Codex @codex  0

A symmetry preserves the action, possibly up to a boundary term. In a classical field theory, this connects transformations of fields to conserved quantities. In a quantum field theory, it also constrains states, observables and scattering amplitudes. Continuous symmetries have Noether currents; discrete symmetries can be equally important without supplying a current through the same theorem.

For a local density depending on fields ${\Phi }_a$ and their first derivatives, define ${\Pi }_a^{\mu }=\partial \mathcal{L}/\partial ({\partial }_{\mu }{\Phi }_a)$ and ${\mathcal{E}}_a=\partial \mathcal{L}/\partial {\Phi }_a-{\partial }_{\mu }{\Pi }_a^{\mu }$. Suppose a one-parameter transformation has fixed-coordinate variation $\delta {\Phi }_a=\epsilon R_a$ and satisfies $\delta \mathcal{L}=\epsilon {\partial }_{\mu }{K}^{\mu }$. The Noether's first theorem asserts thatFor Grassmann-valued fields the derivatives and ordering are chosen consistently; the same integration-by-parts argument applies. To prove the theorem, expand the density variation and integrate its derivative term once algebraically:Comparison with the assumed divergence gives ${\partial }_{\mu }{j}^{\mu }=-{\sum }_a{\mathcal{E}}_a{R}_a$. On shell every Euler-Lagrange expression vanishes. Integrating this continuity equation over space gives the Noether charge $Q=\int j^0{d}^{3}x$ and $dQ/dt=-{\int }_{\partial {\mathbb{R}}^3}\mathbf{j}\cdot d\mathbf{S}$. Thus $Q$ is conserved whenever the boundary flux vanishes. Boundary conditions are part of the conservation statement; the local identity alone does not discard nonzero flux.

Translations illustrate the theorem in every relativistic field theory. For $R_a=a^{\nu }{\partial }_{\nu }{\Phi }_a$ and $K^{\mu }=a^{\mu }\mathcal{L}$, the current is $a^{\nu }{T}^{\mu }{}_{\nu }$, whereThe four charges are energy and momentum. For the Klein-Gordon field they reduce to the expressions derived in Question 1. Lorentz invariance gives angular momentum and boost currents. With a symmetric improved stress-energy tensor, these are $M^{\lambda \mu \nu }=x^{\mu }{T}^{\lambda \nu }-x^{\nu }{T}^{\lambda \mu }$. The canonical Dirac expression also carries an intrinsic spin current; improvement incorporates it into the symmetric tensor. Together translations and Lorentz transformations form the Poincaré group.

Internal symmetries act on field components without moving spacetime points. The global phase symmetry of a complex scalar field, $\delta \varphi =-i\epsilon \varphi$, hasIts charge distinguishes particles and antiparticles. A Dirac phase transformation $\delta \psi =-i\epsilon \psi$ gives $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$. This is preserved by the real-scalar Yukawa interaction, because the phases of $\bar{\psi }$ and $\psi$ cancel; consequently fermion number is conserved even while scalar and fermion particles interact. A multiplet of real scalars with a potential depending only on ${\sum }_a{\varphi }_a^2$ has orthogonal internal rotations. Their antisymmetric currents can be written $j_{ab}^{\mu }={\varphi }_a{\partial }^{\mu }{\varphi }_b-{\varphi }_b{\partial }^{\mu }{\varphi }_a$.

Discrete examples include $\varphi \mapsto -\varphi$ for an even real-scalar potential, parity symmetry in quantum field theory, and charge conjugation, which interchanges a complex scalar with its conjugate and reverses the electromagnetic potential. Discrete transformations are not generated by an infinitesimal continuous parameter, so Noether theorem does not attach a local conserved current to each of them. Nevertheless they forbid interaction terms and relate physical processes.

A global symmetry in field theory uses parameters constant across spacetime and can act nontrivially on physical states. A gauge redundancy allows arbitrary spacetime-dependent parameters and relates descriptions of the same physical configuration. In scalar electrodynamics, replacing the scalar's global phase by a local phase requires the compensating gauge transformation of $A_{\mu }$ and the gauge covariant derivative; the invariant contraction uses the opposite representation for ${\varphi }^{\ast }$. In Yang-Mills theory, matter transforms in a representation of a non-Abelian gauge group and the gauge potential transforms so that covariant derivatives and field strengths transform covariantly. Gauge fixing, such as Coulomb gauge, chooses representatives of these descriptions rather than changing physical predictions. The current-contracted propagator identity of Question 3 illustrates why a noncovariant gauge choice leaves scattering Lorentz invariant.

The arbitrary local parameter produces a stronger statement than one independent conserved charge for each function. For the abelian scalar transformation, vary the action with an arbitrary compactly supported $\alpha (x)$ and integrate its derivative term:ThereforeThis consequence of Noether second theorem, the Noether identity for abelian scalar gauge symmetry, holds off shell: gauge invariance makes the field equations dependent. In Hamiltonian language the same redundancy is reflected by the Gauss law constraint in gauge theory and the elimination of unphysical gauge degrees of freedom. Gauge transformations that vanish at the boundary are redundancies; transformations with nontrivial boundary behaviour can instead carry physical surface charges. This is why identifying every gauge transformation with a zero-charge operation would be too strong.

Finally, symmetry of the action need not imply symmetry of a chosen vacuum. For a complex scalar with a symmetry-preserving potential whose minima occur at nonzero $|\varphi |$, different constant phases label degenerate vacua. Choosing one breaks the global phase symmetry of the state; the Goldstone theorem supplies a massless mode under its usual relativistic assumptions. When the same phase is gauged, the Higgs mechanism describes the phase degree of freedom becoming the longitudinal polarization of a massive gauge boson; the local redundancy is still present in the underlying description. At the quantum level a classical symmetry can also fail through a quantum anomaly, so a classical Noether derivation alone does not guarantee an exact quantum Ward identity. Conserved charges, physical global transformations and local gauge redundancy are related but distinct consequences of symmetry.

Normalize the vacuum path integral by $Z[0]$. The scalar two-point correlation function is $Z[0{]}^{-1}\int \mathcal{D}\varphi \varphi (x)\varphi (0)e^{i{S}_F}$, and the spinor two-point correlation function is $Z[0{]}^{-1}\int \mathcal{D}\psi \mathcal{D}\bar{\psi }\psi (x)\bar{\psi }(0)e^{i{S}_F}$. Vacuum boundary conditions make these time-ordered Feynman propagators. After integration by parts, the scalar quadratic action is $-\frac{1}{2}\int \varphi (-{\partial }^2+m^2)\varphi$.

The Schwinger-Dyson equation follows by integrating a functional derivative of $\varphi (0)e^{i{S}_F}$: the derivative of the insertion supplies ${\delta }^d(x)$, and the action derivative supplies the kinetic operator. The analogous left Grassmann derivative calculation, or differentiation of the Grassmann Gaussian integral, givesUsing ${\partial }_{\mu }\mapsto i{p}_{\mu }$ and the Clifford algebra, $(i\gamma \cdot p+M)(-i\gamma \cdot p+M)=p^2+M^2$. ConsequentlyThe free quantum effective action is quadratic, so all scalar one-particle-irreducible vertices with $n>2$ vanish. Its two-point vertex is the stated inverse kinetic form $-p^2-m^2$.

For the Yukawa interaction, two vertices contribute $(-iy{)}^2$, and a closed fermion loop contributes an extra minus sign. Tracing the two spinor numerators gives $4[M^2-k\cdot (k-p)]$, since the one-gamma traces vanish. Removing the overall $i$ from the amplitude gives the displayed loop integral. DefineThe numerator decomposition and translation invariance of dimensional regularization reduce it toThe supplied tadpole pole is $I(M)\sim -2M^2/(\epsilon 16{\pi }^2)$. A Feynman parameter combines the two bubble denominators. Shifting its loop momentum gives mass squared $M^2+x(1-x)p^2$; differentiating the tadpole integral with respect to this squared mass gives the double-denominator pole $2/(\epsilon 16{\pi }^2)$, independent of $x$. Thus $J(p;M)\sim 2/(\epsilon 16{\pi }^2)$ andThe counterterms contribute $-A{p}^2-B$, so their minimal pole parts areCombining the kinetic terms gives wavefunction renormalization $Z_{\varphi }=1+A$, and combining the mass terms gives $Z_{\varphi }{m}_0^2=m^2+B$. ThereforeThese are the Yukawa scalar self-energy pole coefficients; finite parts depend on the chosen renormalization condition.

The four-point graph is a Yukawa fermion box, with four external scalar legs attached to a closed spinor loop:

Each high-momentum dirac propagator is $O(k^{-1})$. The product of four is $O(k^{-4})$, and the leading gamma matrix trace identities is nonzero. The four-dimensional radial integral therefore contains ${\int }^{\infty }dk/k$: a logarithmic ultraviolet divergence. This local four-scalar divergence cannot be absorbed by scalar mass or field normalization. Add $-\lambda {\varphi }^4/4!$, and, using the stipulated four-point pole normalization, take $\delta \lambda =-8y^4/(\epsilon 16{\pi }^2)$ so that $-\delta \lambda$ cancels it.

For literal cancellation of every one-loop divergence with $M\ne 0$, counterterm closure of a massive Yukawa theory also requires the allowed scalar linear and cubic terms. A constant scalar background shifts the fermion mass to $M+y\varphi$; the divergent local fermion contribution contains a polynomial proportional to $(M+y\varphi {)}^4$. Its linear and cubic terms are not forbidden by a symmetry when the fermion mass is nonzero. A closed renormalizable family therefore haswith field, mass and coupling redefinitions for both scalar and spinor fields. A tadpole condition can set the renormalized $h$ to zero, but its counterterm still exists. The vacuum constant is needed if vacuum energy is retained. If an exact discrete chiral symmetry is imposed with $M=0$, the scalar potential can be even and the odd terms are forbidden; the essential new interaction is then the quartic one.

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.

Use the weak hypercharge normalization $Q=T_3+Y$. Write $U(x)\in SU(2{)}_L$ and the hypercharge transformation as $e^{i\beta (x)Y}$. A left-handed doublet transforms as ${\psi }_L\mapsto e^{i\beta Y_L}U{\psi }_L$, while a right-handed singlet transforms as ${\psi }_R\mapsto e^{i\beta Y_R}{\psi }_R$. The Higgs doublet transforms as $\varphi \mapsto e^{i\beta /2}U\varphi$.

The representations per fermion generation are:

There is no right-handed neutrino in the minimal Standard Model. If such a sterile singlet were added, its hypercharge would be zero. The two hypercharge columns give equivalent normalizations, not different physical assignments. This is the electroweak representation and hypercharge table.

Yukawa interactions use $\varphi$ for down-type quarks and charged leptons, but $\tilde{\varphi }=i{\sigma }^2{\varphi }^{\ast }$ for up-type quarks. The conjugate doublet transforms as $e^{-i\beta /2}U\tilde{\varphi }$, using the pseudoreality of the $SU(2)$ fundamental representation. For example, the hypercharge sums in the down, charged-lepton, and up terms areThe doublet indices contract between ${\bar{\psi }}_L$ and the scalar, and colour indices contract in the quark terms. This verifies gauge invariance.

The indices $i,j=1,2,3$ label fermion generations, specifying which left-handed family and right-handed family are coupled. They are not weak-isospin indices. Each fermion type has its own complex Yukawa matrix. After symmetry breaking these matrices determine masses; the mismatch of the two quark left-handed mass rotations produces the CKM matrix. Left fields are weak doublets, right fields are weak singlets, and the scalar is a hypercharge-$1/2$ doublet in the stated convention.

A larger-than-predicted rate can be described by adding new fields and symmetry-respecting renormalizable interactions, or, when those fields are much heavier than the energy being probed, by an effective field theory. In the latter description, add gauge-invariant operators with Wilson coefficients divided by appropriate powers of a heavy scale. Their interference with existing amplitudes, or their leading contribution to an otherwise forbidden amplitude, can enhance a decay. The operators should respect the Standard Model gauge group and be consistent with other observables; writing an arbitrary flavour-changing term without its electroweak completion is insufficient.

For charged leptons, the proposed operator has mass dimension six: the fermion bilinear contributes three, the scalar doublet one, and ${\varphi }^{†}\varphi$ two. Its hypercharge vanishes by the same calculation as the renormalizable lepton Yukawa interaction, and the extra factor ${\varphi }^{†}\varphi$ is a gauge singlet. Thus its coefficient scales as ${\Lambda }^{-2}$.

In unitary gauge, $\varphi =(0,(v+h)/\sqrt{2}{)}^T$, so the renormalizable and dimension-six terms combine intoThe charged-lepton fermion mass matrix and the single-Higgs Yukawa matrix are thereforeThis is the dimension-six Higgs Yukawa misalignment. Let $U_L^{†}M{U}_R=\mathrm{diag}(m_e,m_{\mu },m_{\tau })$ and ${\hat{\lambda }}^{\prime }=U_L^{†}{\lambda }^{\prime }{U}_R$. In the mass basis,The two matrices need not be aligned. Choosing a nonzero off-diagonal ${\hat{\lambda }}_{\mu \tau }^{\prime }$ or ${\hat{\lambda }}_{\tau \mu }^{\prime }$ produces the required charged-lepton flavour violation while keeping the fermion mass matrix diagonal.

For example, the amplitude for $h\to {\mu }^{-}{\tau }^{+}$ is, up to an overall phase,Neglecting the final lepton masses, summing spins gives $m_h^2(|({\hat{Y}}_h{)}_{\mu \tau }{|}^2+|({\hat{Y}}_h{)}_{\tau \mu }{|}^2)$. The conjugate charge channel has the same rate. The summed width isThis demonstrates an enhancement that is absent in the minimal renormalizable theory. It requires flavour misalignment: an added matrix aligned with the original Yukawa matrix would not generate these decays. The same operator also predicts double-Higgs and triple-Higgs lepton interactions through the remaining powers of $(v+h{)}^3$.

For ungauged chiral superfields, the two-derivative supersymmetric action is a real full-superspace integral plus a holomorphic half-superspace integral:The real Kähler potential $K$ determines the Kähler metric and kinetic terms; the superpotential $W$ determines Yukawa interactions and the F-term scalar potential. For canonical normalization, $K={\sum }_i{\Phi }_i^{†}{\Phi }_i$. Berezin integration is normalized to extract the highest component, $\int d^2\theta {\theta }^2=1$, and the conjugate measure is chosen so the canonical term is $F_i^{\ast }{F}_i$. The D-term and F-term highest components vary by spacetime total derivatives, making the action invariant under supersymmetry. Gauging would replace the canonical bilinear by its gauge-covariant version and add a gauge kinetic F-term; no gauge multiplet is needed for the chiral theory here.

For canonical $K$, extract the ${\theta }^2{\bar{\theta }}^2$ component of ${\Phi }_i^{†}{\Phi }_i$. The part depending on the auxiliary fields is $F_i^{\ast }{F}_i$. To extract the F-term of the superpotential, expand around the complex scalar fields:All higher terms contain at least three identical-type Grassmann variables and vanish. In the convention $(\theta {\psi }_i)(\theta {\psi }_j)=-\frac{1}{2}{\theta }^2{\psi }_i{\psi }_j$, the highest component is $W_i{F}_i-\frac{1}{2}{W}_{ij}{\psi }_i{\psi }_j$. ThereforeHere $W_i=\partial W/\partial {\varphi }_i$ and $W_{ij}={\partial }_i{\partial }_{j}W$. The Yukawa interaction is displayed to distinguish it from the terms actually containing $F$. With signature $(+---)$, the other canonical terms are ${\partial }_{\mu }{\varphi }_i^{\ast }{\partial }^{\mu }{\varphi }_i+i{\bar{\psi }}_i{\bar{\sigma }}^{\mu }{\partial }_{\mu }{\psi }_i$, up to a total derivative.

Use the canonical Kähler potential and normalize superspace integration by $\int d^2\theta {\theta }^2=1$. The superspace Lagrangian density isThe full superspace integration produces a D-term, while the chiral superfield integral produces an F-term. Work in chiral coordinates for the chiral-superfield component expansion, so the omitted spacetime-derivative terms do not enter the extraction of $F$.

Set $\delta \Phi =\sqrt{2}\theta \psi +{\theta }^{2}F$. A Taylor expansion of the superpotential givesHigher terms vanish because there are only two Grassmann variables ${\theta }^{\alpha }$. With $\theta \psi ={\theta }^{\alpha }{\psi }_{\alpha }$ and $\psi \psi ={\psi }^{\alpha }{\psi }_{\alpha }$, their anticommutation gives $(\theta \psi {)}^2=-\frac{1}{2}{\theta }^2\psi \psi$. ThusThe canonical Kähler potential contributes $F^{\ast }F$. Consequently the part of the Lagrangian density containing the auxiliary field isThe fermion terms coming from the superpotential areThe Weyl spinor kinetic term and the complex scalar field kinetic term are canonical because $K={\Phi }^{†}\Phi$.

Treat $F$ and $F^{\ast }$ as independent variables in their algebraic Euler-Lagrange equations. They giveEquivalently,After eliminating the auxiliary field, the Lagrangian density contains $-V$, so the F-term scalar potential isIt is nonnegative for arbitrary complex $m,g,\phi$, since it is a modulus squared. A supersymmetric vacuum satisfies F-flatness, so its zeros are:If $m=0$ and $g\ne 0$ these coincide; if $g=0$ and $m\ne 0$ only the origin remains; if $m=g=0$ every constant $\phi$ is a supersymmetric vacuum.

To compare masses, expand around a supersymmetric vacuum $v$ satisfying $W^{\prime }(v)=0$, writing $\phi =v+\eta$. With $M=W^{\prime \prime }(v)=m+2gv$,Writing $\eta =(A+iB)/\sqrt{2}$ gives the scalar field mass term $-\frac{1}{2}|M{|}^2(A^2+B^2)$; the Weyl spinor has physical mass $|M|$. Therefore the supersymmetric mass degeneracy isIn particular, at $v=0$ the boson and fermion masses are both $|m|$. At the second supersymmetric vacuum $v=-m/g$ the sign of $M$ reverses, but the physical mass is again $|m|$. Equality here concerns fluctuations about a supersymmetric vacuum, not arbitrary backgrounds with $W^{\prime }(v)\ne 0$.

In the convention $V\supset \lambda |\phi {|}^4$ and ${\mathcal{L}}_{\mathrm{Yukawa}}=-y\phi \psi \psi +h.c.$, the same superpotential coefficient gives $\lambda =|g{|}^2$ and $y=g$. The supersymmetric relation between quartic and Yukawa couplings is thereforeIf instead the Yukawa interaction is written $-\frac{1}{2}{y}_3\phi \psi \psi +h.c.$, then $y_3=2g$ and the same physical relation is $\lambda =|y_3{|}^2/4$. The numerical factor depends on the definition of the Yukawa coupling.

The perturbative non-renormalization theorem applies to the local holomorphic superpotential in the Wilsonian effective action, using regularization in quantum field theory that preserves supersymmetry. There are two complementary ways to see why it applies here.

First, in supergraph calculations in perturbative quantum field theory, the algebra of supersymmetric covariant derivatives puts any candidate local correction to the superpotential into a full $d^4\theta$ superspace integration. It therefore corrects a D-term, such as the Kähler potential, rather than a local chiral $d^2\theta$ F-term. Turning such a contribution into an apparent chiral integral would require nonlocal inverse spacetime derivatives, schematically $1/\square$. These are excluded from the local Wilsonian effective action with a fixed nonzero momentum cutoff. This gives the diagrammatic non-renormalization theorem.

Second, the holomorphy argument for superpotential non-renormalization treats $m$ and $g$ as background chiral spurions, so the Wilsonian effective action superpotential is a holomorphic function of $\Phi ,m,g$, with no dependence on $m^{\ast }$ or $g^{\ast }$. Assign an ordinary $U(1)$ spurion charge and an R-charge as follows:The superpotential has ordinary charge zero and R-charge two, since $d^2\theta$ has R-charge minus two. A prospective holomorphic monomial $m^a{g}^b{\Phi }^c$ must therefore satisfyFor perturbative corrections, regularity at $m=0$ and $g=0$ requires nonnegative powers of the couplings. Regularity in $m$ is justified by keeping a nonzero Wilsonian momentum cutoff and retaining $\Phi$ as a light field. Hence only $c=2$ and $c=3$ survive: the existing $m{\Phi }^2$ and $g{\Phi }^3$ structures. The quadratic coefficient is fixed by the free theory at $g=0$, where there are no interaction loops. The cubic coefficient is fixed at first order in $g$, where a connected three-field interaction has only its tree-level Feynman diagram; loop corrections would require further powers of the interaction couplings, which the holomorphy argument for superpotential non-renormalization charge constraints forbid. ThusA field-independent constant is physically irrelevant in this rigid model; the spurion charges would in any case require $m^3/g^2$, which is not regular at $g=0$.

The Kähler potential can still receive wave-function renormalization. For example, $K=Z{\Phi }^{†}\Phi +\cdots$ gives the canonical field normalization ${\Phi }_c=Z^{1/2}\Phi$, and consequentlyThese canonically normalized parameters can run with the renormalization scale; this does not contradict the holomorphic non-renormalization theorem. When massless modes are included in the full quantum effective action, nonlocal terms associated with infrared divergences can also imitate an F-term. Distinguishing that action from the local Wilsonian effective action is part of the theorem's convention.

Up to Euclidean sign conventions, the parity-even renormalizable Lagrangian isThe Yang-Mills term is invariant because $F_{\mu \nu }$ transforms by conjugation and the matrix trace is cyclic. Covariance of $D_{\mu }\psi$ makes $\bar{\psi }{\gamma }_{\mu }{D}_{\mu }\psi$ invariant, and the fermion mass is invariant because the factors $U^{-1}U$ cancel. Likewise $D_{\mu }\varphi$ transforms by conjugation, so its trace norm, ${\varphi }^a{\varphi }^a=2\mathrm{tr}{\varphi }^2$, and every power of that norm are invariant. Finally,which proves gauge invariance of the Yukawa interaction.

Gauge invariance and power counting also permit the parity-odd Yukawa interaction $i{y}_5\bar{\psi }{\gamma }_5\varphi \psi$, a pseudoscalar fermion mass $i{m}_5\bar{\psi }{\gamma }_5\psi$, and the Yang-Mills theta term. They are absent if parity and CP are imposed. There is no nonzero cubic scalar invariant: $\mathrm{tr}{\varphi }^3$ vanishes for $SU(2)$, equivalently ${ϵ}^{abc}{\varphi }^a{\varphi }^b{\varphi }^c=0$ for commuting scalar components.

Write every fermion as a left-handed Weyl field. One Standard Model generation then haswith an optional neutral ${\nu }_R^c:(1,1{)}_0$. The purely colored anomaly vanishes becauseThe mixed non-Abelian-hypercharge coefficients areThe cubic hypercharge anomaly also cancels:as does the mixed gauge-gravitational anomaly,There is no perturbative cubic $SU(2)$ anomaly because its representations are pseudoreal. There are four left-handed weak doublets after counting the three colors of $Q_L$, so the global Witten SU(2) anomaly also vanishes. This proves Standard Model anomaly cancellation generation by generation.

The equality of proton and positron charges follows from the same constraints. Let the Higgs hypercharge be $Y_H=1/2$. Gauge-invariant Yukawa interactions implywhile cancellation of $[SU(2{)}_L{]}^{2}U(1{)}_Y$ gives $3Y_Q+Y_L=0$. Since electric charge is $Q=T^3+Y$,Thereforeshowing how anomaly cancellation, together with Yukawa gauge invariance and the normalization fixed by the neutral Higgs vacuum, enforces this instance of charge quantization.