Superpotential

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In theoretical physics, the superpotential is a function in supersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in the Schrödinger equation. The partner potentials have the same spectrum, apart from a possible eigenvalue of zero, meaning that the physical systems represented by the two potentials have the same characteristic energies, apart from a possible zero-energy ground state.

One-dimensional example

Consider a

so that ${\displaystyle \hbar =1}$. Let W (the superpotential) represent an arbitrary differentiable function of x and define the supersymmetric operators Q₁ and Q₂ as

${\displaystyle Q_{1}={\frac {1}{2}}\left[(p-iW)b+(p+iW)b^{\dagger }\right]}$

${\displaystyle Q_{2}={\frac {i}{2}}\left[(p-iW)b-(p+iW)b^{\dagger }\right]}$

The operators Q₁ and Q₂ are self-adjoint. Let the Hamiltonian be

${\displaystyle H=\{Q_{1},Q_{1}\}=\{Q_{2},Q_{2}\}={\frac {p^{2}}{2}}+{\frac {W^{2}}{2}}+{\frac {W'}{2}}(bb^{\dagger }-b^{\dagger }b)}$

where W' signifies the derivative of W. Also note that {Q₁,Q₂}=0. Under these circumstances, the above system is a

determined by

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

In four spacetime dimensions

In

, which tends to automatically be complex valued. We may identify the complex conjugate of a chiral superfield as an anti-chiral superfield. There are two possible ways to obtain an action from a set of superfields:

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

or

• Integrate a chiral superfield on the chiral half of a superspace, spanned by ${\displaystyle x_{0,1,2,3}}$ and ${\displaystyle \theta }$, not on ${\displaystyle {\bar {\theta }}}$.

The second option tells us that an arbitrary holomorphic function of a set of chiral superfields can show up as a term in a Lagrangian which is invariant under supersymmetry. In this context, holomorphic means that the function can only depend on the chiral superfields, not their complex conjugates. We may call such a function W, the superpotential. The fact that W is holomorphic in the chiral superfields helps explain why supersymmetric theories are relatively tractable, as it allows one to use powerful mathematical tools from complex analysis. Indeed, it is known that W receives no perturbative corrections, a result referred to as the perturbative non-renormalization theorem. Note that non-perturbative processes may correct this, for example through contributions to the beta functions due to instantons.

See also

• Komar superpotential

References

• Stephen P. Martin, A Supersymmetry Primer.
  arXiv:hep-ph/9709356
  .
• B. Mielnik and O. Rosas-Ortiz, "Factorization: Little or great algorithm?", J. Phys. A: Math. Gen. 37: 10007-10035, 2004
• Cooper, Fred; Khare, Avinash; Sukhatme, Uday (1995). "Supersymmetric quantum mechanics". Physics Reports. 251: 267–385.
  doi:10.1016/0370-1573(94)00080-M
  .