Conventions and Formula

The master formulae in QFT:

$$\begin{array}{}\text{(1)}& \int \mathcal{D}{\varphi }^{\ast }\mathcal{D}\varphi \exp \{i\int d^{4}x({\varphi }^{\ast }M\varphi +JM)\}=\frac{\mathcal{N}}{detM}\exp (iJ{M}^{-1}J)\end{array}$$

where $\mathcal{N}$ is some normalization factor.

For fermions,

$$\begin{array}{}\text{(2)}& \int \mathcal{D}\psi \mathcal{D}\bar{\psi }\exp \{i\int d^{4}x(\bar{\psi }M\psi )\}=\mathcal{N}detM.\end{array}$$

Some conventions. Metric:

$$\begin{array}{rl}{g}_{\mu \nu }& =\text{diag}\{1,-1,-1,-1\},\\ {ϵ}_{0123}& =1,\end{array}$$

Fourier transform:

$$\begin{array}{rl}f(x)& =\int \frac{d^{n}k}{(2\pi {)}^n}{e}^{-ip\cdot x}\tilde{f}(p),\\ \tilde{f}(p)& =\int dx{e}^{ip\cdot x}f(x),\end{array}$$

and $\delta$-function:

$$\begin{array}{}\text{(3)}& {\delta }^{(n)}(k)=\frac{1}{(2\pi {)}^n}\int d^{n}x{e}^{ikx}.\end{array}$$

The gauge Lie group $G$ has an underlying Lie algebra $\mathfrak{g}$, whose generator satisfies

$$\begin{array}{}\text{(4)}& [T^a,T^b]=i{f}^{abc}{T}^c\end{array}$$

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We require the generators in the fundamental representations to satisfy normalization condition

$$\begin{array}{}\text{(5)}& \text{Tr }{T}^a{T}^b=\frac{1}{2}{\delta }^{ab}.\end{array}$$

Gauge field is a $\mathfrak{g}$-valued field,

$$\begin{array}{}\text{(6)}& A_{\mu }=A_{\mu }^a{T}^a,T^a\in \mathfrak{g}.\end{array}$$

The $\mathfrak{g}$-valued field strength is

$$\begin{array}{}\text{(7)}& F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }-i[A_{\mu },A_{\nu }],\end{array}$$

which in the language of differential forms is

$$\begin{array}{}\text{(8)}& \boxed{{\displaystyle F=dA-iA\wedge A}}.\end{array}$$

Our convention for covariant derivative acting on a field in the fundamental representation is

$$\begin{array}{}\text{(9)}& D_{\mu }\psi \equiv {\partial }_{\mu }\psi -i{A}_{\mu }\psi ,\end{array}$$

while acting on a field in the adjoint representation is

$$\begin{array}{}\text{(10)}& D_{\mu }\varphi \equiv {\partial }_{\mu }\varphi -i[A_{\mu },\varphi ],\end{array}$$

and

$$\begin{array}{}\text{(11)}& [{\mathcal{D}}_{\mu },{\mathcal{D}}_{\nu }]\psi & =-i{F}_{\mu \nu }\psi ,\text{(12)}& [{\mathcal{D}}_{\mu },{\mathcal{D}}_{\nu }]\varphi & =-i[F_{\mu \nu },\varphi ],\end{array}$$

where again $\psi ,\varphi$ are fields in the fundamental and adjoint representation respectively.

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The gauge transformation for various fields are

• gauge field: $A_{\mu }\to \mathrm{\Omega }(A_{\mu }+i{\partial }_{\mu }){\mathrm{\Omega }}^{†},\mathrm{\Omega }=e^{i{\omega }^i{T}^i}\in SU(N)$
• fundamental scalar field: $\varphi \to \mathrm{\Omega }\varphi$
• fundamental spinor field: $\psi \to \mathrm{\Omega }\psi$
• adjoint scalar field: $\varphi \to \mathrm{\Omega }\varphi {\mathrm{\Omega }}^{†}$

The non-abelian, gauge dependent magnetic field (chromo-magnetic field) is

$$\begin{array}{}\text{(13)}& B_i\equiv -\frac{1}{2}{ϵ}_{ijk}{F}_{jk}\end{array}$$

and the non-abelian electric field

$$\begin{array}{}\text{(14)}& E_i\equiv F_{0i}\end{array}$$

The Weyl (chiral) basis of gamma matrices is

$$\begin{array}{}\text{(15)}& {\gamma }^{\mu }=\left(\begin{array}{cc}0& {\sigma }^{\mu }\\ {\overline{\sigma }}^{\mu }& 0\end{array}\right),{\overline{\sigma }}^{\mu }\equiv (\mathbb{1},{\sigma }^i).\end{array}$$

Going to the Euclidean spacetime:

$$\begin{array}{}\text{(16)}& t\to -i\tau \end{array}$$

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Some notations concerning SUSY.

$$\begin{array}{}\text{(17)}& {\eta }_{\mu \nu }& =\text{diag}1,-1,-1,-1,\text{(18)}& {\sigma }_{a\dot{a}}^{\mu }& =(\mathbb{1},{\sigma }^i)\text{numerically},\text{(19)}& {\bar{\sigma }}^{\mu \dot{a}a}& =(\mathbb{1},-{\sigma }^i)\text{numerically},\text{(20)}& {{\left({\sigma }^{\mu \nu }\right)}_a}^b& \equiv \frac{i}{4}{{\left({\sigma }^{[\mu }{\bar{\sigma }}^{\nu ]}\right)}_a}^b,\text{(21)}& {{\left({\bar{\sigma }}^{\mu \nu }\right)}^{\dot{a}}}_{\dot{b}}& \equiv \frac{i}{4}{{\left({\overline{\sigma }}^{[\mu }{\sigma }^{\nu ]}\right)}^{\dot{a}}}_{\dot{b}}.\end{array}$$

Under Lorentz transformation,

$$\begin{array}{}\text{(22)}& {\psi }_a\mapsto {\exp {(-\frac{i}{2}{\omega }_{\mu \nu }{\sigma }^{\mu \nu })}_a}^b{\psi }_b,{\overline{\psi }}^{\dot{a}}\mapsto {\exp {(-\frac{i}{2}{\omega }_{\mu \nu }{\overline{\sigma }}^{\mu \nu })}^{\dot{a}}}_{\dot{b}}{\overline{\psi }}^{\dot{b}}.\end{array}$$

The contraction works in a peculiar way. The metric is

$$\begin{array}{}\text{(23)}& {ϵ}_{ab}={\sigma }^2(i\to 1)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),{ϵ}^{ab}=-{ϵ}_{ab},{ϵ}^{\dot{a}\dot{b}}=-{ϵ}_{\dot{a}\dot{b}}.\end{array}$$

The metric acts from left on a field, and acts from right to a derivative,

$$\begin{array}{}\text{(24)}& {ϵ}_{ab}{\psi }^b={\psi }_a,{\partial }_a{ϵ}^{ab}={\partial }^b.\end{array}$$

The gamma matrices:

$$\begin{array}{}\text{(25)}& {\gamma }^{\mu }& =\left(\begin{array}{cc}0& {\sigma }^{\mu }\\ {\overline{\sigma }}^{\mu }& 0\end{array}\right),\text{(26)}& {\gamma }^5& =i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3.\end{array}$$ $$\begin{array}{}\text{(27)}& {\chi }^a{\chi }^b& =-\frac{1}{2}{ϵ}^{ab}{\chi }^2,{\chi }_a{\chi }_b=\frac{1}{2}{ϵ}_{ab}{\chi }^2,\text{(28)}& {\overline{\chi }}^{\dot{a}}{\overline{\chi }}^{\dot{b}}& =\frac{1}{2}{ϵ}^{\dot{a}\dot{b}}{\overline{\chi }}^2,{\overline{\chi }}_{\dot{a}}{\overline{\chi }}_{\dot{b}}=-\frac{1}{2}{ϵ}_{\dot{a}\dot{b}}{\overline{\chi }}^2.\end{array}$$

$(\theta {\sigma }^{\mu }\overline{\theta })(\theta {\sigma }^{\nu }\overline{\theta })=\frac{1}{2}{\eta }^{\mu \nu }{\theta }^2{\overline{\theta }}^2.$ $\begin{array}{}\text{(29)}& A^{\mu }& =\frac{1}{2}{A}_{a\dot{b}}{\sigma }^{\mu \dot{b}a},\text{(30)}& A_{a\dot{b}}& =A^{\mu }{\sigma }_{\mu a\dot{b}},\text{(31)}& F_a^b& =F_{\mu \nu }{({\sigma }^{\mu \nu }{)}_a}^b\end{array}$

Note that the complex conjugate reverses the order of Grassmann operators, and

$$\begin{array}{}\text{(32)}& \boxed{{\displaystyle ({\partial }_a{)}^{\ast }=-{\overline{\partial }}_{\dot{a}}}}.\end{array}$$

The chiral field:

$$\begin{array}{}\text{(33)}& x_L& \equiv x^{\mu }-i\theta {\sigma }^{\mu }\overline{\theta },x_R\equiv x^{\mu }+i\theta {\sigma }^{\mu }\overline{\theta },\text{(34)}& D_a& ={\partial }_a-i({\sigma }^{\mu }\overline{\theta }{)}_a{\partial }_{\mu },{\overline{D}}_{\dot{a}}=(D_a{)}^{\ast }=-{\overline{\partial }}_{\dot{a}}+i(\theta {\sigma }^{\mu }{)}_{\dot{a}}{\partial }_{\mu }.\end{array}$$

The supergauge transformation:

$$\begin{array}{}\text{(35)}& \mathrm{\Phi }& \to e^{iq\mathrm{\Lambda }(y,\theta )}\mathrm{\Phi },\text{(36)}& V(x,\theta ,\overline{\theta })& \to V(x,\theta ,\overline{\theta })-i(\mathrm{\Lambda }-\overline{\mathrm{\Lambda }}),\end{array}$$

where the charge $q$ is usually absorbed in the definition of the gauge field $A$. The gauge invariant combination is

$\overline{\mathrm{\Phi }}{e}^V\mathrm{\Phi }.$ The Wess-Zumino supergauge greatly reduces the number of component fields of a real superfield:

$V(x,\theta ,\overline{\theta })=-2\theta {\sigma }^{\mu }\overline{\theta }{A}_{\mu }-2i{\overline{\theta }}^2(\theta \lambda )+2i{\theta }^2\overline{\theta }\overline{\lambda }+{\theta }^2{\overline{\theta }}^{2}D.$ The super-generalization of field strength is a left-chiral superfield, $W_a\equiv \frac{1}{8}{\overline{D}}^2{D}_{a}V,$ to derive it, go to $\{y^{\mu },\theta ,\theta \}$ coordinates and note that

$$\begin{array}{}\text{(37)}& {\overline{D}}_{\dot{a}}\to -{\overline{\partial }}_{\dot{a}}\end{array}$$

and

$$\begin{array}{}\text{(38)}& {\overline{\partial }}^2{\overline{\theta }}^2=-4.\end{array}$$