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Canonical moduli mass matrix and EFT scale-control data

Physical scope

Finite-dimensional retained scalar EFTs in a declared four-dimensional Einstein frame, with a C2 potential and positive target metric; stability criteria are local Hessian or explicitly stated AdS scalar-mode tests, and omitted-mode scale claims require the actual compactification spectrum.

Definition

Let $(M_{4}, g_{4, E})$ be a smooth four-dimensional Lorentzian spacetime in Einstein frame, with signature $(-, +, +, +)$ , and let $U \subseteq R^{N}$ , $N \geq 1$ , be an open field-space coordinate patch with coordinate functions $ϕ^{a} : U \to R$ of mass dimension one. A spacetime scalar configuration in this patch is a smooth map $Φ : M_{4} \to U$ , with component fields $ϕ^{a} \circ Φ$ . Let $G = G_{a b} (ϕ) d ϕ^{a} \otimes d ϕ^{b}$ be a smooth dimensionless symmetric positive-definite metric on $U$ , and let $V : U \to R$ be a $C^{2}$ real function of mass dimension four. Adopt the retained scalar action $S_{s c a l} = \int_{M_{4}} d^{4} x \sqrt{- g_{4, E}} [- \frac{1}{2} G_{a b} (ϕ) g_{4, E}^{μ ν} \partial_{μ} ϕ^{a} \partial_{ν} ϕ^{b} - V (ϕ)] .$ Here the component fields in the action mean $ϕ^{a} \circ Φ$ , and $G$ is the full kinetic coefficient, including all Planck normalization factors; no additional scalar kinetic term is suppressed. For interpreting the eigenvalues as physical scalar masses, use a linear scalar fluctuation sector with gauge redundancy resolved and no quadratic derivative mixing with other fields. An interior critical point is a point $p \in U$ with $d V_{p} = 0$ . Define the Hessian bilinear form $H_{p} (v, w) = (\nabla^{2} V)_{p} (v, w)$ and the mass endomorphism $(M^{2})^{a}_{b} = G^{a c} \nabla_{c} \nabla_{b} V ∣_{p}$ . The Hessian is nondegenerate when its kernel ${v \in T_{p} U : H_{p} (v, w) = 0 for every w \in T_{p} U}$ is ${0}$ . The endomorphism is self-adjoint for $G$ . A local minimum is a point $p$ with a neighborhood $W \subseteq U$ such that $V (q) \geq V (p)$ for every $q \in W$ ; it is strict if there is such a neighborhood with $V (q) > V (p)$ for every $q \neq p$ . At any local minimum the Hessian is positive semidefinite; a positive-definite Hessian is sufficient for a nondegenerate strict local minimum, and at a nondegenerate critical point that is a local minimum it is also necessary. Zero eigenvalues do not by themselves decide whether a critical point is a local minimum; that requires analysis of $V$ beyond its Hessian. For an AdS $_{4}$ scalar mode under standard reflecting boundary conditions, call the mode Breitenlohner–Freedman admissible when $m^{2} L^{2} \geq - 9 / 4$ ; this mode test alone is not a claim of full nonlinear or decay stability. Write $λ_{i}$ for the real eigenvalues of $M^{2}$ and define the nonnegative comparison scales $m_{i} : = \sqrt{∣ λ_{i} ∣}$ . For $λ_{i} < 0$ , this scale is the magnitude of a negative mass-squared eigenvalue, not a real stable-particle mass; the signed eigenvalue itself enters the AdS test, $m^{2} = λ_{i}$ . EFT scale-control data are the explicit inequalities comparing every retained scalar scale $m_{i}$ , the dS Hubble scale $H$ or the AdS curvature scale $L^{- 1}$ , and the external energy range to the lightest actually present omitted bulk Kaluza–Klein, throat, open-string, and string threshold in one common four-dimensional Einstein frame. This definition asserts no hierarchy or metastable vacuum.

Definition references

Baumann and McAllister, §§6.2–6.5

Van Riet and Zoccarato, §2.2 and §§4.3–4.4

Klebanov and Witten, §2.1, equations (2.1)–(2.3)

Facts & Assumptions

Given: A finite-dimensional scalar patch, smooth positive metric $G$ , $V \in C^{2}$ , a declared four-dimensional Einstein frame, and an actual compactification spectrum for any claimed cutoff.

[F1]

Compactification-derived four-dimensional moduli EFT data declares the retained scalar patch, four-dimensional Einstein metric, Planck normalization, truncation domain, cutoff, and omitted sectors.

[F2]

Compactification EFT scales and validity window defines a common-frame energy window and distinguishes a positive omitted-sector gap from the separate KK, warped-throat, open-string, and string thresholds; its carrier has a current escalated readiness record whose stated upstream reason is stale.

[F3]

Planck coefficient from unwarped and warped products gives the unwarped product Planck coefficient after Einstein-frame normalization; its current readiness record remains escalated for an upstream-record issue.

[F4]

Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis gives an orthonormal eigenbasis for a self-adjoint endomorphism of a finite-dimensional real inner-product space.

[F5]

Second-order Taylor expansion $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h + o (∥ h ∥^{2})$ gives the quadratic expansion with $o (∥ h ∥^{2})$ remainder for a $C^{2}$ real function in Euclidean coordinates.

[A1]

The scalar patch is finite-dimensional and smooth, $G$ is positive definite and smooth, and $V$ is $C^{2}$ ; at a critical point the covariant Hessian is evaluated using the Levi-Civita connection of $G$ . The scalar action and full kinetic normalization are those displayed in the Definition. The physical mass interpretation is for fluctuations about a constant-field solution in the stated scalar sector.

[A2]

For the cosmological scales, use the four-dimensional Einstein equations with no additional vacuum-energy contribution: at a constant-field dS $_{4}$ solution $V = 3 M_{P}^{2} H^{2}$ , and at an AdS $_{4}$ solution $V = - 3 M_{P}^{2} / L^{2}$ . The Breitenlohner–Freedman test stated here assumes standard reflecting AdS $_{4}$ boundary conditions.

[A3]

A mass or cutoff comparison is made only after every actually present omitted tower and threshold is listed in the same external Einstein frame; an unknown or gapless omitted sector does not satisfy a positive-gap EFT assumption.

Proof

Given: The hypotheses in [A1]–[A3] and the conditional compactification interfaces [F1]–[F3].

1.1givenA1F4algebra

At a critical point $p$ , the Hessian $H_{a b} = \nabla_{a} \nabla_{b} V ∣_{p}$ is symmetric because the Levi-Civita connection is torsion-free and the commutator of covariant derivatives on a scalar has no curvature term. For tangent vectors $v, w$ , the endomorphism $M^{a}_{b} = G^{a c} H_{c b}$ obeys $G (v, M w) = H (v, w) = H (w, v) = G (M v, w)$ ; hence it is self-adjoint with respect to the positive metric $G$ . Apply [F4] to the finite-dimensional real inner-product space $(T_{p} U, G_{p})$ : it has an orthonormal eigenbasis with real eigenvalues, a finite nonempty list since $N \geq 1$ .

1.2givenA2algebra

Under the Einstein equations assumed in [A2], the constant-field relations define $H = \sqrt{V / (3 M_{P}^{2})}$ for $V > 0$ and $L^{2} = - 3 M_{P}^{2} / V$ for $V < 0$ . The AdS $_{4}$ condition $m^{2} L^{2} \geq - 9 / 4$ is the stated scalar-mode Breitenlohner–Freedman diagnostic under the boundary conditions in [A2]; checking this inequality does not test other fields, nonlinear stability, or decay rates.

2.1givenA1F5step 1.1algebra

Write a fluctuation about a constant-field solution as $ϕ = p + η$ . Since $d V_{p} = 0$ , the Taylor expansion [F5] and the displayed action give its quadratic scalar action $S_{s c a l}^{(2)} = - \frac{1}{2} \int d^{4} x \sqrt{- g_{4, E}} [G_{a b} (p) \partial_{μ} η^{a} \partial^{μ} η^{b} + H_{a b} η^{a} η^{b}]$ . The connection term in the Hessian vanishes at $p$ because it is proportional to $d V_{p}$ . The linearized equations are $□_{4, E} η^{a} - (M^{2})^{a}_{b} η^{b} = 0$ . In the $G_{p}$ -orthonormal eigenbasis of step 1.1, writing $η = \sum_{i} q_{i} e_{i}$ gives $S_{s c a l}^{(2)} = - \frac{1}{2} \sum_{i} \int d^{4} x \sqrt{- g_{4, E}} [(\partial q_{i})^{2} + λ_{i} q_{i}^{2}]$ , so $(□_{4, E} - λ_{i}) q_{i} = 0$ . Thus $λ_{i}$ are the signed canonically normalized scalar mass-squared parameters, with mass dimension two; no unspecified multiple of $G$ can be discarded.

2.2givenA1F4F5step 1.1algebra

Choose a real coordinate chart centered at $p$ . Since $p$ is critical, $d V_{p} = 0$ , so the connection term $Γ_{a b}^{c} \partial_{c} V$ vanishes at $p$ and the covariant Hessian there equals the ordinary chart Hessian. For any tangent vector $v$ , the chart curve with coordinates $t v$ is defined for sufficiently small positive and negative $t$ . At a local minimum, [F5] gives $V (t v) - V (0) = \frac{1}{2} t^{2} H (v, v) + o (t^{2})$ , forcing $H (v, v) \geq 0$ . If $H$ is positive definite, use [F4] to choose a $G_{p}$ -orthonormal eigenbasis for $G_{p}^{- 1} H$ and a linear change of the chart to that basis. Its finite list of positive eigenvalues has a least member $λ > 0$ . The same Taylor formula gives $V (x) - V (0) \geq \frac{1}{2} λ ∣ x ∣^{2} + o (∣ x ∣^{2}) > 0$ for all sufficiently small nonzero $x$ , proving a strict local minimum. Conversely, at a nondegenerate local minimum the eigenvalues are nonnegative by the first argument and nonzero by nondegeneracy, hence positive. The examples $V (x) = x^{4}$ and $V (x) = - x^{4}$ have the same zero Hessian but opposite local-minimum behavior, so zero eigenvalues do not by themselves determine local-minimum status. No geodesic or exponential-map existence is used.

3.1givenF1F2F3A3step 2.1step 1.2∎

Let $M_{o m i t}$ be the infimum of the finite declared list of omitted-sector thresholds from [F2], all measured in the same external Einstein frame; an absent sector has threshold $+ \infty$ , and an empty list has infimum $+ \infty$ . Thus a continuous spectrum needs no attained least mass. Comparisons with $+ \infty$ impose no omitted-sector restriction, but leave the separate gravitational condition in force. The scale-control record lists $m_{i} = \sqrt{∣ λ_{i} ∣} ≪ M_{o m i t}$ for every retained scalar and $H_{b g} ≪ M_{o m i t}$ , where $H_{b g} = H$ in dS $_{4}$ and $H_{b g} = L^{- 1}$ in AdS $_{4}$ , together with $E ≪ M_{o m i t}$ and the Planck-scale condition when gravitational EFT control is claimed. If a required tower is unknown or has zero gap, these inequalities have not been established. This is a record of inputs and comparisons, not an existence or stability proof for a vacuum.

Canonical moduli mass matrix and EFT scale-control data

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources