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Four-Dimensional $N = 1$ F- and D-Term Scalar Potential

Physical scope

Conditional algebraic elimination of the old-minimal chiral, gravity, and vector auxiliary fields in the adopted two-derivative four-dimensional N=1 Einstein-frame model, with no extra auxiliary-independent scalar-potential term; no higher-derivative or nonstandard auxiliary sectors.

Statement

For the two-derivative $N = 1$ framework with positive Kähler metric $K_{i \overset{ˉ}{j}}$ , invertible positive gauge matrix $Re f_{A B}$ , and algebraically eliminated chiral auxiliary fields, the old-minimal gravity auxiliary field, and vector auxiliary fields, assume the zero-derivative scalar-potential term is generated only by this auxiliary sector, with no additional auxiliary-independent term. The Einstein-frame scalar potential is $V = e^{K / M_{P}^{2}} [K^{i \overset{ˉ}{j}} D_{i} W \overline{D_{j} W} - \frac{3 ∣ W ∣^{2}}{M_{P}^{2}}] + \frac{1}{2} (Re f)^{- 1 A B} P_{A} P_{B} .$ Here $D_{i} W = \partial_{i} W + (K_{i} / M_{P}^{2}) W$ , and $P_{A}$ is the real moment map in the declared gauging normalization. A constant Fayet–Iliopoulos (FI) contribution is permitted only when the Kähler-line-bundle lift gauges the appropriate $R$ -symmetry. The result is conditional on the old-minimal off-shell matter content and does not include higher-derivative terms, nonstandard auxiliary sectors, or an unqualified FI term.

Statement references

Hohl, §8.5.5, equations (8.223)–(8.230)

Louis, §9.1, equations (9.2)–(9.4)

Ducrocq, Rausch de Traubenberg, and Valenzuela, §5

Four-Dimensional $N = 1$ Matter-Coupled Supergravity

Four-Dimensional $N = 1$ Kähler–Hodge Line Data

Facts & Assumptions

Given: The raw auxiliary Lagrangian and field conventions of Four-Dimensional $N = 1$ Matter-Coupled Supergravity, $M_{P} > 0$ , positive $K_{i \overset{ˉ}{j}}$ , positive symmetric $R_{A B} : = Re f_{A B}$ , and real $P_{A}$ .

[A2]

The zero-derivative scalar-potential term in this theorem is generated only by the displayed $F^{i}, M, D^{A}$ auxiliary sector; no independent $V_{0} (z, \overset{ˉ}{z})$ is added.

[A1]

Supergravity Conventions And Normalizations fixes four-dimensional Lorentzian Einstein frame, $κ_{4}^{2} = 8 π G_{N} = M_{P}^{- 2}$ , and the Einstein–Hilbert coefficient $M_{P}^{2} / 2$ . The stated dimensionful $M_{P} > 0$ convention is retained.

[F1]

Four-Dimensional $N = 1$ Matter-Coupled Supergravity adopts the old-minimal auxiliary density and the off-shell signs and normalizations displayed in its Statement.

[F2]

Four-Dimensional $N = 1$ Kähler–Hodge Line Data gives $D_{i} W = \partial_{i} W + K_{i} W / M_{P}^{2}$ and the Kähler-line-bundle transition convention. Its covariance makes the displayed F-term patch-independent.

Proof

1.1givenF1F2algebra

Write $A_{i} : = D_{i} W$ , $E : = e^{K / (2 M_{P}^{2})}$ , and $K^{i \overset{ˉ}{j}}$ for the inverse Kähler metric. Expanding the square gives $K_{i \overset{ˉ}{j}} ⁣ (F^{i} + E K^{i \overset{ˉ}{ℓ}} \overline{A_{ℓ}}) ⁣ ({\overset{ˉ}{F}}^{\overset{ˉ}{j}} + E K^{m \overset{ˉ}{j}} A_{m}) = K_{i \overset{ˉ}{j}} F^{i} {\overset{ˉ}{F}}^{\overset{ˉ}{j}} + E (F^{i} A_{i} + {\overset{ˉ}{F}}^{\overset{ˉ}{j}} {\overset{ˉ}{A}}_{\overset{ˉ}{j}}) + E^{2} K^{i \overset{ˉ}{j}} A_{i} {\overset{ˉ}{A}}_{\overset{ˉ}{j}} .$ Therefore the chiral auxiliary part of [F1] is this nonnegative square minus $e^{K / M_{P}^{2}} K^{i \overset{ˉ}{j}} D_{i} W \overline{D_{j} W}$ . Positivity of $K$ makes the algebraic equation solvable and the square vanishes at its unique stationary value.

1.2givenF1algebra

Completing the real vector square gives $\frac{1}{2} R_{A B} D^{A} D^{B} + D^{A} P_{A} = \frac{1}{2} R_{A B} ⁣ (D^{A} + (R^{- 1})^{A C} P_{C}) ⁣ (D^{B} + (R^{- 1})^{B D} P_{D}) - \frac{1}{2} (R^{- 1})^{A B} P_{A} P_{B} .$ Because $R$ is positive definite, its inverse exists and the unique algebraic solution removes the square. Thus this sector contributes $+ \frac{1}{2} (R^{- 1})^{A B} P_{A} P_{B}$ to $V$ .

2.1givenF1step 1.1algebra

Put $a = M_{P}^{2} / 3 > 0$ and $b = E$ . Then $- a ∣ M ∣^{2} - b (M \overset{ˉ}{W} + \overset{ˉ}{M} W) = - a {∣ M + \frac{b}{a} W ∣}^{2} + \frac{b^{2}}{a} ∣ W ∣^{2} .$ Since $b^{2} / a = 3 e^{K / M_{P}^{2}} / M_{P}^{2}$ , eliminating $M$ contributes $+ 3 e^{K / M_{P}^{2}} ∣ W ∣^{2} / M_{P}^{2}$ to $L_{a u x} / \sqrt{- g}$ , equivalently $- 3 e^{K / M_{P}^{2}} ∣ W ∣^{2} / M_{P}^{2}$ to $V$ when the potential enters the Lagrangian as $- V$ .

3.1givenA1A2F1F2step 1.1step 1.2step 2.1algebra∎

At the stationary auxiliary values the three squares vanish. By [A2], the remaining auxiliary Lagrangian is the entire zero-derivative scalar Lagrangian: $- e^{K / M_{P}^{2}} K^{i \overset{ˉ}{j}} D_{i} W \overline{D_{j} W} + 3 e^{K / M_{P}^{2}} ∣ W ∣^{2} / M_{P}^{2} - \frac{1}{2} (R^{- 1})^{A B} P_{A} P_{B} = - V$ . Rearranging yields the stated potential. By [F2], $D_{i} W$ transforms as the section $W$ , so the two F-term contributions have cancelling Kähler weights; the D-term is invariant under the assumed compatible gauging. The FI qualification is part of [F1]'s admitted moment-map data, not an extra free constant.

Unresolved supplier

The convention input is Supergravity Conventions And Normalizations at the exact four-dimensional Einstein-frame and Planck normalization used above. The result remains conditional on its adopted supergravity framework.

Four-Dimensional $N = 1$ F- and D-Term Scalar Potential

Statement

Statement references

Facts & Assumptions

Proof

Unresolved supplier

Depends on

Dependency tree · 2 levels

Sources

Four-Dimensional N=1N=1 F- and D-Term Scalar Potential

Statement

Statement references

Facts & Assumptions

Proof

Unresolved supplier

Depends on

Dependency tree · 2 levels

Sources

Four-Dimensional $N = 1$ F- and D-Term Scalar Potential