场论计算的常用结论

• 原文: https://zhuanlan.zhihu.com/p/513594929
• 发布日期: 2023-10-14
• 分类: 粒子物理 / QFT / 场论计算

其实就是自己平时手动推导的时候懒得翻书又不敢挑战记忆力. 比起直接抄我的结果, 更推荐你算完了来对对, 这才比较万无一失.

持续更新到退圈, 但真的只有结论, 也只收录工作上多次用到过的

所有涉及到强耦合常数 $\mathbf{g}$ 的部分都小心点, 别跟我差个负号还瞎几把套

我一般默认 {D_\mu } = {\partial _\mu } + {\rm{i}g\frac{\lambda _\alpha }{2}A_\mu ^\alpha. DontAskWhy, 我就是一拍脑瓜觉得加号比较爽, 入门看的那帮老东西写的文献也都是加号, 结果现在你吗学术界好像减号占了上风了··· 咋选呢? 明确告诉你, 就看你们组、你周边人怎么选的, 跟他们一致就完了好吧.

目録

0. 常用的 Mathematica代码片段1. Gamma 矩阵的反对易性质2. Gamma 矩阵的自逆性3. 关于${\mathbf{\gamma }}^5,{\sigma }^{\mu \nu }$4. Dirac 与 Weyl 表示下的转置5. 荷共轭算符6. 共轭相关7. 旋量场的 P, C 变换8. δ 函数相关9. d 维 Fourier 变换10. 常用恒等式11. Feynman 传播子12. 固定点规范与协变导数12. 按凝聚量展开的常用完全传播子* 14. 悬挂胶子线16. 其它

0. Mathematica 常用代码片段

https://zhuanlan.zhihu.com/p/136542995https://www.zhihu.com/question/36891138/answer/2164172233https://www.zhihu.com/question/462238545/answer/3255502181- (1). 保存与提取计算结果[1]:

比如说要将变量 Term 中的数据保存下来方便以后随时提取. $\overline{}$导出: Export[NotebookDirectory[] <> "Term.wdx", Term];导入: Term = Import[NotebookDirectory[] <> "Term.wdx"];$\overline{}$这里的 NotebookDirectory[] 是你的 MMA note 所保存的文件夹地址[[2]](#ref_2).你也可以用 "文件所在地址" 代替 NotebookDirectory[]<>"Term.wdx[[3]](#ref_3)". 你会发现写文件地址的时候可以混用 \ 与 /, 但手写请用 , 因为 / 容易混淆. $\overline{}$关于导出: Export[NotebookDirectory[] <> "Term.wdx", Term]; Term.wdx 的部分是可以随便命名的, 输出的文件类型由后缀决定. 文件名好像不区分大小写的, 我也不知道为毛, 反正小心点.

• (2). 在字符串里安置变量:

其实就是要学会使用函数 StringTemplate, 比如你要将所有的 F[m,n] 全都保存下来:Do[Export[StringTemplate[NotebookDirectory[] <> "F[\`,``].wdx"][m, n], F[m, n]], {m, 1, 10}, {n, 1, 7}]` 这个函数的具体特性你可以自己去查. $\overline{}$ 还有一个我个人感觉没那么酷的做法, 就是用 <> 和 ToString 来完成, 有空会给个例子.

• (3). 要保存地址就默认所处文件夹的话能不能写个更简洁的函数呢?

可以是可以, 但没你想的那么简单, 答案如下你直接抄吧: SetAttributes[SaveAndLoad,HoldFirst];SaveAndLoad[Term_,S]:=Export[StringTemplate[NotebookDirectory[]<>"\`.wdx"][SymbolName[Unevaluated[Term]]],Term];SaveAndLoad[Term_,L]:=Term=Import[StringTemplate[NotebookDirectory[]<>"``.wdx"][SymbolName[Unevaluated[Term]]]];`

• (4). 要放文章里的图就别瞎画:

真正的懂哥画图都是先生成表格再用 ListLinePlot, 常用选项如下: ListLinePlot[{SampleA, SampleB}, GridLines -> {0.576, 0.622}, {0.1}, ImageSize -> 700, Mesh -> All, MeshStyle -> PointSize[0.01], PlotStyle -> {Nest[Darker, Green, 2], Thickness[0.003]}, PlotLabels -> {Callout[Style["SampleA", Bold], Before], Callout[Style["SampleB", Bold], Before]}, Frame -> True, FrameStyle -> Directive[Black, Thickness[0.003]], LabelStyle -> {Directive[Bold, Black], FontFamily -> "Times", FontSize -> 15}]

图下如果还要标注什么东西的话:

• (8). 善用 Sequence 函数, 这可以生成没有括号的数列作为函数的自变量.

比如可以定义 {A = Sequence[1, 0], B = Sequence[0, 1]};那么输入 F[B, A] 将得到 F[0,1,1,0].

• (6). 我得看看 Do 循环里边儿啥状况啊, 那就用 Dynamic.

循环外的 Dynamic[变量] 可以显示循环里的变量实时值, 这样循环过程算到哪儿又算出了啥就都很清楚了. 顺带一提, 如果你每次循环的数据能写成一个表格那可以通过 Append 函数把这些表格拼成一个矩阵. 比如可以提前定义好 ResultTable = {"Data1", "Data2", "Data3"}, 然后循环里再放个 ResultTable = Append[ResultTable , NewDataSample].

• (7). 表格的展示形式其实还挺多样的, 请选择你的捍卫者:

当然还有些组合用法:

总之你自己研究下吧, 就那么几个函数反正.

• (8). 请善用 Apply, 这玩意儿能直接用来换 Head:

当然你也可以用 @@ 代替这种写法, 只是我不太喜欢:

• (9). 别太迷信 StandardForm, 真看结构还得 InputForm:

是不是很神秘? 明明一模一样的输入··· 这是因为:

所以 StandardForm 并不总是真东西··· 要 InputForm 都搞不定就试试 FullForm 吧.

1. Gamma 矩阵的反对易性质:

https://zhuanlan.zhihu.com/p/373020066https://zhuanlan.zhihu.com/p/399908095> 任意线性操作 $f$ 都显然跟指标的上下位置没关系, 因为: $f\left({\Gamma }_{\mu \cdots \nu }\right)=f\left(g_{\mu \rho }\cdots g_{\nu \sigma }{\Gamma }^{\rho \cdots \sigma }\right)=g_{\mu \rho }\cdots g_{\nu \sigma }f\left({\Gamma }^{\rho \cdots \sigma }\right).$

• (1). \{\gamma }^{\mu },{\gamma }^{\nu }\}\equiv {\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }=2{g}^{\mu \nu }\mathbf{1}.

2. Gamma 矩阵的泛自逆性

• (1). {({\gamma }^{0})}^{2}={\gamma }^{0}{\gamma }^{0}=\frac{1}{2}\{\gamma }^{0},{\gamma }^{0}\}={g}^{00}=\mathbf{1}.
• (2). {({\gamma }^{i})}^{2}={\gamma }^{i}{\gamma }^{i}=\frac{1}{2}\{\gamma }^{i},{\gamma }^{i}\}={g}^{ii}=-\mathbf{1}.
• (3). ${({\gamma }^5)}^2=\text{i}{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3\text{i}{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3=1.$
• (4). {({\sigma }^{\mu \nu })}^{2}=\frac{\text{i}{2}({\gamma }^{\mu }{\gamma }^{\nu }-{\gamma }^{\nu }{\gamma }^{\mu })\frac{\text{i}{2}({\gamma }^{\mu }{\gamma }^{\nu }-{\gamma }^{\nu }{\gamma }^{\mu })$=-\frac{1}{4}({\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\mu }{\gamma }^{\nu }-{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\nu }{\gamma }^{\mu }-{\gamma }^{\nu }{\gamma }^{\mu }{\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\mu })$$={\gamma }^{\mu }{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\nu }=\pm 1,\mu \ne \nu .$
• (5). ${\left({\sigma }^{0i}\right)}^2={\left({\sigma }^{i0}\right)}^2=-1.$
• (6). ${\left({\sigma }^{ij}\right)}^2={\left({\sigma }^{ji}\right)}^2=1,i\ne j.$

3. 关于 ${\mathbf{\gamma }}^5,{\sigma }^{\mu \nu }$

• (1). {\gamma }^{5}\equiv {\gamma }_{5}\equiv \text{i}{\gamma }^{0}{\gamma }^{1}{\gamma }^{2}{\gamma }^{3}=-\text{i}{\varepsilon }_{0123}{\gamma }^{0}{\gamma }^{1}{\gamma }^{2}{\gamma }^{3}=-\frac{\text{i}{4!}{\varepsilon }_{\mu \nu \rho \sigma }{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\sigma }.
• (2). \{\gamma }^{5},{\gamma }^{\mu }\}=\text{i}({\gamma }^{0}{\gamma }^{1}{\gamma }^{2}{\gamma }^{3}{\gamma }^{\mu }+{\gamma }^{\mu }{\gamma }^{0}{\gamma }^{1}{\gamma }^{2}{\gamma }^{3})$=\text{i}({\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3{\gamma }^{\mu }-{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3{\gamma }^{\mu })=0.$
• (3). $[{\gamma }^5,{\gamma }^{\mu }]={\gamma }^5{\gamma }^{\mu }-{\gamma }^{\mu }{\gamma }^5=2{\gamma }^5{\gamma }^{\mu }=-2{\gamma }^{\mu }{\gamma }^5.$
• (4). {\sigma }^{\mu \nu }\equiv \frac{\text{i}{2}\left[ {\gamma }^{\mu },{\gamma }^{\nu } \right]=\frac{\text{i}{2}({\gamma }^{\mu }{\gamma }^{\nu }-{\gamma }^{\nu }{\gamma }^{\mu }).
• (5). $[{\sigma }^{\mu \nu },{\gamma }^{\rho }]=2\text{i}(g^{\nu \rho }{\gamma }^{\mu }-g^{\mu \rho }{\gamma }^{\nu }).$
• (6). [{\sigma }^{\mu \nu },{\gamma }^{5}]=\frac{\text{i}{2}({\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{5}-{\gamma }^{\nu }{\gamma }^{\mu }{\gamma }^{5}-{\gamma }^{5}{\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{5}{\gamma }^{\nu }{\gamma }^{\mu })=0.
• (7). ${\epsilon }^{\mu \nu \rho \sigma }{\gamma }^5{\gamma }_{\sigma }$
• =-\frac{\rm{i}{4!}{\varepsilon }^{\mu \nu \rho \sigma }{\varepsilon }_{\tau \eta \delta \varepsilon }{\gamma }^{\tau }{\gamma }^{\eta }{\gamma }^{\delta }{\gamma }^{\varepsilon }{\gamma }_{\sigma }
• =\frac{\rm{i}{6}{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }-\frac{\rm{i}{6}{\gamma }^{\mu }{\gamma }^{\rho }{\gamma }^{\nu }-\frac{\rm{i}{6}{\gamma }^{\nu }{\gamma }^{\mu }{\gamma }^{\rho }+\frac{\rm{i}{6}{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\mu }+\frac{\rm{i}{6}{\gamma }^{\rho }{\gamma }^{\mu }{\gamma }^{\nu }-\frac{\rm{i}{6}{\gamma }^{\rho }{\gamma }^{\nu }{\gamma }^{\mu }
• =\frac{\rm{i}{6}{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }+\frac{\rm{i}{6}{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\mu }-\frac{\rm{i}{6}{\gamma }^{\mu }{\gamma }^{\rho }{\gamma }^{\nu }-\frac{\rm{i}{6}{\gamma }^{\nu }{\gamma }^{\mu }{\gamma }^{\rho }+\frac{\rm{i}{6}{\gamma }^{\rho }\left( {\gamma }^{\mu }{\gamma }^{\nu }-{\gamma }^{\nu }{\gamma }^{\mu } \right)
• =\frac{\rm{i}{6}{\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }+\frac{\rm{i}{6}{\gamma }^{\nu }\left( 2{g}^{\rho \mu }-{\gamma }^{\mu }{\gamma }^{\rho } \right)-\frac{\rm{i}{6}{\gamma }^{\mu }\left( 2{g}^{\rho \nu }-{\gamma }^{\nu }{\gamma }^{\rho } \right)-\frac{\rm{i}{6}{\gamma }^{\nu }{\gamma }^{\mu }{\gamma }^{\rho }
• \ \ \ +\frac{\rm{i}{6}{\gamma }^{\rho }\left( {\gamma }^{\mu }{\gamma }^{\nu }-{\gamma }^{\nu }{\gamma }^{\mu } \right)
• =\frac{\rm{i}{3}\left( {\gamma }^{\mu }{\gamma }^{\nu }-{\gamma }^{\nu }{\gamma }^{\mu } \right){\gamma }^{\rho }+\frac{\rm{i}{3}{\gamma }^{\nu }{g}^{\rho \mu }-\frac{\rm{i}{3}{\gamma }^{\mu }{g}^{\rho \nu }+\frac{\rm{i}{6}{\gamma }^{\rho }\left( {\gamma }^{\mu }{\gamma }^{\nu }-{\gamma }^{\nu }{\gamma }^{\mu } \right)
• =\frac{\rm{2}{3}{\sigma }^{\mu \nu }{\gamma }^{\rho }+\frac{\rm{i}{3}{\gamma }^{\nu }{g}^{\rho \mu }-\frac{\rm{i}{3}{\gamma }^{\mu }{g}^{\rho \nu }+\frac{1}{3}{\gamma }^{\rho }{\sigma }^{\mu \nu }
• =\frac{\rm{2}{3}{\sigma }^{\mu \nu }{\gamma }^{\rho }+\frac{\rm{i}{3}{\gamma }^{\nu }{g}^{\rho \mu }-\frac{\rm{i}{3}{\gamma }^{\mu }{g}^{\rho \nu }+\frac{1}{3}\left[ {\gamma }^{\rho },{\sigma }^{\mu \nu } \right]+\frac{1}{3}{\sigma }^{\mu \nu }{\gamma }^{\rho }
• =\frac{\rm{2}{3}{\sigma }^{\mu \nu }{\gamma }^{\rho }+\frac{1}{3}{\sigma }^{\mu \nu }{\gamma }^{\rho }-\frac{\rm{i}{3}\left( {\gamma }^{\mu }{g}^{\nu \rho }-{\gamma }^{\nu }{g}^{\mu \rho } \right)-\frac{2}{3}{\rm{i}\left( {\gamma }^{\mu }{g}^{\nu \rho }-{\gamma }^{\nu }{g}^{\mu \rho } \right)
• ={\sigma }^{\mu \nu }{\gamma }^{\rho }-{\rm{i}\left( {\gamma }^{\mu }{g}^{\nu \rho }-{\gamma }^{\nu }{g}^{\mu \rho } \right).

4. Dirac 与 Weyl 表示下的转置

• (1). {\gamma }^{\mu \text{T}={(-1)}^{\mu }{\gamma }^{\mu }.
• (2). {\sigma }^{\mu \nu \text{T}=\frac{\text{i}{2}{\left( {\gamma }^{\mu }{\gamma }^{\nu }-{\gamma }^{\nu }{\gamma }^{\mu } \right)}^{\text{T}=\frac{\text{i}{2}{\left( -1 \right)}^{\mu +\nu }\left( {\gamma }^{\nu }{\gamma }^{\mu }-{\gamma }^{\mu }{\gamma }^{\nu } \right)={\left( -1 \right)}^{\mu +\nu +1}{\sigma }^{\mu \nu }.\

凡涉及到转置操作都默认选定了表示. 注意这里的 ${\left(-1\right)}^{\mu }$ 可不是指标, 它真的是幂. 其实可以证明有很多结论都是任意表示通用的, 但很复杂, 有空可谈.

5. 荷共轭算符

• (1). $C=\text{i}{\gamma }^2{\gamma }^0.$
• (2). {C}^{\dagger }={C}^{-1}={C}^{\text{T}=-C.
• (3). {C^2} = - {C^{ - 1}C = - {\mathbf{1}.
• (4). $\left\{C,{\gamma }^0\right\}\equiv C{\gamma }^0+{\gamma }^{0}C=0.$
• (5). $C^{†}{\gamma }^0=-\text{i}{\gamma }^{0†}{\gamma }^{2†}{\gamma }^0=-{\gamma }^0\text{i}{\gamma }^0{\gamma }^2={\gamma }^0\text{i}{\gamma }^2{\gamma }^0={\gamma }^{0}C.$
• (6). {\gamma ^{\mu {\rm{T}C = - C{\gamma ^\mu }.
• (7). {\gamma }^{5{\rm{T}C={\rm{i}{\gamma }^{3}{\gamma }^{2}{\gamma }^{1}{\gamma }^{0}{\rm{i}{\gamma }^{2}{\gamma }^{0}=C{\rm{i}{\gamma }^{3}{\gamma }^{2}{\gamma }^{1}{\gamma }^{0}=C{\gamma }^{5}.
• (8). {\sigma ^{\mu \nu {\rm{T}C = - C{\sigma ^{\mu \nu }.
• (9). {\left( {\sigma ^{\mu \nu }{\not \!x} + {\not \!x}{\sigma ^{\mu \nu } \right)^{\rm{T}C = C\left( {\not \!x}{\sigma ^{\mu \nu } + {\sigma ^{\mu \nu }{\not \!x} \right).

6. 共轭相关

Dirac 共轭 $\bar{\psi }\equiv {\psi }^{†}{\gamma }^0.$

• (1). ${\gamma }^{0†}={\gamma }^0,{\gamma }^{i†}=-{\gamma }^i.$
• (2). ${\gamma }^{5†}=-\text{i}{\gamma }^{3†}{\gamma }^{2†}{\gamma }^{1†}{\gamma }^{0†}=\text{i}{\gamma }^3{\gamma }^2{\gamma }^1{\gamma }^0=\text{i}{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3={\gamma }^5.$
• (3). ${\gamma }^{\mu †}{\gamma }^0={\gamma }^0{\gamma }^{\mu }.$
• (4). ${\gamma }^{5†}{\gamma }^0={\gamma }^5{\gamma }^0=-{\gamma }^0{\gamma }^5.$
• (5). {\sigma }^{\mu \nu \dagger }{\gamma }^{0}=-\frac{\text{i}{2}{({\gamma }^{\mu }{\gamma }^{\nu }-{\gamma }^{\nu }{\gamma }^{\mu })}^{\dagger }{\gamma }^{0}=-\frac{\text{i}{2}{\gamma }^{0}({\gamma }^{\nu }{\gamma }^{\mu }-{\gamma }^{\mu }{\gamma }^{\nu })={\gamma }^{0}{\sigma }^{\mu \nu }.
• (6). {\left( {\rm{\bar q}_1}\Gamma {\rm{q}_2} \right)^\dag } = {\rm{q}_2^\dag {\Gamma ^\dag }{\left( {\rm{q}_1^\dag {\gamma ^0} \right)^\dag } = {\rm{q}_2^\dag {\Gamma ^\dag }{\gamma ^0}{\rm{q}_1} = {\rm{q}_2^\dag {\gamma ^0}{\gamma ^0}{\Gamma ^\dag }{\gamma ^0}{\rm{q}_1} = {\rm{\bar q}_2}\left({\gamma ^0}{\Gamma ^\dag }{\gamma ^0}\right){\rm{q}_1}.
• (7). {\left[ {\left( {\rm{u}_a^{\rm{T}C{\Gamma _1}{\rm{d}_b} \right){\Gamma _2}{\rm{s}_c} \right]^\dag }{\gamma ^0} = {\rm{s}_c^\dag {\gamma ^0}{\gamma ^0}\Gamma _2^\dag \left( {\rm{d}_b^\dag {\gamma ^0}{\gamma ^0}\Gamma _1^\dag {C^\dag }{\gamma ^0}{\gamma ^0}{\rm{u}_a^{\rm{T}\dag } \right){\gamma ^0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\rm{\bar s}_c}{\gamma ^0}\Gamma _2^\dag \left[ {\rm{\bar d}_b}{\gamma ^0}\Gamma _1^\dag {C^\dag }{\gamma ^0}{\left( {\rm{u}_a^\dag {\gamma ^{0{\rm{T} \right)}^{\rm{T} \right]{\gamma ^0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\rm{\bar s}_c}{\gamma ^0}\Gamma _2^\dag \left[ {\rm{\bar d}_b}\left( {\gamma ^0}\Gamma _1^\dag {\gamma ^0} \right)C{\rm{\bar u}_a^{\rm{T} \right]{\gamma ^0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\rm{\bar s}_c}\left( {\gamma ^0}\Gamma _2^\dag {\gamma ^0} \right)\left[ {\rm{\bar d}_b}\left( {\gamma ^0}\Gamma _1^\dag {\gamma ^0} \right)C{\rm{\bar u}_a^{\rm{T} \right].
• (8). \Gamma \to {\gamma }^{0}{\Gamma }^{\dagger }{\gamma }^{0}:\left\{ \begin{align} & I\to I,\ \\ & {\gamma }^{\mu }\to {\gamma }^{\mu },\ \\ & {\gamma }_{5}\to -{\gamma }_{5},\ \\ & {\gamma }^{\mu }{\gamma }_{5}\to -{\gamma }_{5}{\gamma }^{\mu }={\gamma }^{\mu }{\gamma }_{5},\ \\ & {\gamma }_{5}{\gamma }^{\mu }=-{\gamma }^{\mu }{\gamma }_{5}\to {\gamma }_{5}{\gamma }^{\mu },\ \\ & {\sigma }^{\mu \nu }\to {\sigma }^{\mu \nu },\ \\ & {\sigma }^{\mu \nu }{\gamma }_{5}\to -{\gamma }_{5}{\sigma }^{\mu \nu }=-{\sigma }^{\mu \nu }{\gamma }_{5},\ \\ & {\gamma }_{5}{\sigma }^{\mu \nu }={\sigma }^{\mu \nu }{\gamma }_{5}\to -{\gamma }_{5}{\sigma }^{\mu \nu }.\ \\ \end{align} \right.

7. 旋量场的 P, C 变换

https://zhuanlan.zhihu.com/p/591789284- P 变换: {A}^{\mu }\to {\mathcal{P}^{\mu }_{\nu }{A}^{\nu }.

• P 变换: \psi (x)\to U_{\mathcal{P}^{\dagger }\psi (x){U}_{\mathcal{P}=\zeta _{P}^{*}{\gamma }^{0}\psi ({\mathcal{P}^{-1}x).
• P 变换: \bar{\psi }(x)\to U_{\mathcal{P}^{\dagger }\bar{\psi }(x){U}_{\mathcal{P}={\zeta }_{P}\bar{\psi }({\mathcal{P}^{-1}x){\gamma }^{0}.
• P 变换: {\psi }^{\rm{T}(x)\to U_{\mathcal{P}^{\dagger }{\psi }^{\rm{T}(x){U}_{\mathcal{P}=\zeta _{P}^{*}{\psi }^{\rm{T}({\mathcal{P}^{-1}x){\gamma }^{0}.
• P 变换: {\bar{\psi }^{\rm{T}(x)\to U_{\mathcal{P}^{\dagger }{\bar{\psi }^{\rm{T}(x){U}_{\mathcal{P}={\zeta }_{P}{\gamma }^{0}{\bar{\psi }^{\rm{T}({\mathcal{P}^{-1}x).
• P 变换: \bar{\psi }\Gamma \psi \to U_{\mathcal{P}^{\dagger }\bar{\psi }\Gamma \psi {U}_{\mathcal{P}=\bar{\psi }{\gamma }^{0}\Gamma {\gamma }^{0}\psi .
• P 变换: {\rm{\bar{q}_{1}\Gamma {\rm{q}_{2}\to U_{\mathcal{P}^{\dagger }{\rm{\bar{q}_{1}\Gamma {\rm{q}_{2}{U}_{\mathcal{P}={\rm{\bar{q}_{1}{\gamma }^{0}\Gamma {\gamma }^{0}{\rm{q}_{2}.
• \boxed{\gamma }^{0}\mathbf{1}{\gamma }^{0}=\mathbf{1}\ \boxed{\gamma }^{0}{\gamma }^{5}{\gamma }^{0}=-{\gamma }^{5}\ \boxed{\gamma }^{0}{\gamma }^{\mu }{\gamma }^{0}={\mathcal{P}^{\mu }_{\nu }{\gamma }^{\nu }\ \boxed{\gamma }^{0}{\sigma }^{\mu \nu }{\gamma }^{0}={\mathcal{P}^{\mu }_{\rho }{\mathcal{P}^{\nu }_{\sigma }{\sigma }^{\rho \sigma }
• C 变换: \psi (x)\to U_{C}^{\dagger }\psi (x){U}_{C}=\zeta _{C}^{*}C{\bar{\psi }^{\rm{T}(x).
• C 变换: \bar{\psi }(x)\to U_{C}^{\dagger }\bar{\psi }(x){U}_{C}={\zeta }_{C}{\psi }^{\rm{T}(x)C.
• C 变换: {\psi }^{\rm{T}(x)\to U_{C}^{\dagger }{\psi }^{\rm{T}(x){U}_{C}=-\zeta _{C}^{*}\bar{\psi }(x)C.
• C 变换: {\bar{\psi }^{\rm{T}(x)\to U_{C}^{\dagger }{\bar{\psi }^{\rm{T}(x){U}_{C}=-{\zeta }_{C}C\psi (x).
• C 变换: \bar{\psi }\Gamma \psi \to U_{C}^{\dagger }\bar{\psi }\Gamma \psi {U}_{C}=\bar{\psi }{C}^{-1}{\Gamma }^{\rm{T}C\psi .
• C 变换: {\rm{\bar{q}_{1}\Gamma {\rm{q}_{2}\to U_{C}^{\dagger }{\rm{\bar{q}_{1}\Gamma {\rm{q}_{2}{U}_{C}={\rm{\bar{q}_{2}{C}^{-1}{\Gamma }^{\rm{T}C{\rm{q}_{1}.
• \boxed{C}^{-1}\mathbf{1}C=\mathbf{1}\ \boxed{C}^{-1}{\gamma }^{5{\rm{T}C={\gamma }^{5}\ \boxed{C}^{-1}{\gamma }^{\mu {\rm{T}C=-{\gamma }^{\mu }\ \boxed{C}^{-1}{\sigma }^{\mu \nu {\rm{T}C=-{\sigma }^{\mu \nu }

\Gamma \to {C}^{-1}{\Gamma }^{\text{T}C:\left\{ \begin{align} & I\to I,\ \\ & {\gamma }^{\mu }\to -{\gamma }^{\mu },\ \\ & {\gamma }_{5}\to {\gamma }_{5},\ \\ & {\sigma }^{\mu \nu }\to -{\sigma }^{\mu \nu },\ \\ & {\gamma }^{\mu }{\gamma }_{5}\to {C}^{-1}\gamma _{5}^{\text{T}C{C}^{-1}{\gamma }^{\mu \text{T}C=-{\gamma }_{5}{\gamma }^{\mu }={\gamma }^{\mu }{\gamma }_{5},\ \\ & {\gamma }_{5}{\gamma }^{\mu }=-{\gamma }^{\mu }{\gamma }_{5}\to {\gamma }_{5}{\gamma }^{\mu } \\ & {\sigma }^{\mu \nu }{\gamma }_{5}\to {C}^{-1}\gamma _{5}^{\text{T}C{C}^{-1}{\sigma }^{\mu \nu \text{T}C=-{\gamma }_{5}{\sigma }^{\mu \nu }=-{\sigma }^{\mu \nu }{\gamma }_{5},\ \\ & {\gamma }_{5}{\sigma }^{\mu \nu }={\sigma }^{\mu \nu }{\gamma }_{5}\to -{\gamma }_{5}{\sigma }^{\mu \nu }.\ \\ \end{align} \right.

8. δ 函数相关

https://zhuanlan.zhihu.com/p/67187240?- (1). 常见来源: \int_{-\infty }^{\infty }{\rm{d}x}\ {\rm{e}^{\pm {\rm{i}px}=2\pi \delta (p).

• (2). 常见来源的 $d$ 维情形: \int{\rm{d}^{d}x}\ {\rm{e}^{\pm {\rm{i}p\cdot x}={\left( 2\pi \right)}^{d}\delta (p).
• (3). 对于存在若干分立零点 $\left\{x_n\right\}$ 的连续实函数 $f(x)$ 有 \delta \left[ f(x) \right]=\sum\limits_{n}{\frac{\delta \left( x-{x}_{n} \right)}{\left| {\left. \frac{\rm{d}{\rm{d}x}f(x) \right|}_{x={x}_{n} \right|}.

这实际上是一个在理论推导中极其常用的式子. 千万 别 忘了 分母上的 绝对值, 要不然你两边积分符号都对不上.

9. d 维 Fourier 变换

• {\rm{i}\int {\rm{d}^d}x} \;{\rm{e}^{\rm{i}p \cdot x}\frac{1}{\left( {x^2} \right)}^a} = {\left( { - 1} \right)^a}{2^{d - 2a}{\pi ^{d/2}{\left( { - {p^2} \right)^{a - d/2}\frac{\Gamma (d/2 - a)}{\Gamma (a)}.
• {\rm{i}\int {\rm{d}^d}x} \;{\rm{e}^{\rm{i}p \cdot x}\frac{x^\mu }{\left( {x^2} \right)}^a} = {\rm{i}{\left( { - 1} \right)^{a + 1}{2^{d - 2a + 1}{\pi ^{d/2}{\left( { - {p^2} \right)^{a - d/2 - 1}{p^\mu }\frac{\Gamma (d/2 - a + 1)}{\Gamma (a)}.
• {\rm{i}\int {\rm{d}^d}x} \;{\rm{e}^{\rm{i}p \cdot x}\frac{x^\mu }{x^\nu }{\left( {x^2} \right)}^a}= {\left( { - 1} \right)^{a + 1}{2^{d - 2a + 1}{\pi ^{d/2}{\left( { - {p^2} \right)^{a - d/2 - 2}
• \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\left[ {2{p^\mu }{p^\nu }\frac{\Gamma (d/2 - a + 2)}{\Gamma (a)} - {p^2}{g^{\mu \nu }\frac{\Gamma (d/2 - a + 1)}{\Gamma (a)} \right].

其实后边儿俩就是对第一行做偏导数, 我是手算的, 你也可以用 FeynCalc.

10. 常用恒等式

下文中所有的 $\epsilon$ 都视为从正半轴趋于零的过程量, 即始终有 $\epsilon >0,\epsilon \to 0.$

• -\frac{\text{1}{\text{2}E}\left[ \theta (t){\text{e}^{-\text{i}Et}\text{+}\theta (-t){\text{e}^{\text{i}Et} \right]=\frac{\text{1}{\text{2}\pi \text{i}\int{\text{d}\omega \frac{\text{e}^{\text{i}\omega \text{t}{\omega }^{\text{2}-{E}^{\text{2}\text{+i}\varepsilon }.
• \operatorname{Im}\frac{1}{x-{x}_{0}-{\rm{i}\varepsilon }=\pi \delta \left( x-{x}_{0} \right),\ \operatorname{Im}\frac{1}{x-{x}_{0}+{\rm{i}\varepsilon }=-\pi \delta \left( x-{x}_{0} \right).
• $\ln \left(-x+\text{i}\epsilon \right)=\ln x+\text{i}\pi ,\ln \left(-x-\text{i}\epsilon \right)=\ln x-\text{i}\pi ,\text{where}x>0.$

https://zhuanlan.zhihu.com/p/409397025https://zhuanlan.zhihu.com/p/42233879311. Feynman 传播子

• \left\langle 0 \right|{\mathsf T}\!\left[ {\rm{q}_{a}^{1}{(x)}^{A}{\rm{\bar{q}_{b}^{2}{(y)}_{B} \right]\!\left| 0 \right\rangle ={\delta }_{12}{\delta }_{ab}\int{\frac{\rm{d}^{4}p}{\left( 2\pi \right)}^{4}\ {\rm{e}^{-{\rm{i}p\cdot \left( x-y \right)}\frac{\rm{i}{\left( \not{p}+{m}_{1} \right)}^{A}_{B}{p}^{2}-m_{1}^{2}+{\rm{i}\varepsilon }.\
• \left\langle 0 \right|{\mathsf T}\!\left[ A_{\mu }^{\alpha }(x)A_{\nu }^{\beta }(y) \right]\!\left| 0 \right\rangle =\int{\frac{\rm{d}^{4}p}{\left( 2\pi \right)}^{4}\ {\rm{e}^{\pm {\rm{i}p\cdot \left( x-y \right)}\frac{-{\rm{i}{g}_{\mu \nu }{\delta }^{\alpha \beta }{p}^{2}+{\rm{i}\varepsilon }.\

12. 固定点规范与协变导数

固定点规范即设定 $x^{\mu }{A}_{\mu }^{\alpha }(x)=0$, 其中 $A_{\mu }^{\alpha }(x)$ 为胶子[4]场.

$x^{\mu }{A}_{\mu }^{\alpha }(x)=0\Rightarrow A_{\mu }^{\alpha }(0)=0.$ $0=\frac{\partial }{\partial {x}^{\mu }\left[ {x}^{\nu }A_{\nu }^{\alpha }(x) \right]={\delta }^{\nu }_{\mu }A_{\nu }^{\alpha }(x)+{x}^{\nu }\frac{\partial }{\partial {x}^{\mu }A_{\nu }^{\alpha }(x)\Rightarrow A_{\mu }^{\alpha }(x)=-{x}^{\nu }\frac{\partial }{\partial {x}^{\mu }A_{\nu }^{\alpha }(x).$

这么一来就能把幂级数展开里的偏微分改成协变微分了:

类似 ${\delta }^{ab},{\delta }_{\beta }^{\alpha }$ 这样的就都只是些单位阵, 可以随便乎来唤去.

• {x}^{\mu }D_{\mu }^{ab}(x)={x}^{\mu }{\delta }^{ab}{\partial }_{\mu }+{x}^{\mu }{\rm{i}g\frac{\lambda _{\alpha }^{ab}{2}A_{\mu }^{\alpha }(x)={x}^{\mu }{\delta }^{ab}{\partial }_{\mu }.
• {x}^{\mu }{\left( {\tilde{D}_{\mu } \right)}^{\alpha }_{\beta }(x)={x}^{\mu }{\delta }^{\alpha }_{\beta }{\partial }_{\mu }+{x}^{\mu }g{f}^{\alpha }_{\beta \gamma }A_{\mu }^{\gamma }(x)={x}^{\mu }{\delta }^{\alpha }_{\beta }{\partial }_{\mu }.
• {\left. D_{\mu }^{ab}(x) \right|}_{x=0}={\left[ {\delta }^{ab}{\partial }_{\mu }+{\rm{i}g\frac{\lambda _{\alpha }^{ab}{2}A_{\mu }^{\alpha }(x) \right]}_{x=0}={\left. {\delta }^{ab}{\partial }_{\mu } \right|}_{x=0}.
• {\left. {\left( {\tilde{D}_{\mu } \right)}^{\alpha }_{\beta }(x) \right|}_{x=0}={\left[ {\delta }^{\alpha }_{\beta }{\partial }_{\mu }+g{f}^{\alpha }_{\beta \gamma }A_{\mu }^{\gamma }(x) \right]}_{x=0}={\delta }^{\alpha }_{\beta }{\left. {\partial }_{\mu } \right|}_{x=0}.
• $\overline{}$
• {\rm{q}_{a}(x)=\sum\limits_{n}{\frac{1}{n!}{x}^{\mu }_{1}\cdots {x}^{\mu }_{n}{\left( {\partial }_{\mu }_{1}\cdots {\partial }_{\mu }_{n} \right)}_{x=0}{\rm{q}_{a}(x)}
• \ \ \ \ \ \ \ \ \ =\sum\limits_{n}{\frac{1}{n!}{x}^{\mu }_{1}\cdots {x}^{\mu }_{n}\left[ {D}_{\mu }_{1}(x)\cdots D{(x)}_{\mu }_{n} \right]_{x=0}^{ab}{\rm{q}_{b}(x)}
• \ \ \ \ \ \ \ \ \ \equiv \sum\limits_{n}{\frac{1}{n!}{x}^{\mu }_{1}\cdots {x}^{\mu }_{n}{\left( {D}_{\mu }_{1}\cdots {D}_{\mu }_{n} \right)}^{ab}{\rm{q}_{b}(0)}.
• {\rm{\bar{q}_{a}(x)=\sum\limits_{n}{\frac{1}{n!}{x}^{\mu }_{1}\cdots {x}^{\mu }_{n}{\rm{\bar{q}_{b}(0){\left( D_{\mu }_{1}^{\dagger }\cdots D_{\mu }_{n}^{\dagger } \right)}^{ab}.

这里需要约定好所有带 $†$ 的微分算子 $D_{\mu }^{†}$ 均向左作用.

常用的还有 {\left( {\tilde{D}_{\nu } \right)}^{\alpha }_{\beta }{\rm{G}_{\rho \mu }^{\beta }(0)=\left[ {\left( {\tilde{D}_{\nu } \right)}^{\alpha }_{\beta }(0),{\rm{G}_{\rho \mu }^{\beta }(0) \right] , 证明如下:

• \ \ \ \left[ {\left( {\tilde{D}_{\nu } \right)}^{\alpha }_{\beta }(0),{\rm{G}_{\rho \mu }^{\beta }(0) \right]*
• $={\left[{\delta }_{\beta }^{\alpha }{\partial }_{\nu }+g{f}_{\beta \gamma }^{\alpha }{A}_{\nu }^{\gamma }(x),{\partial }_{\rho }{A}_{\mu }^{\beta }(x)-{\partial }_{\mu }{A}_{\rho }^{\beta }(x)-g{f}_{\gamma \delta }^{\beta }{A}_{\rho }^{\gamma }(x)A_{\mu }^{\delta }(x)\right]}_{x=0}\ast$
• $=\left[{\delta }_{\beta }^{\alpha }{\partial }_{\nu },{\partial }_{\rho }{A}_{\mu }^{\beta }(0)-{\partial }_{\mu }{A}_{\rho }^{\beta }(0)\right]\ast$
• $=\left[{\partial }_{\nu },{\partial }_{\rho }{A}_{\mu }^{\alpha }(0)-{\partial }_{\mu }{A}_{\rho }^{\alpha }(0)\right]\ast$
• $={\partial }_{\nu }\left\{\left[{\partial }_{\rho }{A}_{\mu }^{\alpha }(0)-{\partial }_{\mu }{A}_{\rho }^{\alpha }(0)\right]\ast \right\}-\left[{\partial }_{\rho }{A}_{\mu }^{\alpha }(0)-{\partial }_{\mu }{A}_{\rho }^{\alpha }(0)\right]{\partial }_{\nu }\ast$
• $={\partial }_{\nu }{\partial }_{\rho }{A}_{\mu }^{\alpha }(0)\ast -{\partial }_{\nu }{\partial }_{\mu }{A}_{\rho }^{\alpha }(0)\ast$
• $+{\partial }_{\rho }{A}_{\mu }^{\alpha }(0){\partial }_{\nu }\ast -{\partial }_{\mu }{A}_{\rho }^{\alpha }(0){\partial }_{\nu }\ast +{\partial }_{\mu }{A}_{\rho }^{\alpha }(0){\partial }_{\nu }\ast -{\partial }_{\rho }{A}_{\mu }^{\alpha }(0){\partial }_{\nu }\ast$
• $={\partial }_{\nu }{\partial }_{\rho }{A}_{\mu }^{\alpha }(0)\ast -{\partial }_{\nu }{\partial }_{\mu }{A}_{\rho }^{\alpha }(0)\ast$
• $={\partial }_{\nu }{\left[{\partial }_{\rho }{A}_{\mu }^{\alpha }(x)-{\partial }_{\mu }{A}_{\rho }^{\alpha }(x)-g{f}_{\beta \gamma }^{\alpha }{A}_{\rho }^{\beta }(x)A_{\mu }^{\gamma }(x)\right]}_{x=0}\ast$
• ={\partial }_{\nu }{\rm{G}_{\rho \mu }^{\alpha }(0)*={\left( {\tilde{D}_{\nu } \right)}^{\alpha }_{\beta }{\rm{G}_{\rho \mu }^{\beta }(0)*.

上面的 $\ast$ 指代被作用的对象, 这对高阶对易子亦成立所以就有:

• {A}_{\mu }(x)=\sum\limits_{n}{\frac{1}{n!\left( n+2 \right)}{x}^{\nu }_{1}\cdots {x}^{\nu }_{n}{x}^{\rho }{\tilde{D}_{\nu }_{1}\cdots {\tilde{D}_{\nu }_{n}{\rm{G}_{\rho \mu }(0)}
• \ \ \ \ \ \ \ \ \ \ =\sum\limits_{n}{\frac{1}{n!\left( n+2 \right)}{x}^{\nu }_{1}\cdots {x}^{\nu }_{n}{x}^{\rho }\left[ {\tilde{D}_{\nu }_{1}(0),\cdots \left[ {\tilde{D}_{\nu }_{n}(0),{\rm{G}_{\rho \mu }(0) \right] \right]}.
• 其中 {A}_{\mu }(x)={\rm i}g\frac{\lambda }_{\alpha }{2}A_{\mu }^{\alpha },\ {\rm{G}_{\mu \nu }={\rm i}g\frac{\lambda }_{\alpha }{2}{\rm{G}_{\mu \nu }^{\alpha }.

不过其实一般能开到第三项的都少见, 所以用的主要就是下面这条: A_{\mu }^{\alpha }(x)=\frac{1}{2}{x}^{\nu }{\rm{G}_{\nu \mu }^{\alpha }(0)+\frac{1}{3}{x}^{\nu }{x}^{\rho }{\left( {\tilde{D}_{\nu } \right)}^{\alpha }_{\beta }{\rm{G}_{\rho \mu }^{\beta }(0)+\cdots

12. 按凝聚量展开的常用完全传播子*

(1). 轻夸克:

• S_{ab}^{\rm{q}(x)\equiv \langle \Omega |\mathsf T[{\rm{q}_{a}(x){\rm{\bar{q}_{b}(0)]\left| \Omega \right\rangle =S_{1ab}^{\rm{q}(x)+S_{2ab}^{\rm{q}(x).
• $\overline{}$
• S_{1ab}^{\rm{q}(x)=\frac{\rm{i}{\delta }_{ab}\not{x}{2{\pi }^{2}{({x}^{2})}^{d/2}-\frac{\delta }_{ab}{12}\left\langle {\rm{\bar{q}q} \right\rangle +\frac{\delta }_{ab}{x}^{2}{192}\left\langle g{\rm{\bar{q}\sigma \cdot {\rm{Gq} \right\rangle -\frac{m}_{\rm{q}{\delta }_{ab}{4{\pi }^{2}{({x}^{2})}^{d/2-1}+\frac{\rm{i}{m}_{\rm{q}{\delta }_{ab}\not{x}{48}\left\langle {\rm{\bar{q}q} \right\rangle
• \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{\rm{i}{m}_{\rm{q}{\delta }_{ab}{x}^{2}\not{x}{1152}\left\langle g{\rm{\bar{q}\sigma \cdot {\rm{Gq} \right\rangle .
• $\overline{}$
• S_{2ab}^{\rm{q}(x) = \frac{\rm i}{32{\pi ^2}g\frac{\lambda _{ab}^\alpha }{2}{\rm{G}_\alpha ^{\mu \nu }\frac{1}{({x^2})}^{d/2 - 1}\left( {\sigma _{\mu \nu }{\not \!x} +{\not \!x}{\sigma _{\mu \nu } \right).
• $\overline{}$
• \ \ \ \ \ CS_{1ab}^{\rm{qT}(x)C
• =\frac{\rm{i}{\delta }_{ab}{x}^{\rho }{2{\pi }^{2}{({x}^{2})}^{d/2}C\gamma _{\rho }^{\rm{T}C+\frac{\delta }_{ab}{12}\left\langle {\rm{\bar{q}q} \right\rangle -\frac{\delta }_{ab}{x}^{2}{192}\left\langle g{\rm{\bar{q}\sigma \cdot {\rm{Gq} \right\rangle +\frac{m}_{\rm{q}{\delta }_{ab}{4{\pi }^{2}{({x}^{2})}^{d/2-1}
• \ \ \ \ +\frac{\rm{i}{m}_{\rm{q}{\delta }_{ab}{x}^{\rho }{48}C\gamma _{\rho }^{\rm{T}C\left\langle {\rm{\bar{q}q} \right\rangle -\frac{\rm{i}{m}_{\rm{q}{\delta }_{ab}{x}^{2}{x}^{\rho }{1152}C\gamma _{\rho }^{\rm{T}C\left\langle g{\rm{\bar{q}\sigma \cdot {\rm{Gq} \right\rangle
• =-\frac{\rm{i}{\delta }_{ab}{x}^{\rho }{2{\pi }^{2}{({x}^{2})}^{d/2}{C}^{2}{\gamma }_{\rho }+\frac{\delta }_{ab}{12}\left\langle {\rm{\bar{q}q} \right\rangle -\frac{\delta }_{ab}{x}^{2}{192}\left\langle g{\rm{\bar{q}\sigma \cdot {\rm{Gq} \right\rangle +\frac{m}_{\rm{q}{\delta }_{ab}{4{\pi }^{2}{({x}^{2})}^{d/2-1}
• \ \ \ \ -\frac{\rm{i}{m}_{\rm{q}{\delta }_{ab}{x}^{\rho }{48}{C}^{2}{\gamma }_{\rho }\left\langle {\rm{\bar{q}q} \right\rangle +\frac{\rm{i}{m}_{\rm{q}{\delta }_{ab}{x}^{2}{x}^{\rho }{1152}{C}^{2}{\gamma }_{\rho }\left\langle g{\rm{\bar{q}\sigma \cdot {\rm{Gq} \right\rangle
• =\frac{\rm{i}{\delta }_{ab}\not{x}{2{\pi }^{2}{({x}^{2})}^{d/2}+\frac{\delta }_{ab}{12}\left\langle {\rm{\bar{q}q} \right\rangle -\frac{\delta }_{ab}{x}^{2}{192}\left\langle g{\rm{\bar{q}\sigma \cdot {\rm{Gq} \right\rangle +\frac{m}_{\rm{q}{\delta }_{ab}{4{\pi }^{2}{({x}^{2})}^{d/2-1}+\frac{\rm{i}{m}_{\rm{q}{\delta }_{ab}\not{x}{48}\left\langle {\rm{\bar{q}q} \right\rangle
• \ \ \ \ -\frac{\rm{i}{m}_{\rm{q}{\delta }_{ab}{x}^{2}\not{x}{1152}\left\langle g{\rm{\bar{q}\sigma \cdot {\rm{Gq} \right\rangle .
• $\overline{}$
• \ \ \ \ \ CS_{2ab}^{\rm{qT}(x)C
• =\frac{\rm{i}{32{\pi }^{2}g\frac{\lambda _{ab}^{\alpha }{2}{\rm{G}_{\alpha }^{\mu \nu }\frac{\rm{1}{\left( {x}^{2} \right)}^{d/2-1}\left( {x}^{\rho }{\rm{C}\sigma _{\mu \nu }^{\rm{T}\gamma _{\rho }^{\rm{T}{\rm{C}+{x}^{\rho }{\rm{C}\gamma _{\rho }^{\rm{T}\sigma _{\mu \nu }^{\rm{T}{\rm{C} \right)
• =\frac{\rm{i}{32{\pi }^{2}g\frac{\lambda _{ab}^{\alpha }{2}{\rm{G}_{\alpha }^{\mu \nu }\frac{\rm{1}{\left( {x}^{2} \right)}^{d/2-1}\left( {x}^{\rho }{\rm{C}^{\rm{2}{\sigma }_{\mu \nu }{\gamma }_{\rho }+{x}^{\rho }{\rm{C}^{\rm{2}{\gamma }_{\rho }{\sigma }_{\mu \nu } \right)
• = - \frac{\rm i}{32{\pi ^2}g\frac{\lambda _{ab}^\alpha }{2}{\rm{G}_\alpha ^{\mu \nu }\frac{\rm{1}{\left( {x^2} \right)}^{d/2 - 1}\left( {\sigma _{\mu \nu }{\not \!x}+ {\rm{ }{\not \!x}{\sigma _{\mu \nu } \right).

但这其实都只是丐版传播子, 下面这个版本稍微严格些:

(1). 轻夸克:

• S_{ab}^{\rm{q}(x)\equiv \langle \Omega |\mathsf T[{\rm{q}_{a}(x){\rm{\bar{q}_{b}(0)]\left| \Omega \right\rangle =S_{1ab}^{\rm{q}(x)+S_{2ab}^{\rm{q}(x).
• $\overline{}$
• S_{1ab}^{\rm{q}(x) = \frac{\rm{i}{\delta _{ab}\Gamma \left( {\frac{d}{2} \right)\not{x}{2{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} + \frac{m_{\rm{q}{\delta _{ab}\Gamma \left( {\frac{d}{2} - 1} \right)}{4{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 1} - \frac{\delta _{ab}{12}\left\langle {\rm{\bar qq} \right\rangle + \frac{\rm{i}{m_{\rm{q}{\delta _{ab}\not{x}{48}\left\langle {\rm{\bar qq} \right\rangle
• \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \frac{\delta _{ab}{x^2}{192}\left\langle {g{\rm{\bar q}\sigma \cdot {\rm{Gq} \right\rangle - \frac{\rm{i}{m_{\rm{q}{\delta _{ab}{x^2}\not{x}{2^7} \cdot {3^2}\left\langle {g{\rm{\bar q}\sigma \cdot {\rm{Gq} \right\rangle
• \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - {\rm{Ext1}\frac{\rm{i}{\delta _{ab}{x^2}\not{x}{2^5} \cdot {3^5}{g^2}{\left\langle {\rm{\bar qq} \right\rangle ^2} + {\rm{Ext2}\frac{m_{\rm{q}{\delta _{ab}\Gamma \left( {\frac{d}{2} - 3} \right)}{2^9} \cdot 3{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 3}\left\langle {g^2}{\rm{G}^2} \right\rangle.
• $\overline{}$
• S_{2ab}^{\rm{q}(x) = - \frac{\rm{i}\Gamma \left( {\frac{d}{2} - 1} \right)\left( {\sigma _{\mu \nu }\not{x} + \not{x}{\sigma _{\mu \nu } \right)}{32{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 1}g{\rm{G}_\alpha ^{\mu \nu }\frac{\lambda _{ab}^\alpha }{2} - {\rm{Ext3}\frac{m_{\rm{q}\Gamma \left( {\frac{d}{2} - 2} \right){\sigma _{\mu \nu }{32{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 2}g{\rm{G}_\alpha ^{\mu \nu }\frac{\lambda _{ab}^\alpha }{2}.
• $\overline{}$
• CS_{1ab}^{\rm{qT}(x)C = \frac{\rm{i}{\delta _{ab}\Gamma \left( {\frac{d}{2} \right){x^\mu }C\gamma _\mu ^{\rm{T}C}{2{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - \frac{m_{\rm{q}{\delta _{ab}\Gamma \left( {\frac{d}{2} - 1} \right)}{4{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 1} + \frac{\delta _{ab}{12}\left\langle {\rm{\bar qq} \right\rangle + \frac{\rm{i}{m_{\rm{q}{\delta _{ab}{x^\mu }C\gamma _\mu ^{\rm{T}C}{48}\left\langle {\rm{\bar qq} \right\rangle
• \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \frac{\delta _{ab}{x^2}{192}\left\langle {g{\rm{\bar q}\sigma \cdot {\rm{Gq} \right\rangle - \frac{\rm{i}{m_{\rm{q}{\delta _{ab}{x^2}{x^\mu }C\gamma _\mu ^{\rm{T}C}{2^7} \cdot {3^2}\left\langle {g{\rm{\bar q}\sigma \cdot {\rm{Gq} \right\rangle
• \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - {\rm{Ext1}\frac{\rm{i}{\delta _{ab}{x^2}{x^\mu }C\gamma _\mu ^{\rm{T}C}{2^5} \cdot {3^5}{g^2}{\left\langle {\rm{\bar qq} \right\rangle ^2} - {\rm{Ext2}\frac{m_{\rm{q}{\delta _{ab}\Gamma \left( {\frac{d}{2} - 3} \right)}{2^9} \cdot 3{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 3}\left\langle {g^2}{\rm{G}^2} \right\rangle
• \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{\rm{i}{\delta _{ab}\Gamma \left( {\frac{d}{2} \right)\not{x}{2{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - \frac{m_{\rm{q}{\delta _{ab}\Gamma \left( {\frac{d}{2} - 1} \right)}{4{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 1} + \frac{\delta _{ab}{12}\left\langle {\rm{\bar qq} \right\rangle + \frac{\rm{i}{m_{\rm{q}{\delta _{ab}\not{x}{48}\left\langle {\rm{\bar qq} \right\rangle
• \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \frac{\delta _{ab}{x^2}{192}\left\langle {g{\rm{\bar q}\sigma \cdot {\rm{Gq} \right\rangle - \frac{\rm{i}{m_{\rm{q}{\delta _{ab}{x^2}\not{x}{2^7} \cdot {3^2}\left\langle {g{\rm{\bar q}\sigma \cdot {\rm{Gq} \right\rangle
• \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - {\rm{Ext1}\frac{\rm{i}{\delta _{ab}{x^2}\not{x}{2^5} \cdot {3^5}{g^2}{\left\langle {\rm{\bar qq} \right\rangle ^2} - {\rm{Ext2}\frac{m_{\rm{q}{\delta _{ab}\Gamma \left( {\frac{d}{2} - 3} \right)}{2^9} \cdot 3{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 3}\left\langle {g^2}{\rm{G}^2} \right\rangle .
• $\overline{}$
• \;\;\;\;CS_{2ab}^{\rm{qT}(x)C
• = - \frac{\rm{i}\Gamma \left( {\frac{d}{2} - 1} \right){x^\rho }\left( {C\gamma _\rho ^{\rm{T}\sigma _{\mu \nu }^{\rm{T}C + C\sigma _{\mu \nu }^{\rm{T}\gamma _\rho ^{\rm{T}C} \right)}{32{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 1}g{\rm{G}_\alpha ^{\mu \nu }\frac{\lambda _{ab}^\alpha }{2} - {\rm{Ext3}\frac{m_{\rm{q}\Gamma \left( {\frac{d}{2} - 2} \right)C\sigma _{\mu \nu }^{\rm{T}C}{32{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 2}g{\rm{G}_\alpha ^{\mu \nu }\frac{\lambda _{ab}^\alpha }{2}
• = \frac{\rm{i}\Gamma \left( {\frac{d}{2} - 1} \right)\left( {\sigma _{\mu \nu }\not{x} + \not{x}{\sigma _{\mu \nu } \right)}{32{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 1}g{\rm{G}_\alpha ^{\mu \nu }\frac{\lambda _{ab}^\alpha }{2} - {\rm{Ext3}\frac{m_{\rm{q}\Gamma \left( {\frac{d}{2} - 2} \right){\sigma _{\mu \nu }{32{\pi ^{\frac{d}{2}{\left( { - {x^2} \right)}^{\frac{d}{2} - 2}g{\rm{G}_\alpha ^{\mu \nu }\frac{\lambda _{ab}^\alpha }{2}.

Ext1 取决于你要不要算次领头阶. Ext2, Ext3 直接等于零算了, 你要能搞懂就用.

(2). 重夸克:

• \ \ \ \ \ S_{ab}^{\rm{Q}(x)\equiv \langle \Omega |\mathsf T\!\left[ {\rm{Q}_{a}(x){\rm{\bar{Q}_{b}(0) \right]\!\left| \Omega \right\rangle
• = \int {\frac{\rm{d}^d}k}{\left( {2\pi } \right)}^d} \;{\rm{e}^{ - {\rm{i}k \cdot x}\frac{\rm{i}\left( {\not k + {m_{\rm{Q} \right)}{k^2} - m_{\rm{Q}^2}{\delta _{ab}
• \ \ \ + \int {\frac{\rm{d}^d}k}{\left( {2\pi } \right)}^d} \;{\rm{e}^{ - {\rm{i}k \cdot x}\frac{\rm{i}{4}g\frac{\lambda _{ab}^\alpha }{2}{\rm{G}_\alpha ^{\mu \nu }\frac{1}{\left( {k^2} - m_{\rm{Q}^2} \right)}^2}\left[ {\sigma _{\mu \nu }\left( {\not k + {m_{\rm{Q} \right) + \left( {\not k + {m_{\rm{Q} \right){\sigma _{\mu \nu } \right]
• \ \ \ - \int {\frac{\rm{d}^d}k}{\left( {2\pi } \right)}^d} \;{\rm{e}^{ - {\rm{i}k \cdot x}\frac{\rm{i}{4}{g^2}\frac{\lambda _{ac}^\alpha }{2}\frac{\lambda _{cb}^\beta }{2}{\rm{G}_\alpha ^{\mu \nu }{\rm{G}_\beta ^{\rho \sigma }\frac{\not k + {m_{\rm{Q}{\left( {k^2} - m_{\rm{Q}^2} \right)}^5}\left( {f_{\mu \nu \rho \sigma } + {f_{\mu \rho \nu \sigma } + {f_{\mu \rho \sigma \nu } \right)\left( {\not k + {m_{\rm{Q} \right)
• \ \ \ - \int {\frac{\rm{d}^d}k}{\left( {2\pi } \right)}^d} \;{\rm{e}^{ - {\rm{i}k \cdot x}\frac{\rm{i}{48}{g^3}{f^{\alpha \beta \gamma }{\rm{G}_\alpha ^{\mu \nu }{\rm{G}_\beta ^{\nu \rho }{\rm{G}_\gamma ^{\rho \mu }
• \ \ \ \ \ \ \ \ \times\frac{\not k + {m_{\rm{Q}{\left( {k^2} - m_{\rm{Q}^2} \right)}^6}\left[ {\not\! k\left( {k^2} - 3m_{\rm{Q}^2} \right) + 2{m_{\rm{Q}\left( {2{k^2} - m_{\rm{Q}^2} \right)} \right]\left( {\not\! k + {m_{\rm{Q} \right){\delta _{ab} + \cdots
• {\sigma _{\mu \nu } \equiv \frac{\rm{i}{2}\left[ {\gamma _\mu },{\gamma _\nu } \right],\;{f_{\mu \nu \rho \sigma } \equiv {\gamma _\mu }\left( {\not \!k} + {m_{\rm{Q} \right){\gamma _\nu }\left( {\not \!k} + {m_{\rm{Q} \right){\gamma _\rho }\left( {\not \!k} + {m_{\rm{Q} \right){\gamma _\sigma }\left( {\not \!k} + {m_{\rm{Q} \right).\;
• $\ast$ 在第三项处可使用公式 \left\langle {\rm{G}_\alpha ^{\mu \nu }{\rm{G}_\beta ^{\rho \sigma } \right\rangle = \frac{1}{96}{g^{ - 2}{\delta _{\alpha \beta }\left( {g_{\mu \rho }{g_{\nu \sigma } - {g_{\mu \sigma }{g_{\nu \rho } \right)\left\langle {g^2}{\rm{G}^2} \right\rangle *
• $\ast$ 一般会忽略高于 $g^2$ 的项 $\ast$
• $\overline{}$
• S_{ab}^{\rm{Q}(x) \equiv \langle \Omega |\mathsf T\!\left[ {\rm{Q}_a}(x){\rm{\bar Q}_b}(0)} \right]\!\left| \Omega \right\rangle = S_{1ab}^{\rm{Q}(x) + S_{2ab}^{\rm{Q}(x).
• S_{1ab}^{\rm{Q}(x) = \int {\frac{\rm{d}^d}k}{\left( {2\pi } \right)}^d} \;{\rm{e}^{ - {\rm{i}k \cdot x}S_{1ab}^{\rm{Q}(k),\;S_{2ab}^{\rm{Q}(x) = \int {\frac{\rm{d}^d}k}{\left( {2\pi } \right)}^d} \;{\rm{e}^{ - {\rm{i}k \cdot x}S_{2ab}^{\rm{Q}(k).\;
• $\overline{}$
• S_{1ab}^{\rm{Q}(k)={\rm{i}\frac{\not{k}+{m}_{\rm{Q}{k}^{2}-m_{\rm{Q}^{2}{\delta }_{ab}+\frac{\rm{i}{m}_{\rm{Q}\left( {m}_{\rm{Q}\not{k}+{k}^{2} \right)}{12{\left( {k}^{2}-m_{\rm{Q}^{2} \right)}^{4}{\delta }_{ab}\left\langle {g}^{2}{\rm{G}^{\rm{2} \right\rangle .
• $\overline{}$
• S_{2ab}^{\rm{Q}(k)=\frac{\rm{i}{4}g\frac{\lambda _{ab}^{\alpha }{2}{\rm{G}_{\alpha }^{\mu \nu }\frac{1}{\left( {k}^{2}-m_{\rm{Q}^{2} \right)}^{2}\left[ {\sigma }_{\mu \nu }\left( \not{k}+{m}_{\rm{Q} \right)+\left( \not{k}+{m}_{\rm{Q} \right){\sigma }_{\mu \nu } \right].
• $\overline{}$
• CS_{1ab}^{\text{QT}(k)C=\text{i}\frac{C\left( {k}^{\mu }\gamma _{\mu }^{\text{T}+{m}_{\text{Q} \right)C}{k}^{2}-m_{\text{Q}^{2}{\delta }_{ab}+\frac{\text{i}{m}_{\text{Q}C\left( {m}_{\text{Q}{k}^{\mu }\gamma _{\mu }^{\text{T}+{k}^{2} \right)C}{12{\left( {k}^{2}-m_{\text{Q}^{2} \right)}^{4}{\delta }_{ab}\left\langle {g}^{2}{\text{G}^{\text{2} \right\rangle
• \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\text{i}\frac{k}^{\mu }C\gamma _{\mu }^{\text{T}C+CC{m}_{\text{Q}{k}^{2}-m_{\text{Q}^{2}{\delta }_{ab}+\frac{\text{i}{m}_{\text{Q}\left( {m}_{\text{Q}{k}^{\mu }C\gamma _{\mu }^{\text{T}C+CC{k}^{2} \right)}{12{\left( {k}^{2}-m_{\text{Q}^{2} \right)}^{4}{\delta }_{ab}\left\langle {g}^{2}{\text{G}^{\text{2} \right\rangle
• \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\text{i}\frac{\not{k}-{m}_{\text{Q}{k}^{2}-m_{\text{Q}^{2}{\delta }_{ab}+\frac{\text{i}{m}_{\text{Q}\left( {m}_{\text{Q}\not{k}-{k}^{2} \right)}{12{\left( {k}^{2}-m_{\text{Q}^{2} \right)}^{4}{\delta }_{ab}\left\langle {g}^{2}{\text{G}^{\text{2} \right\rangle .\
• $\overline{}$
• CS_{2ab}^{\text{QT}(k)C=\frac{\text{i}{4}g\frac{\lambda _{ab}^{\alpha }{2}\text{G}_{\alpha }^{\mu \nu }\frac{1}{\left( {k}^{2}-m_{\text{Q}^{2} \right)}^{2}\left[ C\left( {k}^{\mu }\gamma _{\mu }^{\text{T}+{m}_{\text{Q} \right)\sigma _{\mu \nu }^{\text{T}C+C\sigma _{\mu \nu }^{\text{T}\left( {k}^{\mu }\gamma _{\mu }^{\text{T}+{m}_{\text{Q} \right)C \right]
• \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{\text{i}{4}g\frac{\lambda _{ab}^{\alpha }{2}\text{G}_{\alpha }^{\mu \nu }\frac{1}{\left( {k}^{2}-m_{\text{Q}^{2} \right)}^{2}\left[ -C\left( {k}^{\mu }\gamma _{\mu }^{\text{T}+{m}_{\text{Q} \right)C{\sigma }_{\mu \nu }+C\sigma _{\mu \nu }^{\text{T}C\left( -{k}^{\mu }{\gamma }_{\mu }+{m}_{\text{Q} \right) \right]
• \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{\text{i}{4}g\frac{\lambda _{ab}^{\alpha }{2}\text{G}_{\alpha }^{\mu \nu }\frac{1}{\left( {k}^{2}-m_{\text{Q}^{2} \right)}^{2}\left[ \left( -{k}^{\mu }{\gamma }_{\mu }+{m}_{\text{Q} \right){\sigma }_{\mu \nu }+{\sigma }_{\mu \nu }\left( -{k}^{\mu }{\gamma }_{\mu }+{m}_{\text{Q} \right) \right]
• \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =-\frac{\text{i}{4}g\frac{\lambda _{ab}^{\alpha }{2}\text{G}_{\alpha }^{\mu \nu }\frac{1}{\left( {k}^{2}-m_{\text{Q}^{2} \right)}^{2}\left[ {\sigma }_{\mu \nu }\left( \not{k}-{m}_{\text{Q} \right)+\left( \not{k}-{m}_{\text{Q} \right){\sigma }_{\mu \nu } \right].\

重夸克没考虑夸克凝聚是因为有 \left\langle {\rm{\bar{q}q} \right\rangle \propto {\left\langle {g}^{2}{\rm{GG} \right\rangle }/{m}_{\rm{q}.

14. 悬挂胶子线

简单来讲就是要手动加入两条夸克线之间产生的胶子混杂凝聚项.

比如碰到 \Pi ={\rm{tr}\left[ CS_{ab}^{\rm{uT}(x)C{\Gamma }_{1}S_{ba}^{\rm{d}(-x){\Gamma }_{2} \right] 这样的, 就必须额外考虑这两项:

• (1). {\Pi _1} ={\rm{tr}\left[ {\frac{CS_{2ab}^{\rm{uT}(x)C}{g{\rm{G}_\alpha ^{\mu \nu }{\Gamma _1}\langle \Omega |{\mathsf{T}\!\left[ {\rm{d}_b}(0)g{\rm{G}_\alpha ^{\mu \nu }{\rm{\bar d}_a}(x)} \right]\!\left| \Omega \right\rangle {\Gamma _2} \right].
• (2). {\Pi _2} ={\rm{tr}\left[ {C\langle \Omega |{\mathsf{T}\!\left[ {\rm{u}_a^{\rm{T}(x) \otimes g{\rm{G}_\alpha ^{\mu \nu }{\rm{\bar u}_b^{\rm{T}(0)} \right]\!\left| \Omega \right\rangle C{\Gamma _1}\frac{S_{2ba}^{\rm{d}(-x)}{g{\rm{G}_\alpha ^{\mu \nu }{\Gamma _2} \right].
• 其中 \langle \Omega |{\mathsf{T}\!\left[ {\rm{q}_a}(x)g{\rm{G}_{\mu \nu }^\alpha {\rm{\bar q}_b}(y)} \right]\!\left| \Omega \right\rangle
• \ \ =-\frac{1}{2}^{6}3}\frac{\lambda _{ab}^{\alpha }{2}{\sigma }_{\mu \nu }\left\langle g{\rm{\bar{q}\sigma \cdot {\rm{Gq} \right\rangle +\frac{\rm{i}{m}_{\rm{q}{2}^{8}3}\frac{\lambda _{ab}^{\alpha }{2}\left[ {\sigma }_{\mu \nu }{\left( x-y \right)}_{\rho }{\gamma }^{\rho }+{\left( x-y \right)}_{\rho }{\gamma }^{\rho }{\sigma }_{\mu \nu } \right]\left\langle g{\rm{\bar{q}\sigma \cdot {\rm{Gq} \right\rangle
• \ \ \ \ \ \ +\frac{\left( x-y \right)}^{2}{2}^{10}{3}^{2}\frac{\lambda _{ab}^{\alpha }{2}{\sigma }_{\mu \nu }\left\langle {\rm{\bar{q}q} \right\rangle \left\langle {g}^{2}{\rm{G}^{2} \right\rangle .\

至于 \langle \Omega |{\sf T}\!\left[ {\rm{u}_a^{\rm{T}(x) \otimes gG_\alpha ^{\mu \nu }{\rm{\bar u}_{b}^{\rm{T}(0)} \right]\!{\left| \Omega \right\rangle ^A}_B 则是这样的:

• \ \ \ \ \langle \Omega |{\sf T}\!\left[ {\rm{u}_a^{\rm{T}(x) \otimes gG_\alpha ^{\mu \nu }{\rm{\bar u}_b^{\rm{T}(0)} \right]\!{\left| \Omega \right\rangle ^A}_B
• = \langle \Omega |{\sf{T}\!\left[ {\rm{u}_a^{\rm{T}{(x)}_B}gG_\alpha ^{\mu \nu }{\rm{\bar u}_{\bar a}^{\rm{T}{(0)}^A} \right]\!\left| \Omega \right\rangle
• = \langle \Omega |{\sf{T}\!\left[ {\rm{u}_a}{(x)}^B}gG_\alpha ^{\mu \nu }{\rm{\bar u}_b}{(0)}_A} \right]\!\left| \Omega \right\rangle
• = \langle \Omega |{\sf{T}\!\left[ {\rm{u}_a}(x)gG_\alpha ^{\mu \nu }{\rm{\bar u}_b}(0)} \right]\!{\left| \Omega \right\rangle ^B}_A
• = -\frac{1}{2^6}3}\frac{\lambda _{\alpha ab}{2}{\left( {\sigma ^{\mu \nu } \right)^B}_A\left\langle {g{\rm{\bar u}\sigma \cdot {\rm{Gu} \right\rangle + \frac{\rm{i}{m_{\rm{u}{2^8}3}\frac{\lambda _{\alpha ab}{2}{\left( {\sigma ^{\mu \nu }{\not \!x} + {\not \!x}{\sigma ^{\mu \nu } \right)^B}_A\left\langle {g{\rm{\bar u}\sigma \cdot {\rm{Gu} \right\rangle
• \ \ \ \ + \frac{x^2}{2^{10}{3^2}\frac{\lambda _{\alpha ab}{2}{\left( {\sigma ^{\mu \nu } \right)^B}_A\left\langle {\rm{\bar uu} \right\rangle \left\langle {g^2}{\rm{G}^2} \right\rangle
• =- \frac{1}{2^6}3}\frac{\lambda _{\alpha ab}{2}{\left( {\sigma ^{\mu \nu {\rm{T} \right)^A}_B\left\langle {g{\rm{\bar u}\sigma \cdot {\rm{Gu} \right\rangle + \frac{\rm{i}{m_{\rm{u}{2^8}3}\frac{\lambda _{\alpha ab}{2}{\left[ {\left( {\sigma ^{\mu \nu }{\not \!x} + {\not \!x}{\sigma ^{\mu \nu } \right)}^{\rm{T} \right]^A}_B\left\langle {g{\rm{\bar u}\sigma \cdot {\rm{Gu} \right\rangle
• \ \ \ \ + \frac{x^2}{2^{10}{3^2}\frac{\lambda _{\alpha ab}{2}{\left( {\sigma ^{\mu \nu {\rm{T} \right)^A}_B\left\langle {\rm{\bar uu} \right\rangle \left\langle {g^2}{\rm{G}^2} \right\rangle .

这就说明:

• \ \ \ \ C\langle \Omega |{\sf{T}\!\left[ {\rm{u}_a^{\rm{T}(x) \otimes gG_\alpha ^{\mu \nu }{\rm{\bar u}_{b}^{\rm{T}(0)} \right]\!\left| \Omega \right\rangle C
• =-\frac{1}{2^6}3}\frac{\lambda _{\alpha ab}{2}{\sigma ^{\mu \nu }\left\langle {g{\rm{\bar u}\sigma \cdot {\rm{Gu} \right\rangle -\frac{\rm{i}{m_{\rm{u}{2^8}3}\frac{\lambda _{\alpha ab} }{2}\left( {\sigma ^{\mu \nu }{\not \!x} + {\not \!x}{\sigma ^{\mu \nu } \right)\left\langle {g{\rm{\bar u}\sigma \cdot {\rm{Gu} \right\rangle
• \ \ \ + \frac{x^2}{2^{10}{3^2}\frac{\lambda _{\alpha a\bar a}{2}{\sigma ^{\mu \nu }\left\langle {\rm{\bar uu} \right\rangle \left\langle {g^2}{\rm{G}^2} \right\rangle .

15. 对色散积分的 Borel 变换

https://zhuanlan.zhihu.com/p/396673735https://zhuanlan.zhihu.com/p/416707961- 两点函数的色散积分表达: $\Pi (p^2)=\frac{1}{\pi }{\int }_0^{\infty }\text{d}{m}^2{\frac{\mathrm{Im}\Pi (m^2)}{m}}^2-p^2-\text{i}\epsilon .$

• Borel 变换的定义: {\mathcal{B}_{M}[\Pi ({p}^{2})]=\underset{\begin{matrix} -{p}^{2},n\to \infty ;\ \\ \frac{-{p}^{2}{n}={M}^{2}={\rm{const} \\ \end{matrix}{\mathop{\rm lim}\,\frac{1}{n!}{(-{p}^{2})}^{n+1}{\left( \frac{\rm{d}{\rm{d}{p}^{2} \right)}^{n}\Pi ({p}^{2}).
• 对色散积分的 Borel 变换: {\mathcal{B}_M}[\Pi ({p}^{2})]=\frac{1}{\pi }\int_{0}^{\infty }{\text{d}s{\text{e}^{-\frac{s}{M}^{2}\operatorname{Im}\Pi (s)}.

常用公式:

• {\mathcal{B}_{M}\left[ \frac{1}{\left( {p}^{2}-\alpha \right)}^{\beta } \right]={(-1)}^{\beta }\frac{1}{M}^{2(\beta -1)}(\beta -1)!}{\text{e}^{-\frac{\alpha }{M}^{2},
• {\mathcal{B}_M}\left( {p}^{2m}\ln \frac{1}{-{p}^{2} \right)=m!{M}^{2(m+1)},
• ${\mathcal{B}}_M\left(p^{2m}\right)=0,$ $m$ 为非负整数.
• {\mathcal{B}_M}\left[ \frac{1}{-{p}^{2k}\frac{1}{\left( \ln \frac{-{p}^{2}{\mu }^{2} \right)}^{\varepsilon } \right]=\frac{1}{(k-1)!{M}^{2(k-1)}\frac{1}{\left( \ln \frac{M}^{2}{\mu }^{2} \right)}^{\varepsilon }\left[ 1+O\left( \frac{1}{\ln \frac{M}^{2}{\mu }^{2} \right) \right].

16. 其它

• {\rm{i}{\not{\!\!D}^{ab}{\rm{q}_{b}={m}_{\rm{q}{\rm{q}_{a},\ {\rm{i}{\rm{\bar{q}_{b}{\not{\!\!D}^{\dagger ab}=-{m}_{\rm{q}{\rm{\bar{q}_{a},\ {D}_{\mu }={\partial }_{\mu }+{\rm{i}g\frac{\lambda }_{\alpha }{2}A_{\mu }^{\alpha }.
• \left[ {D}_{\mu },{D}_{\nu } \right]={\rm{i}g\frac{\lambda }_{\alpha }{2}{\rm{G}_{\mu \nu }^{\alpha },\ {\rm{G}_{\mu \nu }^{\alpha }={\partial }_{\mu }A_{\nu }^{\alpha }-{\partial }_{\nu }A_{\mu }^{\alpha }-g{f}^{\alpha }_{\beta \gamma }A_{\mu }^{\beta }A_{\nu }^{\gamma }.
• {\left( {\tilde{D}^{\mu } \right)}^{\alpha }_{\beta }{\rm{G}_{\mu \nu }^{\beta }=g{\rm{\bar{q}{\gamma }^{\nu }\frac{\lambda }^{\alpha }{2}{\rm{q},\ {\left( {\tilde{D}_{\mu } \right)}^{\alpha }_{\beta }={\delta }^{\alpha }_{\beta }{\partial }_{\mu }+g{f}^{\alpha }_{\beta \gamma }A_{\mu }^{\gamma }(x).

T_{\alpha }^{ab}T_{bc}^{\alpha }=\frac{4}{3}{\delta }^{a}_{c},\ {\rm{tr}\left( {T}_{\alpha }{T}^{\alpha } \right)=4,\ {T}_{\alpha }=\frac{\lambda }_{\alpha }{2}.

\left\{ \begin{align} & U\left( t,{t}_{0} \right)={\rm{T}{\rm{e}^{-{\rm{i}\int_{t}_{0}^{t}{\rm{d}\tau }\ H_{1}^{\rm{I}(\tau )}={\rm{T}{\rm{e}^{-{\rm{i}\int{\rm{d}^{4}x}\ \mathcal{H}_{1}^{\rm{I}={\rm{T}{\rm{e}^{\rm{i}\int{\rm{d}^{4}x}\ \mathcal{L}_{1}^{\rm{I}, \\ & {\mathcal{L}_{1}=-g{\bar{\psi }_{a}{\gamma }^{\mu }\frac{\lambda _{\alpha }^{ab}{2}A_{\mu }^{\alpha }{\psi }_{b}=-g{\bar{\psi }_{a}{\gamma }^{\mu }T_{\alpha }^{ab}A_{\mu }^{\alpha }{\psi }_{b}.\ \\ \end{align} \right.

具体而言我们将这样处理指标问题:

https://zhuanlan.zhihu.com/p/591789284- 首先 \psi \to {\psi }^{A},\ \bar{\psi }\to {\bar{\psi }_{A},\ {\psi }^{\rm{T}\to {\left( {\psi }^{\rm{T} \right)}_{A},\ \bar{\psi }\to {\left( {\bar{\psi }^{\rm{T} \right)}^{A}, 箭头右边写的是分量.

• 这里用大写字母是因为, 颜色和八重态指标出来以后, 我他吗实在是找不到能用的字母了[5].
• 然后要有 {\left( {\psi }^{\rm{T} \right)}_{A}={\psi }^{A},\ {\left( {\bar{\psi }^{\rm{T} \right)}^{A}={\bar{\psi }_{A}.
• 这是因为转置操作并不会对分量本身造成影响, 显然无论在行还是列矩阵, 第几分量都一样.
• 最后是 {\left( {\Gamma }^{\rm{T} \right)}^{A}_{B}={\Gamma }^{B}_{A}, 关键点在于指标的上下位置在转置下也必须改变.
• 心里清楚这些是很必要的, 下面举一例:
• 若定义夸克传播子[6]{S_{ab}{\left( {x - y} \right)^A}_B \equiv \left\langle 0 \right|\mathsf T\left[ {\rm{q}_a}{\left( x \right)}^A}{\rm{\bar q}_b}{\left( y \right)}_B} \right]\left| 0 \right\rangle .
• 则必需有 \langle 0|\mathsf T\left[ {\rm{q}_{a}^{\rm{T}{\left( x \right)}_{A}{\rm{\bar{q}_{b}^{\rm{T}{\left( y \right)}^{B} \right]\left| 0 \right\rangle
• \ \ \ \ \ \ \ \ =\langle 0|\mathsf T\left[ {\rm{q}_{a}{\left( x \right)}^{A}{\rm{\bar{q}_{b}{\left( y \right)}_{B} \right]\left| 0 \right\rangle ={S}_{ab}{(x-y)}^{A}_{B}=S_{ab}^{\rm{T}{(x-y)}^{B}_{A}.
• 注意最后一步的转置是必要的, 等你哪天真要算这个你就明白了, 不转置没法还原到矩阵.
• \Rightarrow S_{ab}^{\text{T}{(x-y)}^{B}_{A}=\langle 0|{\sf T}\left[ \text{q}_{a}^{\text{T}\left( x \right)\otimes \bar{\text{q}_{b}^{\text{T}\left( y \right) \right]{\left| 0 \right\rangle }^{B}_{A}
• \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\langle 0|{\mathsf T}\left[ \text{q}_{a}^{\text{T}{\left( x \right)}_{A}\bar{\text{q}_{b}^{\text{T}{\left( y \right)}^{B} \right]\left| 0 \right\rangle .
• 上面和我们原来的约定: [東雲正樹 - 終 · 一步到位的张量指标运算] 是不太一样的, 这是因为我原来给出的约定下连接线性空间与对偶空间的同构映射不是转置而是 dagger, 但旋量空间不会涉及 dagger, 所以从这个意义上来讲或许都不需要区分上下指标, 只要有左右指标就够了.

参考

• ^你该不会每次都重新跑一遍吧? 笑了.
• ^你可以运行一下看看得到什么结果.
• ^保存的文件格式取决于后缀, 比如说你将动画数据保存为 .gif 的话, 那确实是动图. 这里的 .wdx 是 MMA 的万能格式, 只要是 MMA 里产生的数据就都能保存.
• ^更准确来说是背景场, 但, 算了没有但.
• ^所以干脆用大写字母给它明显地标出算了, 反正这种指标也不会出现在运算结果里.
• ^小写的 a,b 是颜色指标.