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Tree-level moduli-mediated visible-sector soft terms

Physical scope

Conditional tree-level formulae in a declared local two-derivative four-dimensional N=1 Einstein-frame supergravity EFT. They do not predict a compactification-independent spectrum, an observed superpartner mass, or the size of loop, anomaly, threshold, or renormalization-group corrections. The retained visible scalar Taylor coefficients through cubic order are auxiliary-induced; V0 is the F-sector value, not an independently shifted total vacuum energy.

Statement

Use reduced Planck units $M_{P} = 1$ , natural units $ℏ = c = 1$ , and a local four-dimensional Einstein-frame patch of the two-derivative old-minimal $N = 1$ supergravity model. Let $Z^{m}$ be holomorphic moduli coordinates and let $α$ range over a finite nonempty visible chiral field index set; $Q^{α}$ are complex visible scalar coordinates expanded about $Q = 0$ . Repeated visible indices are summed and $∥ Q ∥$ is any fixed Euclidean norm on this finite-dimensional coordinate space. Through cubic order take $K = K_{0} (Z, \overset{ˉ}{Z}) + \sum_{α} Z_{α} (Z, \overset{ˉ}{Z}) ∣ Q^{α} ∣^{2} + O (∥ Q ∥^{4}), W = W_{0} (Z) + \frac{1}{6} Y_{α β γ} (Z) Q^{α} Q^{β} Q^{γ} + O (∥ Q ∥^{4}),$ where $K_{0}$ is real and $C^{2}$ with positive-definite Kähler metric $K_{0, m \overset{ˉ}{n}}$ , each diagonal matter metric $Z_{α}$ is real, $C^{2}$ , and strictly positive, and $W_{0}, Y_{α β γ}$ and the gauge-kinetic functions $f_{a}$ are holomorphic on the patch. The visible Kähler potential is $C^{4}$ in the full local coordinates with the displayed Taylor remainders differentiable to the orders used below; $Y_{α β γ}$ is symmetric, and only gauge-invariant Yukawa components are retained. The physical gauge sector has $Re f_{a} > 0$ . Define $D_{m} W_{0} = \partial_{m} W_{0} + (\partial_{m} K_{0}) W_{0}$ , $m_{3 / 2} = e^{K_{0} / 2} ∣ W_{0} ∣$ , $V_{0} = V_{F} (Z, 0)$ , and use the moduli-breaking spurion convention $F^{m} = e^{K_{0} / 2} K_{0}^{m \overset{ˉ}{n}} \overline{D_{n} W_{0}}$ ; this is the negative of the algebraically eliminated old-minimal auxiliary in the plus-linear-term convention of the adopted auxiliary Lagrangian. For a nonzero Yukawa component, define its trilinear coefficient by the displayed $A_{α β γ}$ formula below; at a zero of $Y_{α β γ}$ the regular quantity is the product $A_{α β γ} Y_{α β γ}$ , interpreted by replacing $Y \partial_{m} \log Y$ with $\partial_{m} Y$ .

Assume the retained scalar potential is $V_{r e t} = V_{F} + V_{D} + U (Z, \overset{ˉ}{Z}) + R (Z, \overset{ˉ}{Z}, Q, \overset{ˉ}{Q})$ , where $V_{F}$ and $V_{D}$ are the chiral/gravity and vector auxiliary-elimination contributions of this model, $U$ is independent of the visible fields, and $R$ is $C^{3}$ in the real and imaginary visible coordinates with all visible derivatives of orders zero through three vanishing at $Q = 0$ . Thus no independent retained visible-field potential contributes through cubic order. The symbol $V_{0} = V_{F} (Z, 0)$ retains its F-sector meaning: if $U \neq 0$ , it is not the total vacuum energy. These conditions concern the declared EFT Taylor coefficients, not a proof that an arbitrary compactification has no other contribution.

Assume the moduli $D$ -term potential vanishes at $Q = 0$ and no visible-sector $D$ -term contributes through cubic order in $Q$ . Omit loop/anomaly mediation, thresholds, renormalization-group running, and higher-derivative corrections. Canonical normalization $q^{α} = Z_{α}^{1 / 2} Q^{α}$ and $λ_{a, c} = (Re f_{a})^{1 / 2} λ_{a}$ gives $M_{a} = \frac{F^{m} \partial_{m} f_{a}}{2 Re f_{a}}, m_{α}^{2} = m_{3 / 2}^{2} + V_{0} - F^{m} {\overset{ˉ}{F}}^{\overset{ˉ}{n}} \partial_{m} \partial_{\overset{ˉ}{n}} \log Z_{α},$ $A_{α β γ} = F^{m} [\partial_{m} K_{0} + \partial_{m} \log Y_{α β γ} - \partial_{m} \log (Z_{α} Z_{β} Z_{γ})] .$ The convention for $A_{α β γ}$ is that the canonically normalized scalar potential contains $\frac{1}{6} A_{α β γ} {\hat{Y}}_{α β γ} q^{α} q^{β} q^{γ} + h . c .$ , where ${\hat{Y}}_{α β γ} = e^{K_{0} / 2} Y_{α β γ} / \sqrt{Z_{α} Z_{β} Z_{γ}}$ . These are conditional EFT formulae, not universal spectrum predictions or measured masses.

Statement references

Conlon, Abdussalam, Quevedo, and Suruliz, §§2.1–2.2

McAllister and Quevedo, §4.4

Louis, §§10.3–10.4, full lecture notes

Facts & Assumptions

Given: A local four-dimensional Einstein-frame $N = 1$ moduli EFT with the stated Kähler metric, visible matter expansion, gauge functions, and moduli auxiliary data.

[F1]

Four-Dimensional $N = 1$ F- and D-Term Scalar Potential: for the adopted old-minimal two-derivative model, $V_{F} = e^{K / M_{P}^{2}} (K^{i \overset{ˉ}{j}} D_{i} W \overline{D_{j} W} - 3 ∣ W ∣^{2} / M_{P}^{2})$ after algebraic auxiliary elimination, with $D_{i} W = \partial_{i} W + (K_{i} / M_{P}^{2}) W$ . Its proof is conditional on the stated Einstein/Planck conventions. Its identification with the entire scalar potential requires no independent potential term; its square-completion calculation always identifies the auxiliary contribution. Here [A3] specifies exactly which retained visible Taylor coefficients equal those auxiliary coefficients, without identifying $V_{0}$ with a separate moduli-only contribution.

[F2]

Four-Dimensional $N = 1$ Matter-Coupled Supergravity adopts the local Einstein-frame old-minimal component framework, the Kähler-line-bundle convention, and the raw chiral, gravity, and vector auxiliary density used by [F1].

[F3]

Compactification-derived four-dimensional moduli EFT data defines the compactification-derived EFT input tuple, including the retained moduli chart, $K, W, f_{A B}$ , four-dimensional Einstein frame, Planck normalization, truncation range, and omitted sectors. Its current Step 1 record is escalated because the O3/O7 coordinate and conditional $K / f$ suppliers and the local metric, orientifold, parity, and $N = 2$ period inputs do not all have current ready records; we use only the declared EFT-data interface conditionally.

[F4]

Compactification EFT scales and validity window requires comparisons in a common external Einstein frame and distinguishes an actually established omitted-sector gap from a scale inferred only from total volume. Its current Step 1 record remains escalated on a conventions-readiness rationale that predates the live ready conventions carrier; no scale separation is inferred here, and the owner must refresh that record.

[A1]

The positive spurion $F^{m}$ is the negative of the on-shell auxiliary $F_{a u x}^{m}$ in the sign convention of [F2]. The visible Kähler potential has no terms of degree three in $Q, \overset{ˉ}{Q}$ ; the displayed quadratic diagonal metric is the entire visible contribution needed through the claimed order.

[A2]

Adopt the standard two-derivative component gauge-kinetic convention in which the unnormalized gaugino has kinetic normalization $Re f_{a}$ and the auxiliary-induced bilinear is $- \frac{1}{4} F^{m} (\partial_{m} f_{a}) λ_{a} λ_{a} + h . c .$ . This fermionic term is an adopted component input; it is not inferred from the bosonic potential [F1].

[A3]

The moduli $D$ -term potential vanishes at $Q = 0$ , and its expansion contributes no quadratic or cubic visible soft coefficient. The result is restricted to the stated tree-level truncation. Assume the retained scalar potential is $V_{r e t} = V_{F} + V_{D} + U (Z, \overset{ˉ}{Z}) + R (Z, \overset{ˉ}{Z}, Q, \overset{ˉ}{Q})$ , where $V_{F}$ and $V_{D}$ are the chiral/gravity and vector auxiliary-elimination contributions of this model, $U$ is independent of the visible fields, and $R$ is $C^{3}$ in the real and imaginary visible coordinates with all visible derivatives of orders zero through three vanishing at $Q = 0$ . Thus no independent retained visible-field potential contributes through cubic order. The symbol $V_{0} = V_{F} (Z, 0)$ retains its F-sector meaning: if $U \neq 0$ , it is not the total vacuum energy. These conditions concern the declared EFT Taylor coefficients, not a proof that an arbitrary compactification has no other contribution.

Proof

Given: [F1]–[F4], the expansions and positivity hypotheses in the Statement, the spurion and gauge-component conventions [A1]–[A3], and $M_{P} = 1$ .

1.1givenF1F2A1algebra

At $Q = 0$ the visible auxiliary fields vanish, so [F1] gives $V_{0} = e^{K_{0}} (K_{0}^{m \overset{ˉ}{n}} D_{m} W_{0} \overline{D_{n} W_{0}} - 3 ∣ W_{0} ∣^{2})$ . With the spurion convention in [A1], its positive quadratic norm is $K_{m \overset{ˉ}{n}}^{0} F^{m} {\overset{ˉ}{F}}^{\overset{ˉ}{n}} = e^{K_{0}} K_{0}^{m \overset{ˉ}{n}} D_{m} W_{0} \overline{D_{n} W_{0}}$ , and hence $K_{m \overset{ˉ}{n}}^{0} F^{m} {\overset{ˉ}{F}}^{\overset{ˉ}{n}} = V_{0} + 3 m_{3 / 2}^{2}$ .

1.2givenA2algebra

In the component convention [A2], canonical normalization gives $λ_{a, c} = (Re f_{a})^{1 / 2} λ_{a}$ . Thus the bilinear becomes $- \frac{1}{4} F^{m} (\partial_{m} f_{a}) (Re f_{a})^{- 1} λ_{a, c} λ_{a, c} + h . c .$ .

2.1givenF1F2A1A3step 1.1algebra

To second order in $Q$ , the Schur complement of the visible block $K_{α \overset{ˉ}{β}} = Z_{α} δ_{α β} + O (∥ Q ∥^{2})$ is $K_{0, m \overset{ˉ}{n}} + \sum_{α} R_{m \overset{ˉ}{n} α \overset{ˉ}{α}} ∣ Q^{α} ∣^{2} + O (∥ Q ∥^{3})$ , where $R_{m \overset{ˉ}{n} α \overset{ˉ}{α}} : = \partial_{m} \partial_{\overset{ˉ}{n}} Z_{α} - Z_{α}^{- 1} (\partial_{m} Z_{α}) (\partial_{\overset{ˉ}{n}} Z_{α})$ . Its inverse gives $K^{m \overset{ˉ}{n}} = K_{0}^{m \overset{ˉ}{n}} - \sum_{α} K_{0}^{m \overset{ˉ}{p}} R_{p \overset{ˉ}{q} α \overset{ˉ}{α}} K_{0}^{q \overset{ˉ}{n}} ∣ Q^{α} ∣^{2} + O (∥ Q ∥^{3})$ , while the off-diagonal inverse blocks at first order are $K^{m \overset{ˉ}{α}} = - K_{0}^{m \overset{ˉ}{n}} (\partial_{\overset{ˉ}{n}} Z_{α}) {\overset{ˉ}{Q}}^{\overset{ˉ}{α}} / Z_{α} + O (∥ Q ∥^{2})$ and $K^{α \overset{ˉ}{n}} = - (\partial_{m} Z_{α}) Q^{α} K_{0}^{m \overset{ˉ}{n}} / Z_{α} + O (∥ Q ∥^{2})$ . Also $D_{m} W = D_{m} W_{0} + \sum_{α} (\partial_{m} Z_{α}) W_{0} ∣ Q^{α} ∣^{2} + O (∥ Q ∥^{3})$ and $D_{α} W = Z_{α} {\overset{ˉ}{Q}}^{\overset{ˉ}{α}} W_{0} + O (∥ Q ∥^{2})$ . In $K^{i \overset{ˉ}{j}} D_{i} W \overline{D_{j} W}$ , the cross terms from the $∣ Q^{α} ∣^{2}$ correction to $D_{m} W$ cancel those from $K^{m \overset{ˉ}{α}} D_{m} W_{0} \overline{D_{α} W}$ and its conjugate. The remaining coefficient consists of $Z_{α} V_{0}$ from $e^{K}$ , $Z_{α} m_{3 / 2}^{2}$ from the visible auxiliary norm, and $- F^{m} {\overset{ˉ}{F}}^{\overset{ˉ}{n}} R_{m \overset{ˉ}{n} α \overset{ˉ}{α}}$ from the moduli inverse block. Hence the unnormalized coefficient is $(m_{3 / 2}^{2} + V_{0}) Z_{α} - F^{m} {\overset{ˉ}{F}}^{\overset{ˉ}{n}} R_{m \overset{ˉ}{n} α \overset{ˉ}{α}}$ .

3.1givenA1A3step 2.1algebra

Dividing that coefficient by $Z_{α} > 0$ is the canonical field change $q^{α} = Z_{α}^{1 / 2} Q^{α}$ . Since $R_{m \overset{ˉ}{n} α \overset{ˉ}{α}} / Z_{α} = \partial_{m} \partial_{\overset{ˉ}{n}} \log Z_{α}$ , the physical scalar mass squared is $m_{3 / 2}^{2} + V_{0} - F^{m} {\overset{ˉ}{F}}^{\overset{ˉ}{n}} \partial_{m} \partial_{\overset{ˉ}{n}} \log Z_{α}$ . By [A3], $U$ has no visible derivatives and the quadratic Taylor coefficient of $R$ vanishes, so this auxiliary coefficient is also the retained scalar-mass coefficient; no total-vacuum-energy substitution for $V_{0}$ is used.

3.2givenF1A1step 2.1algebra

The Taylor expansions are $D_{α} W = Z_{α} {\overset{ˉ}{Q}}^{\overset{ˉ}{α}} W_{0} + \frac{1}{2} Y_{α β γ} Q^{β} Q^{γ} + O (∥ Q ∥^{3})$ and $D_{m} W = D_{m} W_{0} + \sum_{δ} (\partial_{m} Z_{δ}) W_{0} ∣ Q^{δ} ∣^{2} + \frac{1}{6} (\partial_{m} Y_{α β γ} + K_{0, m} Y_{α β γ}) Q^{α} Q^{β} Q^{γ} + O (∥ Q ∥^{4})$ . The quadratic term in $D_{m} W$ is included for completeness and contributes no holomorphic cubic. In the moduli block of [F1], the cubic term in $D_{m} W$ contributes $\frac{1}{6} e^{K_{0} / 2} F^{m} (\partial_{m} Y_{α β γ} + K_{0, m} Y_{α β γ}) Q^{α} Q^{β} Q^{γ}$ . The visible diagonal block contributes $+ \frac{1}{2} e^{K_{0}} Y_{α β γ} {\overset{ˉ}{W}}_{0} Q^{α} Q^{β} Q^{γ}$ from the cross product of the linear and quadratic terms of $D_{α} W$ ; this cancels the $- \frac{1}{2} e^{K_{0}} Y_{α β γ} {\overset{ˉ}{W}}_{0} Q^{α} Q^{β} Q^{γ}$ term from $- 3 e^{K} ∣ W ∣^{2}$ . The off-diagonal inverse block $K^{α \overset{ˉ}{n}}$ from step 2.1 paired with the quadratic part of $D_{α} W$ contributes $- \frac{1}{2} e^{K_{0} / 2} F^{m} (\partial_{m} \log Z_{α}) Y_{α β γ} Q^{α} Q^{β} Q^{γ}$ ; relabeling the symmetric indices in the summed expression combines these terms into $- \frac{1}{6} e^{K_{0} / 2} F^{m} Y_{α β γ} (\partial_{m} \log Z_{α} + \partial_{m} \log Z_{β} + \partial_{m} \log Z_{γ}) Q^{α} Q^{β} Q^{γ}$ . Therefore the full holomorphic cubic coefficient is $\frac{1}{6} e^{K_{0} / 2} F^{m} [\partial_{m} Y_{α β γ} + K_{0, m} Y_{α β γ} - Y_{α β γ} (\partial_{m} \log Z_{α} + \partial_{m} \log Z_{β} + \partial_{m} \log Z_{γ})] Q^{α} Q^{β} Q^{γ} + h . c .$ .

4.1givenA1A3step 3.2algebra

The canonical field change divides this coefficient by $\sqrt{Z_{α} Z_{β} Z_{γ}}$ , while the physical supersymmetric Yukawa is ${\hat{Y}}_{α β γ} = e^{K_{0} / 2} Y_{α β γ} / \sqrt{Z_{α} Z_{β} Z_{γ}}$ . Factoring ${\hat{Y}}_{α β γ}$ gives the stated $A_{α β γ}$ when $Y_{α β γ} \neq 0$ ; without division by $Y$ , the product is the regular coefficient from step 3.2. By [A3], neither $U$ nor $R$ has a visible cubic coefficient, so the computed trilinear is also the retained-potential trilinear.

5.1givenF3F4A2step 1.2algebra∎

Comparing with the mass-term convention $- \frac{1}{2} M_{a} λ_{a, c} λ_{a, c} + h . c .$ yields $M_{a} = F^{m} \partial_{m} f_{a} / (2 Re f_{a})$ . By [F3]–[F4], this is a model-level EFT output in its declared frame and cutoff range; it alone implies no hierarchy or measured spectrum.

Unresolved prerequisites

The complete current statements/proofs of the direct suppliers were inspected against their exact uses. The data definition [F3] has a current escalated readiness record with unresolved O3/O7 coordinate, K/f, local-metric, orientifold, parity, and $N = 2$ period inputs. The scale definition [F4] retains an escalated record whose conventions-readiness rationale predates the now-present, ready conventions carrier. The scalar-potential theorem [F1] and $N = 1$ postulate [F2] also retain owner-held escalated records citing that now-ready conventions carrier as absent. These current readiness gaps remain open; this theorem does not refresh or override them. Its local conditional expansion is complete and promotes no EFT assumption to a compactification or empirical claim.

Tree-level moduli-mediated visible-sector soft terms

Statement

Statement references

Facts & Assumptions

Proof

Unresolved prerequisites

Depends on

Dependency tree · 2 levels

Sources