# 巴巴散射

$e^+ e^- \rightarrow e^+ e^-$

## 散射微分截面

$\frac{\mathrm{d} \sigma}{\mathrm{d} (\cos\theta)} = \frac{\pi \alpha^2}{s} \left( u^2 \left( \frac{1}{s} + \frac{1}{t} \right)^2 + \left( \frac{t}{s} \right)^2 + \left( \frac{s}{t} \right)^2 \right) \,$

### 曼德尔斯坦变量

 $s= \,$ $(k+p)^2= \,$ $(k'+p')^2 \approx \,$ $2 k \cdot p \approx\,$ $2 k' \cdot p' \,$ $t= \,$ $(k-k')^2= \,$ $(p-p')^2\approx \,$ $-2 k \cdot k' \approx \,$ $-2 p \cdot p' \,$ $u= \,$ $(k-p')^2= \,$ $(p-k')^2\approx \,$ $-2 k \cdot p' \approx \,$ $-2 k' \cdot p \,$

## 无偏振散射截面的推导

### 矩阵元

 这里变量的含义为: $\gamma^\mu \,$是狄拉克矩阵, $u\,$和$\bar{u}\,$是费米子的四分量旋量， $v\,$和$\bar{v}\,$是反费米子的四分量旋量，参见狄拉克方程。 (散射) (湮灭) $\mathcal{M} = \,$ $-e^2 \left( \bar{v}_{k} \gamma^\mu v_{k'} \right) \frac{1}{(k-k')^2} \left( \bar{u}_{p'} \gamma_\mu u_p \right)$ $+e^2 \left( \bar{v}_{k} \gamma^\nu u_p \right) \frac{1}{(k+p)^2} \left( \bar{u}_{p'} \gamma_\nu v_{k'} \right)$

### 矩阵元的平方

 $\overline{|\mathcal{M}|^2} \,$ $= \frac{1}{(2s_{e-} + 1)(2 s_{e+} + 1)} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,$ $= \frac{1}{4} \sum_{s=1}^2 \sum_{s'=1}^2 \sum_{r=1}^2 \sum_{r'=1}^2 |\mathcal{M}|^2 \,$

 $|\mathcal{M}|^2 \,$= $e^4 \left| \frac{(\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p)}{(k-k')^2} \right|^2 \,$ (散射) ${}- 2 e^4 \left( \frac{ (\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p)}{(k-k')^2} \right)^* \left( \frac{ (\bar{v}_{k} \gamma^\nu u_p )( \bar{u}_{p'} \gamma_\nu v_{k'}) }{(k+p)^2} \right) \,$ (干涉) ${}+ e^4 \left| \frac{(\bar{v}_{k} \gamma^\nu u_p )( \bar{u}_{p'} \gamma_\nu v_{k'} )}{(k+p)^2} \right|^2 \,$ (湮灭)

### 散射项

#### 矩阵元的平方

 $|\mathcal{M}|^2 \,$ $= \frac{e^4}{(k-k')^4} \Big( (\bar{v}_{k} \gamma^\mu v_{k'} )( \bar{u}_{p'} \gamma_\mu u_p) \Big)^* \Big( (\bar{v}_{k} \gamma^\mu v_{k'})( \bar{u}_{p'} \gamma_\mu u_p) \Big) \,$ $(1) \,$ $= \frac{e^4}{(k-k')^4} \Big( (\bar{v}_{k} \gamma^\mu v_{k'} )^* ( \bar{u}_{p'} \gamma_\mu u_p)^* \Big) \Big( (\bar{v}_{k} \gamma^\mu v_{k'})( \bar{u}_{p'} \gamma_\mu u_p) \Big) \,$ $(2) \,$ (复共轭作用到括号内时会交换次序) $= \frac{e^4}{(k-k')^4} \Big( \left(\bar{v}_{k'} \gamma^\mu v_{k} \right) \left( \bar{u}_{p} \gamma_\mu u_{p'} \right) \Big) \Big( \left( \bar{v}_{k} \gamma^\mu v_{k'} \right) \left( \bar{u}_{p'} \gamma_\mu u_p \right) \Big) \,$ $(3) \,$ (将依赖于同一个动量的项写到一起) $= \frac{e^4}{(k-k')^4} \left( \bar{v}_{k'} \gamma^\mu v_{k} \right) \left( \bar{v}_{k} \gamma^\mu v_{k'} \right) \left( \bar{u}_{p} \gamma_\mu u_{p'} \right) \left( \bar{u}_{p'} \gamma_\mu u_p \right) \,$ $(4) \,$

#### 对自旋求和

 $\frac{(k-k')^4}{e^4} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,$ $= \left(\sum_{r'} \bar{v}_{k'} \gamma^\mu (\sum_{r}v_{k} \bar{v}_{k}) \gamma^\mu v_{k'} \right) \left(\sum_{s} \bar{u}_{p} \gamma_\mu (\sum_{s'}{u_{p'} \bar{u}_{p'}}) \gamma_\mu u_p \right) \,$ $(5) \,$ $= \left( \Big(\sum_{r'} v_{k'} \bar{v}_{k'} \Big) \gamma^\mu \Big(\sum_{r}v_{k} \bar{v}_{k} \Big) \gamma^\mu \right) \left( \Big(\sum_{s} u_p \bar{u}_{p} \Big) \gamma_\mu \Big( \sum_{s'}{u_{p'} \bar{u}_{p'}} \Big) \gamma_\mu \right) \,$ $(6) \,$ (下一步推导使用完备性关系) $=\operatorname{Tr}\left( (k\!\!\!/' - m) \gamma^\mu (k\!\!\!/ - m) \gamma^\nu \right) \cdot \operatorname{Tr}\left( (p\!\!\!/' + m) \gamma_\mu (p\!\!\!/ + m) \gamma_\nu \right) \,$ $(7) \,$ (下一步推导使用迹恒等式) $=\left(4 \left( {k'}^\mu k^\nu - \mathbf{k' \cdot k}\eta^{\mu\nu} + k'^\nu k^\mu \right) + 4 m^2 \eta^{\mu\nu} \right) \left( 4 \left( {p'}_\mu p_\nu - \mathbf{p' \cdot p}\eta_{\mu\nu} + p'_\nu p_\mu \right) + 4 m^2 \eta_{\mu\nu} \right) \,$ $(8) \,$ $=32\left( (k' \cdot p') (k \cdot p) + (k' \cdot p) (k \cdot p') -m^2 p' \cdot p - m^2 k' \cdot k + 2m^4 \right) \,$ $(9) \,$

 $\frac{1}{4} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,$ $= \frac{32e^4}{4(k-k')^4} \left( (k' \cdot p') (k \cdot p) + (k' \cdot p) (k \cdot p') \right) \,$ (在相对论极限下使用曼德尔斯坦变量) $=\frac{8e^4}{t^2} \left(\tfrac{1}{2} s \tfrac{1}{2}s + \tfrac{1}{2}u \tfrac{1}{2} u \right) \,$ $= 2 e^4 \frac{s^2 +u^2}{t^2} \,$

### 湮灭项

 $\frac{1}{4} \sum_{\mathrm{spins}} |\mathcal{M}|^2 \,$ $= \frac{32e^4}{4(k+p)^4} \left( (k \cdot k') (p \cdot p') + (k' \cdot p) (k \cdot p') \right) \,$ $=\frac{8e^4}{s^2} \left(\tfrac{1}{2} t \tfrac{1}{2}t + \tfrac{1}{2}u \tfrac{1}{2} u \right) \,$ $= 2 e^4 \frac{t^2 +u^2}{s^2} \,$

### 最终解

$\frac{\overline{|\mathcal{M}|^2}}{2e^4} = \frac{u^2 + s^2}{t^2} + \frac{2 u^2}{st} + \frac{u^2 + t^2}{s^2} \,$

## 简化步骤中用到的关系

### 完备性关系

$\sum_{s=1,2}{u^{(s)}_p \bar{u}^{(s)}_p} = p\!\!\!/ + m \,$
$\sum_{s=1,2}{v^{(s)}_p \bar{v}^{(s)}_p} = p\!\!\!/ - m \,$

$p\!\!\!/ = \gamma^\mu p_\mu \,$      (参见费曼“斜杠”标记en:Feynman Slash Notation ）)
$\bar{u} = u^{\dagger} \gamma^0 \,$

### 迹恒等式

1. 奇数个狄拉克矩阵$\gamma_\mu \,$的乘积的迹为零；
2. $\operatorname{tr} (\gamma^\mu\gamma^\nu) = 4\eta^{\mu\nu}$
3. $\operatorname{Tr}\left( \gamma_\rho \gamma_\mu \gamma_\sigma \gamma_\nu \right) = 4 \left( \eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho\sigma}\eta_{\mu\nu}+\eta_{\rho\nu}\eta_{\mu\sigma} \right) \,$

 $\operatorname{Tr}\left( (p\!\!\!/' + m) \gamma_\mu (p\!\!\!/ + m) \gamma_\nu \right) \,$ $= \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu p\!\!\!/ \gamma_\nu \right) + \operatorname{Tr}\left(m \gamma_\mu p\!\!\!/ \gamma_\nu \right) \,$ $+ \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu m \gamma_\nu \right) + \operatorname{Tr}\left(m^2 \gamma_\mu \gamma_\nu \right) \,$ (根据式（1），中间两项为零) $= \operatorname{Tr}\left( p\!\!\!/' \gamma_\mu p\!\!\!/ \gamma_\nu \right) + m^2 \operatorname{Tr}\left(\gamma_\mu \gamma_\nu \right) \,$ (根据式（2）简化第二项) $= {p'}^{\rho} p^\sigma \operatorname{Tr}\left( \gamma_\rho \gamma_\mu \gamma_\sigma \gamma_\nu \right) + m^2 \cdot 4\eta_{\mu\nu} \,$ (现在用式（3）简化第一项) $= {p'}^{\rho} p^\sigma 4 \left( \eta_{\rho\mu}\eta_{\sigma\nu}-\eta_{\rho\sigma}\eta_{\mu\nu}+\eta_{\rho\nu}\eta_{\mu\sigma} \right) + 4 m^2 \eta_{\mu\nu} \,$ $=4 \left( {p'}_\mu p_\nu - \mathbf{p' \cdot p}\eta_{\mu\nu} + p'_\nu p_\mu \right) + 4 m^2 \eta_{\mu\nu} \,$

## 用途

• 斯坦福大学的大型Z玻色子探测器（Stanford Large Detector）在1993年进行的实验中，小角度的巴巴散射被用来测量实验的光度，测量的相对不确定度低于0.5%[2]
• 位於日本高能加速器研究機構貝爾實驗，其前置量能器(Extreme Forward Calorimeter, [1])即是使用小角度的巴巴散射，來即時地量測該實驗的亮度，並且與中心碘化銫量能器所測得的大角度巴巴散射交互校正。貝爾實驗為目前亮度最高的B介子工廠。
• 正负电子对撞的实验场所是地下的强子共振设备（ 能量约为1GeV至10 GeV），如北京的电子同步加速器BES）、贝尔（Belle）实验和介子的BaBar实验，这些实验利用大角度的巴巴散射作为光度测量的手段。如要达到相对不确定度小于0.1%的测量精确度，实验测量需要和理论计算结果相比较，理论上要求计算到领导项及其下一个高阶项的辐射修正[3]。强子散射截面在这些较低能量下的高精度测量是理论计算μ子反常磁矩的关键条件之一，而计算μ子的反常磁矩能够被用来约束超对称以及其他超越标准模型的粒子理论。

## 参考资料

1. ^
2. ^ White, Sharon Leigh. a Study of Small Angle Radiative Bhabha Scattering and Measurement of the Luminosity at SLD. Thesis (PH.D.)--THE UNIVERSITY OF TENNESSEE, 1995.Source: Dissertation Abstracts International, Volume: 57-02, Section: B, page: 1169.
3. ^ C.M. Carloni Calame, C. Lunardini, G. Montagna, O. Nicrosini, F. Piccinini. Large-angle Bhabha scattering and luminosity at flavour factories. Nucl.Phys. 2000, (B584): 459–479.
• Halzen, Francis; Martin, Alan. Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. 1984. ISBN 0-471-88741-2.
• Peskin, Michael E.; Schroeder, Daniel V. An Introduction to Quantum Field Theory. Perseus Publishing. 1994. ISBN 0-201-50397-2.