MSSM (Minimal Supersymmetric Standard Model)

Question

I have to write the action in superfield for SM with the Higgs, but I have problems with the mass term.

This is what I have (I don't know if it is right):

$$S_{1}=Tr\left(W^{\alpha}W_{\alpha}\right)\vert_{\theta\theta}+Tr\left(\bar{W}_{\dot\alpha}\bar{W}^{\dot\alpha}\right)\vert_{\bar\theta\bar\theta}+Tr\left(V^{\alpha}V_{\alpha}\right)\vert_{\theta\theta}+Tr\left(\bar{V}_{\dot\alpha}\bar{V}^{\dot\alpha}\right)\vert_{\bar\theta\bar\theta}+U^{\alpha}U_{\alpha}\vert_{\theta\theta}+\bar{U}_{\dot\alpha}\bar{U}^{\dot\alpha}\vert_{\bar\theta\bar\theta}.$$

$S_{1}$ is the part of the Lagrangian concerns with the kinetic term of the gauge field. In this case we have 8 gauge fields, gluons ($SU(3)$) in $W_{\alpha}=W_{\alpha}^{a}(t^{a})_{ij}$, $a=1,2,...,8.$ Analogously for $V_{\alpha}$ where we have 3 gauge fields related with weak interaction ($SU(2)$) and a gauge field for the $U(1)$ algebra.
Now we want to introduce matter fields.

$\bullet$(3,2,1/6) left handed quark doublet (A)
$\bullet$($\bar{3}$,1,-2/3) right handed up-type anti-quark (B)
$\bullet$($\bar{3}$,1,1/3) right handed down-type anti-quark (C)
$\bullet$(1,2,-1/2) left handed lepton doublet (D)
$\bullet$(1,1,1) right handed anti-lepton (E)
$\bullet$(1,2,1/2) Higgs ($H_{u}$)
$\bullet$(1,2,-1/2) Higgs ($H_{d}$)

The action related to the matter, Here $N,M=1,2,3$ and $I,J=1,2$.
$\bullet$ $S_{A}=\bar{A}^{NI}\left(e^{V_{W}\oplus V_{V}}\right)_{NI}^{MJ}A_{MJ}\vert_{\theta\theta\bar{\theta}\bar{\theta}}=\bar{A}^{NI}\left[\left(e^{V_{W}}\right)_{N}^{M}\otimes\left(e^{V_{V}}\right)_{I}^{J}\right]A_{MJ}\vert_{\theta\theta\bar{\theta}\bar{\theta}}$

$\bullet$ $S_{B}=B^{N}\left(e^{-V_{W}}\right)_{N}^{M}\bar{B}_{M}\vert_{\theta\theta\bar{\theta}\bar{\theta}}$

$\bullet$$S_{C}=C^{N}\left(e^{-V_{W}}\right)_{N}^{M}\bar{C}_{M}\vert_{\theta\theta\bar{\theta}\bar{\theta}}$

$\bullet$$S_{D}=\bar{D}^{I}\left(e^{V_{V}}\right)_{I}^{J}D_{J}\vert_{\theta\theta\bar{\theta}\bar{\theta}}$

$\bullet$$S_{E}=E\left(e^{-V_{U}}\right)\bar{E}\vert_{\theta\theta\bar{\theta}\bar{\theta}}$

Adding them we get a SUSY Lagrangian without the Higgs. But I don't know how to add the Higgs (Yukawa terms).

Answer 1

$V_W$, $V_V$, $V_U$ are vector superfields, not superpotentials. Superpotential is of the form $$ W=a_i\Phi_i+m_{ij}\Phi_i\Phi_j+y_{ijk}\Phi_i\Phi_j\Phi_k $$ where $a_i$, $m_{ij}$, $y_{ijk}$ are coupling constants ($m$ is mass, $y$ is Yukawa). $\Phi_i$ are chiral superfields. To add these terms to the Lagrangian, $$ \int d^2\theta\;W + h.c.\;\subset\mathcal{L} $$

Take a look at these notes, page 75. Or you can find how to build supersymemtric gauge theories in any SUSY/SUGRA textbook.