Confidence region

In theoretical physics, the label superpotential denotes a concept used in supersymmetric quantum mechanics.

Example of superpotentiality

Consider the example of a one dimensional, non-relativistic particle with a 2D (i.e. two state) internal degree of freedom called "spin". (It does not really spin because "real" spin applies only to particles in three-dimensional space.) Let b signify an operator which transforms a "spin up" particle into a "spin down" particle and its Hermitian adjoint (b^†) transforming a spin down particle into a spin up particle normalized such that the anticommutator (b,b^†) equals 1. And of course, b² equals 0. Let p represent the momentum of the particle and x represent its position with [x,p]=i (let's use natural units where $\hslash =1$). Let W (the superpotential) represent an arbitrary differentiable function of x and let the supersymmetric operators represent:

$Q_1=\frac{1}{2}[(p-iW)b+(p+iW)b^{†}]$

$Q_2=\frac{i}{2}[(p-iW)b-(p+iW)b^{†}]$

Note that Q₁ and Q₂ seem self-adjoint. Let the Hamiltonian

$H=\{Q_1,Q_1\}=\{Q_2,Q_2\}=\frac{p^2}{2}+\frac{W^2}{2}+\frac{W^{\prime }}{2}(b{b}^{†}-b^{†}b)$

where W' signifies the derivative of W. Also note that {Q₁,Q₂}=0. This shows nothing other than N=2 supersymmetry.

Most call the spin down state "bosonic" and the spin up state "fermionic". This exists only in analogy to quantum field theory and should not be taken literally. Then, Q₁ and Q₂ maps "bosonic" states into "fermionic" states and vice versa. Restricting to the bosonic or fermionic sectors gives two partner potentials determined by

$H=\frac{p^2}{2}+\frac{W^2}{2}\pm \frac{W^{\prime }}{2}$

Superpotential in dimension 4

In the theory supersymmetry with four dimensions, which might have some connection to the nature, physicists insist a scalar field as the lowest component of a chiral superfield, which tends to automatically contain complexities. You may identify the complex conjugate of a chiral superfield as an anti-chiral superfield.

To obtain the action from a set of superfields, two choices may occur

• Integrate a superfield on the whole superspace spanned by $x_{0,1,2,3}$ and

$\theta ,\overline{\theta }$

or

• Integrate a chiral superfield on the chiral half of a superspace, spanned by $x_{0,1,2,3}$

and $\theta$, not on $\overline{\theta }$.

Thus, when given a set of chiral superfields and an arbitrary holomorphic function of them, W, a term in the Lagrangian constructed by which invariant under supersymmetry, W cannot depend on the complex conjugates. The function W may be classified as the superpotential. The fact that W tends to have holomorphic properties in the chiral superfields shows the source of the tractability of supersymmetric theories. Indeed, we know W to receive no perturbative corrections, which celebrates the perturbative non-renormalization theorem. The non-perturbative processes may correct it, e.g., by instantons.

References

• Stephen P. Martin, A Supersymmetry Primer. arxiv.org/pdf/hep-ph/9709356v6.pdf.