physics:qft:pion_interaction_from_fermi_theory
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| physics:qft:pion_interaction_from_fermi_theory [2023/02/26 13:22] – mikoto | physics:qft:pion_interaction_from_fermi_theory [2023/02/26 13:54] (現在) – mikoto | ||
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| \] | \] | ||
| + | 散乱振幅は | ||
| + | \begin{align} | ||
| + | \langle f| \mathcal{H}_{int} |i\rangle | ||
| + | &= \frac{1}{V^{2}}\sum_{i} (\bar{v}_{\bar{u}}(p_{\bar{u}}\Gamma_{i} u_{d}(p_{d})) | ||
| + | [\bar{u}_{e}(p_{e}) \Gamma_{i}(C_{i}+C_{i}^{\prime}\gamma_{5})v_{\bar{\nu}}(p_{\bar{\nu}})] \\ | ||
| + | & \times (2\pi)^{4}\delta^{(4)}(p_{e}+p_{\bar{\nu}} - p_{\bar{u}} - p_{d}) | ||
| + | \end{align} | ||
| + | |||
| + | 各項をまとめると | ||
| + | \begin{align} | ||
| + | & | ||
| + | [\bar{u}_{e}(p_{e}) \Gamma_{i}(C_{i}+C_{i}^{\prime}\gamma_{5})v_{\bar{\nu}}(p_{\bar{\nu}})] \\ | ||
| + | &= -(v^{\dagger}_{\bar{u}} u_{d}) | ||
| + | [\bar{u}_{e}(p_{e}) \gamma_{5}(C_{P}+C_{P}^{\prime}\gamma_{5})v_{\bar{\nu}}(p_{\bar{\nu}}) | ||
| + | - \bar{u}_{e}(p_{e}) \gamma_{5}\gamma_{0}(C_{A}+C_{A}^{\prime}\gamma_{5})v_{\bar{\nu}}(p_{\bar{\nu}})] \\ | ||
| + | & -(v^{\dagger}_{\bar{u}} \sigma^{i} u_{d}) | ||
| + | [\bar{u}_{e}(p_{e}) \gamma^{i}(C_{V}+C_{V}^{\prime}\gamma_{5})v_{\bar{\nu}}(p_{\bar{\nu}}) | ||
| + | - 2\bar{u}_{e}(p_{e}) \sigma^{0i}(C_{T}+C_{T}^{\prime}\gamma_{5})v_{\bar{\nu}}(p_{\bar{\nu}})] | ||
| + | \end{align} | ||
| + | |||
| + | $(v^{\dagger}_{\bar{u}} u_{d})$の部分はパイオンと見ることができるが、 | ||
| + | \[ | ||
| + | \mathcal{H}_{int} \propto \partial_{\lambda}\phi_{\pi}[\bar{\psi}_{e}\gamma_{5}\gamma^{\lambda}(1+\gamma_{5})\psi_{\nu}] | ||
| + | \] | ||
| + | の形とは異なっている。 | ||
| + | |||
| + | おそらくはじめにパイオンの静止系を仮定したことによる。 | ||
| ===== 参考資料 ===== | ===== 参考資料 ===== | ||
| ----- | ----- | ||
physics/qft/pion_interaction_from_fermi_theory.1677385367.txt.gz · 最終更新: 2023/02/26 13:22 by mikoto