{c}
{title2=$\Gamma(H\to ZZ)$}

For $m_H>2m_Z$, the <Higgs boson coupling to Z bosons> gives $\mathcal M=(2m_Z^2/v)\epsilon_1^*\cdot\epsilon_2^*$. The <massive vector polarization sum> gives $\sum|\mathcal M|^2=(m_H^4/v^2)(1-4x+12x^2)$, with $x=m_Z^2/m_H^2$. The <two-body decay phase space> and <identical final-state symmetry factor> $1/2$ give
$$
\boxed{\Gamma(H\to ZZ)=\frac{m_H^3}{32\pi v^2}\sqrt{1-4x}(1-4x+12x^2).}
$$
The contraction uses $k_1\cdot k_2=(m_H^2-2m_Z^2)/2$. The on-shell rate vanishes at threshold; below threshold an off-shell calculation is required.

Higgs decay to two Z bosons

Both the <D meson> and <kaon> are spin-zero <pseudoscalars>. The states obey <parity symmetry in quantum field theory>. Consequently an <axial current> <matrix element> between them would have to be a <pseudovector> formed from only $p$ and $k$, which is impossible: an <totally antisymmetric tensor> would require more <linearly independent vectors>. Thus
$$
\langle K^+(k)|\bar c\gamma^\mu\gamma^5s|\bar D^0(p)\rangle=0.
$$
<Lorentz covariance> then leaves two <linearly independent vectors> for the <vector current> <matrix element>, giving the <pseudoscalar-to-pseudoscalar form factors>
$$
\langle K^+(k)|\bar c\gamma^\mu s|\bar D^0(p)\rangle
=(p+k)^\mu f_+(q^2)+(p-k)^\mu f_-(q^2),\qquad q=p-k.
$$
The vanishing <axial current> here follows from <parity symmetry in quantum field theory>, not <parity symmetry in quantum field theory> of the <weak interaction>.

To obtain the requested decay formula, use <naive factorization of a nonleptonic meson decay>: approximate the four-quark <matrix element> by the product of the $\bar D^0\to K^+$ current <matrix element> and the vacuum-to-<pion> <matrix element>. This is an additional hadronic approximation; tree-level weak vertices alone do not establish it, and nonfactorizable <Quantum chromodynamics> effects can change the result. The vacuum-to-<pion> <vector current> <matrix element> vanishes by <parity symmetry in quantum field theory>, while the specified <pion decay constant> normalization gives
$$
\langle\pi^-(q)|\bar d\gamma_\mu(1-\gamma^5)u|0\rangle=+i\sqrt2 F_\pi q_\mu.
$$
The $\sqrt2$ cancels the $1/\sqrt2$ in the <four-fermion interaction>. Up to an irrelevant overall phase, the <scattering amplitude> is therefore
$$
\mathcal M=G_F V_{cs}V_{ud}^*F_\pi\left[(m_D^2-m_K^2)f_+(q^2)+q^2f_-(q^2)\right].
$$
For $m_\pi=0$, $q^2=0$, so only $f_+(0)$ survives. Integrating the <two-body decay phase space> in the <D meson> rest frame gives
$$
|\boldsymbol k|=\frac{m_D^2-m_K^2}{2m_D},\qquad
\Gamma=\frac{|\boldsymbol k|}{8\pi m_D^2}|\mathcal M|^2.
$$
Hence
$$
\boxed{A=\frac{(m_D^2-m_K^2)^3}{16\pi m_D^3},\qquad
\Gamma=\frac{G_F^2(m_D^2-m_K^2)^3}{16\pi m_D^3}\left|V_{ud}^*V_{cs}F_\pi f_+(0)\right|^2.}
$$
This coefficient uses exactly the stated <pion decay constant> convention and the stated massless-<pion> approximation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 305 / 3 / d / Solution

{title2=$\Gamma=\frac{|\boldsymbol k|}{8\pi m_A^2}|\mathcal M|^2$}

This total decay-width formula applies in the parent's rest frame when the squared <scattering amplitude> is independent of direction and the daughters are distinct. For identical daughters an additional symmetry factor is needed.

Codex @codex  0

 Incoming links: Two-body decay phase space