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The local matrix $U(x)$ is a gauge transformation in the representation acting on the matter field. With Hermitian Lie algebra generators it can be written locally in spacetime, and pointwise for $SU(3)$, asThe real functions ${\omega }^a$ can instead include a factor of $g$ by redefining the transformation parameters. The gauge covariant derivative convention in this paper is $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$. Inserting the stated transformed $A_{\mu }$ shows explicitly that the ${\partial }_{\mu }U$ term cancels, giving $D_{\mu }^{\prime }(U\psi )=U{D}_{\mu }\psi$.

In QCD, each quark field is in the colour fundamental representation $3$, with ${\psi }^i\mapsto U^i{}_j{\psi }^j$. Thus its colour generators are eight $3\times 3$ matrices, conventionally $T^a={\lambda }^a/2$ for the Gell-Mann matrices; their number is not their matrix dimension. The eight gluons themselves transform in the adjoint representation of SU(3).

With $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, the pure Yang-Mills theory has Lagrangian ${\mathcal{L}}_{\mathrm{YM}}=-\frac{1}{4}{F}_{\mu \nu }^a{F}^{a\mu \nu }=-\frac{1}{2}\mathrm{tr}(F_{\mu \nu }{F}^{\mu \nu })$. Put $G_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a$ and $C_{\mu \nu }^a=f_{abc}{A}_{\mu }^b{A}_{\nu }^c$. Since $F=G-gC$,The cross term has two derivative contributions. Interchange $\mu ,\nu$ and then $b,c$ in the second; antisymmetry of $f_{abc}$ makes it equal to the first. The cubic and quartic Yang-Mills self-interactions are thereforeThe positive cubic sign follows from the negative $gfAA$ term in the given Yang-Mills field strength; changing the covariant-derivative convention consistently changes intermediate signs.

For the left-handed quark doublet $Q_L=(u_L,d_L{)}^T$, the hypercharge is $Y=1/6$ and the weak generators are ${\sigma }^{\alpha }/2$. Its electroweak term isThe Pauli matrices give diagonal entries $\pm {\mathcal{A}}^3$ and off-diagonal entries ${\mathcal{A}}^1\mp i{\mathcal{A}}^2$. Expanding the term givesIn particular the off-diagonal weak charged current coupling is $-g(W_{\mu }{\bar{u}}_L{\gamma }^{\mu }{d}_L+W_{\mu }^{\ast }{\bar{d}}_L{\gamma }^{\mu }{u}_L)/\sqrt{2}$.

Write $s_W=\sin {\theta }_W$, $c_W=\cos {\theta }_W$. Invert the neutral-field rotation: ${\mathcal{A}}_{\mu }^3=c_W{Z}_{\mu }+s_W{A}_{\mu }$, $B_{\mu }=c_W{A}_{\mu }-s_W{Z}_{\mu }$. The original PDF gives $\tan {\theta }_W=g^{\prime }/g$, soThe photon coefficients inside the derivative are $g{s}_W/2+g^{\prime }{c}_W/6=2e/3$ for $u_L$, and $-g{s}_W/2+g^{\prime }{c}_W/6=-e/3$ for $d_L$. Multiplication by the $i$ in the kinetic Lagrangian gives the requested electromagnetic coupling of a weak multiplet:Equivalently it is $-e{A}_{\mu }{\overline{Q}}_L{\gamma }^{\mu }(T_3+Y)Q_L$, making the electric charges explicit. The remaining neutral terms are $-(g/c_W)Z_{\mu }\{(1/2-2s_W^2/3){\bar{u}}_L{\gamma }^{\mu }{u}_L+(-1/2+s_W^2/3){\bar{d}}_L{\gamma }^{\mu }{d}_L\}$. The TeX's missing prime in the mixing-angle ratio and truncated first Pauli matrix are conversion errors, not physical assumptions.