Compactification EFT scales and validity window

Fix a smooth compact internal space $Y$, a background, a set ${\mathcal{F}}_{\mathrm{r}\mathrm{e}\mathrm{t}}$ of higher-dimensional field species represented in the lower-dimensional model, a set ${\mathcal{F}}_{\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}$ of omitted physical sectors, and an external Einstein-frame metric. For each $\alpha \in {\mathcal{F}}_{\mathrm{r}\mathrm{e}\mathrm{t}}$, let ${\mathrm{\Sigma }}_{\mathrm{K}\mathrm{K}}^{(\alpha )}\subseteq [0,\mathrm{\infty })$ be the mass-squared spectral support of the specified physical internal fluctuation operator for that parent field species, with the zero mode removed. “Retained-field spectrum” here means the internal modes of the declared parent field species, whether or not an individual massive mode is retained. Define $M_{\mathrm{K}\mathrm{K}}:=\inf \{\sqrt{\lambda }:\lambda \in {\bigcup }_{\alpha \in {\mathcal{F}}_{\mathrm{r}\mathrm{e}\mathrm{t}}}{\mathrm{\Sigma }}_{\mathrm{K}\mathrm{K}}^{(\alpha )},\lambda >0\}\mathrm{.}$ This is a spectral threshold, not necessarily an eigenvalue. If the set is empty put $M_{\mathrm{K}\mathrm{K}}=+\mathrm{\infty }$; if positive spectral values accumulate at zero, put $M_{\mathrm{K}\mathrm{K}}=0$. A tachyonic background is outside this stable-spectrum definition.

For each declared omitted physical sector $\sigma$—for example, a bulk Kaluza–Klein tower, a warped-throat tower, an open-string sector, a string-oscillator sector, or a winding sector—let ${\mathrm{\Sigma }}_{\sigma }\subseteq [0,\mathrm{\infty })$ be its physical mass-squared spectral support in the same external Einstein frame, including continuous spectrum. Define its threshold $M_{\sigma }^{\mathrm{t}\mathrm{h}}:=\inf \{\sqrt{\lambda }:\lambda \in {\mathrm{\Sigma }}_{\sigma }\}$, with $M_{\sigma }^{\mathrm{t}\mathrm{h}}=+\mathrm{\infty }$ when that sector is absent. For the finite declared list of omitted sector types ${\mathfrak{S}}_{\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}$, define $M_{\mathrm{g}\mathrm{a}\mathrm{p}}:=\inf \{M_{\sigma }^{\mathrm{t}\mathrm{h}}:\sigma \in {\mathfrak{S}}_{\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}\}\mathrm{.}$ This threshold has units of inverse length. If there is no omitted sector, set $M_{\mathrm{g}\mathrm{a}\mathrm{p}}=+\mathrm{\infty }$; if the infimum is zero, then $M_{\mathrm{g}\mathrm{a}\mathrm{p}}=0$ and there is no positive-gap derivative expansion. An omitted instability supplies no positive gap.

When present, write $M_{\mathrm{K}\mathrm{K},\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{t}}$, $M_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{a}\mathrm{t}}$, $M_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}}$, and $M_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$ for the thresholds $M_{\sigma }^{\mathrm{t}\mathrm{h}}$ of the omitted bulk Kaluza–Klein, warped-throat, open-string, and string-oscillator sectors, respectively. Each is $+\mathrm{\infty }$ when its sector is absent, and $M_{\mathrm{g}\mathrm{a}\mathrm{p}}$ is the infimum over the complete declared omitted-sector list, not just one of these examples.

If the canonically normalized $d$-dimensional Einstein–Hilbert coefficient is $1\mathrm{/}(2{\kappa }_d^2)$ with ${\kappa }_d^{-2}>0$ and $d>2$, define the reduced Planck scale by $M_{d,\mathrm{P}\mathrm{l}}^{d-2}:={\kappa }_d^{-2}$, with the positive real root. Thus $M_{d,\mathrm{P}\mathrm{l}}$ has units of inverse length. On a stated region $\mathcal{O}$ of the smooth time-oriented external spacetime $(M_d,g_E)$, choose a smooth local orthonormal frame $(e_0,\dots ,e_{d-1})$, with $g_E(e_a,e_b)=\mathrm{diag}(-1,1,\dots ,1{)}_{ab}$ and future-directed observer field $U=e_0$. No global frame is assumed. For a future causal momentum covector $k\in T_x^{\ast }{M}_d$, meaning $g_E^{-1}(k,k)\le 0$ and $k(U)<0$ (or $k=0$), its local energy is $E_U(k):=-k(U)$. Declare a momentum domain $\mathcal{P}\subseteq T^{\ast }{M}_d{\mathrm{\mid }}_{\mathcal{O}}$ containing all momentum arguments of the omitted propagators being expanded, including off-shell momenta and all relevant channel sums. Choose a finite $E\ge 0$ with $\mathrm{\mid }p(e_a)\mathrm{\mid }\le E$ for every $p\in \mathcal{P}$ and every $a=0,\dots ,d-1$ throughout $\mathcal{O}$. This bounds local causal energies and every spatial component; a bound on causal energies alone would not control arbitrary spacelike momenta. The frame, region and momentum domain are input data, and $E$ has units of inverse length. For a nonempty stable omitted sector with $0<M_{\mathrm{g}\mathrm{a}\mathrm{p}}<\mathrm{\infty }$, the declared low-momentum derivative-expansion regime requires $E\mathrm{/}{M}_{\mathrm{g}\mathrm{a}\mathrm{p}}\le \delta <1$, with $E\ll M_{\mathrm{g}\mathrm{a}\mathrm{p}}$ denoting a family in which that ratio tends to zero with other declared bounds fixed. It also requires the specified propagator representations, analyticity domains and uniform Taylor remainder estimates of the integrating-out definition; a gap and this ratio alone do not establish locality or error control. For an empty omitted sector there is no omitted propagator to expand; zero gap supplies no such regime. When gravitational EFT control is claimed, impose separately $E\ll M_{d,\mathrm{P}\mathrm{l}}$.

Let ${\alpha }^{\mathrm{\prime }}>0$ have units of length squared. For a fixed unwarped circle, let $R=R_E>0$ be its proper radius in the higher-dimensional Einstein metric, and use that metric's external block as the external comparison metric, with no further lower-dimensional Weyl rescaling. The field-theory momentum contribution in this frame is $\mathrm{\mid }n\mathrm{\mid }\mathrm{/}{R}_E$ for integer $n$; it is distinct from string winding. For the ten-dimensional string comparison assume a constant dilaton ${\mathrm{\Phi }}_0$ and $G_E=e^{-{\mathrm{\Phi }}_0\mathrm{/}2}{G}_s$. The string-frame proper radius is $R_s=e^{{\mathrm{\Phi }}_0\mathrm{/}4}{R}_E$, while rest energies obey $M_E=e^{{\mathrm{\Phi }}_0\mathrm{/}4}{M}_s$. Thus string-frame momentum and winding contributions $\mathrm{\mid }n\mathrm{\mid }\mathrm{/}{R}_s$ and $\mathrm{\mid }w\mathrm{\mid }{R}_s\mathrm{/}{\alpha }^{\mathrm{\prime }}$, for $n,w\in \mathbb{Z}$ and tension $T=1\mathrm{/}(2\pi {\alpha }^{\mathrm{\prime }})$, become $\mathrm{\mid }n\mathrm{\mid }\mathrm{/}{R}_E$ and $e^{{\mathrm{\Phi }}_0\mathrm{/}2}\mathrm{\mid }w\mathrm{\mid }{R}_E\mathrm{/}{\alpha }^{\mathrm{\prime }}$ in the common Einstein frame. These are contributions to the spectrum, not universal formulas for full string-state masses or sector thresholds; oscillator contributions and state constraints must also be supplied. Define the string-frame scale $M_s^{(s)}:={\alpha }^{\mathrm{\prime }-1\mathrm{/}2}$. This is a characteristic scale, not automatically the exact oscillator threshold $M_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}$. In a winding sector, define $M_w$ as the lower mass threshold of the declared winding states in the chosen frame; it is $+\mathrm{\infty }$ if that sector is absent and is the infimum of its spectral support if continuous. A winding-state threshold and a field-theory Fourier momentum threshold are different sectors.

In ten dimensions use the unshifted convention $G_E=e^{-\mathrm{\Phi }\mathrm{/}2}{G}_s$. For a static background dilaton $\mathrm{\Phi }:Y\to \mathbb{R}$, the local Einstein-frame characteristic string scale is $M_s^{(E)}(y)=e^{\mathrm{\Phi }(y)\mathrm{/}4}{M}_s^{(s)},g_s(y):=e^{\mathrm{\Phi }(y)}\mathrm{.}$ If $\mathrm{\Phi }$ is constant, $g_s=e^{{\mathrm{\Phi }}_0}$. A uniform weak-coupling regime on compact $Y$ requires ${\sup }_{y\in Y}{e}^{\mathrm{\Phi }(y)}\ll 1$; a pointwise condition applies only to a stated region.

For a warped Einstein-frame metric $d{s}_E^2=e^{2A(y)}{g}_d+e^{2B(y)}{\widehat{g}}_Y$, a redshift formula requires a static observer convention. Suppose $g_d$ has a chosen unit timelike Killing field $\xi ={\mathrm{\partial }}_t$ and static observers at fixed external spatial coordinates. Their unit tangent in the warped metric is $U=e^{-A(y)}\xi$, and their proper time is $d\tau =e^{A(y)}dt$. A local rest threshold $M^{(E)}(y)$ measured by $U$ therefore has threshold $M^{(t)}(y)=e^{A(y)}{M}^{(E)}(y)$ for frequency conjugate to $t$. If no such normalized static time exists, compare local orthonormal-frame energies and thresholds instead; a global warped KK mass is obtained from the full fluctuation spectral problem, not from a local warp factor.

To define a string-frame curvature/variation length, specify a compact region $K\subseteq Y$, a string-frame orthonormal frame, and a finite list $\mathcal{Q}$ of dimensionless normalized background fields with stated derivative orders $k_q\ge 1$. Use the maximum absolute frame component norm and set $L_q:={\left({\sup }_K\mathrm{\parallel }{\mathrm{\nabla }}_s^{k_q}q\mathrm{\parallel }\right)}^{-1\mathrm{/}{k}_q}$ with $L_q=+\mathrm{\infty }$ when the supremum is zero. Also set $L_R:=({\sup }_K\mathrm{\parallel }\mathrm{R}\mathrm{m}(G_s)\mathrm{\parallel }{)}^{-1\mathrm{/}2}$, with $L_R=+\mathrm{\infty }$ when that supremum is zero, and define $L_s:=\min (\{L_R\}\cup \{L_q:q\in \mathcal{Q}\})$. The region, frame, normalized fields, and derivative orders are part of this convention. If these bounds are not supplied, $L_s$ is undefined. A declared perturbative string expansion may use ${\alpha }^{\mathrm{\prime }}\mathrm{/}{L}_s^2$ and the local $g_s$ as separate dimensionless control parameters; neither is inferred from $E\mathrm{/}{M}_{\mathrm{g}\mathrm{a}\mathrm{p}}$.

Large total internal volume alone implies neither a positive mass gap, a large ratio $M_{\mathrm{g}\mathrm{a}\mathrm{p}}\mathrm{/}E$, small ${\alpha }^{\mathrm{\prime }}\mathrm{/}{L}_s^2$, nor weak local coupling.

Integrating out fields and the Wilsonian effective theory

Kaluza–Klein product and warped reduction data

Planck coefficient from unwarped and warped products