nLab superpotential

Idea

In $D=4$ $\mathcal{N} = 1$ globally supersymmetric field theory, the superpotential $W(\Phi^i)$ is a strictly holomorphic function of the chiral superfields $\Phi^i$ that controls the theory’s non-gauge interaction.

It appears in the action functional as a chiral superspace integral

$$\propto \int d^4 x\, d^2 \theta \, W\big(\Phi^i\big) + \text{h.c.} \,,$$

This determines the F-term contribution to the scalar potential energy:

$$V_F = \sum_i \left\vert \frac{\partial W}{\partial \Phi^i} \right\vert^2 \mathrlap{\,,}$$

while its second derivatives define the Yukawa couplings between the fermions and the complex scalar fields.

The analytic structure of the superpotential restricts allowable quantum corrections; this holomorphy is the physical underpinning of the supersymmetry non-renormalization theorems.

References

Most introductions to global supersymmetry talk about superpotentials, for instance:

• Ulrich Theis; pp. 15 in: An Introduction to Supersymmetry, lecture notes (2007) [pdf, pdf]

See also:

• Wikipedia: Superpotential