F-term scalar potential ( $V_{F}$ )

For canonical global supersymmetry, eliminating the auxiliary fields gives

V_{F} = \sum_{i} \frac{\partial W}{\partial ϕ _{i}}^{2} .

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Table of contents
- Real-slice diagnosis of chiral-scalar vacua F-term scalar potential
- Tree-level supertrace mass sum rule F-term scalar potential
  - Gauge contributions to the F-term supertrace Tree-level supertrace mass sum rule

Real-slice diagnosis of chiral-scalar vacua

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A chiral superfield has a complex scalar. Minimizing its potential only on the real axis can miss supersymmetric vacua. For example

W^{'} (φ) = 1 + φ^{2}

has positive potential

(1 + x^{2})^{2}

for real

x

, yet its complex vacua

φ = \pm i

have zero energy. Supersymmetry breaking must therefore be tested in the full complex field space, not inferred from a positive minimum on an arbitrarily chosen real slice.

Tree-level supertrace mass sum rule ( $STr M^{2} = 0$ )

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For canonical positive kinetic terms in global supersymmetry,

V = \sum_{k} ∣ W_{k} ∣^{2}

has mixed Hessian matrix

V_{i \overset{ˉ}{j}} = \sum_{k} W_{ki} \overline{W_{kj}}

and holomorphic block

V_{ij} = \sum_{k} \overline{W_{k}} W_{kij}

. The

2 n

real scalar squared-mass sum is

2 tr (m^{†} m)

for the chiral-superfield fermion mass matrix

m = W_{ij}

. Each Weyl spinor contributes two states, giving the same fermion weighted sum. Thus their boson-minus-fermion supertrace vanishes even with F-term breaking. The holomorphic Hessian splits real scalar masses without changing their sum. Noncanonical Kähler metrics, supergravity, and radiative corrections require different formulas.

Gauge contributions to the F-term supertrace ( $2 C - 2 (4 C) + 3 (2 C) = 0$ )

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Canonical gauge interactions preserve the tree-level supertrace mass sum rule at a vacuum with vanishing D-terms. Put

C = \sum_{a} g_{a}^{2} ∥ T_{a} v ∥^{2}

for the scalar expectation vector

v

. The extra real-scalar squared-mass trace is

2 C

, the extra Weyl spinor squared-mass trace from gaugino mixing is

4 C

, and the gauge boson squared-mass trace is

2 C

. Their spin-weighted supertrace is

2 C - 2 (4 C) + 3 (2 C) = 0

. The scalar contribution comes from differentiating

\frac{1}{2} \sum_{a} D_{a}^{2}

; the fermionic one counts both blocks of the symmetric mixing matrix with entries

2 g_{a} T_{a} v

; the vector one follows from the covariant scalar kinetic term. Any zero-mass Goldstone bosons in the unreduced scalar Hessian do not change its trace.