Question

Using the couplings in the standard model, calculate $R_\mu$ at $s=M_Z^2$. Use $\sin ^2 \theta_W=\frac{1}{4}, M_Z=90 \mathrm{GeV}$, and $\Gamma_Z=2.5 \mathrm{GeV}$. It is relevant to ask what electroweak effects occur in $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \overline{\mathrm{q}} \mathrm{q}$. They are not the same as in $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \mu^{+} \mu^{-}$, since $c_{V, A}^q \neq c_{V, A^*}^\mu$. However, the calculation proceeds exactly as above. For instance, the counterpart to (13.60) is where $Q_q$ is the charge of the quark and the factor 3 is for color. Following through the calculation, the analogous result to (13.68) is $$ \frac{\sigma\left(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \overline{\mathrm{q}} \mathrm{q}\right)}{\sigma_0} \equiv R_q=3\left[Q_q^2+2 Q_q \operatorname{Re}(r) c_V^q c_V^e+|r|^2\left(c_V^{q 2}+c_A^{q 2}\right)\left(c_V^{e 2}+c_A^{e 2}\right)\right] \text {. } $$

   Using the couplings in the standard model, calculate $R_\mu$ at $s=M_Z^2$. Use $\sin ^2 \theta_W=\frac{1}{4}, M_Z=90 \mathrm{GeV}$, and $\Gamma_Z=2.5 \mathrm{GeV}$.

It is relevant to ask what electroweak effects occur in $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \overline{\mathrm{q}} \mathrm{q}$. They are not the same as in $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \mu^{+} \mu^{-}$, since $c_{V, A}^q \neq c_{V, A^*}^\mu$. However, the calculation proceeds exactly as above. For instance, the counterpart to (13.60) is
where $Q_q$ is the charge of the quark and the factor 3 is for color. Following through the calculation, the analogous result to (13.68) is
$$
\frac{\sigma\left(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \overline{\mathrm{q}} \mathrm{q}\right)}{\sigma_0} \equiv R_q=3\left[Q_q^2+2 Q_q \operatorname{Re}(r) c_V^q c_V^e+|r|^2\left(c_V^{q 2}+c_A^{q 2}\right)\left(c_V^{e 2}+c_A^{e 2}\right)\right] \text {. }
$$
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Quarks and leptons: introductory course in modern particle physics
Quarks and leptons: introductory course in modern particle physics
Francis Halzen, Alan… 1st Edition
Chapter 13, Problem 10 ↓

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This involves considering the electroweak interactions mediated by the Z boson.  Show more…

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Using the couplings in the standard model, calculate $R_\mu$ at $s=M_Z^2$. Use $\sin ^2 \theta_W=\frac{1}{4}, M_Z=90 \mathrm{GeV}$, and $\Gamma_Z=2.5 \mathrm{GeV}$. It is relevant to ask what electroweak effects occur in $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \overline{\mathrm{q}} \mathrm{q}$. They are not the same as in $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \mu^{+} \mu^{-}$, since $c_{V, A}^q \neq c_{V, A^*}^\mu$. However, the calculation proceeds exactly as above. For instance, the counterpart to (13.60) is where $Q_q$ is the charge of the quark and the factor 3 is for color. Following through the calculation, the analogous result to (13.68) is $$ \frac{\sigma\left(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \overline{\mathrm{q}} \mathrm{q}\right)}{\sigma_0} \equiv R_q=3\left[Q_q^2+2 Q_q \operatorname{Re}(r) c_V^q c_V^e+|r|^2\left(c_V^{q 2}+c_A^{q 2}\right)\left(c_V^{e 2}+c_A^{e 2}\right)\right] \text {. } $$
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