Show that 2(k ·p^')(k^' ·p)=(p-k^')^2(k^' ·p)=(m^2-2 m ω^') m ω^' in the muon rest frame, where

step 1 As we all know that the beta plus decay can be given by P changing to D plus 01E. Now the value

Show that 2(k ·p^')(k^' ·p)=(p-k^')^2(k^' ·p)=(m^2-2 m ω^')...

Show that $$ 2\left(k \cdot p^{\prime}\right)\left(k^{\prime} \cdot p\right)=\left(p-k^{\prime}\right)^{2}\left(k^{\prime} \cdot p\right)=\left(m^{2}-2 m \omega^{\prime}\right) m \omega^{\prime} $$ in the muon rest frame, where $p=(m, 0,0,0)$. Gathering these results together, the decay rate in the muon rest frame is $$ \begin{aligned} d \Gamma=& \frac{G^{2}}{2 m \pi^{5}} \frac{d^{3} p^{\prime}}{2 E^{\prime}} \frac{d^{3} k^{\prime}}{2 \omega^{\prime}} m \omega^{\prime}\left(m^{2}-2 m \omega^{\prime}\right) \\ & \times \delta\left(m^{2}-2 m E^{\prime}-2 m \omega^{\prime}+2 E^{\prime} \omega^{\prime}(1-\cos \theta)\right) \end{aligned} $$ and, as for $\beta$-decay, we can replace $d^{3} p^{\prime} d^{3} k^{\prime}$ by $$ 4 \pi E^{\prime 2} d E^{\prime} 2 \pi \omega^{\prime 2} d \omega^{\prime} d \cos \theta $$

Show that
$$
2\left(k \cdot p^{\prime}\right)\left(k^{\prime} \cdot p\right)=\left(p-k^{\prime}\right)^{2}\left(k^{\prime} \cdot p\right)=\left(m^{2}-2 m \omega^{\prime}\right) m \omega^{\prime}
$$
in the muon rest frame, where $p=(m, 0,0,0)$.
Gathering these results together, the decay rate in the muon rest frame is
$$
\begin{aligned}
d \Gamma=& \frac{G^{2}}{2 m \pi^{5}} \frac{d^{3} p^{\prime}}{2 E^{\prime}} \frac{d^{3} k^{\prime}}{2 \omega^{\prime}} m \omega^{\prime}\left(m^{2}-2 m \omega^{\prime}\right) \\
& \times \delta\left(m^{2}-2 m E^{\prime}-2 m \omega^{\prime}+2 E^{\prime} \omega^{\prime}(1-\cos \theta)\right)
\end{aligned}
$$
and, as for $\beta$-decay, we can replace $d^{3} p^{\prime} d^{3} k^{\prime}$ by
$$
4 \pi E^{\prime 2} d E^{\prime} 2 \pi \omega^{\prime 2} d \omega^{\prime} d \cos \theta
$$

Key Concepts

Rest Frame Analysis

Employing the rest frame of the decaying particle simplifies the kinematic analysis of the decay process. In this frame, the initial particle’s spatial momentum is zero, which reduces the complexity of energy and momentum relations. This simplification makes it easier to express and calculate the energy distributions and angular dependencies of the decay products.

Delta Function Constraints

Delta functions are critical in imposing conservation laws in decay processes. They mathematically enforce the conservation of energy and momentum by restricting the integration over phase space to only those final states that satisfy the necessary conservation conditions, thus ensuring physically consistent results.

Four-Momentum Dot Products

The use of four-momentum dot products is fundamental in relativistic calculations. These scalar products combine energy and momentum contributions in a Lorentz-invariant manner, facilitating the simplification of expressions that arise in decay processes. They help relate various kinematic quantities, such as invariant masses and energy transfers, which are important in evaluating the decay amplitude.

Phase Space Integration

Phase space integration is a method used to account for all possible final states in a decay process. It involves integrating over the momentum space variables of the decay products, often by changing variables (such as converting three-dimensional momentum integrals into energy and angular parts) and includes the appropriate density of states, which is crucial for accurately computing the decay rate.

Decay Rate Calculations

In particle physics, decay rate (or lifetime) calculations are essential for understanding how unstable particles transform into other particles. They involve the evaluation of the squared matrix element for the decay process, combined with integrations over the phase space of the final state particles, and are governed by conservation laws and transition probabilities as described by Fermi's golden rule.

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