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Cubic one-field Wess–Zumino F-term potential

Physical scope

A one-field classical rigid N=1 Wess–Zumino model with canonical Kähler metric and cubic superpotential, used to illustrate the F-term elimination formula. No quantum, empirical, or particle-spectrum claim is asserted.

Example

Take one chiral field with canonical Kähler potential $K (\overset{ˉ}{ϕ}, ϕ) = \overset{ˉ}{ϕ} ϕ$ and holomorphic superpotential $W (ϕ) = \frac{1}{2} m ϕ^{2} + \frac{1}{6} y ϕ^{3}$ , where $m \in C$ has mass dimension one and $y \in C$ is dimensionless. In the ordinary complex bosonic sector of the model postulate, take the auxiliary field to be complex-valued; the formal even coefficient space is retained before this restriction. For a smooth compactly supported scalar component $ϕ : M \to C$ , the auxiliary equation and F-term potential are

$F = - \overline{m ϕ + \frac{1}{2} y ϕ^{2}}, V_{F} = {∣ m ϕ + \frac{1}{2} y ϕ^{2} ∣}^{2} .$

This is a classical model calculation, not an empirical claim or a quantum effective potential.

Example references

Stephen P. Martin, A Supersymmetry Primer, §3.2.

Adel Bilal, Introduction to Supersymmetry, §4.3.

Rigid four-dimensional $N = 1$ Wess–Zumino sigma-model action

Kähler-metric F-term scalar potential in the rigid $N = 1$ model

Facts & Assumptions

Given: The one-field canonical model and $W (ϕ)$ in the Example.

[A1]

Rigid four-dimensional $N = 1$ Wess–Zumino sigma-model action: the canonical Kähler choice has $g_{ϕ \overset{ˉ}{ϕ}} = 1$ and its chiral auxiliary sector is $F \overset{ˉ}{F} + F W^{'} (ϕ) + \overset{ˉ}{F} \overline{W^{'} (ϕ)}$ .

[F1]

Kähler-metric F-term scalar potential in the rigid $N = 1$ model: for a positive Kähler metric, $F = - g^{ϕ \overset{ˉ}{ϕ}} \overline{W^{'} (ϕ)}$ and $V_{F} = g^{ϕ \overset{ˉ}{ϕ}} W^{'} (ϕ) \overline{W^{'} (ϕ)}$ .

Verification

Given: [A1], [F1], and the polynomial superpotential in the Example.

1.1givenalgebra

Differentiate $W (ϕ) = \frac{1}{2} m ϕ^{2} + \frac{1}{6} y ϕ^{3}$ to obtain $W^{'} (ϕ) = m ϕ + \frac{1}{2} y ϕ^{2}$ .

2.1A1F1step 1.1algebra∎

Substitute $g_{ϕ \overset{ˉ}{ϕ}} = g^{ϕ \overset{ˉ}{ϕ}} = 1$ and the derivative from step 1.1 into [F1]. This gives $F = - \overline{m ϕ + \frac{1}{2} y ϕ^{2}}$ and $V_{F} = ∣ m ϕ + \frac{1}{2} y ϕ^{2} ∣^{2}$ , as stated.

Cubic one-field Wess–Zumino F-term potential

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources