The result grows linearly with the squared centre-of-momentum energy, $\sigma \propto G_F^{2}s$. This growth eventually violates partial-wave unitarity and signals the breakdown of the pointlike Fermi interaction. At energies comparable to $M_W$ or $M_Z$, the full gauge-boson propagators must replace the contact interaction; the renormalizable electroweak theory then softens the high-energy behavior.

A Fierz rearrangement puts the charged-current operator in the same current ordering as the neutral-current operator. Accounting for the interchange of fermionic fields, the combined amplitude isSum over final spins and average over the initial electron spin. The fermion spin sum and gamma-matrix trace identities giveFor massless two-body scattering in the centre-of-momentum frame, $d\sigma /dt=\overline{|\mathcal{M}{|}^2}/(16\pi s^2)$ and $u=-s-t$. Integrating $-s\le t\le 0$ therefore yieldsConsequently

The first four-fermion operator is the low-energy Fermi interaction obtained by replacing the crossed $W$ propagator in the charged-current diagram by $-g_{\mu \nu }/M_W^2$. The second operator is obtained analogously from the $Z$-exchange weak neutral current; $c_V$ and $c_A$ are its electron vector and axial-vector couplings. Thus the two terms represent respectively the charged-current and neutral-current diagrams in part (v), with their interference retained when the amplitude is squared.

The minima of the scalar potentialsatisfyThe SU(2) group acts transitively on this three-sphere of vacua, and the stabilizer of a nonzero fundamental doublet is trivial. A global transformation may therefore choose ${\varphi }_0=(0,v{)}^T/\sqrt{2}$. Because the symmetry is gauged, unitary gauge removes all three angular Goldstone bosons, leaving only the real radial Higgs mode $h$:Substitution into the gauge-covariant kinetic term giveswithThe interaction terms produce $h^3$, $h^4$, $hBB$, $hhBB$, and the cubic and quartic non-Abelian gauge-boson vertices shown below.

The three broken generators supply the three longitudinal polarizations of the equally massive gauge bosons. No physical massless Goldstone boson remains, and because the unbroken subgroup is trivial there is no massless gauge boson either. The remaining physical spectrum has $3\times 3+1=10$ degrees of freedom, equal to the original $3\times 2+4=10$.

Let $V_i=\partial V/\partial {\varphi }_i$ and $M_{ij}^2={\partial }_i{\partial }_{j}V{|}_{{\varphi }_0}$ be the scalar mass matrix, which is the Hessian matrix of the scalar potential at the vacuum. Invariance under the infinitesimal Lie group action givesDifferentiate with respect to ${\varphi }_j$ and evaluate at the vacuum expectation value ${\varphi }_0$. Since $V_i({\varphi }_0)=0$ at a minimum,Thus every tangent vector $i{t}^a{\varphi }_0$ generated by a broken Lie algebra generator is a zero eigenvalue eigenvector of the scalar mass matrix. The unbroken generators are precisely those in the stabilizer Lie algebra $\mathfrak{h}$ and give the zero tangent vector; independent broken directions span $\mathfrak{g}/\mathfrak{h}$. Hence Goldstone theorem givesmassless scalar modes at the classical level.

The operator $ia\bar{\psi }{\sigma }^{\mu \nu }{\gamma }^5\psi F_{\mu \nu }$ is the relativistic fermion electric dipole moment operator. Its nonrelativistic limit contains $\mathbf{S}\cdot \mathbf{E}$: spin angular momentum is an axial vector, whereas the electric field is a polar vector. It is consequently odd under parity symmetry in quantum field theory. Both the pseudotensor fermion bilinear and the electromagnetic field tensor are odd under charge conjugation, so their product is even. ThereforeThe CPT theorem then makes it odd under time-reversal symmetry. Such an interaction can arise only from CP violation. The Cabibbo-Kobayashi-Maskawa matrix therefore induces a nonzero Standard Model electric dipole moment at sufficiently high loop order, but its flavor structure and loop suppressions make the result extraordinarily small.

The quantum electrodynamics interaction is ${\mathcal{L}}_{\mathrm{int}}=-e{j}^{\mu }{A}_{\mu }$, where the Dirac electromagnetic current is $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$. Under parity symmetry in quantum field theory,Invariance of the interaction therefore requires the electromagnetic four-potential to transform as the same Lorentz four-vector:so $A^0$ is parity even and $\mathbf{A}$ is parity odd. Under charge conjugation, the current is odd, $j^{\mu }\mapsto -j^{\mu }$, so invariance requires

Write ${\psi }^P(x)={\eta }_P{\gamma }^0\psi (x_P)$. The chain rule gives ${\partial }_0\psi (x_P)=({\partial }_0\psi )(x_P)$ and ${\partial }_i\psi (x_P)=-({\partial }_i\psi )(x_P)$. Using the gamma matrix relations $({\gamma }^0{)}^2=1$ and ${\gamma }^i{\gamma }^0=-{\gamma }^0{\gamma }^i$,Thus the Dirac equation is invariant under parity symmetry in quantum field theory.

Apply the parity symmetry in quantum field theory to the given mode expansion of a Dirac field:The stated Dirac spinor identities are equivalently $u^s(p)={\gamma }^0{u}^s(p_P)$ and $v^s(p)=-{\gamma }^0{v}^s(p_P)$. Relabel the momentum sum by $p\mapsto p_P$ and use $p_P\cdot x=p\cdot x_P$ to obtainThe phase $|{\eta }_P|=1$ is the intrinsic parity convention.

In the mode expansion of a Dirac field, $b^s(p)$ is the fermionic annihilation operator for a particle of four-momentum $p$ and spin angular momentum label $s$, while $d^{s†}(p)$ is the fermionic creation operator for the corresponding antiparticle. The Dirac spinor $u^s(p)$ is the positive-frequency particle wavefunction and $v^s(p)$ is the negative-frequency antiparticle wavefunction; they solve $(/p-m)u^s=0$ and $(/p+m)v^s=0$, respectively.

For axial gauge, $G^a[A]=n^{\mu }{A}_{\mu }^a$. Its Faddeev-Popov operator isOn the gauge slice $n\cdot A^c=0$, the second term vanishes. HenceThis functional determinant is independent of the gauge field and may be absorbed into the normalization of the path integral. Any introduced ghosts are free and decouple, so no ghost fields are needed in axial gauge.

The quadratic gauge-field action in momentum space isIn terms of the transverse projector of a vector field and longitudinal projector of a vector field,the operator is $k^2(P_T+{\xi }^{-1}{P}_L)$. Its inverse, the gauge-boson propagator, isTherefore $X=1$ and $Y=\xi -1$.

The variation found above uses the adjoint covariant derivative:For the Lorenz gauge functional $G^a[A]={\partial }^{\mu }{A}_{\mu }^a$,The field-independent factor $1/g$ may be absorbed into normalization. The Grassmann Gaussian integral exponentiates the Faddeev-Popov determinant with anticommuting Faddeev-Popov ghost fields:A Gaussian average over the gauge condition supplies the covariant gauge-fixing term, so