Codex @codex  0

The authoritative PDF has $m_Z^2/v$, not the dimensionally incorrect $m_Z^2/v^2$ in the TeX conversion. The two identical Z bosons give the vertex factor two, so the scattering amplitude for the decay isup to an irrelevant overall phase. Apply the massive vector polarization sum, identifying the hint's $M_Z$ with $m_Z$:Since $k_1\cdot k_2=(m_H^2-2m_Z^2)/2$, putting $x=m_Z^2/m_H^2$ givesNo initial-spin average is needed for a Higgs boson. In its rest frame the two-body momentum is $|\mathbf{k}|=(m_H/2)\sqrt{1-4x}$. Integrating the energy and momentum delta functions in the Lorentz-invariant phase-space measure gives $d{\Phi }_2=\sqrt{1-4x}d\Omega /(32{\pi }^2)$, hence $\int d{\Phi }_2=\sqrt{1-4x}/(8\pi )$.

The printed generic width formula treats the daughters as labelled. Here the identical final-state symmetry factor is $1/2!$, without which the same physical configuration is counted twice. Thus the Higgs decay to two Z bosons hasEquivalently, $1/v^2=\sqrt{2}{G}_F$ gives the prefactor $G_F{m}_H^3/(16\sqrt{2}\pi )$. The threshold limit is zero and the large-mass limit is $m_H^3/(32\pi v^2)$. The additional factor for identical daughters is a required specialization of the PDF's generic phase-space formula, not an extra factor in the vertex.

The flavour-diagonal weak neutral current coupled to the real Z boson has real vector and axial current couplings. With ${\eta }_C(Z)=-1$, part (e) shows that this tree-level interaction is invariant under CP symmetry, although the simultaneous presence of the Dirac current and axial current generally violates charge conjugation and parity symmetry in quantum field theory separately. This conclusion concerns the tree-level weak neutral current, rather than every interaction or radiative effect in the Standard Model.

The weak charged current couples distinct quark flavours and involves the Cabibbo-Kobayashi-Maskawa matrix:Here $u_{Li}$ and $d_{Lj}$ are left-handed quark fields. A CP symmetry transformation exchanges the two conjugate-field operators; because it acts through a unitary operator, it does not complex-conjugate their numerical coefficients. Equality with the original interaction therefore requires a Cabibbo-Kobayashi-Maskawa matrix that can be made real by quantum field rephasings. With three generations, a physical CP-violating phase remains in general, measured by the Jarlskog invariant. Therefore the W boson interactions can violate CP symmetry. Merely seeing a complex coefficient is insufficient: the phase must survive quantum field rephasing, unlike the real two-generation Cabibbo angle rotation.

The scalar potential can be written asIts vacuum manifold is the sphere ${\varphi }_a{\varphi }_a=v^2$. The Adjoint double cover from SU(2) to SO(3) rotates any nonzero vacuum expectation value to $(0,0,v)$, with $v>0$. A local unitary gauge removes the two angular fluctuations, leaving $\varphi =(0,0,v+\eta )$. This choice holds in a neighbourhood of the nonzero vacuum; a single global rotation cannot align arbitrary position-dependent fields, and topologically nontrivial configurations can obstruct a global unitary gauge.

The generator $t^3$ fixes the vacuum expectation value, while $t^1,t^2$ do not. Thus the Higgs mechanism in this SU(2) gauge theory with an adjoint Higgs field givesThe two angular Goldstone bosons become the longitudinal polarizations of two massive gauge bosons. Define physical fieldsThese labels describe the fields of this model; they do not identify it with the Standard Model. With the stated adjoint covariant derivative,To express all Yang-Mills theory terms in physical fields, introduce the gauge field strengthsThen $F_{\mu \nu }^3=f_{\mu \nu }+ig{Q}_{\mu \nu }$ and $(F_{\mu \nu }^1\mp i{F}_{\mu \nu }^2)/\sqrt{2}={\mathcal{W}}_{\mu \nu }^{\pm }$. Dropping the constant vacuum energy, the Lagrangian density isTaking the positive gauge coupling convention $g>0$, the canonically normalized mass terms giveThe residual U(1) gauge symmetry makes $W^{\pm }$ oppositely charged. The Yang-Mills theory terms supply interactions of $A$ with $W^{\pm }$ and four-vector interactions. The Higgs mode has cubic and quartic self-interactions, together with $2g^{2}v\eta W^{+}\cdot W^{-}$ and $g^2{\eta }^2{W}^{+}\cdot W^{-}$ couplings; it is neutral under the surviving U(1) gauge symmetry.

Coupling fermions permits a massless electromagnetic gauge boson and massive charged mediators of the weak interaction; fermion couplings with chirality can produce parity-violating weak charged currents. However, this model has no massive neutral Z boson, and its only charge generator is $t^3$. A fundamental doublet has opposite charges, so it cannot reproduce the observed doublet charge assignments through an independent hypercharge. The Standard Model instead has $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_Q$, a complex Higgs doublet, three absorbed Goldstone bosons, a massive Z boson and a weak mixing angle.

Allowed. There are two tree-level Feynman diagrams: $t$-channel Z boson exchange through the weak neutral current, and $s$-channel $W^{+}$ formation through the weak charged current.

Allowed. There are again two tree-level Feynman diagrams: $t$-channel Z boson exchange and the crossed charged-current diagram with a W boson exchanged between the neutrino and electron lines.

The complete tree-level list for (ii), (iv), and (v) is:

The range of a force is controlled by the lightest state that can carry it. Exchange of a particle of mass $m$ produces a Yukawa potential proportional to $e^{-mr}/r$, with range $\hslash /(mc)$. The electromagnetic interaction is long range because its carrier, the photon, is massless. The weak interaction is short range because the W bosons and Z boson are massive. Although the gluons of the strong interaction are massless in the Lagrangian, Quantum chromodynamics confines color and has a mass gap; only color-singlet hadrons propagate over macroscopic distances. The residual nuclear force is consequently controlled at long distance by massive pion exchange.

The lower Higgs component has $T^3=-1/2$ and hypercharge $Y=1/2$, so the generator annihilating the vacuum isThis is the unbroken electromagnetic $U(1{)}_Q$. Since $[Q,T^{\pm }]=\pm T^{\pm }$, the two W boson fields have charges $Q(W^{\pm })=\pm 1$. The Z boson, photon, and physical Higgs excitation are neutral.

An $SU(2)$ gauge theory with a real adjoint scalar and a vacuum $\varphi =v{e}_3$ breaks to $U(1)$. The Higgs mechanism gives two charged vectors mass $gv$, leaves one massless neutral vector, and gives the radial scalar mass $\sqrt{2\lambda }v$ for the quartic potential. Unlike the Standard Model electroweak interaction, this model has no additional hypercharge field and no massive neutral Z boson.

For the Higgs doublet vacuum $(0,v{)}^T/\sqrt{2}$, the neutral gauge-boson mass matrix in $(W^3,B)$ is $(v^2/4)\left(\begin{array}{cc}{g}^2& -g{g}^{\prime }\\ -g{g}^{\prime }& g^{\prime 2}\end{array}\right)$. Its zero eigenvector is proportional to $(g^{\prime },g)$, the photon; the perpendicular vector is the Z boson. The charged fields $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$ have the stated W boson mass. Thus $m_W=m_Z\cos {\theta }_W$, where ${\theta }_W$ is the Weinberg angle. The unbroken charge generator annihilates the vacuum and explains the zero eigenvalue.