Using the couplings in the standard model, calculate Rμ at s=MZ^2. Use sin^2 θW=(1)/(4), MZ=90 G

Using the couplings in the standard model, calculate Rμ at...

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 $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \overline{q q}$. 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 $$ \frac{d \sigma\left(\mathrm{e}_{L}^{-} \mathrm{e}_{R}^{+} \rightarrow \mathrm{q}_{L}^{-} \mathrm{q}_{R}^{+}\right)}{d \Omega}=3 \frac{\alpha^{2}}{4 s}(1+\cos \theta)^{2}\left|Q_{q}+r c q c_{L}^{e}\right|^{2}, $$ 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 $$ \frac{\sigma\left(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \overline{\mathrm{q}} \mathrm{q}\right)}{\sigma_{0}} \equiv R_{q}=3\left[Q_{q}^{2}+2 Q_{q} \operatorname{Re}(r) c q_{V} c_{V}^{e}+|r|^{2}\left(c q^{q^{2}}+c_{A}^{q 2}\right)\left(c_{V}^{e 2}+c_{A}^{e 2}\right)\right] . $$

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 $\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \overline{q q}$. 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
$$
\frac{d \sigma\left(\mathrm{e}_{L}^{-} \mathrm{e}_{R}^{+} \rightarrow \mathrm{q}_{L}^{-} \mathrm{q}_{R}^{+}\right)}{d \Omega}=3 \frac{\alpha^{2}}{4 s}(1+\cos \theta)^{2}\left|Q_{q}+r c q c_{L}^{e}\right|^{2},
$$
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
$$
\frac{\sigma\left(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \overline{\mathrm{q}} \mathrm{q}\right)}{\sigma_{0}} \equiv R_{q}=3\left[Q_{q}^{2}+2 Q_{q} \operatorname{Re}(r) c q_{V} c_{V}^{e}+|r|^{2}\left(c q^{q^{2}}+c_{A}^{q 2}\right)\left(c_{V}^{e 2}+c_{A}^{e 2}\right)\right] .
$$

Key Concepts

Color Charge in Quark Production

When electron-positron annihilation produces quark-antiquark pairs, the effect of quantum chromodynamics becomes significant. Quarks carry color charge, and the cross section includes an extra color factor (typically 3) to account for the three possible color states. This factor distinguishes quark production from purely leptonic processes and is essential for accurate predictions in the Standard Model.

Interference Effects

In processes like electron-positron annihilation to fermion pairs, interference between photon-mediated and Z boson-mediated amplitudes can occur. This interference is sensitive to the specific couplings and charges of the involved fermions and plays a significant role in shaping the observed cross sections, especially near the Z boson resonance.

Electron-Positron Annihilation Cross Sections

Calculating cross sections for e?e? annihilation into fermion pairs involves summing contributions from different mediators (photon and Z boson) and accounting for their interference. These calculations utilize the electroweak couplings and propagator factors, and they may differ for fermions with different quantum numbers, which is reflected in observable quantities like cross section ratios.

Z Boson Resonance

The Z boson plays a crucial role in processes occurring at energy scales near its mass, as exemplified by resonance phenomena when the center-of-mass energy is close to M_Z. The mass (M_Z) and width (?_Z) of the Z boson determine how the propagator behaves and influence the overall shape and magnitude of the cross section, particularly in electron-positron annihilation into fermion pairs.

Electroweak Couplings

In the Standard Model, the electroweak interaction is governed by the coupling of fermions to the gauge bosons (the photon and the Z boson). These couplings include both electromagnetic interactions, characterized by the electric charge, and weak interactions, characterized by vector (c_V) and axial-vector (c_A) couplings. The differences in these couplings for various fermions result in different strengths and characteristics of the interaction processes, which are pivotal in determining cross sections in scattering events.

Vector and Axial-Vector Currents

Fermion interactions with the Z boson involve both vector and axial-vector currents. The vector (c_V) coupling affects the symmetric part of the interaction, while the axial (c_A) coupling introduces parity-violating effects. The interplay between these two types of couplings is essential for understanding angular distributions and polarization effects in electroweak processes.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that according to the alpha particle model of the nucleus 6 carbon 12, consists of three alpha particles that is 2 aillion 4 nuclei, and 8 oxygen 16 consists of 8 .1 .6.

00:36 Oxygen 16 consists of four alpha particles.

00:45 So with that said, for power a of this problem, we are asked to use the table 15 .1 to evaluate the difference between the total biting energy of carbon and the total body energy of three alpha particles.

01:01 So what we need to do in this case is to calculate the biting energy for 6 carbon, 12, which is the following expression.

01:10 So that will be 6 times the mass of hydrogen that you can find in the table.

01:17 And this plus 6 times the mass of neutron.

01:23 And this minus the mass of carbon.

01:26 And this times 931 .5 mega electron balls per unit of mass.

01:34 So if we do this product in here and substitute the ball.

01:38 That we found in the table in each of these positions, then we will find that the result from this is equal to 92 .1 ,660 mega -electron balls.

01:56 Now, and that of elyon 4, we know that as the problem states, this consists of three alpha particles, so we will have that in this case is, two times the mass of hydrogen.

02:13 We know that this is for elyon, of course, and plus two times the mass of a neutron, and this the mass of elyon.

02:21 And this times 931 .5 megalletron moles per unit of mass.

02:29 Now, if we substitute the values that we found in the table, we obtain a value of 28 .2 ,970 megalletron, now, the difference between these two values, remember that as the problem state and the carbon 12 is composed of three alpha particles, so we will have that that is.

02:55 The energy for carbon 6, carbon 12 minus three times the bounding energy for alien.

03:06 So from this we obtain a value of 7 .275 megalletron volts.

03:14 And this equals the biting energy of carbon on the alpha particle model.

03:20 So that's a solution for par a of this problem.

03:28 Now for part b, we are asked about to evaluate the difference between the total biting energy of oxygen 16 and the total binding energies of four alpha particles.

03:46 So we do this to do something similar as before.

03:50 So we know that for the oxygen, we're going to have that in this case is a times the mass of hydrogen minus, well, plus eight times the mass of neutrons, and this minus the mass of neutrons, and this minus the of oxygen.

04:12 And this, of course, times 931 .5 megalletron balls per unit of mass.

04:19 So using our calculator, we obtain a value of 127 .62 .463 megalletron balls.

04:31 And then the difference between this value that we just have obtained for oxygen minus, as the problem state, this is composed of 4 billion alpha particles.

04:48 So this will give us a value of 14 .44 mega electron volts.

04:55 And is the binding energy of oxygen on the alpha particle model.

05:01 So that's a solution for par b of this problem.

05:06 Now, for par c, we are asked about to draw an schematic diagrams of carbon in oxygen according to the in oxygen...

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