If the vertex factor for the decay of a vector boson X into two spin- (1)/(2) fermions f1 and f2

Step 1: **Identify the decay process and the vertex factor.** The decay process is $X rightarrow

Question

If the vertex factor for the decay of a vector boson $\mathrm{X}$ into two spin- $\frac{1}{2}$ fermions $\mathrm{f}_1$ and $\overline{\mathrm{f}}_2$ is $$ -i g_X \gamma^\mu \frac{1}{2}\left(c_V-c_A \gamma^5\right), $$ then show that $$ \Gamma\left(X \rightarrow f_1 \hat{f}_2\right)=\frac{g_X^2}{48 \pi}\left(c_V^2+c_A^2\right) M_X, $$ where $M_X$ is the mass of the boson and where we have neglected the masses of the fermions.

    If the vertex factor for the decay of a vector boson $\mathrm{X}$ into two spin- $\frac{1}{2}$ fermions $\mathrm{f}_1$ and $\overline{\mathrm{f}}_2$ is
$$
-i g_X \gamma^\mu \frac{1}{2}\left(c_V-c_A \gamma^5\right),
$$
then show that
$$
\Gamma\left(X \rightarrow f_1 \hat{f}_2\right)=\frac{g_X^2}{48 \pi}\left(c_V^2+c_A^2\right) M_X,
$$
where $M_X$ is the mass of the boson and where we have neglected the masses of the fermions.

Quarks and leptons: introductory course in modern particle physics

Francis Halzen, Alan… 1st Edition

Chapter 13, Problem 2

Instant Answer

Step 1

** The decay process is $X \rightarrow f_1 \overline{f}_2$, where $X$ is a vector boson and $f_1$ and $\overline{f}_2$ are spin-$\frac{1}{2}$ fermions. The vertex factor given is $-i g_X \gamma^\mu \frac{1}{2}(c_V - c_A \gamma^5)$. Show more…

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If the vertex factor for the decay of a vector boson $\mathrm{X}$ into two spin- $\frac{1}{2}$ fermions $\mathrm{f}_1$ and $\overline{\mathrm{f}}_2$ is $$ -i g_X \gamma^\mu \frac{1}{2}\left(c_V-c_A \gamma^5\right), $$ then show that $$ \Gamma\left(X \rightarrow f_1 \hat{f}_2\right)=\frac{g_X^2}{48 \pi}\left(c_V^2+c_A^2\right) M_X, $$ where $M_X$ is the mass of the boson and where we have neglected the masses of the fermions.

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