 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-43/2/scalar-potential/solution

e > 0

, and no Fayet–Iliopoulos term, since none is specified. Elimination of the three complex auxiliary fields gives

F_{+}^{*} = - λ φ_{0} φ_{-}, F_{0}^{*} = - λ φ_{+} φ_{-}, F_{-}^{*} = - λ φ_{+} φ_{0} .

The Abelian gauge auxiliary field obeys

D = - e (∣ φ_{+} ∣^{2} - ∣ φ_{-} ∣^{2})

in this normalization. Thus the F-term scalar potential and the gauge D-term give

V = ∣ λ ∣^{2} (∣ φ_{0} ∣^{2} ∣ φ_{-} ∣^{2} + ∣ φ_{+} ∣^{2} ∣ φ_{-} ∣^{2} + ∣ φ_{+} ∣^{2} ∣ φ_{0} ∣^{2}) + \frac{e ^{2}}{2} (∣ φ_{+} ∣^{2} - ∣ φ_{-} ∣^{2})^{2} .

All terms are nonnegative. The normalization of

e

can be changed together with the vector-field normalization, but the relative charges and the zero-potential conditions cannot. A noncanonical Kähler potential would change the inverse-metric factors in the F-term potential; adding a Fayet–Iliopoulos term would shift

D

and define a different model. Neither is silently introduced here.

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