Compton 散射

贡献者：待更新

　　 Consider the process $e^{-} (p) γ (k) \to e^{-} (p^{'}) γ (k^{'})$

\begin{matrix} (1) & \begin{aligned} i M & = (- i e)^{2} ε_{μ}^{*} (k^{'}) ε_{ν} (k) [\bar{u} (p^{'}) γ^{μ} \frac{i (⧸ p + ⧸ k + m)}{(p + k)^{2} - m^{2}} γ^{ν} u (p) + \bar{u} (p^{'}) γ^{ν} \frac{i (⧸ p - ⧸ k^{'} + m)}{(p - k^{'})^{2} - m^{2}} γ^{μ} u (p)] . \end{aligned} \end{matrix}

this process is related to

e^{+} e^{-} \to γ γ

（Bhabha 散射）by crossing symmetry, so we can simply do variable replacements:

k \to - p^{'}, p^{'} \to - k .

Since we change a

e^{+}

of 'in' state to

e^{-}

of 'out' state, we should make the above substitutions, as well as add a factor

- 1

\begin{matrix} (2) & \begin{array}{r} \sum^{―} | M |^{2} = 2 e^{4} [\frac{p \cdot k^{'}}{p \cdot k} + \frac{p \cdot k}{p \cdot k^{'}} + 2 m^{2} (\frac{1}{p \cdot k} - \frac{1}{p \cdot k^{'}}) + m^{4} {(\frac{1}{p \cdot k} - \frac{1}{p \cdot k^{'}})}^{2}] . \end{array} \end{matrix}

in the "lab" frame in which the electron is initially at rest:

k = (ω, ω \hat{z}), p = (m, 0), p^{'} = (E^{'}, p^{'}), k^{'} = (ω^{'}, ω^{'} \sin θ, 0, ω^{'} \cos θ) .

we will express the cross section in terms of

ω, θ

\begin{matrix} (3) & \begin{aligned} m^{2} & = (p^{'})^{2} = (p + k - k^{'})^{2} = p^{2} + 2 p \cdot (k - k^{'}) - 2 k \cdot k^{'} = m^{2} + 2 m (ω - ω^{'}) - 2 ω ω^{'} (1 - \cos θ) \\ \Rightarrow \frac{1}{ω^{'}} - \frac{1}{ω} = \frac{1}{m} (1 - \cos θ) \\ \Rightarrow ω^{'} = \frac{ω}{1 + \frac{ω}{m} (1 - \cos θ)} . \end{aligned} \end{matrix}

the phase space integral:

\begin{matrix} (4) & \begin{aligned} \int d Π_{2} & = \int \frac{d^{3} k^{'}}{(2 π)^{3}} \frac{1}{2 ω^{'}} \frac{d^{3} p^{'}}{(2 π)^{3}} \frac{1}{(2 E^{'})} (2 π)^{4} δ^{4} (k^{'} + p^{'} - k - p) \\ = \int \frac{(ω^{'})^{2} d ω^{'} d Ω}{(2 π)^{3}} \frac{1}{4 ω^{'} E^{'}} \times 2 π δ (ω^{'} + \sqrt{m^{2} + ω^{2} + (ω^{'})^{2} - 2 ω ω^{'} \cos θ} - ω - m) \\ = \int \frac{d \cos θ}{2 π} \frac{ω^{'}}{4 E^{'}} \frac{1}{| 1 + \frac{ω^{'} - ω \cos θ}{E^{'}} |} \\ = \frac{1}{8 π} \int d \cos θ \frac{(ω^{'})^{2}}{ω m} . \end{aligned} \end{matrix}

Using

d σ = \frac{1}{2 E_{A} 2 E_{B} | v_{A} - v_{B} |} d Π_{2} \times \sum^{―} | M |^{2} .

we finally obtain the Klein-Nishina formula:

\begin{matrix} (5) & \begin{aligned} \frac{d σ}{d \cos θ} & = \frac{1}{2 ω 2 m} \cdot \frac{1}{8 π} \frac{(ω^{'})^{2}}{ω m} (\sum^{―} | M |^{2}) \\ = \frac{π α^{2}}{m^{2}} {(\frac{ω^{'}}{ω})}^{2} [\frac{ω^{'}}{ω} + \frac{ω}{ω^{'}} - \sin^{2} θ], ω^{'} = \frac{ω}{1 + \frac{ω}{m} (1 - \cos θ)} . \end{aligned} \end{matrix}

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