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One-modulus no-scale cancellation in the stipulated $N = 1$ EFT

Physical scope

Conditional algebraic example in the one-modulus, two-derivative four-dimensional N=1 effective model with constant W and no gauging; no full string background or empirical claim.

Example

Take a single dimensionless chiral coordinate $ρ$ with $Im ρ > 0$ , dimensionless Kähler potential $K = - 3 \log [- i (ρ - \overset{ˉ}{ρ})]$ , and dimensionful convention $K = M_{P}^{2} K$ . Let $W = W_{0}$ be a constant holomorphic superpotential with $[W_{0}] = {m a s s}^{3}$ , and include no vector multiplet, gauging, or D-term. Adopt the standard old-minimal auxiliary sector and assume that it generates the entire zero-derivative scalar potential, with no auxiliary-independent contribution. In the stipulated two-derivative four-dimensional $N = 1$ model, the F-term potential vanishes at every point of this patch. Consequently this potential does not fix $ρ$ ; the calculation establishes no full Type IIB solution, omitted-mode uplift, or vacuum stability.

Statement references

Giddings, Kachru, and Polchinski, §2.4

Baumann and McAllister, §§6.2–6.4

Facts & Assumptions

Given: The one-modulus patch $Im ρ > 0$ , the coordinate normalization of the GKP modulus interface, $M_{P} > 0$ , and the conditional four-dimensional $N = 1$ F-term potential.

[F1]

GKP single overall Kähler-modulus coordinate and normalization gives $ρ = b / \sqrt{2} + i e^{4 u_{E}}$ , so $- i (ρ - \overset{ˉ}{ρ}) = 2 e^{4 u_{E}} > 0$ , and dimensionless $K = - 3 \log [- i (ρ - \overset{ˉ}{ρ})]$ .

[F2]

Four-Dimensional $N = 1$ F- and D-Term Scalar Potential conditionally gives the Einstein-frame F-term potential after eliminating the stated old-minimal auxiliary fields, provided that their sector generates the entire zero-derivative potential and no auxiliary-independent term is added; its current readiness is escalated because of an upstream readiness record, so this example is conditional on that supplier.

[A1]

There is exactly one chiral field, no vector or D-term sector, $W = W_{0}$ is constant and holomorphic, and $K = M_{P}^{2} K$ with dimensionless $ρ$ . The zero-derivative potential is generated entirely by the old-minimal auxiliary sector of [F2], with no additional auxiliary-independent term.

Verification

1.1givenF1A1algebra

Set $X = - i (ρ - \overset{ˉ}{ρ}) = 2 Im ρ > 0$ . Differentiating $K = - 3 \log X$ gives $K_{ρ} = - 3 / (ρ - \overset{ˉ}{ρ})$ and $K_{ρ \overset{ˉ}{ρ}} = - 3 / (ρ - \overset{ˉ}{ρ})^{2} = 3 / X^{2} > 0$ .

2.1givenA1step 1.1algebra

The inverse metric is $K^{ρ \overset{ˉ}{ρ}} = X^{2} / 3$ , and the dimensionless-coordinate covariant derivative is $D_{ρ} W = \partial_{ρ} W + K_{ρ} W = K_{ρ} W_{0}$ because $\partial_{ρ} W_{0} = 0$ . Therefore $K^{ρ \overset{ˉ}{ρ}} D_{ρ} W \overline{D_{ρ} W} = (X^{2} / 3) (9 / X^{2}) ∣ W_{0} ∣^{2} = 3 ∣ W_{0} ∣^{2}$ .

3.1givenF2A1step 2.1algebra

Since $K_{ρ \overset{ˉ}{ρ}} = M_{P}^{2} K_{ρ \overset{ˉ}{ρ}}$ , its inverse is $M_{P}^{- 2} K^{ρ \overset{ˉ}{ρ}}$ ; inserting step 2.1 into [F2] with the D-term sum empty gives $V = e^{K} M_{P}^{- 2} (3 ∣ W_{0} ∣^{2} - 3 ∣ W_{0} ∣^{2}) = 0$ for every $ρ$ in the patch. This is an algebraic deduction inside the stipulated effective model.

4.1givenF1step 3.1algebra∎

The potential is identically constant, so its first and second derivatives vanish everywhere and it supplies no restoring force or mass for $ρ$ . The result is only a no-scale cancellation in the chosen EFT; no full higher-dimensional equations, omitted-field equations, or empirical observations were used.

One-modulus no-scale cancellation in the stipulated $N = 1$ EFT

Example

Statement references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

One-modulus no-scale cancellation in the stipulated N=1N=1 EFT

Example

Statement references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

One-modulus no-scale cancellation in the stipulated $N = 1$ EFT