for bosonic and fermionic loops. The explicit quadratic cutoff dependence is regulator-dependent, but a heavy physical particle coupled to the Higgs produces a threshold correction of order its mass squared. That heavy-threshold sensitivity is the physical difficulty.

The technical hierarchy problem asks whether a small scale, once chosen, remains stable under such radiative corrections. Maintaining $m_H^2\ll {\Lambda }^2$ by cancelling unrelated bare and loop contributions, and readjusting the cancellation at successive orders, is the naturalness concern. It is different from explaining why the small scale was chosen in the first place. Supersymmetry primarily supplies protection against the technical instability; a complete explanation of the origin of supersymmetry-breaking scales needs additional dynamics.

The relevant comparison of mass types is:

In exact supersymmetry, each supermultiplet has matched bosonic and fermionic degrees of freedom, with their interaction strengths related. Opposite loop signs then give supersymmetric cancellation of quadratic divergences. This is not just equality of state counts: the supersymmetric relation between quartic and Yukawa couplings is also essential. A schematic paired-loop contribution has the high-momentum formFor equal partner masses the displayed contributions cancel. With a small splitting, the difference falls as $(m_F^2-m_B^2)/p_E^4$, so the remaining ultraviolet sensitivity is logarithmic, proportional to the splitting rather than to ${\Lambda }^2$.

Realistic partners need not have exactly equal masses. Soft supersymmetry breaking permits scalar squared masses, gaugino masses and suitable trilinear interactions without restoring the unwanted quadratic sensitivity. Typically,up to coefficients and threshold details. Squarks and sleptons can therefore be heavier than their chiral fermion partners while the scalar sector remains stable against the much larger ultraviolet scale. Nevertheless, very large soft masses, particularly in the Higgs-coupled sector, leave large finite or logarithmic corrections and require tuning. This soft scalar-mass sensitivity remains after the quadratic divergence has cancelled. Supersymmetry solves the quadratic radiative instability when breaking is suitably soft; it does not make arbitrarily heavy superpartners natural or explain the entire hierarchy by itself.

Take real field coordinates ${\varphi }^I$; for complex scalar fields, split them into real and imaginary parts. Assume a smooth scalar potential and positive, canonically normalized kinetic terms in a relativistic theory. At a classical vacuum, $V_I({\varphi }_0)=0$. The Taylor expansion isThus the scalar mass matrix is the Hessian matrix, and its eigenvalues give squared masses of the linearized scalar excitations.

Put $X_a(\varphi )=i{t}^a\varphi$. Infinitesimal invariance of the scalar potential gives the identity $V_I{X}_a^I=0$ at every field value. Differentiate with respect to ${\varphi }^J$ and evaluate at the classical vacuum:Every nonzero infinitesimal symmetry direction therefore lies in the kernel of the scalar mass matrix. This is the classical Goldstone theorem expressed through Goldstone directions in the scalar mass matrix.

Let $H_0$ be the full stabilizer subgroup of ${\varphi }_0$. The linear map from the Lie algebra of $G$ to field space, $T\mapsto iT{\varphi }_0$, has kernel equal to the Lie algebra of $H_0$. The rank-nullity theorem givesThere are consequently at least $\dim G-\dim H_0$ massless scalar directions, namely the tangent directions to the symmetry orbit through the classical vacuum. If the intended $H$ is $H_0$, these are the $\dim G-\dim H$ symmetry-required Goldstone bosons. Exactly that many massless modes occur if the scalar mass matrix is positive definite on a complement of those tangent directions. A nonsingular positive field-space kinetic metric changes normalization, but not the number of zero masses.

Two qualifications are needed for the literal assumptions. A subgroup fixing the classical vacuum need not be the full stabilizer subgroup. For example, take $G=SO(3)$ acting on a real triplet and $V=\lambda (|\varphi {|}^2-v^2{)}^2/8$ with $\lambda ,v>0$. At ${\varphi }_0=v{e}_3$, the scalar mass matrix is $\lambda v^2\mathrm{diag}(0,0,1)$: there are two massless modes. Choosing $H=\{1\}$ satisfies the printed invariance condition but would incorrectly predict three. The full stabilizer subgroup is $SO(2)$.

Even with the full stabilizer subgroup, symmetry does not exclude an accidental massless scalar. Take $G=SO(2)$ rotating $(x,y)$, with an invariant singlet $z$, andAt $(v,0,0)$ the full continuous stabilizer is trivial, but the scalar mass matrix is $\mathrm{diag}(2\lambda v^2,0,0)$. One zero direction is the Goldstone boson; the other is an accidental massless scalar at quadratic order. The proof establishes the symmetry-required count, not unconditional equality with the total number of massless fields. The statement concerns global internal symmetry; gauging it changes the physical interpretation through the Higgs mechanism.

Assume $\lambda >0$ and $v>0$. The vacuum manifold is the sphere $|\varphi |=v$. Choose ${\varphi }_0=v{e}_3$. For the given generators, $t^3{\varphi }_0=0$ whereas $t^1{\varphi }_0$ and $t^2{\varphi }_0$ are nonzero and independent. The Adjoint representation of SU(2) rotates this sphere, and the full connected stabilizer subgroup of $v{e}_3$ is generated by $t^3$. Hence the classical symmetry pattern isThe central element of SU(2) acts trivially on the triplet and belongs to this stabilizer; it does not add a separate broken direction. The gauge-field-free representative has zero gauge field strength and constant scalar magnitude. Locally near this nonzero classical vacuum, unitary gauge aligns the triplet along the third axis.

First make the gauge normalization explicit. Write the gauge kinetic term as $-\mathcal{N}{F}_{\mu \nu }^a{F}^{a\mu \nu }/4$, where $\mathrm{Tr}(t^a{t}^b)=\mathcal{N}{\delta }^{ab}$ if $F=F^a{t}^a$ is the matrix used inside the printed trace. Canonical normalization of a gauge kinetic term givesFor a conventionally normalized component trace, $\mathcal{N}=1$ and $g_c=g$. If the trace is the ordinary matrix trace in the displayed three-dimensional representation, $\mathrm{Tr}(t^a{t}^b)=2{\delta }^{ab}$ and $\mathcal{N}=2$. The PDF does not specify which trace convention is intended; both are covered by this formula.

In unitary gauge, the gauge covariant derivative isExpanding this kinetic term and the scalar potential gives the complete physical-field LagrangianHere a square of a vector means its contraction with the Minkowski metric, in signature $(+---)$. The labels 1 and 2 represent massive vectors; 3 represents the surviving Abelian vector.

To display their charge and all interactions more clearly, put $a_{\mu }={\mathcal{A}}_{\mu }^3$, $W_{\mu }^{\pm }=({\mathcal{A}}_{\mu }^1\mp i{\mathcal{A}}_{\mu }^2)/\sqrt{2}$, and defineWith the sign of gauge field strength printed in the paper, ${\mathcal{F}}^3=f+i{g}_{c}C$ and $G^{\pm }=({\mathcal{F}}^1\mp i{\mathcal{F}}^2)/\sqrt{2}$. Thus the same Lagrangian becomesThe physical masses areIn particular, literal adjoint matrix trace gives $m_W=gv/\sqrt{2}$; the standard canonical component convention gives $m_W=gv$. These are descriptions with differently normalized couplings, not different physical spectra.

The complex vector pair carries opposite charges under the unbroken U(1) gauge symmetry. The gauge field strength terms contain $a{W}^{+}{W}^{-}$ cubic interactions, $aa{W}^{+}{W}^{-}$ quartic interactions, and four-vector interactions involving the charged fields. There is no pure Abelian cubic or quartic self-interaction. The Higgs mode $\eta$ has cubic and quartic scalar interactions and couples through $2g_c^{2}v\eta W^{+}{W}^{-}+g_c^2{\eta }^2{W}^{+}{W}^{-}$. It is neutral and has no tree-level $\eta aa$ interaction. This is the physical charged-vector Lagrangian for an adjoint SU2 Higgs model.

If the angular fields ${\pi }_1$ and ${\pi }_2$ were retained, their vanishing potential masses would identify the two Goldstone bosons of the ungauged triplet. In the gauge theory they mix with the broken-direction gauge fields and can be removed by gauge fixing; the Higgs mechanism uses them as the longitudinal polarizations of the two massive vectors. They are not additional physical massless scalars. The physical degrees of freedom are conserved: $3+3\times 2=9$ before rearrangement, and $1+2\times 3+2=9$ afterwards.

This is not the Standard Model electroweak interaction. It has three original gauge bosons, leaving two massive charged vectors and one massless neutral vector, with no massive neutral Z boson. The Standard Model instead has $SU(2{)}_L\times U(1{)}_Y$, a complex Higgs doublet, and three massive vectors plus the photon. Its charge is $Q=T_3+Y$, not just the surviving $T_3$. Adding fermions cannot supply the missing gauge generator or turn the surviving neutral vector into both a photon and a Z boson. In particular, the usual right-handed fermions in the Standard Model are SU(2) singlets; they would have zero charge if only $T_3$ were available, instead of the charges produced by hypercharge.

Take the Minkowski metric with signature $(+---)$ and a coupling $e\ne 0$. For a charged scalar field use the gauge covariant derivativeA scalar electrodynamics Lagrangian isAny real potential bounded below and depending only on the modulus gives gauge invariance. Indeed,The derivative of the phase cancels the shifted gauge field in the first identity; commuting partial derivatives proves the second. These identities verify invariance of every term, including the interaction hidden in the kinetic term.

For an unbroken example choose $V=m_{\varphi }^2|\varphi {|}^2+\lambda |\varphi {|}^4$ with $m_{\varphi }^2>0$ and $\lambda >0$. The minimum is at zero. There is no vector mass term in the quadratic expansion and no Higgs mechanism. Pure electromagnetic theory has a neutral massless spin-one photon with two physical transverse polarizations, equivalently helicities $+1$ and $-1$; longitudinal and time-component polarizations are gauge redundancies. In the unbroken scalar theory, the same massless photon is accompanied by a spin-zero charged particle and its oppositely charged antiparticle, both of mass $m_{\varphi }$. A complex scalar has two real physical degrees of freedom, rather than two unrelated charged species.

For a Higgs example chooseThe vacuum modulus is nonzero. Around one vacuum representative, unitary gauge removes the phase and writes $\varphi =(v+h)/\sqrt{2}$. ThenThus the quadratic spectrum isThe gauge boson is a massive spin-one particle with three polarizations, and $h$ is a neutral massive spin-zero Higgs boson. The scalar phase supplies the longitudinal vector polarization; it is not an extra physical massless Goldstone boson. The degree count is unchanged: two massless-vector polarizations plus two scalar degrees become three massive-vector polarizations plus one radial scalar degree. The underlying gauge invariance remains a redundancy of the description.

For the two-charge theory, useand retain the same gauge transformation of $a_{\mu }$. Both derivatives transform with the phase of their own field. A manifestly stable two-charge scalar gauge potential is, for positive $m_{\varphi }^2,M^2,{\lambda }_{\varphi },{\lambda }_{\psi }$, $g\ge 0$ and a nonzero complex constant $b$,Since $\psi -b{\varphi }^2$ has charge $2e$, its modulus is gauge-invariant. Expanding its square exhibits the direct couplingIn particular the charge in ${\psi }^{\ast }{\varphi }^2$ is $-2e+2e=0$, which uses the stated charge ratio. The potential is real, bounded below and has the zero-field minimum, while the kinetic terms have the standard positive signs. A complete example is thereforeThere is genuine direct interaction even with $g=0$ because $b\ne 0$. In four spacetime dimensions $b$ has mass dimension $-1$, but the expanded potential contains only quadratic, cubic and quartic field monomials; the cubic coefficient $M^{2}b$ has dimension one. Thus the example is also power-counting renormalizable.

Take $D_{\mu }={\partial }_{\mu }+i{K}_{\mu }$, with the Pauli matrices acting on the Higgs doublet. For electroweak hypercharge $Y=1/2$,This specifies every component of the gauge covariant derivative for the electroweak interaction. Expanding the gauge-covariant kinetic term makes its derivative, trilinear and quartic interactions explicit:where the Pauli matrix multiplication law givesThe antisymmetric Pauli contribution vanishes because $W_{\mu }^a{W}^{b\mu }$ is symmetric in $a,b$. Reversing the sign convention for $D_{\mu }$ reverses the linear gauge interactions consistently, without changing the masses.

A nonzero vacuum expectation value requires ${\mu }^2<0$. Minimizing the Higgs potential gives $v^2=-{\mu }^2/\lambda$. By an gauge transformation choosein unitary gauge. The electroweak doublet gauge-boson mass matrix follows by inserting the vacuum expectation value in the gauge-covariant kinetic term:Define the charged electroweak gauge bosons and the neutral rotation through the Weinberg angle bywith inverse $W_{\mu }^3=c_W{Z}_{\mu }+s_W{A}_{\mu }$ and $B_{\mu }=-s_W{Z}_{\mu }+c_W{A}_{\mu }$. ThenThe factors differ because $W^{+}$ and $W^{-}$ are conjugate fields whereas $Z$ is real. The massless photon corresponds to the unbroken Lie algebra generator $Q=T_3+Y$, which annihilates $⟨\varphi ⟩$. Thus three of the four real gauge bosons acquire mass, with $m_W=m_Z{c}_W$. The three would-be Goldstone bosons provide their longitudinal polarizations; the remaining scalar $h$ is the Higgs boson.

Introduce the left-handed lepton doublet $L=({\nu }_{eL},e_L{)}^T$, with $Y_L=-1/2$, and the right-handed singlet $e_R$, with $Y_R=-1$. Use the chiral projectors $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$ in the course's convention. The minimal Standard Model has no right-handed neutrino. The gauge-invariant fermion terms areThe doublet contraction in the Yukawa interaction is a singlet, and its total hypercharge is $+1/2+1/2-1=0$. A bare term ${\bar{e}}_L{e}_R$ would fail electroweak gauge invariance. The gauge covariant derivatives above already give all the requested fermion-gauge couplings. In terms of mass eigenstates, $e=g{s}_W=g^{\prime }{c}_W$ and they becomeThus the weak charged current is chiral and the neutrino has zero electric charge. The gauge-invariant electron Yukawa mass follows from the Yukawa interaction after electroweak symmetry breaking:Rephase $e_R$ to make $y_e$ real and positive. It givesThe electron Dirac mass is therefore compatible with the original gauge symmetry through the Higgs mechanism. The neutrino remains massless in this minimal renormalizable lepton sector.

For a differentiable curve $R(t)$ in the SO(3) group with $R(0)=I$, differentiating $R(t{)}^{T}R(t)=I$ gives $X^T+X=0$, where $X=R^{\prime }(0)$. Conversely, if $X$ is a real skew-symmetric matrix, $e^{tX}$ is an orthogonal matrix and has determinant $e^{t\mathrm{tr}X}=1$. HenceThe displayed generators of the SO(3) Lie algebra areThey are linearly independent and span the three independent entries of a real skew-symmetric matrix. Equivalently, $T_a\mathbf{v}={\mathbf{e}}_a\times \mathbf{v}$. The vector triple-product identity then givesThus the structure constants of a Lie algebra in this basis are

Use a real, antisymmetric convention throughout the SO(3) vector Higgs model. With gauge coupling $g$, the gauge covariant derivative and gauge field strength areThere is no factor of $i$ in these formulas because $T_a$ are real antisymmetric generators. For a local non-Abelian gauge transformation ${\Phi }^{\prime }=R\Phi$, with $R(x)$ in the SO(3) group, the compatible transformation of the gauge potential isIt gives $D_{\mu }^{\prime }{\Phi }^{\prime }=R{D}_{\mu }\Phi$ and $F_{\mu \nu }^{\prime }=R{F}_{\mu \nu }{R}^{-1}$. These covariance laws fix the relative signs in the gauge covariant derivative and gauge field strength.

An SO(3) invariant quartic scalar potential isSince $(R\Phi {)}^T(R\Phi )={\Phi }^T\Phi$, this scalar potential is gauge-invariant. Its minimum is the sphere ${\Phi }^T\Phi =v^2$, rather than a preferred direction in the internal vector space.

Writing ${\mathbf{A}}_{\mu }=(A_{\mu 1},A_{\mu 2},A_{\mu 3})$, the cross product form of the gauge covariant derivative is $D_{\mu }\Phi ={\partial }_{\mu }\Phi +g{\mathbf{A}}_{\mu }\times \Phi$. In components this isCorrespondingly, the gauge field strength has components

A covariantly constant vector Higgs field has constant norm, becauseThe last equality uses antisymmetry of the gauge potential. On a connected region write this norm as $\varphi \ge 0$. If $\varphi =0$, the vector Higgs field is identically zero. If $\varphi >0$, the SO(3) group acts transitively on unit vectors, so choose a smooth local non-Abelian gauge transformation rotating $\Phi /\varphi$ to ${\mathbf{e}}_3$. In this unitary gauge, $\Phi =(0,0,\varphi {)}^T$ with $\varphi$ constant. This is a local construction, and is global on a trivial contractible region; a nontrivial bundle can require more than one gauge patch.

For nonzero $\varphi$, substituting the constant aligned vector Higgs field into $D_{\mu }\Phi =0$ gives $g\varphi (A_{\mu 2},-A_{\mu 1},0{)}^T=0$. For nonzero gauge coupling, the most general compatible gauge potential and gauge field strength arewhere $a_{\mu }$ is an arbitrary real one-form. Thus the surviving gauge group is the $SO(2)$ stabilizer subgroup of ${\mathbf{e}}_3$. A covariantly constant vector Higgs field does not force the gauge field strength to vanish: it only requires the curvature to annihilate that field.

For the Higgs mechanism, choose a vacuum of the SO(3) invariant quartic scalar potential, so the constant norm is $v$. The canonically normalized kinetic term then containsThe two broken-direction gauge bosons have mass $gv$ (or $|g|v$ if the sign of $g$ is unrestricted), while the $T_3$ gauge boson remains massless. The two angular Goldstone bosons supply the massive vectors' longitudinal gauge-boson polarizations. The radial fluctuation remains a physical scalar field, with $m_h^2=2\lambda v^2$ for this normalization. The condition $D_{\mu }\Phi =0$ selects backgrounds without broken-direction gauge potentials; fluctuations about such a vacuum exhibit the two massive modes. Constancy of the norm alone does not require it to minimize the scalar potential.

Spectrum and symmetry. Choose a vacuum expectation value ${\varphi }_0=(0,v/\sqrt{2}{)}^T$. In unitary gauge the scalar field is $\varphi =(0,(v+h)/\sqrt{2}{)}^T$, with real $h$. The Pauli matrix multiplication law gives $\{{\tau }^a,{\tau }^b\}={\delta }^{ab}I/2$. Since $A^a{A}^b$ is symmetric in gauge indices, only this symmetric product contributes. The kinetic and potential terms therefore becomeComparison with $\frac{1}{2}{m}_A^2{A}_{\mu }^a{A}^{a\mu }$ and $\frac{1}{2}{m}_h^2{h}^2$ givesThese are the scalar interactions after complete SU2 breaking. The scalar mass uses the specified potential normalization: the prefactor $\lambda /2$ is important.

A nonzero complex doublet has trivial stabilizer in $SU(2)$: a matrix fixing $(0,v{)}^T$ must have second column $(0,1{)}^T$, and unitarity and determinant one then fix the first column too. Thus all three gauge generators are broken. Three scalar angular directions are absorbed by the Higgs mechanism, giving each gauge boson a longitudinal polarization. The physical degrees of freedom count isThere is no surviving electromagnetic $U(1)$ in this theory: it contains only the gauged $SU(2)$, not the electroweak product group. The would-be Goldstone bosons are not additional physical massless particles. This is complete breaking by an SU2 scalar doublet.

Scalar interactions. In unitary gauge, all interaction terms containing the physical scalar areThese give $hAA$, $hhAA$, $hhh$, and $hhhh$ Feynman vertices. Their factors are respectively $i{g}^{2}v{\delta }^{ab}{g}_{\mu \nu }/2$, $i{g}^2{\delta }^{ab}{g}_{\mu \nu }/2$, $-3i\lambda v$, and $-3i\lambda$, including the identical-leg multiplicities. Gauge-dependent descriptions also contain the absorbed scalar modes; the above are all physical-scalar interactions in the chosen gauge.

Gauge self-interactions. Write $f_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a$. Expanding the field-strength square gives the cubic and quartic piecesThe sign of the cubic term follows from the minus sign in the given nonlinear field strength. The quartic term can also be written using ${ϵ}^{abc}{ϵ}^{ade}={\delta }^{bd}{\delta }^{ce}-{\delta }^{be}{\delta }^{cd}$. The cubic Feynman vertex joins three wavy gauge lines, has one momentum factor, and is proportional to $g{ϵ}^{abc}$. The quartic Feynman vertex joins four wavy lines, is proportional to $g^2$, and has products of structure constants with metric contractions. There are no higher pure-gauge vertices in this Lagrangian. These are the SU2 gauge self-interaction vertices.

The three real components form the Adjoint representation of $SU(2)$. Around a nonzero vacuum, a local gauge transformation rotates the direction of $\varphi (x)$ to the third axis. In this unitary gauge, the two angular fields are removed from the scalar parametrization and become longitudinal polarizations of two gauge bosons, through the Higgs mechanism. Thus $\varphi =(0,0,v+\eta {)}^T$ describes the perturbative physical fluctuations, with $v^2=-m^2/\lambda$.

The subgroup generated by $t^3$ fixes $v{e}_3$, giving $SU(2)\to U(1)$. Define the physical vector fieldsThe gauge covariant derivative on the radial configuration isso the scalar kinetic term becomes $\frac{1}{2}(\partial \eta {)}^2+g^2(v+\eta {)}^2{W}_{\mu }^{+}{W}^{-\mu }$.

Here is an exact form of the Lagrangian in these physical fields, up to its additive vacuum constant. DefineandThe minus sign in the printed non-Abelian gauge field strength gives precisely these signs. ThenIts quadratic terms give $m_A=0$, $m_{W^{+}}=m_{W^{-}}=gv$, and $m_{\eta }=\sqrt{2\lambda }v$. The two massive vectors have six real polarization degrees of freedom in total; the massless vector has two and the scalar one. Their total of nine equals the original six vector plus three scalar degrees of freedom.

Under the surviving $U(1)$, $W^{+}$ and $W^{-}$ have opposite charges and $\eta$ is neutral. The Yang-Mills theory supplies $A{W}^{+}{W}^{-}$ and $AA{W}^{+}{W}^{-}$ couplings and quartic charged-vector self-interactions. The scalar term supplies $2g^{2}v\eta W^{+}{W}^{-}$ and $g^2{\eta }^2{W}^{+}{W}^{-}$, together with the cubic and quartic scalar self-interactions. There is no direct $\eta AA$ term.

With suitable fermions, this spectrum offers a massless photon and massive charged mediators of short-range weak charged currents, so it captures part of the weak/electromagnetic pattern. It differs crucially from the electroweak interaction of the Standard Model: the latter gauges $SU(2{)}_L\times U(1{)}_Y$, uses a complex Higgs doublet, and breaks to $U(1{)}_{\mathrm{em}}$. Three rather than two generators are broken, leaving a massive neutral Z boson as well as $W^{\pm }$ and a photon. Its charge operator $Q=T^3+Y$ and independent hypercharge coupling give a nontrivial weak mixing angle. The present theory has no massive neutral mediator, no such mixing angle, and does not reproduce the observed chiral fermion charge assignments or their usual Yukawa couplings from the standard doublet/singlet assignments.