Using the above approach, show that Γ(π^- →π^0 e^- ν̅e)=(G^2)/(30 π^3)(Δm)^5, where Δm=m(π^-)-m(

step 1 For this exercise we have that P sub sigma is equal to 1 .602 times 10 to the power of negative

Using the above approach, show that Γ(π^- →π^0 e^-...

Using the above approach, show that $$ \Gamma\left(\pi^{-} \rightarrow \pi^0 \mathrm{e}^{-} \bar{\nu}_e\right)=\frac{G^2}{30 \pi^3}(\Delta m)^5, $$ where $\Delta m=m\left(\pi^{-}\right)-m\left(\pi^0\right)=4.6 \mathrm{MeV}$. Evaluate the decay rate and compare with $\Gamma\left(\pi^{-} \rightarrow \mu^{-} \bar{v}_\mu\right)$. We are now ready to tackle neutrino-quark scattering. As the quarks and lepton weak currents have identical forms, we can carry over the results for $\nu$ e scattering that we obtained in Section 12.7. From (12.59) and (12.62), we obtain in the center-of-mass frame where $\theta$ is defined as in Fig. 12.13. From the figure, it is immediately apparent that the backward $\bar{\nu}_\mu \mathrm{u} \rightarrow \mu^{+} \mathrm{d}$ scattering $(\theta=\pi)$ is forbidden by helicity considerations. The cross sections for scattering from antiquarks, $\bar{\nu}_\mu \overline{\mathrm{d}} \rightarrow \mu^{+} \overline{\mathrm{u}}$ and $\nu_\mu \overline{\mathrm{u}} \rightarrow$ $\mu^{-} \overline{\mathrm{d}}$, are given by (12.69) and (12.70), respectively. We see that, for instance, $\nu_\mu$ does not interact with either $\mathrm{u}$ or $\overline{\mathrm{d}}$ quarks. To compare these results with experiment, we have to embed the constituent cross sections, (12.69) and (12.70), in the overall $\nu N$ inclusive cross section. The procedure is familiar from Chapter 9. We obtain First, note that the angular distributions of the constituent process have been expressed in terms of the dimensionless variable $y$. It is related to $\cos \theta$ by $$ 1-y=\frac{p \cdot k^{\prime}}{p \cdot k}=\frac{1}{2}(1+\cos \theta) $$

Using the above approach, show that
$$
\Gamma\left(\pi^{-} \rightarrow \pi^0 \mathrm{e}^{-} \bar{\nu}_e\right)=\frac{G^2}{30 \pi^3}(\Delta m)^5,
$$
where $\Delta m=m\left(\pi^{-}\right)-m\left(\pi^0\right)=4.6 \mathrm{MeV}$. Evaluate the decay rate and compare with $\Gamma\left(\pi^{-} \rightarrow \mu^{-} \bar{v}_\mu\right)$.
We are now ready to tackle neutrino-quark scattering. As the quarks and lepton weak currents have identical forms, we can carry over the results for $\nu$ e scattering that we obtained in Section 12.7. From (12.59) and (12.62), we obtain in the center-of-mass frame
where $\theta$ is defined as in Fig. 12.13. From the figure, it is immediately apparent that the backward $\bar{\nu}_\mu \mathrm{u} \rightarrow \mu^{+} \mathrm{d}$ scattering $(\theta=\pi)$ is forbidden by helicity considerations. The cross sections for scattering from antiquarks, $\bar{\nu}_\mu \overline{\mathrm{d}} \rightarrow \mu^{+} \overline{\mathrm{u}}$ and $\nu_\mu \overline{\mathrm{u}} \rightarrow$ $\mu^{-} \overline{\mathrm{d}}$, are given by (12.69) and (12.70), respectively. We see that, for instance, $\nu_\mu$ does not interact with either $\mathrm{u}$ or $\overline{\mathrm{d}}$ quarks.

To compare these results with experiment, we have to embed the constituent cross sections, (12.69) and (12.70), in the overall $\nu N$ inclusive cross section. The procedure is familiar from Chapter 9. We obtain First, note that the angular distributions of the constituent process have been expressed in terms of the dimensionless variable $y$. It is related to $\cos \theta$ by
$$
1-y=\frac{p \cdot k^{\prime}}{p \cdot k}=\frac{1}{2}(1+\cos \theta)
$$

Key Concepts

Scattering Kinematics and the Variable y

Scattering kinematics involve the analysis of energy and momentum transfer during interactions and are often simplified by introducing dimensionless variables. The variable y, which relates to the cosine of the scattering angle, serves as a convenient parameter to express angular distributions. It connects the geometry of the scattering event with the underlying dynamics, aiding in the interpretation of experimental data and theoretical predictions.

Helicity Constraints in Weak Processes

Helicity, which is the projection of a particle’s spin along its momentum direction, plays a critical role in weak interactions due to their chiral nature. The restrictions imposed by helicity conservation can forbid certain scattering processes, as seen with backward scattering in neutrino-quark interactions, thereby shaping the angular distribution and overall probability of the reaction.

Neutrino-Quark Scattering and the Parton Model

This concept bridges the gap between fundamental interactions and the composite nature of nucleons. By treating quarks within nucleons as individual scattering centers, results from neutrino–electron scattering can be adapted to neutrino–quark scattering. Embedding constituent cross sections into overall nucleon cross sections allows for a more complete comparison with experimental observations in high-energy processes.

Fermi Theory of Weak Interactions

This theory provides a framework for understanding weak decays and scattering processes at low energies by modeling them with a contact interaction characterized by the Fermi constant, G. It is crucial for calculating decay rates and cross sections, as it underpins the derivation of expressions such as the decay rate of the pion and the scattering amplitudes in neutrino-quark interactions.

Phase Space Factors in Particle Decays

Phase space considerations account for the available energy distribution among decay products and are key to determining decay rates. In processes like the pion decay mentioned, the decay rate scaling as the fifth power of the mass difference (?m) reflects the integration over the multi-body phase space, highlighting the sensitivity of weak decays to the kinematic boundaries.

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